# cpython-withatomic / Python / pymath.c

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247``` ```#include "Python.h" #ifdef X87_DOUBLE_ROUNDING /* On x86 platforms using an x87 FPU, this function is called from the Py_FORCE_DOUBLE macro (defined in pymath.h) to force a floating-point number out of an 80-bit x87 FPU register and into a 64-bit memory location, thus rounding from extended precision to double precision. */ double _Py_force_double(double x) { volatile double y; y = x; return y; } #endif #ifndef HAVE_HYPOT double hypot(double x, double y) { double yx; x = fabs(x); y = fabs(y); if (x < y) { double temp = x; x = y; y = temp; } if (x == 0.) return 0.; else { yx = y/x; return x*sqrt(1.+yx*yx); } } #endif /* HAVE_HYPOT */ #ifndef HAVE_COPYSIGN double copysign(double x, double y) { /* use atan2 to distinguish -0. from 0. */ if (y > 0. || (y == 0. && atan2(y, -1.) > 0.)) { return fabs(x); } else { return -fabs(x); } } #endif /* HAVE_COPYSIGN */ #ifndef HAVE_LOG1P #include double log1p(double x) { /* For x small, we use the following approach. Let y be the nearest float to 1+x, then 1+x = y * (1 - (y-1-x)/y) so log(1+x) = log(y) + log(1-(y-1-x)/y). Since (y-1-x)/y is tiny, the second term is well approximated by (y-1-x)/y. If abs(x) >= DBL_EPSILON/2 or the rounding-mode is some form of round-to-nearest then y-1-x will be exactly representable, and is computed exactly by (y-1)-x. If abs(x) < DBL_EPSILON/2 and the rounding mode is not known to be round-to-nearest then this method is slightly dangerous: 1+x could be rounded up to 1+DBL_EPSILON instead of down to 1, and in that case y-1-x will not be exactly representable any more and the result can be off by many ulps. But this is easily fixed: for a floating-point number |x| < DBL_EPSILON/2., the closest floating-point number to log(1+x) is exactly x. */ double y; if (fabs(x) < DBL_EPSILON/2.) { return x; } else if (-0.5 <= x && x <= 1.) { /* WARNING: it's possible than an overeager compiler will incorrectly optimize the following two lines to the equivalent of "return log(1.+x)". If this happens, then results from log1p will be inaccurate for small x. */ y = 1.+x; return log(y)-((y-1.)-x)/y; } else { /* NaNs and infinities should end up here */ return log(1.+x); } } #endif /* HAVE_LOG1P */ /* * ==================================================== * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * * Developed at SunPro, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== */ static const double ln2 = 6.93147180559945286227E-01; static const double two_pow_m28 = 3.7252902984619141E-09; /* 2**-28 */ static const double two_pow_p28 = 268435456.0; /* 2**28 */ static const double zero = 0.0; /* asinh(x) * Method : * Based on * asinh(x) = sign(x) * log [ |x| + sqrt(x*x+1) ] * we have * asinh(x) := x if 1+x*x=1, * := sign(x)*(log(x)+ln2)) for large |x|, else * := sign(x)*log(2|x|+1/(|x|+sqrt(x*x+1))) if|x|>2, else * := sign(x)*log1p(|x| + x^2/(1 + sqrt(1+x^2))) */ #ifndef HAVE_ASINH double asinh(double x) { double w; double absx = fabs(x); if (Py_IS_NAN(x) || Py_IS_INFINITY(x)) { return x+x; } if (absx < two_pow_m28) { /* |x| < 2**-28 */ return x; /* return x inexact except 0 */ } if (absx > two_pow_p28) { /* |x| > 2**28 */ w = log(absx)+ln2; } else if (absx > 2.0) { /* 2 < |x| < 2**28 */ w = log(2.0*absx + 1.0 / (sqrt(x*x + 1.0) + absx)); } else { /* 2**-28 <= |x| < 2= */ double t = x*x; w = log1p(absx + t / (1.0 + sqrt(1.0 + t))); } return copysign(w, x); } #endif /* HAVE_ASINH */ /* acosh(x) * Method : * Based on * acosh(x) = log [ x + sqrt(x*x-1) ] * we have * acosh(x) := log(x)+ln2, if x is large; else * acosh(x) := log(2x-1/(sqrt(x*x-1)+x)) if x>2; else * acosh(x) := log1p(t+sqrt(2.0*t+t*t)); where t=x-1. * * Special cases: * acosh(x) is NaN with signal if x<1. * acosh(NaN) is NaN without signal. */ #ifndef HAVE_ACOSH double acosh(double x) { if (Py_IS_NAN(x)) { return x+x; } if (x < 1.) { /* x < 1; return a signaling NaN */ errno = EDOM; #ifdef Py_NAN return Py_NAN; #else return (x-x)/(x-x); #endif } else if (x >= two_pow_p28) { /* x > 2**28 */ if (Py_IS_INFINITY(x)) { return x+x; } else { return log(x)+ln2; /* acosh(huge)=log(2x) */ } } else if (x == 1.) { return 0.0; /* acosh(1) = 0 */ } else if (x > 2.) { /* 2 < x < 2**28 */ double t = x*x; return log(2.0*x - 1.0 / (x + sqrt(t - 1.0))); } else { /* 1 < x <= 2 */ double t = x - 1.0; return log1p(t + sqrt(2.0*t + t*t)); } } #endif /* HAVE_ACOSH */ /* atanh(x) * Method : * 1.Reduced x to positive by atanh(-x) = -atanh(x) * 2.For x>=0.5 * 1 2x x * atanh(x) = --- * log(1 + -------) = 0.5 * log1p(2 * --------) * 2 1 - x 1 - x * * For x<0.5 * atanh(x) = 0.5*log1p(2x+2x*x/(1-x)) * * Special cases: * atanh(x) is NaN if |x| >= 1 with signal; * atanh(NaN) is that NaN with no signal; * */ #ifndef HAVE_ATANH double atanh(double x) { double absx; double t; if (Py_IS_NAN(x)) { return x+x; } absx = fabs(x); if (absx >= 1.) { /* |x| >= 1 */ errno = EDOM; #ifdef Py_NAN return Py_NAN; #else return x/zero; #endif } if (absx < two_pow_m28) { /* |x| < 2**-28 */ return x; } if (absx < 0.5) { /* |x| < 0.5 */ t = absx+absx; t = 0.5 * log1p(t + t*absx / (1.0 - absx)); } else { /* 0.5 <= |x| <= 1.0 */ t = 0.5 * log1p((absx + absx) / (1.0 - absx)); } return copysign(t, x); } #endif /* HAVE_ATANH */ ```