/* 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"
/*
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 const double logpi = 1.144729885849400174143427351353058711647;
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 = 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 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 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 */
}
}
/*
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 */
}
}
static PyObject *
math_gcd(PyObject *self, PyObject *args)
{
PyObject *a, *b, *g;
if (!PyArg_ParseTuple(args, "OO:gcd", &a, &b))
return NULL;
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;
}
PyDoc_STRVAR(math_gcd_doc,
"gcd(x, y) -> int\n\
greatest common divisor of x and y");
/* 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_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.
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_1(PyObject *arg, double (*func) (double), int can_overflow)
{
return math_1_to_whatever(arg, func, PyFloat_FromDouble, can_overflow);
}
static PyObject *
math_1_to_int(PyObject *arg, double (*func) (double), int can_overflow)
{
return math_1_to_whatever(arg, func, PyLong_FromDouble, can_overflow);
}
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.")
static PyObject * math_ceil(PyObject *self, PyObject *number) {
_Py_IDENTIFIER(__ceil__);
PyObject *method, *result;
method = _PyObject_LookupSpecial(number, &PyId___ceil__);
if (method == NULL) {
if (PyErr_Occurred())
return NULL;
return math_1_to_int(number, ceil, 0);
}
result = PyObject_CallFunctionObjArgs(method, NULL);
Py_DECREF(method);
return result;
}
PyDoc_STRVAR(math_ceil_doc,
"ceil(x)\n\nReturn the ceiling of x as an Integral.\n"
"This is the smallest integer >= x.");
FUNC2(copysign, copysign,
"copysign(x, y)\n\nReturn a float with the magnitude (absolute value) "
"of x but the sign \nof y. On platforms that support signed zeros, "
"copysign(1.0, -0.0) \nreturns -1.0.\n")
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.")
static PyObject * math_floor(PyObject *self, PyObject *number) {
_Py_IDENTIFIER(__floor__);
PyObject *method, *result;
method = _PyObject_LookupSpecial(number, &PyId___floor__);
if (method == NULL) {
if (PyErr_Occurred())
return NULL;
return math_1_to_int(number, floor, 0);
}
result = PyObject_CallFunctionObjArgs(method, NULL);
Py_DECREF(method);
return result;
}
PyDoc_STRVAR(math_floor_doc,
"floor(x)\n\nReturn the floor of x as an Integral.\n"
"This is the largest integer Integral\n"
"\n"
"Find x!. Raise a ValueError if x is negative or non-integral.");
static PyObject *
math_trunc(PyObject *self, PyObject *number)
{
_Py_IDENTIFIER(__trunc__);
PyObject *trunc, *result;
if (Py_TYPE(number)->tp_dict == NULL) {
if (PyType_Ready(Py_TYPE(number)) < 0)
return NULL;
}
trunc = _PyObject_LookupSpecial(number, &PyId___trunc__);
if (trunc == NULL) {
if (!PyErr_Occurred())
PyErr_Format(PyExc_TypeError,
"type %.100s doesn't define __trunc__ method",
Py_TYPE(number)->tp_name);
return NULL;
}
result = PyObject_CallFunctionObjArgs(trunc, NULL);
Py_DECREF(trunc);
return result;
}
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 bool\n\n\
Return True if x is neither an infinity nor a NaN, and False otherwise.");
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\
Return True if x is a NaN (not a number), and False otherwise.");
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\
Return True if x is a positive or negative infinity, and False otherwise.");
static PyObject *
math_isclose(PyObject *self, PyObject *args, PyObject *kwargs)
{
double a, b;
double rel_tol = 1e-9;
double abs_tol = 0.0;
double diff = 0.0;
long result = 0;
static char *keywords[] = {"a", "b", "rel_tol", "abs_tol", NULL};
if (!PyArg_ParseTupleAndKeywords(args, kwargs, "dd|$dd:isclose",
keywords,
&a, &b, &rel_tol, &abs_tol
))
return NULL;
/* sanity check on the inputs */
if (rel_tol < 0.0 || abs_tol < 0.0 ) {
PyErr_SetString(PyExc_ValueError,
"tolerances must be non-negative");
return NULL;
}
if ( a == b ) {
/* short circuit exact equality -- needed to catch two infinities of
the same sign. And perhaps speeds things up a bit sometimes.
*/
Py_RETURN_TRUE;
}
/* This catches the case of two infinities of opposite sign, or
one infinity and one finite number. Two infinities of opposite
sign would otherwise have an infinite relative tolerance.
Two infinities of the same sign are caught by the equality check
above.
*/
if (Py_IS_INFINITY(a) || Py_IS_INFINITY(b)) {
Py_RETURN_FALSE;
}
/* now do the regular computation
this is essentially the "weak" test from the Boost library
*/
diff = fabs(b - a);
result = (((diff