/* Math module -- standard C math library functions, pi and e */

/* Here are some comments from Tim Peters, extracted from the

discussion attached to http://bugs.python.org/issue1640. They

describe the general aims of the math module with respect to

special values, IEEE-754 floating-point exceptions, and Python

exceptions.

These are the "spirit of 754" rules:

1. If the mathematical result is a real number, but of magnitude too

large to approximate by a machine float, overflow is signaled and the

result is an infinity (with the appropriate sign).

2. If the mathematical result is a real number, but of magnitude too

small to approximate by a machine float, underflow is signaled and the

result is a zero (with the appropriate sign).

3. At a singularity (a value x such that the limit of f(y) as y

approaches x exists and is an infinity), "divide by zero" is signaled

and the result is an infinity (with the appropriate sign). This is

complicated a little by that the left-side and right-side limits may

not be the same; e.g., 1/x approaches +inf or -inf as x approaches 0

from the positive or negative directions. In that specific case, the

sign of the zero determines the result of 1/0.

4. At a point where a function has no defined result in the extended

reals (i.e., the reals plus an infinity or two), invalid operation is

signaled and a NaN is returned.

And these are what Python has historically /tried/ to do (but not

always successfully, as platform libm behavior varies a lot):

For #1, raise OverflowError.

For #2, return a zero (with the appropriate sign if that happens by

accident ;-)).

For #3 and #4, raise ValueError. It may have made sense to raise

Python's ZeroDivisionError in #3, but historically that's only been

raised for division by zero and mod by zero.

*/

/*

In general, on an IEEE-754 platform the aim is to follow the C99

standard, including Annex 'F', whenever possible. Where the

standard recommends raising the 'divide-by-zero' or 'invalid'

floating-point exceptions, Python should raise a ValueError. Where

the standard recommends raising 'overflow', Python should raise an

OverflowError. In all other circumstances a value should be

returned.

*/

#include "Python.h"

#include "_math.h"

#include "clinic/mathmodule.c.h"

/*[clinic input]

module math

[clinic start generated code]*/

/*[clinic end generated code: output=da39a3ee5e6b4b0d input=76bc7002685dd942]*/

/*

sin(pi*x), giving accurate results for all finite x (especially x

integral or close to an integer). This is here for use in the

reflection formula for the gamma function. It conforms to IEEE

754-2008 for finite arguments, but not for infinities or nans.

*/

static const double pi = 3.141592653589793238462643383279502884197;

static const double logpi = 1.144729885849400174143427351353058711647;

#if !defined(HAVE_ERF) || !defined(HAVE_ERFC)

static const double sqrtpi = 1.772453850905516027298167483341145182798;

#endif /* !defined(HAVE_ERF) || !defined(HAVE_ERFC) */

/* Version of PyFloat_AsDouble() with in-line fast paths

for exact floats and integers. Gives a substantial

speed improvement for extracting float arguments.

*/

#define ASSIGN_DOUBLE(target_var, obj, error_label) \

if (PyFloat_CheckExact(obj)) { \

target_var = PyFloat_AS_DOUBLE(obj); \

} \

else if (PyLong_CheckExact(obj)) { \

target_var = PyLong_AsDouble(obj); \

if (target_var == -1.0 && PyErr_Occurred()) { \

goto error_label; \

} \

} \

else { \

target_var = PyFloat_AsDouble(obj); \

if (target_var == -1.0 && PyErr_Occurred()) { \

goto error_label; \

} \

}

static double

sinpi(double x)

{

double y, r;

int n;

/* this function should only ever be called for finite arguments */

assert(Py_IS_FINITE(x));

y = fmod(fabs(x), 2.0);

n = (int)round(2.0*y);

assert(0 <= n && n <= 4);

switch (n) {

case 0:

r = sin(pi*y);

break;

case 1:

r = cos(pi*(y-0.5));

break;

case 2:

/* N.B. -sin(pi*(y-1.0)) is *not* equivalent: it would give

-0.0 instead of 0.0 when y == 1.0. */

r = sin(pi*(1.0-y));

break;

case 3:

r = -cos(pi*(y-1.5));

break;

case 4:

r = sin(pi*(y-2.0));

break;

default:

Py_UNREACHABLE();

}

return copysign(1.0, x)*r;

}

/* Implementation of the real gamma function. In extensive but non-exhaustive

random tests, this function proved accurate to within <= 10 ulps across the

entire float domain. Note that accuracy may depend on the quality of the

system math functions, the pow function in particular. Special cases

follow C99 annex F. The parameters and method are tailored to platforms

whose double format is the IEEE 754 binary64 format.

Method: for x > 0.0 we use the Lanczos approximation with parameters N=13

and g=6.024680040776729583740234375; these parameters are amongst those

used by the Boost library. Following Boost (again), we re-express the

Lanczos sum as a rational function, and compute it that way. The

coefficients below were computed independently using MPFR, and have been

double-checked against the coefficients in the Boost source code.

For x < 0.0 we use the reflection formula.

There's one minor tweak that deserves explanation: Lanczos' formula for

Gamma(x) involves computing pow(x+g-0.5, x-0.5) / exp(x+g-0.5). For many x

values, x+g-0.5 can be represented exactly. However, in cases where it

can't be represented exactly the small error in x+g-0.5 can be magnified

significantly by the pow and exp calls, especially for large x. A cheap

correction is to multiply by (1 + e*g/(x+g-0.5)), where e is the error

involved in the computation of x+g-0.5 (that is, e = computed value of

x+g-0.5 - exact value of x+g-0.5). Here's the proof:

Correction factor

-----------------

Write x+g-0.5 = y-e, where y is exactly representable as an IEEE 754

double, and e is tiny. Then:

pow(x+g-0.5,x-0.5)/exp(x+g-0.5) = pow(y-e, x-0.5)/exp(y-e)

= pow(y, x-0.5)/exp(y) * C,

where the correction_factor C is given by

C = pow(1-e/y, x-0.5) * exp(e)

Since e is tiny, pow(1-e/y, x-0.5) ~ 1-(x-0.5)*e/y, and exp(x) ~ 1+e, so:

C ~ (1-(x-0.5)*e/y) * (1+e) ~ 1 + e*(y-(x-0.5))/y

But y-(x-0.5) = g+e, and g+e ~ g. So we get C ~ 1 + e*g/y, and

pow(x+g-0.5,x-0.5)/exp(x+g-0.5) ~ pow(y, x-0.5)/exp(y) * (1 + e*g/y),

Note that for accuracy, when computing r*C it's better to do

r + e*g/y*r;

than

r * (1 + e*g/y);

since the addition in the latter throws away most of the bits of

information in e*g/y.

*/

#define LANCZOS_N 13

static const double lanczos_g = 6.024680040776729583740234375;

static const double lanczos_g_minus_half = 5.524680040776729583740234375;

static const double lanczos_num_coeffs[LANCZOS_N] = {

23531376880.410759688572007674451636754734846804940,

42919803642.649098768957899047001988850926355848959,

35711959237.355668049440185451547166705960488635843,

17921034426.037209699919755754458931112671403265390,

6039542586.3520280050642916443072979210699388420708,

1439720407.3117216736632230727949123939715485786772,

248874557.86205415651146038641322942321632125127801,

31426415.585400194380614231628318205362874684987640,

2876370.6289353724412254090516208496135991145378768,

186056.26539522349504029498971604569928220784236328,

8071.6720023658162106380029022722506138218516325024,

210.82427775157934587250973392071336271166969580291,

2.5066282746310002701649081771338373386264310793408

};

/* denominator is x*(x+1)*...*(x+LANCZOS_N-2) */

static const double lanczos_den_coeffs[LANCZOS_N] = {

0.0, 39916800.0, 120543840.0, 150917976.0, 105258076.0, 45995730.0,

13339535.0, 2637558.0, 357423.0, 32670.0, 1925.0, 66.0, 1.0};

/* gamma values for small positive integers, 1 though NGAMMA_INTEGRAL */

#define NGAMMA_INTEGRAL 23

static const double gamma_integral[NGAMMA_INTEGRAL] = {

1.0, 1.0, 2.0, 6.0, 24.0, 120.0, 720.0, 5040.0, 40320.0, 362880.0,

3628800.0, 39916800.0, 479001600.0, 6227020800.0, 87178291200.0,

1307674368000.0, 20922789888000.0, 355687428096000.0,

6402373705728000.0, 121645100408832000.0, 2432902008176640000.0,

51090942171709440000.0, 1124000727777607680000.0,

};

/* Lanczos' sum L_g(x), for positive x */

static double

lanczos_sum(double x)

{

double num = 0.0, den = 0.0;

int i;

assert(x > 0.0);

/* evaluate the rational function lanczos_sum(x). For large

x, the obvious algorithm risks overflow, so we instead

rescale the denominator and numerator of the rational

function by x**(1-LANCZOS_N) and treat this as a

rational function in 1/x. This also reduces the error for

larger x values. The choice of cutoff point (5.0 below) is

somewhat arbitrary; in tests, smaller cutoff values than

this resulted in lower accuracy. */

if (x < 5.0) {

for (i = LANCZOS_N; --i >= 0; ) {

num = num * x + lanczos_num_coeffs[i];

den = den * x + lanczos_den_coeffs[i];

}

}

else {

for (i = 0; i < LANCZOS_N; i++) {

num = num / x + lanczos_num_coeffs[i];

den = den / x + lanczos_den_coeffs[i];

}

}

return num/den;

}

/* Constant for +infinity, generated in the same way as float('inf'). */

static double

m_inf(void)

{

#ifndef PY_NO_SHORT_FLOAT_REPR

return _Py_dg_infinity(0);

#else

return Py_HUGE_VAL;

#endif

}

/* Constant nan value, generated in the same way as float('nan'). */

/* We don't currently assume that Py_NAN is defined everywhere. */

#if !defined(PY_NO_SHORT_FLOAT_REPR) || defined(Py_NAN)

static double

m_nan(void)

{

#ifndef PY_NO_SHORT_FLOAT_REPR

return _Py_dg_stdnan(0);

#else

return Py_NAN;

#endif

}

#endif

static double

m_tgamma(double x)

{

double absx, r, y, z, sqrtpow;

/* special cases */

if (!Py_IS_FINITE(x)) {

if (Py_IS_NAN(x) || x > 0.0)

return x; /* tgamma(nan) = nan, tgamma(inf) = inf */

else {

errno = EDOM;

return Py_NAN; /* tgamma(-inf) = nan, invalid */

}

}

if (x == 0.0) {

errno = EDOM;

/* tgamma(+-0.0) = +-inf, divide-by-zero */

return copysign(Py_HUGE_VAL, x);

}

/* integer arguments */

if (x == floor(x)) {

if (x < 0.0) {

errno = EDOM; /* tgamma(n) = nan, invalid for */

return Py_NAN; /* negative integers n */

}

if (x <= NGAMMA_INTEGRAL)

return gamma_integral[(int)x - 1];

}

absx = fabs(x);

/* tiny arguments: tgamma(x) ~ 1/x for x near 0 */

if (absx < 1e-20) {

r = 1.0/x;

if (Py_IS_INFINITY(r))

errno = ERANGE;

return r;

}

/* large arguments: assuming IEEE 754 doubles, tgamma(x) overflows for

x > 200, and underflows to +-0.0 for x < -200, not a negative

integer. */

if (absx > 200.0) {

if (x < 0.0) {

return 0.0/sinpi(x);

}

else {

errno = ERANGE;

return Py_HUGE_VAL;

}

}

y = absx + lanczos_g_minus_half;

/* compute error in sum */

if (absx > lanczos_g_minus_half) {

/* note: the correction can be foiled by an optimizing

compiler that (incorrectly) thinks that an expression like

a + b - a - b can be optimized to 0.0. This shouldn't

happen in a standards-conforming compiler. */

double q = y - absx;

z = q - lanczos_g_minus_half;

}

else {

double q = y - lanczos_g_minus_half;

z = q - absx;

}

z = z * lanczos_g / y;

if (x < 0.0) {

r = -pi / sinpi(absx) / absx * exp(y) / lanczos_sum(absx);

r -= z * r;

if (absx < 140.0) {

r /= pow(y, absx - 0.5);

}

else {

sqrtpow = pow(y, absx / 2.0 - 0.25);

r /= sqrtpow;

r /= sqrtpow;

}

}

else {

r = lanczos_sum(absx) / exp(y);

r += z * r;

if (absx < 140.0) {

r *= pow(y, absx - 0.5);

}

else {

sqrtpow = pow(y, absx / 2.0 - 0.25);

r *= sqrtpow;

r *= sqrtpow;

}

}

if (Py_IS_INFINITY(r))

errno = ERANGE;

return r;

}

/*

lgamma: natural log of the absolute value of the Gamma function.

For large arguments, Lanczos' formula works extremely well here.

*/

static double

m_lgamma(double x)

{

double r;

double absx;

/* special cases */

if (!Py_IS_FINITE(x)) {

if (Py_IS_NAN(x))

return x; /* lgamma(nan) = nan */

else

return Py_HUGE_VAL; /* lgamma(+-inf) = +inf */

}

/* integer arguments */

if (x == floor(x) && x <= 2.0) {

if (x <= 0.0) {

errno = EDOM; /* lgamma(n) = inf, divide-by-zero for */

return Py_HUGE_VAL; /* integers n <= 0 */

}

else {

return 0.0; /* lgamma(1) = lgamma(2) = 0.0 */

}

}

absx = fabs(x);

/* tiny arguments: lgamma(x) ~ -log(fabs(x)) for small x */

if (absx < 1e-20)

return -log(absx);

/* Lanczos' formula. We could save a fraction of a ulp in accuracy by

having a second set of numerator coefficients for lanczos_sum that

absorbed the exp(-lanczos_g) term, and throwing out the lanczos_g

subtraction below; it's probably not worth it. */

r = log(lanczos_sum(absx)) - lanczos_g;

r += (absx - 0.5) * (log(absx + lanczos_g - 0.5) - 1);

if (x < 0.0)

/* Use reflection formula to get value for negative x. */

r = logpi - log(fabs(sinpi(absx))) - log(absx) - r;

if (Py_IS_INFINITY(r))

errno = ERANGE;

return r;

}

#if !defined(HAVE_ERF) || !defined(HAVE_ERFC)

/*

Implementations of the error function erf(x) and the complementary error

function erfc(x).

Method: we use a series approximation for erf for small x, and a continued

fraction approximation for erfc(x) for larger x;

combined with the relations erf(-x) = -erf(x) and erfc(x) = 1.0 - erf(x),

this gives us erf(x) and erfc(x) for all x.

The series expansion used is:

erf(x) = x*exp(-x*x)/sqrt(pi) * [

2/1 + 4/3 x**2 + 8/15 x**4 + 16/105 x**6 + ...]

The coefficient of x**(2k-2) here is 4**k*factorial(k)/factorial(2*k).

This series converges well for smallish x, but slowly for larger x.

The continued fraction expansion used is:

erfc(x) = x*exp(-x*x)/sqrt(pi) * [1/(0.5 + x**2 -) 0.5/(2.5 + x**2 - )

3.0/(4.5 + x**2 - ) 7.5/(6.5 + x**2 - ) ...]

after the first term, the general term has the form:

k*(k-0.5)/(2*k+0.5 + x**2 - ...).

This expansion converges fast for larger x, but convergence becomes

infinitely slow as x approaches 0.0. The (somewhat naive) continued

fraction evaluation algorithm used below also risks overflow for large x;

but for large x, erfc(x) == 0.0 to within machine precision. (For

example, erfc(30.0) is approximately 2.56e-393).

Parameters: use series expansion for abs(x) < ERF_SERIES_CUTOFF and

continued fraction expansion for ERF_SERIES_CUTOFF <= abs(x) <

ERFC_CONTFRAC_CUTOFF. ERFC_SERIES_TERMS and ERFC_CONTFRAC_TERMS are the

numbers of terms to use for the relevant expansions. */

#define ERF_SERIES_CUTOFF 1.5

#define ERF_SERIES_TERMS 25

#define ERFC_CONTFRAC_CUTOFF 30.0

#define ERFC_CONTFRAC_TERMS 50

/*

Error function, via power series.

Given a finite float x, return an approximation to erf(x).

Converges reasonably fast for small x.

*/

static double

m_erf_series(double x)

{

double x2, acc, fk, result;

int i, saved_errno;

x2 = x * x;

acc = 0.0;

fk = (double)ERF_SERIES_TERMS + 0.5;

for (i = 0; i < ERF_SERIES_TERMS; i++) {

acc = 2.0 + x2 * acc / fk;

fk -= 1.0;

}

/* Make sure the exp call doesn't affect errno;

see m_erfc_contfrac for more. */

saved_errno = errno;

result = acc * x * exp(-x2) / sqrtpi;

errno = saved_errno;

return result;

}

/*

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;

}

#endif /* !defined(HAVE_ERF) || !defined(HAVE_ERFC) */

/* Error function erf(x), for general x */

static double

m_erf(double x)

{

#ifdef HAVE_ERF

return erf(x);

#else

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;

}

#endif

}

/* Complementary error function erfc(x), for general x. */

static double

m_erfc(double x)

{

#ifdef HAVE_ERFC

return erfc(x);

#else

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;

}

#endif

}

/*

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);

}

/* IEEE 754-style remainder operation: x - n*y where n*y is the nearest

multiple of y to x, taking n even in the case of a tie. Assuming an IEEE 754

binary floating-point format, the result is always exact. */

static double

m_remainder(double x, double y)

{

/* Deal with most common case first. */

if (Py_IS_FINITE(x) && Py_IS_FINITE(y)) {

double absx, absy, c, m, r;

if (y == 0.0) {

return Py_NAN;

}

absx = fabs(x);

absy = fabs(y);

m = fmod(absx, absy);

/*

Warning: some subtlety here. What we *want* to know at this point is

whether the remainder m is less than, equal to, or greater than half

of absy. However, we can't do that comparison directly because we

can't be sure that 0.5*absy is representable (the mutiplication

might incur precision loss due to underflow). So instead we compare

m with the complement c = absy - m: m < 0.5*absy if and only if m <

c, and so on. The catch is that absy - m might also not be

representable, but it turns out that it doesn't matter:

- if m > 0.5*absy then absy - m is exactly representable, by

Sterbenz's lemma, so m > c

- if m == 0.5*absy then again absy - m is exactly representable

and m == c

- if m < 0.5*absy then either (i) 0.5*absy is exactly representable,

in which case 0.5*absy < absy - m, so 0.5*absy <= c and hence m <

c, or (ii) absy is tiny, either subnormal or in the lowest normal

binade. Then absy - m is exactly representable and again m < c.

*/

c = absy - m;

if (m < c) {

r = m;

}

else if (m > c) {

r = -c;

}

else {

/*

Here absx is exactly halfway between two multiples of absy,

and we need to choose the even multiple. x now has the form

absx = n * absy + m

for some integer n (recalling that m = 0.5*absy at this point).

If n is even we want to return m; if n is odd, we need to

return -m.

So

0.5 * (absx - m) = (n/2) * absy

and now reducing modulo absy gives us:

| m, if n is odd

fmod(0.5 * (absx - m), absy) = |

| 0, if n is even

Now m - 2.0 * fmod(...) gives the desired result: m

if n is even, -m if m is odd.

Note that all steps in fmod(0.5 * (absx - m), absy)

will be computed exactly, with no rounding error

introduced.

*/

assert(m == c);

r = m - 2.0 * fmod(0.5 * (absx - m), absy);

}

return copysign(1.0, x) * r;

}

/* Special values. */

if (Py_IS_NAN(x)) {

return x;

}

if (Py_IS_NAN(y)) {

return y;

}

if (Py_IS_INFINITY(x)) {

return Py_NAN;

}

assert(Py_IS_INFINITY(y));

return 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 */

}

}

/*

log2: log to base 2.

Uses an algorithm that should:

(a) produce exact results for powers of 2, and

(b) give a monotonic log2 (for positive finite floats),

assuming that the system log is monotonic.

*/

static double

m_log2(double x)

{

if (!Py_IS_FINITE(x)) {

if (Py_IS_NAN(x))

return x; /* log2(nan) = nan */

else if (x > 0.0)

return x; /* log2(+inf) = +inf */

else {

errno = EDOM;

return Py_NAN; /* log2(-inf) = nan, invalid-operation */

}

}

if (x > 0.0) {

#ifdef HAVE_LOG2

return log2(x);

#else

double m;

int e;

m = frexp(x, &e);

/* We want log2(m * 2**e) == log(m) / log(2) + e. Care is needed when

* x is just greater than 1.0: in that case e is 1, log(m) is negative,

* and we get significant cancellation error from the addition of

* log(m) / log(2) to e. The slight rewrite of the expression below

* avoids this problem.

*/

if (x >= 1.0) {

return log(2.0 * m) / log(2.0) + (e - 1);

}

else {

return log(m) / log(2.0) + e;

}

#endif

}

else if (x == 0.0) {

errno = EDOM;

return -Py_HUGE_VAL; /* log2(0) = -inf, divide-by-zero */

}

else {

errno = EDOM;

return Py_NAN; /* log2(-inf) = nan, invalid-operation */

}

}

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 */

}

}

/*[clinic input]

math.gcd

x as a: object

y as b: object

/

greatest common divisor of x and y

[clinic start generated code]*/

static PyObject *

math_gcd_impl(PyObject *module, PyObject *a, PyObject *b)

/*[clinic end generated code: output=7b2e0c151bd7a5d8 input=c2691e57fb2a98fa]*/

{

PyObject *g;

a = PyNumber_Index(a);

if (a == NULL)

return NULL;

b = PyNumber_Index(b);

if (b == NULL) {

Py_DECREF(a);

return NULL;

}

g = _PyLong_GCD(a, b);

Py_DECREF(a);

Py_DECREF(b);

return g;

}

/* 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

argument 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_to_whatever(PyObject *arg, double (*func) (double),

PyObject *(*from_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) && !Py_IS_NAN(x)) {

PyErr_SetString(PyExc_ValueError,

"math domain error"); /* invalid arg */

return NULL;

}

if (Py_IS_INFINITY(r) && Py_IS_FINITE(x)) {

if (can_overflow)

PyErr_SetString(PyExc_OverflowError,

"math range error"); /* overflow */

else

PyErr_SetString(PyExc_ValueError,

"math domain error"); /* singularity */

return NULL;

}

if (Py_IS_FINITE(r) && errno && is_error(r))

/* this branch unnecessary on most platforms */

return NULL;

return (*from_double_func)(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.