4710c53dca
These files are a subset of the python-2.7.2.tgz distribution from python.org. Changed files from PyMod-2.7.2 have been copied into the corresponding directories of this tree, replacing the original files in the distribution. Signed-off-by: daryl.mcdaniel@intel.com git-svn-id: https://edk2.svn.sourceforge.net/svnroot/edk2/trunk/edk2@13197 6f19259b-4bc3-4df7-8a09-765794883524
1625 lines
53 KiB
C
1625 lines
53 KiB
C
/* Math module -- standard C math library functions, pi and e */
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/* Here are some comments from Tim Peters, extracted from the
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discussion attached to http://bugs.python.org/issue1640. They
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describe the general aims of the math module with respect to
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special values, IEEE-754 floating-point exceptions, and Python
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exceptions.
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These are the "spirit of 754" rules:
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1. If the mathematical result is a real number, but of magnitude too
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large to approximate by a machine float, overflow is signaled and the
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result is an infinity (with the appropriate sign).
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2. If the mathematical result is a real number, but of magnitude too
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small to approximate by a machine float, underflow is signaled and the
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result is a zero (with the appropriate sign).
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3. At a singularity (a value x such that the limit of f(y) as y
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approaches x exists and is an infinity), "divide by zero" is signaled
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and the result is an infinity (with the appropriate sign). This is
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complicated a little by that the left-side and right-side limits may
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not be the same; e.g., 1/x approaches +inf or -inf as x approaches 0
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from the positive or negative directions. In that specific case, the
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sign of the zero determines the result of 1/0.
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4. At a point where a function has no defined result in the extended
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reals (i.e., the reals plus an infinity or two), invalid operation is
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signaled and a NaN is returned.
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And these are what Python has historically /tried/ to do (but not
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always successfully, as platform libm behavior varies a lot):
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For #1, raise OverflowError.
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For #2, return a zero (with the appropriate sign if that happens by
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accident ;-)).
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For #3 and #4, raise ValueError. It may have made sense to raise
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Python's ZeroDivisionError in #3, but historically that's only been
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raised for division by zero and mod by zero.
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*/
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/*
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In general, on an IEEE-754 platform the aim is to follow the C99
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standard, including Annex 'F', whenever possible. Where the
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standard recommends raising the 'divide-by-zero' or 'invalid'
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floating-point exceptions, Python should raise a ValueError. Where
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the standard recommends raising 'overflow', Python should raise an
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OverflowError. In all other circumstances a value should be
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returned.
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*/
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#include "Python.h"
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#include "_math.h"
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#ifdef _OSF_SOURCE
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/* OSF1 5.1 doesn't make this available with XOPEN_SOURCE_EXTENDED defined */
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extern double copysign(double, double);
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#endif
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/*
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sin(pi*x), giving accurate results for all finite x (especially x
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integral or close to an integer). This is here for use in the
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reflection formula for the gamma function. It conforms to IEEE
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754-2008 for finite arguments, but not for infinities or nans.
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*/
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static const double pi = 3.141592653589793238462643383279502884197;
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static const double sqrtpi = 1.772453850905516027298167483341145182798;
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static double
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sinpi(double x)
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{
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double y, r;
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int n;
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/* this function should only ever be called for finite arguments */
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assert(Py_IS_FINITE(x));
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y = fmod(fabs(x), 2.0);
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n = (int)round(2.0*y);
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assert(0 <= n && n <= 4);
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switch (n) {
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case 0:
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r = sin(pi*y);
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break;
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case 1:
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r = cos(pi*(y-0.5));
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break;
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case 2:
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/* N.B. -sin(pi*(y-1.0)) is *not* equivalent: it would give
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-0.0 instead of 0.0 when y == 1.0. */
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r = sin(pi*(1.0-y));
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break;
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case 3:
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r = -cos(pi*(y-1.5));
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break;
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case 4:
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r = sin(pi*(y-2.0));
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break;
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default:
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assert(0); /* should never get here */
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r = -1.23e200; /* silence gcc warning */
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}
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return copysign(1.0, x)*r;
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}
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/* Implementation of the real gamma function. In extensive but non-exhaustive
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random tests, this function proved accurate to within <= 10 ulps across the
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entire float domain. Note that accuracy may depend on the quality of the
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system math functions, the pow function in particular. Special cases
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follow C99 annex F. The parameters and method are tailored to platforms
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whose double format is the IEEE 754 binary64 format.
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Method: for x > 0.0 we use the Lanczos approximation with parameters N=13
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and g=6.024680040776729583740234375; these parameters are amongst those
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used by the Boost library. Following Boost (again), we re-express the
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Lanczos sum as a rational function, and compute it that way. The
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coefficients below were computed independently using MPFR, and have been
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double-checked against the coefficients in the Boost source code.
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For x < 0.0 we use the reflection formula.
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There's one minor tweak that deserves explanation: Lanczos' formula for
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Gamma(x) involves computing pow(x+g-0.5, x-0.5) / exp(x+g-0.5). For many x
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values, x+g-0.5 can be represented exactly. However, in cases where it
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can't be represented exactly the small error in x+g-0.5 can be magnified
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significantly by the pow and exp calls, especially for large x. A cheap
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correction is to multiply by (1 + e*g/(x+g-0.5)), where e is the error
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involved in the computation of x+g-0.5 (that is, e = computed value of
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x+g-0.5 - exact value of x+g-0.5). Here's the proof:
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Correction factor
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-----------------
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Write x+g-0.5 = y-e, where y is exactly representable as an IEEE 754
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double, and e is tiny. Then:
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pow(x+g-0.5,x-0.5)/exp(x+g-0.5) = pow(y-e, x-0.5)/exp(y-e)
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= pow(y, x-0.5)/exp(y) * C,
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where the correction_factor C is given by
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C = pow(1-e/y, x-0.5) * exp(e)
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Since e is tiny, pow(1-e/y, x-0.5) ~ 1-(x-0.5)*e/y, and exp(x) ~ 1+e, so:
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C ~ (1-(x-0.5)*e/y) * (1+e) ~ 1 + e*(y-(x-0.5))/y
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But y-(x-0.5) = g+e, and g+e ~ g. So we get C ~ 1 + e*g/y, and
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pow(x+g-0.5,x-0.5)/exp(x+g-0.5) ~ pow(y, x-0.5)/exp(y) * (1 + e*g/y),
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Note that for accuracy, when computing r*C it's better to do
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r + e*g/y*r;
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than
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r * (1 + e*g/y);
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since the addition in the latter throws away most of the bits of
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information in e*g/y.
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*/
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#define LANCZOS_N 13
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static const double lanczos_g = 6.024680040776729583740234375;
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static const double lanczos_g_minus_half = 5.524680040776729583740234375;
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static const double lanczos_num_coeffs[LANCZOS_N] = {
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23531376880.410759688572007674451636754734846804940,
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42919803642.649098768957899047001988850926355848959,
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35711959237.355668049440185451547166705960488635843,
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17921034426.037209699919755754458931112671403265390,
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6039542586.3520280050642916443072979210699388420708,
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1439720407.3117216736632230727949123939715485786772,
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248874557.86205415651146038641322942321632125127801,
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31426415.585400194380614231628318205362874684987640,
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2876370.6289353724412254090516208496135991145378768,
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186056.26539522349504029498971604569928220784236328,
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8071.6720023658162106380029022722506138218516325024,
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210.82427775157934587250973392071336271166969580291,
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2.5066282746310002701649081771338373386264310793408
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};
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/* denominator is x*(x+1)*...*(x+LANCZOS_N-2) */
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static const double lanczos_den_coeffs[LANCZOS_N] = {
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0.0, 39916800.0, 120543840.0, 150917976.0, 105258076.0, 45995730.0,
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13339535.0, 2637558.0, 357423.0, 32670.0, 1925.0, 66.0, 1.0};
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/* gamma values for small positive integers, 1 though NGAMMA_INTEGRAL */
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#define NGAMMA_INTEGRAL 23
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static const double gamma_integral[NGAMMA_INTEGRAL] = {
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1.0, 1.0, 2.0, 6.0, 24.0, 120.0, 720.0, 5040.0, 40320.0, 362880.0,
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3628800.0, 39916800.0, 479001600.0, 6227020800.0, 87178291200.0,
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1307674368000.0, 20922789888000.0, 355687428096000.0,
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6402373705728000.0, 121645100408832000.0, 2432902008176640000.0,
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51090942171709440000.0, 1124000727777607680000.0,
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};
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/* Lanczos' sum L_g(x), for positive x */
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static double
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lanczos_sum(double x)
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{
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double num = 0.0, den = 0.0;
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int i;
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assert(x > 0.0);
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/* evaluate the rational function lanczos_sum(x). For large
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x, the obvious algorithm risks overflow, so we instead
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rescale the denominator and numerator of the rational
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function by x**(1-LANCZOS_N) and treat this as a
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rational function in 1/x. This also reduces the error for
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larger x values. The choice of cutoff point (5.0 below) is
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somewhat arbitrary; in tests, smaller cutoff values than
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this resulted in lower accuracy. */
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if (x < 5.0) {
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for (i = LANCZOS_N; --i >= 0; ) {
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num = num * x + lanczos_num_coeffs[i];
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den = den * x + lanczos_den_coeffs[i];
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}
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}
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else {
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for (i = 0; i < LANCZOS_N; i++) {
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num = num / x + lanczos_num_coeffs[i];
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den = den / x + lanczos_den_coeffs[i];
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}
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}
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return num/den;
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}
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static double
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m_tgamma(double x)
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{
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double absx, r, y, z, sqrtpow;
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/* special cases */
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if (!Py_IS_FINITE(x)) {
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if (Py_IS_NAN(x) || x > 0.0)
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return x; /* tgamma(nan) = nan, tgamma(inf) = inf */
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else {
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errno = EDOM;
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return Py_NAN; /* tgamma(-inf) = nan, invalid */
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}
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}
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if (x == 0.0) {
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errno = EDOM;
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return 1.0/x; /* tgamma(+-0.0) = +-inf, divide-by-zero */
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}
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/* integer arguments */
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if (x == floor(x)) {
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if (x < 0.0) {
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errno = EDOM; /* tgamma(n) = nan, invalid for */
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return Py_NAN; /* negative integers n */
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}
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if (x <= NGAMMA_INTEGRAL)
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return gamma_integral[(int)x - 1];
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}
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absx = fabs(x);
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/* tiny arguments: tgamma(x) ~ 1/x for x near 0 */
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if (absx < 1e-20) {
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r = 1.0/x;
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if (Py_IS_INFINITY(r))
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errno = ERANGE;
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return r;
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}
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/* large arguments: assuming IEEE 754 doubles, tgamma(x) overflows for
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x > 200, and underflows to +-0.0 for x < -200, not a negative
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integer. */
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if (absx > 200.0) {
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if (x < 0.0) {
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return 0.0/sinpi(x);
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}
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else {
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errno = ERANGE;
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return Py_HUGE_VAL;
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}
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}
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y = absx + lanczos_g_minus_half;
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/* compute error in sum */
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if (absx > lanczos_g_minus_half) {
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/* note: the correction can be foiled by an optimizing
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compiler that (incorrectly) thinks that an expression like
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a + b - a - b can be optimized to 0.0. This shouldn't
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happen in a standards-conforming compiler. */
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double q = y - absx;
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z = q - lanczos_g_minus_half;
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}
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else {
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double q = y - lanczos_g_minus_half;
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z = q - absx;
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}
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z = z * lanczos_g / y;
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if (x < 0.0) {
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r = -pi / sinpi(absx) / absx * exp(y) / lanczos_sum(absx);
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r -= z * r;
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if (absx < 140.0) {
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r /= pow(y, absx - 0.5);
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}
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else {
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sqrtpow = pow(y, absx / 2.0 - 0.25);
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r /= sqrtpow;
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r /= sqrtpow;
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}
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}
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else {
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r = lanczos_sum(absx) / exp(y);
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r += z * r;
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if (absx < 140.0) {
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r *= pow(y, absx - 0.5);
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}
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else {
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sqrtpow = pow(y, absx / 2.0 - 0.25);
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r *= sqrtpow;
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r *= sqrtpow;
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}
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}
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if (Py_IS_INFINITY(r))
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errno = ERANGE;
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return r;
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}
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/*
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lgamma: natural log of the absolute value of the Gamma function.
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For large arguments, Lanczos' formula works extremely well here.
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*/
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static double
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m_lgamma(double x)
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{
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double r, absx;
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/* special cases */
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if (!Py_IS_FINITE(x)) {
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if (Py_IS_NAN(x))
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return x; /* lgamma(nan) = nan */
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else
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return Py_HUGE_VAL; /* lgamma(+-inf) = +inf */
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}
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/* integer arguments */
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if (x == floor(x) && x <= 2.0) {
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if (x <= 0.0) {
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errno = EDOM; /* lgamma(n) = inf, divide-by-zero for */
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return Py_HUGE_VAL; /* integers n <= 0 */
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}
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else {
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return 0.0; /* lgamma(1) = lgamma(2) = 0.0 */
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}
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}
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absx = fabs(x);
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/* tiny arguments: lgamma(x) ~ -log(fabs(x)) for small x */
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if (absx < 1e-20)
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return -log(absx);
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/* Lanczos' formula */
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if (x > 0.0) {
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/* we could save a fraction of a ulp in accuracy by having a
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second set of numerator coefficients for lanczos_sum that
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absorbed the exp(-lanczos_g) term, and throwing out the
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lanczos_g subtraction below; it's probably not worth it. */
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r = log(lanczos_sum(x)) - lanczos_g +
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(x-0.5)*(log(x+lanczos_g-0.5)-1);
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}
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else {
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r = log(pi) - log(fabs(sinpi(absx))) - log(absx) -
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(log(lanczos_sum(absx)) - lanczos_g +
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(absx-0.5)*(log(absx+lanczos_g-0.5)-1));
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}
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if (Py_IS_INFINITY(r))
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errno = ERANGE;
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return r;
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}
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/*
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Implementations of the error function erf(x) and the complementary error
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function erfc(x).
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Method: following 'Numerical Recipes' by Flannery, Press et. al. (2nd ed.,
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Cambridge University Press), we use a series approximation for erf for
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small x, and a continued fraction approximation for erfc(x) for larger x;
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combined with the relations erf(-x) = -erf(x) and erfc(x) = 1.0 - erf(x),
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this gives us erf(x) and erfc(x) for all x.
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The series expansion used is:
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erf(x) = x*exp(-x*x)/sqrt(pi) * [
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2/1 + 4/3 x**2 + 8/15 x**4 + 16/105 x**6 + ...]
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The coefficient of x**(2k-2) here is 4**k*factorial(k)/factorial(2*k).
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This series converges well for smallish x, but slowly for larger x.
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The continued fraction expansion used is:
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erfc(x) = x*exp(-x*x)/sqrt(pi) * [1/(0.5 + x**2 -) 0.5/(2.5 + x**2 - )
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3.0/(4.5 + x**2 - ) 7.5/(6.5 + x**2 - ) ...]
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after the first term, the general term has the form:
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k*(k-0.5)/(2*k+0.5 + x**2 - ...).
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This expansion converges fast for larger x, but convergence becomes
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infinitely slow as x approaches 0.0. The (somewhat naive) continued
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fraction evaluation algorithm used below also risks overflow for large x;
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but for large x, erfc(x) == 0.0 to within machine precision. (For
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example, erfc(30.0) is approximately 2.56e-393).
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Parameters: use series expansion for abs(x) < ERF_SERIES_CUTOFF and
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continued fraction expansion for ERF_SERIES_CUTOFF <= abs(x) <
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ERFC_CONTFRAC_CUTOFF. ERFC_SERIES_TERMS and ERFC_CONTFRAC_TERMS are the
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numbers of terms to use for the relevant expansions. */
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#define ERF_SERIES_CUTOFF 1.5
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#define ERF_SERIES_TERMS 25
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#define ERFC_CONTFRAC_CUTOFF 30.0
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#define ERFC_CONTFRAC_TERMS 50
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/*
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Error function, via power series.
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Given a finite float x, return an approximation to erf(x).
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Converges reasonably fast for small x.
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*/
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static double
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m_erf_series(double x)
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{
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double x2, acc, fk, result;
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int i, saved_errno;
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x2 = x * x;
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acc = 0.0;
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fk = (double)ERF_SERIES_TERMS + 0.5;
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for (i = 0; i < ERF_SERIES_TERMS; i++) {
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acc = 2.0 + x2 * acc / fk;
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fk -= 1.0;
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}
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/* Make sure the exp call doesn't affect errno;
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see m_erfc_contfrac for more. */
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saved_errno = errno;
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result = acc * x * exp(-x2) / sqrtpi;
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errno = saved_errno;
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return result;
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}
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|
/*
|
|
Complementary error function, via continued fraction expansion.
|
|
|
|
Given a positive float x, return an approximation to erfc(x). Converges
|
|
reasonably fast for x large (say, x > 2.0), and should be safe from
|
|
overflow if x and nterms are not too large. On an IEEE 754 machine, with x
|
|
<= 30.0, we're safe up to nterms = 100. For x >= 30.0, erfc(x) is smaller
|
|
than the smallest representable nonzero float. */
|
|
|
|
static double
|
|
m_erfc_contfrac(double x)
|
|
{
|
|
double x2, a, da, p, p_last, q, q_last, b, result;
|
|
int i, saved_errno;
|
|
|
|
if (x >= ERFC_CONTFRAC_CUTOFF)
|
|
return 0.0;
|
|
|
|
x2 = x*x;
|
|
a = 0.0;
|
|
da = 0.5;
|
|
p = 1.0; p_last = 0.0;
|
|
q = da + x2; q_last = 1.0;
|
|
for (i = 0; i < ERFC_CONTFRAC_TERMS; i++) {
|
|
double temp;
|
|
a += da;
|
|
da += 2.0;
|
|
b = da + x2;
|
|
temp = p; p = b*p - a*p_last; p_last = temp;
|
|
temp = q; q = b*q - a*q_last; q_last = temp;
|
|
}
|
|
/* Issue #8986: On some platforms, exp sets errno on underflow to zero;
|
|
save the current errno value so that we can restore it later. */
|
|
saved_errno = errno;
|
|
result = p / q * x * exp(-x2) / sqrtpi;
|
|
errno = saved_errno;
|
|
return result;
|
|
}
|
|
|
|
/* Error function erf(x), for general x */
|
|
|
|
static double
|
|
m_erf(double x)
|
|
{
|
|
double absx, cf;
|
|
|
|
if (Py_IS_NAN(x))
|
|
return x;
|
|
absx = fabs(x);
|
|
if (absx < ERF_SERIES_CUTOFF)
|
|
return m_erf_series(x);
|
|
else {
|
|
cf = m_erfc_contfrac(absx);
|
|
return x > 0.0 ? 1.0 - cf : cf - 1.0;
|
|
}
|
|
}
|
|
|
|
/* Complementary error function erfc(x), for general x. */
|
|
|
|
static double
|
|
m_erfc(double x)
|
|
{
|
|
double absx, cf;
|
|
|
|
if (Py_IS_NAN(x))
|
|
return x;
|
|
absx = fabs(x);
|
|
if (absx < ERF_SERIES_CUTOFF)
|
|
return 1.0 - m_erf_series(x);
|
|
else {
|
|
cf = m_erfc_contfrac(absx);
|
|
return x > 0.0 ? cf : 2.0 - cf;
|
|
}
|
|
}
|
|
|
|
/*
|
|
wrapper for atan2 that deals directly with special cases before
|
|
delegating to the platform libm for the remaining cases. This
|
|
is necessary to get consistent behaviour across platforms.
|
|
Windows, FreeBSD and alpha Tru64 are amongst platforms that don't
|
|
always follow C99.
|
|
*/
|
|
|
|
static double
|
|
m_atan2(double y, double x)
|
|
{
|
|
if (Py_IS_NAN(x) || Py_IS_NAN(y))
|
|
return Py_NAN;
|
|
if (Py_IS_INFINITY(y)) {
|
|
if (Py_IS_INFINITY(x)) {
|
|
if (copysign(1., x) == 1.)
|
|
/* atan2(+-inf, +inf) == +-pi/4 */
|
|
return copysign(0.25*Py_MATH_PI, y);
|
|
else
|
|
/* atan2(+-inf, -inf) == +-pi*3/4 */
|
|
return copysign(0.75*Py_MATH_PI, y);
|
|
}
|
|
/* atan2(+-inf, x) == +-pi/2 for finite x */
|
|
return copysign(0.5*Py_MATH_PI, y);
|
|
}
|
|
if (Py_IS_INFINITY(x) || y == 0.) {
|
|
if (copysign(1., x) == 1.)
|
|
/* atan2(+-y, +inf) = atan2(+-0, +x) = +-0. */
|
|
return copysign(0., y);
|
|
else
|
|
/* atan2(+-y, -inf) = atan2(+-0., -x) = +-pi. */
|
|
return copysign(Py_MATH_PI, y);
|
|
}
|
|
return atan2(y, x);
|
|
}
|
|
|
|
/*
|
|
Various platforms (Solaris, OpenBSD) do nonstandard things for log(0),
|
|
log(-ve), log(NaN). Here are wrappers for log and log10 that deal with
|
|
special values directly, passing positive non-special values through to
|
|
the system log/log10.
|
|
*/
|
|
|
|
static double
|
|
m_log(double x)
|
|
{
|
|
if (Py_IS_FINITE(x)) {
|
|
if (x > 0.0)
|
|
return log(x);
|
|
errno = EDOM;
|
|
if (x == 0.0)
|
|
return -Py_HUGE_VAL; /* log(0) = -inf */
|
|
else
|
|
return Py_NAN; /* log(-ve) = nan */
|
|
}
|
|
else if (Py_IS_NAN(x))
|
|
return x; /* log(nan) = nan */
|
|
else if (x > 0.0)
|
|
return x; /* log(inf) = inf */
|
|
else {
|
|
errno = EDOM;
|
|
return Py_NAN; /* log(-inf) = nan */
|
|
}
|
|
}
|
|
|
|
static double
|
|
m_log10(double x)
|
|
{
|
|
if (Py_IS_FINITE(x)) {
|
|
if (x > 0.0)
|
|
return log10(x);
|
|
errno = EDOM;
|
|
if (x == 0.0)
|
|
return -Py_HUGE_VAL; /* log10(0) = -inf */
|
|
else
|
|
return Py_NAN; /* log10(-ve) = nan */
|
|
}
|
|
else if (Py_IS_NAN(x))
|
|
return x; /* log10(nan) = nan */
|
|
else if (x > 0.0)
|
|
return x; /* log10(inf) = inf */
|
|
else {
|
|
errno = EDOM;
|
|
return Py_NAN; /* log10(-inf) = nan */
|
|
}
|
|
}
|
|
|
|
|
|
/* Call is_error when errno != 0, and where x is the result libm
|
|
* returned. is_error will usually set up an exception and return
|
|
* true (1), but may return false (0) without setting up an exception.
|
|
*/
|
|
static int
|
|
is_error(double x)
|
|
{
|
|
int result = 1; /* presumption of guilt */
|
|
assert(errno); /* non-zero errno is a precondition for calling */
|
|
if (errno == EDOM)
|
|
PyErr_SetString(PyExc_ValueError, "math domain error");
|
|
|
|
else if (errno == ERANGE) {
|
|
/* ANSI C generally requires libm functions to set ERANGE
|
|
* on overflow, but also generally *allows* them to set
|
|
* ERANGE on underflow too. There's no consistency about
|
|
* the latter across platforms.
|
|
* Alas, C99 never requires that errno be set.
|
|
* Here we suppress the underflow errors (libm functions
|
|
* should return a zero on underflow, and +- HUGE_VAL on
|
|
* overflow, so testing the result for zero suffices to
|
|
* distinguish the cases).
|
|
*
|
|
* On some platforms (Ubuntu/ia64) it seems that errno can be
|
|
* set to ERANGE for subnormal results that do *not* underflow
|
|
* to zero. So to be safe, we'll ignore ERANGE whenever the
|
|
* function result is less than one in absolute value.
|
|
*/
|
|
if (fabs(x) < 1.0)
|
|
result = 0;
|
|
else
|
|
PyErr_SetString(PyExc_OverflowError,
|
|
"math range error");
|
|
}
|
|
else
|
|
/* Unexpected math error */
|
|
PyErr_SetFromErrno(PyExc_ValueError);
|
|
return result;
|
|
}
|
|
|
|
/*
|
|
math_1 is used to wrap a libm function f that takes a double
|
|
arguments and returns a double.
|
|
|
|
The error reporting follows these rules, which are designed to do
|
|
the right thing on C89/C99 platforms and IEEE 754/non IEEE 754
|
|
platforms.
|
|
|
|
- a NaN result from non-NaN inputs causes ValueError to be raised
|
|
- an infinite result from finite inputs causes OverflowError to be
|
|
raised if can_overflow is 1, or raises ValueError if can_overflow
|
|
is 0.
|
|
- if the result is finite and errno == EDOM then ValueError is
|
|
raised
|
|
- if the result is finite and nonzero and errno == ERANGE then
|
|
OverflowError is raised
|
|
|
|
The last rule is used to catch overflow on platforms which follow
|
|
C89 but for which HUGE_VAL is not an infinity.
|
|
|
|
For the majority of one-argument functions these rules are enough
|
|
to ensure that Python's functions behave as specified in 'Annex F'
|
|
of the C99 standard, with the 'invalid' and 'divide-by-zero'
|
|
floating-point exceptions mapping to Python's ValueError and the
|
|
'overflow' floating-point exception mapping to OverflowError.
|
|
math_1 only works for functions that don't have singularities *and*
|
|
the possibility of overflow; fortunately, that covers everything we
|
|
care about right now.
|
|
*/
|
|
|
|
static PyObject *
|
|
math_1(PyObject *arg, double (*func) (double), int can_overflow)
|
|
{
|
|
double x, r;
|
|
x = PyFloat_AsDouble(arg);
|
|
if (x == -1.0 && PyErr_Occurred())
|
|
return NULL;
|
|
errno = 0;
|
|
PyFPE_START_PROTECT("in math_1", return 0);
|
|
r = (*func)(x);
|
|
PyFPE_END_PROTECT(r);
|
|
if (Py_IS_NAN(r)) {
|
|
if (!Py_IS_NAN(x))
|
|
errno = EDOM;
|
|
else
|
|
errno = 0;
|
|
}
|
|
else if (Py_IS_INFINITY(r)) {
|
|
if (Py_IS_FINITE(x))
|
|
errno = can_overflow ? ERANGE : EDOM;
|
|
else
|
|
errno = 0;
|
|
}
|
|
if (errno && is_error(r))
|
|
return NULL;
|
|
else
|
|
return PyFloat_FromDouble(r);
|
|
}
|
|
|
|
/* variant of math_1, to be used when the function being wrapped is known to
|
|
set errno properly (that is, errno = EDOM for invalid or divide-by-zero,
|
|
errno = ERANGE for overflow). */
|
|
|
|
static PyObject *
|
|
math_1a(PyObject *arg, double (*func) (double))
|
|
{
|
|
double x, r;
|
|
x = PyFloat_AsDouble(arg);
|
|
if (x == -1.0 && PyErr_Occurred())
|
|
return NULL;
|
|
errno = 0;
|
|
PyFPE_START_PROTECT("in math_1a", return 0);
|
|
r = (*func)(x);
|
|
PyFPE_END_PROTECT(r);
|
|
if (errno && is_error(r))
|
|
return NULL;
|
|
return PyFloat_FromDouble(r);
|
|
}
|
|
|
|
/*
|
|
math_2 is used to wrap a libm function f that takes two double
|
|
arguments and returns a double.
|
|
|
|
The error reporting follows these rules, which are designed to do
|
|
the right thing on C89/C99 platforms and IEEE 754/non IEEE 754
|
|
platforms.
|
|
|
|
- a NaN result from non-NaN inputs causes ValueError to be raised
|
|
- an infinite result from finite inputs causes OverflowError to be
|
|
raised.
|
|
- if the result is finite and errno == EDOM then ValueError is
|
|
raised
|
|
- if the result is finite and nonzero and errno == ERANGE then
|
|
OverflowError is raised
|
|
|
|
The last rule is used to catch overflow on platforms which follow
|
|
C89 but for which HUGE_VAL is not an infinity.
|
|
|
|
For most two-argument functions (copysign, fmod, hypot, atan2)
|
|
these rules are enough to ensure that Python's functions behave as
|
|
specified in 'Annex F' of the C99 standard, with the 'invalid' and
|
|
'divide-by-zero' floating-point exceptions mapping to Python's
|
|
ValueError and the 'overflow' floating-point exception mapping to
|
|
OverflowError.
|
|
*/
|
|
|
|
static PyObject *
|
|
math_2(PyObject *args, double (*func) (double, double), char *funcname)
|
|
{
|
|
PyObject *ox, *oy;
|
|
double x, y, r;
|
|
if (! PyArg_UnpackTuple(args, funcname, 2, 2, &ox, &oy))
|
|
return NULL;
|
|
x = PyFloat_AsDouble(ox);
|
|
y = PyFloat_AsDouble(oy);
|
|
if ((x == -1.0 || y == -1.0) && PyErr_Occurred())
|
|
return NULL;
|
|
errno = 0;
|
|
PyFPE_START_PROTECT("in math_2", return 0);
|
|
r = (*func)(x, y);
|
|
PyFPE_END_PROTECT(r);
|
|
if (Py_IS_NAN(r)) {
|
|
if (!Py_IS_NAN(x) && !Py_IS_NAN(y))
|
|
errno = EDOM;
|
|
else
|
|
errno = 0;
|
|
}
|
|
else if (Py_IS_INFINITY(r)) {
|
|
if (Py_IS_FINITE(x) && Py_IS_FINITE(y))
|
|
errno = ERANGE;
|
|
else
|
|
errno = 0;
|
|
}
|
|
if (errno && is_error(r))
|
|
return NULL;
|
|
else
|
|
return PyFloat_FromDouble(r);
|
|
}
|
|
|
|
#define FUNC1(funcname, func, can_overflow, docstring) \
|
|
static PyObject * math_##funcname(PyObject *self, PyObject *args) { \
|
|
return math_1(args, func, can_overflow); \
|
|
}\
|
|
PyDoc_STRVAR(math_##funcname##_doc, docstring);
|
|
|
|
#define FUNC1A(funcname, func, docstring) \
|
|
static PyObject * math_##funcname(PyObject *self, PyObject *args) { \
|
|
return math_1a(args, func); \
|
|
}\
|
|
PyDoc_STRVAR(math_##funcname##_doc, docstring);
|
|
|
|
#define FUNC2(funcname, func, docstring) \
|
|
static PyObject * math_##funcname(PyObject *self, PyObject *args) { \
|
|
return math_2(args, func, #funcname); \
|
|
}\
|
|
PyDoc_STRVAR(math_##funcname##_doc, docstring);
|
|
|
|
FUNC1(acos, acos, 0,
|
|
"acos(x)\n\nReturn the arc cosine (measured in radians) of x.")
|
|
FUNC1(acosh, m_acosh, 0,
|
|
"acosh(x)\n\nReturn the hyperbolic arc cosine (measured in radians) of x.")
|
|
FUNC1(asin, asin, 0,
|
|
"asin(x)\n\nReturn the arc sine (measured in radians) of x.")
|
|
FUNC1(asinh, m_asinh, 0,
|
|
"asinh(x)\n\nReturn the hyperbolic arc sine (measured in radians) of x.")
|
|
FUNC1(atan, atan, 0,
|
|
"atan(x)\n\nReturn the arc tangent (measured in radians) of x.")
|
|
FUNC2(atan2, m_atan2,
|
|
"atan2(y, x)\n\nReturn the arc tangent (measured in radians) of y/x.\n"
|
|
"Unlike atan(y/x), the signs of both x and y are considered.")
|
|
FUNC1(atanh, m_atanh, 0,
|
|
"atanh(x)\n\nReturn the hyperbolic arc tangent (measured in radians) of x.")
|
|
FUNC1(ceil, ceil, 0,
|
|
"ceil(x)\n\nReturn the ceiling of x as a float.\n"
|
|
"This is the smallest integral value >= x.")
|
|
FUNC2(copysign, copysign,
|
|
"copysign(x, y)\n\nReturn x with the sign of y.")
|
|
FUNC1(cos, cos, 0,
|
|
"cos(x)\n\nReturn the cosine of x (measured in radians).")
|
|
FUNC1(cosh, cosh, 1,
|
|
"cosh(x)\n\nReturn the hyperbolic cosine of x.")
|
|
FUNC1A(erf, m_erf,
|
|
"erf(x)\n\nError function at x.")
|
|
FUNC1A(erfc, m_erfc,
|
|
"erfc(x)\n\nComplementary error function at x.")
|
|
FUNC1(exp, exp, 1,
|
|
"exp(x)\n\nReturn e raised to the power of x.")
|
|
FUNC1(expm1, m_expm1, 1,
|
|
"expm1(x)\n\nReturn exp(x)-1.\n"
|
|
"This function avoids the loss of precision involved in the direct "
|
|
"evaluation of exp(x)-1 for small x.")
|
|
FUNC1(fabs, fabs, 0,
|
|
"fabs(x)\n\nReturn the absolute value of the float x.")
|
|
FUNC1(floor, floor, 0,
|
|
"floor(x)\n\nReturn the floor of x as a float.\n"
|
|
"This is the largest integral value <= x.")
|
|
FUNC1A(gamma, m_tgamma,
|
|
"gamma(x)\n\nGamma function at x.")
|
|
FUNC1A(lgamma, m_lgamma,
|
|
"lgamma(x)\n\nNatural logarithm of absolute value of Gamma function at x.")
|
|
FUNC1(log1p, m_log1p, 1,
|
|
"log1p(x)\n\nReturn the natural logarithm of 1+x (base e).\n"
|
|
"The result is computed in a way which is accurate for x near zero.")
|
|
FUNC1(sin, sin, 0,
|
|
"sin(x)\n\nReturn the sine of x (measured in radians).")
|
|
FUNC1(sinh, sinh, 1,
|
|
"sinh(x)\n\nReturn the hyperbolic sine of x.")
|
|
FUNC1(sqrt, sqrt, 0,
|
|
"sqrt(x)\n\nReturn the square root of x.")
|
|
FUNC1(tan, tan, 0,
|
|
"tan(x)\n\nReturn the tangent of x (measured in radians).")
|
|
FUNC1(tanh, tanh, 0,
|
|
"tanh(x)\n\nReturn the hyperbolic tangent of x.")
|
|
|
|
/* Precision summation function as msum() by Raymond Hettinger in
|
|
<http://aspn.activestate.com/ASPN/Cookbook/Python/Recipe/393090>,
|
|
enhanced with the exact partials sum and roundoff from Mark
|
|
Dickinson's post at <http://bugs.python.org/file10357/msum4.py>.
|
|
See those links for more details, proofs and other references.
|
|
|
|
Note 1: IEEE 754R floating point semantics are assumed,
|
|
but the current implementation does not re-establish special
|
|
value semantics across iterations (i.e. handling -Inf + Inf).
|
|
|
|
Note 2: No provision is made for intermediate overflow handling;
|
|
therefore, sum([1e+308, 1e-308, 1e+308]) returns 1e+308 while
|
|
sum([1e+308, 1e+308, 1e-308]) raises an OverflowError due to the
|
|
overflow of the first partial sum.
|
|
|
|
Note 3: The intermediate values lo, yr, and hi are declared volatile so
|
|
aggressive compilers won't algebraically reduce lo to always be exactly 0.0.
|
|
Also, the volatile declaration forces the values to be stored in memory as
|
|
regular doubles instead of extended long precision (80-bit) values. This
|
|
prevents double rounding because any addition or subtraction of two doubles
|
|
can be resolved exactly into double-sized hi and lo values. As long as the
|
|
hi value gets forced into a double before yr and lo are computed, the extra
|
|
bits in downstream extended precision operations (x87 for example) will be
|
|
exactly zero and therefore can be losslessly stored back into a double,
|
|
thereby preventing double rounding.
|
|
|
|
Note 4: A similar implementation is in Modules/cmathmodule.c.
|
|
Be sure to update both when making changes.
|
|
|
|
Note 5: The signature of math.fsum() differs from __builtin__.sum()
|
|
because the start argument doesn't make sense in the context of
|
|
accurate summation. Since the partials table is collapsed before
|
|
returning a result, sum(seq2, start=sum(seq1)) may not equal the
|
|
accurate result returned by sum(itertools.chain(seq1, seq2)).
|
|
*/
|
|
|
|
#define NUM_PARTIALS 32 /* initial partials array size, on stack */
|
|
|
|
/* Extend the partials array p[] by doubling its size. */
|
|
static int /* non-zero on error */
|
|
_fsum_realloc(double **p_ptr, Py_ssize_t n,
|
|
double *ps, Py_ssize_t *m_ptr)
|
|
{
|
|
void *v = NULL;
|
|
Py_ssize_t m = *m_ptr;
|
|
|
|
m += m; /* double */
|
|
if (n < m && m < (PY_SSIZE_T_MAX / sizeof(double))) {
|
|
double *p = *p_ptr;
|
|
if (p == ps) {
|
|
v = PyMem_Malloc(sizeof(double) * m);
|
|
if (v != NULL)
|
|
memcpy(v, ps, sizeof(double) * n);
|
|
}
|
|
else
|
|
v = PyMem_Realloc(p, sizeof(double) * m);
|
|
}
|
|
if (v == NULL) { /* size overflow or no memory */
|
|
PyErr_SetString(PyExc_MemoryError, "math.fsum partials");
|
|
return 1;
|
|
}
|
|
*p_ptr = (double*) v;
|
|
*m_ptr = m;
|
|
return 0;
|
|
}
|
|
|
|
/* Full precision summation of a sequence of floats.
|
|
|
|
def msum(iterable):
|
|
partials = [] # sorted, non-overlapping partial sums
|
|
for x in iterable:
|
|
i = 0
|
|
for y in partials:
|
|
if abs(x) < abs(y):
|
|
x, y = y, x
|
|
hi = x + y
|
|
lo = y - (hi - x)
|
|
if lo:
|
|
partials[i] = lo
|
|
i += 1
|
|
x = hi
|
|
partials[i:] = [x]
|
|
return sum_exact(partials)
|
|
|
|
Rounded x+y stored in hi with the roundoff stored in lo. Together hi+lo
|
|
are exactly equal to x+y. The inner loop applies hi/lo summation to each
|
|
partial so that the list of partial sums remains exact.
|
|
|
|
Sum_exact() adds the partial sums exactly and correctly rounds the final
|
|
result (using the round-half-to-even rule). The items in partials remain
|
|
non-zero, non-special, non-overlapping and strictly increasing in
|
|
magnitude, but possibly not all having the same sign.
|
|
|
|
Depends on IEEE 754 arithmetic guarantees and half-even rounding.
|
|
*/
|
|
|
|
static PyObject*
|
|
math_fsum(PyObject *self, PyObject *seq)
|
|
{
|
|
PyObject *item, *iter, *sum = NULL;
|
|
Py_ssize_t i, j, n = 0, m = NUM_PARTIALS;
|
|
double x, y, t, ps[NUM_PARTIALS], *p = ps;
|
|
double xsave, special_sum = 0.0, inf_sum = 0.0;
|
|
volatile double hi, yr, lo;
|
|
|
|
iter = PyObject_GetIter(seq);
|
|
if (iter == NULL)
|
|
return NULL;
|
|
|
|
PyFPE_START_PROTECT("fsum", Py_DECREF(iter); return NULL)
|
|
|
|
for(;;) { /* for x in iterable */
|
|
assert(0 <= n && n <= m);
|
|
assert((m == NUM_PARTIALS && p == ps) ||
|
|
(m > NUM_PARTIALS && p != NULL));
|
|
|
|
item = PyIter_Next(iter);
|
|
if (item == NULL) {
|
|
if (PyErr_Occurred())
|
|
goto _fsum_error;
|
|
break;
|
|
}
|
|
x = PyFloat_AsDouble(item);
|
|
Py_DECREF(item);
|
|
if (PyErr_Occurred())
|
|
goto _fsum_error;
|
|
|
|
xsave = x;
|
|
for (i = j = 0; j < n; j++) { /* for y in partials */
|
|
y = p[j];
|
|
if (fabs(x) < fabs(y)) {
|
|
t = x; x = y; y = t;
|
|
}
|
|
hi = x + y;
|
|
yr = hi - x;
|
|
lo = y - yr;
|
|
if (lo != 0.0)
|
|
p[i++] = lo;
|
|
x = hi;
|
|
}
|
|
|
|
n = i; /* ps[i:] = [x] */
|
|
if (x != 0.0) {
|
|
if (! Py_IS_FINITE(x)) {
|
|
/* a nonfinite x could arise either as
|
|
a result of intermediate overflow, or
|
|
as a result of a nan or inf in the
|
|
summands */
|
|
if (Py_IS_FINITE(xsave)) {
|
|
PyErr_SetString(PyExc_OverflowError,
|
|
"intermediate overflow in fsum");
|
|
goto _fsum_error;
|
|
}
|
|
if (Py_IS_INFINITY(xsave))
|
|
inf_sum += xsave;
|
|
special_sum += xsave;
|
|
/* reset partials */
|
|
n = 0;
|
|
}
|
|
else if (n >= m && _fsum_realloc(&p, n, ps, &m))
|
|
goto _fsum_error;
|
|
else
|
|
p[n++] = x;
|
|
}
|
|
}
|
|
|
|
if (special_sum != 0.0) {
|
|
if (Py_IS_NAN(inf_sum))
|
|
PyErr_SetString(PyExc_ValueError,
|
|
"-inf + inf in fsum");
|
|
else
|
|
sum = PyFloat_FromDouble(special_sum);
|
|
goto _fsum_error;
|
|
}
|
|
|
|
hi = 0.0;
|
|
if (n > 0) {
|
|
hi = p[--n];
|
|
/* sum_exact(ps, hi) from the top, stop when the sum becomes
|
|
inexact. */
|
|
while (n > 0) {
|
|
x = hi;
|
|
y = p[--n];
|
|
assert(fabs(y) < fabs(x));
|
|
hi = x + y;
|
|
yr = hi - x;
|
|
lo = y - yr;
|
|
if (lo != 0.0)
|
|
break;
|
|
}
|
|
/* Make half-even rounding work across multiple partials.
|
|
Needed so that sum([1e-16, 1, 1e16]) will round-up the last
|
|
digit to two instead of down to zero (the 1e-16 makes the 1
|
|
slightly closer to two). With a potential 1 ULP rounding
|
|
error fixed-up, math.fsum() can guarantee commutativity. */
|
|
if (n > 0 && ((lo < 0.0 && p[n-1] < 0.0) ||
|
|
(lo > 0.0 && p[n-1] > 0.0))) {
|
|
y = lo * 2.0;
|
|
x = hi + y;
|
|
yr = x - hi;
|
|
if (y == yr)
|
|
hi = x;
|
|
}
|
|
}
|
|
sum = PyFloat_FromDouble(hi);
|
|
|
|
_fsum_error:
|
|
PyFPE_END_PROTECT(hi)
|
|
Py_DECREF(iter);
|
|
if (p != ps)
|
|
PyMem_Free(p);
|
|
return sum;
|
|
}
|
|
|
|
#undef NUM_PARTIALS
|
|
|
|
PyDoc_STRVAR(math_fsum_doc,
|
|
"fsum(iterable)\n\n\
|
|
Return an accurate floating point sum of values in the iterable.\n\
|
|
Assumes IEEE-754 floating point arithmetic.");
|
|
|
|
static PyObject *
|
|
math_factorial(PyObject *self, PyObject *arg)
|
|
{
|
|
long i, x;
|
|
PyObject *result, *iobj, *newresult;
|
|
|
|
if (PyFloat_Check(arg)) {
|
|
PyObject *lx;
|
|
double dx = PyFloat_AS_DOUBLE((PyFloatObject *)arg);
|
|
if (!(Py_IS_FINITE(dx) && dx == floor(dx))) {
|
|
PyErr_SetString(PyExc_ValueError,
|
|
"factorial() only accepts integral values");
|
|
return NULL;
|
|
}
|
|
lx = PyLong_FromDouble(dx);
|
|
if (lx == NULL)
|
|
return NULL;
|
|
x = PyLong_AsLong(lx);
|
|
Py_DECREF(lx);
|
|
}
|
|
else
|
|
x = PyInt_AsLong(arg);
|
|
|
|
if (x == -1 && PyErr_Occurred())
|
|
return NULL;
|
|
if (x < 0) {
|
|
PyErr_SetString(PyExc_ValueError,
|
|
"factorial() not defined for negative values");
|
|
return NULL;
|
|
}
|
|
|
|
result = (PyObject *)PyInt_FromLong(1);
|
|
if (result == NULL)
|
|
return NULL;
|
|
for (i=1 ; i<=x ; i++) {
|
|
iobj = (PyObject *)PyInt_FromLong(i);
|
|
if (iobj == NULL)
|
|
goto error;
|
|
newresult = PyNumber_Multiply(result, iobj);
|
|
Py_DECREF(iobj);
|
|
if (newresult == NULL)
|
|
goto error;
|
|
Py_DECREF(result);
|
|
result = newresult;
|
|
}
|
|
return result;
|
|
|
|
error:
|
|
Py_DECREF(result);
|
|
return NULL;
|
|
}
|
|
|
|
PyDoc_STRVAR(math_factorial_doc,
|
|
"factorial(x) -> Integral\n"
|
|
"\n"
|
|
"Find x!. Raise a ValueError if x is negative or non-integral.");
|
|
|
|
static PyObject *
|
|
math_trunc(PyObject *self, PyObject *number)
|
|
{
|
|
return PyObject_CallMethod(number, "__trunc__", NULL);
|
|
}
|
|
|
|
PyDoc_STRVAR(math_trunc_doc,
|
|
"trunc(x:Real) -> Integral\n"
|
|
"\n"
|
|
"Truncates x to the nearest Integral toward 0. Uses the __trunc__ magic method.");
|
|
|
|
static PyObject *
|
|
math_frexp(PyObject *self, PyObject *arg)
|
|
{
|
|
int i;
|
|
double x = PyFloat_AsDouble(arg);
|
|
if (x == -1.0 && PyErr_Occurred())
|
|
return NULL;
|
|
/* deal with special cases directly, to sidestep platform
|
|
differences */
|
|
if (Py_IS_NAN(x) || Py_IS_INFINITY(x) || !x) {
|
|
i = 0;
|
|
}
|
|
else {
|
|
PyFPE_START_PROTECT("in math_frexp", return 0);
|
|
x = frexp(x, &i);
|
|
PyFPE_END_PROTECT(x);
|
|
}
|
|
return Py_BuildValue("(di)", x, i);
|
|
}
|
|
|
|
PyDoc_STRVAR(math_frexp_doc,
|
|
"frexp(x)\n"
|
|
"\n"
|
|
"Return the mantissa and exponent of x, as pair (m, e).\n"
|
|
"m is a float and e is an int, such that x = m * 2.**e.\n"
|
|
"If x is 0, m and e are both 0. Else 0.5 <= abs(m) < 1.0.");
|
|
|
|
static PyObject *
|
|
math_ldexp(PyObject *self, PyObject *args)
|
|
{
|
|
double x, r;
|
|
PyObject *oexp;
|
|
long exp;
|
|
int overflow;
|
|
if (! PyArg_ParseTuple(args, "dO:ldexp", &x, &oexp))
|
|
return NULL;
|
|
|
|
if (PyLong_Check(oexp) || PyInt_Check(oexp)) {
|
|
/* on overflow, replace exponent with either LONG_MAX
|
|
or LONG_MIN, depending on the sign. */
|
|
exp = PyLong_AsLongAndOverflow(oexp, &overflow);
|
|
if (exp == -1 && PyErr_Occurred())
|
|
return NULL;
|
|
if (overflow)
|
|
exp = overflow < 0 ? LONG_MIN : LONG_MAX;
|
|
}
|
|
else {
|
|
PyErr_SetString(PyExc_TypeError,
|
|
"Expected an int or long as second argument "
|
|
"to ldexp.");
|
|
return NULL;
|
|
}
|
|
|
|
if (x == 0. || !Py_IS_FINITE(x)) {
|
|
/* NaNs, zeros and infinities are returned unchanged */
|
|
r = x;
|
|
errno = 0;
|
|
} else if (exp > INT_MAX) {
|
|
/* overflow */
|
|
r = copysign(Py_HUGE_VAL, x);
|
|
errno = ERANGE;
|
|
} else if (exp < INT_MIN) {
|
|
/* underflow to +-0 */
|
|
r = copysign(0., x);
|
|
errno = 0;
|
|
} else {
|
|
errno = 0;
|
|
PyFPE_START_PROTECT("in math_ldexp", return 0);
|
|
r = ldexp(x, (int)exp);
|
|
PyFPE_END_PROTECT(r);
|
|
if (Py_IS_INFINITY(r))
|
|
errno = ERANGE;
|
|
}
|
|
|
|
if (errno && is_error(r))
|
|
return NULL;
|
|
return PyFloat_FromDouble(r);
|
|
}
|
|
|
|
PyDoc_STRVAR(math_ldexp_doc,
|
|
"ldexp(x, i)\n\n\
|
|
Return x * (2**i).");
|
|
|
|
static PyObject *
|
|
math_modf(PyObject *self, PyObject *arg)
|
|
{
|
|
double y, x = PyFloat_AsDouble(arg);
|
|
if (x == -1.0 && PyErr_Occurred())
|
|
return NULL;
|
|
/* some platforms don't do the right thing for NaNs and
|
|
infinities, so we take care of special cases directly. */
|
|
if (!Py_IS_FINITE(x)) {
|
|
if (Py_IS_INFINITY(x))
|
|
return Py_BuildValue("(dd)", copysign(0., x), x);
|
|
else if (Py_IS_NAN(x))
|
|
return Py_BuildValue("(dd)", x, x);
|
|
}
|
|
|
|
errno = 0;
|
|
PyFPE_START_PROTECT("in math_modf", return 0);
|
|
x = modf(x, &y);
|
|
PyFPE_END_PROTECT(x);
|
|
return Py_BuildValue("(dd)", x, y);
|
|
}
|
|
|
|
PyDoc_STRVAR(math_modf_doc,
|
|
"modf(x)\n"
|
|
"\n"
|
|
"Return the fractional and integer parts of x. Both results carry the sign\n"
|
|
"of x and are floats.");
|
|
|
|
/* A decent logarithm is easy to compute even for huge longs, but libm can't
|
|
do that by itself -- loghelper can. func is log or log10, and name is
|
|
"log" or "log10". Note that overflow of the result isn't possible: a long
|
|
can contain no more than INT_MAX * SHIFT bits, so has value certainly less
|
|
than 2**(2**64 * 2**16) == 2**2**80, and log2 of that is 2**80, which is
|
|
small enough to fit in an IEEE single. log and log10 are even smaller.
|
|
However, intermediate overflow is possible for a long if the number of bits
|
|
in that long is larger than PY_SSIZE_T_MAX. */
|
|
|
|
static PyObject*
|
|
loghelper(PyObject* arg, double (*func)(double), char *funcname)
|
|
{
|
|
/* If it is long, do it ourselves. */
|
|
if (PyLong_Check(arg)) {
|
|
double x;
|
|
Py_ssize_t e;
|
|
x = _PyLong_Frexp((PyLongObject *)arg, &e);
|
|
if (x == -1.0 && PyErr_Occurred())
|
|
return NULL;
|
|
if (x <= 0.0) {
|
|
PyErr_SetString(PyExc_ValueError,
|
|
"math domain error");
|
|
return NULL;
|
|
}
|
|
/* Special case for log(1), to make sure we get an
|
|
exact result there. */
|
|
if (e == 1 && x == 0.5)
|
|
return PyFloat_FromDouble(0.0);
|
|
/* Value is ~= x * 2**e, so the log ~= log(x) + log(2) * e. */
|
|
x = func(x) + func(2.0) * e;
|
|
return PyFloat_FromDouble(x);
|
|
}
|
|
|
|
/* Else let libm handle it by itself. */
|
|
return math_1(arg, func, 0);
|
|
}
|
|
|
|
static PyObject *
|
|
math_log(PyObject *self, PyObject *args)
|
|
{
|
|
PyObject *arg;
|
|
PyObject *base = NULL;
|
|
PyObject *num, *den;
|
|
PyObject *ans;
|
|
|
|
if (!PyArg_UnpackTuple(args, "log", 1, 2, &arg, &base))
|
|
return NULL;
|
|
|
|
num = loghelper(arg, m_log, "log");
|
|
if (num == NULL || base == NULL)
|
|
return num;
|
|
|
|
den = loghelper(base, m_log, "log");
|
|
if (den == NULL) {
|
|
Py_DECREF(num);
|
|
return NULL;
|
|
}
|
|
|
|
ans = PyNumber_Divide(num, den);
|
|
Py_DECREF(num);
|
|
Py_DECREF(den);
|
|
return ans;
|
|
}
|
|
|
|
PyDoc_STRVAR(math_log_doc,
|
|
"log(x[, base])\n\n\
|
|
Return the logarithm of x to the given base.\n\
|
|
If the base not specified, returns the natural logarithm (base e) of x.");
|
|
|
|
static PyObject *
|
|
math_log10(PyObject *self, PyObject *arg)
|
|
{
|
|
return loghelper(arg, m_log10, "log10");
|
|
}
|
|
|
|
PyDoc_STRVAR(math_log10_doc,
|
|
"log10(x)\n\nReturn the base 10 logarithm of x.");
|
|
|
|
static PyObject *
|
|
math_fmod(PyObject *self, PyObject *args)
|
|
{
|
|
PyObject *ox, *oy;
|
|
double r, x, y;
|
|
if (! PyArg_UnpackTuple(args, "fmod", 2, 2, &ox, &oy))
|
|
return NULL;
|
|
x = PyFloat_AsDouble(ox);
|
|
y = PyFloat_AsDouble(oy);
|
|
if ((x == -1.0 || y == -1.0) && PyErr_Occurred())
|
|
return NULL;
|
|
/* fmod(x, +/-Inf) returns x for finite x. */
|
|
if (Py_IS_INFINITY(y) && Py_IS_FINITE(x))
|
|
return PyFloat_FromDouble(x);
|
|
errno = 0;
|
|
PyFPE_START_PROTECT("in math_fmod", return 0);
|
|
r = fmod(x, y);
|
|
PyFPE_END_PROTECT(r);
|
|
if (Py_IS_NAN(r)) {
|
|
if (!Py_IS_NAN(x) && !Py_IS_NAN(y))
|
|
errno = EDOM;
|
|
else
|
|
errno = 0;
|
|
}
|
|
if (errno && is_error(r))
|
|
return NULL;
|
|
else
|
|
return PyFloat_FromDouble(r);
|
|
}
|
|
|
|
PyDoc_STRVAR(math_fmod_doc,
|
|
"fmod(x, y)\n\nReturn fmod(x, y), according to platform C."
|
|
" x % y may differ.");
|
|
|
|
static PyObject *
|
|
math_hypot(PyObject *self, PyObject *args)
|
|
{
|
|
PyObject *ox, *oy;
|
|
double r, x, y;
|
|
if (! PyArg_UnpackTuple(args, "hypot", 2, 2, &ox, &oy))
|
|
return NULL;
|
|
x = PyFloat_AsDouble(ox);
|
|
y = PyFloat_AsDouble(oy);
|
|
if ((x == -1.0 || y == -1.0) && PyErr_Occurred())
|
|
return NULL;
|
|
/* hypot(x, +/-Inf) returns Inf, even if x is a NaN. */
|
|
if (Py_IS_INFINITY(x))
|
|
return PyFloat_FromDouble(fabs(x));
|
|
if (Py_IS_INFINITY(y))
|
|
return PyFloat_FromDouble(fabs(y));
|
|
errno = 0;
|
|
PyFPE_START_PROTECT("in math_hypot", return 0);
|
|
r = hypot(x, y);
|
|
PyFPE_END_PROTECT(r);
|
|
if (Py_IS_NAN(r)) {
|
|
if (!Py_IS_NAN(x) && !Py_IS_NAN(y))
|
|
errno = EDOM;
|
|
else
|
|
errno = 0;
|
|
}
|
|
else if (Py_IS_INFINITY(r)) {
|
|
if (Py_IS_FINITE(x) && Py_IS_FINITE(y))
|
|
errno = ERANGE;
|
|
else
|
|
errno = 0;
|
|
}
|
|
if (errno && is_error(r))
|
|
return NULL;
|
|
else
|
|
return PyFloat_FromDouble(r);
|
|
}
|
|
|
|
PyDoc_STRVAR(math_hypot_doc,
|
|
"hypot(x, y)\n\nReturn the Euclidean distance, sqrt(x*x + y*y).");
|
|
|
|
/* pow can't use math_2, but needs its own wrapper: the problem is
|
|
that an infinite result can arise either as a result of overflow
|
|
(in which case OverflowError should be raised) or as a result of
|
|
e.g. 0.**-5. (for which ValueError needs to be raised.)
|
|
*/
|
|
|
|
static PyObject *
|
|
math_pow(PyObject *self, PyObject *args)
|
|
{
|
|
PyObject *ox, *oy;
|
|
double r, x, y;
|
|
int odd_y;
|
|
|
|
if (! PyArg_UnpackTuple(args, "pow", 2, 2, &ox, &oy))
|
|
return NULL;
|
|
x = PyFloat_AsDouble(ox);
|
|
y = PyFloat_AsDouble(oy);
|
|
if ((x == -1.0 || y == -1.0) && PyErr_Occurred())
|
|
return NULL;
|
|
|
|
/* deal directly with IEEE specials, to cope with problems on various
|
|
platforms whose semantics don't exactly match C99 */
|
|
r = 0.; /* silence compiler warning */
|
|
if (!Py_IS_FINITE(x) || !Py_IS_FINITE(y)) {
|
|
errno = 0;
|
|
if (Py_IS_NAN(x))
|
|
r = y == 0. ? 1. : x; /* NaN**0 = 1 */
|
|
else if (Py_IS_NAN(y))
|
|
r = x == 1. ? 1. : y; /* 1**NaN = 1 */
|
|
else if (Py_IS_INFINITY(x)) {
|
|
odd_y = Py_IS_FINITE(y) && fmod(fabs(y), 2.0) == 1.0;
|
|
if (y > 0.)
|
|
r = odd_y ? x : fabs(x);
|
|
else if (y == 0.)
|
|
r = 1.;
|
|
else /* y < 0. */
|
|
r = odd_y ? copysign(0., x) : 0.;
|
|
}
|
|
else if (Py_IS_INFINITY(y)) {
|
|
if (fabs(x) == 1.0)
|
|
r = 1.;
|
|
else if (y > 0. && fabs(x) > 1.0)
|
|
r = y;
|
|
else if (y < 0. && fabs(x) < 1.0) {
|
|
r = -y; /* result is +inf */
|
|
if (x == 0.) /* 0**-inf: divide-by-zero */
|
|
errno = EDOM;
|
|
}
|
|
else
|
|
r = 0.;
|
|
}
|
|
}
|
|
else {
|
|
/* let libm handle finite**finite */
|
|
errno = 0;
|
|
PyFPE_START_PROTECT("in math_pow", return 0);
|
|
r = pow(x, y);
|
|
PyFPE_END_PROTECT(r);
|
|
/* a NaN result should arise only from (-ve)**(finite
|
|
non-integer); in this case we want to raise ValueError. */
|
|
if (!Py_IS_FINITE(r)) {
|
|
if (Py_IS_NAN(r)) {
|
|
errno = EDOM;
|
|
}
|
|
/*
|
|
an infinite result here arises either from:
|
|
(A) (+/-0.)**negative (-> divide-by-zero)
|
|
(B) overflow of x**y with x and y finite
|
|
*/
|
|
else if (Py_IS_INFINITY(r)) {
|
|
if (x == 0.)
|
|
errno = EDOM;
|
|
else
|
|
errno = ERANGE;
|
|
}
|
|
}
|
|
}
|
|
|
|
if (errno && is_error(r))
|
|
return NULL;
|
|
else
|
|
return PyFloat_FromDouble(r);
|
|
}
|
|
|
|
PyDoc_STRVAR(math_pow_doc,
|
|
"pow(x, y)\n\nReturn x**y (x to the power of y).");
|
|
|
|
static const double degToRad = Py_MATH_PI / 180.0;
|
|
static const double radToDeg = 180.0 / Py_MATH_PI;
|
|
|
|
static PyObject *
|
|
math_degrees(PyObject *self, PyObject *arg)
|
|
{
|
|
double x = PyFloat_AsDouble(arg);
|
|
if (x == -1.0 && PyErr_Occurred())
|
|
return NULL;
|
|
return PyFloat_FromDouble(x * radToDeg);
|
|
}
|
|
|
|
PyDoc_STRVAR(math_degrees_doc,
|
|
"degrees(x)\n\n\
|
|
Convert angle x from radians to degrees.");
|
|
|
|
static PyObject *
|
|
math_radians(PyObject *self, PyObject *arg)
|
|
{
|
|
double x = PyFloat_AsDouble(arg);
|
|
if (x == -1.0 && PyErr_Occurred())
|
|
return NULL;
|
|
return PyFloat_FromDouble(x * degToRad);
|
|
}
|
|
|
|
PyDoc_STRVAR(math_radians_doc,
|
|
"radians(x)\n\n\
|
|
Convert angle x from degrees to radians.");
|
|
|
|
static PyObject *
|
|
math_isnan(PyObject *self, PyObject *arg)
|
|
{
|
|
double x = PyFloat_AsDouble(arg);
|
|
if (x == -1.0 && PyErr_Occurred())
|
|
return NULL;
|
|
return PyBool_FromLong((long)Py_IS_NAN(x));
|
|
}
|
|
|
|
PyDoc_STRVAR(math_isnan_doc,
|
|
"isnan(x) -> bool\n\n\
|
|
Check if float x is not a number (NaN).");
|
|
|
|
static PyObject *
|
|
math_isinf(PyObject *self, PyObject *arg)
|
|
{
|
|
double x = PyFloat_AsDouble(arg);
|
|
if (x == -1.0 && PyErr_Occurred())
|
|
return NULL;
|
|
return PyBool_FromLong((long)Py_IS_INFINITY(x));
|
|
}
|
|
|
|
PyDoc_STRVAR(math_isinf_doc,
|
|
"isinf(x) -> bool\n\n\
|
|
Check if float x is infinite (positive or negative).");
|
|
|
|
static PyMethodDef math_methods[] = {
|
|
{"acos", math_acos, METH_O, math_acos_doc},
|
|
{"acosh", math_acosh, METH_O, math_acosh_doc},
|
|
{"asin", math_asin, METH_O, math_asin_doc},
|
|
{"asinh", math_asinh, METH_O, math_asinh_doc},
|
|
{"atan", math_atan, METH_O, math_atan_doc},
|
|
{"atan2", math_atan2, METH_VARARGS, math_atan2_doc},
|
|
{"atanh", math_atanh, METH_O, math_atanh_doc},
|
|
{"ceil", math_ceil, METH_O, math_ceil_doc},
|
|
{"copysign", math_copysign, METH_VARARGS, math_copysign_doc},
|
|
{"cos", math_cos, METH_O, math_cos_doc},
|
|
{"cosh", math_cosh, METH_O, math_cosh_doc},
|
|
{"degrees", math_degrees, METH_O, math_degrees_doc},
|
|
{"erf", math_erf, METH_O, math_erf_doc},
|
|
{"erfc", math_erfc, METH_O, math_erfc_doc},
|
|
{"exp", math_exp, METH_O, math_exp_doc},
|
|
{"expm1", math_expm1, METH_O, math_expm1_doc},
|
|
{"fabs", math_fabs, METH_O, math_fabs_doc},
|
|
{"factorial", math_factorial, METH_O, math_factorial_doc},
|
|
{"floor", math_floor, METH_O, math_floor_doc},
|
|
{"fmod", math_fmod, METH_VARARGS, math_fmod_doc},
|
|
{"frexp", math_frexp, METH_O, math_frexp_doc},
|
|
{"fsum", math_fsum, METH_O, math_fsum_doc},
|
|
{"gamma", math_gamma, METH_O, math_gamma_doc},
|
|
{"hypot", math_hypot, METH_VARARGS, math_hypot_doc},
|
|
{"isinf", math_isinf, METH_O, math_isinf_doc},
|
|
{"isnan", math_isnan, METH_O, math_isnan_doc},
|
|
{"ldexp", math_ldexp, METH_VARARGS, math_ldexp_doc},
|
|
{"lgamma", math_lgamma, METH_O, math_lgamma_doc},
|
|
{"log", math_log, METH_VARARGS, math_log_doc},
|
|
{"log1p", math_log1p, METH_O, math_log1p_doc},
|
|
{"log10", math_log10, METH_O, math_log10_doc},
|
|
{"modf", math_modf, METH_O, math_modf_doc},
|
|
{"pow", math_pow, METH_VARARGS, math_pow_doc},
|
|
{"radians", math_radians, METH_O, math_radians_doc},
|
|
{"sin", math_sin, METH_O, math_sin_doc},
|
|
{"sinh", math_sinh, METH_O, math_sinh_doc},
|
|
{"sqrt", math_sqrt, METH_O, math_sqrt_doc},
|
|
{"tan", math_tan, METH_O, math_tan_doc},
|
|
{"tanh", math_tanh, METH_O, math_tanh_doc},
|
|
{"trunc", math_trunc, METH_O, math_trunc_doc},
|
|
{NULL, NULL} /* sentinel */
|
|
};
|
|
|
|
|
|
PyDoc_STRVAR(module_doc,
|
|
"This module is always available. It provides access to the\n"
|
|
"mathematical functions defined by the C standard.");
|
|
|
|
PyMODINIT_FUNC
|
|
initmath(void)
|
|
{
|
|
PyObject *m;
|
|
|
|
m = Py_InitModule3("math", math_methods, module_doc);
|
|
if (m == NULL)
|
|
goto finally;
|
|
|
|
PyModule_AddObject(m, "pi", PyFloat_FromDouble(Py_MATH_PI));
|
|
PyModule_AddObject(m, "e", PyFloat_FromDouble(Py_MATH_E));
|
|
|
|
finally:
|
|
return;
|
|
}
|