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view libphobos/src/std/math.d @ 158:494b0b89df80 default tip
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Mon, 25 May 2020 18:13:55 +0900 |
| parents | 1830386684a0 |
| children |
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// Written in the D programming language. /** * Contains the elementary mathematical functions (powers, roots, * and trigonometric functions), and low-level floating-point operations. * Mathematical special functions are available in $(D std.mathspecial). * $(SCRIPT inhibitQuickIndex = 1;) $(DIVC quickindex, $(BOOKTABLE , $(TR $(TH Category) $(TH Members) ) $(TR $(TDNW Constants) $(TD $(MYREF E) $(MYREF PI) $(MYREF PI_2) $(MYREF PI_4) $(MYREF M_1_PI) $(MYREF M_2_PI) $(MYREF M_2_SQRTPI) $(MYREF LN10) $(MYREF LN2) $(MYREF LOG2) $(MYREF LOG2E) $(MYREF LOG2T) $(MYREF LOG10E) $(MYREF SQRT2) $(MYREF SQRT1_2) )) $(TR $(TDNW Classics) $(TD $(MYREF abs) $(MYREF fabs) $(MYREF sqrt) $(MYREF cbrt) $(MYREF hypot) $(MYREF poly) $(MYREF nextPow2) $(MYREF truncPow2) )) $(TR $(TDNW Trigonometry) $(TD $(MYREF sin) $(MYREF cos) $(MYREF tan) $(MYREF asin) $(MYREF acos) $(MYREF atan) $(MYREF atan2) $(MYREF sinh) $(MYREF cosh) $(MYREF tanh) $(MYREF asinh) $(MYREF acosh) $(MYREF atanh) $(MYREF expi) )) $(TR $(TDNW Rounding) $(TD $(MYREF ceil) $(MYREF floor) $(MYREF round) $(MYREF lround) $(MYREF trunc) $(MYREF rint) $(MYREF lrint) $(MYREF nearbyint) $(MYREF rndtol) $(MYREF quantize) )) $(TR $(TDNW Exponentiation & Logarithms) $(TD $(MYREF pow) $(MYREF exp) $(MYREF exp2) $(MYREF expm1) $(MYREF ldexp) $(MYREF frexp) $(MYREF log) $(MYREF log2) $(MYREF log10) $(MYREF logb) $(MYREF ilogb) $(MYREF log1p) $(MYREF scalbn) )) $(TR $(TDNW Modulus) $(TD $(MYREF fmod) $(MYREF modf) $(MYREF remainder) )) $(TR $(TDNW Floating-point operations) $(TD $(MYREF approxEqual) $(MYREF feqrel) $(MYREF fdim) $(MYREF fmax) $(MYREF fmin) $(MYREF fma) $(MYREF nextDown) $(MYREF nextUp) $(MYREF nextafter) $(MYREF NaN) $(MYREF getNaNPayload) $(MYREF cmp) )) $(TR $(TDNW Introspection) $(TD $(MYREF isFinite) $(MYREF isIdentical) $(MYREF isInfinity) $(MYREF isNaN) $(MYREF isNormal) $(MYREF isSubnormal) $(MYREF signbit) $(MYREF sgn) $(MYREF copysign) $(MYREF isPowerOf2) )) $(TR $(TDNW Complex Numbers) $(TD $(MYREF abs) $(MYREF conj) $(MYREF sin) $(MYREF cos) $(MYREF expi) )) $(TR $(TDNW Hardware Control) $(TD $(MYREF IeeeFlags) $(MYREF FloatingPointControl) )) ) ) * The functionality closely follows the IEEE754-2008 standard for * floating-point arithmetic, including the use of camelCase names rather * than C99-style lower case names. All of these functions behave correctly * when presented with an infinity or NaN. * * The following IEEE 'real' formats are currently supported: * $(UL * $(LI 64 bit Big-endian 'double' (eg PowerPC)) * $(LI 128 bit Big-endian 'quadruple' (eg SPARC)) * $(LI 64 bit Little-endian 'double' (eg x86-SSE2)) * $(LI 80 bit Little-endian, with implied bit 'real80' (eg x87, Itanium)) * $(LI 128 bit Little-endian 'quadruple' (not implemented on any known processor!)) * $(LI Non-IEEE 128 bit Big-endian 'doubledouble' (eg PowerPC) has partial support) * ) * Unlike C, there is no global 'errno' variable. Consequently, almost all of * these functions are pure nothrow. * * Status: * The semantics and names of feqrel and approxEqual will be revised. * * Macros: * TABLE_SV = <table border="1" cellpadding="4" cellspacing="0"> * <caption>Special Values</caption> * $0</table> * SVH = $(TR $(TH $1) $(TH $2)) * SV = $(TR $(TD $1) $(TD $2)) * TH3 = $(TR $(TH $1) $(TH $2) $(TH $3)) * TD3 = $(TR $(TD $1) $(TD $2) $(TD $3)) * TABLE_DOMRG = <table border="1" cellpadding="4" cellspacing="0"> * $(SVH Domain X, Range Y) $(SV $1, $2) * </table> * DOMAIN=$1 * RANGE=$1 * NAN = $(RED NAN) * SUP = <span style="vertical-align:super;font-size:smaller">$0</span> * GAMMA = Γ * THETA = θ * INTEGRAL = ∫ * INTEGRATE = $(BIG ∫<sub>$(SMALL $1)</sub><sup>$2</sup>) * POWER = $1<sup>$2</sup> * SUB = $1<sub>$2</sub> * BIGSUM = $(BIG Σ <sup>$2</sup><sub>$(SMALL $1)</sub>) * CHOOSE = $(BIG () <sup>$(SMALL $1)</sup><sub>$(SMALL $2)</sub> $(BIG )) * PLUSMN = ± * INFIN = ∞ * PLUSMNINF = ±∞ * PI = π * LT = < * GT = > * SQRT = √ * HALF = ½ * * Copyright: Copyright Digital Mars 2000 - 2011. * D implementations of tan, atan, atan2, exp, expm1, exp2, log, log10, log1p, * log2, floor, ceil and lrint functions are based on the CEPHES math library, * which is Copyright (C) 2001 Stephen L. Moshier $(LT)steve@moshier.net$(GT) * and are incorporated herein by permission of the author. The author * reserves the right to distribute this material elsewhere under different * copying permissions. These modifications are distributed here under * the following terms: * License: $(HTTP www.boost.org/LICENSE_1_0.txt, Boost License 1.0). * Authors: $(HTTP digitalmars.com, Walter Bright), Don Clugston, * Conversion of CEPHES math library to D by Iain Buclaw and David Nadlinger * Source: $(PHOBOSSRC std/_math.d) */ /* NOTE: This file has been patched from the original DMD distribution to * work with the GDC compiler. */ module std.math; version (Win64) { version (D_InlineAsm_X86_64) version = Win64_DMD_InlineAsm; } static import core.math; static import core.stdc.math; static import core.stdc.fenv; import std.traits; // CommonType, isFloatingPoint, isIntegral, isSigned, isUnsigned, Largest, Unqual version (LDC) { import ldc.intrinsics; } version (DigitalMars) { version = INLINE_YL2X; // x87 has opcodes for these } version (X86) version = X86_Any; version (X86_64) version = X86_Any; version (PPC) version = PPC_Any; version (PPC64) version = PPC_Any; version (MIPS32) version = MIPS_Any; version (MIPS64) version = MIPS_Any; version (AArch64) version = ARM_Any; version (ARM) version = ARM_Any; version (S390) version = IBMZ_Any; version (SPARC) version = SPARC_Any; version (SPARC64) version = SPARC_Any; version (SystemZ) version = IBMZ_Any; version (RISCV32) version = RISCV_Any; version (RISCV64) version = RISCV_Any; version (D_InlineAsm_X86) { version = InlineAsm_X86_Any; } else version (D_InlineAsm_X86_64) { version = InlineAsm_X86_Any; } version (CRuntime_Microsoft) { version (InlineAsm_X86_Any) version = MSVC_InlineAsm; } version (X86_64) version = StaticallyHaveSSE; version (X86) version (OSX) version = StaticallyHaveSSE; version (StaticallyHaveSSE) { private enum bool haveSSE = true; } else version (X86) { static import core.cpuid; private alias haveSSE = core.cpuid.sse; } version (unittest) { import core.stdc.stdio; // : sprintf; static if (real.sizeof > double.sizeof) enum uint useDigits = 16; else enum uint useDigits = 15; /****************************************** * Compare floating point numbers to n decimal digits of precision. * Returns: * 1 match * 0 nomatch */ private bool equalsDigit(real x, real y, uint ndigits) { if (signbit(x) != signbit(y)) return 0; if (isInfinity(x) && isInfinity(y)) return 1; if (isInfinity(x) || isInfinity(y)) return 0; if (isNaN(x) && isNaN(y)) return 1; if (isNaN(x) || isNaN(y)) return 0; char[30] bufx; char[30] bufy; assert(ndigits < bufx.length); int ix; int iy; version (CRuntime_Microsoft) alias real_t = double; else alias real_t = real; ix = sprintf(bufx.ptr, "%.*Lg", ndigits, cast(real_t) x); iy = sprintf(bufy.ptr, "%.*Lg", ndigits, cast(real_t) y); assert(ix < bufx.length && ix > 0); assert(ix < bufy.length && ix > 0); return bufx[0 .. ix] == bufy[0 .. iy]; } } package: // The following IEEE 'real' formats are currently supported. version (LittleEndian) { static assert(real.mant_dig == 53 || real.mant_dig == 64 || real.mant_dig == 113, "Only 64-bit, 80-bit, and 128-bit reals"~ " are supported for LittleEndian CPUs"); } else { static assert(real.mant_dig == 53 || real.mant_dig == 106 || real.mant_dig == 113, "Only 64-bit and 128-bit reals are supported for BigEndian CPUs."~ " double-double reals have partial support"); } // Underlying format exposed through floatTraits enum RealFormat { ieeeHalf, ieeeSingle, ieeeDouble, ieeeExtended, // x87 80-bit real ieeeExtended53, // x87 real rounded to precision of double. ibmExtended, // IBM 128-bit extended ieeeQuadruple, } // Constants used for extracting the components of the representation. // They supplement the built-in floating point properties. template floatTraits(T) { // EXPMASK is a ushort mask to select the exponent portion (without sign) // EXPSHIFT is the number of bits the exponent is left-shifted by in its ushort // EXPBIAS is the exponent bias - 1 (exp == EXPBIAS yields ×2^-1). // EXPPOS_SHORT is the index of the exponent when represented as a ushort array. // SIGNPOS_BYTE is the index of the sign when represented as a ubyte array. // RECIP_EPSILON is the value such that (smallest_subnormal) * RECIP_EPSILON == T.min_normal enum T RECIP_EPSILON = (1/T.epsilon); static if (T.mant_dig == 24) { // Single precision float enum ushort EXPMASK = 0x7F80; enum ushort EXPSHIFT = 7; enum ushort EXPBIAS = 0x3F00; enum uint EXPMASK_INT = 0x7F80_0000; enum uint MANTISSAMASK_INT = 0x007F_FFFF; enum realFormat = RealFormat.ieeeSingle; version (LittleEndian) { enum EXPPOS_SHORT = 1; enum SIGNPOS_BYTE = 3; } else { enum EXPPOS_SHORT = 0; enum SIGNPOS_BYTE = 0; } } else static if (T.mant_dig == 53) { static if (T.sizeof == 8) { // Double precision float, or real == double enum ushort EXPMASK = 0x7FF0; enum ushort EXPSHIFT = 4; enum ushort EXPBIAS = 0x3FE0; enum uint EXPMASK_INT = 0x7FF0_0000; enum uint MANTISSAMASK_INT = 0x000F_FFFF; // for the MSB only enum realFormat = RealFormat.ieeeDouble; version (LittleEndian) { enum EXPPOS_SHORT = 3; enum SIGNPOS_BYTE = 7; } else { enum EXPPOS_SHORT = 0; enum SIGNPOS_BYTE = 0; } } else static if (T.sizeof == 12) { // Intel extended real80 rounded to double enum ushort EXPMASK = 0x7FFF; enum ushort EXPSHIFT = 0; enum ushort EXPBIAS = 0x3FFE; enum realFormat = RealFormat.ieeeExtended53; version (LittleEndian) { enum EXPPOS_SHORT = 4; enum SIGNPOS_BYTE = 9; } else { enum EXPPOS_SHORT = 0; enum SIGNPOS_BYTE = 0; } } else static assert(false, "No traits support for " ~ T.stringof); } else static if (T.mant_dig == 64) { // Intel extended real80 enum ushort EXPMASK = 0x7FFF; enum ushort EXPSHIFT = 0; enum ushort EXPBIAS = 0x3FFE; enum realFormat = RealFormat.ieeeExtended; version (LittleEndian) { enum EXPPOS_SHORT = 4; enum SIGNPOS_BYTE = 9; } else { enum EXPPOS_SHORT = 0; enum SIGNPOS_BYTE = 0; } } else static if (T.mant_dig == 113) { // Quadruple precision float enum ushort EXPMASK = 0x7FFF; enum ushort EXPSHIFT = 0; enum ushort EXPBIAS = 0x3FFE; enum realFormat = RealFormat.ieeeQuadruple; version (LittleEndian) { enum EXPPOS_SHORT = 7; enum SIGNPOS_BYTE = 15; } else { enum EXPPOS_SHORT = 0; enum SIGNPOS_BYTE = 0; } } else static if (T.mant_dig == 106) { // IBM Extended doubledouble enum ushort EXPMASK = 0x7FF0; enum ushort EXPSHIFT = 4; enum realFormat = RealFormat.ibmExtended; // For IBM doubledouble the larger magnitude double comes first. // It's really a double[2] and arrays don't index differently // between little and big-endian targets. enum DOUBLEPAIR_MSB = 0; enum DOUBLEPAIR_LSB = 1; // The exponent/sign byte is for most significant part. version (LittleEndian) { enum EXPPOS_SHORT = 3; enum SIGNPOS_BYTE = 7; } else { enum EXPPOS_SHORT = 0; enum SIGNPOS_BYTE = 0; } } else static assert(false, "No traits support for " ~ T.stringof); } // These apply to all floating-point types version (LittleEndian) { enum MANTISSA_LSB = 0; enum MANTISSA_MSB = 1; } else { enum MANTISSA_LSB = 1; enum MANTISSA_MSB = 0; } // Common code for math implementations. // Helper for floor/ceil T floorImpl(T)(const T x) @trusted pure nothrow @nogc { alias F = floatTraits!(T); // Take care not to trigger library calls from the compiler, // while ensuring that we don't get defeated by some optimizers. union floatBits { T rv; ushort[T.sizeof/2] vu; // Other kinds of extractors for real formats. static if (F.realFormat == RealFormat.ieeeSingle) int vi; } floatBits y = void; y.rv = x; // Find the exponent (power of 2) // Do this by shifting the raw value so that the exponent lies in the low bits, // then mask out the sign bit, and subtract the bias. static if (F.realFormat == RealFormat.ieeeSingle) { int exp = ((y.vi >> (T.mant_dig - 1)) & 0xff) - 0x7f; } else static if (F.realFormat == RealFormat.ieeeDouble) { int exp = ((y.vu[F.EXPPOS_SHORT] >> 4) & 0x7ff) - 0x3ff; version (LittleEndian) int pos = 0; else int pos = 3; } else static if (F.realFormat == RealFormat.ieeeExtended) { int exp = (y.vu[F.EXPPOS_SHORT] & 0x7fff) - 0x3fff; version (LittleEndian) int pos = 0; else int pos = 4; } else static if (F.realFormat == RealFormat.ieeeQuadruple) { int exp = (y.vu[F.EXPPOS_SHORT] & 0x7fff) - 0x3fff; version (LittleEndian) int pos = 0; else int pos = 7; } else static assert(false, "Not implemented for this architecture"); if (exp < 0) { if (x < 0.0) return -1.0; else return 0.0; } static if (F.realFormat == RealFormat.ieeeSingle) { if (exp < (T.mant_dig - 1)) { // Clear all bits representing the fraction part. const uint fraction_mask = F.MANTISSAMASK_INT >> exp; if ((y.vi & fraction_mask) != 0) { // If 'x' is negative, then first substract 1.0 from the value. if (y.vi < 0) y.vi += 0x00800000 >> exp; y.vi &= ~fraction_mask; } } } else { exp = (T.mant_dig - 1) - exp; // Zero 16 bits at a time. while (exp >= 16) { version (LittleEndian) y.vu[pos++] = 0; else y.vu[pos--] = 0; exp -= 16; } // Clear the remaining bits. if (exp > 0) y.vu[pos] &= 0xffff ^ ((1 << exp) - 1); if ((x < 0.0) && (x != y.rv)) y.rv -= 1.0; } return y.rv; } public: // Values obtained from Wolfram Alpha. 116 bits ought to be enough for anybody. // Wolfram Alpha LLC. 2011. Wolfram|Alpha. http://www.wolframalpha.com/input/?i=e+in+base+16 (access July 6, 2011). enum real E = 0x1.5bf0a8b1457695355fb8ac404e7a8p+1L; /** e = 2.718281... */ enum real LOG2T = 0x1.a934f0979a3715fc9257edfe9b5fbp+1L; /** $(SUB log, 2)10 = 3.321928... */ enum real LOG2E = 0x1.71547652b82fe1777d0ffda0d23a8p+0L; /** $(SUB log, 2)e = 1.442695... */ enum real LOG2 = 0x1.34413509f79fef311f12b35816f92p-2L; /** $(SUB log, 10)2 = 0.301029... */ enum real LOG10E = 0x1.bcb7b1526e50e32a6ab7555f5a67cp-2L; /** $(SUB log, 10)e = 0.434294... */ enum real LN2 = 0x1.62e42fefa39ef35793c7673007e5fp-1L; /** ln 2 = 0.693147... */ enum real LN10 = 0x1.26bb1bbb5551582dd4adac5705a61p+1L; /** ln 10 = 2.302585... */ enum real PI = 0x1.921fb54442d18469898cc51701b84p+1L; /** $(_PI) = 3.141592... */ enum real PI_2 = PI/2; /** $(PI) / 2 = 1.570796... */ enum real PI_4 = PI/4; /** $(PI) / 4 = 0.785398... */ enum real M_1_PI = 0x1.45f306dc9c882a53f84eafa3ea69cp-2L; /** 1 / $(PI) = 0.318309... */ enum real M_2_PI = 2*M_1_PI; /** 2 / $(PI) = 0.636619... */ enum real M_2_SQRTPI = 0x1.20dd750429b6d11ae3a914fed7fd8p+0L; /** 2 / $(SQRT)$(PI) = 1.128379... */ enum real SQRT2 = 0x1.6a09e667f3bcc908b2fb1366ea958p+0L; /** $(SQRT)2 = 1.414213... */ enum real SQRT1_2 = SQRT2/2; /** $(SQRT)$(HALF) = 0.707106... */ // Note: Make sure the magic numbers in compiler backend for x87 match these. /*********************************** * Calculates the absolute value of a number * * Params: * Num = (template parameter) type of number * x = real number value * z = complex number value * y = imaginary number value * * Returns: * The absolute value of the number. If floating-point or integral, * the return type will be the same as the input; if complex or * imaginary, the returned value will be the corresponding floating * point type. * * For complex numbers, abs(z) = sqrt( $(POWER z.re, 2) + $(POWER z.im, 2) ) * = hypot(z.re, z.im). */ Num abs(Num)(Num x) @safe pure nothrow if (is(typeof(Num.init >= 0)) && is(typeof(-Num.init)) && !(is(Num* : const(ifloat*)) || is(Num* : const(idouble*)) || is(Num* : const(ireal*)))) { static if (isFloatingPoint!(Num)) return fabs(x); else return x >= 0 ? x : -x; } /// ditto auto abs(Num)(Num z) @safe pure nothrow @nogc if (is(Num* : const(cfloat*)) || is(Num* : const(cdouble*)) || is(Num* : const(creal*))) { return hypot(z.re, z.im); } /// ditto auto abs(Num)(Num y) @safe pure nothrow @nogc if (is(Num* : const(ifloat*)) || is(Num* : const(idouble*)) || is(Num* : const(ireal*))) { return fabs(y.im); } /// ditto @safe pure nothrow @nogc unittest { assert(isIdentical(abs(-0.0L), 0.0L)); assert(isNaN(abs(real.nan))); assert(abs(-real.infinity) == real.infinity); assert(abs(-3.2Li) == 3.2L); assert(abs(71.6Li) == 71.6L); assert(abs(-56) == 56); assert(abs(2321312L) == 2321312L); assert(abs(-1L+1i) == sqrt(2.0L)); } @safe pure nothrow @nogc unittest { import std.meta : AliasSeq; foreach (T; AliasSeq!(float, double, real)) { T f = 3; assert(abs(f) == f); assert(abs(-f) == f); } foreach (T; AliasSeq!(cfloat, cdouble, creal)) { T f = -12+3i; assert(abs(f) == hypot(f.re, f.im)); assert(abs(-f) == hypot(f.re, f.im)); } } /*********************************** * Complex conjugate * * conj(x + iy) = x - iy * * Note that z * conj(z) = $(POWER z.re, 2) - $(POWER z.im, 2) * is always a real number */ auto conj(Num)(Num z) @safe pure nothrow @nogc if (is(Num* : const(cfloat*)) || is(Num* : const(cdouble*)) || is(Num* : const(creal*))) { //FIXME //Issue 14206 static if (is(Num* : const(cdouble*))) return cast(cdouble) conj(cast(creal) z); else return z.re - z.im*1fi; } /** ditto */ auto conj(Num)(Num y) @safe pure nothrow @nogc if (is(Num* : const(ifloat*)) || is(Num* : const(idouble*)) || is(Num* : const(ireal*))) { return -y; } /// @safe pure nothrow @nogc unittest { creal c = 7 + 3Li; assert(conj(c) == 7-3Li); ireal z = -3.2Li; assert(conj(z) == -z); } //Issue 14206 @safe pure nothrow @nogc unittest { cdouble c = 7 + 3i; assert(conj(c) == 7-3i); idouble z = -3.2i; assert(conj(z) == -z); } //Issue 14206 @safe pure nothrow @nogc unittest { cfloat c = 7f + 3fi; assert(conj(c) == 7f-3fi); ifloat z = -3.2fi; assert(conj(z) == -z); } /*********************************** * Returns cosine of x. x is in radians. * * $(TABLE_SV * $(TR $(TH x) $(TH cos(x)) $(TH invalid?)) * $(TR $(TD $(NAN)) $(TD $(NAN)) $(TD yes) ) * $(TR $(TD $(PLUSMN)$(INFIN)) $(TD $(NAN)) $(TD yes) ) * ) * Bugs: * Results are undefined if |x| >= $(POWER 2,64). */ real cos(real x) @safe pure nothrow @nogc { pragma(inline, true); return core.math.cos(x); } //FIXME ///ditto double cos(double x) @safe pure nothrow @nogc { return cos(cast(real) x); } //FIXME ///ditto float cos(float x) @safe pure nothrow @nogc { return cos(cast(real) x); } @safe unittest { real function(real) pcos = &cos; assert(pcos != null); } /*********************************** * Returns $(HTTP en.wikipedia.org/wiki/Sine, sine) of x. x is in $(HTTP en.wikipedia.org/wiki/Radian, radians). * * $(TABLE_SV * $(TH3 x , sin(x) , invalid?) * $(TD3 $(NAN) , $(NAN) , yes ) * $(TD3 $(PLUSMN)0.0, $(PLUSMN)0.0, no ) * $(TD3 $(PLUSMNINF), $(NAN) , yes ) * ) * * Params: * x = angle in radians (not degrees) * Returns: * sine of x * See_Also: * $(MYREF cos), $(MYREF tan), $(MYREF asin) * Bugs: * Results are undefined if |x| >= $(POWER 2,64). */ real sin(real x) @safe pure nothrow @nogc { pragma(inline, true); return core.math.sin(x); } //FIXME ///ditto double sin(double x) @safe pure nothrow @nogc { return sin(cast(real) x); } //FIXME ///ditto float sin(float x) @safe pure nothrow @nogc { return sin(cast(real) x); } /// @safe unittest { import std.math : sin, PI; import std.stdio : writefln; void someFunc() { real x = 30.0; auto result = sin(x * (PI / 180)); // convert degrees to radians writefln("The sine of %s degrees is %s", x, result); } } @safe unittest { real function(real) psin = &sin; assert(psin != null); } /*********************************** * Returns sine for complex and imaginary arguments. * * sin(z) = sin(z.re)*cosh(z.im) + cos(z.re)*sinh(z.im)i * * If both sin($(THETA)) and cos($(THETA)) are required, * it is most efficient to use expi($(THETA)). */ creal sin(creal z) @safe pure nothrow @nogc { const creal cs = expi(z.re); const creal csh = coshisinh(z.im); return cs.im * csh.re + cs.re * csh.im * 1i; } /** ditto */ ireal sin(ireal y) @safe pure nothrow @nogc { return cosh(y.im)*1i; } /// @safe pure nothrow @nogc unittest { assert(sin(0.0+0.0i) == 0.0); assert(sin(2.0+0.0i) == sin(2.0L) ); } /*********************************** * cosine, complex and imaginary * * cos(z) = cos(z.re)*cosh(z.im) - sin(z.re)*sinh(z.im)i */ creal cos(creal z) @safe pure nothrow @nogc { const creal cs = expi(z.re); const creal csh = coshisinh(z.im); return cs.re * csh.re - cs.im * csh.im * 1i; } /** ditto */ real cos(ireal y) @safe pure nothrow @nogc { return cosh(y.im); } /// @safe pure nothrow @nogc unittest { assert(cos(0.0+0.0i)==1.0); assert(cos(1.3L+0.0i)==cos(1.3L)); assert(cos(5.2Li)== cosh(5.2L)); } /**************************************************************************** * Returns tangent of x. x is in radians. * * $(TABLE_SV * $(TR $(TH x) $(TH tan(x)) $(TH invalid?)) * $(TR $(TD $(NAN)) $(TD $(NAN)) $(TD yes)) * $(TR $(TD $(PLUSMN)0.0) $(TD $(PLUSMN)0.0) $(TD no)) * $(TR $(TD $(PLUSMNINF)) $(TD $(NAN)) $(TD yes)) * ) */ real tan(real x) @trusted pure nothrow @nogc { version (D_InlineAsm_X86) { asm pure nothrow @nogc { fld x[EBP] ; // load theta fxam ; // test for oddball values fstsw AX ; sahf ; jc trigerr ; // x is NAN, infinity, or empty // 387's can handle subnormals SC18: fptan ; fstsw AX ; sahf ; jnp Clear1 ; // C2 = 1 (x is out of range) // Do argument reduction to bring x into range fldpi ; fxch ; SC17: fprem1 ; fstsw AX ; sahf ; jp SC17 ; fstp ST(1) ; // remove pi from stack jmp SC18 ; trigerr: jnp Lret ; // if theta is NAN, return theta fstp ST(0) ; // dump theta } return real.nan; Clear1: asm pure nothrow @nogc{ fstp ST(0) ; // dump X, which is always 1 } Lret: {} } else version (D_InlineAsm_X86_64) { version (Win64) { asm pure nothrow @nogc { fld real ptr [RCX] ; // load theta } } else { asm pure nothrow @nogc { fld x[RBP] ; // load theta } } asm pure nothrow @nogc { fxam ; // test for oddball values fstsw AX ; test AH,1 ; jnz trigerr ; // x is NAN, infinity, or empty // 387's can handle subnormals SC18: fptan ; fstsw AX ; test AH,4 ; jz Clear1 ; // C2 = 1 (x is out of range) // Do argument reduction to bring x into range fldpi ; fxch ; SC17: fprem1 ; fstsw AX ; test AH,4 ; jnz SC17 ; fstp ST(1) ; // remove pi from stack jmp SC18 ; trigerr: test AH,4 ; jz Lret ; // if theta is NAN, return theta fstp ST(0) ; // dump theta } return real.nan; Clear1: asm pure nothrow @nogc{ fstp ST(0) ; // dump X, which is always 1 } Lret: {} } else { // Coefficients for tan(x) and PI/4 split into three parts. static if (floatTraits!real.realFormat == RealFormat.ieeeQuadruple) { static immutable real[6] P = [ 2.883414728874239697964612246732416606301E10L, -2.307030822693734879744223131873392503321E9L, 5.160188250214037865511600561074819366815E7L, -4.249691853501233575668486667664718192660E5L, 1.272297782199996882828849455156962260810E3L, -9.889929415807650724957118893791829849557E-1L ]; static immutable real[7] Q = [ 8.650244186622719093893836740197250197602E10L, -4.152206921457208101480801635640958361612E10L, 2.758476078803232151774723646710890525496E9L, -5.733709132766856723608447733926138506824E7L, 4.529422062441341616231663543669583527923E5L, -1.317243702830553658702531997959756728291E3L, 1.0 ]; enum real P1 = 7.853981633974483067550664827649598009884357452392578125E-1L; enum real P2 = 2.8605943630549158983813312792950660807511260829685741796657E-18L; enum real P3 = 2.1679525325309452561992610065108379921905808E-35L; } else { static immutable real[3] P = [ -1.7956525197648487798769E7L, 1.1535166483858741613983E6L, -1.3093693918138377764608E4L, ]; static immutable real[5] Q = [ -5.3869575592945462988123E7L, 2.5008380182335791583922E7L, -1.3208923444021096744731E6L, 1.3681296347069295467845E4L, 1.0000000000000000000000E0L, ]; enum real P1 = 7.853981554508209228515625E-1L; enum real P2 = 7.946627356147928367136046290398E-9L; enum real P3 = 3.061616997868382943065164830688E-17L; } // Special cases. if (x == 0.0 || isNaN(x)) return x; if (isInfinity(x)) return real.nan; // Make argument positive but save the sign. bool sign = false; if (signbit(x)) { sign = true; x = -x; } // Compute x mod PI/4. real y = floor(x / PI_4); // Strip high bits of integer part. real z = ldexp(y, -4); // Compute y - 16 * (y / 16). z = y - ldexp(floor(z), 4); // Integer and fraction part modulo one octant. int j = cast(int)(z); // Map zeros and singularities to origin. if (j & 1) { j += 1; y += 1.0; } z = ((x - y * P1) - y * P2) - y * P3; const real zz = z * z; if (zz > 1.0e-20L) y = z + z * (zz * poly(zz, P) / poly(zz, Q)); else y = z; if (j & 2) y = -1.0 / y; return (sign) ? -y : y; } } @safe nothrow @nogc unittest { static real[2][] vals = // angle,tan [ [ 0, 0], [ .5, .5463024898], [ 1, 1.557407725], [ 1.5, 14.10141995], [ 2, -2.185039863], [ 2.5,-.7470222972], [ 3, -.1425465431], [ 3.5, .3745856402], [ 4, 1.157821282], [ 4.5, 4.637332055], [ 5, -3.380515006], [ 5.5,-.9955840522], [ 6, -.2910061914], [ 6.5, .2202772003], [ 10, .6483608275], // special angles [ PI_4, 1], //[ PI_2, real.infinity], // PI_2 is not _exactly_ pi/2. [ 3*PI_4, -1], [ PI, 0], [ 5*PI_4, 1], //[ 3*PI_2, -real.infinity], [ 7*PI_4, -1], [ 2*PI, 0], ]; int i; for (i = 0; i < vals.length; i++) { real x = vals[i][0]; real r = vals[i][1]; real t = tan(x); //printf("tan(%Lg) = %Lg, should be %Lg\n", x, t, r); if (!isIdentical(r, t)) assert(fabs(r-t) <= .0000001); x = -x; r = -r; t = tan(x); //printf("tan(%Lg) = %Lg, should be %Lg\n", x, t, r); if (!isIdentical(r, t) && !(r != r && t != t)) assert(fabs(r-t) <= .0000001); } // overflow assert(isNaN(tan(real.infinity))); assert(isNaN(tan(-real.infinity))); // NaN propagation assert(isIdentical( tan(NaN(0x0123L)), NaN(0x0123L) )); } @system unittest { assert(equalsDigit(tan(PI / 3), std.math.sqrt(3.0), useDigits)); } /*************** * Calculates the arc cosine of x, * returning a value ranging from 0 to $(PI). * * $(TABLE_SV * $(TR $(TH x) $(TH acos(x)) $(TH invalid?)) * $(TR $(TD $(GT)1.0) $(TD $(NAN)) $(TD yes)) * $(TR $(TD $(LT)-1.0) $(TD $(NAN)) $(TD yes)) * $(TR $(TD $(NAN)) $(TD $(NAN)) $(TD yes)) * ) */ real acos(real x) @safe pure nothrow @nogc { return atan2(sqrt(1-x*x), x); } /// ditto double acos(double x) @safe pure nothrow @nogc { return acos(cast(real) x); } /// ditto float acos(float x) @safe pure nothrow @nogc { return acos(cast(real) x); } @system unittest { assert(equalsDigit(acos(0.5), std.math.PI / 3, useDigits)); } /*************** * Calculates the arc sine of x, * returning a value ranging from -$(PI)/2 to $(PI)/2. * * $(TABLE_SV * $(TR $(TH x) $(TH asin(x)) $(TH invalid?)) * $(TR $(TD $(PLUSMN)0.0) $(TD $(PLUSMN)0.0) $(TD no)) * $(TR $(TD $(GT)1.0) $(TD $(NAN)) $(TD yes)) * $(TR $(TD $(LT)-1.0) $(TD $(NAN)) $(TD yes)) * ) */ real asin(real x) @safe pure nothrow @nogc { return atan2(x, sqrt(1-x*x)); } /// ditto double asin(double x) @safe pure nothrow @nogc { return asin(cast(real) x); } /// ditto float asin(float x) @safe pure nothrow @nogc { return asin(cast(real) x); } @system unittest { assert(equalsDigit(asin(0.5), PI / 6, useDigits)); } /*************** * Calculates the arc tangent of x, * returning a value ranging from -$(PI)/2 to $(PI)/2. * * $(TABLE_SV * $(TR $(TH x) $(TH atan(x)) $(TH invalid?)) * $(TR $(TD $(PLUSMN)0.0) $(TD $(PLUSMN)0.0) $(TD no)) * $(TR $(TD $(PLUSMN)$(INFIN)) $(TD $(NAN)) $(TD yes)) * ) */ real atan(real x) @safe pure nothrow @nogc { version (InlineAsm_X86_Any) { return atan2(x, 1.0L); } else { // Coefficients for atan(x) static if (floatTraits!real.realFormat == RealFormat.ieeeQuadruple) { static immutable real[9] P = [ -6.880597774405940432145577545328795037141E2L, -2.514829758941713674909996882101723647996E3L, -3.696264445691821235400930243493001671932E3L, -2.792272753241044941703278827346430350236E3L, -1.148164399808514330375280133523543970854E3L, -2.497759878476618348858065206895055957104E2L, -2.548067867495502632615671450650071218995E1L, -8.768423468036849091777415076702113400070E-1L, -6.635810778635296712545011270011752799963E-4L ]; static immutable real[9] Q = [ 2.064179332321782129643673263598686441900E3L, 8.782996876218210302516194604424986107121E3L, 1.547394317752562611786521896296215170819E4L, 1.458510242529987155225086911411015961174E4L, 7.928572347062145288093560392463784743935E3L, 2.494680540950601626662048893678584497900E3L, 4.308348370818927353321556740027020068897E2L, 3.566239794444800849656497338030115886153E1L, 1.0 ]; } else { static immutable real[5] P = [ -5.0894116899623603312185E1L, -9.9988763777265819915721E1L, -6.3976888655834347413154E1L, -1.4683508633175792446076E1L, -8.6863818178092187535440E-1L, ]; static immutable real[6] Q = [ 1.5268235069887081006606E2L, 3.9157570175111990631099E2L, 3.6144079386152023162701E2L, 1.4399096122250781605352E2L, 2.2981886733594175366172E1L, 1.0000000000000000000000E0L, ]; } // tan(PI/8) enum real TAN_PI_8 = 0.414213562373095048801688724209698078569672L; // tan(3 * PI/8) enum real TAN3_PI_8 = 2.414213562373095048801688724209698078569672L; // Special cases. if (x == 0.0) return x; if (isInfinity(x)) return copysign(PI_2, x); // Make argument positive but save the sign. bool sign = false; if (signbit(x)) { sign = true; x = -x; } // Range reduction. real y; if (x > TAN3_PI_8) { y = PI_2; x = -(1.0 / x); } else if (x > TAN_PI_8) { y = PI_4; x = (x - 1.0)/(x + 1.0); } else y = 0.0; // Rational form in x^^2. const real z = x * x; y = y + (poly(z, P) / poly(z, Q)) * z * x + x; return (sign) ? -y : y; } } /// ditto double atan(double x) @safe pure nothrow @nogc { return atan(cast(real) x); } /// ditto float atan(float x) @safe pure nothrow @nogc { return atan(cast(real) x); } @system unittest { assert(equalsDigit(atan(std.math.sqrt(3.0)), PI / 3, useDigits)); } /*************** * Calculates the arc tangent of y / x, * returning a value ranging from -$(PI) to $(PI). * * $(TABLE_SV * $(TR $(TH y) $(TH x) $(TH atan(y, x))) * $(TR $(TD $(NAN)) $(TD anything) $(TD $(NAN)) ) * $(TR $(TD anything) $(TD $(NAN)) $(TD $(NAN)) ) * $(TR $(TD $(PLUSMN)0.0) $(TD $(GT)0.0) $(TD $(PLUSMN)0.0) ) * $(TR $(TD $(PLUSMN)0.0) $(TD +0.0) $(TD $(PLUSMN)0.0) ) * $(TR $(TD $(PLUSMN)0.0) $(TD $(LT)0.0) $(TD $(PLUSMN)$(PI))) * $(TR $(TD $(PLUSMN)0.0) $(TD -0.0) $(TD $(PLUSMN)$(PI))) * $(TR $(TD $(GT)0.0) $(TD $(PLUSMN)0.0) $(TD $(PI)/2) ) * $(TR $(TD $(LT)0.0) $(TD $(PLUSMN)0.0) $(TD -$(PI)/2) ) * $(TR $(TD $(GT)0.0) $(TD $(INFIN)) $(TD $(PLUSMN)0.0) ) * $(TR $(TD $(PLUSMN)$(INFIN)) $(TD anything) $(TD $(PLUSMN)$(PI)/2)) * $(TR $(TD $(GT)0.0) $(TD -$(INFIN)) $(TD $(PLUSMN)$(PI)) ) * $(TR $(TD $(PLUSMN)$(INFIN)) $(TD $(INFIN)) $(TD $(PLUSMN)$(PI)/4)) * $(TR $(TD $(PLUSMN)$(INFIN)) $(TD -$(INFIN)) $(TD $(PLUSMN)3$(PI)/4)) * ) */ real atan2(real y, real x) @trusted pure nothrow @nogc { version (InlineAsm_X86_Any) { version (Win64) { asm pure nothrow @nogc { naked; fld real ptr [RDX]; // y fld real ptr [RCX]; // x fpatan; ret; } } else { asm pure nothrow @nogc { fld y; fld x; fpatan; } } } else { // Special cases. if (isNaN(x) || isNaN(y)) return real.nan; if (y == 0.0) { if (x >= 0 && !signbit(x)) return copysign(0, y); else return copysign(PI, y); } if (x == 0.0) return copysign(PI_2, y); if (isInfinity(x)) { if (signbit(x)) { if (isInfinity(y)) return copysign(3*PI_4, y); else return copysign(PI, y); } else { if (isInfinity(y)) return copysign(PI_4, y); else return copysign(0.0, y); } } if (isInfinity(y)) return copysign(PI_2, y); // Call atan and determine the quadrant. real z = atan(y / x); if (signbit(x)) { if (signbit(y)) z = z - PI; else z = z + PI; } if (z == 0.0) return copysign(z, y); return z; } } /// ditto double atan2(double y, double x) @safe pure nothrow @nogc { return atan2(cast(real) y, cast(real) x); } /// ditto float atan2(float y, float x) @safe pure nothrow @nogc { return atan2(cast(real) y, cast(real) x); } @system unittest { assert(equalsDigit(atan2(1.0L, std.math.sqrt(3.0L)), PI / 6, useDigits)); } /*********************************** * Calculates the hyperbolic cosine of x. * * $(TABLE_SV * $(TR $(TH x) $(TH cosh(x)) $(TH invalid?)) * $(TR $(TD $(PLUSMN)$(INFIN)) $(TD $(PLUSMN)0.0) $(TD no) ) * ) */ real cosh(real x) @safe pure nothrow @nogc { // cosh = (exp(x)+exp(-x))/2. // The naive implementation works correctly. const real y = exp(x); return (y + 1.0/y) * 0.5; } /// ditto double cosh(double x) @safe pure nothrow @nogc { return cosh(cast(real) x); } /// ditto float cosh(float x) @safe pure nothrow @nogc { return cosh(cast(real) x); } @system unittest { assert(equalsDigit(cosh(1.0), (E + 1.0 / E) / 2, useDigits)); } /*********************************** * Calculates the hyperbolic sine of x. * * $(TABLE_SV * $(TR $(TH x) $(TH sinh(x)) $(TH invalid?)) * $(TR $(TD $(PLUSMN)0.0) $(TD $(PLUSMN)0.0) $(TD no)) * $(TR $(TD $(PLUSMN)$(INFIN)) $(TD $(PLUSMN)$(INFIN)) $(TD no)) * ) */ real sinh(real x) @safe pure nothrow @nogc { // sinh(x) = (exp(x)-exp(-x))/2; // Very large arguments could cause an overflow, but // the maximum value of x for which exp(x) + exp(-x)) != exp(x) // is x = 0.5 * (real.mant_dig) * LN2. // = 22.1807 for real80. if (fabs(x) > real.mant_dig * LN2) { return copysign(0.5 * exp(fabs(x)), x); } const real y = expm1(x); return 0.5 * y / (y+1) * (y+2); } /// ditto double sinh(double x) @safe pure nothrow @nogc { return sinh(cast(real) x); } /// ditto float sinh(float x) @safe pure nothrow @nogc { return sinh(cast(real) x); } @system unittest { assert(equalsDigit(sinh(1.0), (E - 1.0 / E) / 2, useDigits)); } /*********************************** * Calculates the hyperbolic tangent of x. * * $(TABLE_SV * $(TR $(TH x) $(TH tanh(x)) $(TH invalid?)) * $(TR $(TD $(PLUSMN)0.0) $(TD $(PLUSMN)0.0) $(TD no) ) * $(TR $(TD $(PLUSMN)$(INFIN)) $(TD $(PLUSMN)1.0) $(TD no)) * ) */ real tanh(real x) @safe pure nothrow @nogc { // tanh(x) = (exp(x) - exp(-x))/(exp(x)+exp(-x)) if (fabs(x) > real.mant_dig * LN2) { return copysign(1, x); } const real y = expm1(2*x); return y / (y + 2); } /// ditto double tanh(double x) @safe pure nothrow @nogc { return tanh(cast(real) x); } /// ditto float tanh(float x) @safe pure nothrow @nogc { return tanh(cast(real) x); } @system unittest { assert(equalsDigit(tanh(1.0), sinh(1.0) / cosh(1.0), 15)); } package: /* Returns cosh(x) + I * sinh(x) * Only one call to exp() is performed. */ creal coshisinh(real x) @safe pure nothrow @nogc { // See comments for cosh, sinh. if (fabs(x) > real.mant_dig * LN2) { const real y = exp(fabs(x)); return y * 0.5 + 0.5i * copysign(y, x); } else { const real y = expm1(x); return (y + 1.0 + 1.0/(y + 1.0)) * 0.5 + 0.5i * y / (y+1) * (y+2); } } @safe pure nothrow @nogc unittest { creal c = coshisinh(3.0L); assert(c.re == cosh(3.0L)); assert(c.im == sinh(3.0L)); } public: /*********************************** * Calculates the inverse hyperbolic cosine of x. * * Mathematically, acosh(x) = log(x + sqrt( x*x - 1)) * * $(TABLE_DOMRG * $(DOMAIN 1..$(INFIN)), * $(RANGE 0..$(INFIN)) * ) * * $(TABLE_SV * $(SVH x, acosh(x) ) * $(SV $(NAN), $(NAN) ) * $(SV $(LT)1, $(NAN) ) * $(SV 1, 0 ) * $(SV +$(INFIN),+$(INFIN)) * ) */ real acosh(real x) @safe pure nothrow @nogc { if (x > 1/real.epsilon) return LN2 + log(x); else return log(x + sqrt(x*x - 1)); } /// ditto double acosh(double x) @safe pure nothrow @nogc { return acosh(cast(real) x); } /// ditto float acosh(float x) @safe pure nothrow @nogc { return acosh(cast(real) x); } @system unittest { assert(isNaN(acosh(0.9))); assert(isNaN(acosh(real.nan))); assert(acosh(1.0)==0.0); assert(acosh(real.infinity) == real.infinity); assert(isNaN(acosh(0.5))); assert(equalsDigit(acosh(cosh(3.0)), 3, useDigits)); } /*********************************** * Calculates the inverse hyperbolic sine of x. * * Mathematically, * --------------- * asinh(x) = log( x + sqrt( x*x + 1 )) // if x >= +0 * asinh(x) = -log(-x + sqrt( x*x + 1 )) // if x <= -0 * ------------- * * $(TABLE_SV * $(SVH x, asinh(x) ) * $(SV $(NAN), $(NAN) ) * $(SV $(PLUSMN)0, $(PLUSMN)0 ) * $(SV $(PLUSMN)$(INFIN),$(PLUSMN)$(INFIN)) * ) */ real asinh(real x) @safe pure nothrow @nogc { return (fabs(x) > 1 / real.epsilon) // beyond this point, x*x + 1 == x*x ? copysign(LN2 + log(fabs(x)), x) // sqrt(x*x + 1) == 1 + x * x / ( 1 + sqrt(x*x + 1) ) : copysign(log1p(fabs(x) + x*x / (1 + sqrt(x*x + 1)) ), x); } /// ditto double asinh(double x) @safe pure nothrow @nogc { return asinh(cast(real) x); } /// ditto float asinh(float x) @safe pure nothrow @nogc { return asinh(cast(real) x); } @system unittest { assert(isIdentical(asinh(0.0), 0.0)); assert(isIdentical(asinh(-0.0), -0.0)); assert(asinh(real.infinity) == real.infinity); assert(asinh(-real.infinity) == -real.infinity); assert(isNaN(asinh(real.nan))); assert(equalsDigit(asinh(sinh(3.0)), 3, useDigits)); } /*********************************** * Calculates the inverse hyperbolic tangent of x, * returning a value from ranging from -1 to 1. * * Mathematically, atanh(x) = log( (1+x)/(1-x) ) / 2 * * $(TABLE_DOMRG * $(DOMAIN -$(INFIN)..$(INFIN)), * $(RANGE -1 .. 1) * ) * $(BR) * $(TABLE_SV * $(SVH x, acosh(x) ) * $(SV $(NAN), $(NAN) ) * $(SV $(PLUSMN)0, $(PLUSMN)0) * $(SV -$(INFIN), -0) * ) */ real atanh(real x) @safe pure nothrow @nogc { // log( (1+x)/(1-x) ) == log ( 1 + (2*x)/(1-x) ) return 0.5 * log1p( 2 * x / (1 - x) ); } /// ditto double atanh(double x) @safe pure nothrow @nogc { return atanh(cast(real) x); } /// ditto float atanh(float x) @safe pure nothrow @nogc { return atanh(cast(real) x); } @system unittest { assert(isIdentical(atanh(0.0), 0.0)); assert(isIdentical(atanh(-0.0),-0.0)); assert(isNaN(atanh(real.nan))); assert(isNaN(atanh(-real.infinity))); assert(atanh(0.0) == 0); assert(equalsDigit(atanh(tanh(0.5L)), 0.5, useDigits)); } /***************************************** * Returns x rounded to a long value using the current rounding mode. * If the integer value of x is * greater than long.max, the result is * indeterminate. */ long rndtol(real x) @nogc @safe pure nothrow { pragma(inline, true); return core.math.rndtol(x); } //FIXME ///ditto long rndtol(double x) @safe pure nothrow @nogc { return rndtol(cast(real) x); } //FIXME ///ditto long rndtol(float x) @safe pure nothrow @nogc { return rndtol(cast(real) x); } @safe unittest { long function(real) prndtol = &rndtol; assert(prndtol != null); } /***************************************** * Returns x rounded to a long value using the FE_TONEAREST rounding mode. * If the integer value of x is * greater than long.max, the result is * indeterminate. */ extern (C) real rndtonl(real x); /*************************************** * Compute square root of x. * * $(TABLE_SV * $(TR $(TH x) $(TH sqrt(x)) $(TH invalid?)) * $(TR $(TD -0.0) $(TD -0.0) $(TD no)) * $(TR $(TD $(LT)0.0) $(TD $(NAN)) $(TD yes)) * $(TR $(TD +$(INFIN)) $(TD +$(INFIN)) $(TD no)) * ) */ float sqrt(float x) @nogc @safe pure nothrow { pragma(inline, true); return core.math.sqrt(x); } /// ditto double sqrt(double x) @nogc @safe pure nothrow { pragma(inline, true); return core.math.sqrt(x); } /// ditto real sqrt(real x) @nogc @safe pure nothrow { pragma(inline, true); return core.math.sqrt(x); } @safe pure nothrow @nogc unittest { //ctfe enum ZX80 = sqrt(7.0f); enum ZX81 = sqrt(7.0); enum ZX82 = sqrt(7.0L); assert(isNaN(sqrt(-1.0f))); assert(isNaN(sqrt(-1.0))); assert(isNaN(sqrt(-1.0L))); } @safe unittest { float function(float) psqrtf = &sqrt; assert(psqrtf != null); double function(double) psqrtd = &sqrt; assert(psqrtd != null); real function(real) psqrtr = &sqrt; assert(psqrtr != null); } creal sqrt(creal z) @nogc @safe pure nothrow { creal c; real x,y,w,r; if (z == 0) { c = 0 + 0i; } else { const real z_re = z.re; const real z_im = z.im; x = fabs(z_re); y = fabs(z_im); if (x >= y) { r = y / x; w = sqrt(x) * sqrt(0.5 * (1 + sqrt(1 + r * r))); } else { r = x / y; w = sqrt(y) * sqrt(0.5 * (r + sqrt(1 + r * r))); } if (z_re >= 0) { c = w + (z_im / (w + w)) * 1.0i; } else { if (z_im < 0) w = -w; c = z_im / (w + w) + w * 1.0i; } } return c; } /** * Calculates e$(SUPERSCRIPT x). * * $(TABLE_SV * $(TR $(TH x) $(TH e$(SUPERSCRIPT x)) ) * $(TR $(TD +$(INFIN)) $(TD +$(INFIN)) ) * $(TR $(TD -$(INFIN)) $(TD +0.0) ) * $(TR $(TD $(NAN)) $(TD $(NAN)) ) * ) */ real exp(real x) @trusted pure nothrow @nogc { version (D_InlineAsm_X86) { // e^^x = 2^^(LOG2E*x) // (This is valid because the overflow & underflow limits for exp // and exp2 are so similar). return exp2(LOG2E*x); } else version (D_InlineAsm_X86_64) { // e^^x = 2^^(LOG2E*x) // (This is valid because the overflow & underflow limits for exp // and exp2 are so similar). return exp2(LOG2E*x); } else { alias F = floatTraits!real; static if (F.realFormat == RealFormat.ieeeDouble) { // Coefficients for exp(x) static immutable real[3] P = [ 9.99999999999999999910E-1L, 3.02994407707441961300E-2L, 1.26177193074810590878E-4L, ]; static immutable real[4] Q = [ 2.00000000000000000009E0L, 2.27265548208155028766E-1L, 2.52448340349684104192E-3L, 3.00198505138664455042E-6L, ]; // C1 + C2 = LN2. enum real C1 = 6.93145751953125E-1; enum real C2 = 1.42860682030941723212E-6; // Overflow and Underflow limits. enum real OF = 7.09782712893383996732E2; // ln((1-2^-53) * 2^1024) enum real UF = -7.451332191019412076235E2; // ln(2^-1075) } else static if (F.realFormat == RealFormat.ieeeExtended) { // Coefficients for exp(x) static immutable real[3] P = [ 9.9999999999999999991025E-1L, 3.0299440770744196129956E-2L, 1.2617719307481059087798E-4L, ]; static immutable real[4] Q = [ 2.0000000000000000000897E0L, 2.2726554820815502876593E-1L, 2.5244834034968410419224E-3L, 3.0019850513866445504159E-6L, ]; // C1 + C2 = LN2. enum real C1 = 6.9314575195312500000000E-1L; enum real C2 = 1.4286068203094172321215E-6L; // Overflow and Underflow limits. enum real OF = 1.1356523406294143949492E4L; // ln((1-2^-64) * 2^16384) enum real UF = -1.13994985314888605586758E4L; // ln(2^-16446) } else static if (F.realFormat == RealFormat.ieeeQuadruple) { // Coefficients for exp(x) - 1 static immutable real[5] P = [ 9.999999999999999999999999999999999998502E-1L, 3.508710990737834361215404761139478627390E-2L, 2.708775201978218837374512615596512792224E-4L, 6.141506007208645008909088812338454698548E-7L, 3.279723985560247033712687707263393506266E-10L ]; static immutable real[6] Q = [ 2.000000000000000000000000000000000000150E0, 2.368408864814233538909747618894558968880E-1L, 3.611828913847589925056132680618007270344E-3L, 1.504792651814944826817779302637284053660E-5L, 1.771372078166251484503904874657985291164E-8L, 2.980756652081995192255342779918052538681E-12L ]; // C1 + C2 = LN2. enum real C1 = 6.93145751953125E-1L; enum real C2 = 1.428606820309417232121458176568075500134E-6L; // Overflow and Underflow limits. enum real OF = 1.135583025911358400418251384584930671458833e4L; enum real UF = -1.143276959615573793352782661133116431383730e4L; } else static assert(0, "Not implemented for this architecture"); // Special cases. Raises an overflow or underflow flag accordingly, // except in the case for CTFE, where there are no hardware controls. if (isNaN(x)) return x; if (x > OF) { if (__ctfe) return real.infinity; else return real.max * copysign(real.max, real.infinity); } if (x < UF) { if (__ctfe) return 0.0; else return real.min_normal * copysign(real.min_normal, 0.0); } // Express: e^^x = e^^g * 2^^n // = e^^g * e^^(n * LOG2E) // = e^^(g + n * LOG2E) int n = cast(int) floor(LOG2E * x + 0.5); x -= n * C1; x -= n * C2; // Rational approximation for exponential of the fractional part: // e^^x = 1 + 2x P(x^^2) / (Q(x^^2) - P(x^^2)) const real xx = x * x; const real px = x * poly(xx, P); x = px / (poly(xx, Q) - px); x = 1.0 + ldexp(x, 1); // Scale by power of 2. x = ldexp(x, n); return x; } } /// ditto double exp(double x) @safe pure nothrow @nogc { return exp(cast(real) x); } /// ditto float exp(float x) @safe pure nothrow @nogc { return exp(cast(real) x); } @system unittest { assert(equalsDigit(exp(3.0L), E * E * E, useDigits)); } /** * Calculates the value of the natural logarithm base (e) * raised to the power of x, minus 1. * * For very small x, expm1(x) is more accurate * than exp(x)-1. * * $(TABLE_SV * $(TR $(TH x) $(TH e$(SUPERSCRIPT x)-1) ) * $(TR $(TD $(PLUSMN)0.0) $(TD $(PLUSMN)0.0) ) * $(TR $(TD +$(INFIN)) $(TD +$(INFIN)) ) * $(TR $(TD -$(INFIN)) $(TD -1.0) ) * $(TR $(TD $(NAN)) $(TD $(NAN)) ) * ) */ real expm1(real x) @trusted pure nothrow @nogc { version (D_InlineAsm_X86) { enum PARAMSIZE = (real.sizeof+3)&(0xFFFF_FFFC); // always a multiple of 4 asm pure nothrow @nogc { /* expm1() for x87 80-bit reals, IEEE754-2008 conformant. * Author: Don Clugston. * * expm1(x) = 2^^(rndint(y))* 2^^(y-rndint(y)) - 1 where y = LN2*x. * = 2rndy * 2ym1 + 2rndy - 1, where 2rndy = 2^^(rndint(y)) * and 2ym1 = (2^^(y-rndint(y))-1). * If 2rndy < 0.5*real.epsilon, result is -1. * Implementation is otherwise the same as for exp2() */ naked; fld real ptr [ESP+4] ; // x mov AX, [ESP+4+8]; // AX = exponent and sign sub ESP, 12+8; // Create scratch space on the stack // [ESP,ESP+2] = scratchint // [ESP+4..+6, +8..+10, +10] = scratchreal // set scratchreal mantissa = 1.0 mov dword ptr [ESP+8], 0; mov dword ptr [ESP+8+4], 0x80000000; and AX, 0x7FFF; // drop sign bit cmp AX, 0x401D; // avoid InvalidException in fist jae L_extreme; fldl2e; fmulp ST(1), ST; // y = x*log2(e) fist dword ptr [ESP]; // scratchint = rndint(y) fisub dword ptr [ESP]; // y - rndint(y) // and now set scratchreal exponent mov EAX, [ESP]; add EAX, 0x3fff; jle short L_largenegative; cmp EAX,0x8000; jge short L_largepositive; mov [ESP+8+8],AX; f2xm1; // 2ym1 = 2^^(y-rndint(y)) -1 fld real ptr [ESP+8] ; // 2rndy = 2^^rndint(y) fmul ST(1), ST; // ST=2rndy, ST(1)=2rndy*2ym1 fld1; fsubp ST(1), ST; // ST = 2rndy-1, ST(1) = 2rndy * 2ym1 - 1 faddp ST(1), ST; // ST = 2rndy * 2ym1 + 2rndy - 1 add ESP,12+8; ret PARAMSIZE; L_extreme: // Extreme exponent. X is very large positive, very // large negative, infinity, or NaN. fxam; fstsw AX; test AX, 0x0400; // NaN_or_zero, but we already know x != 0 jz L_was_nan; // if x is NaN, returns x test AX, 0x0200; jnz L_largenegative; L_largepositive: // Set scratchreal = real.max. // squaring it will create infinity, and set overflow flag. mov word ptr [ESP+8+8], 0x7FFE; fstp ST(0); fld real ptr [ESP+8]; // load scratchreal fmul ST(0), ST; // square it, to create havoc! L_was_nan: add ESP,12+8; ret PARAMSIZE; L_largenegative: fstp ST(0); fld1; fchs; // return -1. Underflow flag is not set. add ESP,12+8; ret PARAMSIZE; } } else version (D_InlineAsm_X86_64) { asm pure nothrow @nogc { naked; } version (Win64) { asm pure nothrow @nogc { fld real ptr [RCX]; // x mov AX,[RCX+8]; // AX = exponent and sign } } else { asm pure nothrow @nogc { fld real ptr [RSP+8]; // x mov AX,[RSP+8+8]; // AX = exponent and sign } } asm pure nothrow @nogc { /* expm1() for x87 80-bit reals, IEEE754-2008 conformant. * Author: Don Clugston. * * expm1(x) = 2^(rndint(y))* 2^(y-rndint(y)) - 1 where y = LN2*x. * = 2rndy * 2ym1 + 2rndy - 1, where 2rndy = 2^(rndint(y)) * and 2ym1 = (2^(y-rndint(y))-1). * If 2rndy < 0.5*real.epsilon, result is -1. * Implementation is otherwise the same as for exp2() */ sub RSP, 24; // Create scratch space on the stack // [RSP,RSP+2] = scratchint // [RSP+4..+6, +8..+10, +10] = scratchreal // set scratchreal mantissa = 1.0 mov dword ptr [RSP+8], 0; mov dword ptr [RSP+8+4], 0x80000000; and AX, 0x7FFF; // drop sign bit cmp AX, 0x401D; // avoid InvalidException in fist jae L_extreme; fldl2e; fmul ; // y = x*log2(e) fist dword ptr [RSP]; // scratchint = rndint(y) fisub dword ptr [RSP]; // y - rndint(y) // and now set scratchreal exponent mov EAX, [RSP]; add EAX, 0x3fff; jle short L_largenegative; cmp EAX,0x8000; jge short L_largepositive; mov [RSP+8+8],AX; f2xm1; // 2^(y-rndint(y)) -1 fld real ptr [RSP+8] ; // 2^rndint(y) fmul ST(1), ST; fld1; fsubp ST(1), ST; fadd; add RSP,24; ret; L_extreme: // Extreme exponent. X is very large positive, very // large negative, infinity, or NaN. fxam; fstsw AX; test AX, 0x0400; // NaN_or_zero, but we already know x != 0 jz L_was_nan; // if x is NaN, returns x test AX, 0x0200; jnz L_largenegative; L_largepositive: // Set scratchreal = real.max. // squaring it will create infinity, and set overflow flag. mov word ptr [RSP+8+8], 0x7FFE; fstp ST(0); fld real ptr [RSP+8]; // load scratchreal fmul ST(0), ST; // square it, to create havoc! L_was_nan: add RSP,24; ret; L_largenegative: fstp ST(0); fld1; fchs; // return -1. Underflow flag is not set. add RSP,24; ret; } } else { // Coefficients for exp(x) - 1 and overflow/underflow limits. static if (floatTraits!real.realFormat == RealFormat.ieeeQuadruple) { static immutable real[8] P = [ 2.943520915569954073888921213330863757240E8L, -5.722847283900608941516165725053359168840E7L, 8.944630806357575461578107295909719817253E6L, -7.212432713558031519943281748462837065308E5L, 4.578962475841642634225390068461943438441E4L, -1.716772506388927649032068540558788106762E3L, 4.401308817383362136048032038528753151144E1L, -4.888737542888633647784737721812546636240E-1L ]; static immutable real[9] Q = [ 1.766112549341972444333352727998584753865E9L, -7.848989743695296475743081255027098295771E8L, 1.615869009634292424463780387327037251069E8L, -2.019684072836541751428967854947019415698E7L, 1.682912729190313538934190635536631941751E6L, -9.615511549171441430850103489315371768998E4L, 3.697714952261803935521187272204485251835E3L, -8.802340681794263968892934703309274564037E1L, 1.0 ]; enum real OF = 1.1356523406294143949491931077970764891253E4L; enum real UF = -1.143276959615573793352782661133116431383730e4L; } else { static immutable real[5] P = [ -1.586135578666346600772998894928250240826E4L, 2.642771505685952966904660652518429479531E3L, -3.423199068835684263987132888286791620673E2L, 1.800826371455042224581246202420972737840E1L, -5.238523121205561042771939008061958820811E-1L, ]; static immutable real[6] Q = [ -9.516813471998079611319047060563358064497E4L, 3.964866271411091674556850458227710004570E4L, -7.207678383830091850230366618190187434796E3L, 7.206038318724600171970199625081491823079E2L, -4.002027679107076077238836622982900945173E1L, 1.0 ]; enum real OF = 1.1356523406294143949492E4L; enum real UF = -4.5054566736396445112120088E1L; } // C1 + C2 = LN2. enum real C1 = 6.9314575195312500000000E-1L; enum real C2 = 1.428606820309417232121458176568075500134E-6L; // Special cases. Raises an overflow flag, except in the case // for CTFE, where there are no hardware controls. if (x > OF) { if (__ctfe) return real.infinity; else return real.max * copysign(real.max, real.infinity); } if (x == 0.0) return x; if (x < UF) return -1.0; // Express x = LN2 (n + remainder), remainder not exceeding 1/2. int n = cast(int) floor(0.5 + x / LN2); x -= n * C1; x -= n * C2; // Rational approximation: // exp(x) - 1 = x + 0.5 x^^2 + x^^3 P(x) / Q(x) real px = x * poly(x, P); real qx = poly(x, Q); const real xx = x * x; qx = x + (0.5 * xx + xx * px / qx); // We have qx = exp(remainder LN2) - 1, so: // exp(x) - 1 = 2^^n (qx + 1) - 1 = 2^^n qx + 2^^n - 1. px = ldexp(1.0, n); x = px * qx + (px - 1.0); return x; } } /** * Calculates 2$(SUPERSCRIPT x). * * $(TABLE_SV * $(TR $(TH x) $(TH exp2(x)) ) * $(TR $(TD +$(INFIN)) $(TD +$(INFIN)) ) * $(TR $(TD -$(INFIN)) $(TD +0.0) ) * $(TR $(TD $(NAN)) $(TD $(NAN)) ) * ) */ pragma(inline, true) real exp2(real x) @nogc @trusted pure nothrow { version (InlineAsm_X86_Any) { if (!__ctfe) return exp2Asm(x); else return exp2Impl(x); } else { return exp2Impl(x); } } version (InlineAsm_X86_Any) private real exp2Asm(real x) @nogc @trusted pure nothrow { version (D_InlineAsm_X86) { enum PARAMSIZE = (real.sizeof+3)&(0xFFFF_FFFC); // always a multiple of 4 asm pure nothrow @nogc { /* exp2() for x87 80-bit reals, IEEE754-2008 conformant. * Author: Don Clugston. * * exp2(x) = 2^^(rndint(x))* 2^^(y-rndint(x)) * The trick for high performance is to avoid the fscale(28cycles on core2), * frndint(19 cycles), leaving f2xm1(19 cycles) as the only slow instruction. * * We can do frndint by using fist. BUT we can't use it for huge numbers, * because it will set the Invalid Operation flag if overflow or NaN occurs. * Fortunately, whenever this happens the result would be zero or infinity. * * We can perform fscale by directly poking into the exponent. BUT this doesn't * work for the (very rare) cases where the result is subnormal. So we fall back * to the slow method in that case. */ naked; fld real ptr [ESP+4] ; // x mov AX, [ESP+4+8]; // AX = exponent and sign sub ESP, 12+8; // Create scratch space on the stack // [ESP,ESP+2] = scratchint // [ESP+4..+6, +8..+10, +10] = scratchreal // set scratchreal mantissa = 1.0 mov dword ptr [ESP+8], 0; mov dword ptr [ESP+8+4], 0x80000000; and AX, 0x7FFF; // drop sign bit cmp AX, 0x401D; // avoid InvalidException in fist jae L_extreme; fist dword ptr [ESP]; // scratchint = rndint(x) fisub dword ptr [ESP]; // x - rndint(x) // and now set scratchreal exponent mov EAX, [ESP]; add EAX, 0x3fff; jle short L_subnormal; cmp EAX,0x8000; jge short L_overflow; mov [ESP+8+8],AX; L_normal: f2xm1; fld1; faddp ST(1), ST; // 2^^(x-rndint(x)) fld real ptr [ESP+8] ; // 2^^rndint(x) add ESP,12+8; fmulp ST(1), ST; ret PARAMSIZE; L_subnormal: // Result will be subnormal. // In this rare case, the simple poking method doesn't work. // The speed doesn't matter, so use the slow fscale method. fild dword ptr [ESP]; // scratchint fld1; fscale; fstp real ptr [ESP+8]; // scratchreal = 2^^scratchint fstp ST(0); // drop scratchint jmp L_normal; L_extreme: // Extreme exponent. X is very large positive, very // large negative, infinity, or NaN. fxam; fstsw AX; test AX, 0x0400; // NaN_or_zero, but we already know x != 0 jz L_was_nan; // if x is NaN, returns x // set scratchreal = real.min_normal // squaring it will return 0, setting underflow flag mov word ptr [ESP+8+8], 1; test AX, 0x0200; jnz L_waslargenegative; L_overflow: // Set scratchreal = real.max. // squaring it will create infinity, and set overflow flag. mov word ptr [ESP+8+8], 0x7FFE; L_waslargenegative: fstp ST(0); fld real ptr [ESP+8]; // load scratchreal fmul ST(0), ST; // square it, to create havoc! L_was_nan: add ESP,12+8; ret PARAMSIZE; } } else version (D_InlineAsm_X86_64) { asm pure nothrow @nogc { naked; } version (Win64) { asm pure nothrow @nogc { fld real ptr [RCX]; // x mov AX,[RCX+8]; // AX = exponent and sign } } else { asm pure nothrow @nogc { fld real ptr [RSP+8]; // x mov AX,[RSP+8+8]; // AX = exponent and sign } } asm pure nothrow @nogc { /* exp2() for x87 80-bit reals, IEEE754-2008 conformant. * Author: Don Clugston. * * exp2(x) = 2^(rndint(x))* 2^(y-rndint(x)) * The trick for high performance is to avoid the fscale(28cycles on core2), * frndint(19 cycles), leaving f2xm1(19 cycles) as the only slow instruction. * * We can do frndint by using fist. BUT we can't use it for huge numbers, * because it will set the Invalid Operation flag is overflow or NaN occurs. * Fortunately, whenever this happens the result would be zero or infinity. * * We can perform fscale by directly poking into the exponent. BUT this doesn't * work for the (very rare) cases where the result is subnormal. So we fall back * to the slow method in that case. */ sub RSP, 24; // Create scratch space on the stack // [RSP,RSP+2] = scratchint // [RSP+4..+6, +8..+10, +10] = scratchreal // set scratchreal mantissa = 1.0 mov dword ptr [RSP+8], 0; mov dword ptr [RSP+8+4], 0x80000000; and AX, 0x7FFF; // drop sign bit cmp AX, 0x401D; // avoid InvalidException in fist jae L_extreme; fist dword ptr [RSP]; // scratchint = rndint(x) fisub dword ptr [RSP]; // x - rndint(x) // and now set scratchreal exponent mov EAX, [RSP]; add EAX, 0x3fff; jle short L_subnormal; cmp EAX,0x8000; jge short L_overflow; mov [RSP+8+8],AX; L_normal: f2xm1; fld1; fadd; // 2^(x-rndint(x)) fld real ptr [RSP+8] ; // 2^rndint(x) add RSP,24; fmulp ST(1), ST; ret; L_subnormal: // Result will be subnormal. // In this rare case, the simple poking method doesn't work. // The speed doesn't matter, so use the slow fscale method. fild dword ptr [RSP]; // scratchint fld1; fscale; fstp real ptr [RSP+8]; // scratchreal = 2^scratchint fstp ST(0); // drop scratchint jmp L_normal; L_extreme: // Extreme exponent. X is very large positive, very // large negative, infinity, or NaN. fxam; fstsw AX; test AX, 0x0400; // NaN_or_zero, but we already know x != 0 jz L_was_nan; // if x is NaN, returns x // set scratchreal = real.min // squaring it will return 0, setting underflow flag mov word ptr [RSP+8+8], 1; test AX, 0x0200; jnz L_waslargenegative; L_overflow: // Set scratchreal = real.max. // squaring it will create infinity, and set overflow flag. mov word ptr [RSP+8+8], 0x7FFE; L_waslargenegative: fstp ST(0); fld real ptr [RSP+8]; // load scratchreal fmul ST(0), ST; // square it, to create havoc! L_was_nan: add RSP,24; ret; } } else static assert(0); } private real exp2Impl(real x) @nogc @trusted pure nothrow { // Coefficients for exp2(x) static if (floatTraits!real.realFormat == RealFormat.ieeeQuadruple) { static immutable real[5] P = [ 9.079594442980146270952372234833529694788E12L, 1.530625323728429161131811299626419117557E11L, 5.677513871931844661829755443994214173883E8L, 6.185032670011643762127954396427045467506E5L, 1.587171580015525194694938306936721666031E2L ]; static immutable real[6] Q = [ 2.619817175234089411411070339065679229869E13L, 1.490560994263653042761789432690793026977E12L, 1.092141473886177435056423606755843616331E10L, 2.186249607051644894762167991800811827835E7L, 1.236602014442099053716561665053645270207E4L, 1.0 ]; } else { static immutable real[3] P = [ 2.0803843631901852422887E6L, 3.0286971917562792508623E4L, 6.0614853552242266094567E1L, ]; static immutable real[4] Q = [ 6.0027204078348487957118E6L, 3.2772515434906797273099E5L, 1.7492876999891839021063E3L, 1.0000000000000000000000E0L, ]; } // Overflow and Underflow limits. enum real OF = 16_384.0L; enum real UF = -16_382.0L; // Special cases. Raises an overflow or underflow flag accordingly, // except in the case for CTFE, where there are no hardware controls. if (isNaN(x)) return x; if (x > OF) { if (__ctfe) return real.infinity; else return real.max * copysign(real.max, real.infinity); } if (x < UF) { if (__ctfe) return 0.0; else return real.min_normal * copysign(real.min_normal, 0.0); } // Separate into integer and fractional parts. int n = cast(int) floor(x + 0.5); x -= n; // Rational approximation: // exp2(x) = 1.0 + 2x P(x^^2) / (Q(x^^2) - P(x^^2)) const real xx = x * x; const real px = x * poly(xx, P); x = px / (poly(xx, Q) - px); x = 1.0 + ldexp(x, 1); // Scale by power of 2. x = ldexp(x, n); return x; } /// @safe unittest { assert(feqrel(exp2(0.5L), SQRT2) >= real.mant_dig -1); assert(exp2(8.0L) == 256.0); assert(exp2(-9.0L)== 1.0L/512.0); } @safe unittest { version (CRuntime_Microsoft) {} else // aexp2/exp2f/exp2l not implemented { assert( core.stdc.math.exp2f(0.0f) == 1 ); assert( core.stdc.math.exp2 (0.0) == 1 ); assert( core.stdc.math.exp2l(0.0L) == 1 ); } } @system unittest { FloatingPointControl ctrl; if (FloatingPointControl.hasExceptionTraps) ctrl.disableExceptions(FloatingPointControl.allExceptions); ctrl.rounding = FloatingPointControl.roundToNearest; static if (real.mant_dig == 113) { static immutable real[2][] exptestpoints = [ // x exp(x) [ 1.0L, E ], [ 0.5L, 0x1.a61298e1e069bc972dfefab6df34p+0L ], [ 3.0L, E*E*E ], [ 0x1.6p+13L, 0x1.6e509d45728655cdb4840542acb5p+16250L ], // near overflow [ 0x1.7p+13L, real.infinity ], // close overflow [ 0x1p+80L, real.infinity ], // far overflow [ real.infinity, real.infinity ], [-0x1.18p+13L, 0x1.5e4bf54b4807034ea97fef0059a6p-12927L ], // near underflow [-0x1.625p+13L, 0x1.a6bd68a39d11fec3a250cd97f524p-16358L ], // ditto [-0x1.62dafp+13L, 0x0.cb629e9813b80ed4d639e875be6cp-16382L ], // near underflow - subnormal [-0x1.6549p+13L, 0x0.0000000000000000000000000001p-16382L ], // ditto [-0x1.655p+13L, 0 ], // close underflow [-0x1p+30L, 0 ], // far underflow ]; } else static if (real.mant_dig == 64) // 80-bit reals { static immutable real[2][] exptestpoints = [ // x exp(x) [ 1.0L, E ], [ 0.5L, 0x1.a61298e1e069bc97p+0L ], [ 3.0L, E*E*E ], [ 0x1.1p+13L, 0x1.29aeffefc8ec645p+12557L ], // near overflow [ 0x1.7p+13L, real.infinity ], // close overflow [ 0x1p+80L, real.infinity ], // far overflow [ real.infinity, real.infinity ], [-0x1.18p+13L, 0x1.5e4bf54b4806db9p-12927L ], // near underflow [-0x1.625p+13L, 0x1.a6bd68a39d11f35cp-16358L ], // ditto [-0x1.62dafp+13L, 0x1.96c53d30277021dp-16383L ], // near underflow - subnormal [-0x1.643p+13L, 0x1p-16444L ], // ditto [-0x1.645p+13L, 0 ], // close underflow [-0x1p+30L, 0 ], // far underflow ]; } else static if (real.mant_dig == 53) // 64-bit reals { static immutable real[2][] exptestpoints = [ // x, exp(x) [ 1.0L, E ], [ 0.5L, 0x1.a61298e1e069cp+0L ], [ 3.0L, E*E*E ], [ 0x1.6p+9L, 0x1.93bf4ec282efbp+1015L ], // near overflow [ 0x1.7p+9L, real.infinity ], // close overflow [ 0x1p+80L, real.infinity ], // far overflow [ real.infinity, real.infinity ], [-0x1.6p+9L, 0x1.44a3824e5285fp-1016L ], // near underflow [-0x1.64p+9L, 0x0.06f84920bb2d3p-1022L ], // near underflow - subnormal [-0x1.743p+9L, 0x0.0000000000001p-1022L ], // ditto [-0x1.8p+9L, 0 ], // close underflow [-0x1p30L, 0 ], // far underflow ]; } else static assert(0, "No exp() tests for real type!"); const minEqualDecimalDigits = real.dig - 3; real x; IeeeFlags f; foreach (ref pair; exptestpoints) { resetIeeeFlags(); x = exp(pair[0]); f = ieeeFlags; assert(equalsDigit(x, pair[1], minEqualDecimalDigits)); version (IeeeFlagsSupport) { // Check the overflow bit if (x == real.infinity) { // don't care about the overflow bit if input was inf // (e.g., the LLVM intrinsic doesn't set it on Linux x86_64) assert(pair[0] == real.infinity || f.overflow); } else assert(!f.overflow); // Check the underflow bit assert(f.underflow == (fabs(x) < real.min_normal)); // Invalid and div by zero shouldn't be affected. assert(!f.invalid); assert(!f.divByZero); } } // Ideally, exp(0) would not set the inexact flag. // Unfortunately, fldl2e sets it! // So it's not realistic to avoid setting it. assert(exp(0.0L) == 1.0); // NaN propagation. Doesn't set flags, bcos was already NaN. resetIeeeFlags(); x = exp(real.nan); f = ieeeFlags; assert(isIdentical(abs(x), real.nan)); assert(f.flags == 0); resetIeeeFlags(); x = exp(-real.nan); f = ieeeFlags; assert(isIdentical(abs(x), real.nan)); assert(f.flags == 0); x = exp(NaN(0x123)); assert(isIdentical(x, NaN(0x123))); // High resolution test (verified against GNU MPFR/Mathematica). assert(exp(0.5L) == 0x1.A612_98E1_E069_BC97_2DFE_FAB6_DF34p+0L); } /** * Calculate cos(y) + i sin(y). * * On many CPUs (such as x86), this is a very efficient operation; * almost twice as fast as calculating sin(y) and cos(y) separately, * and is the preferred method when both are required. */ creal expi(real y) @trusted pure nothrow @nogc { version (InlineAsm_X86_Any) { version (Win64) { asm pure nothrow @nogc { naked; fld real ptr [ECX]; fsincos; fxch ST(1), ST(0); ret; } } else { asm pure nothrow @nogc { fld y; fsincos; fxch ST(1), ST(0); } } } else { return cos(y) + sin(y)*1i; } } /// @safe pure nothrow @nogc unittest { assert(expi(1.3e5L) == cos(1.3e5L) + sin(1.3e5L) * 1i); assert(expi(0.0L) == 1L + 0.0Li); } /********************************************************************* * Separate floating point value into significand and exponent. * * Returns: * Calculate and return $(I x) and $(I exp) such that * value =$(I x)*2$(SUPERSCRIPT exp) and * .5 $(LT)= |$(I x)| $(LT) 1.0 * * $(I x) has same sign as value. * * $(TABLE_SV * $(TR $(TH value) $(TH returns) $(TH exp)) * $(TR $(TD $(PLUSMN)0.0) $(TD $(PLUSMN)0.0) $(TD 0)) * $(TR $(TD +$(INFIN)) $(TD +$(INFIN)) $(TD int.max)) * $(TR $(TD -$(INFIN)) $(TD -$(INFIN)) $(TD int.min)) * $(TR $(TD $(PLUSMN)$(NAN)) $(TD $(PLUSMN)$(NAN)) $(TD int.min)) * ) */ T frexp(T)(const T value, out int exp) @trusted pure nothrow @nogc if (isFloatingPoint!T) { Unqual!T vf = value; ushort* vu = cast(ushort*)&vf; static if (is(Unqual!T == float)) int* vi = cast(int*)&vf; else long* vl = cast(long*)&vf; int ex; alias F = floatTraits!T; ex = vu[F.EXPPOS_SHORT] & F.EXPMASK; static if (F.realFormat == RealFormat.ieeeExtended) { if (ex) { // If exponent is non-zero if (ex == F.EXPMASK) // infinity or NaN { if (*vl & 0x7FFF_FFFF_FFFF_FFFF) // NaN { *vl |= 0xC000_0000_0000_0000; // convert NaNS to NaNQ exp = int.min; } else if (vu[F.EXPPOS_SHORT] & 0x8000) // negative infinity exp = int.min; else // positive infinity exp = int.max; } else { exp = ex - F.EXPBIAS; vu[F.EXPPOS_SHORT] = (0x8000 & vu[F.EXPPOS_SHORT]) | 0x3FFE; } } else if (!*vl) { // vf is +-0.0 exp = 0; } else { // subnormal vf *= F.RECIP_EPSILON; ex = vu[F.EXPPOS_SHORT] & F.EXPMASK; exp = ex - F.EXPBIAS - T.mant_dig + 1; vu[F.EXPPOS_SHORT] = ((-1 - F.EXPMASK) & vu[F.EXPPOS_SHORT]) | 0x3FFE; } return vf; } else static if (F.realFormat == RealFormat.ieeeQuadruple) { if (ex) // If exponent is non-zero { if (ex == F.EXPMASK) { // infinity or NaN if (vl[MANTISSA_LSB] | (vl[MANTISSA_MSB] & 0x0000_FFFF_FFFF_FFFF)) // NaN { // convert NaNS to NaNQ vl[MANTISSA_MSB] |= 0x0000_8000_0000_0000; exp = int.min; } else if (vu[F.EXPPOS_SHORT] & 0x8000) // negative infinity exp = int.min; else // positive infinity exp = int.max; } else { exp = ex - F.EXPBIAS; vu[F.EXPPOS_SHORT] = F.EXPBIAS | (0x8000 & vu[F.EXPPOS_SHORT]); } } else if ((vl[MANTISSA_LSB] | (vl[MANTISSA_MSB] & 0x0000_FFFF_FFFF_FFFF)) == 0) { // vf is +-0.0 exp = 0; } else { // subnormal vf *= F.RECIP_EPSILON; ex = vu[F.EXPPOS_SHORT] & F.EXPMASK; exp = ex - F.EXPBIAS - T.mant_dig + 1; vu[F.EXPPOS_SHORT] = F.EXPBIAS | (0x8000 & vu[F.EXPPOS_SHORT]); } return vf; } else static if (F.realFormat == RealFormat.ieeeDouble) { if (ex) // If exponent is non-zero { if (ex == F.EXPMASK) // infinity or NaN { if (*vl == 0x7FF0_0000_0000_0000) // positive infinity { exp = int.max; } else if (*vl == 0xFFF0_0000_0000_0000) // negative infinity exp = int.min; else { // NaN *vl |= 0x0008_0000_0000_0000; // convert NaNS to NaNQ exp = int.min; } } else { exp = (ex - F.EXPBIAS) >> 4; vu[F.EXPPOS_SHORT] = cast(ushort)((0x800F & vu[F.EXPPOS_SHORT]) | 0x3FE0); } } else if (!(*vl & 0x7FFF_FFFF_FFFF_FFFF)) { // vf is +-0.0 exp = 0; } else { // subnormal vf *= F.RECIP_EPSILON; ex = vu[F.EXPPOS_SHORT] & F.EXPMASK; exp = ((ex - F.EXPBIAS) >> 4) - T.mant_dig + 1; vu[F.EXPPOS_SHORT] = cast(ushort)(((-1 - F.EXPMASK) & vu[F.EXPPOS_SHORT]) | 0x3FE0); } return vf; } else static if (F.realFormat == RealFormat.ieeeSingle) { if (ex) // If exponent is non-zero { if (ex == F.EXPMASK) // infinity or NaN { if (*vi == 0x7F80_0000) // positive infinity { exp = int.max; } else if (*vi == 0xFF80_0000) // negative infinity exp = int.min; else { // NaN *vi |= 0x0040_0000; // convert NaNS to NaNQ exp = int.min; } } else { exp = (ex - F.EXPBIAS) >> 7; vu[F.EXPPOS_SHORT] = cast(ushort)((0x807F & vu[F.EXPPOS_SHORT]) | 0x3F00); } } else if (!(*vi & 0x7FFF_FFFF)) { // vf is +-0.0 exp = 0; } else { // subnormal vf *= F.RECIP_EPSILON; ex = vu[F.EXPPOS_SHORT] & F.EXPMASK; exp = ((ex - F.EXPBIAS) >> 7) - T.mant_dig + 1; vu[F.EXPPOS_SHORT] = cast(ushort)(((-1 - F.EXPMASK) & vu[F.EXPPOS_SHORT]) | 0x3F00); } return vf; } else // static if (F.realFormat == RealFormat.ibmExtended) { assert(0, "frexp not implemented"); } } /// @system unittest { int exp; real mantissa = frexp(123.456L, exp); // check if values are equal to 19 decimal digits of precision assert(equalsDigit(mantissa * pow(2.0L, cast(real) exp), 123.456L, 19)); assert(frexp(-real.nan, exp) && exp == int.min); assert(frexp(real.nan, exp) && exp == int.min); assert(frexp(-real.infinity, exp) == -real.infinity && exp == int.min); assert(frexp(real.infinity, exp) == real.infinity && exp == int.max); assert(frexp(-0.0, exp) == -0.0 && exp == 0); assert(frexp(0.0, exp) == 0.0 && exp == 0); } @safe unittest { import std.meta : AliasSeq; import std.typecons : tuple, Tuple; foreach (T; AliasSeq!(real, double, float)) { Tuple!(T, T, int)[] vals = // x,frexp,exp [ tuple(T(0.0), T( 0.0 ), 0), tuple(T(-0.0), T( -0.0), 0), tuple(T(1.0), T( .5 ), 1), tuple(T(-1.0), T( -.5 ), 1), tuple(T(2.0), T( .5 ), 2), tuple(T(float.min_normal/2.0f), T(.5), -126), tuple(T.infinity, T.infinity, int.max), tuple(-T.infinity, -T.infinity, int.min), tuple(T.nan, T.nan, int.min), tuple(-T.nan, -T.nan, int.min), // Phobos issue #16026: tuple(3 * (T.min_normal * T.epsilon), T( .75), (T.min_exp - T.mant_dig) + 2) ]; foreach (elem; vals) { T x = elem[0]; T e = elem[1]; int exp = elem[2]; int eptr; T v = frexp(x, eptr); assert(isIdentical(e, v)); assert(exp == eptr); } static if (floatTraits!(T).realFormat == RealFormat.ieeeExtended) { static T[3][] extendedvals = [ // x,frexp,exp [0x1.a5f1c2eb3fe4efp+73L, 0x1.A5F1C2EB3FE4EFp-1L, 74], // normal [0x1.fa01712e8f0471ap-1064L, 0x1.fa01712e8f0471ap-1L, -1063], [T.min_normal, .5, -16381], [T.min_normal/2.0L, .5, -16382] // subnormal ]; foreach (elem; extendedvals) { T x = elem[0]; T e = elem[1]; int exp = cast(int) elem[2]; int eptr; T v = frexp(x, eptr); assert(isIdentical(e, v)); assert(exp == eptr); } } } } @safe unittest { import std.meta : AliasSeq; void foo() { foreach (T; AliasSeq!(real, double, float)) { int exp; const T a = 1; immutable T b = 2; auto c = frexp(a, exp); auto d = frexp(b, exp); } } } /****************************************** * Extracts the exponent of x as a signed integral value. * * If x is not a special value, the result is the same as * $(D cast(int) logb(x)). * * $(TABLE_SV * $(TR $(TH x) $(TH ilogb(x)) $(TH Range error?)) * $(TR $(TD 0) $(TD FP_ILOGB0) $(TD yes)) * $(TR $(TD $(PLUSMN)$(INFIN)) $(TD int.max) $(TD no)) * $(TR $(TD $(NAN)) $(TD FP_ILOGBNAN) $(TD no)) * ) */ int ilogb(T)(const T x) @trusted pure nothrow @nogc if (isFloatingPoint!T) { import core.bitop : bsr; alias F = floatTraits!T; union floatBits { T rv; ushort[T.sizeof/2] vu; uint[T.sizeof/4] vui; static if (T.sizeof >= 8) ulong[T.sizeof/8] vul; } floatBits y = void; y.rv = x; int ex = y.vu[F.EXPPOS_SHORT] & F.EXPMASK; static if (F.realFormat == RealFormat.ieeeExtended) { if (ex) { // If exponent is non-zero if (ex == F.EXPMASK) // infinity or NaN { if (y.vul[0] & 0x7FFF_FFFF_FFFF_FFFF) // NaN return FP_ILOGBNAN; else // +-infinity return int.max; } else { return ex - F.EXPBIAS - 1; } } else if (!y.vul[0]) { // vf is +-0.0 return FP_ILOGB0; } else { // subnormal return ex - F.EXPBIAS - T.mant_dig + 1 + bsr(y.vul[0]); } } else static if (F.realFormat == RealFormat.ieeeQuadruple) { if (ex) // If exponent is non-zero { if (ex == F.EXPMASK) { // infinity or NaN if (y.vul[MANTISSA_LSB] | ( y.vul[MANTISSA_MSB] & 0x0000_FFFF_FFFF_FFFF)) // NaN return FP_ILOGBNAN; else // +- infinity return int.max; } else { return ex - F.EXPBIAS - 1; } } else if ((y.vul[MANTISSA_LSB] | (y.vul[MANTISSA_MSB] & 0x0000_FFFF_FFFF_FFFF)) == 0) { // vf is +-0.0 return FP_ILOGB0; } else { // subnormal const ulong msb = y.vul[MANTISSA_MSB] & 0x0000_FFFF_FFFF_FFFF; const ulong lsb = y.vul[MANTISSA_LSB]; if (msb) return ex - F.EXPBIAS - T.mant_dig + 1 + bsr(msb) + 64; else return ex - F.EXPBIAS - T.mant_dig + 1 + bsr(lsb); } } else static if (F.realFormat == RealFormat.ieeeDouble) { if (ex) // If exponent is non-zero { if (ex == F.EXPMASK) // infinity or NaN { if ((y.vul[0] & 0x7FFF_FFFF_FFFF_FFFF) == 0x7FF0_0000_0000_0000) // +- infinity return int.max; else // NaN return FP_ILOGBNAN; } else { return ((ex - F.EXPBIAS) >> 4) - 1; } } else if (!(y.vul[0] & 0x7FFF_FFFF_FFFF_FFFF)) { // vf is +-0.0 return FP_ILOGB0; } else { // subnormal enum MANTISSAMASK_64 = ((cast(ulong) F.MANTISSAMASK_INT) << 32) | 0xFFFF_FFFF; return ((ex - F.EXPBIAS) >> 4) - T.mant_dig + 1 + bsr(y.vul[0] & MANTISSAMASK_64); } } else static if (F.realFormat == RealFormat.ieeeSingle) { if (ex) // If exponent is non-zero { if (ex == F.EXPMASK) // infinity or NaN { if ((y.vui[0] & 0x7FFF_FFFF) == 0x7F80_0000) // +- infinity return int.max; else // NaN return FP_ILOGBNAN; } else { return ((ex - F.EXPBIAS) >> 7) - 1; } } else if (!(y.vui[0] & 0x7FFF_FFFF)) { // vf is +-0.0 return FP_ILOGB0; } else { // subnormal const uint mantissa = y.vui[0] & F.MANTISSAMASK_INT; return ((ex - F.EXPBIAS) >> 7) - T.mant_dig + 1 + bsr(mantissa); } } else // static if (F.realFormat == RealFormat.ibmExtended) { core.stdc.math.ilogbl(x); } } /// ditto int ilogb(T)(const T x) @safe pure nothrow @nogc if (isIntegral!T && isUnsigned!T) { import core.bitop : bsr; if (x == 0) return FP_ILOGB0; else { static assert(T.sizeof <= ulong.sizeof, "integer size too large for the current ilogb implementation"); return bsr(x); } } /// ditto int ilogb(T)(const T x) @safe pure nothrow @nogc if (isIntegral!T && isSigned!T) { import std.traits : Unsigned; // Note: abs(x) can not be used because the return type is not Unsigned and // the return value would be wrong for x == int.min Unsigned!T absx = x >= 0 ? x : -x; return ilogb(absx); } alias FP_ILOGB0 = core.stdc.math.FP_ILOGB0; alias FP_ILOGBNAN = core.stdc.math.FP_ILOGBNAN; @system nothrow @nogc unittest { import std.meta : AliasSeq; import std.typecons : Tuple; foreach (F; AliasSeq!(float, double, real)) { alias T = Tuple!(F, int); T[13] vals = // x, ilogb(x) [ T( F.nan , FP_ILOGBNAN ), T( -F.nan , FP_ILOGBNAN ), T( F.infinity, int.max ), T( -F.infinity, int.max ), T( 0.0 , FP_ILOGB0 ), T( -0.0 , FP_ILOGB0 ), T( 2.0 , 1 ), T( 2.0001 , 1 ), T( 1.9999 , 0 ), T( 0.5 , -1 ), T( 123.123 , 6 ), T( -123.123 , 6 ), T( 0.123 , -4 ), ]; foreach (elem; vals) { assert(ilogb(elem[0]) == elem[1]); } } // min_normal and subnormals assert(ilogb(-float.min_normal) == -126); assert(ilogb(nextUp(-float.min_normal)) == -127); assert(ilogb(nextUp(-float(0.0))) == -149); assert(ilogb(-double.min_normal) == -1022); assert(ilogb(nextUp(-double.min_normal)) == -1023); assert(ilogb(nextUp(-double(0.0))) == -1074); static if (floatTraits!(real).realFormat == RealFormat.ieeeExtended) { assert(ilogb(-real.min_normal) == -16382); assert(ilogb(nextUp(-real.min_normal)) == -16383); assert(ilogb(nextUp(-real(0.0))) == -16445); } else static if (floatTraits!(real).realFormat == RealFormat.ieeeDouble) { assert(ilogb(-real.min_normal) == -1022); assert(ilogb(nextUp(-real.min_normal)) == -1023); assert(ilogb(nextUp(-real(0.0))) == -1074); } // test integer types assert(ilogb(0) == FP_ILOGB0); assert(ilogb(int.max) == 30); assert(ilogb(int.min) == 31); assert(ilogb(uint.max) == 31); assert(ilogb(long.max) == 62); assert(ilogb(long.min) == 63); assert(ilogb(ulong.max) == 63); } /******************************************* * Compute n * 2$(SUPERSCRIPT exp) * References: frexp */ real ldexp(real n, int exp) @nogc @safe pure nothrow { pragma(inline, true); return core.math.ldexp(n, exp); } //FIXME ///ditto double ldexp(double n, int exp) @safe pure nothrow @nogc { return ldexp(cast(real) n, exp); } //FIXME ///ditto float ldexp(float n, int exp) @safe pure nothrow @nogc { return ldexp(cast(real) n, exp); } /// @nogc @safe pure nothrow unittest { import std.meta : AliasSeq; foreach (T; AliasSeq!(float, double, real)) { T r; r = ldexp(3.0L, 3); assert(r == 24); r = ldexp(cast(T) 3.0, cast(int) 3); assert(r == 24); T n = 3.0; int exp = 3; r = ldexp(n, exp); assert(r == 24); } } @safe pure nothrow @nogc unittest { static if (floatTraits!(real).realFormat == RealFormat.ieeeExtended || floatTraits!(real).realFormat == RealFormat.ieeeQuadruple) { assert(ldexp(1.0L, -16384) == 0x1p-16384L); assert(ldexp(1.0L, -16382) == 0x1p-16382L); int x; real n = frexp(0x1p-16384L, x); assert(n == 0.5L); assert(x==-16383); assert(ldexp(n, x)==0x1p-16384L); } else static if (floatTraits!(real).realFormat == RealFormat.ieeeDouble) { assert(ldexp(1.0L, -1024) == 0x1p-1024L); assert(ldexp(1.0L, -1022) == 0x1p-1022L); int x; real n = frexp(0x1p-1024L, x); assert(n == 0.5L); assert(x==-1023); assert(ldexp(n, x)==0x1p-1024L); } else static assert(false, "Floating point type real not supported"); } /* workaround Issue 14718, float parsing depends on platform strtold @safe pure nothrow @nogc unittest { assert(ldexp(1.0, -1024) == 0x1p-1024); assert(ldexp(1.0, -1022) == 0x1p-1022); int x; double n = frexp(0x1p-1024, x); assert(n == 0.5); assert(x==-1023); assert(ldexp(n, x)==0x1p-1024); } @safe pure nothrow @nogc unittest { assert(ldexp(1.0f, -128) == 0x1p-128f); assert(ldexp(1.0f, -126) == 0x1p-126f); int x; float n = frexp(0x1p-128f, x); assert(n == 0.5f); assert(x==-127); assert(ldexp(n, x)==0x1p-128f); } */ @system unittest { static real[3][] vals = // value,exp,ldexp [ [ 0, 0, 0], [ 1, 0, 1], [ -1, 0, -1], [ 1, 1, 2], [ 123, 10, 125952], [ real.max, int.max, real.infinity], [ real.max, -int.max, 0], [ real.min_normal, -int.max, 0], ]; int i; for (i = 0; i < vals.length; i++) { real x = vals[i][0]; int exp = cast(int) vals[i][1]; real z = vals[i][2]; real l = ldexp(x, exp); assert(equalsDigit(z, l, 7)); } real function(real, int) pldexp = &ldexp; assert(pldexp != null); } private { version (INLINE_YL2X) {} else { static if (floatTraits!real.realFormat == RealFormat.ieeeQuadruple) { // Coefficients for log(1 + x) = x - x**2/2 + x**3 P(x)/Q(x) static immutable real[13] logCoeffsP = [ 1.313572404063446165910279910527789794488E4L, 7.771154681358524243729929227226708890930E4L, 2.014652742082537582487669938141683759923E5L, 3.007007295140399532324943111654767187848E5L, 2.854829159639697837788887080758954924001E5L, 1.797628303815655343403735250238293741397E5L, 7.594356839258970405033155585486712125861E4L, 2.128857716871515081352991964243375186031E4L, 3.824952356185897735160588078446136783779E3L, 4.114517881637811823002128927449878962058E2L, 2.321125933898420063925789532045674660756E1L, 4.998469661968096229986658302195402690910E-1L, 1.538612243596254322971797716843006400388E-6L ]; static immutable real[13] logCoeffsQ = [ 3.940717212190338497730839731583397586124E4L, 2.626900195321832660448791748036714883242E5L, 7.777690340007566932935753241556479363645E5L, 1.347518538384329112529391120390701166528E6L, 1.514882452993549494932585972882995548426E6L, 1.158019977462989115839826904108208787040E6L, 6.132189329546557743179177159925690841200E5L, 2.248234257620569139969141618556349415120E5L, 5.605842085972455027590989944010492125825E4L, 9.147150349299596453976674231612674085381E3L, 9.104928120962988414618126155557301584078E2L, 4.839208193348159620282142911143429644326E1L, 1.0 ]; // Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2) // where z = 2(x-1)/(x+1) static immutable real[6] logCoeffsR = [ -8.828896441624934385266096344596648080902E-1L, 8.057002716646055371965756206836056074715E1L, -2.024301798136027039250415126250455056397E3L, 2.048819892795278657810231591630928516206E4L, -8.977257995689735303686582344659576526998E4L, 1.418134209872192732479751274970992665513E5L ]; static immutable real[6] logCoeffsS = [ 1.701761051846631278975701529965589676574E6L -1.332535117259762928288745111081235577029E6L, 4.001557694070773974936904547424676279307E5L, -5.748542087379434595104154610899551484314E4L, 3.998526750980007367835804959888064681098E3L, -1.186359407982897997337150403816839480438E2L, 1.0 ]; } else { // Coefficients for log(1 + x) = x - x**2/2 + x**3 P(x)/Q(x) static immutable real[7] logCoeffsP = [ 2.0039553499201281259648E1L, 5.7112963590585538103336E1L, 6.0949667980987787057556E1L, 2.9911919328553073277375E1L, 6.5787325942061044846969E0L, 4.9854102823193375972212E-1L, 4.5270000862445199635215E-5L, ]; static immutable real[7] logCoeffsQ = [ 6.0118660497603843919306E1L, 2.1642788614495947685003E2L, 3.0909872225312059774938E2L, 2.2176239823732856465394E2L, 8.3047565967967209469434E1L, 1.5062909083469192043167E1L, 1.0000000000000000000000E0L, ]; // Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2) // where z = 2(x-1)/(x+1) static immutable real[4] logCoeffsR = [ -3.5717684488096787370998E1L, 1.0777257190312272158094E1L, -7.1990767473014147232598E-1L, 1.9757429581415468984296E-3L, ]; static immutable real[4] logCoeffsS = [ -4.2861221385716144629696E2L, 1.9361891836232102174846E2L, -2.6201045551331104417768E1L, 1.0000000000000000000000E0L, ]; } } } /************************************** * Calculate the natural logarithm of x. * * $(TABLE_SV * $(TR $(TH x) $(TH log(x)) $(TH divide by 0?) $(TH invalid?)) * $(TR $(TD $(PLUSMN)0.0) $(TD -$(INFIN)) $(TD yes) $(TD no)) * $(TR $(TD $(LT)0.0) $(TD $(NAN)) $(TD no) $(TD yes)) * $(TR $(TD +$(INFIN)) $(TD +$(INFIN)) $(TD no) $(TD no)) * ) */ real log(real x) @safe pure nothrow @nogc { version (INLINE_YL2X) return core.math.yl2x(x, LN2); else { // C1 + C2 = LN2. enum real C1 = 6.93145751953125E-1L; enum real C2 = 1.428606820309417232121458176568075500134E-6L; // Special cases. if (isNaN(x)) return x; if (isInfinity(x) && !signbit(x)) return x; if (x == 0.0) return -real.infinity; if (x < 0.0) return real.nan; // Separate mantissa from exponent. // Note, frexp is used so that denormal numbers will be handled properly. real y, z; int exp; x = frexp(x, exp); // Logarithm using log(x) = z + z^^3 R(z) / S(z), // where z = 2(x - 1)/(x + 1) if ((exp > 2) || (exp < -2)) { if (x < SQRT1_2) { // 2(2x - 1)/(2x + 1) exp -= 1; z = x - 0.5; y = 0.5 * z + 0.5; } else { // 2(x - 1)/(x + 1) z = x - 0.5; z -= 0.5; y = 0.5 * x + 0.5; } x = z / y; z = x * x; z = x * (z * poly(z, logCoeffsR) / poly(z, logCoeffsS)); z += exp * C2; z += x; z += exp * C1; return z; } // Logarithm using log(1 + x) = x - .5x^^2 + x^^3 P(x) / Q(x) if (x < SQRT1_2) { // 2x - 1 exp -= 1; x = ldexp(x, 1) - 1.0; } else { x = x - 1.0; } z = x * x; y = x * (z * poly(x, logCoeffsP) / poly(x, logCoeffsQ)); y += exp * C2; z = y - ldexp(z, -1); // Note, the sum of above terms does not exceed x/4, // so it contributes at most about 1/4 lsb to the error. z += x; z += exp * C1; return z; } } /// @safe pure nothrow @nogc unittest { assert(log(E) == 1); } /************************************** * Calculate the base-10 logarithm of x. * * $(TABLE_SV * $(TR $(TH x) $(TH log10(x)) $(TH divide by 0?) $(TH invalid?)) * $(TR $(TD $(PLUSMN)0.0) $(TD -$(INFIN)) $(TD yes) $(TD no)) * $(TR $(TD $(LT)0.0) $(TD $(NAN)) $(TD no) $(TD yes)) * $(TR $(TD +$(INFIN)) $(TD +$(INFIN)) $(TD no) $(TD no)) * ) */ real log10(real x) @safe pure nothrow @nogc { version (INLINE_YL2X) return core.math.yl2x(x, LOG2); else { // log10(2) split into two parts. enum real L102A = 0.3125L; enum real L102B = -1.14700043360188047862611052755069732318101185E-2L; // log10(e) split into two parts. enum real L10EA = 0.5L; enum real L10EB = -6.570551809674817234887108108339491770560299E-2L; // Special cases are the same as for log. if (isNaN(x)) return x; if (isInfinity(x) && !signbit(x)) return x; if (x == 0.0) return -real.infinity; if (x < 0.0) return real.nan; // Separate mantissa from exponent. // Note, frexp is used so that denormal numbers will be handled properly. real y, z; int exp; x = frexp(x, exp); // Logarithm using log(x) = z + z^^3 R(z) / S(z), // where z = 2(x - 1)/(x + 1) if ((exp > 2) || (exp < -2)) { if (x < SQRT1_2) { // 2(2x - 1)/(2x + 1) exp -= 1; z = x - 0.5; y = 0.5 * z + 0.5; } else { // 2(x - 1)/(x + 1) z = x - 0.5; z -= 0.5; y = 0.5 * x + 0.5; } x = z / y; z = x * x; y = x * (z * poly(z, logCoeffsR) / poly(z, logCoeffsS)); goto Ldone; } // Logarithm using log(1 + x) = x - .5x^^2 + x^^3 P(x) / Q(x) if (x < SQRT1_2) { // 2x - 1 exp -= 1; x = ldexp(x, 1) - 1.0; } else x = x - 1.0; z = x * x; y = x * (z * poly(x, logCoeffsP) / poly(x, logCoeffsQ)); y = y - ldexp(z, -1); // Multiply log of fraction by log10(e) and base 2 exponent by log10(2). // This sequence of operations is critical and it may be horribly // defeated by some compiler optimizers. Ldone: z = y * L10EB; z += x * L10EB; z += exp * L102B; z += y * L10EA; z += x * L10EA; z += exp * L102A; return z; } } /// @safe pure nothrow @nogc unittest { assert(fabs(log10(1000) - 3) < .000001); } /****************************************** * Calculates the natural logarithm of 1 + x. * * For very small x, log1p(x) will be more accurate than * log(1 + x). * * $(TABLE_SV * $(TR $(TH x) $(TH log1p(x)) $(TH divide by 0?) $(TH invalid?)) * $(TR $(TD $(PLUSMN)0.0) $(TD $(PLUSMN)0.0) $(TD no) $(TD no)) * $(TR $(TD -1.0) $(TD -$(INFIN)) $(TD yes) $(TD no)) * $(TR $(TD $(LT)-1.0) $(TD $(NAN)) $(TD no) $(TD yes)) * $(TR $(TD +$(INFIN)) $(TD -$(INFIN)) $(TD no) $(TD no)) * ) */ real log1p(real x) @safe pure nothrow @nogc { version (INLINE_YL2X) { // On x87, yl2xp1 is valid if and only if -0.5 <= lg(x) <= 0.5, // ie if -0.29 <= x <= 0.414 return (fabs(x) <= 0.25) ? core.math.yl2xp1(x, LN2) : core.math.yl2x(x+1, LN2); } else { // Special cases. if (isNaN(x) || x == 0.0) return x; if (isInfinity(x) && !signbit(x)) return x; if (x == -1.0) return -real.infinity; if (x < -1.0) return real.nan; return log(x + 1.0); } } /*************************************** * Calculates the base-2 logarithm of x: * $(SUB log, 2)x * * $(TABLE_SV * $(TR $(TH x) $(TH log2(x)) $(TH divide by 0?) $(TH invalid?)) * $(TR $(TD $(PLUSMN)0.0) $(TD -$(INFIN)) $(TD yes) $(TD no) ) * $(TR $(TD $(LT)0.0) $(TD $(NAN)) $(TD no) $(TD yes) ) * $(TR $(TD +$(INFIN)) $(TD +$(INFIN)) $(TD no) $(TD no) ) * ) */ real log2(real x) @safe pure nothrow @nogc { version (INLINE_YL2X) return core.math.yl2x(x, 1); else { // Special cases are the same as for log. if (isNaN(x)) return x; if (isInfinity(x) && !signbit(x)) return x; if (x == 0.0) return -real.infinity; if (x < 0.0) return real.nan; // Separate mantissa from exponent. // Note, frexp is used so that denormal numbers will be handled properly. real y, z; int exp; x = frexp(x, exp); // Logarithm using log(x) = z + z^^3 R(z) / S(z), // where z = 2(x - 1)/(x + 1) if ((exp > 2) || (exp < -2)) { if (x < SQRT1_2) { // 2(2x - 1)/(2x + 1) exp -= 1; z = x - 0.5; y = 0.5 * z + 0.5; } else { // 2(x - 1)/(x + 1) z = x - 0.5; z -= 0.5; y = 0.5 * x + 0.5; } x = z / y; z = x * x; y = x * (z * poly(z, logCoeffsR) / poly(z, logCoeffsS)); goto Ldone; } // Logarithm using log(1 + x) = x - .5x^^2 + x^^3 P(x) / Q(x) if (x < SQRT1_2) { // 2x - 1 exp -= 1; x = ldexp(x, 1) - 1.0; } else x = x - 1.0; z = x * x; y = x * (z * poly(x, logCoeffsP) / poly(x, logCoeffsQ)); y = y - ldexp(z, -1); // Multiply log of fraction by log10(e) and base 2 exponent by log10(2). // This sequence of operations is critical and it may be horribly // defeated by some compiler optimizers. Ldone: z = y * (LOG2E - 1.0); z += x * (LOG2E - 1.0); z += y; z += x; z += exp; return z; } } /// @system unittest { // check if values are equal to 19 decimal digits of precision assert(equalsDigit(log2(1024.0L), 10, 19)); } /***************************************** * Extracts the exponent of x as a signed integral value. * * If x is subnormal, it is treated as if it were normalized. * For a positive, finite x: * * 1 $(LT)= $(I x) * FLT_RADIX$(SUPERSCRIPT -logb(x)) $(LT) FLT_RADIX * * $(TABLE_SV * $(TR $(TH x) $(TH logb(x)) $(TH divide by 0?) ) * $(TR $(TD $(PLUSMN)$(INFIN)) $(TD +$(INFIN)) $(TD no)) * $(TR $(TD $(PLUSMN)0.0) $(TD -$(INFIN)) $(TD yes) ) * ) */ real logb(real x) @trusted nothrow @nogc { version (Win64_DMD_InlineAsm) { asm pure nothrow @nogc { naked ; fld real ptr [RCX] ; fxtract ; fstp ST(0) ; ret ; } } else version (MSVC_InlineAsm) { asm pure nothrow @nogc { fld x ; fxtract ; fstp ST(0) ; } } else return core.stdc.math.logbl(x); } /************************************ * Calculates the remainder from the calculation x/y. * Returns: * The value of x - i * y, where i is the number of times that y can * be completely subtracted from x. The result has the same sign as x. * * $(TABLE_SV * $(TR $(TH x) $(TH y) $(TH fmod(x, y)) $(TH invalid?)) * $(TR $(TD $(PLUSMN)0.0) $(TD not 0.0) $(TD $(PLUSMN)0.0) $(TD no)) * $(TR $(TD $(PLUSMNINF)) $(TD anything) $(TD $(NAN)) $(TD yes)) * $(TR $(TD anything) $(TD $(PLUSMN)0.0) $(TD $(NAN)) $(TD yes)) * $(TR $(TD !=$(PLUSMNINF)) $(TD $(PLUSMNINF)) $(TD x) $(TD no)) * ) */ real fmod(real x, real y) @trusted nothrow @nogc { version (CRuntime_Microsoft) { return x % y; } else return core.stdc.math.fmodl(x, y); } /************************************ * Breaks x into an integral part and a fractional part, each of which has * the same sign as x. The integral part is stored in i. * Returns: * The fractional part of x. * * $(TABLE_SV * $(TR $(TH x) $(TH i (on input)) $(TH modf(x, i)) $(TH i (on return))) * $(TR $(TD $(PLUSMNINF)) $(TD anything) $(TD $(PLUSMN)0.0) $(TD $(PLUSMNINF))) * ) */ real modf(real x, ref real i) @trusted nothrow @nogc { version (CRuntime_Microsoft) { i = trunc(x); return copysign(isInfinity(x) ? 0.0 : x - i, x); } else return core.stdc.math.modfl(x,&i); } /************************************* * Efficiently calculates x * 2$(SUPERSCRIPT n). * * scalbn handles underflow and overflow in * the same fashion as the basic arithmetic operators. * * $(TABLE_SV * $(TR $(TH x) $(TH scalb(x))) * $(TR $(TD $(PLUSMNINF)) $(TD $(PLUSMNINF)) ) * $(TR $(TD $(PLUSMN)0.0) $(TD $(PLUSMN)0.0) ) * ) */ real scalbn(real x, int n) @trusted nothrow @nogc { version (InlineAsm_X86_Any) { // scalbnl is not supported on DMD-Windows, so use asm pure nothrow @nogc. version (Win64) { asm pure nothrow @nogc { naked ; mov 16[RSP],RCX ; fild word ptr 16[RSP] ; fld real ptr [RDX] ; fscale ; fstp ST(1) ; ret ; } } else { asm pure nothrow @nogc { fild n; fld x; fscale; fstp ST(1); } } } else { return core.stdc.math.scalbnl(x, n); } } /// @safe nothrow @nogc unittest { assert(scalbn(-real.infinity, 5) == -real.infinity); } /*************** * Calculates the cube root of x. * * $(TABLE_SV * $(TR $(TH $(I x)) $(TH cbrt(x)) $(TH invalid?)) * $(TR $(TD $(PLUSMN)0.0) $(TD $(PLUSMN)0.0) $(TD no) ) * $(TR $(TD $(NAN)) $(TD $(NAN)) $(TD yes) ) * $(TR $(TD $(PLUSMN)$(INFIN)) $(TD $(PLUSMN)$(INFIN)) $(TD no) ) * ) */ real cbrt(real x) @trusted nothrow @nogc { version (CRuntime_Microsoft) { version (INLINE_YL2X) return copysign(exp2(core.math.yl2x(fabs(x), 1.0L/3.0L)), x); else return core.stdc.math.cbrtl(x); } else return core.stdc.math.cbrtl(x); } /******************************* * Returns |x| * * $(TABLE_SV * $(TR $(TH x) $(TH fabs(x))) * $(TR $(TD $(PLUSMN)0.0) $(TD +0.0) ) * $(TR $(TD $(PLUSMN)$(INFIN)) $(TD +$(INFIN)) ) * ) */ real fabs(real x) @safe pure nothrow @nogc { pragma(inline, true); return core.math.fabs(x); } //FIXME ///ditto double fabs(double x) @safe pure nothrow @nogc { return fabs(cast(real) x); } //FIXME ///ditto float fabs(float x) @safe pure nothrow @nogc { return fabs(cast(real) x); } @safe unittest { real function(real) pfabs = &fabs; assert(pfabs != null); } /*********************************************************************** * Calculates the length of the * hypotenuse of a right-angled triangle with sides of length x and y. * The hypotenuse is the value of the square root of * the sums of the squares of x and y: * * sqrt($(POWER x, 2) + $(POWER y, 2)) * * Note that hypot(x, y), hypot(y, x) and * hypot(x, -y) are equivalent. * * $(TABLE_SV * $(TR $(TH x) $(TH y) $(TH hypot(x, y)) $(TH invalid?)) * $(TR $(TD x) $(TD $(PLUSMN)0.0) $(TD |x|) $(TD no)) * $(TR $(TD $(PLUSMNINF)) $(TD y) $(TD +$(INFIN)) $(TD no)) * $(TR $(TD $(PLUSMNINF)) $(TD $(NAN)) $(TD +$(INFIN)) $(TD no)) * ) */ real hypot(real x, real y) @safe pure nothrow @nogc { // Scale x and y to avoid underflow and overflow. // If one is huge and the other tiny, return the larger. // If both are huge, avoid overflow by scaling by 1/sqrt(real.max/2). // If both are tiny, avoid underflow by scaling by sqrt(real.min_normal*real.epsilon). enum real SQRTMIN = 0.5 * sqrt(real.min_normal); // This is a power of 2. enum real SQRTMAX = 1.0L / SQRTMIN; // 2^^((max_exp)/2) = nextUp(sqrt(real.max)) static assert(2*(SQRTMAX/2)*(SQRTMAX/2) <= real.max); // Proves that sqrt(real.max) ~~ 0.5/sqrt(real.min_normal) static assert(real.min_normal*real.max > 2 && real.min_normal*real.max <= 4); real u = fabs(x); real v = fabs(y); if (!(u >= v)) // check for NaN as well. { v = u; u = fabs(y); if (u == real.infinity) return u; // hypot(inf, nan) == inf if (v == real.infinity) return v; // hypot(nan, inf) == inf } // Now u >= v, or else one is NaN. if (v >= SQRTMAX*0.5) { // hypot(huge, huge) -- avoid overflow u *= SQRTMIN*0.5; v *= SQRTMIN*0.5; return sqrt(u*u + v*v) * SQRTMAX * 2.0; } if (u <= SQRTMIN) { // hypot (tiny, tiny) -- avoid underflow // This is only necessary to avoid setting the underflow // flag. u *= SQRTMAX / real.epsilon; v *= SQRTMAX / real.epsilon; return sqrt(u*u + v*v) * SQRTMIN * real.epsilon; } if (u * real.epsilon > v) { // hypot (huge, tiny) = huge return u; } // both are in the normal range return sqrt(u*u + v*v); } @safe unittest { static real[3][] vals = // x,y,hypot [ [ 0.0, 0.0, 0.0], [ 0.0, -0.0, 0.0], [ -0.0, -0.0, 0.0], [ 3.0, 4.0, 5.0], [ -300, -400, 500], [0.0, 7.0, 7.0], [9.0, 9*real.epsilon, 9.0], [88/(64*sqrt(real.min_normal)), 105/(64*sqrt(real.min_normal)), 137/(64*sqrt(real.min_normal))], [88/(128*sqrt(real.min_normal)), 105/(128*sqrt(real.min_normal)), 137/(128*sqrt(real.min_normal))], [3*real.min_normal*real.epsilon, 4*real.min_normal*real.epsilon, 5*real.min_normal*real.epsilon], [ real.min_normal, real.min_normal, sqrt(2.0L)*real.min_normal], [ real.max/sqrt(2.0L), real.max/sqrt(2.0L), real.max], [ real.infinity, real.nan, real.infinity], [ real.nan, real.infinity, real.infinity], [ real.nan, real.nan, real.nan], [ real.nan, real.max, real.nan], [ real.max, real.nan, real.nan], ]; for (int i = 0; i < vals.length; i++) { real x = vals[i][0]; real y = vals[i][1]; real z = vals[i][2]; real h = hypot(x, y); assert(isIdentical(z,h) || feqrel(z, h) >= real.mant_dig - 1); } } /************************************** * Returns the value of x rounded upward to the next integer * (toward positive infinity). */ real ceil(real x) @trusted pure nothrow @nogc { version (Win64_DMD_InlineAsm) { asm pure nothrow @nogc { naked ; fld real ptr [RCX] ; fstcw 8[RSP] ; mov AL,9[RSP] ; mov DL,AL ; and AL,0xC3 ; or AL,0x08 ; // round to +infinity mov 9[RSP],AL ; fldcw 8[RSP] ; frndint ; mov 9[RSP],DL ; fldcw 8[RSP] ; ret ; } } else version (MSVC_InlineAsm) { short cw; asm pure nothrow @nogc { fld x ; fstcw cw ; mov AL,byte ptr cw+1 ; mov DL,AL ; and AL,0xC3 ; or AL,0x08 ; // round to +infinity mov byte ptr cw+1,AL ; fldcw cw ; frndint ; mov byte ptr cw+1,DL ; fldcw cw ; } } else { // Special cases. if (isNaN(x) || isInfinity(x)) return x; real y = floorImpl(x); if (y < x) y += 1.0; return y; } } /// @safe pure nothrow @nogc unittest { assert(ceil(+123.456L) == +124); assert(ceil(-123.456L) == -123); assert(ceil(-1.234L) == -1); assert(ceil(-0.123L) == 0); assert(ceil(0.0L) == 0); assert(ceil(+0.123L) == 1); assert(ceil(+1.234L) == 2); assert(ceil(real.infinity) == real.infinity); assert(isNaN(ceil(real.nan))); assert(isNaN(ceil(real.init))); } // ditto double ceil(double x) @trusted pure nothrow @nogc { // Special cases. if (isNaN(x) || isInfinity(x)) return x; double y = floorImpl(x); if (y < x) y += 1.0; return y; } @safe pure nothrow @nogc unittest { assert(ceil(+123.456) == +124); assert(ceil(-123.456) == -123); assert(ceil(-1.234) == -1); assert(ceil(-0.123) == 0); assert(ceil(0.0) == 0); assert(ceil(+0.123) == 1); assert(ceil(+1.234) == 2); assert(ceil(double.infinity) == double.infinity); assert(isNaN(ceil(double.nan))); assert(isNaN(ceil(double.init))); } // ditto float ceil(float x) @trusted pure nothrow @nogc { // Special cases. if (isNaN(x) || isInfinity(x)) return x; float y = floorImpl(x); if (y < x) y += 1.0; return y; } @safe pure nothrow @nogc unittest { assert(ceil(+123.456f) == +124); assert(ceil(-123.456f) == -123); assert(ceil(-1.234f) == -1); assert(ceil(-0.123f) == 0); assert(ceil(0.0f) == 0); assert(ceil(+0.123f) == 1); assert(ceil(+1.234f) == 2); assert(ceil(float.infinity) == float.infinity); assert(isNaN(ceil(float.nan))); assert(isNaN(ceil(float.init))); } /************************************** * Returns the value of x rounded downward to the next integer * (toward negative infinity). */ real floor(real x) @trusted pure nothrow @nogc { version (Win64_DMD_InlineAsm) { asm pure nothrow @nogc { naked ; fld real ptr [RCX] ; fstcw 8[RSP] ; mov AL,9[RSP] ; mov DL,AL ; and AL,0xC3 ; or AL,0x04 ; // round to -infinity mov 9[RSP],AL ; fldcw 8[RSP] ; frndint ; mov 9[RSP],DL ; fldcw 8[RSP] ; ret ; } } else version (MSVC_InlineAsm) { short cw; asm pure nothrow @nogc { fld x ; fstcw cw ; mov AL,byte ptr cw+1 ; mov DL,AL ; and AL,0xC3 ; or AL,0x04 ; // round to -infinity mov byte ptr cw+1,AL ; fldcw cw ; frndint ; mov byte ptr cw+1,DL ; fldcw cw ; } } else { // Special cases. if (isNaN(x) || isInfinity(x) || x == 0.0) return x; return floorImpl(x); } } /// @safe pure nothrow @nogc unittest { assert(floor(+123.456L) == +123); assert(floor(-123.456L) == -124); assert(floor(-1.234L) == -2); assert(floor(-0.123L) == -1); assert(floor(0.0L) == 0); assert(floor(+0.123L) == 0); assert(floor(+1.234L) == 1); assert(floor(real.infinity) == real.infinity); assert(isNaN(floor(real.nan))); assert(isNaN(floor(real.init))); } // ditto double floor(double x) @trusted pure nothrow @nogc { // Special cases. if (isNaN(x) || isInfinity(x) || x == 0.0) return x; return floorImpl(x); } @safe pure nothrow @nogc unittest { assert(floor(+123.456) == +123); assert(floor(-123.456) == -124); assert(floor(-1.234) == -2); assert(floor(-0.123) == -1); assert(floor(0.0) == 0); assert(floor(+0.123) == 0); assert(floor(+1.234) == 1); assert(floor(double.infinity) == double.infinity); assert(isNaN(floor(double.nan))); assert(isNaN(floor(double.init))); } // ditto float floor(float x) @trusted pure nothrow @nogc { // Special cases. if (isNaN(x) || isInfinity(x) || x == 0.0) return x; return floorImpl(x); } @safe pure nothrow @nogc unittest { assert(floor(+123.456f) == +123); assert(floor(-123.456f) == -124); assert(floor(-1.234f) == -2); assert(floor(-0.123f) == -1); assert(floor(0.0f) == 0); assert(floor(+0.123f) == 0); assert(floor(+1.234f) == 1); assert(floor(float.infinity) == float.infinity); assert(isNaN(floor(float.nan))); assert(isNaN(floor(float.init))); } /** * Round `val` to a multiple of `unit`. `rfunc` specifies the rounding * function to use; by default this is `rint`, which uses the current * rounding mode. */ Unqual!F quantize(alias rfunc = rint, F)(const F val, const F unit) if (is(typeof(rfunc(F.init)) : F) && isFloatingPoint!F) { typeof(return) ret = val; if (unit != 0) { const scaled = val / unit; if (!scaled.isInfinity) ret = rfunc(scaled) * unit; } return ret; } /// @safe pure nothrow @nogc unittest { assert(12345.6789L.quantize(0.01L) == 12345.68L); assert(12345.6789L.quantize!floor(0.01L) == 12345.67L); assert(12345.6789L.quantize(22.0L) == 12342.0L); } /// @safe pure nothrow @nogc unittest { assert(12345.6789L.quantize(0) == 12345.6789L); assert(12345.6789L.quantize(real.infinity).isNaN); assert(12345.6789L.quantize(real.nan).isNaN); assert(real.infinity.quantize(0.01L) == real.infinity); assert(real.infinity.quantize(real.nan).isNaN); assert(real.nan.quantize(0.01L).isNaN); assert(real.nan.quantize(real.infinity).isNaN); assert(real.nan.quantize(real.nan).isNaN); } /** * Round `val` to a multiple of `pow(base, exp)`. `rfunc` specifies the * rounding function to use; by default this is `rint`, which uses the * current rounding mode. */ Unqual!F quantize(real base, alias rfunc = rint, F, E)(const F val, const E exp) if (is(typeof(rfunc(F.init)) : F) && isFloatingPoint!F && isIntegral!E) { // TODO: Compile-time optimization for power-of-two bases? return quantize!rfunc(val, pow(cast(F) base, exp)); } /// ditto Unqual!F quantize(real base, long exp = 1, alias rfunc = rint, F)(const F val) if (is(typeof(rfunc(F.init)) : F) && isFloatingPoint!F) { enum unit = cast(F) pow(base, exp); return quantize!rfunc(val, unit); } /// @safe pure nothrow @nogc unittest { assert(12345.6789L.quantize!10(-2) == 12345.68L); assert(12345.6789L.quantize!(10, -2) == 12345.68L); assert(12345.6789L.quantize!(10, floor)(-2) == 12345.67L); assert(12345.6789L.quantize!(10, -2, floor) == 12345.67L); assert(12345.6789L.quantize!22(1) == 12342.0L); assert(12345.6789L.quantize!22 == 12342.0L); } @safe pure nothrow @nogc unittest { import std.meta : AliasSeq; foreach (F; AliasSeq!(real, double, float)) { const maxL10 = cast(int) F.max.log10.floor; const maxR10 = pow(cast(F) 10, maxL10); assert((cast(F) 0.9L * maxR10).quantize!10(maxL10) == maxR10); assert((cast(F)-0.9L * maxR10).quantize!10(maxL10) == -maxR10); assert(F.max.quantize(F.min_normal) == F.max); assert((-F.max).quantize(F.min_normal) == -F.max); assert(F.min_normal.quantize(F.max) == 0); assert((-F.min_normal).quantize(F.max) == 0); assert(F.min_normal.quantize(F.min_normal) == F.min_normal); assert((-F.min_normal).quantize(F.min_normal) == -F.min_normal); } } /****************************************** * Rounds x to the nearest integer value, using the current rounding * mode. * * Unlike the rint functions, nearbyint does not raise the * FE_INEXACT exception. */ real nearbyint(real x) @trusted nothrow @nogc { version (CRuntime_Microsoft) { assert(0); // not implemented in C library } else return core.stdc.math.nearbyintl(x); } /********************************** * Rounds x to the nearest integer value, using the current rounding * mode. * If the return value is not equal to x, the FE_INEXACT * exception is raised. * $(B nearbyint) performs * the same operation, but does not set the FE_INEXACT exception. */ real rint(real x) @safe pure nothrow @nogc { pragma(inline, true); return core.math.rint(x); } //FIXME ///ditto double rint(double x) @safe pure nothrow @nogc { return rint(cast(real) x); } //FIXME ///ditto float rint(float x) @safe pure nothrow @nogc { return rint(cast(real) x); } @safe unittest { real function(real) print = &rint; assert(print != null); } /*************************************** * Rounds x to the nearest integer value, using the current rounding * mode. * * This is generally the fastest method to convert a floating-point number * to an integer. Note that the results from this function * depend on the rounding mode, if the fractional part of x is exactly 0.5. * If using the default rounding mode (ties round to even integers) * lrint(4.5) == 4, lrint(5.5)==6. */ long lrint(real x) @trusted pure nothrow @nogc { version (InlineAsm_X86_Any) { version (Win64) { asm pure nothrow @nogc { naked; fld real ptr [RCX]; fistp qword ptr 8[RSP]; mov RAX,8[RSP]; ret; } } else { long n; asm pure nothrow @nogc { fld x; fistp n; } return n; } } else { alias F = floatTraits!(real); static if (F.realFormat == RealFormat.ieeeDouble) { long result; // Rounding limit when casting from real(double) to ulong. enum real OF = 4.50359962737049600000E15L; uint* vi = cast(uint*)(&x); // Find the exponent and sign uint msb = vi[MANTISSA_MSB]; uint lsb = vi[MANTISSA_LSB]; int exp = ((msb >> 20) & 0x7ff) - 0x3ff; const int sign = msb >> 31; msb &= 0xfffff; msb |= 0x100000; if (exp < 63) { if (exp >= 52) result = (cast(long) msb << (exp - 20)) | (lsb << (exp - 52)); else { // Adjust x and check result. const real j = sign ? -OF : OF; x = (j + x) - j; msb = vi[MANTISSA_MSB]; lsb = vi[MANTISSA_LSB]; exp = ((msb >> 20) & 0x7ff) - 0x3ff; msb &= 0xfffff; msb |= 0x100000; if (exp < 0) result = 0; else if (exp < 20) result = cast(long) msb >> (20 - exp); else if (exp == 20) result = cast(long) msb; else result = (cast(long) msb << (exp - 20)) | (lsb >> (52 - exp)); } } else { // It is left implementation defined when the number is too large. return cast(long) x; } return sign ? -result : result; } else static if (F.realFormat == RealFormat.ieeeExtended) { long result; // Rounding limit when casting from real(80-bit) to ulong. enum real OF = 9.22337203685477580800E18L; ushort* vu = cast(ushort*)(&x); uint* vi = cast(uint*)(&x); // Find the exponent and sign int exp = (vu[F.EXPPOS_SHORT] & 0x7fff) - 0x3fff; const int sign = (vu[F.EXPPOS_SHORT] >> 15) & 1; if (exp < 63) { // Adjust x and check result. const real j = sign ? -OF : OF; x = (j + x) - j; exp = (vu[F.EXPPOS_SHORT] & 0x7fff) - 0x3fff; version (LittleEndian) { if (exp < 0) result = 0; else if (exp <= 31) result = vi[1] >> (31 - exp); else result = (cast(long) vi[1] << (exp - 31)) | (vi[0] >> (63 - exp)); } else { if (exp < 0) result = 0; else if (exp <= 31) result = vi[1] >> (31 - exp); else result = (cast(long) vi[1] << (exp - 31)) | (vi[2] >> (63 - exp)); } } else { // It is left implementation defined when the number is too large // to fit in a 64bit long. return cast(long) x; } return sign ? -result : result; } else static if (F.realFormat == RealFormat.ieeeQuadruple) { const vu = cast(ushort*)(&x); // Find the exponent and sign const sign = (vu[F.EXPPOS_SHORT] >> 15) & 1; if ((vu[F.EXPPOS_SHORT] & F.EXPMASK) - (F.EXPBIAS + 1) > 63) { // The result is left implementation defined when the number is // too large to fit in a 64 bit long. return cast(long) x; } // Force rounding of lower bits according to current rounding // mode by adding ±2^-112 and subtracting it again. enum OF = 5.19229685853482762853049632922009600E33L; const j = sign ? -OF : OF; x = (j + x) - j; const exp = (vu[F.EXPPOS_SHORT] & F.EXPMASK) - (F.EXPBIAS + 1); const implicitOne = 1UL << 48; auto vl = cast(ulong*)(&x); vl[MANTISSA_MSB] &= implicitOne - 1; vl[MANTISSA_MSB] |= implicitOne; long result; if (exp < 0) result = 0; else if (exp <= 48) result = vl[MANTISSA_MSB] >> (48 - exp); else result = (vl[MANTISSA_MSB] << (exp - 48)) | (vl[MANTISSA_LSB] >> (112 - exp)); return sign ? -result : result; } else { static assert(false, "real type not supported by lrint()"); } } } /// @safe pure nothrow @nogc unittest { assert(lrint(4.5) == 4); assert(lrint(5.5) == 6); assert(lrint(-4.5) == -4); assert(lrint(-5.5) == -6); assert(lrint(int.max - 0.5) == 2147483646L); assert(lrint(int.max + 0.5) == 2147483648L); assert(lrint(int.min - 0.5) == -2147483648L); assert(lrint(int.min + 0.5) == -2147483648L); } static if (real.mant_dig >= long.sizeof * 8) { @safe pure nothrow @nogc unittest { assert(lrint(long.max - 1.5L) == long.max - 1); assert(lrint(long.max - 0.5L) == long.max - 1); assert(lrint(long.min + 0.5L) == long.min); assert(lrint(long.min + 1.5L) == long.min + 2); } } /******************************************* * Return the value of x rounded to the nearest integer. * If the fractional part of x is exactly 0.5, the return value is * rounded away from zero. */ real round(real x) @trusted nothrow @nogc { version (CRuntime_Microsoft) { auto old = FloatingPointControl.getControlState(); FloatingPointControl.setControlState( (old & ~FloatingPointControl.roundingMask) | FloatingPointControl.roundToZero ); x = rint((x >= 0) ? x + 0.5 : x - 0.5); FloatingPointControl.setControlState(old); return x; } else return core.stdc.math.roundl(x); } /********************************************** * Return the value of x rounded to the nearest integer. * * If the fractional part of x is exactly 0.5, the return value is rounded * away from zero. * * $(BLUE This function is Posix-Only.) */ long lround(real x) @trusted nothrow @nogc { version (Posix) return core.stdc.math.llroundl(x); else assert(0, "lround not implemented"); } version (Posix) { @safe nothrow @nogc unittest { assert(lround(0.49) == 0); assert(lround(0.5) == 1); assert(lround(1.5) == 2); } } /**************************************************** * Returns the integer portion of x, dropping the fractional portion. * * This is also known as "chop" rounding. */ real trunc(real x) @trusted nothrow @nogc { version (Win64_DMD_InlineAsm) { asm pure nothrow @nogc { naked ; fld real ptr [RCX] ; fstcw 8[RSP] ; mov AL,9[RSP] ; mov DL,AL ; and AL,0xC3 ; or AL,0x0C ; // round to 0 mov 9[RSP],AL ; fldcw 8[RSP] ; frndint ; mov 9[RSP],DL ; fldcw 8[RSP] ; ret ; } } else version (MSVC_InlineAsm) { short cw; asm pure nothrow @nogc { fld x ; fstcw cw ; mov AL,byte ptr cw+1 ; mov DL,AL ; and AL,0xC3 ; or AL,0x0C ; // round to 0 mov byte ptr cw+1,AL ; fldcw cw ; frndint ; mov byte ptr cw+1,DL ; fldcw cw ; } } else return core.stdc.math.truncl(x); } /**************************************************** * Calculate the remainder x REM y, following IEC 60559. * * REM is the value of x - y * n, where n is the integer nearest the exact * value of x / y. * If |n - x / y| == 0.5, n is even. * If the result is zero, it has the same sign as x. * Otherwise, the sign of the result is the sign of x / y. * Precision mode has no effect on the remainder functions. * * remquo returns n in the parameter n. * * $(TABLE_SV * $(TR $(TH x) $(TH y) $(TH remainder(x, y)) $(TH n) $(TH invalid?)) * $(TR $(TD $(PLUSMN)0.0) $(TD not 0.0) $(TD $(PLUSMN)0.0) $(TD 0.0) $(TD no)) * $(TR $(TD $(PLUSMNINF)) $(TD anything) $(TD $(NAN)) $(TD ?) $(TD yes)) * $(TR $(TD anything) $(TD $(PLUSMN)0.0) $(TD $(NAN)) $(TD ?) $(TD yes)) * $(TR $(TD != $(PLUSMNINF)) $(TD $(PLUSMNINF)) $(TD x) $(TD ?) $(TD no)) * ) * * $(BLUE `remquo` and `remainder` not supported on Windows.) */ real remainder(real x, real y) @trusted nothrow @nogc { version (CRuntime_Microsoft) { int n; return remquo(x, y, n); } else return core.stdc.math.remainderl(x, y); } real remquo(real x, real y, out int n) @trusted nothrow @nogc /// ditto { version (Posix) return core.stdc.math.remquol(x, y, &n); else assert(0, "remquo not implemented"); } /** IEEE exception status flags ('sticky bits') These flags indicate that an exceptional floating-point condition has occurred. They indicate that a NaN or an infinity has been generated, that a result is inexact, or that a signalling NaN has been encountered. If floating-point exceptions are enabled (unmasked), a hardware exception will be generated instead of setting these flags. */ struct IeeeFlags { private: // The x87 FPU status register is 16 bits. // The Pentium SSE2 status register is 32 bits. // The ARM and PowerPC FPSCR is a 32-bit register. // The SPARC FSR is a 32bit register (64 bits for SPARC 7 & 8, but high bits are uninteresting). // The RISC-V (32 & 64 bit) fcsr is 32-bit register. uint flags; version (CRuntime_Microsoft) { // Microsoft uses hardware-incompatible custom constants in fenv.h (core.stdc.fenv). // Applies to both x87 status word (16 bits) and SSE2 status word(32 bits). enum : int { INEXACT_MASK = 0x20, UNDERFLOW_MASK = 0x10, OVERFLOW_MASK = 0x08, DIVBYZERO_MASK = 0x04, INVALID_MASK = 0x01, EXCEPTIONS_MASK = 0b11_1111 } // Don't bother about subnormals, they are not supported on most CPUs. // SUBNORMAL_MASK = 0x02; } else { enum : int { INEXACT_MASK = core.stdc.fenv.FE_INEXACT, UNDERFLOW_MASK = core.stdc.fenv.FE_UNDERFLOW, OVERFLOW_MASK = core.stdc.fenv.FE_OVERFLOW, DIVBYZERO_MASK = core.stdc.fenv.FE_DIVBYZERO, INVALID_MASK = core.stdc.fenv.FE_INVALID, EXCEPTIONS_MASK = core.stdc.fenv.FE_ALL_EXCEPT, } } private: static uint getIeeeFlags() { version (GNU) { version (X86_Any) { ushort sw; asm pure nothrow @nogc { "fstsw %0" : "=a" (sw); } // OR the result with the SSE2 status register (MXCSR). if (haveSSE) { uint mxcsr; asm pure nothrow @nogc { "stmxcsr %0" : "=m" (mxcsr); } return (sw | mxcsr) & EXCEPTIONS_MASK; } else return sw & EXCEPTIONS_MASK; } else version (ARM) { version (ARM_SoftFloat) return 0; else { uint result = void; asm pure nothrow @nogc { "vmrs %0, FPSCR; and %0, %0, #0x1F;" : "=r" result; } return result; } } else version (RISCV_Any) { version (D_SoftFloat) return 0; else { uint result = void; asm pure nothrow @nogc { "frflags %0" : "=r" result; } return result; } } else assert(0, "Not yet supported"); } else version (InlineAsm_X86_Any) { ushort sw; asm pure nothrow @nogc { fstsw sw; } // OR the result with the SSE2 status register (MXCSR). if (haveSSE) { uint mxcsr; asm pure nothrow @nogc { stmxcsr mxcsr; } return (sw | mxcsr) & EXCEPTIONS_MASK; } else return sw & EXCEPTIONS_MASK; } else version (SPARC) { /* int retval; asm pure nothrow @nogc { st %fsr, retval; } return retval; */ assert(0, "Not yet supported"); } else version (ARM) { assert(false, "Not yet supported."); } else assert(0, "Not yet supported"); } static void resetIeeeFlags() @nogc { version (GNU) { version (X86_Any) { asm pure nothrow @nogc { "fnclex"; } // Also clear exception flags in MXCSR, SSE's control register. if (haveSSE) { uint mxcsr; asm pure nothrow @nogc { "stmxcsr %0" : "=m" (mxcsr); } mxcsr &= ~EXCEPTIONS_MASK; asm pure nothrow @nogc { "ldmxcsr %0" : : "m" (mxcsr); } } } else version (ARM) { version (ARM_SoftFloat) return; else { uint old = FloatingPointControl.getControlState(); old &= ~0b11111; // http://infocenter.arm.com/help/topic/com.arm.doc.ddi0408i/Chdfifdc.html asm pure nothrow @nogc { "vmsr FPSCR, %0" : : "r" (old); } } } else version (RISCV_Any) { version (D_SoftFloat) return; else { uint newValues = 0x0; asm pure nothrow @nogc { "fsflags %0" : : "r" newValues; } } } else assert(0, "Not yet supported"); } else version (InlineAsm_X86_Any) { asm pure nothrow @nogc { fnclex; } // Also clear exception flags in MXCSR, SSE's control register. if (haveSSE) { uint mxcsr; asm nothrow @nogc { stmxcsr mxcsr; } mxcsr &= ~EXCEPTIONS_MASK; asm nothrow @nogc { ldmxcsr mxcsr; } } } else { /* SPARC: int tmpval; asm pure nothrow @nogc { st %fsr, tmpval; } tmpval &=0xFFFF_FC00; asm pure nothrow @nogc { ld tmpval, %fsr; } */ assert(0, "Not yet supported"); } } public: version (IeeeFlagsSupport) { /** * The result cannot be represented exactly, so rounding occurred. * Example: `x = sin(0.1);` */ @property bool inexact() const { return (flags & INEXACT_MASK) != 0; } /** * A zero was generated by underflow * Example: `x = real.min*real.epsilon/2;` */ @property bool underflow() const { return (flags & UNDERFLOW_MASK) != 0; } /** * An infinity was generated by overflow * Example: `x = real.max*2;` */ @property bool overflow() const { return (flags & OVERFLOW_MASK) != 0; } /** * An infinity was generated by division by zero * Example: `x = 3/0.0;` */ @property bool divByZero() const { return (flags & DIVBYZERO_MASK) != 0; } /** * A machine NaN was generated. * Example: `x = real.infinity * 0.0;` */ @property bool invalid() const { return (flags & INVALID_MASK) != 0; } } } /// version (GNU) { // ieeeFlags test disabled, see LDC Issue #888. } else @system unittest { static void func() { int a = 10 * 10; } real a=3.5; // Set all the flags to zero resetIeeeFlags(); assert(!ieeeFlags.divByZero); // Perform a division by zero. a/=0.0L; assert(a == real.infinity); assert(ieeeFlags.divByZero); // Create a NaN a*=0.0L; assert(ieeeFlags.invalid); assert(isNaN(a)); // Check that calling func() has no effect on the // status flags. IeeeFlags f = ieeeFlags; func(); assert(ieeeFlags == f); } version (GNU) { // ieeeFlags test disabled, see LDC Issue #888. } else @system unittest { import std.meta : AliasSeq; static struct Test { void delegate() action; bool function() ieeeCheck; } foreach (T; AliasSeq!(float, double, real)) { T x; /* Needs to be here to trick -O. It would optimize away the calculations if x were local to the function literals. */ auto tests = [ Test( () { x = 1; x += 0.1; }, () => ieeeFlags.inexact ), Test( () { x = T.min_normal; x /= T.max; }, () => ieeeFlags.underflow ), Test( () { x = T.max; x += T.max; }, () => ieeeFlags.overflow ), Test( () { x = 1; x /= 0; }, () => ieeeFlags.divByZero ), Test( () { x = 0; x /= 0; }, () => ieeeFlags.invalid ) ]; foreach (test; tests) { resetIeeeFlags(); assert(!test.ieeeCheck()); test.action(); assert(test.ieeeCheck()); } } } version (X86_Any) { version = IeeeFlagsSupport; } else version (PPC_Any) { version = IeeeFlagsSupport; } else version (RISCV_Any) { version = IeeeFlagsSupport; } else version (MIPS_Any) { version = IeeeFlagsSupport; } else version (ARM_Any) { version = IeeeFlagsSupport; } /// Set all of the floating-point status flags to false. void resetIeeeFlags() @nogc { IeeeFlags.resetIeeeFlags(); } /// Returns: snapshot of the current state of the floating-point status flags @property IeeeFlags ieeeFlags() { return IeeeFlags(IeeeFlags.getIeeeFlags()); } /** Control the Floating point hardware Change the IEEE754 floating-point rounding mode and the floating-point hardware exceptions. By default, the rounding mode is roundToNearest and all hardware exceptions are disabled. For most applications, debugging is easier if the $(I division by zero), $(I overflow), and $(I invalid operation) exceptions are enabled. These three are combined into a $(I severeExceptions) value for convenience. Note in particular that if $(I invalidException) is enabled, a hardware trap will be generated whenever an uninitialized floating-point variable is used. All changes are temporary. The previous state is restored at the end of the scope. Example: ---- { FloatingPointControl fpctrl; // Enable hardware exceptions for division by zero, overflow to infinity, // invalid operations, and uninitialized floating-point variables. fpctrl.enableExceptions(FloatingPointControl.severeExceptions); // This will generate a hardware exception, if x is a // default-initialized floating point variable: real x; // Add `= 0` or even `= real.nan` to not throw the exception. real y = x * 3.0; // The exception is only thrown for default-uninitialized NaN-s. // NaN-s with other payload are valid: real z = y * real.nan; // ok // Changing the rounding mode: fpctrl.rounding = FloatingPointControl.roundUp; assert(rint(1.1) == 2); // The set hardware exceptions will be disabled when leaving this scope. // The original rounding mode will also be restored. } // Ensure previous values are returned: assert(!FloatingPointControl.enabledExceptions); assert(FloatingPointControl.rounding == FloatingPointControl.roundToNearest); assert(rint(1.1) == 1); ---- */ struct FloatingPointControl { alias RoundingMode = uint; /// version (StdDdoc) { enum : RoundingMode { /** IEEE rounding modes. * The default mode is roundToNearest. * * roundingMask = A mask of all rounding modes. */ roundToNearest, roundDown, /// ditto roundUp, /// ditto roundToZero, /// ditto roundingMask, /// ditto } } else version (CRuntime_Microsoft) { // Microsoft uses hardware-incompatible custom constants in fenv.h (core.stdc.fenv). enum : RoundingMode { roundToNearest = 0x0000, roundDown = 0x0400, roundUp = 0x0800, roundToZero = 0x0C00, roundingMask = roundToNearest | roundDown | roundUp | roundToZero, } } else { enum : RoundingMode { roundToNearest = core.stdc.fenv.FE_TONEAREST, roundDown = core.stdc.fenv.FE_DOWNWARD, roundUp = core.stdc.fenv.FE_UPWARD, roundToZero = core.stdc.fenv.FE_TOWARDZERO, roundingMask = roundToNearest | roundDown | roundUp | roundToZero, } } //// Change the floating-point hardware rounding mode @property void rounding(RoundingMode newMode) @nogc { initialize(); setControlState(cast(ushort)((getControlState() & (-1 - roundingMask)) | (newMode & roundingMask))); } /// Returns: the currently active rounding mode @property static RoundingMode rounding() @nogc { return cast(RoundingMode)(getControlState() & roundingMask); } alias ExceptionMask = uint; /// version (StdDdoc) { enum : ExceptionMask { /** IEEE hardware exceptions. * By default, all exceptions are masked (disabled). * * severeExceptions = The overflow, division by zero, and invalid * exceptions. */ subnormalException, inexactException, /// ditto underflowException, /// ditto overflowException, /// ditto divByZeroException, /// ditto invalidException, /// ditto severeExceptions, /// ditto allExceptions, /// ditto } } else version (ARM_Any) { enum : ExceptionMask { subnormalException = 0x8000, inexactException = 0x1000, underflowException = 0x0800, overflowException = 0x0400, divByZeroException = 0x0200, invalidException = 0x0100, severeExceptions = overflowException | divByZeroException | invalidException, allExceptions = severeExceptions | underflowException | inexactException | subnormalException, } } else version (PPC_Any) { enum : ExceptionMask { inexactException = 0x0008, divByZeroException = 0x0010, underflowException = 0x0020, overflowException = 0x0040, invalidException = 0x0080, severeExceptions = overflowException | divByZeroException | invalidException, allExceptions = severeExceptions | underflowException | inexactException, } } else version (HPPA) { enum : ExceptionMask { inexactException = 0x01, underflowException = 0x02, overflowException = 0x04, divByZeroException = 0x08, invalidException = 0x10, severeExceptions = overflowException | divByZeroException | invalidException, allExceptions = severeExceptions | underflowException | inexactException, } } else version (MIPS_Any) { enum : ExceptionMask { inexactException = 0x0080, divByZeroException = 0x0400, overflowException = 0x0200, underflowException = 0x0100, invalidException = 0x0800, severeExceptions = overflowException | divByZeroException | invalidException, allExceptions = severeExceptions | underflowException | inexactException, } } else version (SPARC_Any) { enum : ExceptionMask { inexactException = 0x0800000, divByZeroException = 0x1000000, overflowException = 0x4000000, underflowException = 0x2000000, invalidException = 0x8000000, severeExceptions = overflowException | divByZeroException | invalidException, allExceptions = severeExceptions | underflowException | inexactException, } } else version (IBMZ_Any) { enum : ExceptionMask { inexactException = 0x08000000, divByZeroException = 0x40000000, overflowException = 0x20000000, underflowException = 0x10000000, invalidException = 0x80000000, severeExceptions = overflowException | divByZeroException | invalidException, allExceptions = severeExceptions | underflowException | inexactException, } } else version (RISCV_Any) { enum : ExceptionMask { inexactException = 0x01, divByZeroException = 0x02, underflowException = 0x04, overflowException = 0x08, invalidException = 0x10, severeExceptions = overflowException | divByZeroException | invalidException, allExceptions = severeExceptions | underflowException | inexactException, } } else version (X86_Any) { enum : ExceptionMask { inexactException = 0x20, underflowException = 0x10, overflowException = 0x08, divByZeroException = 0x04, subnormalException = 0x02, invalidException = 0x01, severeExceptions = overflowException | divByZeroException | invalidException, allExceptions = severeExceptions | underflowException | inexactException | subnormalException, } } else static assert(false, "Not implemented for this architecture"); public: /// Returns: true if the current FPU supports exception trapping @property static bool hasExceptionTraps() @safe nothrow @nogc { version (X86_Any) return true; else version (PPC_Any) return true; else version (MIPS_Any) return true; else version (ARM_Any) { auto oldState = getControlState(); // If exceptions are not supported, we set the bit but read it back as zero // https://sourceware.org/ml/libc-ports/2012-06/msg00091.html setControlState(oldState | divByZeroException); immutable result = (getControlState() & allExceptions) != 0; setControlState(oldState); return result; } else assert(0, "Not yet supported"); } /// Enable (unmask) specific hardware exceptions. Multiple exceptions may be ORed together. void enableExceptions(ExceptionMask exceptions) @nogc { assert(hasExceptionTraps); initialize(); version (X86_Any) setControlState(getControlState() & ~(exceptions & allExceptions)); else setControlState(getControlState() | (exceptions & allExceptions)); } /// Disable (mask) specific hardware exceptions. Multiple exceptions may be ORed together. void disableExceptions(ExceptionMask exceptions) @nogc { assert(hasExceptionTraps); initialize(); version (X86_Any) setControlState(getControlState() | (exceptions & allExceptions)); else setControlState(getControlState() & ~(exceptions & allExceptions)); } /// Returns: the exceptions which are currently enabled (unmasked) @property static ExceptionMask enabledExceptions() @nogc { assert(hasExceptionTraps); version (X86_Any) return (getControlState() & allExceptions) ^ allExceptions; else return (getControlState() & allExceptions); } /// Clear all pending exceptions, then restore the original exception state and rounding mode. ~this() @nogc { clearExceptions(); if (initialized) setControlState(savedState); } private: ControlState savedState; bool initialized = false; version (ARM_Any) { alias ControlState = uint; } else version (HPPA) { alias ControlState = uint; } else version (PPC_Any) { alias ControlState = uint; } else version (MIPS_Any) { alias ControlState = uint; } else version (SPARC_Any) { alias ControlState = ulong; } else version (IBMZ_Any) { alias ControlState = uint; } else version (RISCV_Any) { alias ControlState = uint; } else version (X86_Any) { alias ControlState = ushort; } else static assert(false, "Not implemented for this architecture"); void initialize() @nogc { // BUG: This works around the absence of this() constructors. if (initialized) return; clearExceptions(); savedState = getControlState(); initialized = true; } // Clear all pending exceptions static void clearExceptions() @nogc { resetIeeeFlags(); } // Read from the control register static ControlState getControlState() @trusted nothrow @nogc { version (GNU) { version (X86_Any) { ControlState cont; asm pure nothrow @nogc { "fstcw %0" : "=m" cont; } return cont; } else version (AArch64) { ControlState cont; asm pure nothrow @nogc { "mrs %0, FPCR;" : "=r" cont; } return cont; } else version (ARM) { ControlState cont; version (ARM_SoftFloat) cont = 0; else { asm pure nothrow @nogc { "vmrs %0, FPSCR" : "=r" cont; } } return cont; } else version (RISCV_Any) { version (D_SoftFloat) return 0; else { ControlState cont; asm pure nothrow @nogc { "frcsr %0" : "=r" cont; } return cont; } } else assert(0, "Not yet supported"); } else version (D_InlineAsm_X86) { short cont; asm nothrow @nogc { xor EAX, EAX; fstcw cont; } return cont; } else version (D_InlineAsm_X86_64) { short cont; asm nothrow @nogc { xor RAX, RAX; fstcw cont; } return cont; } else assert(0, "Not yet supported"); } // Set the control register static void setControlState(ControlState newState) @trusted nothrow @nogc { version (GNU) { version (X86_Any) { asm pure nothrow @nogc { "fclex; fldcw %0" : : "m" newState; } // Also update MXCSR, SSE's control register. if (haveSSE) { uint mxcsr; asm pure nothrow @nogc { "stmxcsr %0" : "=m" mxcsr; } /* In the FPU control register, rounding mode is in bits 10 and 11. In MXCSR it's in bits 13 and 14. */ mxcsr &= ~(roundingMask << 3); // delete old rounding mode mxcsr |= (newState & roundingMask) << 3; // write new rounding mode /* In the FPU control register, masks are bits 0 through 5. In MXCSR they're 7 through 12. */ mxcsr &= ~(allExceptions << 7); // delete old masks mxcsr |= (newState & allExceptions) << 7; // write new exception masks asm pure nothrow @nogc { "ldmxcsr %0" : : "m" mxcsr; } } } else version (AArch64) { asm pure nothrow @nogc { "msr FPCR, %0;" : : "r" (newState); } } else version (ARM) { version (ARM_SoftFloat) return; else { asm pure nothrow @nogc { "vmsr FPSCR, %0" : : "r" (newState); } } } else version (RISCV_Any) { version (D_SoftFloat) return; else { asm pure nothrow @nogc { "fscsr %0" : : "r" (newState); } } } else assert(0, "Not yet supported"); } else version (InlineAsm_X86_Any) { asm nothrow @nogc { fclex; fldcw newState; } // Also update MXCSR, SSE's control register. if (haveSSE) { uint mxcsr; asm nothrow @nogc { stmxcsr mxcsr; } /* In the FPU control register, rounding mode is in bits 10 and 11. In MXCSR it's in bits 13 and 14. */ mxcsr &= ~(roundingMask << 3); // delete old rounding mode mxcsr |= (newState & roundingMask) << 3; // write new rounding mode /* In the FPU control register, masks are bits 0 through 5. In MXCSR they're 7 through 12. */ mxcsr &= ~(allExceptions << 7); // delete old masks mxcsr |= (newState & allExceptions) << 7; // write new exception masks asm nothrow @nogc { ldmxcsr mxcsr; } } } else assert(0, "Not yet supported"); } } version (D_HardFloat) @system unittest { void ensureDefaults() { assert(FloatingPointControl.rounding == FloatingPointControl.roundToNearest); if (FloatingPointControl.hasExceptionTraps) assert(FloatingPointControl.enabledExceptions == 0); } { FloatingPointControl ctrl; } ensureDefaults(); { FloatingPointControl ctrl; ctrl.rounding = FloatingPointControl.roundDown; assert(FloatingPointControl.rounding == FloatingPointControl.roundDown); } ensureDefaults(); if (FloatingPointControl.hasExceptionTraps) { FloatingPointControl ctrl; ctrl.enableExceptions(FloatingPointControl.divByZeroException | FloatingPointControl.overflowException); assert(ctrl.enabledExceptions == (FloatingPointControl.divByZeroException | FloatingPointControl.overflowException)); ctrl.rounding = FloatingPointControl.roundUp; assert(FloatingPointControl.rounding == FloatingPointControl.roundUp); } ensureDefaults(); } version (D_HardFloat) @system unittest // rounding { import std.meta : AliasSeq; foreach (T; AliasSeq!(float, double, real)) { FloatingPointControl fpctrl; fpctrl.rounding = FloatingPointControl.roundUp; T u = 1; u += 0.1; fpctrl.rounding = FloatingPointControl.roundDown; T d = 1; d += 0.1; fpctrl.rounding = FloatingPointControl.roundToZero; T z = 1; z += 0.1; assert(u > d); assert(z == d); fpctrl.rounding = FloatingPointControl.roundUp; u = -1; u -= 0.1; fpctrl.rounding = FloatingPointControl.roundDown; d = -1; d -= 0.1; fpctrl.rounding = FloatingPointControl.roundToZero; z = -1; z -= 0.1; assert(u > d); assert(z == u); } } /********************************* * Determines if $(D_PARAM x) is NaN. * Params: * x = a floating point number. * Returns: * $(D true) if $(D_PARAM x) is Nan. */ bool isNaN(X)(X x) @nogc @trusted pure nothrow if (isFloatingPoint!(X)) { alias F = floatTraits!(X); static if (F.realFormat == RealFormat.ieeeSingle) { const uint p = *cast(uint *)&x; return ((p & 0x7F80_0000) == 0x7F80_0000) && p & 0x007F_FFFF; // not infinity } else static if (F.realFormat == RealFormat.ieeeDouble) { const ulong p = *cast(ulong *)&x; return ((p & 0x7FF0_0000_0000_0000) == 0x7FF0_0000_0000_0000) && p & 0x000F_FFFF_FFFF_FFFF; // not infinity } else static if (F.realFormat == RealFormat.ieeeExtended) { const ushort e = F.EXPMASK & (cast(ushort *)&x)[F.EXPPOS_SHORT]; const ulong ps = *cast(ulong *)&x; return e == F.EXPMASK && ps & 0x7FFF_FFFF_FFFF_FFFF; // not infinity } else static if (F.realFormat == RealFormat.ieeeQuadruple) { const ushort e = F.EXPMASK & (cast(ushort *)&x)[F.EXPPOS_SHORT]; const ulong psLsb = (cast(ulong *)&x)[MANTISSA_LSB]; const ulong psMsb = (cast(ulong *)&x)[MANTISSA_MSB]; return e == F.EXPMASK && (psLsb | (psMsb& 0x0000_FFFF_FFFF_FFFF)) != 0; } else { return x != x; } } /// @safe pure nothrow @nogc unittest { assert( isNaN(float.init)); assert( isNaN(-double.init)); assert( isNaN(real.nan)); assert( isNaN(-real.nan)); assert(!isNaN(cast(float) 53.6)); assert(!isNaN(cast(real)-53.6)); } @safe pure nothrow @nogc unittest { import std.meta : AliasSeq; foreach (T; AliasSeq!(float, double, real)) { // CTFE-able tests assert(isNaN(T.init)); assert(isNaN(-T.init)); assert(isNaN(T.nan)); assert(isNaN(-T.nan)); assert(!isNaN(T.infinity)); assert(!isNaN(-T.infinity)); assert(!isNaN(cast(T) 53.6)); assert(!isNaN(cast(T)-53.6)); // Runtime tests shared T f; f = T.init; assert(isNaN(f)); assert(isNaN(-f)); f = T.nan; assert(isNaN(f)); assert(isNaN(-f)); f = T.infinity; assert(!isNaN(f)); assert(!isNaN(-f)); f = cast(T) 53.6; assert(!isNaN(f)); assert(!isNaN(-f)); } } /********************************* * Determines if $(D_PARAM x) is finite. * Params: * x = a floating point number. * Returns: * $(D true) if $(D_PARAM x) is finite. */ bool isFinite(X)(X x) @trusted pure nothrow @nogc { alias F = floatTraits!(X); ushort* pe = cast(ushort *)&x; return (pe[F.EXPPOS_SHORT] & F.EXPMASK) != F.EXPMASK; } /// @safe pure nothrow @nogc unittest { assert( isFinite(1.23f)); assert( isFinite(float.max)); assert( isFinite(float.min_normal)); assert(!isFinite(float.nan)); assert(!isFinite(float.infinity)); } @safe pure nothrow @nogc unittest { assert(isFinite(1.23)); assert(isFinite(double.max)); assert(isFinite(double.min_normal)); assert(!isFinite(double.nan)); assert(!isFinite(double.infinity)); assert(isFinite(1.23L)); assert(isFinite(real.max)); assert(isFinite(real.min_normal)); assert(!isFinite(real.nan)); assert(!isFinite(real.infinity)); } /********************************* * Determines if $(D_PARAM x) is normalized. * * A normalized number must not be zero, subnormal, infinite nor $(NAN). * * Params: * x = a floating point number. * Returns: * $(D true) if $(D_PARAM x) is normalized. */ /* Need one for each format because subnormal floats might * be converted to normal reals. */ bool isNormal(X)(X x) @trusted pure nothrow @nogc { alias F = floatTraits!(X); static if (F.realFormat == RealFormat.ibmExtended) { // doubledouble is normal if the least significant part is normal. return isNormal((cast(double*)&x)[MANTISSA_LSB]); } else { ushort e = F.EXPMASK & (cast(ushort *)&x)[F.EXPPOS_SHORT]; return (e != F.EXPMASK && e != 0); } } /// @safe pure nothrow @nogc unittest { float f = 3; double d = 500; real e = 10e+48; assert(isNormal(f)); assert(isNormal(d)); assert(isNormal(e)); f = d = e = 0; assert(!isNormal(f)); assert(!isNormal(d)); assert(!isNormal(e)); assert(!isNormal(real.infinity)); assert(isNormal(-real.max)); assert(!isNormal(real.min_normal/4)); } /********************************* * Determines if $(D_PARAM x) is subnormal. * * Subnormals (also known as "denormal number"), have a 0 exponent * and a 0 most significant mantissa bit. * * Params: * x = a floating point number. * Returns: * $(D true) if $(D_PARAM x) is a denormal number. */ bool isSubnormal(X)(X x) @trusted pure nothrow @nogc { /* Need one for each format because subnormal floats might be converted to normal reals. */ alias F = floatTraits!(X); static if (F.realFormat == RealFormat.ieeeSingle) { uint *p = cast(uint *)&x; return (*p & F.EXPMASK_INT) == 0 && *p & F.MANTISSAMASK_INT; } else static if (F.realFormat == RealFormat.ieeeDouble) { uint *p = cast(uint *)&x; return (p[MANTISSA_MSB] & F.EXPMASK_INT) == 0 && (p[MANTISSA_LSB] || p[MANTISSA_MSB] & F.MANTISSAMASK_INT); } else static if (F.realFormat == RealFormat.ieeeQuadruple) { ushort e = F.EXPMASK & (cast(ushort *)&x)[F.EXPPOS_SHORT]; long* ps = cast(long *)&x; return (e == 0 && ((ps[MANTISSA_LSB]|(ps[MANTISSA_MSB]& 0x0000_FFFF_FFFF_FFFF)) != 0)); } else static if (F.realFormat == RealFormat.ieeeExtended) { ushort* pe = cast(ushort *)&x; long* ps = cast(long *)&x; return (pe[F.EXPPOS_SHORT] & F.EXPMASK) == 0 && *ps > 0; } else static if (F.realFormat == RealFormat.ibmExtended) { return isSubnormal((cast(double*)&x)[MANTISSA_MSB]); } else { static assert(false, "Not implemented for this architecture"); } } /// @safe pure nothrow @nogc unittest { import std.meta : AliasSeq; foreach (T; AliasSeq!(float, double, real)) { T f; for (f = 1.0; !isSubnormal(f); f /= 2) assert(f != 0); } } /********************************* * Determines if $(D_PARAM x) is $(PLUSMN)$(INFIN). * Params: * x = a floating point number. * Returns: * $(D true) if $(D_PARAM x) is $(PLUSMN)$(INFIN). */ bool isInfinity(X)(X x) @nogc @trusted pure nothrow if (isFloatingPoint!(X)) { alias F = floatTraits!(X); static if (F.realFormat == RealFormat.ieeeSingle) { return ((*cast(uint *)&x) & 0x7FFF_FFFF) == 0x7F80_0000; } else static if (F.realFormat == RealFormat.ieeeDouble) { return ((*cast(ulong *)&x) & 0x7FFF_FFFF_FFFF_FFFF) == 0x7FF0_0000_0000_0000; } else static if (F.realFormat == RealFormat.ieeeExtended) { const ushort e = cast(ushort)(F.EXPMASK & (cast(ushort *)&x)[F.EXPPOS_SHORT]); const ulong ps = *cast(ulong *)&x; // On Motorola 68K, infinity can have hidden bit = 1 or 0. On x86, it is always 1. return e == F.EXPMASK && (ps & 0x7FFF_FFFF_FFFF_FFFF) == 0; } else static if (F.realFormat == RealFormat.ibmExtended) { return (((cast(ulong *)&x)[MANTISSA_MSB]) & 0x7FFF_FFFF_FFFF_FFFF) == 0x7FF8_0000_0000_0000; } else static if (F.realFormat == RealFormat.ieeeQuadruple) { const long psLsb = (cast(long *)&x)[MANTISSA_LSB]; const long psMsb = (cast(long *)&x)[MANTISSA_MSB]; return (psLsb == 0) && (psMsb & 0x7FFF_FFFF_FFFF_FFFF) == 0x7FFF_0000_0000_0000; } else { return (x < -X.max) || (X.max < x); } } /// @nogc @safe pure nothrow unittest { assert(!isInfinity(float.init)); assert(!isInfinity(-float.init)); assert(!isInfinity(float.nan)); assert(!isInfinity(-float.nan)); assert(isInfinity(float.infinity)); assert(isInfinity(-float.infinity)); assert(isInfinity(-1.0f / 0.0f)); } @safe pure nothrow @nogc unittest { // CTFE-able tests assert(!isInfinity(double.init)); assert(!isInfinity(-double.init)); assert(!isInfinity(double.nan)); assert(!isInfinity(-double.nan)); assert(isInfinity(double.infinity)); assert(isInfinity(-double.infinity)); assert(isInfinity(-1.0 / 0.0)); assert(!isInfinity(real.init)); assert(!isInfinity(-real.init)); assert(!isInfinity(real.nan)); assert(!isInfinity(-real.nan)); assert(isInfinity(real.infinity)); assert(isInfinity(-real.infinity)); assert(isInfinity(-1.0L / 0.0L)); // Runtime tests shared float f; f = float.init; assert(!isInfinity(f)); assert(!isInfinity(-f)); f = float.nan; assert(!isInfinity(f)); assert(!isInfinity(-f)); f = float.infinity; assert(isInfinity(f)); assert(isInfinity(-f)); f = (-1.0f / 0.0f); assert(isInfinity(f)); shared double d; d = double.init; assert(!isInfinity(d)); assert(!isInfinity(-d)); d = double.nan; assert(!isInfinity(d)); assert(!isInfinity(-d)); d = double.infinity; assert(isInfinity(d)); assert(isInfinity(-d)); d = (-1.0 / 0.0); assert(isInfinity(d)); shared real e; e = real.init; assert(!isInfinity(e)); assert(!isInfinity(-e)); e = real.nan; assert(!isInfinity(e)); assert(!isInfinity(-e)); e = real.infinity; assert(isInfinity(e)); assert(isInfinity(-e)); e = (-1.0L / 0.0L); assert(isInfinity(e)); } /********************************* * Is the binary representation of x identical to y? * * Same as ==, except that positive and negative zero are not identical, * and two $(NAN)s are identical if they have the same 'payload'. */ bool isIdentical(real x, real y) @trusted pure nothrow @nogc { // We're doing a bitwise comparison so the endianness is irrelevant. long* pxs = cast(long *)&x; long* pys = cast(long *)&y; alias F = floatTraits!(real); static if (F.realFormat == RealFormat.ieeeDouble) { return pxs[0] == pys[0]; } else static if (F.realFormat == RealFormat.ieeeQuadruple || F.realFormat == RealFormat.ibmExtended) { return pxs[0] == pys[0] && pxs[1] == pys[1]; } else { ushort* pxe = cast(ushort *)&x; ushort* pye = cast(ushort *)&y; return pxe[4] == pye[4] && pxs[0] == pys[0]; } } /********************************* * Return 1 if sign bit of e is set, 0 if not. */ int signbit(X)(X x) @nogc @trusted pure nothrow { alias F = floatTraits!(X); return ((cast(ubyte *)&x)[F.SIGNPOS_BYTE] & 0x80) != 0; } /// @nogc @safe pure nothrow unittest { assert(!signbit(float.nan)); assert(signbit(-float.nan)); assert(!signbit(168.1234f)); assert(signbit(-168.1234f)); assert(!signbit(0.0f)); assert(signbit(-0.0f)); assert(signbit(-float.max)); assert(!signbit(float.max)); assert(!signbit(double.nan)); assert(signbit(-double.nan)); assert(!signbit(168.1234)); assert(signbit(-168.1234)); assert(!signbit(0.0)); assert(signbit(-0.0)); assert(signbit(-double.max)); assert(!signbit(double.max)); assert(!signbit(real.nan)); assert(signbit(-real.nan)); assert(!signbit(168.1234L)); assert(signbit(-168.1234L)); assert(!signbit(0.0L)); assert(signbit(-0.0L)); assert(signbit(-real.max)); assert(!signbit(real.max)); } /********************************* * Return a value composed of to with from's sign bit. */ R copysign(R, X)(R to, X from) @trusted pure nothrow @nogc if (isFloatingPoint!(R) && isFloatingPoint!(X)) { ubyte* pto = cast(ubyte *)&to; const ubyte* pfrom = cast(ubyte *)&from; alias T = floatTraits!(R); alias F = floatTraits!(X); pto[T.SIGNPOS_BYTE] &= 0x7F; pto[T.SIGNPOS_BYTE] |= pfrom[F.SIGNPOS_BYTE] & 0x80; return to; } // ditto R copysign(R, X)(X to, R from) @trusted pure nothrow @nogc if (isIntegral!(X) && isFloatingPoint!(R)) { return copysign(cast(R) to, from); } @safe pure nothrow @nogc unittest { import std.meta : AliasSeq; foreach (X; AliasSeq!(float, double, real, int, long)) { foreach (Y; AliasSeq!(float, double, real)) (){ // avoid slow optimizations for large functions @@@BUG@@@ 2396 X x = 21; Y y = 23.8; Y e = void; e = copysign(x, y); assert(e == 21.0); e = copysign(-x, y); assert(e == 21.0); e = copysign(x, -y); assert(e == -21.0); e = copysign(-x, -y); assert(e == -21.0); static if (isFloatingPoint!X) { e = copysign(X.nan, y); assert(isNaN(e) && !signbit(e)); e = copysign(X.nan, -y); assert(isNaN(e) && signbit(e)); } }(); } } /********************************* Returns $(D -1) if $(D x < 0), $(D x) if $(D x == 0), $(D 1) if $(D x > 0), and $(NAN) if x==$(NAN). */ F sgn(F)(F x) @safe pure nothrow @nogc { // @@@TODO@@@: make this faster return x > 0 ? 1 : x < 0 ? -1 : x; } /// @safe pure nothrow @nogc unittest { assert(sgn(168.1234) == 1); assert(sgn(-168.1234) == -1); assert(sgn(0.0) == 0); assert(sgn(-0.0) == 0); } // Functions for NaN payloads /* * A 'payload' can be stored in the significand of a $(NAN). One bit is required * to distinguish between a quiet and a signalling $(NAN). This leaves 22 bits * of payload for a float; 51 bits for a double; 62 bits for an 80-bit real; * and 111 bits for a 128-bit quad. */ /** * Create a quiet $(NAN), storing an integer inside the payload. * * For floats, the largest possible payload is 0x3F_FFFF. * For doubles, it is 0x3_FFFF_FFFF_FFFF. * For 80-bit or 128-bit reals, it is 0x3FFF_FFFF_FFFF_FFFF. */ real NaN(ulong payload) @trusted pure nothrow @nogc { alias F = floatTraits!(real); static if (F.realFormat == RealFormat.ieeeExtended) { // real80 (in x86 real format, the implied bit is actually // not implied but a real bit which is stored in the real) ulong v = 3; // implied bit = 1, quiet bit = 1 } else { ulong v = 1; // no implied bit. quiet bit = 1 } ulong a = payload; // 22 Float bits ulong w = a & 0x3F_FFFF; a -= w; v <<=22; v |= w; a >>=22; // 29 Double bits v <<=29; w = a & 0xFFF_FFFF; v |= w; a -= w; a >>=29; static if (F.realFormat == RealFormat.ieeeDouble) { v |= 0x7FF0_0000_0000_0000; real x; * cast(ulong *)(&x) = v; return x; } else { v <<=11; a &= 0x7FF; v |= a; real x = real.nan; // Extended real bits static if (F.realFormat == RealFormat.ieeeQuadruple) { v <<= 1; // there's no implicit bit version (LittleEndian) { *cast(ulong*)(6+cast(ubyte*)(&x)) = v; } else { *cast(ulong*)(2+cast(ubyte*)(&x)) = v; } } else { *cast(ulong *)(&x) = v; } return x; } } @system pure nothrow @nogc unittest // not @safe because taking address of local. { static if (floatTraits!(real).realFormat == RealFormat.ieeeDouble) { auto x = NaN(1); auto xl = *cast(ulong*)&x; assert(xl & 0x8_0000_0000_0000UL); //non-signaling bit, bit 52 assert((xl & 0x7FF0_0000_0000_0000UL) == 0x7FF0_0000_0000_0000UL); //all exp bits set } } /** * Extract an integral payload from a $(NAN). * * Returns: * the integer payload as a ulong. * * For floats, the largest possible payload is 0x3F_FFFF. * For doubles, it is 0x3_FFFF_FFFF_FFFF. * For 80-bit or 128-bit reals, it is 0x3FFF_FFFF_FFFF_FFFF. */ ulong getNaNPayload(real x) @trusted pure nothrow @nogc { // assert(isNaN(x)); alias F = floatTraits!(real); static if (F.realFormat == RealFormat.ieeeDouble) { ulong m = *cast(ulong *)(&x); // Make it look like an 80-bit significand. // Skip exponent, and quiet bit m &= 0x0007_FFFF_FFFF_FFFF; m <<= 11; } else static if (F.realFormat == RealFormat.ieeeQuadruple) { version (LittleEndian) { ulong m = *cast(ulong*)(6+cast(ubyte*)(&x)); } else { ulong m = *cast(ulong*)(2+cast(ubyte*)(&x)); } m >>= 1; // there's no implicit bit } else { ulong m = *cast(ulong *)(&x); } // ignore implicit bit and quiet bit const ulong f = m & 0x3FFF_FF00_0000_0000L; ulong w = f >>> 40; w |= (m & 0x00FF_FFFF_F800L) << (22 - 11); w |= (m & 0x7FF) << 51; return w; } debug(UnitTest) { @safe pure nothrow @nogc unittest { real nan4 = NaN(0x789_ABCD_EF12_3456); static if (floatTraits!(real).realFormat == RealFormat.ieeeExtended || floatTraits!(real).realFormat == RealFormat.ieeeQuadruple) { assert(getNaNPayload(nan4) == 0x789_ABCD_EF12_3456); } else { assert(getNaNPayload(nan4) == 0x1_ABCD_EF12_3456); } double nan5 = nan4; assert(getNaNPayload(nan5) == 0x1_ABCD_EF12_3456); float nan6 = nan4; assert(getNaNPayload(nan6) == 0x12_3456); nan4 = NaN(0xFABCD); assert(getNaNPayload(nan4) == 0xFABCD); nan6 = nan4; assert(getNaNPayload(nan6) == 0xFABCD); nan5 = NaN(0x100_0000_0000_3456); assert(getNaNPayload(nan5) == 0x0000_0000_3456); } } /** * Calculate the next largest floating point value after x. * * Return the least number greater than x that is representable as a real; * thus, it gives the next point on the IEEE number line. * * $(TABLE_SV * $(SVH x, nextUp(x) ) * $(SV -$(INFIN), -real.max ) * $(SV $(PLUSMN)0.0, real.min_normal*real.epsilon ) * $(SV real.max, $(INFIN) ) * $(SV $(INFIN), $(INFIN) ) * $(SV $(NAN), $(NAN) ) * ) */ real nextUp(real x) @trusted pure nothrow @nogc { alias F = floatTraits!(real); static if (F.realFormat == RealFormat.ieeeDouble) { return nextUp(cast(double) x); } else static if (F.realFormat == RealFormat.ieeeQuadruple) { ushort e = F.EXPMASK & (cast(ushort *)&x)[F.EXPPOS_SHORT]; if (e == F.EXPMASK) { // NaN or Infinity if (x == -real.infinity) return -real.max; return x; // +Inf and NaN are unchanged. } auto ps = cast(ulong *)&x; if (ps[MANTISSA_MSB] & 0x8000_0000_0000_0000) { // Negative number if (ps[MANTISSA_LSB] == 0 && ps[MANTISSA_MSB] == 0x8000_0000_0000_0000) { // it was negative zero, change to smallest subnormal ps[MANTISSA_LSB] = 1; ps[MANTISSA_MSB] = 0; return x; } if (ps[MANTISSA_LSB] == 0) --ps[MANTISSA_MSB]; --ps[MANTISSA_LSB]; } else { // Positive number ++ps[MANTISSA_LSB]; if (ps[MANTISSA_LSB] == 0) ++ps[MANTISSA_MSB]; } return x; } else static if (F.realFormat == RealFormat.ieeeExtended) { // For 80-bit reals, the "implied bit" is a nuisance... ushort *pe = cast(ushort *)&x; ulong *ps = cast(ulong *)&x; if ((pe[F.EXPPOS_SHORT] & F.EXPMASK) == F.EXPMASK) { // First, deal with NANs and infinity if (x == -real.infinity) return -real.max; return x; // +Inf and NaN are unchanged. } if (pe[F.EXPPOS_SHORT] & 0x8000) { // Negative number -- need to decrease the significand --*ps; // Need to mask with 0x7FFF... so subnormals are treated correctly. if ((*ps & 0x7FFF_FFFF_FFFF_FFFF) == 0x7FFF_FFFF_FFFF_FFFF) { if (pe[F.EXPPOS_SHORT] == 0x8000) // it was negative zero { *ps = 1; pe[F.EXPPOS_SHORT] = 0; // smallest subnormal. return x; } --pe[F.EXPPOS_SHORT]; if (pe[F.EXPPOS_SHORT] == 0x8000) return x; // it's become a subnormal, implied bit stays low. *ps = 0xFFFF_FFFF_FFFF_FFFF; // set the implied bit return x; } return x; } else { // Positive number -- need to increase the significand. // Works automatically for positive zero. ++*ps; if ((*ps & 0x7FFF_FFFF_FFFF_FFFF) == 0) { // change in exponent ++pe[F.EXPPOS_SHORT]; *ps = 0x8000_0000_0000_0000; // set the high bit } } return x; } else // static if (F.realFormat == RealFormat.ibmExtended) { assert(0, "nextUp not implemented"); } } /** ditto */ double nextUp(double x) @trusted pure nothrow @nogc { ulong *ps = cast(ulong *)&x; if ((*ps & 0x7FF0_0000_0000_0000) == 0x7FF0_0000_0000_0000) { // First, deal with NANs and infinity if (x == -x.infinity) return -x.max; return x; // +INF and NAN are unchanged. } if (*ps & 0x8000_0000_0000_0000) // Negative number { if (*ps == 0x8000_0000_0000_0000) // it was negative zero { *ps = 0x0000_0000_0000_0001; // change to smallest subnormal return x; } --*ps; } else { // Positive number ++*ps; } return x; } /** ditto */ float nextUp(float x) @trusted pure nothrow @nogc { uint *ps = cast(uint *)&x; if ((*ps & 0x7F80_0000) == 0x7F80_0000) { // First, deal with NANs and infinity if (x == -x.infinity) return -x.max; return x; // +INF and NAN are unchanged. } if (*ps & 0x8000_0000) // Negative number { if (*ps == 0x8000_0000) // it was negative zero { *ps = 0x0000_0001; // change to smallest subnormal return x; } --*ps; } else { // Positive number ++*ps; } return x; } /** * Calculate the next smallest floating point value before x. * * Return the greatest number less than x that is representable as a real; * thus, it gives the previous point on the IEEE number line. * * $(TABLE_SV * $(SVH x, nextDown(x) ) * $(SV $(INFIN), real.max ) * $(SV $(PLUSMN)0.0, -real.min_normal*real.epsilon ) * $(SV -real.max, -$(INFIN) ) * $(SV -$(INFIN), -$(INFIN) ) * $(SV $(NAN), $(NAN) ) * ) */ real nextDown(real x) @safe pure nothrow @nogc { return -nextUp(-x); } /** ditto */ double nextDown(double x) @safe pure nothrow @nogc { return -nextUp(-x); } /** ditto */ float nextDown(float x) @safe pure nothrow @nogc { return -nextUp(-x); } /// @safe pure nothrow @nogc unittest { assert( nextDown(1.0 + real.epsilon) == 1.0); } @safe pure nothrow @nogc unittest { static if (floatTraits!(real).realFormat == RealFormat.ieeeExtended) { // Tests for 80-bit reals assert(isIdentical(nextUp(NaN(0xABC)), NaN(0xABC))); // negative numbers assert( nextUp(-real.infinity) == -real.max ); assert( nextUp(-1.0L-real.epsilon) == -1.0 ); assert( nextUp(-2.0L) == -2.0 + real.epsilon); // subnormals and zero assert( nextUp(-real.min_normal) == -real.min_normal*(1-real.epsilon) ); assert( nextUp(-real.min_normal*(1-real.epsilon)) == -real.min_normal*(1-2*real.epsilon) ); assert( isIdentical(-0.0L, nextUp(-real.min_normal*real.epsilon)) ); assert( nextUp(-0.0L) == real.min_normal*real.epsilon ); assert( nextUp(0.0L) == real.min_normal*real.epsilon ); assert( nextUp(real.min_normal*(1-real.epsilon)) == real.min_normal ); assert( nextUp(real.min_normal) == real.min_normal*(1+real.epsilon) ); // positive numbers assert( nextUp(1.0L) == 1.0 + real.epsilon ); assert( nextUp(2.0L-real.epsilon) == 2.0 ); assert( nextUp(real.max) == real.infinity ); assert( nextUp(real.infinity)==real.infinity ); } double n = NaN(0xABC); assert(isIdentical(nextUp(n), n)); // negative numbers assert( nextUp(-double.infinity) == -double.max ); assert( nextUp(-1-double.epsilon) == -1.0 ); assert( nextUp(-2.0) == -2.0 + double.epsilon); // subnormals and zero assert( nextUp(-double.min_normal) == -double.min_normal*(1-double.epsilon) ); assert( nextUp(-double.min_normal*(1-double.epsilon)) == -double.min_normal*(1-2*double.epsilon) ); assert( isIdentical(-0.0, nextUp(-double.min_normal*double.epsilon)) ); assert( nextUp(0.0) == double.min_normal*double.epsilon ); assert( nextUp(-0.0) == double.min_normal*double.epsilon ); assert( nextUp(double.min_normal*(1-double.epsilon)) == double.min_normal ); assert( nextUp(double.min_normal) == double.min_normal*(1+double.epsilon) ); // positive numbers assert( nextUp(1.0) == 1.0 + double.epsilon ); assert( nextUp(2.0-double.epsilon) == 2.0 ); assert( nextUp(double.max) == double.infinity ); float fn = NaN(0xABC); assert(isIdentical(nextUp(fn), fn)); float f = -float.min_normal*(1-float.epsilon); float f1 = -float.min_normal; assert( nextUp(f1) == f); f = 1.0f+float.epsilon; f1 = 1.0f; assert( nextUp(f1) == f ); f1 = -0.0f; assert( nextUp(f1) == float.min_normal*float.epsilon); assert( nextUp(float.infinity)==float.infinity ); assert(nextDown(1.0L+real.epsilon)==1.0); assert(nextDown(1.0+double.epsilon)==1.0); f = 1.0f+float.epsilon; assert(nextDown(f)==1.0); assert(nextafter(1.0+real.epsilon, -real.infinity)==1.0); } /****************************************** * Calculates the next representable value after x in the direction of y. * * If y > x, the result will be the next largest floating-point value; * if y < x, the result will be the next smallest value. * If x == y, the result is y. * * Remarks: * This function is not generally very useful; it's almost always better to use * the faster functions nextUp() or nextDown() instead. * * The FE_INEXACT and FE_OVERFLOW exceptions will be raised if x is finite and * the function result is infinite. The FE_INEXACT and FE_UNDERFLOW * exceptions will be raised if the function value is subnormal, and x is * not equal to y. */ T nextafter(T)(const T x, const T y) @safe pure nothrow @nogc { if (x == y) return y; return ((y>x) ? nextUp(x) : nextDown(x)); } /// @safe pure nothrow @nogc unittest { float a = 1; assert(is(typeof(nextafter(a, a)) == float)); assert(nextafter(a, a.infinity) > a); double b = 2; assert(is(typeof(nextafter(b, b)) == double)); assert(nextafter(b, b.infinity) > b); real c = 3; assert(is(typeof(nextafter(c, c)) == real)); assert(nextafter(c, c.infinity) > c); } //real nexttoward(real x, real y) { return core.stdc.math.nexttowardl(x, y); } /******************************************* * Returns the positive difference between x and y. * Returns: * $(TABLE_SV * $(TR $(TH x, y) $(TH fdim(x, y))) * $(TR $(TD x $(GT) y) $(TD x - y)) * $(TR $(TD x $(LT)= y) $(TD +0.0)) * ) */ real fdim(real x, real y) @safe pure nothrow @nogc { return (x > y) ? x - y : +0.0; } /**************************************** * Returns the larger of x and y. */ real fmax(real x, real y) @safe pure nothrow @nogc { return x > y ? x : y; } /**************************************** * Returns the smaller of x and y. */ real fmin(real x, real y) @safe pure nothrow @nogc { return x < y ? x : y; } /************************************** * Returns (x * y) + z, rounding only once according to the * current rounding mode. * * BUGS: Not currently implemented - rounds twice. */ real fma(real x, real y, real z) @safe pure nothrow @nogc { return (x * y) + z; } /******************************************************************* * Compute the value of x $(SUPERSCRIPT n), where n is an integer */ Unqual!F pow(F, G)(F x, G n) @nogc @trusted pure nothrow if (isFloatingPoint!(F) && isIntegral!(G)) { import std.traits : Unsigned; real p = 1.0, v = void; Unsigned!(Unqual!G) m = n; if (n < 0) { switch (n) { case -1: return 1 / x; case -2: return 1 / (x * x); default: } m = cast(typeof(m))(0 - n); v = p / x; } else { switch (n) { case 0: return 1.0; case 1: return x; case 2: return x * x; default: } v = x; } while (1) { if (m & 1) p *= v; m >>= 1; if (!m) break; v *= v; } return p; } @safe pure nothrow @nogc unittest { // Make sure it instantiates and works properly on immutable values and // with various integer and float types. immutable real x = 46; immutable float xf = x; immutable double xd = x; immutable uint one = 1; immutable ushort two = 2; immutable ubyte three = 3; immutable ulong eight = 8; immutable int neg1 = -1; immutable short neg2 = -2; immutable byte neg3 = -3; immutable long neg8 = -8; assert(pow(x,0) == 1.0); assert(pow(xd,one) == x); assert(pow(xf,two) == x * x); assert(pow(x,three) == x * x * x); assert(pow(x,eight) == (x * x) * (x * x) * (x * x) * (x * x)); assert(pow(x, neg1) == 1 / x); // Test disabled on most targets. // See https://issues.dlang.org/show_bug.cgi?id=5628 version (X86_64) enum BUG5628 = false; else version (ARM) enum BUG5628 = false; else version (GNU) enum BUG5628 = false; else enum BUG5628 = true; static if (BUG5628) { assert(pow(xd, neg2) == 1 / (x * x)); assert(pow(xf, neg8) == 1 / ((x * x) * (x * x) * (x * x) * (x * x))); } assert(feqrel(pow(x, neg3), 1 / (x * x * x)) >= real.mant_dig - 1); } @system unittest { assert(equalsDigit(pow(2.0L, 10.0L), 1024, 19)); } /** Compute the value of an integer x, raised to the power of a positive * integer n. * * If both x and n are 0, the result is 1. * If n is negative, an integer divide error will occur at runtime, * regardless of the value of x. */ typeof(Unqual!(F).init * Unqual!(G).init) pow(F, G)(F x, G n) @nogc @trusted pure nothrow if (isIntegral!(F) && isIntegral!(G)) { if (n<0) return x/0; // Only support positive powers typeof(return) p, v = void; Unqual!G m = n; switch (m) { case 0: p = 1; break; case 1: p = x; break; case 2: p = x * x; break; default: v = x; p = 1; while (1) { if (m & 1) p *= v; m >>= 1; if (!m) break; v *= v; } break; } return p; } /// @safe pure nothrow @nogc unittest { immutable int one = 1; immutable byte two = 2; immutable ubyte three = 3; immutable short four = 4; immutable long ten = 10; assert(pow(two, three) == 8); assert(pow(two, ten) == 1024); assert(pow(one, ten) == 1); assert(pow(ten, four) == 10_000); assert(pow(four, 10) == 1_048_576); assert(pow(three, four) == 81); } /**Computes integer to floating point powers.*/ real pow(I, F)(I x, F y) @nogc @trusted pure nothrow if (isIntegral!I && isFloatingPoint!F) { return pow(cast(real) x, cast(Unqual!F) y); } /********************************************* * Calculates x$(SUPERSCRIPT y). * * $(TABLE_SV * $(TR $(TH x) $(TH y) $(TH pow(x, y)) * $(TH div 0) $(TH invalid?)) * $(TR $(TD anything) $(TD $(PLUSMN)0.0) $(TD 1.0) * $(TD no) $(TD no) ) * $(TR $(TD |x| $(GT) 1) $(TD +$(INFIN)) $(TD +$(INFIN)) * $(TD no) $(TD no) ) * $(TR $(TD |x| $(LT) 1) $(TD +$(INFIN)) $(TD +0.0) * $(TD no) $(TD no) ) * $(TR $(TD |x| $(GT) 1) $(TD -$(INFIN)) $(TD +0.0) * $(TD no) $(TD no) ) * $(TR $(TD |x| $(LT) 1) $(TD -$(INFIN)) $(TD +$(INFIN)) * $(TD no) $(TD no) ) * $(TR $(TD +$(INFIN)) $(TD $(GT) 0.0) $(TD +$(INFIN)) * $(TD no) $(TD no) ) * $(TR $(TD +$(INFIN)) $(TD $(LT) 0.0) $(TD +0.0) * $(TD no) $(TD no) ) * $(TR $(TD -$(INFIN)) $(TD odd integer $(GT) 0.0) $(TD -$(INFIN)) * $(TD no) $(TD no) ) * $(TR $(TD -$(INFIN)) $(TD $(GT) 0.0, not odd integer) $(TD +$(INFIN)) * $(TD no) $(TD no)) * $(TR $(TD -$(INFIN)) $(TD odd integer $(LT) 0.0) $(TD -0.0) * $(TD no) $(TD no) ) * $(TR $(TD -$(INFIN)) $(TD $(LT) 0.0, not odd integer) $(TD +0.0) * $(TD no) $(TD no) ) * $(TR $(TD $(PLUSMN)1.0) $(TD $(PLUSMN)$(INFIN)) $(TD $(NAN)) * $(TD no) $(TD yes) ) * $(TR $(TD $(LT) 0.0) $(TD finite, nonintegral) $(TD $(NAN)) * $(TD no) $(TD yes)) * $(TR $(TD $(PLUSMN)0.0) $(TD odd integer $(LT) 0.0) $(TD $(PLUSMNINF)) * $(TD yes) $(TD no) ) * $(TR $(TD $(PLUSMN)0.0) $(TD $(LT) 0.0, not odd integer) $(TD +$(INFIN)) * $(TD yes) $(TD no)) * $(TR $(TD $(PLUSMN)0.0) $(TD odd integer $(GT) 0.0) $(TD $(PLUSMN)0.0) * $(TD no) $(TD no) ) * $(TR $(TD $(PLUSMN)0.0) $(TD $(GT) 0.0, not odd integer) $(TD +0.0) * $(TD no) $(TD no) ) * ) */ Unqual!(Largest!(F, G)) pow(F, G)(F x, G y) @nogc @trusted pure nothrow if (isFloatingPoint!(F) && isFloatingPoint!(G)) { alias Float = typeof(return); static real impl(real x, real y) @nogc pure nothrow { // Special cases. if (isNaN(y)) return y; if (isNaN(x) && y != 0.0) return x; // Even if x is NaN. if (y == 0.0) return 1.0; if (y == 1.0) return x; if (isInfinity(y)) { if (fabs(x) > 1) { if (signbit(y)) return +0.0; else return F.infinity; } else if (fabs(x) == 1) { return y * 0; // generate NaN. } else // < 1 { if (signbit(y)) return F.infinity; else return +0.0; } } if (isInfinity(x)) { if (signbit(x)) { long i = cast(long) y; if (y > 0.0) { if (i == y && i & 1) return -F.infinity; else return F.infinity; } else if (y < 0.0) { if (i == y && i & 1) return -0.0; else return +0.0; } } else { if (y > 0.0) return F.infinity; else if (y < 0.0) return +0.0; } } if (x == 0.0) { if (signbit(x)) { long i = cast(long) y; if (y > 0.0) { if (i == y && i & 1) return -0.0; else return +0.0; } else if (y < 0.0) { if (i == y && i & 1) return -F.infinity; else return F.infinity; } } else { if (y > 0.0) return +0.0; else if (y < 0.0) return F.infinity; } } if (x == 1.0) return 1.0; if (y >= F.max) { if ((x > 0.0 && x < 1.0) || (x > -1.0 && x < 0.0)) return 0.0; if (x > 1.0 || x < -1.0) return F.infinity; } if (y <= -F.max) { if ((x > 0.0 && x < 1.0) || (x > -1.0 && x < 0.0)) return F.infinity; if (x > 1.0 || x < -1.0) return 0.0; } if (x >= F.max) { if (y > 0.0) return F.infinity; else return 0.0; } if (x <= -F.max) { long i = cast(long) y; if (y > 0.0) { if (i == y && i & 1) return -F.infinity; else return F.infinity; } else if (y < 0.0) { if (i == y && i & 1) return -0.0; else return +0.0; } } // Integer power of x. long iy = cast(long) y; if (iy == y && fabs(y) < 32_768.0) return pow(x, iy); real sign = 1.0; if (x < 0) { // Result is real only if y is an integer // Check for a non-zero fractional part enum maxOdd = pow(2.0L, real.mant_dig) - 1.0L; static if (maxOdd > ulong.max) { // Generic method, for any FP type if (floor(y) != y) return sqrt(x); // Complex result -- create a NaN const hy = ldexp(y, -1); if (floor(hy) != hy) sign = -1.0; } else { // Much faster, if ulong has enough precision const absY = fabs(y); if (absY <= maxOdd) { const uy = cast(ulong) absY; if (uy != absY) return sqrt(x); // Complex result -- create a NaN if (uy & 1) sign = -1.0; } } x = -x; } version (INLINE_YL2X) { // If x > 0, x ^^ y == 2 ^^ ( y * log2(x) ) // TODO: This is not accurate in practice. A fast and accurate // (though complicated) method is described in: // "An efficient rounding boundary test for pow(x, y) // in double precision", C.Q. Lauter and V. Lefèvre, INRIA (2007). return sign * exp2( core.math.yl2x(x, y) ); } else { // If x > 0, x ^^ y == 2 ^^ ( y * log2(x) ) // TODO: This is not accurate in practice. A fast and accurate // (though complicated) method is described in: // "An efficient rounding boundary test for pow(x, y) // in double precision", C.Q. Lauter and V. Lefèvre, INRIA (2007). Float w = exp2(y * log2(x)); return sign * w; } } return impl(x, y); } @safe pure nothrow @nogc unittest { // Test all the special values. These unittests can be run on Windows // by temporarily changing the version (linux) to version (all). immutable float zero = 0; immutable real one = 1; immutable double two = 2; immutable float three = 3; immutable float fnan = float.nan; immutable double dnan = double.nan; immutable real rnan = real.nan; immutable dinf = double.infinity; immutable rninf = -real.infinity; assert(pow(fnan, zero) == 1); assert(pow(dnan, zero) == 1); assert(pow(rnan, zero) == 1); assert(pow(two, dinf) == double.infinity); assert(isIdentical(pow(0.2f, dinf), +0.0)); assert(pow(0.99999999L, rninf) == real.infinity); assert(isIdentical(pow(1.000000001, rninf), +0.0)); assert(pow(dinf, 0.001) == dinf); assert(isIdentical(pow(dinf, -0.001), +0.0)); assert(pow(rninf, 3.0L) == rninf); assert(pow(rninf, 2.0L) == real.infinity); assert(isIdentical(pow(rninf, -3.0), -0.0)); assert(isIdentical(pow(rninf, -2.0), +0.0)); // @@@BUG@@@ somewhere version (OSX) {} else assert(isNaN(pow(one, dinf))); version (OSX) {} else assert(isNaN(pow(-one, dinf))); assert(isNaN(pow(-0.2, PI))); // boundary cases. Note that epsilon == 2^^-n for some n, // so 1/epsilon == 2^^n is always even. assert(pow(-1.0L, 1/real.epsilon - 1.0L) == -1.0L); assert(pow(-1.0L, 1/real.epsilon) == 1.0L); assert(isNaN(pow(-1.0L, 1/real.epsilon-0.5L))); assert(isNaN(pow(-1.0L, -1/real.epsilon+0.5L))); assert(pow(0.0, -3.0) == double.infinity); assert(pow(-0.0, -3.0) == -double.infinity); assert(pow(0.0, -PI) == double.infinity); assert(pow(-0.0, -PI) == double.infinity); assert(isIdentical(pow(0.0, 5.0), 0.0)); assert(isIdentical(pow(-0.0, 5.0), -0.0)); assert(isIdentical(pow(0.0, 6.0), 0.0)); assert(isIdentical(pow(-0.0, 6.0), 0.0)); // Issue #14786 fixed immutable real maxOdd = pow(2.0L, real.mant_dig) - 1.0L; assert(pow(-1.0L, maxOdd) == -1.0L); assert(pow(-1.0L, -maxOdd) == -1.0L); assert(pow(-1.0L, maxOdd + 1.0L) == 1.0L); assert(pow(-1.0L, -maxOdd + 1.0L) == 1.0L); assert(pow(-1.0L, maxOdd - 1.0L) == 1.0L); assert(pow(-1.0L, -maxOdd - 1.0L) == 1.0L); // Now, actual numbers. assert(approxEqual(pow(two, three), 8.0)); assert(approxEqual(pow(two, -2.5), 0.1767767)); // Test integer to float power. immutable uint twoI = 2; assert(approxEqual(pow(twoI, three), 8.0)); } /************************************** * To what precision is x equal to y? * * Returns: the number of mantissa bits which are equal in x and y. * eg, 0x1.F8p+60 and 0x1.F1p+60 are equal to 5 bits of precision. * * $(TABLE_SV * $(TR $(TH x) $(TH y) $(TH feqrel(x, y))) * $(TR $(TD x) $(TD x) $(TD real.mant_dig)) * $(TR $(TD x) $(TD $(GT)= 2*x) $(TD 0)) * $(TR $(TD x) $(TD $(LT)= x/2) $(TD 0)) * $(TR $(TD $(NAN)) $(TD any) $(TD 0)) * $(TR $(TD any) $(TD $(NAN)) $(TD 0)) * ) */ int feqrel(X)(const X x, const X y) @trusted pure nothrow @nogc if (isFloatingPoint!(X)) { /* Public Domain. Author: Don Clugston, 18 Aug 2005. */ alias F = floatTraits!(X); static if (F.realFormat == RealFormat.ibmExtended) { if (cast(double*)(&x)[MANTISSA_MSB] == cast(double*)(&y)[MANTISSA_MSB]) { return double.mant_dig + feqrel(cast(double*)(&x)[MANTISSA_LSB], cast(double*)(&y)[MANTISSA_LSB]); } else { return feqrel(cast(double*)(&x)[MANTISSA_MSB], cast(double*)(&y)[MANTISSA_MSB]); } } else { static assert(F.realFormat == RealFormat.ieeeSingle || F.realFormat == RealFormat.ieeeDouble || F.realFormat == RealFormat.ieeeExtended || F.realFormat == RealFormat.ieeeQuadruple); if (x == y) return X.mant_dig; // ensure diff != 0, cope with INF. Unqual!X diff = fabs(x - y); ushort *pa = cast(ushort *)(&x); ushort *pb = cast(ushort *)(&y); ushort *pd = cast(ushort *)(&diff); // The difference in abs(exponent) between x or y and abs(x-y) // is equal to the number of significand bits of x which are // equal to y. If negative, x and y have different exponents. // If positive, x and y are equal to 'bitsdiff' bits. // AND with 0x7FFF to form the absolute value. // To avoid out-by-1 errors, we subtract 1 so it rounds down // if the exponents were different. This means 'bitsdiff' is // always 1 lower than we want, except that if bitsdiff == 0, // they could have 0 or 1 bits in common. int bitsdiff = ((( (pa[F.EXPPOS_SHORT] & F.EXPMASK) + (pb[F.EXPPOS_SHORT] & F.EXPMASK) - (1 << F.EXPSHIFT)) >> 1) - (pd[F.EXPPOS_SHORT] & F.EXPMASK)) >> F.EXPSHIFT; if ( (pd[F.EXPPOS_SHORT] & F.EXPMASK) == 0) { // Difference is subnormal // For subnormals, we need to add the number of zeros that // lie at the start of diff's significand. // We do this by multiplying by 2^^real.mant_dig diff *= F.RECIP_EPSILON; return bitsdiff + X.mant_dig - ((pd[F.EXPPOS_SHORT] & F.EXPMASK) >> F.EXPSHIFT); } if (bitsdiff > 0) return bitsdiff + 1; // add the 1 we subtracted before // Avoid out-by-1 errors when factor is almost 2. if (bitsdiff == 0 && ((pa[F.EXPPOS_SHORT] ^ pb[F.EXPPOS_SHORT]) & F.EXPMASK) == 0) { return 1; } else return 0; } } @safe pure nothrow @nogc unittest { void testFeqrel(F)() { // Exact equality assert(feqrel(F.max, F.max) == F.mant_dig); assert(feqrel!(F)(0.0, 0.0) == F.mant_dig); assert(feqrel(F.infinity, F.infinity) == F.mant_dig); // a few bits away from exact equality F w=1; for (int i = 1; i < F.mant_dig - 1; ++i) { assert(feqrel!(F)(1.0 + w * F.epsilon, 1.0) == F.mant_dig-i); assert(feqrel!(F)(1.0 - w * F.epsilon, 1.0) == F.mant_dig-i); assert(feqrel!(F)(1.0, 1 + (w-1) * F.epsilon) == F.mant_dig - i + 1); w*=2; } assert(feqrel!(F)(1.5+F.epsilon, 1.5) == F.mant_dig-1); assert(feqrel!(F)(1.5-F.epsilon, 1.5) == F.mant_dig-1); assert(feqrel!(F)(1.5-F.epsilon, 1.5+F.epsilon) == F.mant_dig-2); // Numbers that are close assert(feqrel!(F)(0x1.Bp+84, 0x1.B8p+84) == 5); assert(feqrel!(F)(0x1.8p+10, 0x1.Cp+10) == 2); assert(feqrel!(F)(1.5 * (1 - F.epsilon), 1.0L) == 2); assert(feqrel!(F)(1.5, 1.0) == 1); assert(feqrel!(F)(2 * (1 - F.epsilon), 1.0L) == 1); // Factors of 2 assert(feqrel(F.max, F.infinity) == 0); assert(feqrel!(F)(2 * (1 - F.epsilon), 1.0L) == 1); assert(feqrel!(F)(1.0, 2.0) == 0); assert(feqrel!(F)(4.0, 1.0) == 0); // Extreme inequality assert(feqrel(F.nan, F.nan) == 0); assert(feqrel!(F)(0.0L, -F.nan) == 0); assert(feqrel(F.nan, F.infinity) == 0); assert(feqrel(F.infinity, -F.infinity) == 0); assert(feqrel(F.max, -F.max) == 0); assert(feqrel(F.min_normal / 8, F.min_normal / 17) == 3); const F Const = 2; immutable F Immutable = 2; auto Compiles = feqrel(Const, Immutable); } assert(feqrel(7.1824L, 7.1824L) == real.mant_dig); testFeqrel!(real)(); testFeqrel!(double)(); testFeqrel!(float)(); } package: // Not public yet /* Return the value that lies halfway between x and y on the IEEE number line. * * Formally, the result is the arithmetic mean of the binary significands of x * and y, multiplied by the geometric mean of the binary exponents of x and y. * x and y must have the same sign, and must not be NaN. * Note: this function is useful for ensuring O(log n) behaviour in algorithms * involving a 'binary chop'. * * Special cases: * If x and y are within a factor of 2, (ie, feqrel(x, y) > 0), the return value * is the arithmetic mean (x + y) / 2. * If x and y are even powers of 2, the return value is the geometric mean, * ieeeMean(x, y) = sqrt(x * y). * */ T ieeeMean(T)(const T x, const T y) @trusted pure nothrow @nogc in { // both x and y must have the same sign, and must not be NaN. assert(signbit(x) == signbit(y)); assert(x == x && y == y); } body { // Runtime behaviour for contract violation: // If signs are opposite, or one is a NaN, return 0. if (!((x >= 0 && y >= 0) || (x <= 0 && y <= 0))) return 0.0; // The implementation is simple: cast x and y to integers, // average them (avoiding overflow), and cast the result back to a floating-point number. alias F = floatTraits!(T); T u; static if (F.realFormat == RealFormat.ieeeExtended) { // There's slight additional complexity because they are actually // 79-bit reals... ushort *ue = cast(ushort *)&u; ulong *ul = cast(ulong *)&u; ushort *xe = cast(ushort *)&x; ulong *xl = cast(ulong *)&x; ushort *ye = cast(ushort *)&y; ulong *yl = cast(ulong *)&y; // Ignore the useless implicit bit. (Bonus: this prevents overflows) ulong m = ((*xl) & 0x7FFF_FFFF_FFFF_FFFFL) + ((*yl) & 0x7FFF_FFFF_FFFF_FFFFL); // @@@ BUG? @@@ // Cast shouldn't be here ushort e = cast(ushort) ((xe[F.EXPPOS_SHORT] & F.EXPMASK) + (ye[F.EXPPOS_SHORT] & F.EXPMASK)); if (m & 0x8000_0000_0000_0000L) { ++e; m &= 0x7FFF_FFFF_FFFF_FFFFL; } // Now do a multi-byte right shift const uint c = e & 1; // carry e >>= 1; m >>>= 1; if (c) m |= 0x4000_0000_0000_0000L; // shift carry into significand if (e) *ul = m | 0x8000_0000_0000_0000L; // set implicit bit... else *ul = m; // ... unless exponent is 0 (subnormal or zero). ue[4]= e | (xe[F.EXPPOS_SHORT]& 0x8000); // restore sign bit } else static if (F.realFormat == RealFormat.ieeeQuadruple) { // This would be trivial if 'ucent' were implemented... ulong *ul = cast(ulong *)&u; ulong *xl = cast(ulong *)&x; ulong *yl = cast(ulong *)&y; // Multi-byte add, then multi-byte right shift. import core.checkedint : addu; bool carry; ulong ml = addu(xl[MANTISSA_LSB], yl[MANTISSA_LSB], carry); ulong mh = carry + (xl[MANTISSA_MSB] & 0x7FFF_FFFF_FFFF_FFFFL) + (yl[MANTISSA_MSB] & 0x7FFF_FFFF_FFFF_FFFFL); ul[MANTISSA_MSB] = (mh >>> 1) | (xl[MANTISSA_MSB] & 0x8000_0000_0000_0000); ul[MANTISSA_LSB] = (ml >>> 1) | (mh & 1) << 63; } else static if (F.realFormat == RealFormat.ieeeDouble) { ulong *ul = cast(ulong *)&u; ulong *xl = cast(ulong *)&x; ulong *yl = cast(ulong *)&y; ulong m = (((*xl) & 0x7FFF_FFFF_FFFF_FFFFL) + ((*yl) & 0x7FFF_FFFF_FFFF_FFFFL)) >>> 1; m |= ((*xl) & 0x8000_0000_0000_0000L); *ul = m; } else static if (F.realFormat == RealFormat.ieeeSingle) { uint *ul = cast(uint *)&u; uint *xl = cast(uint *)&x; uint *yl = cast(uint *)&y; uint m = (((*xl) & 0x7FFF_FFFF) + ((*yl) & 0x7FFF_FFFF)) >>> 1; m |= ((*xl) & 0x8000_0000); *ul = m; } else { assert(0, "Not implemented"); } return u; } @safe pure nothrow @nogc unittest { assert(ieeeMean(-0.0,-1e-20)<0); assert(ieeeMean(0.0,1e-20)>0); assert(ieeeMean(1.0L,4.0L)==2L); assert(ieeeMean(2.0*1.013,8.0*1.013)==4*1.013); assert(ieeeMean(-1.0L,-4.0L)==-2L); assert(ieeeMean(-1.0,-4.0)==-2); assert(ieeeMean(-1.0f,-4.0f)==-2f); assert(ieeeMean(-1.0,-2.0)==-1.5); assert(ieeeMean(-1*(1+8*real.epsilon),-2*(1+8*real.epsilon)) ==-1.5*(1+5*real.epsilon)); assert(ieeeMean(0x1p60,0x1p-10)==0x1p25); static if (floatTraits!(real).realFormat == RealFormat.ieeeExtended) { assert(ieeeMean(1.0L,real.infinity)==0x1p8192L); assert(ieeeMean(0.0L,real.infinity)==1.5); } assert(ieeeMean(0.5*real.min_normal*(1-4*real.epsilon),0.5*real.min_normal) == 0.5*real.min_normal*(1-2*real.epsilon)); } public: /*********************************** * Evaluate polynomial A(x) = $(SUB a, 0) + $(SUB a, 1)x + $(SUB a, 2)$(POWER x,2) * + $(SUB a,3)$(POWER x,3); ... * * Uses Horner's rule A(x) = $(SUB a, 0) + x($(SUB a, 1) + x($(SUB a, 2) * + x($(SUB a, 3) + ...))) * Params: * x = the value to evaluate. * A = array of coefficients $(SUB a, 0), $(SUB a, 1), etc. */ Unqual!(CommonType!(T1, T2)) poly(T1, T2)(T1 x, in T2[] A) @trusted pure nothrow @nogc if (isFloatingPoint!T1 && isFloatingPoint!T2) in { assert(A.length > 0); } body { static if (is(Unqual!T2 == real)) { return polyImpl(x, A); } else { return polyImplBase(x, A); } } /// @safe nothrow @nogc unittest { real x = 3.1; static real[] pp = [56.1, 32.7, 6]; assert(poly(x, pp) == (56.1L + (32.7L + 6.0L * x) * x)); } @safe nothrow @nogc unittest { double x = 3.1; static double[] pp = [56.1, 32.7, 6]; double y = x; y *= 6.0; y += 32.7; y *= x; y += 56.1; assert(poly(x, pp) == y); } @safe unittest { static assert(poly(3.0, [1.0, 2.0, 3.0]) == 34); } private Unqual!(CommonType!(T1, T2)) polyImplBase(T1, T2)(T1 x, in T2[] A) @trusted pure nothrow @nogc if (isFloatingPoint!T1 && isFloatingPoint!T2) { ptrdiff_t i = A.length - 1; typeof(return) r = A[i]; while (--i >= 0) { r *= x; r += A[i]; } return r; } private real polyImpl(real x, in real[] A) @trusted pure nothrow @nogc { version (D_InlineAsm_X86) { if (__ctfe) { return polyImplBase(x, A); } version (Windows) { // BUG: This code assumes a frame pointer in EBP. asm pure nothrow @nogc // assembler by W. Bright { // EDX = (A.length - 1) * real.sizeof mov ECX,A[EBP] ; // ECX = A.length dec ECX ; lea EDX,[ECX][ECX*8] ; add EDX,ECX ; add EDX,A+4[EBP] ; fld real ptr [EDX] ; // ST0 = coeff[ECX] jecxz return_ST ; fld x[EBP] ; // ST0 = x fxch ST(1) ; // ST1 = x, ST0 = r align 4 ; L2: fmul ST,ST(1) ; // r *= x fld real ptr -10[EDX] ; sub EDX,10 ; // deg-- faddp ST(1),ST ; dec ECX ; jne L2 ; fxch ST(1) ; // ST1 = r, ST0 = x fstp ST(0) ; // dump x align 4 ; return_ST: ; ; } } else version (linux) { asm pure nothrow @nogc // assembler by W. Bright { // EDX = (A.length - 1) * real.sizeof mov ECX,A[EBP] ; // ECX = A.length dec ECX ; lea EDX,[ECX*8] ; lea EDX,[EDX][ECX*4] ; add EDX,A+4[EBP] ; fld real ptr [EDX] ; // ST0 = coeff[ECX] jecxz return_ST ; fld x[EBP] ; // ST0 = x fxch ST(1) ; // ST1 = x, ST0 = r align 4 ; L2: fmul ST,ST(1) ; // r *= x fld real ptr -12[EDX] ; sub EDX,12 ; // deg-- faddp ST(1),ST ; dec ECX ; jne L2 ; fxch ST(1) ; // ST1 = r, ST0 = x fstp ST(0) ; // dump x align 4 ; return_ST: ; ; } } else version (OSX) { asm pure nothrow @nogc // assembler by W. Bright { // EDX = (A.length - 1) * real.sizeof mov ECX,A[EBP] ; // ECX = A.length dec ECX ; lea EDX,[ECX*8] ; add EDX,EDX ; add EDX,A+4[EBP] ; fld real ptr [EDX] ; // ST0 = coeff[ECX] jecxz return_ST ; fld x[EBP] ; // ST0 = x fxch ST(1) ; // ST1 = x, ST0 = r align 4 ; L2: fmul ST,ST(1) ; // r *= x fld real ptr -16[EDX] ; sub EDX,16 ; // deg-- faddp ST(1),ST ; dec ECX ; jne L2 ; fxch ST(1) ; // ST1 = r, ST0 = x fstp ST(0) ; // dump x align 4 ; return_ST: ; ; } } else version (FreeBSD) { asm pure nothrow @nogc // assembler by W. Bright { // EDX = (A.length - 1) * real.sizeof mov ECX,A[EBP] ; // ECX = A.length dec ECX ; lea EDX,[ECX*8] ; lea EDX,[EDX][ECX*4] ; add EDX,A+4[EBP] ; fld real ptr [EDX] ; // ST0 = coeff[ECX] jecxz return_ST ; fld x[EBP] ; // ST0 = x fxch ST(1) ; // ST1 = x, ST0 = r align 4 ; L2: fmul ST,ST(1) ; // r *= x fld real ptr -12[EDX] ; sub EDX,12 ; // deg-- faddp ST(1),ST ; dec ECX ; jne L2 ; fxch ST(1) ; // ST1 = r, ST0 = x fstp ST(0) ; // dump x align 4 ; return_ST: ; ; } } else version (Solaris) { asm pure nothrow @nogc // assembler by W. Bright { // EDX = (A.length - 1) * real.sizeof mov ECX,A[EBP] ; // ECX = A.length dec ECX ; lea EDX,[ECX*8] ; lea EDX,[EDX][ECX*4] ; add EDX,A+4[EBP] ; fld real ptr [EDX] ; // ST0 = coeff[ECX] jecxz return_ST ; fld x[EBP] ; // ST0 = x fxch ST(1) ; // ST1 = x, ST0 = r align 4 ; L2: fmul ST,ST(1) ; // r *= x fld real ptr -12[EDX] ; sub EDX,12 ; // deg-- faddp ST(1),ST ; dec ECX ; jne L2 ; fxch ST(1) ; // ST1 = r, ST0 = x fstp ST(0) ; // dump x align 4 ; return_ST: ; ; } } else version (DragonFlyBSD) { asm pure nothrow @nogc // assembler by W. Bright { // EDX = (A.length - 1) * real.sizeof mov ECX,A[EBP] ; // ECX = A.length dec ECX ; lea EDX,[ECX*8] ; lea EDX,[EDX][ECX*4] ; add EDX,A+4[EBP] ; fld real ptr [EDX] ; // ST0 = coeff[ECX] jecxz return_ST ; fld x[EBP] ; // ST0 = x fxch ST(1) ; // ST1 = x, ST0 = r align 4 ; L2: fmul ST,ST(1) ; // r *= x fld real ptr -12[EDX] ; sub EDX,12 ; // deg-- faddp ST(1),ST ; dec ECX ; jne L2 ; fxch ST(1) ; // ST1 = r, ST0 = x fstp ST(0) ; // dump x align 4 ; return_ST: ; ; } } else { static assert(0); } } else { return polyImplBase(x, A); } } /** Computes whether two values are approximately equal, admitting a maximum relative difference, and a maximum absolute difference. Params: lhs = First item to compare. rhs = Second item to compare. maxRelDiff = Maximum allowable difference relative to `rhs`. maxAbsDiff = Maximum absolute difference. Returns: `true` if the two items are approximately equal under either criterium. If one item is a range, and the other is a single value, then the result is the logical and-ing of calling `approxEqual` on each element of the ranged item against the single item. If both items are ranges, then `approxEqual` returns `true` if and only if the ranges have the same number of elements and if `approxEqual` evaluates to `true` for each pair of elements. */ bool approxEqual(T, U, V)(T lhs, U rhs, V maxRelDiff, V maxAbsDiff = 1e-5) { import std.range.primitives : empty, front, isInputRange, popFront; static if (isInputRange!T) { static if (isInputRange!U) { // Two ranges for (;; lhs.popFront(), rhs.popFront()) { if (lhs.empty) return rhs.empty; if (rhs.empty) return lhs.empty; if (!approxEqual(lhs.front, rhs.front, maxRelDiff, maxAbsDiff)) return false; } } else static if (isIntegral!U) { // convert rhs to real return approxEqual(lhs, real(rhs), maxRelDiff, maxAbsDiff); } else { // lhs is range, rhs is number for (; !lhs.empty; lhs.popFront()) { if (!approxEqual(lhs.front, rhs, maxRelDiff, maxAbsDiff)) return false; } return true; } } else { static if (isInputRange!U) { // lhs is number, rhs is range for (; !rhs.empty; rhs.popFront()) { if (!approxEqual(lhs, rhs.front, maxRelDiff, maxAbsDiff)) return false; } return true; } else static if (isIntegral!T || isIntegral!U) { // convert both lhs and rhs to real return approxEqual(real(lhs), real(rhs), maxRelDiff, maxAbsDiff); } else { // two numbers //static assert(is(T : real) && is(U : real)); if (rhs == 0) { return fabs(lhs) <= maxAbsDiff; } static if (is(typeof(lhs.infinity)) && is(typeof(rhs.infinity))) { if (lhs == lhs.infinity && rhs == rhs.infinity || lhs == -lhs.infinity && rhs == -rhs.infinity) return true; } return fabs((lhs - rhs) / rhs) <= maxRelDiff || maxAbsDiff != 0 && fabs(lhs - rhs) <= maxAbsDiff; } } } /** Returns $(D approxEqual(lhs, rhs, 1e-2, 1e-5)). */ bool approxEqual(T, U)(T lhs, U rhs) { return approxEqual(lhs, rhs, 1e-2, 1e-5); } /// @safe pure nothrow unittest { assert(approxEqual(1.0, 1.0099)); assert(!approxEqual(1.0, 1.011)); float[] arr1 = [ 1.0, 2.0, 3.0 ]; double[] arr2 = [ 1.001, 1.999, 3 ]; assert(approxEqual(arr1, arr2)); real num = real.infinity; assert(num == real.infinity); // Passes. assert(approxEqual(num, real.infinity)); // Fails. num = -real.infinity; assert(num == -real.infinity); // Passes. assert(approxEqual(num, -real.infinity)); // Fails. assert(!approxEqual(3, 0)); assert(approxEqual(3, 3)); assert(approxEqual(3.0, 3)); assert(approxEqual([3, 3, 3], 3.0)); assert(approxEqual([3.0, 3.0, 3.0], 3)); int a = 10; assert(approxEqual(10, a)); } @safe pure nothrow @nogc unittest { real num = real.infinity; assert(num == real.infinity); // Passes. assert(approxEqual(num, real.infinity)); // Fails. } @safe pure nothrow @nogc unittest { float f = sqrt(2.0f); assert(fabs(f * f - 2.0f) < .00001); double d = sqrt(2.0); assert(fabs(d * d - 2.0) < .00001); real r = sqrt(2.0L); assert(fabs(r * r - 2.0) < .00001); } @safe pure nothrow @nogc unittest { float f = fabs(-2.0f); assert(f == 2); double d = fabs(-2.0); assert(d == 2); real r = fabs(-2.0L); assert(r == 2); } @safe pure nothrow @nogc unittest { float f = sin(-2.0f); assert(fabs(f - -0.909297f) < .00001); double d = sin(-2.0); assert(fabs(d - -0.909297f) < .00001); real r = sin(-2.0L); assert(fabs(r - -0.909297f) < .00001); } @safe pure nothrow @nogc unittest { float f = cos(-2.0f); assert(fabs(f - -0.416147f) < .00001); double d = cos(-2.0); assert(fabs(d - -0.416147f) < .00001); real r = cos(-2.0L); assert(fabs(r - -0.416147f) < .00001); } @safe pure nothrow @nogc unittest { float f = tan(-2.0f); assert(fabs(f - 2.18504f) < .00001); double d = tan(-2.0); assert(fabs(d - 2.18504f) < .00001); real r = tan(-2.0L); assert(fabs(r - 2.18504f) < .00001); // Verify correct behavior for large inputs assert(!isNaN(tan(0x1p63))); assert(!isNaN(tan(0x1p300L))); assert(!isNaN(tan(-0x1p63))); assert(!isNaN(tan(-0x1p300L))); } @safe pure nothrow unittest { // issue 6381: floor/ceil should be usable in pure function. auto x = floor(1.2); auto y = ceil(1.2); } @safe pure nothrow unittest { // relative comparison depends on rhs, make sure proper side is used when // comparing range to single value. Based on bugzilla issue 15763 auto a = [2e-3 - 1e-5]; auto b = 2e-3 + 1e-5; assert(a[0].approxEqual(b)); assert(!b.approxEqual(a[0])); assert(a.approxEqual(b)); assert(!b.approxEqual(a)); } /*********************************** * Defines a total order on all floating-point numbers. * * The order is defined as follows: * $(UL * $(LI All numbers in [-$(INFIN), +$(INFIN)] are ordered * the same way as by built-in comparison, with the exception of * -0.0, which is less than +0.0;) * $(LI If the sign bit is set (that is, it's 'negative'), $(NAN) is less * than any number; if the sign bit is not set (it is 'positive'), * $(NAN) is greater than any number;) * $(LI $(NAN)s of the same sign are ordered by the payload ('negative' * ones - in reverse order).) * ) * * Returns: * negative value if $(D x) precedes $(D y) in the order specified above; * 0 if $(D x) and $(D y) are identical, and positive value otherwise. * * See_Also: * $(MYREF isIdentical) * Standards: Conforms to IEEE 754-2008 */ int cmp(T)(const(T) x, const(T) y) @nogc @trusted pure nothrow if (isFloatingPoint!T) { alias F = floatTraits!T; static if (F.realFormat == RealFormat.ieeeSingle || F.realFormat == RealFormat.ieeeDouble) { static if (T.sizeof == 4) alias UInt = uint; else alias UInt = ulong; union Repainter { T number; UInt bits; } enum msb = ~(UInt.max >>> 1); import std.typecons : Tuple; Tuple!(Repainter, Repainter) vars = void; vars[0].number = x; vars[1].number = y; foreach (ref var; vars) if (var.bits & msb) var.bits = ~var.bits; else var.bits |= msb; if (vars[0].bits < vars[1].bits) return -1; else if (vars[0].bits > vars[1].bits) return 1; else return 0; } else static if (F.realFormat == RealFormat.ieeeExtended53 || F.realFormat == RealFormat.ieeeExtended || F.realFormat == RealFormat.ieeeQuadruple) { static if (F.realFormat == RealFormat.ieeeQuadruple) alias RemT = ulong; else alias RemT = ushort; struct Bits { ulong bulk; RemT rem; } union Repainter { T number; Bits bits; ubyte[T.sizeof] bytes; } import std.typecons : Tuple; Tuple!(Repainter, Repainter) vars = void; vars[0].number = x; vars[1].number = y; foreach (ref var; vars) if (var.bytes[F.SIGNPOS_BYTE] & 0x80) { var.bits.bulk = ~var.bits.bulk; var.bits.rem = cast(typeof(var.bits.rem))(-1 - var.bits.rem); // ~var.bits.rem } else { var.bytes[F.SIGNPOS_BYTE] |= 0x80; } version (LittleEndian) { if (vars[0].bits.rem < vars[1].bits.rem) return -1; else if (vars[0].bits.rem > vars[1].bits.rem) return 1; else if (vars[0].bits.bulk < vars[1].bits.bulk) return -1; else if (vars[0].bits.bulk > vars[1].bits.bulk) return 1; else return 0; } else { if (vars[0].bits.bulk < vars[1].bits.bulk) return -1; else if (vars[0].bits.bulk > vars[1].bits.bulk) return 1; else if (vars[0].bits.rem < vars[1].bits.rem) return -1; else if (vars[0].bits.rem > vars[1].bits.rem) return 1; else return 0; } } else { // IBM Extended doubledouble does not follow the general // sign-exponent-significand layout, so has to be handled generically const int xSign = signbit(x), ySign = signbit(y); if (xSign == 1 && ySign == 1) return cmp(-y, -x); else if (xSign == 1) return -1; else if (ySign == 1) return 1; else if (x < y) return -1; else if (x == y) return 0; else if (x > y) return 1; else if (isNaN(x) && !isNaN(y)) return 1; else if (isNaN(y) && !isNaN(x)) return -1; else if (getNaNPayload(x) < getNaNPayload(y)) return -1; else if (getNaNPayload(x) > getNaNPayload(y)) return 1; else return 0; } } /// Most numbers are ordered naturally. @safe unittest { assert(cmp(-double.infinity, -double.max) < 0); assert(cmp(-double.max, -100.0) < 0); assert(cmp(-100.0, -0.5) < 0); assert(cmp(-0.5, 0.0) < 0); assert(cmp(0.0, 0.5) < 0); assert(cmp(0.5, 100.0) < 0); assert(cmp(100.0, double.max) < 0); assert(cmp(double.max, double.infinity) < 0); assert(cmp(1.0, 1.0) == 0); } /// Positive and negative zeroes are distinct. @safe unittest { assert(cmp(-0.0, +0.0) < 0); assert(cmp(+0.0, -0.0) > 0); } /// Depending on the sign, $(NAN)s go to either end of the spectrum. @safe unittest { assert(cmp(-double.nan, -double.infinity) < 0); assert(cmp(double.infinity, double.nan) < 0); assert(cmp(-double.nan, double.nan) < 0); } /// $(NAN)s of the same sign are ordered by the payload. @safe unittest { assert(cmp(NaN(10), NaN(20)) < 0); assert(cmp(-NaN(20), -NaN(10)) < 0); } @safe unittest { import std.meta : AliasSeq; foreach (T; AliasSeq!(float, double, real)) { T[] values = [-cast(T) NaN(20), -cast(T) NaN(10), -T.nan, -T.infinity, -T.max, -T.max / 2, T(-16.0), T(-1.0).nextDown, T(-1.0), T(-1.0).nextUp, T(-0.5), -T.min_normal, (-T.min_normal).nextUp, -2 * T.min_normal * T.epsilon, -T.min_normal * T.epsilon, T(-0.0), T(0.0), T.min_normal * T.epsilon, 2 * T.min_normal * T.epsilon, T.min_normal.nextDown, T.min_normal, T(0.5), T(1.0).nextDown, T(1.0), T(1.0).nextUp, T(16.0), T.max / 2, T.max, T.infinity, T.nan, cast(T) NaN(10), cast(T) NaN(20)]; foreach (i, x; values) { foreach (y; values[i + 1 .. $]) { assert(cmp(x, y) < 0); assert(cmp(y, x) > 0); } assert(cmp(x, x) == 0); } } } private enum PowType { floor, ceil } pragma(inline, true) private T powIntegralImpl(PowType type, T)(T val) { import core.bitop : bsr; if (val == 0 || (type == PowType.ceil && (val > T.max / 2 || val == T.min))) return 0; else { static if (isSigned!T) return cast(Unqual!T) (val < 0 ? -(T(1) << bsr(0 - val) + type) : T(1) << bsr(val) + type); else return cast(Unqual!T) (T(1) << bsr(val) + type); } } private T powFloatingPointImpl(PowType type, T)(T x) { if (!x.isFinite) return x; if (!x) return x; int exp; auto y = frexp(x, exp); static if (type == PowType.ceil) y = ldexp(cast(T) 0.5, exp + 1); else y = ldexp(cast(T) 0.5, exp); if (!y.isFinite) return cast(T) 0.0; y = copysign(y, x); return y; } /** * Gives the next power of two after $(D val). `T` can be any built-in * numerical type. * * If the operation would lead to an over/underflow, this function will * return `0`. * * Params: * val = any number * * Returns: * the next power of two after $(D val) */ T nextPow2(T)(const T val) if (isIntegral!T) { return powIntegralImpl!(PowType.ceil)(val); } /// ditto T nextPow2(T)(const T val) if (isFloatingPoint!T) { return powFloatingPointImpl!(PowType.ceil)(val); } /// @safe @nogc pure nothrow unittest { assert(nextPow2(2) == 4); assert(nextPow2(10) == 16); assert(nextPow2(4000) == 4096); assert(nextPow2(-2) == -4); assert(nextPow2(-10) == -16); assert(nextPow2(uint.max) == 0); assert(nextPow2(uint.min) == 0); assert(nextPow2(size_t.max) == 0); assert(nextPow2(size_t.min) == 0); assert(nextPow2(int.max) == 0); assert(nextPow2(int.min) == 0); assert(nextPow2(long.max) == 0); assert(nextPow2(long.min) == 0); } /// @safe @nogc pure nothrow unittest { assert(nextPow2(2.1) == 4.0); assert(nextPow2(-2.0) == -4.0); assert(nextPow2(0.25) == 0.5); assert(nextPow2(-4.0) == -8.0); assert(nextPow2(double.max) == 0.0); assert(nextPow2(double.infinity) == double.infinity); } @safe @nogc pure nothrow unittest { assert(nextPow2(ubyte(2)) == 4); assert(nextPow2(ubyte(10)) == 16); assert(nextPow2(byte(2)) == 4); assert(nextPow2(byte(10)) == 16); assert(nextPow2(short(2)) == 4); assert(nextPow2(short(10)) == 16); assert(nextPow2(short(4000)) == 4096); assert(nextPow2(ushort(2)) == 4); assert(nextPow2(ushort(10)) == 16); assert(nextPow2(ushort(4000)) == 4096); } @safe @nogc pure nothrow unittest { foreach (ulong i; 1 .. 62) { assert(nextPow2(1UL << i) == 2UL << i); assert(nextPow2((1UL << i) - 1) == 1UL << i); assert(nextPow2((1UL << i) + 1) == 2UL << i); assert(nextPow2((1UL << i) + (1UL<<(i-1))) == 2UL << i); } } @safe @nogc pure nothrow unittest { import std.meta : AliasSeq; foreach (T; AliasSeq!(float, double, real)) { enum T subNormal = T.min_normal / 2; static if (subNormal) assert(nextPow2(subNormal) == T.min_normal); assert(nextPow2(T(0.0)) == 0.0); assert(nextPow2(T(2.0)) == 4.0); assert(nextPow2(T(2.1)) == 4.0); assert(nextPow2(T(3.1)) == 4.0); assert(nextPow2(T(4.0)) == 8.0); assert(nextPow2(T(0.25)) == 0.5); assert(nextPow2(T(-2.0)) == -4.0); assert(nextPow2(T(-2.1)) == -4.0); assert(nextPow2(T(-3.1)) == -4.0); assert(nextPow2(T(-4.0)) == -8.0); assert(nextPow2(T(-0.25)) == -0.5); assert(nextPow2(T.max) == 0); assert(nextPow2(-T.max) == 0); assert(nextPow2(T.infinity) == T.infinity); assert(nextPow2(T.init).isNaN); } } @safe @nogc pure nothrow unittest // Issue 15973 { assert(nextPow2(uint.max / 2) == uint.max / 2 + 1); assert(nextPow2(uint.max / 2 + 2) == 0); assert(nextPow2(int.max / 2) == int.max / 2 + 1); assert(nextPow2(int.max / 2 + 2) == 0); assert(nextPow2(int.min + 1) == int.min); } /** * Gives the last power of two before $(D val). $(T) can be any built-in * numerical type. * * Params: * val = any number * * Returns: * the last power of two before $(D val) */ T truncPow2(T)(const T val) if (isIntegral!T) { return powIntegralImpl!(PowType.floor)(val); } /// ditto T truncPow2(T)(const T val) if (isFloatingPoint!T) { return powFloatingPointImpl!(PowType.floor)(val); } /// @safe @nogc pure nothrow unittest { assert(truncPow2(3) == 2); assert(truncPow2(4) == 4); assert(truncPow2(10) == 8); assert(truncPow2(4000) == 2048); assert(truncPow2(-5) == -4); assert(truncPow2(-20) == -16); assert(truncPow2(uint.max) == int.max + 1); assert(truncPow2(uint.min) == 0); assert(truncPow2(ulong.max) == long.max + 1); assert(truncPow2(ulong.min) == 0); assert(truncPow2(int.max) == (int.max / 2) + 1); assert(truncPow2(int.min) == int.min); assert(truncPow2(long.max) == (long.max / 2) + 1); assert(truncPow2(long.min) == long.min); } /// @safe @nogc pure nothrow unittest { assert(truncPow2(2.1) == 2.0); assert(truncPow2(7.0) == 4.0); assert(truncPow2(-1.9) == -1.0); assert(truncPow2(0.24) == 0.125); assert(truncPow2(-7.0) == -4.0); assert(truncPow2(double.infinity) == double.infinity); } @safe @nogc pure nothrow unittest { assert(truncPow2(ubyte(3)) == 2); assert(truncPow2(ubyte(4)) == 4); assert(truncPow2(ubyte(10)) == 8); assert(truncPow2(byte(3)) == 2); assert(truncPow2(byte(4)) == 4); assert(truncPow2(byte(10)) == 8); assert(truncPow2(ushort(3)) == 2); assert(truncPow2(ushort(4)) == 4); assert(truncPow2(ushort(10)) == 8); assert(truncPow2(ushort(4000)) == 2048); assert(truncPow2(short(3)) == 2); assert(truncPow2(short(4)) == 4); assert(truncPow2(short(10)) == 8); assert(truncPow2(short(4000)) == 2048); } @safe @nogc pure nothrow unittest { foreach (ulong i; 1 .. 62) { assert(truncPow2(2UL << i) == 2UL << i); assert(truncPow2((2UL << i) + 1) == 2UL << i); assert(truncPow2((2UL << i) - 1) == 1UL << i); assert(truncPow2((2UL << i) - (2UL<<(i-1))) == 1UL << i); } } @safe @nogc pure nothrow unittest { import std.meta : AliasSeq; foreach (T; AliasSeq!(float, double, real)) { assert(truncPow2(T(0.0)) == 0.0); assert(truncPow2(T(4.0)) == 4.0); assert(truncPow2(T(2.1)) == 2.0); assert(truncPow2(T(3.5)) == 2.0); assert(truncPow2(T(7.0)) == 4.0); assert(truncPow2(T(0.24)) == 0.125); assert(truncPow2(T(-2.0)) == -2.0); assert(truncPow2(T(-2.1)) == -2.0); assert(truncPow2(T(-3.1)) == -2.0); assert(truncPow2(T(-7.0)) == -4.0); assert(truncPow2(T(-0.24)) == -0.125); assert(truncPow2(T.infinity) == T.infinity); assert(truncPow2(T.init).isNaN); } } /** Check whether a number is an integer power of two. Note that only positive numbers can be integer powers of two. This function always return `false` if `x` is negative or zero. Params: x = the number to test Returns: `true` if `x` is an integer power of two. */ bool isPowerOf2(X)(const X x) pure @safe nothrow @nogc if (isNumeric!X) { static if (isFloatingPoint!X) { int exp; const X sig = frexp(x, exp); return (exp != int.min) && (sig is cast(X) 0.5L); } else { static if (isSigned!X) { auto y = cast(typeof(x + 0))x; return y > 0 && !(y & (y - 1)); } else { auto y = cast(typeof(x + 0u))x; return (y & -y) > (y - 1); } } } /// @safe unittest { assert( isPowerOf2(1.0L)); assert( isPowerOf2(2.0L)); assert( isPowerOf2(0.5L)); assert( isPowerOf2(pow(2.0L, 96))); assert( isPowerOf2(pow(2.0L, -77))); assert(!isPowerOf2(-2.0L)); assert(!isPowerOf2(-0.5L)); assert(!isPowerOf2(0.0L)); assert(!isPowerOf2(4.315)); assert(!isPowerOf2(1.0L / 3.0L)); assert(!isPowerOf2(real.nan)); assert(!isPowerOf2(real.infinity)); } /// @safe unittest { assert( isPowerOf2(1)); assert( isPowerOf2(2)); assert( isPowerOf2(1uL << 63)); assert(!isPowerOf2(-4)); assert(!isPowerOf2(0)); assert(!isPowerOf2(1337u)); } @safe unittest { import std.meta : AliasSeq; immutable smallP2 = pow(2.0L, -62); immutable bigP2 = pow(2.0L, 50); immutable smallP7 = pow(7.0L, -35); immutable bigP7 = pow(7.0L, 30); foreach (X; AliasSeq!(float, double, real)) { immutable min_sub = X.min_normal * X.epsilon; foreach (x; AliasSeq!(smallP2, min_sub, X.min_normal, .25L, 0.5L, 1.0L, 2.0L, 8.0L, pow(2.0L, X.max_exp - 1), bigP2)) { assert( isPowerOf2(cast(X) x)); assert(!isPowerOf2(cast(X)-x)); } foreach (x; AliasSeq!(0.0L, 3 * min_sub, smallP7, 0.1L, 1337.0L, bigP7, X.max, real.nan, real.infinity)) { assert(!isPowerOf2(cast(X) x)); assert(!isPowerOf2(cast(X)-x)); } } foreach (X; AliasSeq!(byte, ubyte, short, ushort, int, uint, long, ulong)) { foreach (x; [1, 2, 4, 8, (X.max >>> 1) + 1]) { assert( isPowerOf2(cast(X) x)); static if (isSigned!X) assert(!isPowerOf2(cast(X)-x)); } foreach (x; [0, 3, 5, 13, 77, X.min, X.max]) assert(!isPowerOf2(cast(X) x)); } }