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145
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1 /* Return arc hyperbolic sine for a complex float type, with the
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2 imaginary part of the result possibly adjusted for use in
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3 computing other functions.
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4 Copyright (C) 1997-2018 Free Software Foundation, Inc.
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5 This file is part of the GNU C Library.
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6
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7 The GNU C Library is free software; you can redistribute it and/or
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8 modify it under the terms of the GNU Lesser General Public
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9 License as published by the Free Software Foundation; either
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10 version 2.1 of the License, or (at your option) any later version.
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11
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12 The GNU C Library is distributed in the hope that it will be useful,
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13 but WITHOUT ANY WARRANTY; without even the implied warranty of
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14 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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15 Lesser General Public License for more details.
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16
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17 You should have received a copy of the GNU Lesser General Public
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18 License along with the GNU C Library; if not, see
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19 <http://www.gnu.org/licenses/>. */
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20
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21 #include "quadmath-imp.h"
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22
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23 /* Return the complex inverse hyperbolic sine of finite nonzero Z,
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24 with the imaginary part of the result subtracted from pi/2 if ADJ
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25 is nonzero. */
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26
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27 __complex128
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28 __quadmath_kernel_casinhq (__complex128 x, int adj)
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29 {
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30 __complex128 res;
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31 __float128 rx, ix;
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32 __complex128 y;
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33
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34 /* Avoid cancellation by reducing to the first quadrant. */
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35 rx = fabsq (__real__ x);
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36 ix = fabsq (__imag__ x);
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37
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38 if (rx >= 1 / FLT128_EPSILON || ix >= 1 / FLT128_EPSILON)
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39 {
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40 /* For large x in the first quadrant, x + csqrt (1 + x * x)
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41 is sufficiently close to 2 * x to make no significant
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42 difference to the result; avoid possible overflow from
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43 the squaring and addition. */
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44 __real__ y = rx;
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45 __imag__ y = ix;
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46
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47 if (adj)
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48 {
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49 __float128 t = __real__ y;
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50 __real__ y = copysignq (__imag__ y, __imag__ x);
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51 __imag__ y = t;
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52 }
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53
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54 res = clogq (y);
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55 __real__ res += (__float128) M_LN2q;
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56 }
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57 else if (rx >= 0.5Q && ix < FLT128_EPSILON / 8)
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58 {
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59 __float128 s = hypotq (1, rx);
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60
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61 __real__ res = logq (rx + s);
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62 if (adj)
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63 __imag__ res = atan2q (s, __imag__ x);
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64 else
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65 __imag__ res = atan2q (ix, s);
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66 }
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67 else if (rx < FLT128_EPSILON / 8 && ix >= 1.5Q)
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68 {
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69 __float128 s = sqrtq ((ix + 1) * (ix - 1));
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70
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71 __real__ res = logq (ix + s);
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72 if (adj)
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73 __imag__ res = atan2q (rx, copysignq (s, __imag__ x));
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74 else
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75 __imag__ res = atan2q (s, rx);
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76 }
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77 else if (ix > 1 && ix < 1.5Q && rx < 0.5Q)
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78 {
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79 if (rx < FLT128_EPSILON * FLT128_EPSILON)
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80 {
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81 __float128 ix2m1 = (ix + 1) * (ix - 1);
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82 __float128 s = sqrtq (ix2m1);
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83
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84 __real__ res = log1pq (2 * (ix2m1 + ix * s)) / 2;
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85 if (adj)
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86 __imag__ res = atan2q (rx, copysignq (s, __imag__ x));
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87 else
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88 __imag__ res = atan2q (s, rx);
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89 }
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90 else
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91 {
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92 __float128 ix2m1 = (ix + 1) * (ix - 1);
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93 __float128 rx2 = rx * rx;
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94 __float128 f = rx2 * (2 + rx2 + 2 * ix * ix);
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95 __float128 d = sqrtq (ix2m1 * ix2m1 + f);
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96 __float128 dp = d + ix2m1;
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97 __float128 dm = f / dp;
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98 __float128 r1 = sqrtq ((dm + rx2) / 2);
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99 __float128 r2 = rx * ix / r1;
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100
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101 __real__ res = log1pq (rx2 + dp + 2 * (rx * r1 + ix * r2)) / 2;
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102 if (adj)
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103 __imag__ res = atan2q (rx + r1, copysignq (ix + r2, __imag__ x));
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104 else
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105 __imag__ res = atan2q (ix + r2, rx + r1);
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106 }
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107 }
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108 else if (ix == 1 && rx < 0.5Q)
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109 {
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110 if (rx < FLT128_EPSILON / 8)
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111 {
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112 __real__ res = log1pq (2 * (rx + sqrtq (rx))) / 2;
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113 if (adj)
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114 __imag__ res = atan2q (sqrtq (rx), copysignq (1, __imag__ x));
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115 else
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116 __imag__ res = atan2q (1, sqrtq (rx));
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117 }
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118 else
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119 {
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120 __float128 d = rx * sqrtq (4 + rx * rx);
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121 __float128 s1 = sqrtq ((d + rx * rx) / 2);
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122 __float128 s2 = sqrtq ((d - rx * rx) / 2);
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123
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124 __real__ res = log1pq (rx * rx + d + 2 * (rx * s1 + s2)) / 2;
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125 if (adj)
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126 __imag__ res = atan2q (rx + s1, copysignq (1 + s2, __imag__ x));
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127 else
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128 __imag__ res = atan2q (1 + s2, rx + s1);
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129 }
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130 }
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131 else if (ix < 1 && rx < 0.5Q)
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132 {
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133 if (ix >= FLT128_EPSILON)
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134 {
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135 if (rx < FLT128_EPSILON * FLT128_EPSILON)
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136 {
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137 __float128 onemix2 = (1 + ix) * (1 - ix);
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138 __float128 s = sqrtq (onemix2);
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139
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140 __real__ res = log1pq (2 * rx / s) / 2;
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141 if (adj)
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142 __imag__ res = atan2q (s, __imag__ x);
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143 else
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144 __imag__ res = atan2q (ix, s);
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145 }
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146 else
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147 {
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148 __float128 onemix2 = (1 + ix) * (1 - ix);
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149 __float128 rx2 = rx * rx;
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150 __float128 f = rx2 * (2 + rx2 + 2 * ix * ix);
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151 __float128 d = sqrtq (onemix2 * onemix2 + f);
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152 __float128 dp = d + onemix2;
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153 __float128 dm = f / dp;
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154 __float128 r1 = sqrtq ((dp + rx2) / 2);
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155 __float128 r2 = rx * ix / r1;
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156
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157 __real__ res = log1pq (rx2 + dm + 2 * (rx * r1 + ix * r2)) / 2;
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158 if (adj)
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159 __imag__ res = atan2q (rx + r1, copysignq (ix + r2,
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160 __imag__ x));
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161 else
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162 __imag__ res = atan2q (ix + r2, rx + r1);
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163 }
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164 }
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165 else
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166 {
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167 __float128 s = hypotq (1, rx);
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168
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169 __real__ res = log1pq (2 * rx * (rx + s)) / 2;
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170 if (adj)
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171 __imag__ res = atan2q (s, __imag__ x);
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172 else
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173 __imag__ res = atan2q (ix, s);
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174 }
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175 math_check_force_underflow_nonneg (__real__ res);
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176 }
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177 else
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178 {
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179 __real__ y = (rx - ix) * (rx + ix) + 1;
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180 __imag__ y = 2 * rx * ix;
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181
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182 y = csqrtq (y);
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183
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184 __real__ y += rx;
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185 __imag__ y += ix;
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186
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187 if (adj)
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188 {
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189 __float128 t = __real__ y;
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190 __real__ y = copysignq (__imag__ y, __imag__ x);
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191 __imag__ y = t;
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192 }
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193
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194 res = clogq (y);
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195 }
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196
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197 /* Give results the correct sign for the original argument. */
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198 __real__ res = copysignq (__real__ res, __real__ x);
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199 __imag__ res = copysignq (__imag__ res, (adj ? 1 : __imag__ x));
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200
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201 return res;
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202 }
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