/* 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
m_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/m_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 / m_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 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) {
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 *
math_2(PyObject *const *args, Py_ssize_t nargs,
double (*func) (double, double), const char *funcname)
{
double x, y, r;
if (!_PyArg_CheckPositional(funcname, nargs, 2, 2))
return NULL;
x = PyFloat_AsDouble(args[0]);
if (x == -1.0 && PyErr_Occurred()) {
return NULL;
}
y = PyFloat_AsDouble(args[1]);
if (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 *const *args, Py_ssize_t nargs) { \
return math_2(args, nargs, func, #funcname); \
}\
PyDoc_STRVAR(math_##funcname##_doc, docstring);
FUNC1(acos, acos, 0,
"acos($module, x, /)\n--\n\n"
"Return the arc cosine (measured in radians) of x.")
FUNC1(acosh, m_acosh, 0,
"acosh($module, x, /)\n--\n\n"
"Return the inverse hyperbolic cosine of x.")
FUNC1(asin, asin, 0,
"asin($module, x, /)\n--\n\n"
"Return the arc sine (measured in radians) of x.")
FUNC1(asinh, m_asinh, 0,
"asinh($module, x, /)\n--\n\n"
"Return the inverse hyperbolic sine of x.")
FUNC1(atan, atan, 0,
"atan($module, x, /)\n--\n\n"
"Return the arc tangent (measured in radians) of x.")
FUNC2(atan2, m_atan2,
"atan2($module, y, x, /)\n--\n\n"
"Return the arc tangent (measured in radians) of y/x.\n\n"
"Unlike atan(y/x), the signs of both x and y are considered.")
FUNC1(atanh, m_atanh, 0,
"atanh($module, x, /)\n--\n\n"
"Return the inverse hyperbolic tangent of x.")
/*[clinic input]
math.ceil
x as number: object
/
Return the ceiling of x as an Integral.
This is the smallest integer >= x.
[clinic start generated code]*/
static PyObject *
math_ceil(PyObject *module, PyObject *number)
/*[clinic end generated code: output=6c3b8a78bc201c67 input=2725352806399cab]*/
{
_Py_IDENTIFIER(__ceil__);
PyObject *method, *result;
method = _PyObject_LookupSpecial(number, &PyId___ceil__);
if (method == NULL) {
if (PyErr_Occurred())
return NULL;
return math_1_to_int(number, ceil, 0);
}
result = _PyObject_CallNoArg(method);
Py_DECREF(method);
return result;
}
FUNC2(copysign, copysign,
"copysign($module, x, y, /)\n--\n\n"
"Return a float with the magnitude (absolute value) of x but the sign of y.\n\n"
"On platforms that support signed zeros, copysign(1.0, -0.0)\n"
"returns -1.0.\n")
FUNC1(cos, cos, 0,
"cos($module, x, /)\n--\n\n"
"Return the cosine of x (measured in radians).")
FUNC1(cosh, cosh, 1,
"cosh($module, x, /)\n--\n\n"
"Return the hyperbolic cosine of x.")
FUNC1A(erf, m_erf,
"erf($module, x, /)\n--\n\n"
"Error function at x.")
FUNC1A(erfc, m_erfc,
"erfc($module, x, /)\n--\n\n"
"Complementary error function at x.")
FUNC1(exp, exp, 1,
"exp($module, x, /)\n--\n\n"
"Return e raised to the power of x.")
FUNC1(expm1, m_expm1, 1,
"expm1($module, x, /)\n--\n\n"
"Return exp(x)-1.\n\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($module, x, /)\n--\n\n"
"Return the absolute value of the float x.")
/*[clinic input]
math.floor
x as number: object
/
Return the floor of x as an Integral.
This is the largest integer >= 1;
}
return len;
}
static unsigned long
count_set_bits(unsigned long n)
{
unsigned long count = 0;
while (n != 0) {
++count;
n &= n - 1; /* clear least significant bit */
}
return count;
}
/* Integer square root
Given a nonnegative integer `n`, we want to compute the largest integer
`a` for which `a * a > 2*(c - d)) < (a+1)**2
e = d
d = c >> s
a = (a > 2*c - e - d + 1) // a
return a - (a*a > n)
Sketch of proof of correctness
------------------------------
The delicate part of the correctness proof is showing that the loop invariant
is preserved from one iteration to the next. That is, just before the line
a = (a > 2*c - e - d + 1) // a
is executed in the above code, we know that
(1) (a - 1)**2 < (n >> 2*(c - e)) < (a + 1)**2.
(since `e` is always the value of `d` from the previous iteration). We must
prove that after that line is executed, we have
(a - 1)**2 < (n >> 2*(c - d)) < (a + 1)**2
To facilitate the proof, we make some changes of notation. Write `m` for
`n >> 2*(c-d)`, and write `b` for the new value of `a`, so
b = (a > 2*c - e - d + 1) // a
or equivalently:
(2) b = (a > d - e + 1) // a
Then we can rewrite (1) as:
(3) (a - 1)**2 < (m >> 2*(d - e)) < (a + 1)**2
and we must show that (b - 1)**2 < m < (b + 1)**2.
From this point on, we switch to mathematical notation, so `/` means exact
division rather than integer division and `^` is used for exponentiation. We
use the `` symbol for the exact square root. In (3), we can remove the
implicit floor operation to give:
(4) (a - 1)^2 < m / 4^(d - e) < (a + 1)^2
Taking square roots throughout (4), scaling by `2^(d-e)`, and rearranging gives
(5) 0 = m;
return PyLong_FromUnsignedLongLong((unsigned long long)u);
}
/* Slow path: n >= 2**64. We perform the first five iterations in C integer
arithmetic, then switch to using Python long integers. */
/* From n >= 2**64 it follows that c.bit_length() >= 6. */
c_bit_length = 6;
while ((c >> c_bit_length) > 0U) {
++c_bit_length;
}
/* Initialise d and a. */
d = c >> (c_bit_length - 5);
b = _PyLong_Rshift(n, 2U*c - 62U);
if (b == NULL) {
goto error;
}
m = (uint64_t)PyLong_AsUnsignedLongLong(b);
Py_DECREF(b);
if (m == (uint64_t)(-1) && PyErr_Occurred()) {
goto error;
}
u = _approximate_isqrt(m) >> (31U - d);
a = PyLong_FromUnsignedLongLong((unsigned long long)u);
if (a == NULL) {
goto error;
}
for (int s = c_bit_length - 6; s >= 0; --s) {
PyObject *q;
size_t e = d;
d = c >> s;
/* q = (n >> 2*c - e - d + 1) // a */
q = _PyLong_Rshift(n, 2U*c - d - e + 1U);
if (q == NULL) {
goto error;
}
Py_SETREF(q, PyNumber_FloorDivide(q, a));
if (q == NULL) {
goto error;
}
/* a = (a start.
* max_bits must be >= bit_length(stop - 2). */
static PyObject *
factorial_partial_product(unsigned long start, unsigned long stop,
unsigned long max_bits)
{
unsigned long midpoint, num_operands;
PyObject *left = NULL, *right = NULL, *result = NULL;
/* If the return value will fit an unsigned long, then we can
* multiply in a tight, fast loop where each multiply is O(1).
* Compute an upper bound on the number of bits required to store
* the answer.
*
* Storing some integer z requires floor(lg(z))+1 bits, which is
* conveniently the value returned by bit_length(z). The
* product x*y will require at most
* bit_length(x) + bit_length(y) bits to store, based
* on the idea that lg product = lg x + lg y.
*
* We know that stop - 2 is the largest number to be multiplied. From
* there, we have: bit_length(answer) i;
if (v = 8
6227020800, 87178291200, 1307674368000,
20922789888000, 355687428096000, 6402373705728000,
121645100408832000, 2432902008176640000
#endif
};
/*[clinic input]
math.factorial
x as arg: object
/
Find x!.
Raise a ValueError if x is negative or non-integral.
[clinic start generated code]*/
static PyObject *
math_factorial(PyObject *module, PyObject *arg)
/*[clinic end generated code: output=6686f26fae00e9ca input=6d1c8105c0d91fb4]*/
{
long x, two_valuation;
int overflow;
PyObject *result, *odd_part, *pyint_form;
if (PyFloat_Check(arg)) {
PyObject *lx;
double dx = PyFloat_AS_DOUBLE((PyFloatObject *)arg);
if (!(Py_IS_FINITE(dx) && dx == floor(dx))) {
PyErr_SetString(PyExc_ValueError,
"factorial() only accepts integral values");
return NULL;
}
lx = PyLong_FromDouble(dx);
if (lx == NULL)
return NULL;
x = PyLong_AsLongAndOverflow(lx, &overflow);
Py_DECREF(lx);
}
else {
pyint_form = PyNumber_Index(arg);
if (pyint_form == NULL) {
return NULL;
}
x = PyLong_AsLongAndOverflow(pyint_form, &overflow);
Py_DECREF(pyint_form);
}
if (x == -1 && PyErr_Occurred()) {
return NULL;
}
else if (overflow == 1) {
PyErr_Format(PyExc_OverflowError,
"factorial() argument should not exceed %ld",
LONG_MAX);
return NULL;
}
else if (overflow == -1 || x < 0) {
PyErr_SetString(PyExc_ValueError,
"factorial() not defined for negative values");
return NULL;
}
/* use lookup table if x is small */
if (x < (long)Py_ARRAY_LENGTH(SmallFactorials))
return PyLong_FromUnsignedLong(SmallFactorials[x]);
/* else express in the form odd_part * 2**two_valuation, and compute as
odd_part tp_dict == NULL) {
if (PyType_Ready(Py_TYPE(x)) < 0)
return NULL;
}
trunc = _PyObject_LookupSpecial(x, &PyId___trunc__);
if (trunc == NULL) {
if (!PyErr_Occurred())
PyErr_Format(PyExc_TypeError,
"type %.100s doesn't define __trunc__ method",
Py_TYPE(x)->tp_name);
return NULL;
}
result = _PyObject_CallNoArg(trunc);
Py_DECREF(trunc);
return result;
}
/*[clinic input]
math.frexp
x: double
/
Return the mantissa and exponent of x, as pair (m, e).
m is a float and e is an int, such that x = m * 2.**e.
If x is 0, m and e are both 0. Else 0.5 INT_MAX) {
/* overflow */
r = copysign(Py_HUGE_VAL, x);
errno = ERANGE;
} else if (exp < INT_MIN) {
/* underflow to +-0 */
r = copysign(0., x);
errno = 0;
} else {
errno = 0;
PyFPE_START_PROTECT("in math_ldexp", return 0);
r = ldexp(x, (int)exp);
PyFPE_END_PROTECT(r);
if (Py_IS_INFINITY(r))
errno = ERANGE;
}
if (errno && is_error(r))
return NULL;
return PyFloat_FromDouble(r);
}
/*[clinic input]
math.modf
x: double
/
Return the fractional and integer parts of x.
Both results carry the sign of x and are floats.
[clinic start generated code]*/
static PyObject *
math_modf_impl(PyObject *module, double x)
/*[clinic end generated code: output=90cee0260014c3c0 input=b4cfb6786afd9035]*/
{
double y;
/* some platforms don't do the right thing for NaNs and
infinities, so we take care of special cases directly. */
if (!Py_IS_FINITE(x)) {
if (Py_IS_INFINITY(x))
return Py_BuildValue("(dd)", copysign(0., x), x);
else if (Py_IS_NAN(x))
return Py_BuildValue("(dd)", x, x);
}
errno = 0;
PyFPE_START_PROTECT("in math_modf", return 0);
x = modf(x, &y);
PyFPE_END_PROTECT(x);
return Py_BuildValue("(dd)", x, y);
}
/* A decent logarithm is easy to compute even for huge ints, but libm can't
do that by itself -- loghelper can. func is log or log10, and name is
"log" or "log10". Note that overflow of the result isn't possible: an int
can contain no more than INT_MAX * SHIFT bits, so has value certainly less
than 2**(2**64 * 2**16) == 2**2**80, and log2 of that is 2**80, which is
small enough to fit in an IEEE single. log and log10 are even smaller.
However, intermediate overflow is possible for an int if the number of bits
in that int is larger than PY_SSIZE_T_MAX. */
static PyObject*
loghelper(PyObject* arg, double (*func)(double), const char *funcname)
{
/* If it is int, do it ourselves. */
if (PyLong_Check(arg)) {
double x, result;
Py_ssize_t e;
/* Negative or zero inputs give a ValueError. */
if (Py_SIZE(arg) 0) {
result = PyLong_FromLong(0);
goto done;
}
goto error;
}
factors = PyLong_AsLongLongAndOverflow(k, &overflow);
if (overflow > 0) {
PyErr_Format(PyExc_OverflowError,
"k must not exceed %lld",
LLONG_MAX);
goto error;
}
else if (factors == -1) {
/* k is nonnegative, so a return value of -1 can only indicate error */
goto error;
}
if (factors == 0) {
result = PyLong_FromLong(1);
goto done;
}
result = n;
Py_INCREF(result);
if (factors == 1) {
goto done;
}
factor = n;
Py_INCREF(factor);
for (i = 1; i < factors; ++i) {
Py_SETREF(factor, PyNumber_Subtract(factor, _PyLong_One));
if (factor == NULL) {
goto error;
}
Py_SETREF(result, PyNumber_Multiply(result, factor));
if (result == NULL) {
goto error;
}
}
Py_DECREF(factor);
done:
Py_DECREF(n);
Py_DECREF(k);
return result;
error:
Py_XDECREF(factor);
Py_XDECREF(result);
Py_DECREF(n);
Py_DECREF(k);
return NULL;
}
/*[clinic input]
math.comb
n: object
k: object
/
Number of ways to choose k items from n items without repetition and without order.
Evaluates to n! / (k! * (n - k)!) when k n.
Also called the binomial coefficient because it is equivalent
to the coefficient of k-th term in polynomial expansion of the
expression (1 + x)**n.
Raises TypeError if either of the arguments are not integers.
Raises ValueError if either of the arguments are negative.
[clinic start generated code]*/
static PyObject *
math_comb_impl(PyObject *module, PyObject *n, PyObject *k)
/*[clinic end generated code: output=bd2cec8d854f3493 input=9a05315af2518709]*/
{
PyObject *result = NULL, *factor = NULL, *temp;
int overflow, cmp;
long long i, factors;
n = PyNumber_Index(n);
if (n == NULL) {
return NULL;
}
if (!PyLong_CheckExact(n)) {
Py_SETREF(n, _PyLong_Copy((PyLongObject *)n));
if (n == NULL) {
return NULL;
}
}
k = PyNumber_Index(k);
if (k == NULL) {
Py_DECREF(n);
return NULL;
}
if (!PyLong_CheckExact(k)) {
Py_SETREF(k, _PyLong_Copy((PyLongObject *)k));
if (k == NULL) {
Py_DECREF(n);
return NULL;
}
}
if (Py_SIZE(n) < 0) {
PyErr_SetString(PyExc_ValueError,
"n must be a non-negative integer");
goto error;
}
if (Py_SIZE(k) < 0) {
PyErr_SetString(PyExc_ValueError,
"k must be a non-negative integer");
goto error;
}
/* k = min(k, n - k) */
temp = PyNumber_Subtract(n, k);
if (temp == NULL) {
goto error;
}
if (Py_SIZE(temp) < 0) {
Py_DECREF(temp);
result = PyLong_FromLong(0);
goto done;
}
cmp = PyObject_RichCompareBool(temp, k, Py_LT);
if (cmp > 0) {
Py_SETREF(k, temp);
}
else {
Py_DECREF(temp);
if (cmp < 0) {
goto error;
}
}
factors = PyLong_AsLongLongAndOverflow(k, &overflow);
if (overflow > 0) {
PyErr_Format(PyExc_OverflowError,
"min(n - k, k) must not exceed %lld",
LLONG_MAX);
goto error;
}
if (factors == -1) {
/* k is nonnegative, so a return value of -1 can only indicate error */
goto error;
}
if (factors == 0) {
result = PyLong_FromLong(1);
goto done;
}
result = n;
Py_INCREF(result);
if (factors == 1) {
goto done;
}
factor = n;
Py_INCREF(factor);
for (i = 1; i < factors; ++i) {
Py_SETREF(factor, PyNumber_Subtract(factor, _PyLong_One));
if (factor == NULL) {
goto error;
}
Py_SETREF(result, PyNumber_Multiply(result, factor));
if (result == NULL) {
goto error;
}
temp = PyLong_FromUnsignedLongLong((unsigned long long)i + 1);
if (temp == NULL) {
goto error;
}
Py_SETREF(result, PyNumber_FloorDivide(result, temp));
Py_DECREF(temp);
if (result == NULL) {
goto error;
}
}
Py_DECREF(factor);
done:
Py_DECREF(n);
Py_DECREF(k);
return result;
error:
Py_XDECREF(factor);
Py_XDECREF(result);
Py_DECREF(n);
Py_DECREF(k);
return NULL;
}
static PyMethodDef math_methods[] = {
{"acos", math_acos, METH_O, math_acos_doc},
{"acosh", math_acosh, METH_O, math_acosh_doc},
{"asin", math_asin, METH_O, math_asin_doc},
{"asinh", math_asinh, METH_O, math_asinh_doc},
{"atan", math_atan, METH_O, math_atan_doc},
{"atan2", (PyCFunction)(void(*)(void))math_atan2, METH_FASTCALL, math_atan2_doc},
{"atanh", math_atanh, METH_O, math_atanh_doc},
MATH_CEIL_METHODDEF
{"copysign", (PyCFunction)(void(*)(void))math_copysign, METH_FASTCALL, math_copysign_doc},
{"cos", math_cos, METH_O, math_cos_doc},
{"cosh", math_cosh, METH_O, math_cosh_doc},
MATH_DEGREES_METHODDEF
MATH_DIST_METHODDEF
{"erf", math_erf, METH_O, math_erf_doc},
{"erfc", math_erfc, METH_O, math_erfc_doc},
{"exp", math_exp, METH_O, math_exp_doc},
{"expm1", math_expm1, METH_O, math_expm1_doc},
{"fabs", math_fabs, METH_O, math_fabs_doc},
MATH_FACTORIAL_METHODDEF
MATH_FLOOR_METHODDEF
MATH_FMOD_METHODDEF
MATH_FREXP_METHODDEF
MATH_FSUM_METHODDEF
{"gamma", math_gamma, METH_O, math_gamma_doc},
MATH_GCD_METHODDEF
{"hypot", (PyCFunction)(void(*)(void))math_hypot, METH_FASTCALL, math_hypot_doc},
MATH_ISCLOSE_METHODDEF
MATH_ISFINITE_METHODDEF
MATH_ISINF_METHODDEF
MATH_ISNAN_METHODDEF
MATH_ISQRT_METHODDEF
MATH_LDEXP_METHODDEF
{"lgamma", math_lgamma, METH_O, math_lgamma_doc},
MATH_LOG_METHODDEF
{"log1p", math_log1p, METH_O, math_log1p_doc},
MATH_LOG10_METHODDEF
MATH_LOG2_METHODDEF
MATH_MODF_METHODDEF
MATH_POW_METHODDEF
MATH_RADIANS_METHODDEF
{"remainder", (PyCFunction)(void(*)(void))math_remainder, METH_FASTCALL, math_remainder_doc},
{"sin", math_sin, METH_O, math_sin_doc},
{"sinh", math_sinh, METH_O, math_sinh_doc},
{"sqrt", math_sqrt, METH_O, math_sqrt_doc},
{"tan", math_tan, METH_O, math_tan_doc},
{"tanh", math_tanh, METH_O, math_tanh_doc},
MATH_TRUNC_METHODDEF
MATH_PROD_METHODDEF
MATH_PERM_METHODDEF
MATH_COMB_METHODDEF
{NULL, NULL} /* sentinel */
};
PyDoc_STRVAR(module_doc,
"This module provides access to the mathematical functions\n"
"defined by the C standard.");
static struct PyModuleDef mathmodule = {
PyModuleDef_HEAD_INIT,
"math",
module_doc,
-1,
math_methods,
NULL,
NULL,
NULL,
NULL
};
PyMODINIT_FUNC
PyInit_math(void)
{
PyObject *m;
m = PyModule_Create(&mathmodule);
if (m == NULL)
goto finally;
PyModule_AddObject(m, "pi", PyFloat_FromDouble(Py_MATH_PI));
PyModule_AddObject(m, "e", PyFloat_FromDouble(Py_MATH_E));
PyModule_AddObject(m, "tau", PyFloat_FromDouble(Py_MATH_TAU)); /* 2pi */
PyModule_AddObject(m, "inf", PyFloat_FromDouble(m_inf()));
#if !defined(PY_NO_SHORT_FLOAT_REPR) || defined(Py_NAN)
PyModule_AddObject(m, "nan", PyFloat_FromDouble(m_nan()));
#endif
finally:
return m;
}