/* 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"

#ifdef _OSF_SOURCE

/* OSF1 5.1 doesn't make this available with XOPEN_SOURCE_EXTENDED defined */

extern double copysign(double, double);

#endif

/*

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 sqrtpi = 1.772453850905516027298167483341145182798;

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:

assert(0); /* should never get here */

r = -1.23e200; /* silence gcc warning */

}

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;

}

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;

return 1.0/x; /* tgamma(+-0.0) = +-inf, divide-by-zero */

}

/* 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, 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 */

if (x > 0.0) {

/* 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(x)) - lanczos_g +

(x-0.5)*(log(x+lanczos_g-0.5)-1);

}

else {

r = log(pi) - log(fabs(sinpi(absx))) - log(absx) -

(log(lanczos_sum(absx)) - lanczos_g +

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

}

if (Py_IS_INFINITY(r))

errno = ERANGE;

return r;

}

/*

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

function erfc(x).

Method: following 'Numerical Recipes' by Flannery, Press et. al. (2nd ed.,

Cambridge University Press), 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;

}

/* 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 inverse hyperbolic cosine 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 inverse hyperbolic sine 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 inverse hyperbolic tangent 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;