diff options
Diffstat (limited to 'Modules/_math.c')
| -rw-r--r-- | Modules/_math.c | 199 |
1 files changed, 198 insertions, 1 deletions
diff --git a/Modules/_math.c b/Modules/_math.c index 9d330aa..e27c100 100644 --- a/Modules/_math.c +++ b/Modules/_math.c @@ -1,8 +1,161 @@ /* Definitions of some C99 math library functions, for those platforms that don't implement these functions already. */ +#include "Python.h" #include <float.h> -#include <math.h> + +/* The following copyright notice applies to the original + implementations of acosh, asinh and atanh. */ + +/* + * ==================================================== + * 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; + +/* 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. + */ + +double +_Py_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)); + } +} + + +/* 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))) + */ + +double +_Py_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); + +} + +/* 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; + * + */ + +double +_Py_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); +} /* Mathematically, expm1(x) = exp(x) - 1. The expm1 function is designed to avoid the significant loss of precision that arises from direct @@ -29,3 +182,47 @@ _Py_expm1(double x) else return exp(x) - 1.0; } + +/* log1p(x) = log(1+x). The log1p function is designed to avoid the + significant loss of precision that arises from direct evaluation when x is + small. */ + +double +_Py_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); + } +} |