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/* 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) */
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
;
}
/* 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.
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
*
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