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Diffstat (limited to 'AppPkg/Applications/Python/Python-2.7.2/Modules/mathmodule.c')
| -rw-r--r-- | AppPkg/Applications/Python/Python-2.7.2/Modules/mathmodule.c | 1624 |
1 files changed, 1624 insertions, 0 deletions
diff --git a/AppPkg/Applications/Python/Python-2.7.2/Modules/mathmodule.c b/AppPkg/Applications/Python/Python-2.7.2/Modules/mathmodule.c new file mode 100644 index 0000000000..04b66f4162 --- /dev/null +++ b/AppPkg/Applications/Python/Python-2.7.2/Modules/mathmodule.c @@ -0,0 +1,1624 @@ +/* Math module -- standard C math library functions, pi and e */
+
+/* Here are some comments from Tim Peters, extracted from the
+ discussion attached to http://bugs.python.org/issue1640. They
+ describe the general aims of the math module with respect to
+ special values, IEEE-754 floating-point exceptions, and Python
+ exceptions.
+
+These are the "spirit of 754" rules:
+
+1. If the mathematical result is a real number, but of magnitude too
+large to approximate by a machine float, overflow is signaled and the
+result is an infinity (with the appropriate sign).
+
+2. If the mathematical result is a real number, but of magnitude too
+small to approximate by a machine float, underflow is signaled and the
+result is a zero (with the appropriate sign).
+
+3. At a singularity (a value x such that the limit of f(y) as y
+approaches x exists and is an infinity), "divide by zero" is signaled
+and the result is an infinity (with the appropriate sign). This is
+complicated a little by that the left-side and right-side limits may
+not be the same; e.g., 1/x approaches +inf or -inf as x approaches 0
+from the positive or negative directions. In that specific case, the
+sign of the zero determines the result of 1/0.
+
+4. At a point where a function has no defined result in the extended
+reals (i.e., the reals plus an infinity or two), invalid operation is
+signaled and a NaN is returned.
+
+And these are what Python has historically /tried/ to do (but not
+always successfully, as platform libm behavior varies a lot):
+
+For #1, raise OverflowError.
+
+For #2, return a zero (with the appropriate sign if that happens by
+accident ;-)).
+
+For #3 and #4, raise ValueError. It may have made sense to raise
+Python's ZeroDivisionError in #3, but historically that's only been
+raised for division by zero and mod by zero.
+
+*/
+
+/*
+ In general, on an IEEE-754 platform the aim is to follow the C99
+ standard, including Annex 'F', whenever possible. Where the
+ standard recommends raising the 'divide-by-zero' or 'invalid'
+ floating-point exceptions, Python should raise a ValueError. Where
+ the standard recommends raising 'overflow', Python should raise an
+ OverflowError. In all other circumstances a value should be
+ returned.
+ */
+
+#include "Python.h"
+#include "_math.h"
+
+#ifdef _OSF_SOURCE
+/* OSF1 5.1 doesn't make this available with XOPEN_SOURCE_EXTENDED defined */
+extern double copysign(double, double);
+#endif
+
+/*
+ sin(pi*x), giving accurate results for all finite x (especially x
+ integral or close to an integer). This is here for use in the
+ reflection formula for the gamma function. It conforms to IEEE
+ 754-2008 for finite arguments, but not for infinities or nans.
+*/
+
+static const double pi = 3.141592653589793238462643383279502884197;
+static const double sqrtpi = 1.772453850905516027298167483341145182798;
+
+static double
+sinpi(double x)
+{
+ double y, r;
+ int n;
+ /* this function should only ever be called for finite arguments */
+ assert(Py_IS_FINITE(x));
+ y = fmod(fabs(x), 2.0);
+ n = (int)round(2.0*y);
+ assert(0 <= n && n <= 4);
+ switch (n) {
+ case 0:
+ r = sin(pi*y);
+ break;
+ case 1:
+ r = cos(pi*(y-0.5));
+ break;
+ case 2:
+ /* N.B. -sin(pi*(y-1.0)) is *not* equivalent: it would give
+ -0.0 instead of 0.0 when y == 1.0. */
+ r = sin(pi*(1.0-y));
+ break;
+ case 3:
+ r = -cos(pi*(y-1.5));
+ break;
+ case 4:
+ r = sin(pi*(y-2.0));
+ break;
+ default:
+ assert(0); /* should never get here */
+ r = -1.23e200; /* silence gcc warning */
+ }
+ return copysign(1.0, x)*r;
+}
+
+/* Implementation of the real gamma function. In extensive but non-exhaustive
+ random tests, this function proved accurate to within <= 10 ulps across the
+ entire float domain. Note that accuracy may depend on the quality of the
+ system math functions, the pow function in particular. Special cases
+ follow C99 annex F. The parameters and method are tailored to platforms
+ whose double format is the IEEE 754 binary64 format.
+
+ Method: for x > 0.0 we use the Lanczos approximation with parameters N=13
+ and g=6.024680040776729583740234375; these parameters are amongst those
+ used by the Boost library. Following Boost (again), we re-express the
+ Lanczos sum as a rational function, and compute it that way. The
+ coefficients below were computed independently using MPFR, and have been
+ double-checked against the coefficients in the Boost source code.
+
+ For x < 0.0 we use the reflection formula.
+
+ There's one minor tweak that deserves explanation: Lanczos' formula for
+ Gamma(x) involves computing pow(x+g-0.5, x-0.5) / exp(x+g-0.5). For many x
+ values, x+g-0.5 can be represented exactly. However, in cases where it
+ can't be represented exactly the small error in x+g-0.5 can be magnified
+ significantly by the pow and exp calls, especially for large x. A cheap
+ correction is to multiply by (1 + e*g/(x+g-0.5)), where e is the error
+ involved in the computation of x+g-0.5 (that is, e = computed value of
+ x+g-0.5 - exact value of x+g-0.5). Here's the proof:
+
+ Correction factor
+ -----------------
+ Write x+g-0.5 = y-e, where y is exactly representable as an IEEE 754
+ double, and e is tiny. Then:
+
+ pow(x+g-0.5,x-0.5)/exp(x+g-0.5) = pow(y-e, x-0.5)/exp(y-e)
+ = pow(y, x-0.5)/exp(y) * C,
+
+ where the correction_factor C is given by
+
+ C = pow(1-e/y, x-0.5) * exp(e)
+
+ Since e is tiny, pow(1-e/y, x-0.5) ~ 1-(x-0.5)*e/y, and exp(x) ~ 1+e, so:
+
+ C ~ (1-(x-0.5)*e/y) * (1+e) ~ 1 + e*(y-(x-0.5))/y
+
+ But y-(x-0.5) = g+e, and g+e ~ g. So we get C ~ 1 + e*g/y, and
+
+ pow(x+g-0.5,x-0.5)/exp(x+g-0.5) ~ pow(y, x-0.5)/exp(y) * (1 + e*g/y),
+
+ Note that for accuracy, when computing r*C it's better to do
+
+ r + e*g/y*r;
+
+ than
+
+ r * (1 + e*g/y);
+
+ since the addition in the latter throws away most of the bits of
+ information in e*g/y.
+*/
+
+#define LANCZOS_N 13
+static const double lanczos_g = 6.024680040776729583740234375;
+static const double lanczos_g_minus_half = 5.524680040776729583740234375;
+static const double lanczos_num_coeffs[LANCZOS_N] = {
+ 23531376880.410759688572007674451636754734846804940,
+ 42919803642.649098768957899047001988850926355848959,
+ 35711959237.355668049440185451547166705960488635843,
+ 17921034426.037209699919755754458931112671403265390,
+ 6039542586.3520280050642916443072979210699388420708,
+ 1439720407.3117216736632230727949123939715485786772,
+ 248874557.86205415651146038641322942321632125127801,
+ 31426415.585400194380614231628318205362874684987640,
+ 2876370.6289353724412254090516208496135991145378768,
+ 186056.26539522349504029498971604569928220784236328,
+ 8071.6720023658162106380029022722506138218516325024,
+ 210.82427775157934587250973392071336271166969580291,
+ 2.5066282746310002701649081771338373386264310793408
+};
+
+/* denominator is x*(x+1)*...*(x+LANCZOS_N-2) */
+static const double lanczos_den_coeffs[LANCZOS_N] = {
+ 0.0, 39916800.0, 120543840.0, 150917976.0, 105258076.0, 45995730.0,
+ 13339535.0, 2637558.0, 357423.0, 32670.0, 1925.0, 66.0, 1.0};
+
+/* gamma values for small positive integers, 1 though NGAMMA_INTEGRAL */
+#define NGAMMA_INTEGRAL 23
+static const double gamma_integral[NGAMMA_INTEGRAL] = {
+ 1.0, 1.0, 2.0, 6.0, 24.0, 120.0, 720.0, 5040.0, 40320.0, 362880.0,
+ 3628800.0, 39916800.0, 479001600.0, 6227020800.0, 87178291200.0,
+ 1307674368000.0, 20922789888000.0, 355687428096000.0,
+ 6402373705728000.0, 121645100408832000.0, 2432902008176640000.0,
+ 51090942171709440000.0, 1124000727777607680000.0,
+};
+
+/* Lanczos' sum L_g(x), for positive x */
+
+static double
+lanczos_sum(double x)
+{
+ double num = 0.0, den = 0.0;
+ int i;
+ assert(x > 0.0);
+ /* evaluate the rational function lanczos_sum(x). For large
+ x, the obvious algorithm risks overflow, so we instead
+ rescale the denominator and numerator of the rational
+ function by x**(1-LANCZOS_N) and treat this as a
+ rational function in 1/x. This also reduces the error for
+ larger x values. The choice of cutoff point (5.0 below) is
+ somewhat arbitrary; in tests, smaller cutoff values than
+ this resulted in lower accuracy. */
+ if (x < 5.0) {
+ for (i = LANCZOS_N; --i >= 0; ) {
+ num = num * x + lanczos_num_coeffs[i];
+ den = den * x + lanczos_den_coeffs[i];
+ }
+ }
+ else {
+ for (i = 0; i < LANCZOS_N; i++) {
+ num = num / x + lanczos_num_coeffs[i];
+ den = den / x + lanczos_den_coeffs[i];
+ }
+ }
+ return num/den;
+}
+
+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;
+}
+
+/*
+ lgamma: natural log of the absolute value of the Gamma function.
+ For large arguments, Lanczos' formula works extremely well here.
+*/
+
+static double
+m_lgamma(double x)
+{
+ double r, absx;
+
+ /* special cases */
+ if (!Py_IS_FINITE(x)) {
+ if (Py_IS_NAN(x))
+ return x; /* lgamma(nan) = nan */
+ else
+ return Py_HUGE_VAL; /* lgamma(+-inf) = +inf */
+ }
+
+ /* integer arguments */
+ if (x == floor(x) && x <= 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 */
+ if (x > 0.0) {
+ /* 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(x)) - lanczos_g +
+ (x-0.5)*(log(x+lanczos_g-0.5)-1);
+ }
+ else {
+ r = log(pi) - log(fabs(sinpi(absx))) - log(absx) -
+ (log(lanczos_sum(absx)) - lanczos_g +
+ (absx-0.5)*(log(absx+lanczos_g-0.5)-1));
+ }
+ if (Py_IS_INFINITY(r))
+ errno = ERANGE;
+ return r;
+}
+
+/*
+ Implementations of the error function erf(x) and the complementary error
+ function erfc(x).
+
+ Method: following 'Numerical Recipes' by Flannery, Press et. al. (2nd ed.,
+ Cambridge University Press), we use a series approximation for erf for
+ small x, and a continued fraction approximation for erfc(x) for larger x;
+ combined with the relations erf(-x) = -erf(x) and erfc(x) = 1.0 - erf(x),
+ this gives us erf(x) and erfc(x) for all x.
+
+ The series expansion used is:
+
+ erf(x) = x*exp(-x*x)/sqrt(pi) * [
+ 2/1 + 4/3 x**2 + 8/15 x**4 + 16/105 x**6 + ...]
+
+ The coefficient of x**(2k-2) here is 4**k*factorial(k)/factorial(2*k).
+ This series converges well for smallish x, but slowly for larger x.
+
+ The continued fraction expansion used is:
+
+ erfc(x) = x*exp(-x*x)/sqrt(pi) * [1/(0.5 + x**2 -) 0.5/(2.5 + x**2 - )
+ 3.0/(4.5 + x**2 - ) 7.5/(6.5 + x**2 - ) ...]
+
+ after the first term, the general term has the form:
+
+ k*(k-0.5)/(2*k+0.5 + x**2 - ...).
+
+ This expansion converges fast for larger x, but convergence becomes
+ infinitely slow as x approaches 0.0. The (somewhat naive) continued
+ fraction evaluation algorithm used below also risks overflow for large x;
+ but for large x, erfc(x) == 0.0 to within machine precision. (For
+ example, erfc(30.0) is approximately 2.56e-393).
+
+ Parameters: use series expansion for abs(x) < ERF_SERIES_CUTOFF and
+ continued fraction expansion for ERF_SERIES_CUTOFF <= abs(x) <
+ ERFC_CONTFRAC_CUTOFF. ERFC_SERIES_TERMS and ERFC_CONTFRAC_TERMS are the
+ numbers of terms to use for the relevant expansions. */
+
+#define ERF_SERIES_CUTOFF 1.5
+#define ERF_SERIES_TERMS 25
+#define ERFC_CONTFRAC_CUTOFF 30.0
+#define ERFC_CONTFRAC_TERMS 50
+
+/*
+ Error function, via power series.
+
+ Given a finite float x, return an approximation to erf(x).
+ Converges reasonably fast for small x.
+*/
+
+static double
+m_erf_series(double x)
+{
+ double x2, acc, fk, result;
+ int i, saved_errno;
+
+ x2 = x * x;
+ acc = 0.0;
+ fk = (double)ERF_SERIES_TERMS + 0.5;
+ for (i = 0; i < ERF_SERIES_TERMS; i++) {
+ acc = 2.0 + x2 * acc / fk;
+ fk -= 1.0;
+ }
+ /* Make sure the exp call doesn't affect errno;
+ see m_erfc_contfrac for more. */
+ saved_errno = errno;
+ result = acc * x * exp(-x2) / sqrtpi;
+ errno = saved_errno;
+ return result;
+}
+
+/*
+ Complementary error function, via continued fraction expansion.
+
+ Given a positive float x, return an approximation to erfc(x). Converges
+ reasonably fast for x large (say, x > 2.0), and should be safe from
+ overflow if x and nterms are not too large. On an IEEE 754 machine, with x
+ <= 30.0, we're safe up to nterms = 100. For x >= 30.0, erfc(x) is smaller
+ than the smallest representable nonzero float. */
+
+static double
+m_erfc_contfrac(double x)
+{
+ double x2, a, da, p, p_last, q, q_last, b, result;
+ int i, saved_errno;
+
+ if (x >= ERFC_CONTFRAC_CUTOFF)
+ return 0.0;
+
+ x2 = x*x;
+ a = 0.0;
+ da = 0.5;
+ p = 1.0; p_last = 0.0;
+ q = da + x2; q_last = 1.0;
+ for (i = 0; i < ERFC_CONTFRAC_TERMS; i++) {
+ double temp;
+ a += da;
+ da += 2.0;
+ b = da + x2;
+ temp = p; p = b*p - a*p_last; p_last = temp;
+ temp = q; q = b*q - a*q_last; q_last = temp;
+ }
+ /* Issue #8986: On some platforms, exp sets errno on underflow to zero;
+ save the current errno value so that we can restore it later. */
+ saved_errno = errno;
+ result = p / q * x * exp(-x2) / sqrtpi;
+ errno = saved_errno;
+ return result;
+}
+
+/* Error function erf(x), for general x */
+
+static double
+m_erf(double x)
+{
+ double absx, cf;
+
+ if (Py_IS_NAN(x))
+ return x;
+ absx = fabs(x);
+ if (absx < ERF_SERIES_CUTOFF)
+ return m_erf_series(x);
+ else {
+ cf = m_erfc_contfrac(absx);
+ return x > 0.0 ? 1.0 - cf : cf - 1.0;
+ }
+}
+
+/* Complementary error function erfc(x), for general x. */
+
+static double
+m_erfc(double x)
+{
+ double absx, cf;
+
+ if (Py_IS_NAN(x))
+ return x;
+ absx = fabs(x);
+ if (absx < ERF_SERIES_CUTOFF)
+ return 1.0 - m_erf_series(x);
+ else {
+ cf = m_erfc_contfrac(absx);
+ return x > 0.0 ? cf : 2.0 - cf;
+ }
+}
+
+/*
+ wrapper for atan2 that deals directly with special cases before
+ delegating to the platform libm for the remaining cases. This
+ is necessary to get consistent behaviour across platforms.
+ Windows, FreeBSD and alpha Tru64 are amongst platforms that don't
+ always follow C99.
+*/
+
+static double
+m_atan2(double y, double x)
+{
+ if (Py_IS_NAN(x) || Py_IS_NAN(y))
+ return Py_NAN;
+ if (Py_IS_INFINITY(y)) {
+ if (Py_IS_INFINITY(x)) {
+ if (copysign(1., x) == 1.)
+ /* atan2(+-inf, +inf) == +-pi/4 */
+ return copysign(0.25*Py_MATH_PI, y);
+ else
+ /* atan2(+-inf, -inf) == +-pi*3/4 */
+ return copysign(0.75*Py_MATH_PI, y);
+ }
+ /* atan2(+-inf, x) == +-pi/2 for finite x */
+ return copysign(0.5*Py_MATH_PI, y);
+ }
+ if (Py_IS_INFINITY(x) || y == 0.) {
+ if (copysign(1., x) == 1.)
+ /* atan2(+-y, +inf) = atan2(+-0, +x) = +-0. */
+ return copysign(0., y);
+ else
+ /* atan2(+-y, -inf) = atan2(+-0., -x) = +-pi. */
+ return copysign(Py_MATH_PI, y);
+ }
+ return atan2(y, x);
+}
+
+/*
+ Various platforms (Solaris, OpenBSD) do nonstandard things for log(0),
+ log(-ve), log(NaN). Here are wrappers for log and log10 that deal with
+ special values directly, passing positive non-special values through to
+ the system log/log10.
+ */
+
+static double
+m_log(double x)
+{
+ if (Py_IS_FINITE(x)) {
+ if (x > 0.0)
+ return log(x);
+ errno = EDOM;
+ if (x == 0.0)
+ return -Py_HUGE_VAL; /* log(0) = -inf */
+ else
+ return Py_NAN; /* log(-ve) = nan */
+ }
+ else if (Py_IS_NAN(x))
+ return x; /* log(nan) = nan */
+ else if (x > 0.0)
+ return x; /* log(inf) = inf */
+ else {
+ errno = EDOM;
+ return Py_NAN; /* log(-inf) = nan */
+ }
+}
+
+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 */
+ }
+}
+
+
+/* Call is_error when errno != 0, and where x is the result libm
+ * returned. is_error will usually set up an exception and return
+ * true (1), but may return false (0) without setting up an exception.
+ */
+static int
+is_error(double x)
+{
+ int result = 1; /* presumption of guilt */
+ assert(errno); /* non-zero errno is a precondition for calling */
+ if (errno == EDOM)
+ PyErr_SetString(PyExc_ValueError, "math domain error");
+
+ else if (errno == ERANGE) {
+ /* ANSI C generally requires libm functions to set ERANGE
+ * on overflow, but also generally *allows* them to set
+ * ERANGE on underflow too. There's no consistency about
+ * the latter across platforms.
+ * Alas, C99 never requires that errno be set.
+ * Here we suppress the underflow errors (libm functions
+ * should return a zero on underflow, and +- HUGE_VAL on
+ * overflow, so testing the result for zero suffices to
+ * distinguish the cases).
+ *
+ * On some platforms (Ubuntu/ia64) it seems that errno can be
+ * set to ERANGE for subnormal results that do *not* underflow
+ * to zero. So to be safe, we'll ignore ERANGE whenever the
+ * function result is less than one in absolute value.
+ */
+ if (fabs(x) < 1.0)
+ result = 0;
+ else
+ PyErr_SetString(PyExc_OverflowError,
+ "math range error");
+ }
+ else
+ /* Unexpected math error */
+ PyErr_SetFromErrno(PyExc_ValueError);
+ return result;
+}
+
+/*
+ math_1 is used to wrap a libm function f that takes a double
+ arguments and returns a double.
+
+ The error reporting follows these rules, which are designed to do
+ the right thing on C89/C99 platforms and IEEE 754/non IEEE 754
+ platforms.
+
+ - a NaN result from non-NaN inputs causes ValueError to be raised
+ - an infinite result from finite inputs causes OverflowError to be
+ raised if can_overflow is 1, or raises ValueError if can_overflow
+ is 0.
+ - if the result is finite and errno == EDOM then ValueError is
+ raised
+ - if the result is finite and nonzero and errno == ERANGE then
+ OverflowError is raised
+
+ The last rule is used to catch overflow on platforms which follow
+ C89 but for which HUGE_VAL is not an infinity.
+
+ For the majority of one-argument functions these rules are enough
+ to ensure that Python's functions behave as specified in 'Annex F'
+ of the C99 standard, with the 'invalid' and 'divide-by-zero'
+ floating-point exceptions mapping to Python's ValueError and the
+ 'overflow' floating-point exception mapping to OverflowError.
+ math_1 only works for functions that don't have singularities *and*
+ the possibility of overflow; fortunately, that covers everything we
+ care about right now.
+*/
+
+static PyObject *
+math_1(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;
+ PyFPE_START_PROTECT("in math_1", return 0);
+ r = (*func)(x);
+ PyFPE_END_PROTECT(r);
+ if (Py_IS_NAN(r)) {
+ if (!Py_IS_NAN(x))
+ errno = EDOM;
+ else
+ errno = 0;
+ }
+ else if (Py_IS_INFINITY(r)) {
+ if (Py_IS_FINITE(x))
+ errno = can_overflow ? ERANGE : EDOM;
+ else
+ errno = 0;
+ }
+ if (errno && is_error(r))
+ return NULL;
+ else
+ 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.
+
+ 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 *args, double (*func) (double, double), char *funcname)
+{
+ PyObject *ox, *oy;
+ double x, y, r;
+ if (! PyArg_UnpackTuple(args, funcname, 2, 2, &ox, &oy))
+ return NULL;
+ x = PyFloat_AsDouble(ox);
+ y = PyFloat_AsDouble(oy);
+ if ((x == -1.0 || y == -1.0) && PyErr_Occurred())
+ return NULL;
+ errno = 0;
+ PyFPE_START_PROTECT("in math_2", return 0);
+ r = (*func)(x, y);
+ PyFPE_END_PROTECT(r);
+ if (Py_IS_NAN(r)) {
+ if (!Py_IS_NAN(x) && !Py_IS_NAN(y))
+ errno = EDOM;
+ else
+ errno = 0;
+ }
+ else if (Py_IS_INFINITY(r)) {
+ if (Py_IS_FINITE(x) && Py_IS_FINITE(y))
+ errno = ERANGE;
+ else
+ errno = 0;
+ }
+ if (errno && is_error(r))
+ return NULL;
+ else
+ return PyFloat_FromDouble(r);
+}
+
+#define FUNC1(funcname, func, can_overflow, docstring) \
+ static PyObject * math_##funcname(PyObject *self, PyObject *args) { \
+ return math_1(args, func, can_overflow); \
+ }\
+ PyDoc_STRVAR(math_##funcname##_doc, docstring);
+
+#define FUNC1A(funcname, func, docstring) \
+ static PyObject * math_##funcname(PyObject *self, PyObject *args) { \
+ return math_1a(args, func); \
+ }\
+ PyDoc_STRVAR(math_##funcname##_doc, docstring);
+
+#define FUNC2(funcname, func, docstring) \
+ static PyObject * math_##funcname(PyObject *self, PyObject *args) { \
+ return math_2(args, func, #funcname); \
+ }\
+ PyDoc_STRVAR(math_##funcname##_doc, docstring);
+
+FUNC1(acos, acos, 0,
+ "acos(x)\n\nReturn the arc cosine (measured in radians) of x.")
+FUNC1(acosh, m_acosh, 0,
+ "acosh(x)\n\nReturn the hyperbolic arc cosine (measured in radians) of x.")
+FUNC1(asin, asin, 0,
+ "asin(x)\n\nReturn the arc sine (measured in radians) of x.")
+FUNC1(asinh, m_asinh, 0,
+ "asinh(x)\n\nReturn the hyperbolic arc sine (measured in radians) of x.")
+FUNC1(atan, atan, 0,
+ "atan(x)\n\nReturn the arc tangent (measured in radians) of x.")
+FUNC2(atan2, m_atan2,
+ "atan2(y, x)\n\nReturn the arc tangent (measured in radians) of y/x.\n"
+ "Unlike atan(y/x), the signs of both x and y are considered.")
+FUNC1(atanh, m_atanh, 0,
+ "atanh(x)\n\nReturn the hyperbolic arc tangent (measured in radians) of x.")
+FUNC1(ceil, ceil, 0,
+ "ceil(x)\n\nReturn the ceiling of x as a float.\n"
+ "This is the smallest integral value >= x.")
+FUNC2(copysign, copysign,
+ "copysign(x, y)\n\nReturn x with the sign of y.")
+FUNC1(cos, cos, 0,
+ "cos(x)\n\nReturn the cosine of x (measured in radians).")
+FUNC1(cosh, cosh, 1,
+ "cosh(x)\n\nReturn the hyperbolic cosine of x.")
+FUNC1A(erf, m_erf,
+ "erf(x)\n\nError function at x.")
+FUNC1A(erfc, m_erfc,
+ "erfc(x)\n\nComplementary error function at x.")
+FUNC1(exp, exp, 1,
+ "exp(x)\n\nReturn e raised to the power of x.")
+FUNC1(expm1, m_expm1, 1,
+ "expm1(x)\n\nReturn exp(x)-1.\n"
+ "This function avoids the loss of precision involved in the direct "
+ "evaluation of exp(x)-1 for small x.")
+FUNC1(fabs, fabs, 0,
+ "fabs(x)\n\nReturn the absolute value of the float x.")
+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.")
+FUNC1A(lgamma, m_lgamma,
+ "lgamma(x)\n\nNatural logarithm of absolute value of Gamma function at x.")
+FUNC1(log1p, m_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.")
+FUNC1(sin, sin, 0,
+ "sin(x)\n\nReturn the sine of x (measured in radians).")
+FUNC1(sinh, sinh, 1,
+ "sinh(x)\n\nReturn the hyperbolic sine of x.")
+FUNC1(sqrt, sqrt, 0,
+ "sqrt(x)\n\nReturn the square root of x.")
+FUNC1(tan, tan, 0,
+ "tan(x)\n\nReturn the tangent of x (measured in radians).")
+FUNC1(tanh, tanh, 0,
+ "tanh(x)\n\nReturn 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 754R floating point semantics are assumed,
+ but the current implementation does not re-establish special
+ value semantics across iterations (i.e. handling -Inf + Inf).
+
+ Note 2: No provision is made for intermediate overflow handling;
+ therefore, sum([1e+308, 1e-308, 1e+308]) returns 1e+308 while
+ sum([1e+308, 1e+308, 1e-308]) raises an OverflowError due to the
+ overflow of the first partial sum.
+
+ Note 3: The intermediate values lo, yr, and hi are declared volatile so
+ aggressive compilers won't algebraically reduce lo to always be exactly 0.0.
+ Also, the volatile declaration forces the values to be stored in memory as
+ regular doubles instead of extended long precision (80-bit) values. This
+ prevents double rounding because any addition or subtraction of two doubles
+ can be resolved exactly into double-sized hi and lo values. As long as the
+ hi value gets forced into a double before yr and lo are computed, the extra
+ bits in downstream extended precision operations (x87 for example) will be
+ exactly zero and therefore can be losslessly stored back into a double,
+ thereby preventing double rounding.
+
+ Note 4: A similar implementation is in Modules/cmathmodule.c.
+ Be sure to update both when making changes.
+
+ Note 5: The signature of math.fsum() differs from __builtin__.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 && m < (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.
+*/
+
+static PyObject*
+math_fsum(PyObject *self, PyObject *seq)
+{
+ 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;
+ volatile double hi, yr, lo;
+
+ iter = PyObject_GetIter(seq);
+ if (iter == NULL)
+ return NULL;
+
+ PyFPE_START_PROTECT("fsum", Py_DECREF(iter); 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;
+ }
+ x = PyFloat_AsDouble(item);
+ Py_DECREF(item);
+ if (PyErr_Occurred())
+ goto _fsum_error;
+
+ 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:
+ PyFPE_END_PROTECT(hi)
+ Py_DECREF(iter);
+ if (p != ps)
+ PyMem_Free(p);
+ return sum;
+}
+
+#undef NUM_PARTIALS
+
+PyDoc_STRVAR(math_fsum_doc,
+"fsum(iterable)\n\n\
+Return an accurate floating point sum of values in the iterable.\n\
+Assumes IEEE-754 floating point arithmetic.");
+
+static PyObject *
+math_factorial(PyObject *self, PyObject *arg)
+{
+ long i, x;
+ PyObject *result, *iobj, *newresult;
+
+ if (PyFloat_Check(arg)) {
+ PyObject *lx;
+ double dx = PyFloat_AS_DOUBLE((PyFloatObject *)arg);
+ if (!(Py_IS_FINITE(dx) && dx == floor(dx))) {
+ PyErr_SetString(PyExc_ValueError,
+ "factorial() only accepts integral values");
+ return NULL;
+ }
+ lx = PyLong_FromDouble(dx);
+ if (lx == NULL)
+ return NULL;
+ x = PyLong_AsLong(lx);
+ Py_DECREF(lx);
+ }
+ else
+ x = PyInt_AsLong(arg);
+
+ if (x == -1 && PyErr_Occurred())
+ return NULL;
+ if (x < 0) {
+ PyErr_SetString(PyExc_ValueError,
+ "factorial() not defined for negative values");
+ return NULL;
+ }
+
+ result = (PyObject *)PyInt_FromLong(1);
+ if (result == NULL)
+ return NULL;
+ for (i=1 ; i<=x ; i++) {
+ iobj = (PyObject *)PyInt_FromLong(i);
+ if (iobj == NULL)
+ goto error;
+ newresult = PyNumber_Multiply(result, iobj);
+ Py_DECREF(iobj);
+ if (newresult == NULL)
+ goto error;
+ Py_DECREF(result);
+ result = newresult;
+ }
+ return result;
+
+error:
+ Py_DECREF(result);
+ return NULL;
+}
+
+PyDoc_STRVAR(math_factorial_doc,
+"factorial(x) -> Integral\n"
+"\n"
+"Find x!. Raise a ValueError if x is negative or non-integral.");
+
+static PyObject *
+math_trunc(PyObject *self, PyObject *number)
+{
+ return PyObject_CallMethod(number, "__trunc__", NULL);
+}
+
+PyDoc_STRVAR(math_trunc_doc,
+"trunc(x:Real) -> Integral\n"
+"\n"
+"Truncates x to the nearest Integral toward 0. Uses the __trunc__ magic method.");
+
+static PyObject *
+math_frexp(PyObject *self, PyObject *arg)
+{
+ int i;
+ double x = PyFloat_AsDouble(arg);
+ if (x == -1.0 && PyErr_Occurred())
+ return NULL;
+ /* deal with special cases directly, to sidestep platform
+ differences */
+ if (Py_IS_NAN(x) || Py_IS_INFINITY(x) || !x) {
+ i = 0;
+ }
+ else {
+ PyFPE_START_PROTECT("in math_frexp", return 0);
+ x = frexp(x, &i);
+ PyFPE_END_PROTECT(x);
+ }
+ return Py_BuildValue("(di)", x, i);
+}
+
+PyDoc_STRVAR(math_frexp_doc,
+"frexp(x)\n"
+"\n"
+"Return the mantissa and exponent of x, as pair (m, e).\n"
+"m is a float and e is an int, such that x = m * 2.**e.\n"
+"If x is 0, m and e are both 0. Else 0.5 <= abs(m) < 1.0.");
+
+static PyObject *
+math_ldexp(PyObject *self, PyObject *args)
+{
+ double x, r;
+ PyObject *oexp;
+ long exp;
+ int overflow;
+ if (! PyArg_ParseTuple(args, "dO:ldexp", &x, &oexp))
+ return NULL;
+
+ if (PyLong_Check(oexp) || PyInt_Check(oexp)) {
+ /* on overflow, replace exponent with either LONG_MAX
+ or LONG_MIN, depending on the sign. */
+ exp = PyLong_AsLongAndOverflow(oexp, &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 or long 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;
+ PyFPE_START_PROTECT("in math_ldexp", return 0);
+ r = ldexp(x, (int)exp);
+ PyFPE_END_PROTECT(r);
+ if (Py_IS_INFINITY(r))
+ errno = ERANGE;
+ }
+
+ if (errno && is_error(r))
+ return NULL;
+ return PyFloat_FromDouble(r);
+}
+
+PyDoc_STRVAR(math_ldexp_doc,
+"ldexp(x, i)\n\n\
+Return x * (2**i).");
+
+static PyObject *
+math_modf(PyObject *self, PyObject *arg)
+{
+ double y, x = PyFloat_AsDouble(arg);
+ if (x == -1.0 && PyErr_Occurred())
+ return NULL;
+ /* some platforms don't do the right thing for NaNs and
+ infinities, so we take care of special cases directly. */
+ if (!Py_IS_FINITE(x)) {
+ if (Py_IS_INFINITY(x))
+ return Py_BuildValue("(dd)", copysign(0., x), x);
+ else if (Py_IS_NAN(x))
+ return Py_BuildValue("(dd)", x, x);
+ }
+
+ errno = 0;
+ PyFPE_START_PROTECT("in math_modf", return 0);
+ x = modf(x, &y);
+ PyFPE_END_PROTECT(x);
+ return Py_BuildValue("(dd)", x, y);
+}
+
+PyDoc_STRVAR(math_modf_doc,
+"modf(x)\n"
+"\n"
+"Return the fractional and integer parts of x. Both results carry the sign\n"
+"of x and are floats.");
+
+/* A decent logarithm is easy to compute even for huge longs, 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: a long
+ 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 a long if the number of bits
+ in that long is larger than PY_SSIZE_T_MAX. */
+
+static PyObject*
+loghelper(PyObject* arg, double (*func)(double), char *funcname)
+{
+ /* If it is long, do it ourselves. */
+ if (PyLong_Check(arg)) {
+ double x;
+ Py_ssize_t e;
+ x = _PyLong_Frexp((PyLongObject *)arg, &e);
+ if (x == -1.0 && PyErr_Occurred())
+ return NULL;
+ if (x <= 0.0) {
+ PyErr_SetString(PyExc_ValueError,
+ "math domain error");
+ return NULL;
+ }
+ /* Special case for log(1), to make sure we get an
+ exact result there. */
+ if (e == 1 && x == 0.5)
+ return PyFloat_FromDouble(0.0);
+ /* Value is ~= x * 2**e, so the log ~= log(x) + log(2) * e. */
+ x = func(x) + func(2.0) * e;
+ return PyFloat_FromDouble(x);
+ }
+
+ /* Else let libm handle it by itself. */
+ return math_1(arg, func, 0);
+}
+
+static PyObject *
+math_log(PyObject *self, PyObject *args)
+{
+ PyObject *arg;
+ PyObject *base = NULL;
+ PyObject *num, *den;
+ PyObject *ans;
+
+ if (!PyArg_UnpackTuple(args, "log", 1, 2, &arg, &base))
+ return NULL;
+
+ num = loghelper(arg, m_log, "log");
+ if (num == NULL || base == NULL)
+ return num;
+
+ den = loghelper(base, m_log, "log");
+ if (den == NULL) {
+ Py_DECREF(num);
+ return NULL;
+ }
+
+ ans = PyNumber_Divide(num, den);
+ Py_DECREF(num);
+ Py_DECREF(den);
+ return ans;
+}
+
+PyDoc_STRVAR(math_log_doc,
+"log(x[, base])\n\n\
+Return the logarithm of x to the given base.\n\
+If the base not specified, returns the natural logarithm (base e) of x.");
+
+static PyObject *
+math_log10(PyObject *self, PyObject *arg)
+{
+ return loghelper(arg, m_log10, "log10");
+}
+
+PyDoc_STRVAR(math_log10_doc,
+"log10(x)\n\nReturn the base 10 logarithm of x.");
+
+static PyObject *
+math_fmod(PyObject *self, PyObject *args)
+{
+ PyObject *ox, *oy;
+ double r, x, y;
+ if (! PyArg_UnpackTuple(args, "fmod", 2, 2, &ox, &oy))
+ return NULL;
+ x = PyFloat_AsDouble(ox);
+ y = PyFloat_AsDouble(oy);
+ if ((x == -1.0 || y == -1.0) && PyErr_Occurred())
+ return NULL;
+ /* fmod(x, +/-Inf) returns x for finite x. */
+ if (Py_IS_INFINITY(y) && Py_IS_FINITE(x))
+ return PyFloat_FromDouble(x);
+ errno = 0;
+ PyFPE_START_PROTECT("in math_fmod", return 0);
+ r = fmod(x, y);
+ PyFPE_END_PROTECT(r);
+ if (Py_IS_NAN(r)) {
+ if (!Py_IS_NAN(x) && !Py_IS_NAN(y))
+ errno = EDOM;
+ else
+ errno = 0;
+ }
+ if (errno && is_error(r))
+ return NULL;
+ else
+ return PyFloat_FromDouble(r);
+}
+
+PyDoc_STRVAR(math_fmod_doc,
+"fmod(x, y)\n\nReturn fmod(x, y), according to platform C."
+" x % y may differ.");
+
+static PyObject *
+math_hypot(PyObject *self, PyObject *args)
+{
+ PyObject *ox, *oy;
+ double r, x, y;
+ if (! PyArg_UnpackTuple(args, "hypot", 2, 2, &ox, &oy))
+ return NULL;
+ x = PyFloat_AsDouble(ox);
+ y = PyFloat_AsDouble(oy);
+ if ((x == -1.0 || y == -1.0) && PyErr_Occurred())
+ return NULL;
+ /* hypot(x, +/-Inf) returns Inf, even if x is a NaN. */
+ if (Py_IS_INFINITY(x))
+ return PyFloat_FromDouble(fabs(x));
+ if (Py_IS_INFINITY(y))
+ return PyFloat_FromDouble(fabs(y));
+ errno = 0;
+ PyFPE_START_PROTECT("in math_hypot", return 0);
+ r = hypot(x, y);
+ PyFPE_END_PROTECT(r);
+ if (Py_IS_NAN(r)) {
+ if (!Py_IS_NAN(x) && !Py_IS_NAN(y))
+ errno = EDOM;
+ else
+ errno = 0;
+ }
+ else if (Py_IS_INFINITY(r)) {
+ if (Py_IS_FINITE(x) && Py_IS_FINITE(y))
+ errno = ERANGE;
+ else
+ errno = 0;
+ }
+ if (errno && is_error(r))
+ return NULL;
+ else
+ return PyFloat_FromDouble(r);
+}
+
+PyDoc_STRVAR(math_hypot_doc,
+"hypot(x, y)\n\nReturn the Euclidean distance, sqrt(x*x + y*y).");
+
+/* 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.)
+*/
+
+static PyObject *
+math_pow(PyObject *self, PyObject *args)
+{
+ PyObject *ox, *oy;
+ double r, x, y;
+ int odd_y;
+
+ if (! PyArg_UnpackTuple(args, "pow", 2, 2, &ox, &oy))
+ return NULL;
+ x = PyFloat_AsDouble(ox);
+ y = PyFloat_AsDouble(oy);
+ if ((x == -1.0 || y == -1.0) && PyErr_Occurred())
+ return NULL;
+
+ /* 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 */
+ if (x == 0.) /* 0**-inf: divide-by-zero */
+ errno = EDOM;
+ }
+ else
+ r = 0.;
+ }
+ }
+ else {
+ /* let libm handle finite**finite */
+ errno = 0;
+ PyFPE_START_PROTECT("in math_pow", return 0);
+ r = pow(x, y);
+ PyFPE_END_PROTECT(r);
+ /* 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);
+}
+
+PyDoc_STRVAR(math_pow_doc,
+"pow(x, y)\n\nReturn x**y (x to the power of y).");
+
+static const double degToRad = Py_MATH_PI / 180.0;
+static const double radToDeg = 180.0 / Py_MATH_PI;
+
+static PyObject *
+math_degrees(PyObject *self, PyObject *arg)
+{
+ double x = PyFloat_AsDouble(arg);
+ if (x == -1.0 && PyErr_Occurred())
+ return NULL;
+ return PyFloat_FromDouble(x * radToDeg);
+}
+
+PyDoc_STRVAR(math_degrees_doc,
+"degrees(x)\n\n\
+Convert angle x from radians to degrees.");
+
+static PyObject *
+math_radians(PyObject *self, PyObject *arg)
+{
+ double x = PyFloat_AsDouble(arg);
+ if (x == -1.0 && PyErr_Occurred())
+ return NULL;
+ return PyFloat_FromDouble(x * degToRad);
+}
+
+PyDoc_STRVAR(math_radians_doc,
+"radians(x)\n\n\
+Convert angle x from degrees to radians.");
+
+static PyObject *
+math_isnan(PyObject *self, PyObject *arg)
+{
+ double x = PyFloat_AsDouble(arg);
+ if (x == -1.0 && PyErr_Occurred())
+ return NULL;
+ return PyBool_FromLong((long)Py_IS_NAN(x));
+}
+
+PyDoc_STRVAR(math_isnan_doc,
+"isnan(x) -> bool\n\n\
+Check if float x is not a number (NaN).");
+
+static PyObject *
+math_isinf(PyObject *self, PyObject *arg)
+{
+ double x = PyFloat_AsDouble(arg);
+ if (x == -1.0 && PyErr_Occurred())
+ return NULL;
+ return PyBool_FromLong((long)Py_IS_INFINITY(x));
+}
+
+PyDoc_STRVAR(math_isinf_doc,
+"isinf(x) -> bool\n\n\
+Check if float x is infinite (positive or negative).");
+
+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", math_atan2, METH_VARARGS, math_atan2_doc},
+ {"atanh", math_atanh, METH_O, math_atanh_doc},
+ {"ceil", math_ceil, METH_O, math_ceil_doc},
+ {"copysign", math_copysign, METH_VARARGS, math_copysign_doc},
+ {"cos", math_cos, METH_O, math_cos_doc},
+ {"cosh", math_cosh, METH_O, math_cosh_doc},
+ {"degrees", math_degrees, METH_O, math_degrees_doc},
+ {"erf", math_erf, METH_O, math_erf_doc},
+ {"erfc", math_erfc, METH_O, math_erfc_doc},
+ {"exp", math_exp, METH_O, math_exp_doc},
+ {"expm1", math_expm1, METH_O, math_expm1_doc},
+ {"fabs", math_fabs, METH_O, math_fabs_doc},
+ {"factorial", math_factorial, METH_O, math_factorial_doc},
+ {"floor", math_floor, METH_O, math_floor_doc},
+ {"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},
+ {"ldexp", math_ldexp, METH_VARARGS, math_ldexp_doc},
+ {"lgamma", math_lgamma, METH_O, math_lgamma_doc},
+ {"log", math_log, METH_VARARGS, math_log_doc},
+ {"log1p", math_log1p, METH_O, math_log1p_doc},
+ {"log10", math_log10, METH_O, math_log10_doc},
+ {"modf", math_modf, METH_O, math_modf_doc},
+ {"pow", math_pow, METH_VARARGS, math_pow_doc},
+ {"radians", math_radians, METH_O, math_radians_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},
+ {"trunc", math_trunc, METH_O, math_trunc_doc},
+ {NULL, NULL} /* sentinel */
+};
+
+
+PyDoc_STRVAR(module_doc,
+"This module is always available. It provides access to the\n"
+"mathematical functions defined by the C standard.");
+
+PyMODINIT_FUNC
+initmath(void)
+{
+ PyObject *m;
+
+ m = Py_InitModule3("math", math_methods, module_doc);
+ if (m == NULL)
+ goto finally;
+
+ PyModule_AddObject(m, "pi", PyFloat_FromDouble(Py_MATH_PI));
+ PyModule_AddObject(m, "e", PyFloat_FromDouble(Py_MATH_E));
+
+ finally:
+ return;
+}
|