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annotate libquadmath/math/expm1q.c @ 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
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
rev   line source
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1 /* expm1q.c
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Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
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2 *
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3 * Exponential function, minus 1
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4 * 128-bit long double precision
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5 *
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6 *
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7 *
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Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
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8 * SYNOPSIS:
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9 *
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10 * long double x, y, expm1q();
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11 *
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12 * y = expm1q( x );
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13 *
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14 *
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15 *
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16 * DESCRIPTION:
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17 *
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18 * Returns e (2.71828...) raised to the x power, minus one.
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19 *
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20 * Range reduction is accomplished by separating the argument
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21 * into an integer k and fraction f such that
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22 *
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23 * x k f
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24 * e = 2 e.
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25 *
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26 * An expansion x + .5 x^2 + x^3 R(x) approximates exp(f) - 1
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27 * in the basic range [-0.5 ln 2, 0.5 ln 2].
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28 *
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29 *
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30 * ACCURACY:
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31 *
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32 * Relative error:
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33 * arithmetic domain # trials peak rms
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34 * IEEE -79,+MAXLOG 100,000 1.7e-34 4.5e-35
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35 *
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36 */
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37
111
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38 /* Copyright 2001 by Stephen L. Moshier
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39
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Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
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40 This library is free software; you can redistribute it and/or
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41 modify it under the terms of the GNU Lesser General Public
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42 License as published by the Free Software Foundation; either
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43 version 2.1 of the License, or (at your option) any later version.
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44
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45 This library is distributed in the hope that it will be useful,
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46 but WITHOUT ANY WARRANTY; without even the implied warranty of
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47 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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48 Lesser General Public License for more details.
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49
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50 You should have received a copy of the GNU Lesser General Public
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51 License along with this library; if not, see
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52 <http://www.gnu.org/licenses/>. */
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53
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54 #include "quadmath-imp.h"
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55
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56 /* exp(x) - 1 = x + 0.5 x^2 + x^3 P(x)/Q(x)
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57 -.5 ln 2 < x < .5 ln 2
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58 Theoretical peak relative error = 8.1e-36 */
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59
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60 static const __float128
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61 P0 = 2.943520915569954073888921213330863757240E8Q,
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62 P1 = -5.722847283900608941516165725053359168840E7Q,
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63 P2 = 8.944630806357575461578107295909719817253E6Q,
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64 P3 = -7.212432713558031519943281748462837065308E5Q,
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65 P4 = 4.578962475841642634225390068461943438441E4Q,
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66 P5 = -1.716772506388927649032068540558788106762E3Q,
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67 P6 = 4.401308817383362136048032038528753151144E1Q,
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68 P7 = -4.888737542888633647784737721812546636240E-1Q,
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69 Q0 = 1.766112549341972444333352727998584753865E9Q,
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70 Q1 = -7.848989743695296475743081255027098295771E8Q,
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71 Q2 = 1.615869009634292424463780387327037251069E8Q,
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72 Q3 = -2.019684072836541751428967854947019415698E7Q,
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73 Q4 = 1.682912729190313538934190635536631941751E6Q,
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74 Q5 = -9.615511549171441430850103489315371768998E4Q,
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75 Q6 = 3.697714952261803935521187272204485251835E3Q,
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76 Q7 = -8.802340681794263968892934703309274564037E1Q,
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77 /* Q8 = 1.000000000000000000000000000000000000000E0 */
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78 /* C1 + C2 = ln 2 */
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79
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80 C1 = 6.93145751953125E-1Q,
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81 C2 = 1.428606820309417232121458176568075500134E-6Q,
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82 /* ln 2^-114 */
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83 minarg = -7.9018778583833765273564461846232128760607E1Q, big = 1e4932Q;
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84
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85
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86 __float128
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87 expm1q (__float128 x)
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88 {
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89 __float128 px, qx, xx;
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90 int32_t ix, sign;
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91 ieee854_float128 u;
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92 int k;
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93
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94 /* Detect infinity and NaN. */
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95 u.value = x;
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96 ix = u.words32.w0;
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97 sign = ix & 0x80000000;
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98 ix &= 0x7fffffff;
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99 if (!sign && ix >= 0x40060000)
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100 {
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101 /* If num is positive and exp >= 6 use plain exp. */
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102 return expq (x);
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103 }
68
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104 if (ix >= 0x7fff0000)
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105 {
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106 /* Infinity (which must be negative infinity). */
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107 if (((ix & 0xffff) | u.words32.w1 | u.words32.w2 | u.words32.w3) == 0)
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108 return -1;
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109 /* NaN. Invalid exception if signaling. */
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110 return x + x;
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111 }
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112
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113 /* expm1(+- 0) = +- 0. */
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114 if ((ix == 0) && (u.words32.w1 | u.words32.w2 | u.words32.w3) == 0)
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115 return x;
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116
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117 /* Minimum value. */
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118 if (x < minarg)
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119 return (4.0/big - 1);
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120
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121 /* Avoid internal underflow when result does not underflow, while
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122 ensuring underflow (without returning a zero of the wrong sign)
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123 when the result does underflow. */
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124 if (fabsq (x) < 0x1p-113Q)
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125 {
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126 math_check_force_underflow (x);
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127 return x;
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128 }
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129
68
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130 /* Express x = ln 2 (k + remainder), remainder not exceeding 1/2. */
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131 xx = C1 + C2; /* ln 2. */
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132 px = floorq (0.5 + x / xx);
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133 k = px;
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134 /* remainder times ln 2 */
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135 x -= px * C1;
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136 x -= px * C2;
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Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
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137
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Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
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138 /* Approximate exp(remainder ln 2). */
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Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
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139 px = (((((((P7 * x
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Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
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140 + P6) * x
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Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
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141 + P5) * x + P4) * x + P3) * x + P2) * x + P1) * x + P0) * x;
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Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
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142
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Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
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143 qx = (((((((x
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Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
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144 + Q7) * x
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Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
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145 + Q6) * x + Q5) * x + Q4) * x + Q3) * x + Q2) * x + Q1) * x + Q0;
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Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
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146
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Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
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147 xx = x * x;
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Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
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148 qx = x + (0.5 * xx + xx * px / qx);
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Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
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149
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Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
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150 /* exp(x) = exp(k ln 2) exp(remainder ln 2) = 2^k exp(remainder ln 2).
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Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
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151
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Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
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152 We have qx = exp(remainder ln 2) - 1, so
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Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
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153 exp(x) - 1 = 2^k (qx + 1) - 1
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Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
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154 = 2^k qx + 2^k - 1. */
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Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
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155
145
1830386684a0 gcc-9.2.0
anatofuz
parents: 111
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156 px = ldexpq (1, k);
68
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Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
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157 x = px * qx + (px - 1.0);
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Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
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158 return x;
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Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
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159 }