| /* 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: 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 inverse hyperbolic cosine 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 inverse hyperbolic sine 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 inverse hyperbolic tangent 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 (_PyAnyInt_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, result; |
| Py_ssize_t e; |
| |
| /* Negative or zero inputs give a ValueError. */ |
| if (Py_SIZE(arg) <= 0) { |
| 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); |
| } |
| |
| 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; |
| } |
