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4161 lines
129 KiB
C
4161 lines
129 KiB
C
/* Math module -- standard C math library functions, pi and e */
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/* Here are some comments from Tim Peters, extracted from the
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discussion attached to http://bugs.python.org/issue1640. They
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describe the general aims of the math module with respect to
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special values, IEEE-754 floating-point exceptions, and Python
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exceptions.
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These are the "spirit of 754" rules:
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1. If the mathematical result is a real number, but of magnitude too
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large to approximate by a machine float, overflow is signaled and the
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result is an infinity (with the appropriate sign).
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2. If the mathematical result is a real number, but of magnitude too
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small to approximate by a machine float, underflow is signaled and the
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result is a zero (with the appropriate sign).
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3. At a singularity (a value x such that the limit of f(y) as y
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approaches x exists and is an infinity), "divide by zero" is signaled
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and the result is an infinity (with the appropriate sign). This is
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complicated a little by that the left-side and right-side limits may
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not be the same; e.g., 1/x approaches +inf or -inf as x approaches 0
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from the positive or negative directions. In that specific case, the
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sign of the zero determines the result of 1/0.
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4. At a point where a function has no defined result in the extended
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reals (i.e., the reals plus an infinity or two), invalid operation is
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signaled and a NaN is returned.
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And these are what Python has historically /tried/ to do (but not
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always successfully, as platform libm behavior varies a lot):
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For #1, raise OverflowError.
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For #2, return a zero (with the appropriate sign if that happens by
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accident ;-)).
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For #3 and #4, raise ValueError. It may have made sense to raise
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Python's ZeroDivisionError in #3, but historically that's only been
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raised for division by zero and mod by zero.
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*/
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/*
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In general, on an IEEE-754 platform the aim is to follow the C99
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standard, including Annex 'F', whenever possible. Where the
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standard recommends raising the 'divide-by-zero' or 'invalid'
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floating-point exceptions, Python should raise a ValueError. Where
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the standard recommends raising 'overflow', Python should raise an
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OverflowError. In all other circumstances a value should be
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returned.
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*/
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#ifndef Py_BUILD_CORE_BUILTIN
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# define Py_BUILD_CORE_MODULE 1
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#endif
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#include "Python.h"
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#include "pycore_bitutils.h" // _Py_bit_length()
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#include "pycore_call.h" // _PyObject_CallNoArgs()
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#include "pycore_long.h" // _PyLong_GetZero()
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#include "pycore_moduleobject.h" // _PyModule_GetState()
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#include "pycore_object.h" // _PyObject_LookupSpecial()
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#include "pycore_pymath.h" // _PY_SHORT_FLOAT_REPR
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/* For DBL_EPSILON in _math.h */
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#include <float.h>
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/* For _Py_log1p with workarounds for buggy handling of zeros. */
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#include "_math.h"
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#include <stdbool.h>
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#include "clinic/mathmodule.c.h"
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/*[clinic input]
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module math
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[clinic start generated code]*/
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/*[clinic end generated code: output=da39a3ee5e6b4b0d input=76bc7002685dd942]*/
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typedef struct {
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PyObject *str___ceil__;
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PyObject *str___floor__;
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PyObject *str___trunc__;
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} math_module_state;
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static inline math_module_state*
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get_math_module_state(PyObject *module)
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{
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void *state = _PyModule_GetState(module);
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assert(state != NULL);
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return (math_module_state *)state;
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}
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/*
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Double and triple length extended precision algorithms from:
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Accurate Sum and Dot Product
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by Takeshi Ogita, Siegfried M. Rump, and Shin’Ichi Oishi
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https://doi.org/10.1137/030601818
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https://www.tuhh.de/ti3/paper/rump/OgRuOi05.pdf
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*/
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typedef struct{ double hi; double lo; } DoubleLength;
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static DoubleLength
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dl_fast_sum(double a, double b)
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{
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/* Algorithm 1.1. Compensated summation of two floating point numbers. */
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assert(fabs(a) >= fabs(b));
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double x = a + b;
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double y = (a - x) + b;
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return (DoubleLength) {x, y};
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}
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static DoubleLength
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dl_sum(double a, double b)
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{
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/* Algorithm 3.1 Error-free transformation of the sum */
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double x = a + b;
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double z = x - a;
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double y = (a - (x - z)) + (b - z);
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return (DoubleLength) {x, y};
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}
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#ifndef UNRELIABLE_FMA
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static DoubleLength
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dl_mul(double x, double y)
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{
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/* Algorithm 3.5. Error-free transformation of a product */
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double z = x * y;
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double zz = fma(x, y, -z);
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return (DoubleLength) {z, zz};
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}
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#else
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/*
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The default implementation of dl_mul() depends on the C math library
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having an accurate fma() function as required by § 7.12.13.1 of the
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C99 standard.
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The UNRELIABLE_FMA option is provided as a slower but accurate
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alternative for builds where the fma() function is found wanting.
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The speed penalty may be modest (17% slower on an Apple M1 Max),
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so don't hesitate to enable this build option.
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The algorithms are from the T. J. Dekker paper:
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A Floating-Point Technique for Extending the Available Precision
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https://csclub.uwaterloo.ca/~pbarfuss/dekker1971.pdf
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*/
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static DoubleLength
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dl_split(double x) {
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// Dekker (5.5) and (5.6).
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double t = x * 134217729.0; // Veltkamp constant = 2.0 ** 27 + 1
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double hi = t - (t - x);
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double lo = x - hi;
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return (DoubleLength) {hi, lo};
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}
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static DoubleLength
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dl_mul(double x, double y)
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{
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// Dekker (5.12) and mul12()
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DoubleLength xx = dl_split(x);
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DoubleLength yy = dl_split(y);
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double p = xx.hi * yy.hi;
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double q = xx.hi * yy.lo + xx.lo * yy.hi;
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double z = p + q;
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double zz = p - z + q + xx.lo * yy.lo;
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return (DoubleLength) {z, zz};
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}
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#endif
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typedef struct { double hi; double lo; double tiny; } TripleLength;
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static const TripleLength tl_zero = {0.0, 0.0, 0.0};
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static TripleLength
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tl_fma(double x, double y, TripleLength total)
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{
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/* Algorithm 5.10 with SumKVert for K=3 */
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DoubleLength pr = dl_mul(x, y);
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DoubleLength sm = dl_sum(total.hi, pr.hi);
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DoubleLength r1 = dl_sum(total.lo, pr.lo);
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DoubleLength r2 = dl_sum(r1.hi, sm.lo);
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return (TripleLength) {sm.hi, r2.hi, total.tiny + r1.lo + r2.lo};
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}
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static double
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tl_to_d(TripleLength total)
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{
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DoubleLength last = dl_sum(total.lo, total.hi);
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return total.tiny + last.lo + last.hi;
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}
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/*
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sin(pi*x), giving accurate results for all finite x (especially x
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integral or close to an integer). This is here for use in the
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reflection formula for the gamma function. It conforms to IEEE
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754-2008 for finite arguments, but not for infinities or nans.
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*/
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static const double pi = 3.141592653589793238462643383279502884197;
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static const double logpi = 1.144729885849400174143427351353058711647;
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/* Version of PyFloat_AsDouble() with in-line fast paths
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for exact floats and integers. Gives a substantial
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speed improvement for extracting float arguments.
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*/
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#define ASSIGN_DOUBLE(target_var, obj, error_label) \
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if (PyFloat_CheckExact(obj)) { \
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target_var = PyFloat_AS_DOUBLE(obj); \
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} \
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else if (PyLong_CheckExact(obj)) { \
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target_var = PyLong_AsDouble(obj); \
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if (target_var == -1.0 && PyErr_Occurred()) { \
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goto error_label; \
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} \
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} \
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else { \
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target_var = PyFloat_AsDouble(obj); \
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if (target_var == -1.0 && PyErr_Occurred()) { \
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goto error_label; \
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} \
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}
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static double
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m_sinpi(double x)
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{
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double y, r;
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int n;
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/* this function should only ever be called for finite arguments */
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assert(Py_IS_FINITE(x));
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y = fmod(fabs(x), 2.0);
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n = (int)round(2.0*y);
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assert(0 <= n && n <= 4);
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switch (n) {
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case 0:
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r = sin(pi*y);
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break;
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case 1:
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r = cos(pi*(y-0.5));
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break;
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case 2:
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/* N.B. -sin(pi*(y-1.0)) is *not* equivalent: it would give
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-0.0 instead of 0.0 when y == 1.0. */
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r = sin(pi*(1.0-y));
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break;
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case 3:
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r = -cos(pi*(y-1.5));
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break;
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case 4:
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r = sin(pi*(y-2.0));
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break;
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default:
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Py_UNREACHABLE();
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}
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return copysign(1.0, x)*r;
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}
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/* Implementation of the real gamma function. Kept here to work around
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issues (see e.g. gh-70309) with quality of libm's tgamma/lgamma implementations
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on various platforms (Windows, MacOS). In extensive but non-exhaustive
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random tests, this function proved accurate to within <= 10 ulps across the
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entire float domain. Note that accuracy may depend on the quality of the
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system math functions, the pow function in particular. Special cases
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follow C99 annex F. The parameters and method are tailored to platforms
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whose double format is the IEEE 754 binary64 format.
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Method: for x > 0.0 we use the Lanczos approximation with parameters N=13
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and g=6.024680040776729583740234375; these parameters are amongst those
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used by the Boost library. Following Boost (again), we re-express the
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Lanczos sum as a rational function, and compute it that way. The
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coefficients below were computed independently using MPFR, and have been
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double-checked against the coefficients in the Boost source code.
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For x < 0.0 we use the reflection formula.
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There's one minor tweak that deserves explanation: Lanczos' formula for
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Gamma(x) involves computing pow(x+g-0.5, x-0.5) / exp(x+g-0.5). For many x
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values, x+g-0.5 can be represented exactly. However, in cases where it
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can't be represented exactly the small error in x+g-0.5 can be magnified
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significantly by the pow and exp calls, especially for large x. A cheap
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correction is to multiply by (1 + e*g/(x+g-0.5)), where e is the error
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involved in the computation of x+g-0.5 (that is, e = computed value of
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x+g-0.5 - exact value of x+g-0.5). Here's the proof:
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Correction factor
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-----------------
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Write x+g-0.5 = y-e, where y is exactly representable as an IEEE 754
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double, and e is tiny. Then:
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pow(x+g-0.5,x-0.5)/exp(x+g-0.5) = pow(y-e, x-0.5)/exp(y-e)
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= pow(y, x-0.5)/exp(y) * C,
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where the correction_factor C is given by
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C = pow(1-e/y, x-0.5) * exp(e)
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Since e is tiny, pow(1-e/y, x-0.5) ~ 1-(x-0.5)*e/y, and exp(x) ~ 1+e, so:
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C ~ (1-(x-0.5)*e/y) * (1+e) ~ 1 + e*(y-(x-0.5))/y
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But y-(x-0.5) = g+e, and g+e ~ g. So we get C ~ 1 + e*g/y, and
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pow(x+g-0.5,x-0.5)/exp(x+g-0.5) ~ pow(y, x-0.5)/exp(y) * (1 + e*g/y),
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Note that for accuracy, when computing r*C it's better to do
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r + e*g/y*r;
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than
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r * (1 + e*g/y);
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since the addition in the latter throws away most of the bits of
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information in e*g/y.
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*/
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#define LANCZOS_N 13
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static const double lanczos_g = 6.024680040776729583740234375;
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static const double lanczos_g_minus_half = 5.524680040776729583740234375;
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static const double lanczos_num_coeffs[LANCZOS_N] = {
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23531376880.410759688572007674451636754734846804940,
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42919803642.649098768957899047001988850926355848959,
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35711959237.355668049440185451547166705960488635843,
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17921034426.037209699919755754458931112671403265390,
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6039542586.3520280050642916443072979210699388420708,
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1439720407.3117216736632230727949123939715485786772,
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248874557.86205415651146038641322942321632125127801,
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31426415.585400194380614231628318205362874684987640,
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2876370.6289353724412254090516208496135991145378768,
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186056.26539522349504029498971604569928220784236328,
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8071.6720023658162106380029022722506138218516325024,
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210.82427775157934587250973392071336271166969580291,
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2.5066282746310002701649081771338373386264310793408
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};
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/* denominator is x*(x+1)*...*(x+LANCZOS_N-2) */
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static const double lanczos_den_coeffs[LANCZOS_N] = {
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0.0, 39916800.0, 120543840.0, 150917976.0, 105258076.0, 45995730.0,
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13339535.0, 2637558.0, 357423.0, 32670.0, 1925.0, 66.0, 1.0};
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/* gamma values for small positive integers, 1 though NGAMMA_INTEGRAL */
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#define NGAMMA_INTEGRAL 23
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static const double gamma_integral[NGAMMA_INTEGRAL] = {
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1.0, 1.0, 2.0, 6.0, 24.0, 120.0, 720.0, 5040.0, 40320.0, 362880.0,
|
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3628800.0, 39916800.0, 479001600.0, 6227020800.0, 87178291200.0,
|
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1307674368000.0, 20922789888000.0, 355687428096000.0,
|
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6402373705728000.0, 121645100408832000.0, 2432902008176640000.0,
|
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51090942171709440000.0, 1124000727777607680000.0,
|
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};
|
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|
||
/* Lanczos' sum L_g(x), for positive x */
|
||
|
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static double
|
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lanczos_sum(double x)
|
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{
|
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double num = 0.0, den = 0.0;
|
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int i;
|
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assert(x > 0.0);
|
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/* evaluate the rational function lanczos_sum(x). For large
|
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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; ) {
|
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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;
|
||
/* tgamma(+-0.0) = +-inf, divide-by-zero */
|
||
return copysign(Py_INFINITY, 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/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 <= 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(m_sinpi(absx))) - log(absx) - r;
|
||
if (Py_IS_INFINITY(r))
|
||
errno = ERANGE;
|
||
return r;
|
||
}
|
||
|
||
/*
|
||
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 multiplication
|
||
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) {
|
||
return log2(x);
|
||
}
|
||
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 *module, PyObject * const *args, Py_ssize_t nargs)
|
||
{
|
||
PyObject *res, *x;
|
||
Py_ssize_t i;
|
||
|
||
if (nargs == 0) {
|
||
return PyLong_FromLong(0);
|
||
}
|
||
res = PyNumber_Index(args[0]);
|
||
if (res == NULL) {
|
||
return NULL;
|
||
}
|
||
if (nargs == 1) {
|
||
Py_SETREF(res, PyNumber_Absolute(res));
|
||
return res;
|
||
}
|
||
|
||
PyObject *one = _PyLong_GetOne(); // borrowed ref
|
||
for (i = 1; i < nargs; i++) {
|
||
x = _PyNumber_Index(args[i]);
|
||
if (x == NULL) {
|
||
Py_DECREF(res);
|
||
return NULL;
|
||
}
|
||
if (res == one) {
|
||
/* Fast path: just check arguments.
|
||
It is okay to use identity comparison here. */
|
||
Py_DECREF(x);
|
||
continue;
|
||
}
|
||
Py_SETREF(res, _PyLong_GCD(res, x));
|
||
Py_DECREF(x);
|
||
if (res == NULL) {
|
||
return NULL;
|
||
}
|
||
}
|
||
return res;
|
||
}
|
||
|
||
PyDoc_STRVAR(math_gcd_doc,
|
||
"gcd($module, *integers)\n"
|
||
"--\n"
|
||
"\n"
|
||
"Greatest Common Divisor.");
|
||
|
||
|
||
static PyObject *
|
||
long_lcm(PyObject *a, PyObject *b)
|
||
{
|
||
PyObject *g, *m, *f, *ab;
|
||
|
||
if (_PyLong_IsZero((PyLongObject *)a) || _PyLong_IsZero((PyLongObject *)b)) {
|
||
return PyLong_FromLong(0);
|
||
}
|
||
g = _PyLong_GCD(a, b);
|
||
if (g == NULL) {
|
||
return NULL;
|
||
}
|
||
f = PyNumber_FloorDivide(a, g);
|
||
Py_DECREF(g);
|
||
if (f == NULL) {
|
||
return NULL;
|
||
}
|
||
m = PyNumber_Multiply(f, b);
|
||
Py_DECREF(f);
|
||
if (m == NULL) {
|
||
return NULL;
|
||
}
|
||
ab = PyNumber_Absolute(m);
|
||
Py_DECREF(m);
|
||
return ab;
|
||
}
|
||
|
||
|
||
static PyObject *
|
||
math_lcm(PyObject *module, PyObject * const *args, Py_ssize_t nargs)
|
||
{
|
||
PyObject *res, *x;
|
||
Py_ssize_t i;
|
||
|
||
if (nargs == 0) {
|
||
return PyLong_FromLong(1);
|
||
}
|
||
res = PyNumber_Index(args[0]);
|
||
if (res == NULL) {
|
||
return NULL;
|
||
}
|
||
if (nargs == 1) {
|
||
Py_SETREF(res, PyNumber_Absolute(res));
|
||
return res;
|
||
}
|
||
|
||
PyObject *zero = _PyLong_GetZero(); // borrowed ref
|
||
for (i = 1; i < nargs; i++) {
|
||
x = PyNumber_Index(args[i]);
|
||
if (x == NULL) {
|
||
Py_DECREF(res);
|
||
return NULL;
|
||
}
|
||
if (res == zero) {
|
||
/* Fast path: just check arguments.
|
||
It is okay to use identity comparison here. */
|
||
Py_DECREF(x);
|
||
continue;
|
||
}
|
||
Py_SETREF(res, long_lcm(res, x));
|
||
Py_DECREF(x);
|
||
if (res == NULL) {
|
||
return NULL;
|
||
}
|
||
}
|
||
return res;
|
||
}
|
||
|
||
|
||
PyDoc_STRVAR(math_lcm_doc,
|
||
"lcm($module, *integers)\n"
|
||
"--\n"
|
||
"\n"
|
||
"Least Common Multiple.");
|
||
|
||
|
||
/* 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 1.5 in absolute value.
|
||
*
|
||
* bpo-46018: Changed to 1.5 to ensure underflows in expm1()
|
||
* are correctly detected, since the function may underflow
|
||
* toward -1.0 rather than 0.0.
|
||
*/
|
||
if (fabs(x) < 1.5)
|
||
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(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;
|
||
r = (*func)(x);
|
||
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 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;
|
||
r = (*func)(x);
|
||
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 *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;
|
||
r = (*func)(x, y);
|
||
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($module, x, /)\n--\n\n"
|
||
"Return the arc cosine (measured in radians) of x.\n\n"
|
||
"The result is between 0 and pi.")
|
||
FUNC1(acosh, acosh, 0,
|
||
"acosh($module, x, /)\n--\n\n"
|
||
"Return the inverse hyperbolic cosine of x.")
|
||
FUNC1(asin, asin, 0,
|
||
"asin($module, x, /)\n--\n\n"
|
||
"Return the arc sine (measured in radians) of x.\n\n"
|
||
"The result is between -pi/2 and pi/2.")
|
||
FUNC1(asinh, asinh, 0,
|
||
"asinh($module, x, /)\n--\n\n"
|
||
"Return the inverse hyperbolic sine of x.")
|
||
FUNC1(atan, atan, 0,
|
||
"atan($module, x, /)\n--\n\n"
|
||
"Return the arc tangent (measured in radians) of x.\n\n"
|
||
"The result is between -pi/2 and pi/2.")
|
||
FUNC2(atan2, m_atan2,
|
||
"atan2($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, atanh, 0,
|
||
"atanh($module, x, /)\n--\n\n"
|
||
"Return the inverse hyperbolic tangent of x.")
|
||
FUNC1(cbrt, cbrt, 0,
|
||
"cbrt($module, x, /)\n--\n\n"
|
||
"Return the cube root 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]*/
|
||
{
|
||
|
||
if (!PyFloat_CheckExact(number)) {
|
||
math_module_state *state = get_math_module_state(module);
|
||
PyObject *method = _PyObject_LookupSpecial(number, state->str___ceil__);
|
||
if (method != NULL) {
|
||
PyObject *result = _PyObject_CallNoArgs(method);
|
||
Py_DECREF(method);
|
||
return result;
|
||
}
|
||
if (PyErr_Occurred())
|
||
return NULL;
|
||
}
|
||
double x = PyFloat_AsDouble(number);
|
||
if (x == -1.0 && PyErr_Occurred())
|
||
return NULL;
|
||
|
||
return PyLong_FromDouble(ceil(x));
|
||
}
|
||
|
||
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($module, x, /)\n--\n\n"
|
||
"Return the hyperbolic cosine of x.")
|
||
FUNC1A(erf, erf,
|
||
"erf($module, x, /)\n--\n\n"
|
||
"Error function at x.")
|
||
FUNC1A(erfc, 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(exp2, exp2, 1,
|
||
"exp2($module, x, /)\n--\n\n"
|
||
"Return 2 raised to the power of x.")
|
||
FUNC1(expm1, expm1, 1,
|
||
"expm1($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($module, x, /)\n--\n\n"
|
||
"Return the absolute value of the float x.")
|
||
|
||
/*[clinic input]
|
||
math.floor
|
||
|
||
x as number: object
|
||
/
|
||
|
||
Return the floor of x as an Integral.
|
||
|
||
This is the largest integer <= x.
|
||
[clinic start generated code]*/
|
||
|
||
static PyObject *
|
||
math_floor(PyObject *module, PyObject *number)
|
||
/*[clinic end generated code: output=c6a65c4884884b8a input=63af6b5d7ebcc3d6]*/
|
||
{
|
||
double x;
|
||
|
||
if (PyFloat_CheckExact(number)) {
|
||
x = PyFloat_AS_DOUBLE(number);
|
||
}
|
||
else
|
||
{
|
||
math_module_state *state = get_math_module_state(module);
|
||
PyObject *method = _PyObject_LookupSpecial(number, state->str___floor__);
|
||
if (method != NULL) {
|
||
PyObject *result = _PyObject_CallNoArgs(method);
|
||
Py_DECREF(method);
|
||
return result;
|
||
}
|
||
if (PyErr_Occurred())
|
||
return NULL;
|
||
x = PyFloat_AsDouble(number);
|
||
if (x == -1.0 && PyErr_Occurred())
|
||
return NULL;
|
||
}
|
||
return PyLong_FromDouble(floor(x));
|
||
}
|
||
|
||
FUNC1A(gamma, m_tgamma,
|
||
"gamma($module, x, /)\n--\n\n"
|
||
"Gamma function at x.")
|
||
FUNC1A(lgamma, m_lgamma,
|
||
"lgamma($module, x, /)\n--\n\n"
|
||
"Natural logarithm of absolute value of Gamma function at x.")
|
||
FUNC1(log1p, m_log1p, 0,
|
||
"log1p($module, x, /)\n--\n\n"
|
||
"Return the natural logarithm of 1+x (base e).\n\n"
|
||
"The result is computed in a way which is accurate for x near zero.")
|
||
FUNC2(remainder, m_remainder,
|
||
"remainder($module, x, y, /)\n--\n\n"
|
||
"Difference between x and the closest integer multiple of y.\n\n"
|
||
"Return x - n*y where n*y is the closest integer multiple of y.\n"
|
||
"In the case where x is exactly halfway between two multiples of\n"
|
||
"y, the nearest even value of n is used. The result is always exact.")
|
||
FUNC1(sin, sin, 0,
|
||
"sin($module, x, /)\n--\n\n"
|
||
"Return the sine of x (measured in radians).")
|
||
FUNC1(sinh, sinh, 1,
|
||
"sinh($module, x, /)\n--\n\n"
|
||
"Return the hyperbolic sine of x.")
|
||
FUNC1(sqrt, sqrt, 0,
|
||
"sqrt($module, x, /)\n--\n\n"
|
||
"Return the square root of x.")
|
||
FUNC1(tan, tan, 0,
|
||
"tan($module, x, /)\n--\n\n"
|
||
"Return the tangent of x (measured in radians).")
|
||
FUNC1(tanh, tanh, 0,
|
||
"tanh($module, x, /)\n--\n\n"
|
||
"Return 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 754 floating-point semantics with a rounding mode of
|
||
roundTiesToEven are assumed.
|
||
|
||
Note 2: No provision is made for intermediate overflow handling;
|
||
therefore, fsum([1e+308, -1e+308, 1e+308]) returns 1e+308 while
|
||
fsum([1e+308, 1e+308, -1e+308]) raises an OverflowError due to the
|
||
overflow of the first partial sum.
|
||
|
||
Note 3: The algorithm has two potential sources of fragility. First, C
|
||
permits arithmetic operations on `double`s to be performed in an
|
||
intermediate format whose range and precision may be greater than those of
|
||
`double` (see for example C99 §5.2.4.2.2, paragraph 8). This can happen for
|
||
example on machines using the now largely historical x87 FPUs. In this case,
|
||
`fsum` can produce incorrect results. If `FLT_EVAL_METHOD` is `0` or `1`, or
|
||
`FLT_EVAL_METHOD` is `2` and `long double` is identical to `double`, then we
|
||
should be safe from this source of errors. Second, an aggressively
|
||
optimizing compiler can re-associate operations so that (for example) the
|
||
statement `yr = hi - x;` is treated as `yr = (x + y) - x` and then
|
||
re-associated as `yr = y + (x - x)`, giving `y = yr` and `lo = 0.0`. That
|
||
re-association would be in violation of the C standard, and should not occur
|
||
except possibly in the presence of unsafe optimizations (e.g., -ffast-math,
|
||
-fassociative-math). Such optimizations should be avoided for this module.
|
||
|
||
Note 4: The signature of math.fsum() differs from builtins.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 && (size_t)m < ((size_t)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.
|
||
*/
|
||
|
||
/*[clinic input]
|
||
math.fsum
|
||
|
||
seq: object
|
||
/
|
||
|
||
Return an accurate floating point sum of values in the iterable seq.
|
||
|
||
Assumes IEEE-754 floating point arithmetic.
|
||
[clinic start generated code]*/
|
||
|
||
static PyObject *
|
||
math_fsum(PyObject *module, PyObject *seq)
|
||
/*[clinic end generated code: output=ba5c672b87fe34fc input=c51b7d8caf6f6e82]*/
|
||
{
|
||
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;
|
||
double hi, yr, lo = 0.0;
|
||
|
||
iter = PyObject_GetIter(seq);
|
||
if (iter == NULL)
|
||
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;
|
||
}
|
||
ASSIGN_DOUBLE(x, item, error_with_item);
|
||
Py_DECREF(item);
|
||
|
||
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;
|
||
lo = y - yr;
|
||
if (lo != 0.0)
|
||
p[i++] = lo;
|
||
x = hi;
|
||
}
|
||
|
||
n = i; /* ps[i:] = [x] */
|
||
if (x != 0.0) {
|
||
if (! Py_IS_FINITE(x)) {
|
||
/* a nonfinite x could arise either as
|
||
a result of intermediate overflow, or
|
||
as a result of a nan or inf in the
|
||
summands */
|
||
if (Py_IS_FINITE(xsave)) {
|
||
PyErr_SetString(PyExc_OverflowError,
|
||
"intermediate overflow in fsum");
|
||
goto _fsum_error;
|
||
}
|
||
if (Py_IS_INFINITY(xsave))
|
||
inf_sum += xsave;
|
||
special_sum += xsave;
|
||
/* reset partials */
|
||
n = 0;
|
||
}
|
||
else if (n >= m && _fsum_realloc(&p, n, ps, &m))
|
||
goto _fsum_error;
|
||
else
|
||
p[n++] = x;
|
||
}
|
||
}
|
||
|
||
if (special_sum != 0.0) {
|
||
if (Py_IS_NAN(inf_sum))
|
||
PyErr_SetString(PyExc_ValueError,
|
||
"-inf + inf in fsum");
|
||
else
|
||
sum = PyFloat_FromDouble(special_sum);
|
||
goto _fsum_error;
|
||
}
|
||
|
||
hi = 0.0;
|
||
if (n > 0) {
|
||
hi = p[--n];
|
||
/* sum_exact(ps, hi) from the top, stop when the sum becomes
|
||
inexact. */
|
||
while (n > 0) {
|
||
x = hi;
|
||
y = p[--n];
|
||
assert(fabs(y) < fabs(x));
|
||
hi = x + y;
|
||
yr = hi - x;
|
||
lo = y - yr;
|
||
if (lo != 0.0)
|
||
break;
|
||
}
|
||
/* Make half-even rounding work across multiple partials.
|
||
Needed so that sum([1e-16, 1, 1e16]) will round-up the last
|
||
digit to two instead of down to zero (the 1e-16 makes the 1
|
||
slightly closer to two). With a potential 1 ULP rounding
|
||
error fixed-up, math.fsum() can guarantee commutativity. */
|
||
if (n > 0 && ((lo < 0.0 && p[n-1] < 0.0) ||
|
||
(lo > 0.0 && p[n-1] > 0.0))) {
|
||
y = lo * 2.0;
|
||
x = hi + y;
|
||
yr = x - hi;
|
||
if (y == yr)
|
||
hi = x;
|
||
}
|
||
}
|
||
sum = PyFloat_FromDouble(hi);
|
||
|
||
_fsum_error:
|
||
Py_DECREF(iter);
|
||
if (p != ps)
|
||
PyMem_Free(p);
|
||
return sum;
|
||
|
||
error_with_item:
|
||
Py_DECREF(item);
|
||
goto _fsum_error;
|
||
}
|
||
|
||
#undef NUM_PARTIALS
|
||
|
||
|
||
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 <= n`, or equivalently the integer part of the exact
|
||
square root of `n`.
|
||
|
||
We use an adaptive-precision pure-integer version of Newton's iteration. Given
|
||
a positive integer `n`, the algorithm produces at each iteration an integer
|
||
approximation `a` to the square root of `n >> s` for some even integer `s`,
|
||
with `s` decreasing as the iterations progress. On the final iteration, `s` is
|
||
zero and we have an approximation to the square root of `n` itself.
|
||
|
||
At every step, the approximation `a` is strictly within 1.0 of the true square
|
||
root, so we have
|
||
|
||
(a - 1)**2 < (n >> s) < (a + 1)**2
|
||
|
||
After the final iteration, a check-and-correct step is needed to determine
|
||
whether `a` or `a - 1` gives the desired integer square root of `n`.
|
||
|
||
The algorithm is remarkable in its simplicity. There's no need for a
|
||
per-iteration check-and-correct step, and termination is straightforward: the
|
||
number of iterations is known in advance (it's exactly `floor(log2(log2(n)))`
|
||
for `n > 1`). The only tricky part of the correctness proof is in establishing
|
||
that the bound `(a - 1)**2 < (n >> s) < (a + 1)**2` is maintained from one
|
||
iteration to the next. A sketch of the proof of this is given below.
|
||
|
||
In addition to the proof sketch, a formal, computer-verified proof
|
||
of correctness (using Lean) of an equivalent recursive algorithm can be found
|
||
here:
|
||
|
||
https://github.com/mdickinson/snippets/blob/master/proofs/isqrt/src/isqrt.lean
|
||
|
||
|
||
Here's Python code equivalent to the C implementation below:
|
||
|
||
def isqrt(n):
|
||
"""
|
||
Return the integer part of the square root of the input.
|
||
"""
|
||
n = operator.index(n)
|
||
|
||
if n < 0:
|
||
raise ValueError("isqrt() argument must be nonnegative")
|
||
if n == 0:
|
||
return 0
|
||
|
||
c = (n.bit_length() - 1) // 2
|
||
a = 1
|
||
d = 0
|
||
for s in reversed(range(c.bit_length())):
|
||
# Loop invariant: (a-1)**2 < (n >> 2*(c - d)) < (a+1)**2
|
||
e = d
|
||
d = c >> s
|
||
a = (a << d - e - 1) + (n >> 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 << d - e - 1) + (n >> 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 << d - e - 1) + (n >> 2*c - e - d + 1) // a
|
||
|
||
or equivalently:
|
||
|
||
(2) b = (a << d - e - 1) + (m >> 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 <= | 2^(d-e)a - √m | < 2^(d-e)
|
||
|
||
Squaring and dividing through by `2^(d-e+1) a` gives
|
||
|
||
(6) 0 <= 2^(d-e-1) a + m / (2^(d-e+1) a) - √m < 2^(d-e-1) / a
|
||
|
||
We'll show below that `2^(d-e-1) <= a`. Given that, we can replace the
|
||
right-hand side of (6) with `1`, and now replacing the central
|
||
term `m / (2^(d-e+1) a)` with its floor in (6) gives
|
||
|
||
(7) -1 < 2^(d-e-1) a + m // 2^(d-e+1) a - √m < 1
|
||
|
||
Or equivalently, from (2):
|
||
|
||
(7) -1 < b - √m < 1
|
||
|
||
and rearranging gives that `(b-1)^2 < m < (b+1)^2`, which is what we needed
|
||
to prove.
|
||
|
||
We're not quite done: we still have to prove the inequality `2^(d - e - 1) <=
|
||
a` that was used to get line (7) above. From the definition of `c`, we have
|
||
`4^c <= n`, which implies
|
||
|
||
(8) 4^d <= m
|
||
|
||
also, since `e == d >> 1`, `d` is at most `2e + 1`, from which it follows
|
||
that `2d - 2e - 1 <= d` and hence that
|
||
|
||
(9) 4^(2d - 2e - 1) <= m
|
||
|
||
Dividing both sides by `4^(d - e)` gives
|
||
|
||
(10) 4^(d - e - 1) <= m / 4^(d - e)
|
||
|
||
But we know from (4) that `m / 4^(d-e) < (a + 1)^2`, hence
|
||
|
||
(11) 4^(d - e - 1) < (a + 1)^2
|
||
|
||
Now taking square roots of both sides and observing that both `2^(d-e-1)` and
|
||
`a` are integers gives `2^(d - e - 1) <= a`, which is what we needed. This
|
||
completes the proof sketch.
|
||
|
||
*/
|
||
|
||
/*
|
||
The _approximate_isqrt_tab table provides approximate square roots for
|
||
16-bit integers. For any n in the range 2**14 <= n < 2**16, the value
|
||
|
||
a = _approximate_isqrt_tab[(n >> 8) - 64]
|
||
|
||
is an approximate square root of n, satisfying (a - 1)**2 < n < (a + 1)**2.
|
||
|
||
The table was computed in Python using the expression:
|
||
|
||
[min(round(sqrt(256*n + 128)), 255) for n in range(64, 256)]
|
||
*/
|
||
|
||
static const uint8_t _approximate_isqrt_tab[192] = {
|
||
128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139,
|
||
140, 141, 142, 143, 144, 144, 145, 146, 147, 148, 149, 150,
|
||
151, 151, 152, 153, 154, 155, 156, 156, 157, 158, 159, 160,
|
||
160, 161, 162, 163, 164, 164, 165, 166, 167, 167, 168, 169,
|
||
170, 170, 171, 172, 173, 173, 174, 175, 176, 176, 177, 178,
|
||
179, 179, 180, 181, 181, 182, 183, 183, 184, 185, 186, 186,
|
||
187, 188, 188, 189, 190, 190, 191, 192, 192, 193, 194, 194,
|
||
195, 196, 196, 197, 198, 198, 199, 200, 200, 201, 201, 202,
|
||
203, 203, 204, 205, 205, 206, 206, 207, 208, 208, 209, 210,
|
||
210, 211, 211, 212, 213, 213, 214, 214, 215, 216, 216, 217,
|
||
217, 218, 219, 219, 220, 220, 221, 221, 222, 223, 223, 224,
|
||
224, 225, 225, 226, 227, 227, 228, 228, 229, 229, 230, 230,
|
||
231, 232, 232, 233, 233, 234, 234, 235, 235, 236, 237, 237,
|
||
238, 238, 239, 239, 240, 240, 241, 241, 242, 242, 243, 243,
|
||
244, 244, 245, 246, 246, 247, 247, 248, 248, 249, 249, 250,
|
||
250, 251, 251, 252, 252, 253, 253, 254, 254, 255, 255, 255,
|
||
};
|
||
|
||
/* Approximate square root of a large 64-bit integer.
|
||
|
||
Given `n` satisfying `2**62 <= n < 2**64`, return `a`
|
||
satisfying `(a - 1)**2 < n < (a + 1)**2`. */
|
||
|
||
static inline uint32_t
|
||
_approximate_isqrt(uint64_t n)
|
||
{
|
||
uint32_t u = _approximate_isqrt_tab[(n >> 56) - 64];
|
||
u = (u << 7) + (uint32_t)(n >> 41) / u;
|
||
return (u << 15) + (uint32_t)((n >> 17) / u);
|
||
}
|
||
|
||
/*[clinic input]
|
||
math.isqrt
|
||
|
||
n: object
|
||
/
|
||
|
||
Return the integer part of the square root of the input.
|
||
[clinic start generated code]*/
|
||
|
||
static PyObject *
|
||
math_isqrt(PyObject *module, PyObject *n)
|
||
/*[clinic end generated code: output=35a6f7f980beab26 input=5b6e7ae4fa6c43d6]*/
|
||
{
|
||
int a_too_large, c_bit_length;
|
||
size_t c, d;
|
||
uint64_t m;
|
||
uint32_t u;
|
||
PyObject *a = NULL, *b;
|
||
|
||
n = _PyNumber_Index(n);
|
||
if (n == NULL) {
|
||
return NULL;
|
||
}
|
||
|
||
if (_PyLong_IsNegative((PyLongObject *)n)) {
|
||
PyErr_SetString(
|
||
PyExc_ValueError,
|
||
"isqrt() argument must be nonnegative");
|
||
goto error;
|
||
}
|
||
if (_PyLong_IsZero((PyLongObject *)n)) {
|
||
Py_DECREF(n);
|
||
return PyLong_FromLong(0);
|
||
}
|
||
|
||
/* c = (n.bit_length() - 1) // 2 */
|
||
c = _PyLong_NumBits(n);
|
||
if (c == (size_t)(-1)) {
|
||
goto error;
|
||
}
|
||
c = (c - 1U) / 2U;
|
||
|
||
/* Fast path: if c <= 31 then n < 2**64 and we can compute directly with a
|
||
fast, almost branch-free algorithm. */
|
||
if (c <= 31U) {
|
||
int shift = 31 - (int)c;
|
||
m = (uint64_t)PyLong_AsUnsignedLongLong(n);
|
||
Py_DECREF(n);
|
||
if (m == (uint64_t)(-1) && PyErr_Occurred()) {
|
||
return NULL;
|
||
}
|
||
u = _approximate_isqrt(m << 2*shift) >> shift;
|
||
u -= (uint64_t)u * u > m;
|
||
return PyLong_FromUnsignedLong(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_FromUnsignedLong(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 << d - 1 - e) + q */
|
||
Py_SETREF(a, _PyLong_Lshift(a, d - 1U - e));
|
||
if (a == NULL) {
|
||
Py_DECREF(q);
|
||
goto error;
|
||
}
|
||
Py_SETREF(a, PyNumber_Add(a, q));
|
||
Py_DECREF(q);
|
||
if (a == NULL) {
|
||
goto error;
|
||
}
|
||
}
|
||
|
||
/* The correct result is either a or a - 1. Figure out which, and
|
||
decrement a if necessary. */
|
||
|
||
/* a_too_large = n < a * a */
|
||
b = PyNumber_Multiply(a, a);
|
||
if (b == NULL) {
|
||
goto error;
|
||
}
|
||
a_too_large = PyObject_RichCompareBool(n, b, Py_LT);
|
||
Py_DECREF(b);
|
||
if (a_too_large == -1) {
|
||
goto error;
|
||
}
|
||
|
||
if (a_too_large) {
|
||
Py_SETREF(a, PyNumber_Subtract(a, _PyLong_GetOne()));
|
||
}
|
||
Py_DECREF(n);
|
||
return a;
|
||
|
||
error:
|
||
Py_XDECREF(a);
|
||
Py_DECREF(n);
|
||
return NULL;
|
||
}
|
||
|
||
/* Divide-and-conquer factorial algorithm
|
||
*
|
||
* Based on the formula and pseudo-code provided at:
|
||
* http://www.luschny.de/math/factorial/binarysplitfact.html
|
||
*
|
||
* Faster algorithms exist, but they're more complicated and depend on
|
||
* a fast prime factorization algorithm.
|
||
*
|
||
* Notes on the algorithm
|
||
* ----------------------
|
||
*
|
||
* factorial(n) is written in the form 2**k * m, with m odd. k and m are
|
||
* computed separately, and then combined using a left shift.
|
||
*
|
||
* The function factorial_odd_part computes the odd part m (i.e., the greatest
|
||
* odd divisor) of factorial(n), using the formula:
|
||
*
|
||
* factorial_odd_part(n) =
|
||
*
|
||
* product_{i >= 0} product_{0 < j <= n / 2**i, j odd} j
|
||
*
|
||
* Example: factorial_odd_part(20) =
|
||
*
|
||
* (1) *
|
||
* (1) *
|
||
* (1 * 3 * 5) *
|
||
* (1 * 3 * 5 * 7 * 9) *
|
||
* (1 * 3 * 5 * 7 * 9 * 11 * 13 * 15 * 17 * 19)
|
||
*
|
||
* Here i goes from large to small: the first term corresponds to i=4 (any
|
||
* larger i gives an empty product), and the last term corresponds to i=0.
|
||
* Each term can be computed from the last by multiplying by the extra odd
|
||
* numbers required: e.g., to get from the penultimate term to the last one,
|
||
* we multiply by (11 * 13 * 15 * 17 * 19).
|
||
*
|
||
* To see a hint of why this formula works, here are the same numbers as above
|
||
* but with the even parts (i.e., the appropriate powers of 2) included. For
|
||
* each subterm in the product for i, we multiply that subterm by 2**i:
|
||
*
|
||
* factorial(20) =
|
||
*
|
||
* (16) *
|
||
* (8) *
|
||
* (4 * 12 * 20) *
|
||
* (2 * 6 * 10 * 14 * 18) *
|
||
* (1 * 3 * 5 * 7 * 9 * 11 * 13 * 15 * 17 * 19)
|
||
*
|
||
* The factorial_partial_product function computes the product of all odd j in
|
||
* range(start, stop) for given start and stop. It's used to compute the
|
||
* partial products like (11 * 13 * 15 * 17 * 19) in the example above. It
|
||
* operates recursively, repeatedly splitting the range into two roughly equal
|
||
* pieces until the subranges are small enough to be computed using only C
|
||
* integer arithmetic.
|
||
*
|
||
* The two-valuation k (i.e., the exponent of the largest power of 2 dividing
|
||
* the factorial) is computed independently in the main math_factorial
|
||
* function. By standard results, its value is:
|
||
*
|
||
* two_valuation = n//2 + n//4 + n//8 + ....
|
||
*
|
||
* It can be shown (e.g., by complete induction on n) that two_valuation is
|
||
* equal to n - count_set_bits(n), where count_set_bits(n) gives the number of
|
||
* '1'-bits in the binary expansion of n.
|
||
*/
|
||
|
||
/* factorial_partial_product: Compute product(range(start, stop, 2)) using
|
||
* divide and conquer. Assumes start and stop are odd and stop > 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) <= num_operands *
|
||
* bit_length(stop - 2)
|
||
*/
|
||
|
||
num_operands = (stop - start) / 2;
|
||
/* The "num_operands <= 8 * SIZEOF_LONG" check guards against the
|
||
* unlikely case of an overflow in num_operands * max_bits. */
|
||
if (num_operands <= 8 * SIZEOF_LONG &&
|
||
num_operands * max_bits <= 8 * SIZEOF_LONG) {
|
||
unsigned long j, total;
|
||
for (total = start, j = start + 2; j < stop; j += 2)
|
||
total *= j;
|
||
return PyLong_FromUnsignedLong(total);
|
||
}
|
||
|
||
/* find midpoint of range(start, stop), rounded up to next odd number. */
|
||
midpoint = (start + num_operands) | 1;
|
||
left = factorial_partial_product(start, midpoint,
|
||
_Py_bit_length(midpoint - 2));
|
||
if (left == NULL)
|
||
goto error;
|
||
right = factorial_partial_product(midpoint, stop, max_bits);
|
||
if (right == NULL)
|
||
goto error;
|
||
result = PyNumber_Multiply(left, right);
|
||
|
||
error:
|
||
Py_XDECREF(left);
|
||
Py_XDECREF(right);
|
||
return result;
|
||
}
|
||
|
||
/* factorial_odd_part: compute the odd part of factorial(n). */
|
||
|
||
static PyObject *
|
||
factorial_odd_part(unsigned long n)
|
||
{
|
||
long i;
|
||
unsigned long v, lower, upper;
|
||
PyObject *partial, *tmp, *inner, *outer;
|
||
|
||
inner = PyLong_FromLong(1);
|
||
if (inner == NULL)
|
||
return NULL;
|
||
outer = Py_NewRef(inner);
|
||
|
||
upper = 3;
|
||
for (i = _Py_bit_length(n) - 2; i >= 0; i--) {
|
||
v = n >> i;
|
||
if (v <= 2)
|
||
continue;
|
||
lower = upper;
|
||
/* (v + 1) | 1 = least odd integer strictly larger than n / 2**i */
|
||
upper = (v + 1) | 1;
|
||
/* Here inner is the product of all odd integers j in the range (0,
|
||
n/2**(i+1)]. The factorial_partial_product call below gives the
|
||
product of all odd integers j in the range (n/2**(i+1), n/2**i]. */
|
||
partial = factorial_partial_product(lower, upper, _Py_bit_length(upper-2));
|
||
/* inner *= partial */
|
||
if (partial == NULL)
|
||
goto error;
|
||
tmp = PyNumber_Multiply(inner, partial);
|
||
Py_DECREF(partial);
|
||
if (tmp == NULL)
|
||
goto error;
|
||
Py_SETREF(inner, tmp);
|
||
/* Now inner is the product of all odd integers j in the range (0,
|
||
n/2**i], giving the inner product in the formula above. */
|
||
|
||
/* outer *= inner; */
|
||
tmp = PyNumber_Multiply(outer, inner);
|
||
if (tmp == NULL)
|
||
goto error;
|
||
Py_SETREF(outer, tmp);
|
||
}
|
||
Py_DECREF(inner);
|
||
return outer;
|
||
|
||
error:
|
||
Py_DECREF(outer);
|
||
Py_DECREF(inner);
|
||
return NULL;
|
||
}
|
||
|
||
|
||
/* Lookup table for small factorial values */
|
||
|
||
static const unsigned long SmallFactorials[] = {
|
||
1, 1, 2, 6, 24, 120, 720, 5040, 40320,
|
||
362880, 3628800, 39916800, 479001600,
|
||
#if SIZEOF_LONG >= 8
|
||
6227020800, 87178291200, 1307674368000,
|
||
20922789888000, 355687428096000, 6402373705728000,
|
||
121645100408832000, 2432902008176640000
|
||
#endif
|
||
};
|
||
|
||
/*[clinic input]
|
||
math.factorial
|
||
|
||
n as arg: object
|
||
/
|
||
|
||
Find n!.
|
||
|
||
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=713fb771677e8c31]*/
|
||
{
|
||
long x, two_valuation;
|
||
int overflow;
|
||
PyObject *result, *odd_part;
|
||
|
||
x = PyLong_AsLongAndOverflow(arg, &overflow);
|
||
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 << two_valuation. */
|
||
odd_part = factorial_odd_part(x);
|
||
if (odd_part == NULL)
|
||
return NULL;
|
||
two_valuation = x - count_set_bits(x);
|
||
result = _PyLong_Lshift(odd_part, two_valuation);
|
||
Py_DECREF(odd_part);
|
||
return result;
|
||
}
|
||
|
||
|
||
/*[clinic input]
|
||
math.trunc
|
||
|
||
x: object
|
||
/
|
||
|
||
Truncates the Real x to the nearest Integral toward 0.
|
||
|
||
Uses the __trunc__ magic method.
|
||
[clinic start generated code]*/
|
||
|
||
static PyObject *
|
||
math_trunc(PyObject *module, PyObject *x)
|
||
/*[clinic end generated code: output=34b9697b707e1031 input=2168b34e0a09134d]*/
|
||
{
|
||
PyObject *trunc, *result;
|
||
|
||
if (PyFloat_CheckExact(x)) {
|
||
return PyFloat_Type.tp_as_number->nb_int(x);
|
||
}
|
||
|
||
if (!_PyType_IsReady(Py_TYPE(x))) {
|
||
if (PyType_Ready(Py_TYPE(x)) < 0)
|
||
return NULL;
|
||
}
|
||
|
||
math_module_state *state = get_math_module_state(module);
|
||
trunc = _PyObject_LookupSpecial(x, state->str___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_CallNoArgs(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 <= abs(m) < 1.0.
|
||
[clinic start generated code]*/
|
||
|
||
static PyObject *
|
||
math_frexp_impl(PyObject *module, double x)
|
||
/*[clinic end generated code: output=03e30d252a15ad4a input=96251c9e208bc6e9]*/
|
||
{
|
||
int i;
|
||
/* deal with special cases directly, to sidestep platform
|
||
differences */
|
||
if (Py_IS_NAN(x) || Py_IS_INFINITY(x) || !x) {
|
||
i = 0;
|
||
}
|
||
else {
|
||
x = frexp(x, &i);
|
||
}
|
||
return Py_BuildValue("(di)", x, i);
|
||
}
|
||
|
||
|
||
/*[clinic input]
|
||
math.ldexp
|
||
|
||
x: double
|
||
i: object
|
||
/
|
||
|
||
Return x * (2**i).
|
||
|
||
This is essentially the inverse of frexp().
|
||
[clinic start generated code]*/
|
||
|
||
static PyObject *
|
||
math_ldexp_impl(PyObject *module, double x, PyObject *i)
|
||
/*[clinic end generated code: output=b6892f3c2df9cc6a input=17d5970c1a40a8c1]*/
|
||
{
|
||
double r;
|
||
long exp;
|
||
int overflow;
|
||
|
||
if (PyLong_Check(i)) {
|
||
/* on overflow, replace exponent with either LONG_MAX
|
||
or LONG_MIN, depending on the sign. */
|
||
exp = PyLong_AsLongAndOverflow(i, &overflow);
|
||
if (exp == -1 && PyErr_Occurred())
|
||
return NULL;
|
||
if (overflow)
|
||
exp = overflow < 0 ? LONG_MIN : LONG_MAX;
|
||
}
|
||
else {
|
||
PyErr_SetString(PyExc_TypeError,
|
||
"Expected an int as second argument to ldexp.");
|
||
return NULL;
|
||
}
|
||
|
||
if (x == 0. || !Py_IS_FINITE(x)) {
|
||
/* NaNs, zeros and infinities are returned unchanged */
|
||
r = x;
|
||
errno = 0;
|
||
} else if (exp > 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;
|
||
r = ldexp(x, (int)exp);
|
||
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;
|
||
x = modf(x, &y);
|
||
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))
|
||
{
|
||
/* 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 (!_PyLong_IsPositive((PyLongObject *)arg)) {
|
||
PyErr_SetString(PyExc_ValueError,
|
||
"math domain error");
|
||
return NULL;
|
||
}
|
||
|
||
x = PyLong_AsDouble(arg);
|
||
if (x == -1.0 && PyErr_Occurred()) {
|
||
if (!PyErr_ExceptionMatches(PyExc_OverflowError))
|
||
return NULL;
|
||
/* Here the conversion to double overflowed, but it's possible
|
||
to compute the log anyway. Clear the exception and continue. */
|
||
PyErr_Clear();
|
||
x = _PyLong_Frexp((PyLongObject *)arg, &e);
|
||
if (x == -1.0 && PyErr_Occurred())
|
||
return NULL;
|
||
/* Value is ~= x * 2**e, so the log ~= log(x) + log(2) * e. */
|
||
result = func(x) + func(2.0) * e;
|
||
}
|
||
else
|
||
/* Successfully converted x to a double. */
|
||
result = func(x);
|
||
return PyFloat_FromDouble(result);
|
||
}
|
||
|
||
/* Else let libm handle it by itself. */
|
||
return math_1(arg, func, 0);
|
||
}
|
||
|
||
|
||
/* AC: cannot convert yet, see gh-102839 and gh-89381, waiting
|
||
for support of multiple signatures */
|
||
static PyObject *
|
||
math_log(PyObject *module, PyObject * const *args, Py_ssize_t nargs)
|
||
{
|
||
PyObject *num, *den;
|
||
PyObject *ans;
|
||
|
||
if (!_PyArg_CheckPositional("log", nargs, 1, 2))
|
||
return NULL;
|
||
|
||
num = loghelper(args[0], m_log);
|
||
if (num == NULL || nargs == 1)
|
||
return num;
|
||
|
||
den = loghelper(args[1], m_log);
|
||
if (den == NULL) {
|
||
Py_DECREF(num);
|
||
return NULL;
|
||
}
|
||
|
||
ans = PyNumber_TrueDivide(num, den);
|
||
Py_DECREF(num);
|
||
Py_DECREF(den);
|
||
return ans;
|
||
}
|
||
|
||
PyDoc_STRVAR(math_log_doc,
|
||
"log(x, [base=math.e])\n\
|
||
Return the logarithm of x to the given base.\n\n\
|
||
If the base is not specified, returns the natural logarithm (base e) of x.");
|
||
|
||
/*[clinic input]
|
||
math.log2
|
||
|
||
x: object
|
||
/
|
||
|
||
Return the base 2 logarithm of x.
|
||
[clinic start generated code]*/
|
||
|
||
static PyObject *
|
||
math_log2(PyObject *module, PyObject *x)
|
||
/*[clinic end generated code: output=5425899a4d5d6acb input=08321262bae4f39b]*/
|
||
{
|
||
return loghelper(x, m_log2);
|
||
}
|
||
|
||
|
||
/*[clinic input]
|
||
math.log10
|
||
|
||
x: object
|
||
/
|
||
|
||
Return the base 10 logarithm of x.
|
||
[clinic start generated code]*/
|
||
|
||
static PyObject *
|
||
math_log10(PyObject *module, PyObject *x)
|
||
/*[clinic end generated code: output=be72a64617df9c6f input=b2469d02c6469e53]*/
|
||
{
|
||
return loghelper(x, m_log10);
|
||
}
|
||
|
||
|
||
/*[clinic input]
|
||
math.fmod
|
||
|
||
x: double
|
||
y: double
|
||
/
|
||
|
||
Return fmod(x, y), according to platform C.
|
||
|
||
x % y may differ.
|
||
[clinic start generated code]*/
|
||
|
||
static PyObject *
|
||
math_fmod_impl(PyObject *module, double x, double y)
|
||
/*[clinic end generated code: output=7559d794343a27b5 input=4f84caa8cfc26a03]*/
|
||
{
|
||
double r;
|
||
/* fmod(x, +/-Inf) returns x for finite x. */
|
||
if (Py_IS_INFINITY(y) && Py_IS_FINITE(x))
|
||
return PyFloat_FromDouble(x);
|
||
errno = 0;
|
||
r = fmod(x, y);
|
||
if (Py_IS_NAN(r)) {
|
||
if (!Py_IS_NAN(x) && !Py_IS_NAN(y))
|
||
errno = EDOM;
|
||
else
|
||
errno = 0;
|
||
}
|
||
if (errno && is_error(r))
|
||
return NULL;
|
||
else
|
||
return PyFloat_FromDouble(r);
|
||
}
|
||
|
||
/*
|
||
Given a *vec* of values, compute the vector norm:
|
||
|
||
sqrt(sum(x ** 2 for x in vec))
|
||
|
||
The *max* variable should be equal to the largest fabs(x).
|
||
The *n* variable is the length of *vec*.
|
||
If n==0, then *max* should be 0.0.
|
||
If an infinity is present in the vec, *max* should be INF.
|
||
The *found_nan* variable indicates whether some member of
|
||
the *vec* is a NaN.
|
||
|
||
To avoid overflow/underflow and to achieve high accuracy giving results
|
||
that are almost always correctly rounded, four techniques are used:
|
||
|
||
* lossless scaling using a power-of-two scaling factor
|
||
* accurate squaring using Veltkamp-Dekker splitting [1]
|
||
or an equivalent with an fma() call
|
||
* compensated summation using a variant of the Neumaier algorithm [2]
|
||
* differential correction of the square root [3]
|
||
|
||
The usual presentation of the Neumaier summation algorithm has an
|
||
expensive branch depending on which operand has the larger
|
||
magnitude. We avoid this cost by arranging the calculation so that
|
||
fabs(csum) is always as large as fabs(x).
|
||
|
||
To establish the invariant, *csum* is initialized to 1.0 which is
|
||
always larger than x**2 after scaling or after division by *max*.
|
||
After the loop is finished, the initial 1.0 is subtracted out for a
|
||
net zero effect on the final sum. Since *csum* will be greater than
|
||
1.0, the subtraction of 1.0 will not cause fractional digits to be
|
||
dropped from *csum*.
|
||
|
||
To get the full benefit from compensated summation, the largest
|
||
addend should be in the range: 0.5 <= |x| <= 1.0. Accordingly,
|
||
scaling or division by *max* should not be skipped even if not
|
||
otherwise needed to prevent overflow or loss of precision.
|
||
|
||
The assertion that hi*hi <= 1.0 is a bit subtle. Each vector element
|
||
gets scaled to a magnitude below 1.0. The Veltkamp-Dekker splitting
|
||
algorithm gives a *hi* value that is correctly rounded to half
|
||
precision. When a value at or below 1.0 is correctly rounded, it
|
||
never goes above 1.0. And when values at or below 1.0 are squared,
|
||
they remain at or below 1.0, thus preserving the summation invariant.
|
||
|
||
Another interesting assertion is that csum+lo*lo == csum. In the loop,
|
||
each scaled vector element has a magnitude less than 1.0. After the
|
||
Veltkamp split, *lo* has a maximum value of 2**-27. So the maximum
|
||
value of *lo* squared is 2**-54. The value of ulp(1.0)/2.0 is 2**-53.
|
||
Given that csum >= 1.0, we have:
|
||
lo**2 <= 2**-54 < 2**-53 == 1/2*ulp(1.0) <= ulp(csum)/2
|
||
Since lo**2 is less than 1/2 ulp(csum), we have csum+lo*lo == csum.
|
||
|
||
To minimize loss of information during the accumulation of fractional
|
||
values, each term has a separate accumulator. This also breaks up
|
||
sequential dependencies in the inner loop so the CPU can maximize
|
||
floating point throughput. [4] On an Apple M1 Max, hypot(*vec)
|
||
takes only 3.33 µsec when len(vec) == 1000.
|
||
|
||
The square root differential correction is needed because a
|
||
correctly rounded square root of a correctly rounded sum of
|
||
squares can still be off by as much as one ulp.
|
||
|
||
The differential correction starts with a value *x* that is
|
||
the difference between the square of *h*, the possibly inaccurately
|
||
rounded square root, and the accurately computed sum of squares.
|
||
The correction is the first order term of the Maclaurin series
|
||
expansion of sqrt(h**2 + x) == h + x/(2*h) + O(x**2). [5]
|
||
|
||
Essentially, this differential correction is equivalent to one
|
||
refinement step in Newton's divide-and-average square root
|
||
algorithm, effectively doubling the number of accurate bits.
|
||
This technique is used in Dekker's SQRT2 algorithm and again in
|
||
Borges' ALGORITHM 4 and 5.
|
||
|
||
The hypot() function is faithfully rounded (less than 1 ulp error)
|
||
and usually correctly rounded (within 1/2 ulp). The squaring
|
||
step is exact. The Neumaier summation computes as if in doubled
|
||
precision (106 bits) and has the advantage that its input squares
|
||
are non-negative so that the condition number of the sum is one.
|
||
The square root with a differential correction is likewise computed
|
||
as if in doubled precision.
|
||
|
||
For n <= 1000, prior to the final addition that rounds the overall
|
||
result, the internal accuracy of "h" together with its correction of
|
||
"x / (2.0 * h)" is at least 100 bits. [6] Also, hypot() was tested
|
||
against a Decimal implementation with prec=300. After 100 million
|
||
trials, no incorrectly rounded examples were found. In addition,
|
||
perfect commutativity (all permutations are exactly equal) was
|
||
verified for 1 billion random inputs with n=5. [7]
|
||
|
||
References:
|
||
|
||
1. Veltkamp-Dekker splitting: http://csclub.uwaterloo.ca/~pbarfuss/dekker1971.pdf
|
||
2. Compensated summation: http://www.ti3.tu-harburg.de/paper/rump/Ru08b.pdf
|
||
3. Square root differential correction: https://arxiv.org/pdf/1904.09481.pdf
|
||
4. Data dependency graph: https://bugs.python.org/file49439/hypot.png
|
||
5. https://www.wolframalpha.com/input/?i=Maclaurin+series+sqrt%28h**2+%2B+x%29+at+x%3D0
|
||
6. Analysis of internal accuracy: https://bugs.python.org/file49484/best_frac.py
|
||
7. Commutativity test: https://bugs.python.org/file49448/test_hypot_commutativity.py
|
||
|
||
*/
|
||
|
||
static inline double
|
||
vector_norm(Py_ssize_t n, double *vec, double max, int found_nan)
|
||
{
|
||
double x, h, scale, csum = 1.0, frac1 = 0.0, frac2 = 0.0;
|
||
DoubleLength pr, sm;
|
||
int max_e;
|
||
Py_ssize_t i;
|
||
|
||
if (Py_IS_INFINITY(max)) {
|
||
return max;
|
||
}
|
||
if (found_nan) {
|
||
return Py_NAN;
|
||
}
|
||
if (max == 0.0 || n <= 1) {
|
||
return max;
|
||
}
|
||
frexp(max, &max_e);
|
||
if (max_e < -1023) {
|
||
/* When max_e < -1023, ldexp(1.0, -max_e) would overflow. */
|
||
for (i=0 ; i < n ; i++) {
|
||
vec[i] /= DBL_MIN; // convert subnormals to normals
|
||
}
|
||
return DBL_MIN * vector_norm(n, vec, max / DBL_MIN, found_nan);
|
||
}
|
||
scale = ldexp(1.0, -max_e);
|
||
assert(max * scale >= 0.5);
|
||
assert(max * scale < 1.0);
|
||
for (i=0 ; i < n ; i++) {
|
||
x = vec[i];
|
||
assert(Py_IS_FINITE(x) && fabs(x) <= max);
|
||
x *= scale; // lossless scaling
|
||
assert(fabs(x) < 1.0);
|
||
pr = dl_mul(x, x); // lossless squaring
|
||
assert(pr.hi <= 1.0);
|
||
sm = dl_fast_sum(csum, pr.hi); // lossless addition
|
||
csum = sm.hi;
|
||
frac1 += pr.lo; // lossy addition
|
||
frac2 += sm.lo; // lossy addition
|
||
}
|
||
h = sqrt(csum - 1.0 + (frac1 + frac2));
|
||
pr = dl_mul(-h, h);
|
||
sm = dl_fast_sum(csum, pr.hi);
|
||
csum = sm.hi;
|
||
frac1 += pr.lo;
|
||
frac2 += sm.lo;
|
||
x = csum - 1.0 + (frac1 + frac2);
|
||
h += x / (2.0 * h); // differential correction
|
||
return h / scale;
|
||
}
|
||
|
||
#define NUM_STACK_ELEMS 16
|
||
|
||
/*[clinic input]
|
||
math.dist
|
||
|
||
p: object
|
||
q: object
|
||
/
|
||
|
||
Return the Euclidean distance between two points p and q.
|
||
|
||
The points should be specified as sequences (or iterables) of
|
||
coordinates. Both inputs must have the same dimension.
|
||
|
||
Roughly equivalent to:
|
||
sqrt(sum((px - qx) ** 2.0 for px, qx in zip(p, q)))
|
||
[clinic start generated code]*/
|
||
|
||
static PyObject *
|
||
math_dist_impl(PyObject *module, PyObject *p, PyObject *q)
|
||
/*[clinic end generated code: output=56bd9538d06bbcfe input=74e85e1b6092e68e]*/
|
||
{
|
||
PyObject *item;
|
||
double max = 0.0;
|
||
double x, px, qx, result;
|
||
Py_ssize_t i, m, n;
|
||
int found_nan = 0, p_allocated = 0, q_allocated = 0;
|
||
double diffs_on_stack[NUM_STACK_ELEMS];
|
||
double *diffs = diffs_on_stack;
|
||
|
||
if (!PyTuple_Check(p)) {
|
||
p = PySequence_Tuple(p);
|
||
if (p == NULL) {
|
||
return NULL;
|
||
}
|
||
p_allocated = 1;
|
||
}
|
||
if (!PyTuple_Check(q)) {
|
||
q = PySequence_Tuple(q);
|
||
if (q == NULL) {
|
||
if (p_allocated) {
|
||
Py_DECREF(p);
|
||
}
|
||
return NULL;
|
||
}
|
||
q_allocated = 1;
|
||
}
|
||
|
||
m = PyTuple_GET_SIZE(p);
|
||
n = PyTuple_GET_SIZE(q);
|
||
if (m != n) {
|
||
PyErr_SetString(PyExc_ValueError,
|
||
"both points must have the same number of dimensions");
|
||
goto error_exit;
|
||
}
|
||
if (n > NUM_STACK_ELEMS) {
|
||
diffs = (double *) PyObject_Malloc(n * sizeof(double));
|
||
if (diffs == NULL) {
|
||
PyErr_NoMemory();
|
||
goto error_exit;
|
||
}
|
||
}
|
||
for (i=0 ; i<n ; i++) {
|
||
item = PyTuple_GET_ITEM(p, i);
|
||
ASSIGN_DOUBLE(px, item, error_exit);
|
||
item = PyTuple_GET_ITEM(q, i);
|
||
ASSIGN_DOUBLE(qx, item, error_exit);
|
||
x = fabs(px - qx);
|
||
diffs[i] = x;
|
||
found_nan |= Py_IS_NAN(x);
|
||
if (x > max) {
|
||
max = x;
|
||
}
|
||
}
|
||
result = vector_norm(n, diffs, max, found_nan);
|
||
if (diffs != diffs_on_stack) {
|
||
PyObject_Free(diffs);
|
||
}
|
||
if (p_allocated) {
|
||
Py_DECREF(p);
|
||
}
|
||
if (q_allocated) {
|
||
Py_DECREF(q);
|
||
}
|
||
return PyFloat_FromDouble(result);
|
||
|
||
error_exit:
|
||
if (diffs != diffs_on_stack) {
|
||
PyObject_Free(diffs);
|
||
}
|
||
if (p_allocated) {
|
||
Py_DECREF(p);
|
||
}
|
||
if (q_allocated) {
|
||
Py_DECREF(q);
|
||
}
|
||
return NULL;
|
||
}
|
||
|
||
/* AC: cannot convert yet, waiting for *args support */
|
||
static PyObject *
|
||
math_hypot(PyObject *self, PyObject *const *args, Py_ssize_t nargs)
|
||
{
|
||
Py_ssize_t i;
|
||
PyObject *item;
|
||
double max = 0.0;
|
||
double x, result;
|
||
int found_nan = 0;
|
||
double coord_on_stack[NUM_STACK_ELEMS];
|
||
double *coordinates = coord_on_stack;
|
||
|
||
if (nargs > NUM_STACK_ELEMS) {
|
||
coordinates = (double *) PyObject_Malloc(nargs * sizeof(double));
|
||
if (coordinates == NULL) {
|
||
return PyErr_NoMemory();
|
||
}
|
||
}
|
||
for (i = 0; i < nargs; i++) {
|
||
item = args[i];
|
||
ASSIGN_DOUBLE(x, item, error_exit);
|
||
x = fabs(x);
|
||
coordinates[i] = x;
|
||
found_nan |= Py_IS_NAN(x);
|
||
if (x > max) {
|
||
max = x;
|
||
}
|
||
}
|
||
result = vector_norm(nargs, coordinates, max, found_nan);
|
||
if (coordinates != coord_on_stack) {
|
||
PyObject_Free(coordinates);
|
||
}
|
||
return PyFloat_FromDouble(result);
|
||
|
||
error_exit:
|
||
if (coordinates != coord_on_stack) {
|
||
PyObject_Free(coordinates);
|
||
}
|
||
return NULL;
|
||
}
|
||
|
||
#undef NUM_STACK_ELEMS
|
||
|
||
PyDoc_STRVAR(math_hypot_doc,
|
||
"hypot(*coordinates) -> value\n\n\
|
||
Multidimensional Euclidean distance from the origin to a point.\n\
|
||
\n\
|
||
Roughly equivalent to:\n\
|
||
sqrt(sum(x**2 for x in coordinates))\n\
|
||
\n\
|
||
For a two dimensional point (x, y), gives the hypotenuse\n\
|
||
using the Pythagorean theorem: sqrt(x*x + y*y).\n\
|
||
\n\
|
||
For example, the hypotenuse of a 3/4/5 right triangle is:\n\
|
||
\n\
|
||
>>> hypot(3.0, 4.0)\n\
|
||
5.0\n\
|
||
");
|
||
|
||
/** sumprod() ***************************************************************/
|
||
|
||
/* Forward declaration */
|
||
static inline int _check_long_mult_overflow(long a, long b);
|
||
|
||
static inline bool
|
||
long_add_would_overflow(long a, long b)
|
||
{
|
||
return (a > 0) ? (b > LONG_MAX - a) : (b < LONG_MIN - a);
|
||
}
|
||
|
||
/*[clinic input]
|
||
math.sumprod
|
||
|
||
p: object
|
||
q: object
|
||
/
|
||
|
||
Return the sum of products of values from two iterables p and q.
|
||
|
||
Roughly equivalent to:
|
||
|
||
sum(itertools.starmap(operator.mul, zip(p, q, strict=True)))
|
||
|
||
For float and mixed int/float inputs, the intermediate products
|
||
and sums are computed with extended precision.
|
||
[clinic start generated code]*/
|
||
|
||
static PyObject *
|
||
math_sumprod_impl(PyObject *module, PyObject *p, PyObject *q)
|
||
/*[clinic end generated code: output=6722dbfe60664554 input=82be54fe26f87e30]*/
|
||
{
|
||
PyObject *p_i = NULL, *q_i = NULL, *term_i = NULL, *new_total = NULL;
|
||
PyObject *p_it, *q_it, *total;
|
||
iternextfunc p_next, q_next;
|
||
bool p_stopped = false, q_stopped = false;
|
||
bool int_path_enabled = true, int_total_in_use = false;
|
||
bool flt_path_enabled = true, flt_total_in_use = false;
|
||
long int_total = 0;
|
||
TripleLength flt_total = tl_zero;
|
||
|
||
p_it = PyObject_GetIter(p);
|
||
if (p_it == NULL) {
|
||
return NULL;
|
||
}
|
||
q_it = PyObject_GetIter(q);
|
||
if (q_it == NULL) {
|
||
Py_DECREF(p_it);
|
||
return NULL;
|
||
}
|
||
total = PyLong_FromLong(0);
|
||
if (total == NULL) {
|
||
Py_DECREF(p_it);
|
||
Py_DECREF(q_it);
|
||
return NULL;
|
||
}
|
||
p_next = *Py_TYPE(p_it)->tp_iternext;
|
||
q_next = *Py_TYPE(q_it)->tp_iternext;
|
||
while (1) {
|
||
bool finished;
|
||
|
||
assert (p_i == NULL);
|
||
assert (q_i == NULL);
|
||
assert (term_i == NULL);
|
||
assert (new_total == NULL);
|
||
|
||
assert (p_it != NULL);
|
||
assert (q_it != NULL);
|
||
assert (total != NULL);
|
||
|
||
p_i = p_next(p_it);
|
||
if (p_i == NULL) {
|
||
if (PyErr_Occurred()) {
|
||
if (!PyErr_ExceptionMatches(PyExc_StopIteration)) {
|
||
goto err_exit;
|
||
}
|
||
PyErr_Clear();
|
||
}
|
||
p_stopped = true;
|
||
}
|
||
q_i = q_next(q_it);
|
||
if (q_i == NULL) {
|
||
if (PyErr_Occurred()) {
|
||
if (!PyErr_ExceptionMatches(PyExc_StopIteration)) {
|
||
goto err_exit;
|
||
}
|
||
PyErr_Clear();
|
||
}
|
||
q_stopped = true;
|
||
}
|
||
if (p_stopped != q_stopped) {
|
||
PyErr_Format(PyExc_ValueError, "Inputs are not the same length");
|
||
goto err_exit;
|
||
}
|
||
finished = p_stopped & q_stopped;
|
||
|
||
if (int_path_enabled) {
|
||
|
||
if (!finished && PyLong_CheckExact(p_i) & PyLong_CheckExact(q_i)) {
|
||
int overflow;
|
||
long int_p, int_q, int_prod;
|
||
|
||
int_p = PyLong_AsLongAndOverflow(p_i, &overflow);
|
||
if (overflow) {
|
||
goto finalize_int_path;
|
||
}
|
||
int_q = PyLong_AsLongAndOverflow(q_i, &overflow);
|
||
if (overflow) {
|
||
goto finalize_int_path;
|
||
}
|
||
if (_check_long_mult_overflow(int_p, int_q)) {
|
||
goto finalize_int_path;
|
||
}
|
||
int_prod = int_p * int_q;
|
||
if (long_add_would_overflow(int_total, int_prod)) {
|
||
goto finalize_int_path;
|
||
}
|
||
int_total += int_prod;
|
||
int_total_in_use = true;
|
||
Py_CLEAR(p_i);
|
||
Py_CLEAR(q_i);
|
||
continue;
|
||
}
|
||
|
||
finalize_int_path:
|
||
// We're finished, overflowed, or have a non-int
|
||
int_path_enabled = false;
|
||
if (int_total_in_use) {
|
||
term_i = PyLong_FromLong(int_total);
|
||
if (term_i == NULL) {
|
||
goto err_exit;
|
||
}
|
||
new_total = PyNumber_Add(total, term_i);
|
||
if (new_total == NULL) {
|
||
goto err_exit;
|
||
}
|
||
Py_SETREF(total, new_total);
|
||
new_total = NULL;
|
||
Py_CLEAR(term_i);
|
||
int_total = 0; // An ounce of prevention, ...
|
||
int_total_in_use = false;
|
||
}
|
||
}
|
||
|
||
if (flt_path_enabled) {
|
||
|
||
if (!finished) {
|
||
double flt_p, flt_q;
|
||
bool p_type_float = PyFloat_CheckExact(p_i);
|
||
bool q_type_float = PyFloat_CheckExact(q_i);
|
||
if (p_type_float && q_type_float) {
|
||
flt_p = PyFloat_AS_DOUBLE(p_i);
|
||
flt_q = PyFloat_AS_DOUBLE(q_i);
|
||
} else if (p_type_float && (PyLong_CheckExact(q_i) || PyBool_Check(q_i))) {
|
||
/* We care about float/int pairs and int/float pairs because
|
||
they arise naturally in several use cases such as price
|
||
times quantity, measurements with integer weights, or
|
||
data selected by a vector of bools. */
|
||
flt_p = PyFloat_AS_DOUBLE(p_i);
|
||
flt_q = PyLong_AsDouble(q_i);
|
||
if (flt_q == -1.0 && PyErr_Occurred()) {
|
||
PyErr_Clear();
|
||
goto finalize_flt_path;
|
||
}
|
||
} else if (q_type_float && (PyLong_CheckExact(p_i) || PyBool_Check(q_i))) {
|
||
flt_q = PyFloat_AS_DOUBLE(q_i);
|
||
flt_p = PyLong_AsDouble(p_i);
|
||
if (flt_p == -1.0 && PyErr_Occurred()) {
|
||
PyErr_Clear();
|
||
goto finalize_flt_path;
|
||
}
|
||
} else {
|
||
goto finalize_flt_path;
|
||
}
|
||
TripleLength new_flt_total = tl_fma(flt_p, flt_q, flt_total);
|
||
if (isfinite(new_flt_total.hi)) {
|
||
flt_total = new_flt_total;
|
||
flt_total_in_use = true;
|
||
Py_CLEAR(p_i);
|
||
Py_CLEAR(q_i);
|
||
continue;
|
||
}
|
||
}
|
||
|
||
finalize_flt_path:
|
||
// We're finished, overflowed, have a non-float, or got a non-finite value
|
||
flt_path_enabled = false;
|
||
if (flt_total_in_use) {
|
||
term_i = PyFloat_FromDouble(tl_to_d(flt_total));
|
||
if (term_i == NULL) {
|
||
goto err_exit;
|
||
}
|
||
new_total = PyNumber_Add(total, term_i);
|
||
if (new_total == NULL) {
|
||
goto err_exit;
|
||
}
|
||
Py_SETREF(total, new_total);
|
||
new_total = NULL;
|
||
Py_CLEAR(term_i);
|
||
flt_total = tl_zero;
|
||
flt_total_in_use = false;
|
||
}
|
||
}
|
||
|
||
assert(!int_total_in_use);
|
||
assert(!flt_total_in_use);
|
||
if (finished) {
|
||
goto normal_exit;
|
||
}
|
||
term_i = PyNumber_Multiply(p_i, q_i);
|
||
if (term_i == NULL) {
|
||
goto err_exit;
|
||
}
|
||
new_total = PyNumber_Add(total, term_i);
|
||
if (new_total == NULL) {
|
||
goto err_exit;
|
||
}
|
||
Py_SETREF(total, new_total);
|
||
new_total = NULL;
|
||
Py_CLEAR(p_i);
|
||
Py_CLEAR(q_i);
|
||
Py_CLEAR(term_i);
|
||
}
|
||
|
||
normal_exit:
|
||
Py_DECREF(p_it);
|
||
Py_DECREF(q_it);
|
||
return total;
|
||
|
||
err_exit:
|
||
Py_DECREF(p_it);
|
||
Py_DECREF(q_it);
|
||
Py_DECREF(total);
|
||
Py_XDECREF(p_i);
|
||
Py_XDECREF(q_i);
|
||
Py_XDECREF(term_i);
|
||
Py_XDECREF(new_total);
|
||
return NULL;
|
||
}
|
||
|
||
|
||
/* pow can't use math_2, but needs its own wrapper: the problem is
|
||
that an infinite result can arise either as a result of overflow
|
||
(in which case OverflowError should be raised) or as a result of
|
||
e.g. 0.**-5. (for which ValueError needs to be raised.)
|
||
*/
|
||
|
||
/*[clinic input]
|
||
math.pow
|
||
|
||
x: double
|
||
y: double
|
||
/
|
||
|
||
Return x**y (x to the power of y).
|
||
[clinic start generated code]*/
|
||
|
||
static PyObject *
|
||
math_pow_impl(PyObject *module, double x, double y)
|
||
/*[clinic end generated code: output=fff93e65abccd6b0 input=c26f1f6075088bfd]*/
|
||
{
|
||
double r;
|
||
int odd_y;
|
||
|
||
/* deal directly with IEEE specials, to cope with problems on various
|
||
platforms whose semantics don't exactly match C99 */
|
||
r = 0.; /* silence compiler warning */
|
||
if (!Py_IS_FINITE(x) || !Py_IS_FINITE(y)) {
|
||
errno = 0;
|
||
if (Py_IS_NAN(x))
|
||
r = y == 0. ? 1. : x; /* NaN**0 = 1 */
|
||
else if (Py_IS_NAN(y))
|
||
r = x == 1. ? 1. : y; /* 1**NaN = 1 */
|
||
else if (Py_IS_INFINITY(x)) {
|
||
odd_y = Py_IS_FINITE(y) && fmod(fabs(y), 2.0) == 1.0;
|
||
if (y > 0.)
|
||
r = odd_y ? x : fabs(x);
|
||
else if (y == 0.)
|
||
r = 1.;
|
||
else /* y < 0. */
|
||
r = odd_y ? copysign(0., x) : 0.;
|
||
}
|
||
else if (Py_IS_INFINITY(y)) {
|
||
if (fabs(x) == 1.0)
|
||
r = 1.;
|
||
else if (y > 0. && fabs(x) > 1.0)
|
||
r = y;
|
||
else if (y < 0. && fabs(x) < 1.0) {
|
||
r = -y; /* result is +inf */
|
||
}
|
||
else
|
||
r = 0.;
|
||
}
|
||
}
|
||
else {
|
||
/* let libm handle finite**finite */
|
||
errno = 0;
|
||
r = pow(x, y);
|
||
/* a NaN result should arise only from (-ve)**(finite
|
||
non-integer); in this case we want to raise ValueError. */
|
||
if (!Py_IS_FINITE(r)) {
|
||
if (Py_IS_NAN(r)) {
|
||
errno = EDOM;
|
||
}
|
||
/*
|
||
an infinite result here arises either from:
|
||
(A) (+/-0.)**negative (-> divide-by-zero)
|
||
(B) overflow of x**y with x and y finite
|
||
*/
|
||
else if (Py_IS_INFINITY(r)) {
|
||
if (x == 0.)
|
||
errno = EDOM;
|
||
else
|
||
errno = ERANGE;
|
||
}
|
||
}
|
||
}
|
||
|
||
if (errno && is_error(r))
|
||
return NULL;
|
||
else
|
||
return PyFloat_FromDouble(r);
|
||
}
|
||
|
||
|
||
static const double degToRad = Py_MATH_PI / 180.0;
|
||
static const double radToDeg = 180.0 / Py_MATH_PI;
|
||
|
||
/*[clinic input]
|
||
math.degrees
|
||
|
||
x: double
|
||
/
|
||
|
||
Convert angle x from radians to degrees.
|
||
[clinic start generated code]*/
|
||
|
||
static PyObject *
|
||
math_degrees_impl(PyObject *module, double x)
|
||
/*[clinic end generated code: output=7fea78b294acd12f input=81e016555d6e3660]*/
|
||
{
|
||
return PyFloat_FromDouble(x * radToDeg);
|
||
}
|
||
|
||
|
||
/*[clinic input]
|
||
math.radians
|
||
|
||
x: double
|
||
/
|
||
|
||
Convert angle x from degrees to radians.
|
||
[clinic start generated code]*/
|
||
|
||
static PyObject *
|
||
math_radians_impl(PyObject *module, double x)
|
||
/*[clinic end generated code: output=34daa47caf9b1590 input=91626fc489fe3d63]*/
|
||
{
|
||
return PyFloat_FromDouble(x * degToRad);
|
||
}
|
||
|
||
|
||
/*[clinic input]
|
||
math.isfinite
|
||
|
||
x: double
|
||
/
|
||
|
||
Return True if x is neither an infinity nor a NaN, and False otherwise.
|
||
[clinic start generated code]*/
|
||
|
||
static PyObject *
|
||
math_isfinite_impl(PyObject *module, double x)
|
||
/*[clinic end generated code: output=8ba1f396440c9901 input=46967d254812e54a]*/
|
||
{
|
||
return PyBool_FromLong((long)Py_IS_FINITE(x));
|
||
}
|
||
|
||
|
||
/*[clinic input]
|
||
math.isnan
|
||
|
||
x: double
|
||
/
|
||
|
||
Return True if x is a NaN (not a number), and False otherwise.
|
||
[clinic start generated code]*/
|
||
|
||
static PyObject *
|
||
math_isnan_impl(PyObject *module, double x)
|
||
/*[clinic end generated code: output=f537b4d6df878c3e input=935891e66083f46a]*/
|
||
{
|
||
return PyBool_FromLong((long)Py_IS_NAN(x));
|
||
}
|
||
|
||
|
||
/*[clinic input]
|
||
math.isinf
|
||
|
||
x: double
|
||
/
|
||
|
||
Return True if x is a positive or negative infinity, and False otherwise.
|
||
[clinic start generated code]*/
|
||
|
||
static PyObject *
|
||
math_isinf_impl(PyObject *module, double x)
|
||
/*[clinic end generated code: output=9f00cbec4de7b06b input=32630e4212cf961f]*/
|
||
{
|
||
return PyBool_FromLong((long)Py_IS_INFINITY(x));
|
||
}
|
||
|
||
|
||
/*[clinic input]
|
||
math.isclose -> bool
|
||
|
||
a: double
|
||
b: double
|
||
*
|
||
rel_tol: double = 1e-09
|
||
maximum difference for being considered "close", relative to the
|
||
magnitude of the input values
|
||
abs_tol: double = 0.0
|
||
maximum difference for being considered "close", regardless of the
|
||
magnitude of the input values
|
||
|
||
Determine whether two floating point numbers are close in value.
|
||
|
||
Return True if a is close in value to b, and False otherwise.
|
||
|
||
For the values to be considered close, the difference between them
|
||
must be smaller than at least one of the tolerances.
|
||
|
||
-inf, inf and NaN behave similarly to the IEEE 754 Standard. That
|
||
is, NaN is not close to anything, even itself. inf and -inf are
|
||
only close to themselves.
|
||
[clinic start generated code]*/
|
||
|
||
static int
|
||
math_isclose_impl(PyObject *module, double a, double b, double rel_tol,
|
||
double abs_tol)
|
||
/*[clinic end generated code: output=b73070207511952d input=f28671871ea5bfba]*/
|
||
{
|
||
double diff = 0.0;
|
||
|
||
/* sanity check on the inputs */
|
||
if (rel_tol < 0.0 || abs_tol < 0.0 ) {
|
||
PyErr_SetString(PyExc_ValueError,
|
||
"tolerances must be non-negative");
|
||
return -1;
|
||
}
|
||
|
||
if ( a == b ) {
|
||
/* short circuit exact equality -- needed to catch two infinities of
|
||
the same sign. And perhaps speeds things up a bit sometimes.
|
||
*/
|
||
return 1;
|
||
}
|
||
|
||
/* 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)) {
|
||
return 0;
|
||
}
|
||
|
||
/* now do the regular computation
|
||
this is essentially the "weak" test from the Boost library
|
||
*/
|
||
|
||
diff = fabs(b - a);
|
||
|
||
return (((diff <= fabs(rel_tol * b)) ||
|
||
(diff <= fabs(rel_tol * a))) ||
|
||
(diff <= abs_tol));
|
||
}
|
||
|
||
static inline int
|
||
_check_long_mult_overflow(long a, long b) {
|
||
|
||
/* From Python2's int_mul code:
|
||
|
||
Integer overflow checking for * is painful: Python tried a couple ways, but
|
||
they didn't work on all platforms, or failed in endcases (a product of
|
||
-sys.maxint-1 has been a particular pain).
|
||
|
||
Here's another way:
|
||
|
||
The native long product x*y is either exactly right or *way* off, being
|
||
just the last n bits of the true product, where n is the number of bits
|
||
in a long (the delivered product is the true product plus i*2**n for
|
||
some integer i).
|
||
|
||
The native double product (double)x * (double)y is subject to three
|
||
rounding errors: on a sizeof(long)==8 box, each cast to double can lose
|
||
info, and even on a sizeof(long)==4 box, the multiplication can lose info.
|
||
But, unlike the native long product, it's not in *range* trouble: even
|
||
if sizeof(long)==32 (256-bit longs), the product easily fits in the
|
||
dynamic range of a double. So the leading 50 (or so) bits of the double
|
||
product are correct.
|
||
|
||
We check these two ways against each other, and declare victory if they're
|
||
approximately the same. Else, because the native long product is the only
|
||
one that can lose catastrophic amounts of information, it's the native long
|
||
product that must have overflowed.
|
||
|
||
*/
|
||
|
||
long longprod = (long)((unsigned long)a * b);
|
||
double doubleprod = (double)a * (double)b;
|
||
double doubled_longprod = (double)longprod;
|
||
|
||
if (doubled_longprod == doubleprod) {
|
||
return 0;
|
||
}
|
||
|
||
const double diff = doubled_longprod - doubleprod;
|
||
const double absdiff = diff >= 0.0 ? diff : -diff;
|
||
const double absprod = doubleprod >= 0.0 ? doubleprod : -doubleprod;
|
||
|
||
if (32.0 * absdiff <= absprod) {
|
||
return 0;
|
||
}
|
||
|
||
return 1;
|
||
}
|
||
|
||
/*[clinic input]
|
||
math.prod
|
||
|
||
iterable: object
|
||
/
|
||
*
|
||
start: object(c_default="NULL") = 1
|
||
|
||
Calculate the product of all the elements in the input iterable.
|
||
|
||
The default start value for the product is 1.
|
||
|
||
When the iterable is empty, return the start value. This function is
|
||
intended specifically for use with numeric values and may reject
|
||
non-numeric types.
|
||
[clinic start generated code]*/
|
||
|
||
static PyObject *
|
||
math_prod_impl(PyObject *module, PyObject *iterable, PyObject *start)
|
||
/*[clinic end generated code: output=36153bedac74a198 input=4c5ab0682782ed54]*/
|
||
{
|
||
PyObject *result = start;
|
||
PyObject *temp, *item, *iter;
|
||
|
||
iter = PyObject_GetIter(iterable);
|
||
if (iter == NULL) {
|
||
return NULL;
|
||
}
|
||
|
||
if (result == NULL) {
|
||
result = _PyLong_GetOne();
|
||
}
|
||
Py_INCREF(result);
|
||
#ifndef SLOW_PROD
|
||
/* Fast paths for integers keeping temporary products in C.
|
||
* Assumes all inputs are the same type.
|
||
* If the assumption fails, default to use PyObjects instead.
|
||
*/
|
||
if (PyLong_CheckExact(result)) {
|
||
int overflow;
|
||
long i_result = PyLong_AsLongAndOverflow(result, &overflow);
|
||
/* If this already overflowed, don't even enter the loop. */
|
||
if (overflow == 0) {
|
||
Py_SETREF(result, NULL);
|
||
}
|
||
/* Loop over all the items in the iterable until we finish, we overflow
|
||
* or we found a non integer element */
|
||
while (result == NULL) {
|
||
item = PyIter_Next(iter);
|
||
if (item == NULL) {
|
||
Py_DECREF(iter);
|
||
if (PyErr_Occurred()) {
|
||
return NULL;
|
||
}
|
||
return PyLong_FromLong(i_result);
|
||
}
|
||
if (PyLong_CheckExact(item)) {
|
||
long b = PyLong_AsLongAndOverflow(item, &overflow);
|
||
if (overflow == 0 && !_check_long_mult_overflow(i_result, b)) {
|
||
long x = i_result * b;
|
||
i_result = x;
|
||
Py_DECREF(item);
|
||
continue;
|
||
}
|
||
}
|
||
/* Either overflowed or is not an int.
|
||
* Restore real objects and process normally */
|
||
result = PyLong_FromLong(i_result);
|
||
if (result == NULL) {
|
||
Py_DECREF(item);
|
||
Py_DECREF(iter);
|
||
return NULL;
|
||
}
|
||
temp = PyNumber_Multiply(result, item);
|
||
Py_DECREF(result);
|
||
Py_DECREF(item);
|
||
result = temp;
|
||
if (result == NULL) {
|
||
Py_DECREF(iter);
|
||
return NULL;
|
||
}
|
||
}
|
||
}
|
||
|
||
/* Fast paths for floats keeping temporary products in C.
|
||
* Assumes all inputs are the same type.
|
||
* If the assumption fails, default to use PyObjects instead.
|
||
*/
|
||
if (PyFloat_CheckExact(result)) {
|
||
double f_result = PyFloat_AS_DOUBLE(result);
|
||
Py_SETREF(result, NULL);
|
||
while(result == NULL) {
|
||
item = PyIter_Next(iter);
|
||
if (item == NULL) {
|
||
Py_DECREF(iter);
|
||
if (PyErr_Occurred()) {
|
||
return NULL;
|
||
}
|
||
return PyFloat_FromDouble(f_result);
|
||
}
|
||
if (PyFloat_CheckExact(item)) {
|
||
f_result *= PyFloat_AS_DOUBLE(item);
|
||
Py_DECREF(item);
|
||
continue;
|
||
}
|
||
if (PyLong_CheckExact(item)) {
|
||
long value;
|
||
int overflow;
|
||
value = PyLong_AsLongAndOverflow(item, &overflow);
|
||
if (!overflow) {
|
||
f_result *= (double)value;
|
||
Py_DECREF(item);
|
||
continue;
|
||
}
|
||
}
|
||
result = PyFloat_FromDouble(f_result);
|
||
if (result == NULL) {
|
||
Py_DECREF(item);
|
||
Py_DECREF(iter);
|
||
return NULL;
|
||
}
|
||
temp = PyNumber_Multiply(result, item);
|
||
Py_DECREF(result);
|
||
Py_DECREF(item);
|
||
result = temp;
|
||
if (result == NULL) {
|
||
Py_DECREF(iter);
|
||
return NULL;
|
||
}
|
||
}
|
||
}
|
||
#endif
|
||
/* Consume rest of the iterable (if any) that could not be handled
|
||
* by specialized functions above.*/
|
||
for(;;) {
|
||
item = PyIter_Next(iter);
|
||
if (item == NULL) {
|
||
/* error, or end-of-sequence */
|
||
if (PyErr_Occurred()) {
|
||
Py_SETREF(result, NULL);
|
||
}
|
||
break;
|
||
}
|
||
temp = PyNumber_Multiply(result, item);
|
||
Py_DECREF(result);
|
||
Py_DECREF(item);
|
||
result = temp;
|
||
if (result == NULL)
|
||
break;
|
||
}
|
||
Py_DECREF(iter);
|
||
return result;
|
||
}
|
||
|
||
|
||
/* least significant 64 bits of the odd part of factorial(n), for n in range(128).
|
||
|
||
Python code to generate the values:
|
||
|
||
import math
|
||
|
||
for n in range(128):
|
||
fac = math.factorial(n)
|
||
fac_odd_part = fac // (fac & -fac)
|
||
reduced_fac_odd_part = fac_odd_part % (2**64)
|
||
print(f"{reduced_fac_odd_part:#018x}u")
|
||
*/
|
||
static const uint64_t reduced_factorial_odd_part[] = {
|
||
0x0000000000000001u, 0x0000000000000001u, 0x0000000000000001u, 0x0000000000000003u,
|
||
0x0000000000000003u, 0x000000000000000fu, 0x000000000000002du, 0x000000000000013bu,
|
||
0x000000000000013bu, 0x0000000000000b13u, 0x000000000000375fu, 0x0000000000026115u,
|
||
0x000000000007233fu, 0x00000000005cca33u, 0x0000000002898765u, 0x00000000260eeeebu,
|
||
0x00000000260eeeebu, 0x0000000286fddd9bu, 0x00000016beecca73u, 0x000001b02b930689u,
|
||
0x00000870d9df20adu, 0x0000b141df4dae31u, 0x00079dd498567c1bu, 0x00af2e19afc5266du,
|
||
0x020d8a4d0f4f7347u, 0x335281867ec241efu, 0x9b3093d46fdd5923u, 0x5e1f9767cc5866b1u,
|
||
0x92dd23d6966aced7u, 0xa30d0f4f0a196e5bu, 0x8dc3e5a1977d7755u, 0x2ab8ce915831734bu,
|
||
0x2ab8ce915831734bu, 0x81d2a0bc5e5fdcabu, 0x9efcac82445da75bu, 0xbc8b95cf58cde171u,
|
||
0xa0e8444a1f3cecf9u, 0x4191deb683ce3ffdu, 0xddd3878bc84ebfc7u, 0xcb39a64b83ff3751u,
|
||
0xf8203f7993fc1495u, 0xbd2a2a78b35f4bddu, 0x84757be6b6d13921u, 0x3fbbcfc0b524988bu,
|
||
0xbd11ed47c8928df9u, 0x3c26b59e41c2f4c5u, 0x677a5137e883fdb3u, 0xff74e943b03b93ddu,
|
||
0xfe5ebbcb10b2bb97u, 0xb021f1de3235e7e7u, 0x33509eb2e743a58fu, 0x390f9da41279fb7du,
|
||
0xe5cb0154f031c559u, 0x93074695ba4ddb6du, 0x81c471caa636247fu, 0xe1347289b5a1d749u,
|
||
0x286f21c3f76ce2ffu, 0x00be84a2173e8ac7u, 0x1595065ca215b88bu, 0xf95877595b018809u,
|
||
0x9c2efe3c5516f887u, 0x373294604679382bu, 0xaf1ff7a888adcd35u, 0x18ddf279a2c5800bu,
|
||
0x18ddf279a2c5800bu, 0x505a90e2542582cbu, 0x5bacad2cd8d5dc2bu, 0xfe3152bcbff89f41u,
|
||
0xe1467e88bf829351u, 0xb8001adb9e31b4d5u, 0x2803ac06a0cbb91fu, 0x1904b5d698805799u,
|
||
0xe12a648b5c831461u, 0x3516abbd6160cfa9u, 0xac46d25f12fe036du, 0x78bfa1da906b00efu,
|
||
0xf6390338b7f111bdu, 0x0f25f80f538255d9u, 0x4ec8ca55b8db140fu, 0x4ff670740b9b30a1u,
|
||
0x8fd032443a07f325u, 0x80dfe7965c83eeb5u, 0xa3dc1714d1213afdu, 0x205b7bbfcdc62007u,
|
||
0xa78126bbe140a093u, 0x9de1dc61ca7550cfu, 0x84f0046d01b492c5u, 0x2d91810b945de0f3u,
|
||
0xf5408b7f6008aa71u, 0x43707f4863034149u, 0xdac65fb9679279d5u, 0xc48406e7d1114eb7u,
|
||
0xa7dc9ed3c88e1271u, 0xfb25b2efdb9cb30du, 0x1bebda0951c4df63u, 0x5c85e975580ee5bdu,
|
||
0x1591bc60082cb137u, 0x2c38606318ef25d7u, 0x76ca72f7c5c63e27u, 0xf04a75d17baa0915u,
|
||
0x77458175139ae30du, 0x0e6c1330bc1b9421u, 0xdf87d2b5797e8293u, 0xefa5c703e1e68925u,
|
||
0x2b6b1b3278b4f6e1u, 0xceee27b382394249u, 0xd74e3829f5dab91du, 0xfdb17989c26b5f1fu,
|
||
0xc1b7d18781530845u, 0x7b4436b2105a8561u, 0x7ba7c0418372a7d7u, 0x9dbc5c67feb6c639u,
|
||
0x502686d7f6ff6b8fu, 0x6101855406be7a1fu, 0x9956afb5806930e7u, 0xe1f0ee88af40f7c5u,
|
||
0x984b057bda5c1151u, 0x9a49819acc13ea05u, 0x8ef0dead0896ef27u, 0x71f7826efe292b21u,
|
||
0xad80a480e46986efu, 0x01cdc0ebf5e0c6f7u, 0x6e06f839968f68dbu, 0xdd5943ab56e76139u,
|
||
0xcdcf31bf8604c5e7u, 0x7e2b4a847054a1cbu, 0x0ca75697a4d3d0f5u, 0x4703f53ac514a98bu,
|
||
};
|
||
|
||
/* inverses of reduced_factorial_odd_part values modulo 2**64.
|
||
|
||
Python code to generate the values:
|
||
|
||
import math
|
||
|
||
for n in range(128):
|
||
fac = math.factorial(n)
|
||
fac_odd_part = fac // (fac & -fac)
|
||
inverted_fac_odd_part = pow(fac_odd_part, -1, 2**64)
|
||
print(f"{inverted_fac_odd_part:#018x}u")
|
||
*/
|
||
static const uint64_t inverted_factorial_odd_part[] = {
|
||
0x0000000000000001u, 0x0000000000000001u, 0x0000000000000001u, 0xaaaaaaaaaaaaaaabu,
|
||
0xaaaaaaaaaaaaaaabu, 0xeeeeeeeeeeeeeeefu, 0x4fa4fa4fa4fa4fa5u, 0x2ff2ff2ff2ff2ff3u,
|
||
0x2ff2ff2ff2ff2ff3u, 0x938cc70553e3771bu, 0xb71c27cddd93e49fu, 0xb38e3229fcdee63du,
|
||
0xe684bb63544a4cbfu, 0xc2f684917ca340fbu, 0xf747c9cba417526du, 0xbb26eb51d7bd49c3u,
|
||
0xbb26eb51d7bd49c3u, 0xb0a7efb985294093u, 0xbe4b8c69f259eabbu, 0x6854d17ed6dc4fb9u,
|
||
0xe1aa904c915f4325u, 0x3b8206df131cead1u, 0x79c6009fea76fe13u, 0xd8c5d381633cd365u,
|
||
0x4841f12b21144677u, 0x4a91ff68200b0d0fu, 0x8f9513a58c4f9e8bu, 0x2b3e690621a42251u,
|
||
0x4f520f00e03c04e7u, 0x2edf84ee600211d3u, 0xadcaa2764aaacdfdu, 0x161f4f9033f4fe63u,
|
||
0x161f4f9033f4fe63u, 0xbada2932ea4d3e03u, 0xcec189f3efaa30d3u, 0xf7475bb68330bf91u,
|
||
0x37eb7bf7d5b01549u, 0x46b35660a4e91555u, 0xa567c12d81f151f7u, 0x4c724007bb2071b1u,
|
||
0x0f4a0cce58a016bdu, 0xfa21068e66106475u, 0x244ab72b5a318ae1u, 0x366ce67e080d0f23u,
|
||
0xd666fdae5dd2a449u, 0xd740ddd0acc06a0du, 0xb050bbbb28e6f97bu, 0x70b003fe890a5c75u,
|
||
0xd03aabff83037427u, 0x13ec4ca72c783bd7u, 0x90282c06afdbd96fu, 0x4414ddb9db4a95d5u,
|
||
0xa2c68735ae6832e9u, 0xbf72d71455676665u, 0xa8469fab6b759b7fu, 0xc1e55b56e606caf9u,
|
||
0x40455630fc4a1cffu, 0x0120a7b0046d16f7u, 0xa7c3553b08faef23u, 0x9f0bfd1b08d48639u,
|
||
0xa433ffce9a304d37u, 0xa22ad1d53915c683u, 0xcb6cbc723ba5dd1du, 0x547fb1b8ab9d0ba3u,
|
||
0x547fb1b8ab9d0ba3u, 0x8f15a826498852e3u, 0x32e1a03f38880283u, 0x3de4cce63283f0c1u,
|
||
0x5dfe6667e4da95b1u, 0xfda6eeeef479e47du, 0xf14de991cc7882dfu, 0xe68db79247630ca9u,
|
||
0xa7d6db8207ee8fa1u, 0x255e1f0fcf034499u, 0xc9a8990e43dd7e65u, 0x3279b6f289702e0fu,
|
||
0xe7b5905d9b71b195u, 0x03025ba41ff0da69u, 0xb7df3d6d3be55aefu, 0xf89b212ebff2b361u,
|
||
0xfe856d095996f0adu, 0xd6e533e9fdf20f9du, 0xf8c0e84a63da3255u, 0xa677876cd91b4db7u,
|
||
0x07ed4f97780d7d9bu, 0x90a8705f258db62fu, 0xa41bbb2be31b1c0du, 0x6ec28690b038383bu,
|
||
0xdb860c3bb2edd691u, 0x0838286838a980f9u, 0x558417a74b36f77du, 0x71779afc3646ef07u,
|
||
0x743cda377ccb6e91u, 0x7fdf9f3fe89153c5u, 0xdc97d25df49b9a4bu, 0x76321a778eb37d95u,
|
||
0x7cbb5e27da3bd487u, 0x9cff4ade1a009de7u, 0x70eb166d05c15197u, 0xdcf0460b71d5fe3du,
|
||
0x5ac1ee5260b6a3c5u, 0xc922dedfdd78efe1u, 0xe5d381dc3b8eeb9bu, 0xd57e5347bafc6aadu,
|
||
0x86939040983acd21u, 0x395b9d69740a4ff9u, 0x1467299c8e43d135u, 0x5fe440fcad975cdfu,
|
||
0xcaa9a39794a6ca8du, 0xf61dbd640868dea1u, 0xac09d98d74843be7u, 0x2b103b9e1a6b4809u,
|
||
0x2ab92d16960f536fu, 0x6653323d5e3681dfu, 0xefd48c1c0624e2d7u, 0xa496fefe04816f0du,
|
||
0x1754a7b07bbdd7b1u, 0x23353c829a3852cdu, 0xbf831261abd59097u, 0x57a8e656df0618e1u,
|
||
0x16e9206c3100680fu, 0xadad4c6ee921dac7u, 0x635f2b3860265353u, 0xdd6d0059f44b3d09u,
|
||
0xac4dd6b894447dd7u, 0x42ea183eeaa87be3u, 0x15612d1550ee5b5du, 0x226fa19d656cb623u,
|
||
};
|
||
|
||
/* exponent of the largest power of 2 dividing factorial(n), for n in range(68)
|
||
|
||
Python code to generate the values:
|
||
|
||
import math
|
||
|
||
for n in range(128):
|
||
fac = math.factorial(n)
|
||
fac_trailing_zeros = (fac & -fac).bit_length() - 1
|
||
print(fac_trailing_zeros)
|
||
*/
|
||
|
||
static const uint8_t factorial_trailing_zeros[] = {
|
||
0, 0, 1, 1, 3, 3, 4, 4, 7, 7, 8, 8, 10, 10, 11, 11, // 0-15
|
||
15, 15, 16, 16, 18, 18, 19, 19, 22, 22, 23, 23, 25, 25, 26, 26, // 16-31
|
||
31, 31, 32, 32, 34, 34, 35, 35, 38, 38, 39, 39, 41, 41, 42, 42, // 32-47
|
||
46, 46, 47, 47, 49, 49, 50, 50, 53, 53, 54, 54, 56, 56, 57, 57, // 48-63
|
||
63, 63, 64, 64, 66, 66, 67, 67, 70, 70, 71, 71, 73, 73, 74, 74, // 64-79
|
||
78, 78, 79, 79, 81, 81, 82, 82, 85, 85, 86, 86, 88, 88, 89, 89, // 80-95
|
||
94, 94, 95, 95, 97, 97, 98, 98, 101, 101, 102, 102, 104, 104, 105, 105, // 96-111
|
||
109, 109, 110, 110, 112, 112, 113, 113, 116, 116, 117, 117, 119, 119, 120, 120, // 112-127
|
||
};
|
||
|
||
/* Number of permutations and combinations.
|
||
* P(n, k) = n! / (n-k)!
|
||
* C(n, k) = P(n, k) / k!
|
||
*/
|
||
|
||
/* Calculate C(n, k) for n in the 63-bit range. */
|
||
static PyObject *
|
||
perm_comb_small(unsigned long long n, unsigned long long k, int iscomb)
|
||
{
|
||
if (k == 0) {
|
||
return PyLong_FromLong(1);
|
||
}
|
||
|
||
/* For small enough n and k the result fits in the 64-bit range and can
|
||
* be calculated without allocating intermediate PyLong objects. */
|
||
if (iscomb) {
|
||
/* Maps k to the maximal n so that 2*k-1 <= n <= 127 and C(n, k)
|
||
* fits into a uint64_t. Exclude k = 1, because the second fast
|
||
* path is faster for this case.*/
|
||
static const unsigned char fast_comb_limits1[] = {
|
||
0, 0, 127, 127, 127, 127, 127, 127, // 0-7
|
||
127, 127, 127, 127, 127, 127, 127, 127, // 8-15
|
||
116, 105, 97, 91, 86, 82, 78, 76, // 16-23
|
||
74, 72, 71, 70, 69, 68, 68, 67, // 24-31
|
||
67, 67, 67, // 32-34
|
||
};
|
||
if (k < Py_ARRAY_LENGTH(fast_comb_limits1) && n <= fast_comb_limits1[k]) {
|
||
/*
|
||
comb(n, k) fits into a uint64_t. We compute it as
|
||
|
||
comb_odd_part << shift
|
||
|
||
where 2**shift is the largest power of two dividing comb(n, k)
|
||
and comb_odd_part is comb(n, k) >> shift. comb_odd_part can be
|
||
calculated efficiently via arithmetic modulo 2**64, using three
|
||
lookups and two uint64_t multiplications.
|
||
*/
|
||
uint64_t comb_odd_part = reduced_factorial_odd_part[n]
|
||
* inverted_factorial_odd_part[k]
|
||
* inverted_factorial_odd_part[n - k];
|
||
int shift = factorial_trailing_zeros[n]
|
||
- factorial_trailing_zeros[k]
|
||
- factorial_trailing_zeros[n - k];
|
||
return PyLong_FromUnsignedLongLong(comb_odd_part << shift);
|
||
}
|
||
|
||
/* Maps k to the maximal n so that 2*k-1 <= n <= 127 and C(n, k)*k
|
||
* fits into a long long (which is at least 64 bit). Only contains
|
||
* items larger than in fast_comb_limits1. */
|
||
static const unsigned long long fast_comb_limits2[] = {
|
||
0, ULLONG_MAX, 4294967296ULL, 3329022, 102570, 13467, 3612, 1449, // 0-7
|
||
746, 453, 308, 227, 178, 147, // 8-13
|
||
};
|
||
if (k < Py_ARRAY_LENGTH(fast_comb_limits2) && n <= fast_comb_limits2[k]) {
|
||
/* C(n, k) = C(n, k-1) * (n-k+1) / k */
|
||
unsigned long long result = n;
|
||
for (unsigned long long i = 1; i < k;) {
|
||
result *= --n;
|
||
result /= ++i;
|
||
}
|
||
return PyLong_FromUnsignedLongLong(result);
|
||
}
|
||
}
|
||
else {
|
||
/* Maps k to the maximal n so that k <= n and P(n, k)
|
||
* fits into a long long (which is at least 64 bit). */
|
||
static const unsigned long long fast_perm_limits[] = {
|
||
0, ULLONG_MAX, 4294967296ULL, 2642246, 65537, 7133, 1627, 568, // 0-7
|
||
259, 142, 88, 61, 45, 36, 30, 26, // 8-15
|
||
24, 22, 21, 20, 20, // 16-20
|
||
};
|
||
if (k < Py_ARRAY_LENGTH(fast_perm_limits) && n <= fast_perm_limits[k]) {
|
||
if (n <= 127) {
|
||
/* P(n, k) fits into a uint64_t. */
|
||
uint64_t perm_odd_part = reduced_factorial_odd_part[n]
|
||
* inverted_factorial_odd_part[n - k];
|
||
int shift = factorial_trailing_zeros[n]
|
||
- factorial_trailing_zeros[n - k];
|
||
return PyLong_FromUnsignedLongLong(perm_odd_part << shift);
|
||
}
|
||
|
||
/* P(n, k) = P(n, k-1) * (n-k+1) */
|
||
unsigned long long result = n;
|
||
for (unsigned long long i = 1; i < k;) {
|
||
result *= --n;
|
||
++i;
|
||
}
|
||
return PyLong_FromUnsignedLongLong(result);
|
||
}
|
||
}
|
||
|
||
/* For larger n use recursive formulas:
|
||
*
|
||
* P(n, k) = P(n, j) * P(n-j, k-j)
|
||
* C(n, k) = C(n, j) * C(n-j, k-j) // C(k, j)
|
||
*/
|
||
unsigned long long j = k / 2;
|
||
PyObject *a, *b;
|
||
a = perm_comb_small(n, j, iscomb);
|
||
if (a == NULL) {
|
||
return NULL;
|
||
}
|
||
b = perm_comb_small(n - j, k - j, iscomb);
|
||
if (b == NULL) {
|
||
goto error;
|
||
}
|
||
Py_SETREF(a, PyNumber_Multiply(a, b));
|
||
Py_DECREF(b);
|
||
if (iscomb && a != NULL) {
|
||
b = perm_comb_small(k, j, 1);
|
||
if (b == NULL) {
|
||
goto error;
|
||
}
|
||
Py_SETREF(a, PyNumber_FloorDivide(a, b));
|
||
Py_DECREF(b);
|
||
}
|
||
return a;
|
||
|
||
error:
|
||
Py_DECREF(a);
|
||
return NULL;
|
||
}
|
||
|
||
/* Calculate P(n, k) or C(n, k) using recursive formulas.
|
||
* It is more efficient than sequential multiplication thanks to
|
||
* Karatsuba multiplication.
|
||
*/
|
||
static PyObject *
|
||
perm_comb(PyObject *n, unsigned long long k, int iscomb)
|
||
{
|
||
if (k == 0) {
|
||
return PyLong_FromLong(1);
|
||
}
|
||
if (k == 1) {
|
||
return Py_NewRef(n);
|
||
}
|
||
|
||
/* P(n, k) = P(n, j) * P(n-j, k-j) */
|
||
/* C(n, k) = C(n, j) * C(n-j, k-j) // C(k, j) */
|
||
unsigned long long j = k / 2;
|
||
PyObject *a, *b;
|
||
a = perm_comb(n, j, iscomb);
|
||
if (a == NULL) {
|
||
return NULL;
|
||
}
|
||
PyObject *t = PyLong_FromUnsignedLongLong(j);
|
||
if (t == NULL) {
|
||
goto error;
|
||
}
|
||
n = PyNumber_Subtract(n, t);
|
||
Py_DECREF(t);
|
||
if (n == NULL) {
|
||
goto error;
|
||
}
|
||
b = perm_comb(n, k - j, iscomb);
|
||
Py_DECREF(n);
|
||
if (b == NULL) {
|
||
goto error;
|
||
}
|
||
Py_SETREF(a, PyNumber_Multiply(a, b));
|
||
Py_DECREF(b);
|
||
if (iscomb && a != NULL) {
|
||
b = perm_comb_small(k, j, 1);
|
||
if (b == NULL) {
|
||
goto error;
|
||
}
|
||
Py_SETREF(a, PyNumber_FloorDivide(a, b));
|
||
Py_DECREF(b);
|
||
}
|
||
return a;
|
||
|
||
error:
|
||
Py_DECREF(a);
|
||
return NULL;
|
||
}
|
||
|
||
/*[clinic input]
|
||
math.perm
|
||
|
||
n: object
|
||
k: object = None
|
||
/
|
||
|
||
Number of ways to choose k items from n items without repetition and with order.
|
||
|
||
Evaluates to n! / (n - k)! when k <= n and evaluates
|
||
to zero when k > n.
|
||
|
||
If k is not specified or is None, then k defaults to n
|
||
and the function returns 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_perm_impl(PyObject *module, PyObject *n, PyObject *k)
|
||
/*[clinic end generated code: output=e021a25469653e23 input=5311c5a00f359b53]*/
|
||
{
|
||
PyObject *result = NULL;
|
||
int overflow, cmp;
|
||
long long ki, ni;
|
||
|
||
if (k == Py_None) {
|
||
return math_factorial(module, n);
|
||
}
|
||
n = PyNumber_Index(n);
|
||
if (n == NULL) {
|
||
return NULL;
|
||
}
|
||
k = PyNumber_Index(k);
|
||
if (k == NULL) {
|
||
Py_DECREF(n);
|
||
return NULL;
|
||
}
|
||
assert(PyLong_CheckExact(n) && PyLong_CheckExact(k));
|
||
|
||
if (_PyLong_IsNegative((PyLongObject *)n)) {
|
||
PyErr_SetString(PyExc_ValueError,
|
||
"n must be a non-negative integer");
|
||
goto error;
|
||
}
|
||
if (_PyLong_IsNegative((PyLongObject *)k)) {
|
||
PyErr_SetString(PyExc_ValueError,
|
||
"k must be a non-negative integer");
|
||
goto error;
|
||
}
|
||
|
||
cmp = PyObject_RichCompareBool(n, k, Py_LT);
|
||
if (cmp != 0) {
|
||
if (cmp > 0) {
|
||
result = PyLong_FromLong(0);
|
||
goto done;
|
||
}
|
||
goto error;
|
||
}
|
||
|
||
ki = PyLong_AsLongLongAndOverflow(k, &overflow);
|
||
assert(overflow >= 0 && !PyErr_Occurred());
|
||
if (overflow > 0) {
|
||
PyErr_Format(PyExc_OverflowError,
|
||
"k must not exceed %lld",
|
||
LLONG_MAX);
|
||
goto error;
|
||
}
|
||
assert(ki >= 0);
|
||
|
||
ni = PyLong_AsLongLongAndOverflow(n, &overflow);
|
||
assert(overflow >= 0 && !PyErr_Occurred());
|
||
if (!overflow && ki > 1) {
|
||
assert(ni >= 0);
|
||
result = perm_comb_small((unsigned long long)ni,
|
||
(unsigned long long)ki, 0);
|
||
}
|
||
else {
|
||
result = perm_comb(n, (unsigned long long)ki, 0);
|
||
}
|
||
|
||
done:
|
||
Py_DECREF(n);
|
||
Py_DECREF(k);
|
||
return result;
|
||
|
||
error:
|
||
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 and evaluates
|
||
to zero 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, *temp;
|
||
int overflow, cmp;
|
||
long long ki, ni;
|
||
|
||
n = PyNumber_Index(n);
|
||
if (n == NULL) {
|
||
return NULL;
|
||
}
|
||
k = PyNumber_Index(k);
|
||
if (k == NULL) {
|
||
Py_DECREF(n);
|
||
return NULL;
|
||
}
|
||
assert(PyLong_CheckExact(n) && PyLong_CheckExact(k));
|
||
|
||
if (_PyLong_IsNegative((PyLongObject *)n)) {
|
||
PyErr_SetString(PyExc_ValueError,
|
||
"n must be a non-negative integer");
|
||
goto error;
|
||
}
|
||
if (_PyLong_IsNegative((PyLongObject *)k)) {
|
||
PyErr_SetString(PyExc_ValueError,
|
||
"k must be a non-negative integer");
|
||
goto error;
|
||
}
|
||
|
||
ni = PyLong_AsLongLongAndOverflow(n, &overflow);
|
||
assert(overflow >= 0 && !PyErr_Occurred());
|
||
if (!overflow) {
|
||
assert(ni >= 0);
|
||
ki = PyLong_AsLongLongAndOverflow(k, &overflow);
|
||
assert(overflow >= 0 && !PyErr_Occurred());
|
||
if (overflow || ki > ni) {
|
||
result = PyLong_FromLong(0);
|
||
goto done;
|
||
}
|
||
assert(ki >= 0);
|
||
|
||
ki = Py_MIN(ki, ni - ki);
|
||
if (ki > 1) {
|
||
result = perm_comb_small((unsigned long long)ni,
|
||
(unsigned long long)ki, 1);
|
||
goto done;
|
||
}
|
||
/* For k == 1 just return the original n in perm_comb(). */
|
||
}
|
||
else {
|
||
/* k = min(k, n - k) */
|
||
temp = PyNumber_Subtract(n, k);
|
||
if (temp == NULL) {
|
||
goto error;
|
||
}
|
||
assert(PyLong_Check(temp));
|
||
if (_PyLong_IsNegative((PyLongObject *)temp)) {
|
||
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;
|
||
}
|
||
}
|
||
|
||
ki = PyLong_AsLongLongAndOverflow(k, &overflow);
|
||
assert(overflow >= 0 && !PyErr_Occurred());
|
||
if (overflow) {
|
||
PyErr_Format(PyExc_OverflowError,
|
||
"min(n - k, k) must not exceed %lld",
|
||
LLONG_MAX);
|
||
goto error;
|
||
}
|
||
assert(ki >= 0);
|
||
}
|
||
|
||
result = perm_comb(n, (unsigned long long)ki, 1);
|
||
|
||
done:
|
||
Py_DECREF(n);
|
||
Py_DECREF(k);
|
||
return result;
|
||
|
||
error:
|
||
Py_DECREF(n);
|
||
Py_DECREF(k);
|
||
return NULL;
|
||
}
|
||
|
||
|
||
/*[clinic input]
|
||
math.nextafter
|
||
|
||
x: double
|
||
y: double
|
||
/
|
||
*
|
||
steps: object = None
|
||
|
||
Return the floating-point value the given number of steps after x towards y.
|
||
|
||
If steps is not specified or is None, it defaults to 1.
|
||
|
||
Raises a TypeError, if x or y is not a double, or if steps is not an integer.
|
||
Raises ValueError if steps is negative.
|
||
[clinic start generated code]*/
|
||
|
||
static PyObject *
|
||
math_nextafter_impl(PyObject *module, double x, double y, PyObject *steps)
|
||
/*[clinic end generated code: output=cc6511f02afc099e input=7f2a5842112af2b4]*/
|
||
{
|
||
#if defined(_AIX)
|
||
if (x == y) {
|
||
/* On AIX 7.1, libm nextafter(-0.0, +0.0) returns -0.0.
|
||
Bug fixed in bos.adt.libm 7.2.2.0 by APAR IV95512. */
|
||
return PyFloat_FromDouble(y);
|
||
}
|
||
if (Py_IS_NAN(x)) {
|
||
return PyFloat_FromDouble(x);
|
||
}
|
||
if (Py_IS_NAN(y)) {
|
||
return PyFloat_FromDouble(y);
|
||
}
|
||
#endif
|
||
if (steps == Py_None) {
|
||
// fast path: we default to one step.
|
||
return PyFloat_FromDouble(nextafter(x, y));
|
||
}
|
||
steps = PyNumber_Index(steps);
|
||
if (steps == NULL) {
|
||
return NULL;
|
||
}
|
||
assert(PyLong_CheckExact(steps));
|
||
if (_PyLong_IsNegative((PyLongObject *)steps)) {
|
||
PyErr_SetString(PyExc_ValueError,
|
||
"steps must be a non-negative integer");
|
||
Py_DECREF(steps);
|
||
return NULL;
|
||
}
|
||
|
||
unsigned long long usteps_ull = PyLong_AsUnsignedLongLong(steps);
|
||
// Conveniently, uint64_t and double have the same number of bits
|
||
// on all the platforms we care about.
|
||
// So if an overflow occurs, we can just use UINT64_MAX.
|
||
Py_DECREF(steps);
|
||
if (usteps_ull >= UINT64_MAX) {
|
||
// This branch includes the case where an error occurred, since
|
||
// (unsigned long long)(-1) = ULLONG_MAX >= UINT64_MAX. Note that
|
||
// usteps_ull can be strictly larger than UINT64_MAX on a machine
|
||
// where unsigned long long has width > 64 bits.
|
||
if (PyErr_Occurred()) {
|
||
if (PyErr_ExceptionMatches(PyExc_OverflowError)) {
|
||
PyErr_Clear();
|
||
}
|
||
else {
|
||
return NULL;
|
||
}
|
||
}
|
||
usteps_ull = UINT64_MAX;
|
||
}
|
||
assert(usteps_ull <= UINT64_MAX);
|
||
uint64_t usteps = (uint64_t)usteps_ull;
|
||
|
||
if (usteps == 0) {
|
||
return PyFloat_FromDouble(x);
|
||
}
|
||
if (Py_IS_NAN(x)) {
|
||
return PyFloat_FromDouble(x);
|
||
}
|
||
if (Py_IS_NAN(y)) {
|
||
return PyFloat_FromDouble(y);
|
||
}
|
||
|
||
// We assume that double and uint64_t have the same endianness.
|
||
// This is not guaranteed by the C-standard, but it is true for
|
||
// all platforms we care about. (The most likely form of violation
|
||
// would be a "mixed-endian" double.)
|
||
union pun {double f; uint64_t i;};
|
||
union pun ux = {x}, uy = {y};
|
||
if (ux.i == uy.i) {
|
||
return PyFloat_FromDouble(x);
|
||
}
|
||
|
||
const uint64_t sign_bit = 1ULL<<63;
|
||
|
||
uint64_t ax = ux.i & ~sign_bit;
|
||
uint64_t ay = uy.i & ~sign_bit;
|
||
|
||
// opposite signs
|
||
if (((ux.i ^ uy.i) & sign_bit)) {
|
||
// NOTE: ax + ay can never overflow, because their most significant bit
|
||
// ain't set.
|
||
if (ax + ay <= usteps) {
|
||
return PyFloat_FromDouble(uy.f);
|
||
// This comparison has to use <, because <= would get +0.0 vs -0.0
|
||
// wrong.
|
||
} else if (ax < usteps) {
|
||
union pun result = {.i = (uy.i & sign_bit) | (usteps - ax)};
|
||
return PyFloat_FromDouble(result.f);
|
||
} else {
|
||
ux.i -= usteps;
|
||
return PyFloat_FromDouble(ux.f);
|
||
}
|
||
// same sign
|
||
} else if (ax > ay) {
|
||
if (ax - ay >= usteps) {
|
||
ux.i -= usteps;
|
||
return PyFloat_FromDouble(ux.f);
|
||
} else {
|
||
return PyFloat_FromDouble(uy.f);
|
||
}
|
||
} else {
|
||
if (ay - ax >= usteps) {
|
||
ux.i += usteps;
|
||
return PyFloat_FromDouble(ux.f);
|
||
} else {
|
||
return PyFloat_FromDouble(uy.f);
|
||
}
|
||
}
|
||
}
|
||
|
||
|
||
/*[clinic input]
|
||
math.ulp -> double
|
||
|
||
x: double
|
||
/
|
||
|
||
Return the value of the least significant bit of the float x.
|
||
[clinic start generated code]*/
|
||
|
||
static double
|
||
math_ulp_impl(PyObject *module, double x)
|
||
/*[clinic end generated code: output=f5207867a9384dd4 input=31f9bfbbe373fcaa]*/
|
||
{
|
||
if (Py_IS_NAN(x)) {
|
||
return x;
|
||
}
|
||
x = fabs(x);
|
||
if (Py_IS_INFINITY(x)) {
|
||
return x;
|
||
}
|
||
double inf = Py_INFINITY;
|
||
double x2 = nextafter(x, inf);
|
||
if (Py_IS_INFINITY(x2)) {
|
||
/* special case: x is the largest positive representable float */
|
||
x2 = nextafter(x, -inf);
|
||
return x - x2;
|
||
}
|
||
return x2 - x;
|
||
}
|
||
|
||
static int
|
||
math_exec(PyObject *module)
|
||
{
|
||
|
||
math_module_state *state = get_math_module_state(module);
|
||
state->str___ceil__ = PyUnicode_InternFromString("__ceil__");
|
||
if (state->str___ceil__ == NULL) {
|
||
return -1;
|
||
}
|
||
state->str___floor__ = PyUnicode_InternFromString("__floor__");
|
||
if (state->str___floor__ == NULL) {
|
||
return -1;
|
||
}
|
||
state->str___trunc__ = PyUnicode_InternFromString("__trunc__");
|
||
if (state->str___trunc__ == NULL) {
|
||
return -1;
|
||
}
|
||
if (PyModule_Add(module, "pi", PyFloat_FromDouble(Py_MATH_PI)) < 0) {
|
||
return -1;
|
||
}
|
||
if (PyModule_Add(module, "e", PyFloat_FromDouble(Py_MATH_E)) < 0) {
|
||
return -1;
|
||
}
|
||
// 2pi
|
||
if (PyModule_Add(module, "tau", PyFloat_FromDouble(Py_MATH_TAU)) < 0) {
|
||
return -1;
|
||
}
|
||
if (PyModule_Add(module, "inf", PyFloat_FromDouble(Py_INFINITY)) < 0) {
|
||
return -1;
|
||
}
|
||
if (PyModule_Add(module, "nan", PyFloat_FromDouble(fabs(Py_NAN))) < 0) {
|
||
return -1;
|
||
}
|
||
return 0;
|
||
}
|
||
|
||
static int
|
||
math_clear(PyObject *module)
|
||
{
|
||
math_module_state *state = get_math_module_state(module);
|
||
Py_CLEAR(state->str___ceil__);
|
||
Py_CLEAR(state->str___floor__);
|
||
Py_CLEAR(state->str___trunc__);
|
||
return 0;
|
||
}
|
||
|
||
static void
|
||
math_free(void *module)
|
||
{
|
||
math_clear((PyObject *)module);
|
||
}
|
||
|
||
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_CAST(math_atan2), METH_FASTCALL, math_atan2_doc},
|
||
{"atanh", math_atanh, METH_O, math_atanh_doc},
|
||
{"cbrt", math_cbrt, METH_O, math_cbrt_doc},
|
||
MATH_CEIL_METHODDEF
|
||
{"copysign", _PyCFunction_CAST(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},
|
||
{"exp2", math_exp2, METH_O, math_exp2_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},
|
||
{"gcd", _PyCFunction_CAST(math_gcd), METH_FASTCALL, math_gcd_doc},
|
||
{"hypot", _PyCFunction_CAST(math_hypot), METH_FASTCALL, math_hypot_doc},
|
||
MATH_ISCLOSE_METHODDEF
|
||
MATH_ISFINITE_METHODDEF
|
||
MATH_ISINF_METHODDEF
|
||
MATH_ISNAN_METHODDEF
|
||
MATH_ISQRT_METHODDEF
|
||
{"lcm", _PyCFunction_CAST(math_lcm), METH_FASTCALL, math_lcm_doc},
|
||
MATH_LDEXP_METHODDEF
|
||
{"lgamma", math_lgamma, METH_O, math_lgamma_doc},
|
||
{"log", _PyCFunction_CAST(math_log), METH_FASTCALL, math_log_doc},
|
||
{"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_CAST(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_SUMPROD_METHODDEF
|
||
MATH_TRUNC_METHODDEF
|
||
MATH_PROD_METHODDEF
|
||
MATH_PERM_METHODDEF
|
||
MATH_COMB_METHODDEF
|
||
MATH_NEXTAFTER_METHODDEF
|
||
MATH_ULP_METHODDEF
|
||
{NULL, NULL} /* sentinel */
|
||
};
|
||
|
||
static PyModuleDef_Slot math_slots[] = {
|
||
{Py_mod_exec, math_exec},
|
||
{Py_mod_multiple_interpreters, Py_MOD_PER_INTERPRETER_GIL_SUPPORTED},
|
||
{0, NULL}
|
||
};
|
||
|
||
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,
|
||
.m_name = "math",
|
||
.m_doc = module_doc,
|
||
.m_size = sizeof(math_module_state),
|
||
.m_methods = math_methods,
|
||
.m_slots = math_slots,
|
||
.m_clear = math_clear,
|
||
.m_free = math_free,
|
||
};
|
||
|
||
PyMODINIT_FUNC
|
||
PyInit_math(void)
|
||
{
|
||
return PyModuleDef_Init(&mathmodule);
|
||
}
|