Issue #3366: Add gamma function to math module.
(lgamma, erf and erfc to follow).
diff --git a/Modules/mathmodule.c b/Modules/mathmodule.c
index 952d56a..b2fee3f 100644
--- a/Modules/mathmodule.c
+++ b/Modules/mathmodule.c
@@ -60,44 +60,265 @@
extern double copysign(double, double);
#endif
-/* 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");
+/*
+ sin(pi*x), giving accurate results for all finite x (especially x
+ integral or close to an integer). This is here for use in the
+ reflection formula for the gamma function. It conforms to IEEE
+ 754-2008 for finite arguments, but not for infinities or nans.
+*/
- else if (errno == ERANGE) {
- /* ANSI C generally requires libm functions to set ERANGE
- * on overflow, but also generally *allows* them to set
- * ERANGE on underflow too. There's no consistency about
- * the latter across platforms.
- * Alas, C99 never requires that errno be set.
- * Here we suppress the underflow errors (libm functions
- * should return a zero on underflow, and +- HUGE_VAL on
- * overflow, so testing the result for zero suffices to
- * distinguish the cases).
- *
- * On some platforms (Ubuntu/ia64) it seems that errno can be
- * set to ERANGE for subnormal results that do *not* underflow
- * to zero. So to be safe, we'll ignore ERANGE whenever the
- * function result is less than one in absolute value.
- */
- if (fabs(x) < 1.0)
- result = 0;
- else
- PyErr_SetString(PyExc_OverflowError,
- "math range error");
+static const double pi = 3.141592653589793238462643383279502884197;
+
+static double
+sinpi(double x)
+{
+ double y, r;
+ int n;
+ /* this function should only ever be called for finite arguments */
+ assert(Py_IS_FINITE(x));
+ y = fmod(fabs(x), 2.0);
+ n = (int)round(2.0*y);
+ assert(0 <= n && n <= 4);
+ switch (n) {
+ case 0:
+ r = sin(pi*y);
+ break;
+ case 1:
+ r = cos(pi*(y-0.5));
+ break;
+ case 2:
+ /* N.B. -sin(pi*(y-1.0)) is *not* equivalent: it would give
+ -0.0 instead of 0.0 when y == 1.0. */
+ r = sin(pi*(1.0-y));
+ break;
+ case 3:
+ r = -cos(pi*(y-1.5));
+ break;
+ case 4:
+ r = sin(pi*(y-2.0));
+ break;
+ default:
+ assert(0); /* should never get here */
+ r = -1.23e200; /* silence gcc warning */
}
- else
- /* Unexpected math error */
- PyErr_SetFromErrno(PyExc_ValueError);
- return result;
+ return copysign(1.0, x)*r;
+}
+
+/* Implementation of the real gamma function. In extensive but non-exhaustive
+ random tests, this function proved accurate to within <= 10 ulps across the
+ entire float domain. Note that accuracy may depend on the quality of the
+ system math functions, the pow function in particular. Special cases
+ follow C99 annex F. The parameters and method are tailored to platforms
+ whose double format is the IEEE 754 binary64 format.
+
+ Method: for x > 0.0 we use the Lanczos approximation with parameters N=13
+ and g=6.024680040776729583740234375; these parameters are amongst those
+ used by the Boost library. Following Boost (again), we re-express the
+ Lanczos sum as a rational function, and compute it that way. The
+ coefficients below were computed independently using MPFR, and have been
+ double-checked against the coefficients in the Boost source code.
+
+ For x < 0.0 we use the reflection formula.
+
+ There's one minor tweak that deserves explanation: Lanczos' formula for
+ Gamma(x) involves computing pow(x+g-0.5, x-0.5) / exp(x+g-0.5). For many x
+ values, x+g-0.5 can be represented exactly. However, in cases where it
+ can't be represented exactly the small error in x+g-0.5 can be magnified
+ significantly by the pow and exp calls, especially for large x. A cheap
+ correction is to multiply by (1 + e*g/(x+g-0.5)), where e is the error
+ involved in the computation of x+g-0.5 (that is, e = computed value of
+ x+g-0.5 - exact value of x+g-0.5). Here's the proof:
+
+ Correction factor
+ -----------------
+ Write x+g-0.5 = y-e, where y is exactly representable as an IEEE 754
+ double, and e is tiny. Then:
+
+ pow(x+g-0.5,x-0.5)/exp(x+g-0.5) = pow(y-e, x-0.5)/exp(y-e)
+ = pow(y, x-0.5)/exp(y) * C,
+
+ where the correction_factor C is given by
+
+ C = pow(1-e/y, x-0.5) * exp(e)
+
+ Since e is tiny, pow(1-e/y, x-0.5) ~ 1-(x-0.5)*e/y, and exp(x) ~ 1+e, so:
+
+ C ~ (1-(x-0.5)*e/y) * (1+e) ~ 1 + e*(y-(x-0.5))/y
+
+ But y-(x-0.5) = g+e, and g+e ~ g. So we get C ~ 1 + e*g/y, and
+
+ pow(x+g-0.5,x-0.5)/exp(x+g-0.5) ~ pow(y, x-0.5)/exp(y) * (1 + e*g/y),
+
+ Note that for accuracy, when computing r*C it's better to do
+
+ r + e*g/y*r;
+
+ than
+
+ r * (1 + e*g/y);
+
+ since the addition in the latter throws away most of the bits of
+ information in e*g/y.
+*/
+
+#define LANCZOS_N 13
+static const double lanczos_g = 6.024680040776729583740234375;
+static const double lanczos_g_minus_half = 5.524680040776729583740234375;
+static const double lanczos_num_coeffs[LANCZOS_N] = {
+ 23531376880.410759688572007674451636754734846804940,
+ 42919803642.649098768957899047001988850926355848959,
+ 35711959237.355668049440185451547166705960488635843,
+ 17921034426.037209699919755754458931112671403265390,
+ 6039542586.3520280050642916443072979210699388420708,
+ 1439720407.3117216736632230727949123939715485786772,
+ 248874557.86205415651146038641322942321632125127801,
+ 31426415.585400194380614231628318205362874684987640,
+ 2876370.6289353724412254090516208496135991145378768,
+ 186056.26539522349504029498971604569928220784236328,
+ 8071.6720023658162106380029022722506138218516325024,
+ 210.82427775157934587250973392071336271166969580291,
+ 2.5066282746310002701649081771338373386264310793408
+};
+
+/* denominator is x*(x+1)*...*(x+LANCZOS_N-2) */
+static const double lanczos_den_coeffs[LANCZOS_N] = {
+ 0.0, 39916800.0, 120543840.0, 150917976.0, 105258076.0, 45995730.0,
+ 13339535.0, 2637558.0, 357423.0, 32670.0, 1925.0, 66.0, 1.0};
+
+/* gamma values for small positive integers, 1 though NGAMMA_INTEGRAL */
+#define NGAMMA_INTEGRAL 23
+static const double gamma_integral[NGAMMA_INTEGRAL] = {
+ 1.0, 1.0, 2.0, 6.0, 24.0, 120.0, 720.0, 5040.0, 40320.0, 362880.0,
+ 3628800.0, 39916800.0, 479001600.0, 6227020800.0, 87178291200.0,
+ 1307674368000.0, 20922789888000.0, 355687428096000.0,
+ 6402373705728000.0, 121645100408832000.0, 2432902008176640000.0,
+ 51090942171709440000.0, 1124000727777607680000.0,
+};
+
+/* Lanczos' sum L_g(x), for positive x */
+
+static double
+lanczos_sum(double x)
+{
+ double num = 0.0, den = 0.0;
+ int i;
+ assert(x > 0.0);
+ /* evaluate the rational function lanczos_sum(x). For large
+ x, the obvious algorithm risks overflow, so we instead
+ rescale the denominator and numerator of the rational
+ function by x**(1-LANCZOS_N) and treat this as a
+ rational function in 1/x. This also reduces the error for
+ larger x values. The choice of cutoff point (5.0 below) is
+ somewhat arbitrary; in tests, smaller cutoff values than
+ this resulted in lower accuracy. */
+ if (x < 5.0) {
+ for (i = LANCZOS_N; --i >= 0; ) {
+ num = num * x + lanczos_num_coeffs[i];
+ den = den * x + lanczos_den_coeffs[i];
+ }
+ }
+ else {
+ for (i = 0; i < LANCZOS_N; i++) {
+ num = num / x + lanczos_num_coeffs[i];
+ den = den / x + lanczos_den_coeffs[i];
+ }
+ }
+ return num/den;
+}
+
+static double
+m_tgamma(double x)
+{
+ double absx, r, y, z, sqrtpow;
+
+ /* special cases */
+ if (!Py_IS_FINITE(x)) {
+ if (Py_IS_NAN(x) || x > 0.0)
+ return x; /* tgamma(nan) = nan, tgamma(inf) = inf */
+ else {
+ errno = EDOM;
+ return Py_NAN; /* tgamma(-inf) = nan, invalid */
+ }
+ }
+ if (x == 0.0) {
+ errno = EDOM;
+ return 1.0/x; /* tgamma(+-0.0) = +-inf, divide-by-zero */
+ }
+
+ /* integer arguments */
+ if (x == floor(x)) {
+ if (x < 0.0) {
+ errno = EDOM; /* tgamma(n) = nan, invalid for */
+ return Py_NAN; /* negative integers n */
+ }
+ if (x <= NGAMMA_INTEGRAL)
+ return gamma_integral[(int)x - 1];
+ }
+ absx = fabs(x);
+
+ /* tiny arguments: tgamma(x) ~ 1/x for x near 0 */
+ if (absx < 1e-20) {
+ r = 1.0/x;
+ if (Py_IS_INFINITY(r))
+ errno = ERANGE;
+ return r;
+ }
+
+ /* large arguments: assuming IEEE 754 doubles, tgamma(x) overflows for
+ x > 200, and underflows to +-0.0 for x < -200, not a negative
+ integer. */
+ if (absx > 200.0) {
+ if (x < 0.0) {
+ return 0.0/sinpi(x);
+ }
+ else {
+ errno = ERANGE;
+ return Py_HUGE_VAL;
+ }
+ }
+
+ y = absx + lanczos_g_minus_half;
+ /* compute error in sum */
+ if (absx > lanczos_g_minus_half) {
+ /* note: the correction can be foiled by an optimizing
+ compiler that (incorrectly) thinks that an expression like
+ a + b - a - b can be optimized to 0.0. This shouldn't
+ happen in a standards-conforming compiler. */
+ double q = y - absx;
+ z = q - lanczos_g_minus_half;
+ }
+ else {
+ double q = y - lanczos_g_minus_half;
+ z = q - absx;
+ }
+ z = z * lanczos_g / y;
+ if (x < 0.0) {
+ r = -pi / sinpi(absx) / absx * exp(y) / lanczos_sum(absx);
+ r -= z * r;
+ if (absx < 140.0) {
+ r /= pow(y, absx - 0.5);
+ }
+ else {
+ sqrtpow = pow(y, absx / 2.0 - 0.25);
+ r /= sqrtpow;
+ r /= sqrtpow;
+ }
+ }
+ else {
+ r = lanczos_sum(absx) / exp(y);
+ r += z * r;
+ if (absx < 140.0) {
+ r *= pow(y, absx - 0.5);
+ }
+ else {
+ sqrtpow = pow(y, absx / 2.0 - 0.25);
+ r *= sqrtpow;
+ r *= sqrtpow;
+ }
+ }
+ if (Py_IS_INFINITY(r))
+ errno = ERANGE;
+ return r;
}
/*
@@ -188,6 +409,46 @@
}
+/* Call is_error when errno != 0, and where x is the result libm
+ * returned. is_error will usually set up an exception and return
+ * true (1), but may return false (0) without setting up an exception.
+ */
+static int
+is_error(double x)
+{
+ int result = 1; /* presumption of guilt */
+ assert(errno); /* non-zero errno is a precondition for calling */
+ if (errno == EDOM)
+ PyErr_SetString(PyExc_ValueError, "math domain error");
+
+ else if (errno == ERANGE) {
+ /* ANSI C generally requires libm functions to set ERANGE
+ * on overflow, but also generally *allows* them to set
+ * ERANGE on underflow too. There's no consistency about
+ * the latter across platforms.
+ * Alas, C99 never requires that errno be set.
+ * Here we suppress the underflow errors (libm functions
+ * should return a zero on underflow, and +- HUGE_VAL on
+ * overflow, so testing the result for zero suffices to
+ * distinguish the cases).
+ *
+ * On some platforms (Ubuntu/ia64) it seems that errno can be
+ * set to ERANGE for subnormal results that do *not* underflow
+ * to zero. So to be safe, we'll ignore ERANGE whenever the
+ * function result is less than one in absolute value.
+ */
+ if (fabs(x) < 1.0)
+ result = 0;
+ else
+ PyErr_SetString(PyExc_OverflowError,
+ "math range error");
+ }
+ else
+ /* Unexpected math error */
+ PyErr_SetFromErrno(PyExc_ValueError);
+ return result;
+}
+
/*
math_1 is used to wrap a libm function f that takes a double
arguments and returns a double.
@@ -247,6 +508,26 @@
return PyFloat_FromDouble(r);
}
+/* variant of math_1, to be used when the function being wrapped is known to
+ set errno properly (that is, errno = EDOM for invalid or divide-by-zero,
+ errno = ERANGE for overflow). */
+
+static PyObject *
+math_1a(PyObject *arg, double (*func) (double))
+{
+ double x, r;
+ x = PyFloat_AsDouble(arg);
+ if (x == -1.0 && PyErr_Occurred())
+ return NULL;
+ errno = 0;
+ PyFPE_START_PROTECT("in math_1a", return 0);
+ r = (*func)(x);
+ PyFPE_END_PROTECT(r);
+ if (errno && is_error(r))
+ return NULL;
+ return PyFloat_FromDouble(r);
+}
+
/*
math_2 is used to wrap a libm function f that takes two double
arguments and returns a double.
@@ -313,6 +594,12 @@
}\
PyDoc_STRVAR(math_##funcname##_doc, docstring);
+#define FUNC1A(funcname, func, docstring) \
+ static PyObject * math_##funcname(PyObject *self, PyObject *args) { \
+ return math_1a(args, func); \
+ }\
+ PyDoc_STRVAR(math_##funcname##_doc, docstring);
+
#define FUNC2(funcname, func, docstring) \
static PyObject * math_##funcname(PyObject *self, PyObject *args) { \
return math_2(args, func, #funcname); \
@@ -350,6 +637,8 @@
FUNC1(floor, floor, 0,
"floor(x)\n\nReturn the floor of x as a float.\n"
"This is the largest integral value <= x.")
+FUNC1A(gamma, m_tgamma,
+ "gamma(x)\n\nGamma function at x.")
FUNC1(log1p, log1p, 1,
"log1p(x)\n\nReturn the natural logarithm of 1+x (base e).\n\
The result is computed in a way which is accurate for x near zero.")
@@ -1077,6 +1366,7 @@
{"fmod", math_fmod, METH_VARARGS, math_fmod_doc},
{"frexp", math_frexp, METH_O, math_frexp_doc},
{"fsum", math_fsum, METH_O, math_fsum_doc},
+ {"gamma", math_gamma, METH_O, math_gamma_doc},
{"hypot", math_hypot, METH_VARARGS, math_hypot_doc},
{"isinf", math_isinf, METH_O, math_isinf_doc},
{"isnan", math_isnan, METH_O, math_isnan_doc},