/* 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 m_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 = m_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 * m_log1p(t + t*absx / (1.0 - absx)); | |
} | |
else { /* 0.5 <= |x| <= 1.0 */ | |
t = 0.5 * m_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 | |
evaluation of the expression exp(x) - 1, for x near 0. */ | |
double | |
_Py_expm1(double x) | |
{ | |
/* For abs(x) >= log(2), it's safe to evaluate exp(x) - 1 directly; this | |
also works fine for infinities and nans. | |
For smaller x, we can use a method due to Kahan that achieves close to | |
full accuracy. | |
*/ | |
if (fabs(x) < 0.7) { | |
double u; | |
u = exp(x); | |
if (u == 1.0) | |
return x; | |
else | |
return (u - 1.0) * x / log(u); | |
} | |
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); | |
} | |
} |