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rhettinger
bpo-37691: Let math.dist() accept sequences and iterables for coordin…
6b5f1b4
Jul 28, 2019
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Sign up| /* Math module -- standard C math library functions, pi and e */ | |
| /* Here are some comments from Tim Peters, extracted from the | |
| discussion attached to http://bugs.python.org/issue1640. They | |
| describe the general aims of the math module with respect to | |
| special values, IEEE-754 floating-point exceptions, and Python | |
| exceptions. | |
| These are the "spirit of 754" rules: | |
| 1. If the mathematical result is a real number, but of magnitude too | |
| large to approximate by a machine float, overflow is signaled and the | |
| result is an infinity (with the appropriate sign). | |
| 2. If the mathematical result is a real number, but of magnitude too | |
| small to approximate by a machine float, underflow is signaled and the | |
| result is a zero (with the appropriate sign). | |
| 3. At a singularity (a value x such that the limit of f(y) as y | |
| approaches x exists and is an infinity), "divide by zero" is signaled | |
| and the result is an infinity (with the appropriate sign). This is | |
| complicated a little by that the left-side and right-side limits may | |
| not be the same; e.g., 1/x approaches +inf or -inf as x approaches 0 | |
| from the positive or negative directions. In that specific case, the | |
| sign of the zero determines the result of 1/0. | |
| 4. At a point where a function has no defined result in the extended | |
| reals (i.e., the reals plus an infinity or two), invalid operation is | |
| signaled and a NaN is returned. | |
| And these are what Python has historically /tried/ to do (but not | |
| always successfully, as platform libm behavior varies a lot): | |
| For #1, raise OverflowError. | |
| For #2, return a zero (with the appropriate sign if that happens by | |
| accident ;-)). | |
| For #3 and #4, raise ValueError. It may have made sense to raise | |
| Python's ZeroDivisionError in #3, but historically that's only been | |
| raised for division by zero and mod by zero. | |
| */ | |
| /* | |
| In general, on an IEEE-754 platform the aim is to follow the C99 | |
| standard, including Annex 'F', whenever possible. Where the | |
| standard recommends raising the 'divide-by-zero' or 'invalid' | |
| floating-point exceptions, Python should raise a ValueError. Where | |
| the standard recommends raising 'overflow', Python should raise an | |
| OverflowError. In all other circumstances a value should be | |
| returned. | |
| */ | |
| #include "Python.h" | |
| #include "_math.h" | |
| #include "clinic/mathmodule.c.h" | |
| /*[clinic input] | |
| module math | |
| [clinic start generated code]*/ | |
| /*[clinic end generated code: output=da39a3ee5e6b4b0d input=76bc7002685dd942]*/ | |
| /* | |
| sin(pi*x), giving accurate results for all finite x (especially x | |
| integral or close to an integer). This is here for use in the | |
| reflection formula for the gamma function. It conforms to IEEE | |
| 754-2008 for finite arguments, but not for infinities or nans. | |
| */ | |
| static const double pi = 3.141592653589793238462643383279502884197; | |
| static const double logpi = 1.144729885849400174143427351353058711647; | |
| #if !defined(HAVE_ERF) || !defined(HAVE_ERFC) | |
| static const double sqrtpi = 1.772453850905516027298167483341145182798; | |
| #endif /* !defined(HAVE_ERF) || !defined(HAVE_ERFC) */ | |
| /* Version of PyFloat_AsDouble() with in-line fast paths | |
| for exact floats and integers. Gives a substantial | |
| speed improvement for extracting float arguments. | |
| */ | |
| #define ASSIGN_DOUBLE(target_var, obj, error_label) \ | |
| if (PyFloat_CheckExact(obj)) { \ | |
| target_var = PyFloat_AS_DOUBLE(obj); \ | |
| } \ | |
| else if (PyLong_CheckExact(obj)) { \ | |
| target_var = PyLong_AsDouble(obj); \ | |
| if (target_var == -1.0 && PyErr_Occurred()) { \ | |
| goto error_label; \ | |
| } \ | |
| } \ | |
| else { \ | |
| target_var = PyFloat_AsDouble(obj); \ | |
| if (target_var == -1.0 && PyErr_Occurred()) { \ | |
| goto error_label; \ | |
| } \ | |
| } | |
| static double | |
| m_sinpi(double x) | |
| { | |
| double y, r; | |
| int n; | |
| /* this function should only ever be called for finite arguments */ | |
| assert(Py_IS_FINITE(x)); | |
| y = fmod(fabs(x), 2.0); | |
| n = (int)round(2.0*y); | |
| assert(0 <= 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: | |
| Py_UNREACHABLE(); | |
| } | |
| 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; | |
| } | |
| /* Constant for +infinity, generated in the same way as float('inf'). */ | |
| static double | |
| m_inf(void) | |
| { | |
| #ifndef PY_NO_SHORT_FLOAT_REPR | |
| return _Py_dg_infinity(0); | |
| #else | |
| return Py_HUGE_VAL; | |
| #endif | |
| } | |
| /* Constant nan value, generated in the same way as float('nan'). */ | |
| /* We don't currently assume that Py_NAN is defined everywhere. */ | |
| #if !defined(PY_NO_SHORT_FLOAT_REPR) || defined(Py_NAN) | |
| static double | |
| m_nan(void) | |
| { | |
| #ifndef PY_NO_SHORT_FLOAT_REPR | |
| return _Py_dg_stdnan(0); | |
| #else | |
| return Py_NAN; | |
| #endif | |
| } | |
| #endif | |
| static double | |
| m_tgamma(double x) | |
| { | |
| double absx, r, y, z, sqrtpow; | |
| /* special cases */ | |
| if (!Py_IS_FINITE(x)) { | |
| if (Py_IS_NAN(x) || x > 0.0) | |
| return x; /* tgamma(nan) = nan, tgamma(inf) = inf */ | |
| else { | |
| errno = EDOM; | |
| return Py_NAN; /* tgamma(-inf) = nan, invalid */ | |
| } | |
| } | |
| if (x == 0.0) { | |
| errno = EDOM; | |
| /* tgamma(+-0.0) = +-inf, divide-by-zero */ | |
| return copysign(Py_HUGE_VAL, x); | |
| } | |
| /* integer arguments */ | |
| if (x == floor(x)) { | |
| if (x < 0.0) { | |
| errno = EDOM; /* tgamma(n) = nan, invalid for */ | |
| return Py_NAN; /* negative integers n */ | |
| } | |
| if (x <= NGAMMA_INTEGRAL) | |
| return gamma_integral[(int)x - 1]; | |
| } | |
| absx = fabs(x); | |
| /* tiny arguments: tgamma(x) ~ 1/x for x near 0 */ | |
| if (absx < 1e-20) { | |
| r = 1.0/x; | |
| if (Py_IS_INFINITY(r)) | |
| errno = ERANGE; | |
| return r; | |
| } | |
| /* large arguments: assuming IEEE 754 doubles, tgamma(x) overflows for | |
| x > 200, and underflows to +-0.0 for x < -200, not a negative | |
| integer. */ | |
| if (absx > 200.0) { | |
| if (x < 0.0) { | |
| return 0.0/m_sinpi(x); | |
| } | |
| else { | |
| errno = ERANGE; | |
| return Py_HUGE_VAL; | |
| } | |
| } | |
| y = absx + lanczos_g_minus_half; | |
| /* compute error in sum */ | |
| if (absx > lanczos_g_minus_half) { | |
| /* note: the correction can be foiled by an optimizing | |
| compiler that (incorrectly) thinks that an expression like | |
| a + b - a - b can be optimized to 0.0. This shouldn't | |
| happen in a standards-conforming compiler. */ | |
| double q = y - absx; | |
| z = q - lanczos_g_minus_half; | |
| } | |
| else { | |
| double q = y - lanczos_g_minus_half; | |
| z = q - absx; | |
| } | |
| z = z * lanczos_g / y; | |
| if (x < 0.0) { | |
| r = -pi / m_sinpi(absx) / absx * exp(y) / lanczos_sum(absx); | |
| r -= z * r; | |
| if (absx < 140.0) { | |
| r /= pow(y, absx - 0.5); | |
| } | |
| else { | |
| sqrtpow = pow(y, absx / 2.0 - 0.25); | |
| r /= sqrtpow; | |
| r /= sqrtpow; | |
| } | |
| } | |
| else { | |
| r = lanczos_sum(absx) / exp(y); | |
| r += z * r; | |
| if (absx < 140.0) { | |
| r *= pow(y, absx - 0.5); | |
| } | |
| else { | |
| sqrtpow = pow(y, absx / 2.0 - 0.25); | |
| r *= sqrtpow; | |
| r *= sqrtpow; | |
| } | |
| } | |
| if (Py_IS_INFINITY(r)) | |
| errno = ERANGE; | |
| return r; | |
| } | |
| /* | |
| lgamma: natural log of the absolute value of the Gamma function. | |
| For large arguments, Lanczos' formula works extremely well here. | |
| */ | |
| static double | |
| m_lgamma(double x) | |
| { | |
| double r; | |
| double absx; | |
| /* special cases */ | |
| if (!Py_IS_FINITE(x)) { | |
| if (Py_IS_NAN(x)) | |
| return x; /* lgamma(nan) = nan */ | |
| else | |
| return Py_HUGE_VAL; /* lgamma(+-inf) = +inf */ | |
| } | |
| /* integer arguments */ | |
| if (x == floor(x) && x <= 2.0) { | |
| if (x <= 0.0) { | |
| errno = EDOM; /* lgamma(n) = inf, divide-by-zero for */ | |
| return Py_HUGE_VAL; /* integers n <= 0 */ | |
| } | |
| else { | |
| return 0.0; /* lgamma(1) = lgamma(2) = 0.0 */ | |
| } | |
| } | |
| absx = fabs(x); | |
| /* tiny arguments: lgamma(x) ~ -log(fabs(x)) for small x */ | |
| if (absx < 1e-20) | |
| return -log(absx); | |
| /* Lanczos' formula. We could save a fraction of a ulp in accuracy by | |
| having a second set of numerator coefficients for lanczos_sum that | |
| absorbed the exp(-lanczos_g) term, and throwing out the lanczos_g | |
| subtraction below; it's probably not worth it. */ | |
| r = log(lanczos_sum(absx)) - lanczos_g; | |
| r += (absx - 0.5) * (log(absx + lanczos_g - 0.5) - 1); | |
| if (x < 0.0) | |
| /* Use reflection formula to get value for negative x. */ | |
| r = logpi - log(fabs(m_sinpi(absx))) - log(absx) - r; | |
| if (Py_IS_INFINITY(r)) | |
| errno = ERANGE; | |
| return r; | |
| } | |
| #if !defined(HAVE_ERF) || !defined(HAVE_ERFC) | |
| /* | |
| 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; | |
| } | |
| #endif /* !defined(HAVE_ERF) || !defined(HAVE_ERFC) */ | |
| /* Error function erf(x), for general x */ | |
| static double | |
| m_erf(double x) | |
| { | |
| #ifdef HAVE_ERF | |
| return erf(x); | |
| #else | |
| 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; | |
| } | |
| #endif | |
| } | |
| /* Complementary error function erfc(x), for general x. */ | |
| static double | |
| m_erfc(double x) | |
| { | |
| #ifdef HAVE_ERFC | |
| return erfc(x); | |
| #else | |
| double absx, cf; | |
| if (Py_IS_NAN(x)) | |
| return x; | |
| absx = fabs(x); | |
| if (absx < ERF_SERIES_CUTOFF) | |
| return 1.0 - m_erf_series(x); | |
| else { | |
| cf = m_erfc_contfrac(absx); | |
| return x > 0.0 ? cf : 2.0 - cf; | |
| } | |
| #endif | |
| } | |
| /* | |
| wrapper for atan2 that deals directly with special cases before | |
| delegating to the platform libm for the remaining cases. This | |
| is necessary to get consistent behaviour across platforms. | |
| Windows, FreeBSD and alpha Tru64 are amongst platforms that don't | |
| always follow C99. | |
| */ | |
| static double | |
| m_atan2(double y, double x) | |
| { | |
| if (Py_IS_NAN(x) || Py_IS_NAN(y)) | |
| return Py_NAN; | |
| if (Py_IS_INFINITY(y)) { | |
| if (Py_IS_INFINITY(x)) { | |
| if (copysign(1., x) == 1.) | |
| /* atan2(+-inf, +inf) == +-pi/4 */ | |
| return copysign(0.25*Py_MATH_PI, y); | |
| else | |
| /* atan2(+-inf, -inf) == +-pi*3/4 */ | |
| return copysign(0.75*Py_MATH_PI, y); | |
| } | |
| /* atan2(+-inf, x) == +-pi/2 for finite x */ | |
| return copysign(0.5*Py_MATH_PI, y); | |
| } | |
| if (Py_IS_INFINITY(x) || y == 0.) { | |
| if (copysign(1., x) == 1.) | |
| /* atan2(+-y, +inf) = atan2(+-0, +x) = +-0. */ | |
| return copysign(0., y); | |
| else | |
| /* atan2(+-y, -inf) = atan2(+-0., -x) = +-pi. */ | |
| return copysign(Py_MATH_PI, y); | |
| } | |
| return atan2(y, x); | |
| } | |
| /* IEEE 754-style remainder operation: x - n*y where n*y is the nearest | |
| multiple of y to x, taking n even in the case of a tie. Assuming an IEEE 754 | |
| binary floating-point format, the result is always exact. */ | |
| static double | |
| m_remainder(double x, double y) | |
| { | |
| /* Deal with most common case first. */ | |
| if (Py_IS_FINITE(x) && Py_IS_FINITE(y)) { | |
| double absx, absy, c, m, r; | |
| if (y == 0.0) { | |
| return Py_NAN; | |
| } | |
| absx = fabs(x); | |
| absy = fabs(y); | |
| m = fmod(absx, absy); | |
| /* | |
| Warning: some subtlety here. What we *want* to know at this point is | |
| whether the remainder m is less than, equal to, or greater than half | |
| of absy. However, we can't do that comparison directly because we | |
| can't be sure that 0.5*absy is representable (the multiplication | |
| might incur precision loss due to underflow). So instead we compare | |
| m with the complement c = absy - m: m < 0.5*absy if and only if m < | |
| c, and so on. The catch is that absy - m might also not be | |
| representable, but it turns out that it doesn't matter: | |
| - if m > 0.5*absy then absy - m is exactly representable, by | |
| Sterbenz's lemma, so m > c | |
| - if m == 0.5*absy then again absy - m is exactly representable | |
| and m == c | |
| - if m < 0.5*absy then either (i) 0.5*absy is exactly representable, | |
| in which case 0.5*absy < absy - m, so 0.5*absy <= c and hence m < | |
| c, or (ii) absy is tiny, either subnormal or in the lowest normal | |
| binade. Then absy - m is exactly representable and again m < c. | |
| */ | |
| c = absy - m; | |
| if (m < c) { | |
| r = m; | |
| } | |
| else if (m > c) { | |
| r = -c; | |
| } | |
| else { | |
| /* | |
| Here absx is exactly halfway between two multiples of absy, | |
| and we need to choose the even multiple. x now has the form | |
| absx = n * absy + m | |
| for some integer n (recalling that m = 0.5*absy at this point). | |
| If n is even we want to return m; if n is odd, we need to | |
| return -m. | |
| So | |
| 0.5 * (absx - m) = (n/2) * absy | |
| and now reducing modulo absy gives us: | |
| | m, if n is odd | |
| fmod(0.5 * (absx - m), absy) = | | |
| | 0, if n is even | |
| Now m - 2.0 * fmod(...) gives the desired result: m | |
| if n is even, -m if m is odd. | |
| Note that all steps in fmod(0.5 * (absx - m), absy) | |
| will be computed exactly, with no rounding error | |
| introduced. | |
| */ | |
| assert(m == c); | |
| r = m - 2.0 * fmod(0.5 * (absx - m), absy); | |
| } | |
| return copysign(1.0, x) * r; | |
| } | |
| /* Special values. */ | |
| if (Py_IS_NAN(x)) { | |
| return x; | |
| } | |
| if (Py_IS_NAN(y)) { | |
| return y; | |
| } | |
| if (Py_IS_INFINITY(x)) { | |
| return Py_NAN; | |
| } | |
| assert(Py_IS_INFINITY(y)); | |
| return x; | |
| } | |
| /* | |
| Various platforms (Solaris, OpenBSD) do nonstandard things for log(0), | |
| log(-ve), log(NaN). Here are wrappers for log and log10 that deal with | |
| special values directly, passing positive non-special values through to | |
| the system log/log10. | |
| */ | |
| static double | |
| m_log(double x) | |
| { | |
| if (Py_IS_FINITE(x)) { | |
| if (x > 0.0) | |
| return log(x); | |
| errno = EDOM; | |
| if (x == 0.0) | |
| return -Py_HUGE_VAL; /* log(0) = -inf */ | |
| else | |
| return Py_NAN; /* log(-ve) = nan */ | |
| } | |
| else if (Py_IS_NAN(x)) | |
| return x; /* log(nan) = nan */ | |
| else if (x > 0.0) | |
| return x; /* log(inf) = inf */ | |
| else { | |
| errno = EDOM; | |
| return Py_NAN; /* log(-inf) = nan */ | |
| } | |
| } | |
| /* | |
| log2: log to base 2. | |
| Uses an algorithm that should: | |
| (a) produce exact results for powers of 2, and | |
| (b) give a monotonic log2 (for positive finite floats), | |
| assuming that the system log is monotonic. | |
| */ | |
| static double | |
| m_log2(double x) | |
| { | |
| if (!Py_IS_FINITE(x)) { | |
| if (Py_IS_NAN(x)) | |
| return x; /* log2(nan) = nan */ | |
| else if (x > 0.0) | |
| return x; /* log2(+inf) = +inf */ | |
| else { | |
| errno = EDOM; | |
| return Py_NAN; /* log2(-inf) = nan, invalid-operation */ | |
| } | |
| } | |
| if (x > 0.0) { | |
| #ifdef HAVE_LOG2 | |
| return log2(x); | |
| #else | |
| double m; | |
| int e; | |
| m = frexp(x, &e); | |
| /* We want log2(m * 2**e) == log(m) / log(2) + e. Care is needed when | |
| * x is just greater than 1.0: in that case e is 1, log(m) is negative, | |
| * and we get significant cancellation error from the addition of | |
| * log(m) / log(2) to e. The slight rewrite of the expression below | |
| * avoids this problem. | |
| */ | |
| if (x >= 1.0) { | |
| return log(2.0 * m) / log(2.0) + (e - 1); | |
| } | |
| else { | |
| return log(m) / log(2.0) + e; | |
| } | |
| #endif | |
| } | |
| else if (x == 0.0) { | |
| errno = EDOM; | |
| return -Py_HUGE_VAL; /* log2(0) = -inf, divide-by-zero */ | |
| } | |
| else { | |
| errno = EDOM; | |
| return Py_NAN; /* log2(-inf) = nan, invalid-operation */ | |
| } | |
| } | |
| static double | |
| m_log10(double x) | |
| { | |
| if (Py_IS_FINITE(x)) { | |
| if (x > 0.0) | |
| return log10(x); | |
| errno = EDOM; | |
| if (x == 0.0) | |
| return -Py_HUGE_VAL; /* log10(0) = -inf */ | |
| else | |
| return Py_NAN; /* log10(-ve) = nan */ | |
| } | |
| else if (Py_IS_NAN(x)) | |
| return x; /* log10(nan) = nan */ | |
| else if (x > 0.0) | |
| return x; /* log10(inf) = inf */ | |
| else { | |
| errno = EDOM; | |
| return Py_NAN; /* log10(-inf) = nan */ | |
| } | |
| } | |
| /*[clinic input] | |
| math.gcd | |
| x as a: object | |
| y as b: object | |
| / | |
| greatest common divisor of x and y | |
| [clinic start generated code]*/ | |
| static PyObject * | |
| math_gcd_impl(PyObject *module, PyObject *a, PyObject *b) | |
| /*[clinic end generated code: output=7b2e0c151bd7a5d8 input=c2691e57fb2a98fa]*/ | |
| { | |
| PyObject *g; | |
| a = PyNumber_Index(a); | |
| if (a == NULL) | |
| return NULL; | |
| b = PyNumber_Index(b); | |
| if (b == NULL) { | |
| Py_DECREF(a); | |
| return NULL; | |
| } | |
| g = _PyLong_GCD(a, b); | |
| Py_DECREF(a); | |
| Py_DECREF(b); | |
| return g; | |
| } | |
| /* Call is_error when errno != 0, and where x is the result libm | |
| * returned. is_error will usually set up an exception and return | |
| * true (1), but may return false (0) without setting up an exception. | |
| */ | |
| static int | |
| is_error(double x) | |
| { | |
| int result = 1; /* presumption of guilt */ | |
| assert(errno); /* non-zero errno is a precondition for calling */ | |
| if (errno == EDOM) | |
| PyErr_SetString(PyExc_ValueError, "math domain error"); | |
| else if (errno == ERANGE) { | |
| /* ANSI C generally requires libm functions to set ERANGE | |
| * on overflow, but also generally *allows* them to set | |
| * ERANGE on underflow too. There's no consistency about | |
| * the latter across platforms. | |
| * Alas, C99 never requires that errno be set. | |
| * Here we suppress the underflow errors (libm functions | |
| * should return a zero on underflow, and +- HUGE_VAL on | |
| * overflow, so testing the result for zero suffices to | |
| * distinguish the cases). | |
| * | |
| * On some platforms (Ubuntu/ia64) it seems that errno can be | |
| * set to ERANGE for subnormal results that do *not* underflow | |
| * to zero. So to be safe, we'll ignore ERANGE whenever the | |
| * function result is less than one in absolute value. | |
| */ | |
| if (fabs(x) < 1.0) | |
| result = 0; | |
| else | |
| PyErr_SetString(PyExc_OverflowError, | |
| "math range error"); | |
| } | |
| else | |
| /* Unexpected math error */ | |
| PyErr_SetFromErrno(PyExc_ValueError); | |
| return result; | |
| } | |
| /* | |
| math_1 is used to wrap a libm function f that takes a double | |
| argument and returns a double. | |
| The error reporting follows these rules, which are designed to do | |
| the right thing on C89/C99 platforms and IEEE 754/non IEEE 754 | |
| platforms. | |
| - a NaN result from non-NaN inputs causes ValueError to be raised | |
| - an infinite result from finite inputs causes OverflowError to be | |
| raised if can_overflow is 1, or raises ValueError if can_overflow | |
| is 0. | |
| - if the result is finite and errno == EDOM then ValueError is | |
| raised | |
| - if the result is finite and nonzero and errno == ERANGE then | |
| OverflowError is raised | |
| The last rule is used to catch overflow on platforms which follow | |
| C89 but for which HUGE_VAL is not an infinity. | |
| For the majority of one-argument functions these rules are enough | |
| to ensure that Python's functions behave as specified in 'Annex F' | |
| of the C99 standard, with the 'invalid' and 'divide-by-zero' | |
| floating-point exceptions mapping to Python's ValueError and the | |
| 'overflow' floating-point exception mapping to OverflowError. | |
| math_1 only works for functions that don't have singularities *and* | |
| the possibility of overflow; fortunately, that covers everything we | |
| care about right now. | |
| */ | |
| static PyObject * | |
| math_1_to_whatever(PyObject *arg, double (*func) (double), | |
| PyObject *(*from_double_func) (double), | |
| int can_overflow) | |
| { | |
| double x, r; | |
| x = PyFloat_AsDouble(arg); | |
| if (x == -1.0 && PyErr_Occurred()) | |
| return NULL; | |
| errno = 0; | |
| PyFPE_START_PROTECT("in math_1", return 0); | |
| r = (*func)(x); | |
| PyFPE_END_PROTECT(r); | |
| if (Py_IS_NAN(r) && !Py_IS_NAN(x)) { | |
| PyErr_SetString(PyExc_ValueError, | |
| "math domain error"); /* invalid arg */ | |
| return NULL; | |
| } | |
| if (Py_IS_INFINITY(r) && Py_IS_FINITE(x)) { | |
| if (can_overflow) | |
| PyErr_SetString(PyExc_OverflowError, | |
| "math range error"); /* overflow */ | |
| else | |
| PyErr_SetString(PyExc_ValueError, | |
| "math domain error"); /* singularity */ | |
| return NULL; | |
| } | |
| if (Py_IS_FINITE(r) && errno && is_error(r)) | |
| /* this branch unnecessary on most platforms */ | |
| return NULL; | |
| return (*from_double_func)(r); | |
| } | |
| /* variant of math_1, to be used when the function being wrapped is known to | |
| set errno properly (that is, errno = EDOM for invalid or divide-by-zero, | |
| errno = ERANGE for overflow). */ | |
| static PyObject * | |
| math_1a(PyObject *arg, double (*func) (double)) | |
| { | |
| double x, r; | |
| x = PyFloat_AsDouble(arg); | |
| if (x == -1.0 && PyErr_Occurred()) | |
| return NULL; | |
| errno = 0; | |
| PyFPE_START_PROTECT("in math_1a", return 0); | |
| r = (*func)(x); | |
| PyFPE_END_PROTECT(r); | |
| if (errno && is_error(r)) | |
| return NULL; | |
| return PyFloat_FromDouble(r); | |
| } | |
| /* | |
| math_2 is used to wrap a libm function f that takes two double | |
| arguments and returns a double. | |
| The error reporting follows these rules, which are designed to do | |
| the right thing on C89/C99 platforms and IEEE 754/non IEEE 754 | |
| platforms. | |
| - a NaN result from non-NaN inputs causes ValueError to be raised | |
| - an infinite result from finite inputs causes OverflowError to be | |
| raised. | |
| - if the result is finite and errno == EDOM then ValueError is | |
| raised | |
| - if the result is finite and nonzero and errno == ERANGE then | |
| OverflowError is raised | |
| The last rule is used to catch overflow on platforms which follow | |
| C89 but for which HUGE_VAL is not an infinity. | |
| For most two-argument functions (copysign, fmod, hypot, atan2) | |
| these rules are enough to ensure that Python's functions behave as | |
| specified in 'Annex F' of the C99 standard, with the 'invalid' and | |
| 'divide-by-zero' floating-point exceptions mapping to Python's | |
| ValueError and the 'overflow' floating-point exception mapping to | |
| OverflowError. | |
| */ | |
| static PyObject * | |
| math_1(PyObject *arg, double (*func) (double), int can_overflow) | |
| { | |
| return math_1_to_whatever(arg, func, PyFloat_FromDouble, can_overflow); | |
| } | |
| static PyObject * | |
| math_1_to_int(PyObject *arg, double (*func) (double), int can_overflow) | |
| { | |
| return math_1_to_whatever(arg, func, PyLong_FromDouble, can_overflow); | |
| } | |
| static PyObject * | |
| math_2(PyObject *const *args, Py_ssize_t nargs, | |
| double (*func) (double, double), const char *funcname) | |
| { | |
| double x, y, r; | |
| if (!_PyArg_CheckPositional(funcname, nargs, 2, 2)) | |
| return NULL; | |
| x = PyFloat_AsDouble(args[0]); | |
| y = PyFloat_AsDouble(args[1]); | |
| 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 *const *args, Py_ssize_t nargs) { \ | |
| return math_2(args, nargs, func, #funcname); \ | |
| }\ | |
| PyDoc_STRVAR(math_##funcname##_doc, docstring); | |
| FUNC1(acos, acos, 0, | |
| "acos($module, x, /)\n--\n\n" | |
| "Return the arc cosine (measured in radians) of x.\n\n" | |
| "The result is between 0 and pi.") | |
| FUNC1(acosh, m_acosh, 0, | |
| "acosh($module, x, /)\n--\n\n" | |
| "Return the inverse hyperbolic cosine of x.") | |
| FUNC1(asin, asin, 0, | |
| "asin($module, x, /)\n--\n\n" | |
| "Return the arc sine (measured in radians) of x.\n\n" | |
| "The result is between -pi/2 and pi/2.") | |
| FUNC1(asinh, m_asinh, 0, | |
| "asinh($module, x, /)\n--\n\n" | |
| "Return the inverse hyperbolic sine of x.") | |
| FUNC1(atan, atan, 0, | |
| "atan($module, x, /)\n--\n\n" | |
| "Return the arc tangent (measured in radians) of x.\n\n" | |
| "The result is between -pi/2 and pi/2.") | |
| FUNC2(atan2, m_atan2, | |
| "atan2($module, y, x, /)\n--\n\n" | |
| "Return the arc tangent (measured in radians) of y/x.\n\n" | |
| "Unlike atan(y/x), the signs of both x and y are considered.") | |
| FUNC1(atanh, m_atanh, 0, | |
| "atanh($module, x, /)\n--\n\n" | |
| "Return the inverse hyperbolic tangent of x.") | |
| /*[clinic input] | |
| math.ceil | |
| x as number: object | |
| / | |
| Return the ceiling of x as an Integral. | |
| This is the smallest integer >= x. | |
| [clinic start generated code]*/ | |
| static PyObject * | |
| math_ceil(PyObject *module, PyObject *number) | |
| /*[clinic end generated code: output=6c3b8a78bc201c67 input=2725352806399cab]*/ | |
| { | |
| _Py_IDENTIFIER(__ceil__); | |
| PyObject *method, *result; | |
| method = _PyObject_LookupSpecial(number, &PyId___ceil__); | |
| if (method == NULL) { | |
| if (PyErr_Occurred()) | |
| return NULL; | |
| return math_1_to_int(number, ceil, 0); | |
| } | |
| result = _PyObject_CallNoArg(method); | |
| Py_DECREF(method); | |
| return result; | |
| } | |
| FUNC2(copysign, copysign, | |
| "copysign($module, x, y, /)\n--\n\n" | |
| "Return a float with the magnitude (absolute value) of x but the sign of y.\n\n" | |
| "On platforms that support signed zeros, copysign(1.0, -0.0)\n" | |
| "returns -1.0.\n") | |
| FUNC1(cos, cos, 0, | |
| "cos($module, x, /)\n--\n\n" | |
| "Return the cosine of x (measured in radians).") | |
| FUNC1(cosh, cosh, 1, | |
| "cosh($module, x, /)\n--\n\n" | |
| "Return the hyperbolic cosine of x.") | |
| FUNC1A(erf, m_erf, | |
| "erf($module, x, /)\n--\n\n" | |
| "Error function at x.") | |
| FUNC1A(erfc, m_erfc, | |
| "erfc($module, x, /)\n--\n\n" | |
| "Complementary error function at x.") | |
| FUNC1(exp, exp, 1, | |
| "exp($module, x, /)\n--\n\n" | |
| "Return e raised to the power of x.") | |
| FUNC1(expm1, m_expm1, 1, | |
| "expm1($module, x, /)\n--\n\n" | |
| "Return exp(x)-1.\n\n" | |
| "This function avoids the loss of precision involved in the direct " | |
| "evaluation of exp(x)-1 for small x.") | |
| FUNC1(fabs, fabs, 0, | |
| "fabs($module, x, /)\n--\n\n" | |
| "Return the absolute value of the float x.") | |
| /*[clinic input] | |
| math.floor | |
| x as number: object | |
| / | |
| Return the floor of x as an Integral. | |
| This is the largest integer <= x. | |
| [clinic start generated code]*/ | |
| static PyObject * | |
| math_floor(PyObject *module, PyObject *number) | |
| /*[clinic end generated code: output=c6a65c4884884b8a input=63af6b5d7ebcc3d6]*/ | |
| { | |
| _Py_IDENTIFIER(__floor__); | |
| PyObject *method, *result; | |
| method = _PyObject_LookupSpecial(number, &PyId___floor__); | |
| if (method == NULL) { | |
| if (PyErr_Occurred()) | |
| return NULL; | |
| return math_1_to_int(number, floor, 0); | |
| } | |
| result = _PyObject_CallNoArg(method); | |
| Py_DECREF(method); | |
| return result; | |
| } | |
| FUNC1A(gamma, m_tgamma, | |
| "gamma($module, x, /)\n--\n\n" | |
| "Gamma function at x.") | |
| FUNC1A(lgamma, m_lgamma, | |
| "lgamma($module, x, /)\n--\n\n" | |
| "Natural logarithm of absolute value of Gamma function at x.") | |
| FUNC1(log1p, m_log1p, 0, | |
| "log1p($module, x, /)\n--\n\n" | |
| "Return the natural logarithm of 1+x (base e).\n\n" | |
| "The result is computed in a way which is accurate for x near zero.") | |
| FUNC2(remainder, m_remainder, | |
| "remainder($module, x, y, /)\n--\n\n" | |
| "Difference between x and the closest integer multiple of y.\n\n" | |
| "Return x - n*y where n*y is the closest integer multiple of y.\n" | |
| "In the case where x is exactly halfway between two multiples of\n" | |
| "y, the nearest even value of n is used. The result is always exact.") | |
| FUNC1(sin, sin, 0, | |
| "sin($module, x, /)\n--\n\n" | |
| "Return the sine of x (measured in radians).") | |
| FUNC1(sinh, sinh, 1, | |
| "sinh($module, x, /)\n--\n\n" | |
| "Return the hyperbolic sine of x.") | |
| FUNC1(sqrt, sqrt, 0, | |
| "sqrt($module, x, /)\n--\n\n" | |
| "Return the square root of x.") | |
| FUNC1(tan, tan, 0, | |
| "tan($module, x, /)\n--\n\n" | |
| "Return the tangent of x (measured in radians).") | |
| FUNC1(tanh, tanh, 0, | |
| "tanh($module, x, /)\n--\n\n" | |
| "Return the hyperbolic tangent of x.") | |
| /* Precision summation function as msum() by Raymond Hettinger in | |
| <http://aspn.activestate.com/ASPN/Cookbook/Python/Recipe/393090>, | |
| enhanced with the exact partials sum and roundoff from Mark | |
| Dickinson's post at <http://bugs.python.org/file10357/msum4.py>. | |
| See those links for more details, proofs and other references. | |
| Note 1: IEEE 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 builtins.sum() | |
| because the start argument doesn't make sense in the context of | |
| accurate summation. Since the partials table is collapsed before | |
| returning a result, sum(seq2, start=sum(seq1)) may not equal the | |
| accurate result returned by sum(itertools.chain(seq1, seq2)). | |
| */ | |
| #define NUM_PARTIALS 32 /* initial partials array size, on stack */ | |
| /* Extend the partials array p[] by doubling its size. */ | |
| static int /* non-zero on error */ | |
| _fsum_realloc(double **p_ptr, Py_ssize_t n, | |
| double *ps, Py_ssize_t *m_ptr) | |
| { | |
| void *v = NULL; | |
| Py_ssize_t m = *m_ptr; | |
| m += m; /* double */ | |
| if (n < m && (size_t)m < ((size_t)PY_SSIZE_T_MAX / sizeof(double))) { | |
| double *p = *p_ptr; | |
| if (p == ps) { | |
| v = PyMem_Malloc(sizeof(double) * m); | |
| if (v != NULL) | |
| memcpy(v, ps, sizeof(double) * n); | |
| } | |
| else | |
| v = PyMem_Realloc(p, sizeof(double) * m); | |
| } | |
| if (v == NULL) { /* size overflow or no memory */ | |
| PyErr_SetString(PyExc_MemoryError, "math.fsum partials"); | |
| return 1; | |
| } | |
| *p_ptr = (double*) v; | |
| *m_ptr = m; | |
| return 0; | |
| } | |
| /* Full precision summation of a sequence of floats. | |
| def msum(iterable): | |
| partials = [] # sorted, non-overlapping partial sums | |
| for x in iterable: | |
| i = 0 | |
| for y in partials: | |
| if abs(x) < abs(y): | |
| x, y = y, x | |
| hi = x + y | |
| lo = y - (hi - x) | |
| if lo: | |
| partials[i] = lo | |
| i += 1 | |
| x = hi | |
| partials[i:] = [x] | |
| return sum_exact(partials) | |
| Rounded x+y stored in hi with the roundoff stored in lo. Together hi+lo | |
| are exactly equal to x+y. The inner loop applies hi/lo summation to each | |
| partial so that the list of partial sums remains exact. | |
| Sum_exact() adds the partial sums exactly and correctly rounds the final | |
| result (using the round-half-to-even rule). The items in partials remain | |
| non-zero, non-special, non-overlapping and strictly increasing in | |
| magnitude, but possibly not all having the same sign. | |
| Depends on IEEE 754 arithmetic guarantees and half-even rounding. | |
| */ | |
| /*[clinic input] | |
| math.fsum | |
| seq: object | |
| / | |
| Return an accurate floating point sum of values in the iterable seq. | |
| Assumes IEEE-754 floating point arithmetic. | |
| [clinic start generated code]*/ | |
| static PyObject * | |
| math_fsum(PyObject *module, PyObject *seq) | |
| /*[clinic end generated code: output=ba5c672b87fe34fc input=c51b7d8caf6f6e82]*/ | |
| { | |
| PyObject *item, *iter, *sum = NULL; | |
| Py_ssize_t i, j, n = 0, m = NUM_PARTIALS; | |
| double x, y, t, ps[NUM_PARTIALS], *p = ps; | |
| double xsave, special_sum = 0.0, inf_sum = 0.0; | |
| 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; | |
| } | |
| ASSIGN_DOUBLE(x, item, error_with_item); | |
| Py_DECREF(item); | |
| xsave = x; | |
| for (i = j = 0; j < n; j++) { /* for y in partials */ | |
| y = p[j]; | |
| if (fabs(x) < fabs(y)) { | |
| t = x; x = y; y = t; | |
| } | |
| hi = x + y; | |
| yr = hi - x; | |
| lo = y - yr; | |
| if (lo != 0.0) | |
| p[i++] = lo; | |
| x = hi; | |
| } | |
| n = i; /* ps[i:] = [x] */ | |
| if (x != 0.0) { | |
| if (! Py_IS_FINITE(x)) { | |
| /* a nonfinite x could arise either as | |
| a result of intermediate overflow, or | |
| as a result of a nan or inf in the | |
| summands */ | |
| if (Py_IS_FINITE(xsave)) { | |
| PyErr_SetString(PyExc_OverflowError, | |
| "intermediate overflow in fsum"); | |
| goto _fsum_error; | |
| } | |
| if (Py_IS_INFINITY(xsave)) | |
| inf_sum += xsave; | |
| special_sum += xsave; | |
| /* reset partials */ | |
| n = 0; | |
| } | |
| else if (n >= m && _fsum_realloc(&p, n, ps, &m)) | |
| goto _fsum_error; | |
| else | |
| p[n++] = x; | |
| } | |
| } | |
| if (special_sum != 0.0) { | |
| if (Py_IS_NAN(inf_sum)) | |
| PyErr_SetString(PyExc_ValueError, | |
| "-inf + inf in fsum"); | |
| else | |
| sum = PyFloat_FromDouble(special_sum); | |
| goto _fsum_error; | |
| } | |
| hi = 0.0; | |
| if (n > 0) { | |
| hi = p[--n]; | |
| /* sum_exact(ps, hi) from the top, stop when the sum becomes | |
| inexact. */ | |
| while (n > 0) { | |
| x = hi; | |
| y = p[--n]; | |
| assert(fabs(y) < fabs(x)); | |
| hi = x + y; | |
| yr = hi - x; | |
| lo = y - yr; | |
| if (lo != 0.0) | |
| break; | |
| } | |
| /* Make half-even rounding work across multiple partials. | |
| Needed so that sum([1e-16, 1, 1e16]) will round-up the last | |
| digit to two instead of down to zero (the 1e-16 makes the 1 | |
| slightly closer to two). With a potential 1 ULP rounding | |
| error fixed-up, math.fsum() can guarantee commutativity. */ | |
| if (n > 0 && ((lo < 0.0 && p[n-1] < 0.0) || | |
| (lo > 0.0 && p[n-1] > 0.0))) { | |
| y = lo * 2.0; | |
| x = hi + y; | |
| yr = x - hi; | |
| if (y == yr) | |
| hi = x; | |
| } | |
| } | |
| sum = PyFloat_FromDouble(hi); | |
| _fsum_error: | |
| PyFPE_END_PROTECT(hi) | |
| Py_DECREF(iter); | |
| if (p != ps) | |
| PyMem_Free(p); | |
| return sum; | |
| error_with_item: | |
| Py_DECREF(item); | |
| goto _fsum_error; | |
| } | |
| #undef NUM_PARTIALS | |
| /* Return the smallest integer k such that n < 2**k, or 0 if n == 0. | |
| * Equivalent to floor(lg(x))+1. Also equivalent to: bitwidth_of_type - | |
| * count_leading_zero_bits(x) | |
| */ | |
| /* XXX: This routine does more or less the same thing as | |
| * bits_in_digit() in Objects/longobject.c. Someday it would be nice to | |
| * consolidate them. On BSD, there's a library function called fls() | |
| * that we could use, and GCC provides __builtin_clz(). | |
| */ | |
| static unsigned long | |
| bit_length(unsigned long n) | |
| { | |
| unsigned long len = 0; | |
| while (n != 0) { | |
| ++len; | |
| n >>= 1; | |
| } | |
| return len; | |
| } | |
| static unsigned long | |
| count_set_bits(unsigned long n) | |
| { | |
| unsigned long count = 0; | |
| while (n != 0) { | |
| ++count; | |
| n &= n - 1; /* clear least significant bit */ | |
| } | |
| return count; | |
| } | |
| /* Integer square root | |
| Given a nonnegative integer `n`, we want to compute the largest integer | |
| `a` for which `a * a <= n`, or equivalently the integer part of the exact | |
| square root of `n`. | |
| We use an adaptive-precision pure-integer version of Newton's iteration. Given | |
| a positive integer `n`, the algorithm produces at each iteration an integer | |
| approximation `a` to the square root of `n >> s` for some even integer `s`, | |
| with `s` decreasing as the iterations progress. On the final iteration, `s` is | |
| zero and we have an approximation to the square root of `n` itself. | |
| At every step, the approximation `a` is strictly within 1.0 of the true square | |
| root, so we have | |
| (a - 1)**2 < (n >> s) < (a + 1)**2 | |
| After the final iteration, a check-and-correct step is needed to determine | |
| whether `a` or `a - 1` gives the desired integer square root of `n`. | |
| The algorithm is remarkable in its simplicity. There's no need for a | |
| per-iteration check-and-correct step, and termination is straightforward: the | |
| number of iterations is known in advance (it's exactly `floor(log2(log2(n)))` | |
| for `n > 1`). The only tricky part of the correctness proof is in establishing | |
| that the bound `(a - 1)**2 < (n >> s) < (a + 1)**2` is maintained from one | |
| iteration to the next. A sketch of the proof of this is given below. | |
| In addition to the proof sketch, a formal, computer-verified proof | |
| of correctness (using Lean) of an equivalent recursive algorithm can be found | |
| here: | |
| https://github.com/mdickinson/snippets/blob/master/proofs/isqrt/src/isqrt.lean | |
| Here's Python code equivalent to the C implementation below: | |
| def isqrt(n): | |
| """ | |
| Return the integer part of the square root of the input. | |
| """ | |
| n = operator.index(n) | |
| if n < 0: | |
| raise ValueError("isqrt() argument must be nonnegative") | |
| if n == 0: | |
| return 0 | |
| c = (n.bit_length() - 1) // 2 | |
| a = 1 | |
| d = 0 | |
| for s in reversed(range(c.bit_length())): | |
| # Loop invariant: (a-1)**2 < (n >> 2*(c - d)) < (a+1)**2 | |
| e = d | |
| d = c >> s | |
| a = (a << d - e - 1) + (n >> 2*c - e - d + 1) // a | |
| return a - (a*a > n) | |
| Sketch of proof of correctness | |
| ------------------------------ | |
| The delicate part of the correctness proof is showing that the loop invariant | |
| is preserved from one iteration to the next. That is, just before the line | |
| a = (a << d - e - 1) + (n >> 2*c - e - d + 1) // a | |
| is executed in the above code, we know that | |
| (1) (a - 1)**2 < (n >> 2*(c - e)) < (a + 1)**2. | |
| (since `e` is always the value of `d` from the previous iteration). We must | |
| prove that after that line is executed, we have | |
| (a - 1)**2 < (n >> 2*(c - d)) < (a + 1)**2 | |
| To facilitate the proof, we make some changes of notation. Write `m` for | |
| `n >> 2*(c-d)`, and write `b` for the new value of `a`, so | |
| b = (a << d - e - 1) + (n >> 2*c - e - d + 1) // a | |
| or equivalently: | |
| (2) b = (a << d - e - 1) + (m >> d - e + 1) // a | |
| Then we can rewrite (1) as: | |
| (3) (a - 1)**2 < (m >> 2*(d - e)) < (a + 1)**2 | |
| and we must show that (b - 1)**2 < m < (b + 1)**2. | |
| From this point on, we switch to mathematical notation, so `/` means exact | |
| division rather than integer division and `^` is used for exponentiation. We | |
| use the `√` symbol for the exact square root. In (3), we can remove the | |
| implicit floor operation to give: | |
| (4) (a - 1)^2 < m / 4^(d - e) < (a + 1)^2 | |
| Taking square roots throughout (4), scaling by `2^(d-e)`, and rearranging gives | |
| (5) 0 <= | 2^(d-e)a - √m | < 2^(d-e) | |
| Squaring and dividing through by `2^(d-e+1) a` gives | |
| (6) 0 <= 2^(d-e-1) a + m / (2^(d-e+1) a) - √m < 2^(d-e-1) / a | |
| We'll show below that `2^(d-e-1) <= a`. Given that, we can replace the | |
| right-hand side of (6) with `1`, and now replacing the central | |
| term `m / (2^(d-e+1) a)` with its floor in (6) gives | |
| (7) -1 < 2^(d-e-1) a + m // 2^(d-e+1) a - √m < 1 | |
| Or equivalently, from (2): | |
| (7) -1 < b - √m < 1 | |
| and rearranging gives that `(b-1)^2 < m < (b+1)^2`, which is what we needed | |
| to prove. | |
| We're not quite done: we still have to prove the inequality `2^(d - e - 1) <= | |
| a` that was used to get line (7) above. From the definition of `c`, we have | |
| `4^c <= n`, which implies | |
| (8) 4^d <= m | |
| also, since `e == d >> 1`, `d` is at most `2e + 1`, from which it follows | |
| that `2d - 2e - 1 <= d` and hence that | |
| (9) 4^(2d - 2e - 1) <= m | |
| Dividing both sides by `4^(d - e)` gives | |
| (10) 4^(d - e - 1) <= m / 4^(d - e) | |
| But we know from (4) that `m / 4^(d-e) < (a + 1)^2`, hence | |
| (11) 4^(d - e - 1) < (a + 1)^2 | |
| Now taking square roots of both sides and observing that both `2^(d-e-1)` and | |
| `a` are integers gives `2^(d - e - 1) <= a`, which is what we needed. This | |
| completes the proof sketch. | |
| */ | |
| /* Approximate square root of a large 64-bit integer. | |
| Given `n` satisfying `2**62 <= n < 2**64`, return `a` | |
| satisfying `(a - 1)**2 < n < (a + 1)**2`. */ | |
| static uint64_t | |
| _approximate_isqrt(uint64_t n) | |
| { | |
| uint32_t u = 1U + (n >> 62); | |
| u = (u << 1) + (n >> 59) / u; | |
| u = (u << 3) + (n >> 53) / u; | |
| u = (u << 7) + (n >> 41) / u; | |
| return (u << 15) + (n >> 17) / u; | |
| } | |
| /*[clinic input] | |
| math.isqrt | |
| n: object | |
| / | |
| Return the integer part of the square root of the input. | |
| [clinic start generated code]*/ | |
| static PyObject * | |
| math_isqrt(PyObject *module, PyObject *n) | |
| /*[clinic end generated code: output=35a6f7f980beab26 input=5b6e7ae4fa6c43d6]*/ | |
| { | |
| int a_too_large, c_bit_length; | |
| size_t c, d; | |
| uint64_t m, u; | |
| PyObject *a = NULL, *b; | |
| n = PyNumber_Index(n); | |
| if (n == NULL) { | |
| return NULL; | |
| } | |
| if (_PyLong_Sign(n) < 0) { | |
| PyErr_SetString( | |
| PyExc_ValueError, | |
| "isqrt() argument must be nonnegative"); | |
| goto error; | |
| } | |
| if (_PyLong_Sign(n) == 0) { | |
| Py_DECREF(n); | |
| return PyLong_FromLong(0); | |
| } | |
| /* c = (n.bit_length() - 1) // 2 */ | |
| c = _PyLong_NumBits(n); | |
| if (c == (size_t)(-1)) { | |
| goto error; | |
| } | |
| c = (c - 1U) / 2U; | |
| /* Fast path: if c <= 31 then n < 2**64 and we can compute directly with a | |
| fast, almost branch-free algorithm. In the final correction, we use `u*u | |
| - 1 >= m` instead of the simpler `u*u > m` in order to get the correct | |
| result in the corner case where `u=2**32`. */ | |
| if (c <= 31U) { | |
| m = (uint64_t)PyLong_AsUnsignedLongLong(n); | |
| Py_DECREF(n); | |
| if (m == (uint64_t)(-1) && PyErr_Occurred()) { | |
| return NULL; | |
| } | |
| u = _approximate_isqrt(m << (62U - 2U*c)) >> (31U - c); | |
| u -= u * u - 1U >= m; | |
| return PyLong_FromUnsignedLongLong((unsigned long long)u); | |
| } | |
| /* Slow path: n >= 2**64. We perform the first five iterations in C integer | |
| arithmetic, then switch to using Python long integers. */ | |
| /* From n >= 2**64 it follows that c.bit_length() >= 6. */ | |
| c_bit_length = 6; | |
| while ((c >> c_bit_length) > 0U) { | |
| ++c_bit_length; | |
| } | |
| /* Initialise d and a. */ | |
| d = c >> (c_bit_length - 5); | |
| b = _PyLong_Rshift(n, 2U*c - 62U); | |
| if (b == NULL) { | |
| goto error; | |
| } | |
| m = (uint64_t)PyLong_AsUnsignedLongLong(b); | |
| Py_DECREF(b); | |
| if (m == (uint64_t)(-1) && PyErr_Occurred()) { | |
| goto error; | |
| } | |
| u = _approximate_isqrt(m) >> (31U - d); | |
| a = PyLong_FromUnsignedLongLong((unsigned long long)u); | |
| if (a == NULL) { | |
| goto error; | |
| } | |
| for (int s = c_bit_length - 6; s >= 0; --s) { | |
| PyObject *q; | |
| size_t e = d; | |
| d = c >> s; | |
| /* q = (n >> 2*c - e - d + 1) // a */ | |
| q = _PyLong_Rshift(n, 2U*c - d - e + 1U); | |
| if (q == NULL) { | |
| goto error; | |
| } | |
| Py_SETREF(q, PyNumber_FloorDivide(q, a)); | |
| if (q == NULL) { | |
| goto error; | |
| } | |
| /* a = (a << d - 1 - e) + q */ | |
| Py_SETREF(a, _PyLong_Lshift(a, d - 1U - e)); | |
| if (a == NULL) { | |
| Py_DECREF(q); | |
| goto error; | |
| } | |
| Py_SETREF(a, PyNumber_Add(a, q)); | |
| Py_DECREF(q); | |
| if (a == NULL) { | |
| goto error; | |
| } | |
| } | |
| /* The correct result is either a or a - 1. Figure out which, and | |
| decrement a if necessary. */ | |
| /* a_too_large = n < a * a */ | |
| b = PyNumber_Multiply(a, a); | |
| if (b == NULL) { | |
| goto error; | |
| } | |
| a_too_large = PyObject_RichCompareBool(n, b, Py_LT); | |
| Py_DECREF(b); | |
| if (a_too_large == -1) { | |
| goto error; | |
| } | |
| if (a_too_large) { | |
| Py_SETREF(a, PyNumber_Subtract(a, _PyLong_One)); | |
| } | |
| Py_DECREF(n); | |
| return a; | |
| error: | |
| Py_XDECREF(a); | |
| Py_DECREF(n); | |
| return NULL; | |
| } | |
| /* Divide-and-conquer factorial algorithm | |
| * | |
| * Based on the formula and pseudo-code provided at: | |
| * http://www.luschny.de/math/factorial/binarysplitfact.html | |
| * | |
| * Faster algorithms exist, but they're more complicated and depend on | |
| * a fast prime factorization algorithm. | |
| * | |
| * Notes on the algorithm | |
| * ---------------------- | |
| * | |
| * factorial(n) is written in the form 2**k * m, with m odd. k and m are | |
| * computed separately, and then combined using a left shift. | |
| * | |
| * The function factorial_odd_part computes the odd part m (i.e., the greatest | |
| * odd divisor) of factorial(n), using the formula: | |
| * | |
| * factorial_odd_part(n) = | |
| * | |
| * product_{i >= 0} product_{0 < j <= n / 2**i, j odd} j | |
| * | |
| * Example: factorial_odd_part(20) = | |
| * | |
| * (1) * | |
| * (1) * | |
| * (1 * 3 * 5) * | |
| * (1 * 3 * 5 * 7 * 9) | |
| * (1 * 3 * 5 * 7 * 9 * 11 * 13 * 15 * 17 * 19) | |
| * | |
| * Here i goes from large to small: the first term corresponds to i=4 (any | |
| * larger i gives an empty product), and the last term corresponds to i=0. | |
| * Each term can be computed from the last by multiplying by the extra odd | |
| * numbers required: e.g., to get from the penultimate term to the last one, | |
| * we multiply by (11 * 13 * 15 * 17 * 19). | |
| * | |
| * To see a hint of why this formula works, here are the same numbers as above | |
| * but with the even parts (i.e., the appropriate powers of 2) included. For | |
| * each subterm in the product for i, we multiply that subterm by 2**i: | |
| * | |
| * factorial(20) = | |
| * | |
| * (16) * | |
| * (8) * | |
| * (4 * 12 * 20) * | |
| * (2 * 6 * 10 * 14 * 18) * | |
| * (1 * 3 * 5 * 7 * 9 * 11 * 13 * 15 * 17 * 19) | |
| * | |
| * The factorial_partial_product function computes the product of all odd j in | |
| * range(start, stop) for given start and stop. It's used to compute the | |
| * partial products like (11 * 13 * 15 * 17 * 19) in the example above. It | |
| * operates recursively, repeatedly splitting the range into two roughly equal | |
| * pieces until the subranges are small enough to be computed using only C | |
| * integer arithmetic. | |
| * | |
| * The two-valuation k (i.e., the exponent of the largest power of 2 dividing | |
| * the factorial) is computed independently in the main math_factorial | |
| * function. By standard results, its value is: | |
| * | |
| * two_valuation = n//2 + n//4 + n//8 + .... | |
| * | |
| * It can be shown (e.g., by complete induction on n) that two_valuation is | |
| * equal to n - count_set_bits(n), where count_set_bits(n) gives the number of | |
| * '1'-bits in the binary expansion of n. | |
| */ | |
| /* factorial_partial_product: Compute product(range(start, stop, 2)) using | |
| * divide and conquer. Assumes start and stop are odd and stop > start. | |
| * max_bits must be >= bit_length(stop - 2). */ | |
| static PyObject * | |
| factorial_partial_product(unsigned long start, unsigned long stop, | |
| unsigned long max_bits) | |
| { | |
| unsigned long midpoint, num_operands; | |
| PyObject *left = NULL, *right = NULL, *result = NULL; | |
| /* If the return value will fit an unsigned long, then we can | |
| * multiply in a tight, fast loop where each multiply is O(1). | |
| * Compute an upper bound on the number of bits required to store | |
| * the answer. | |
| * | |
| * Storing some integer z requires floor(lg(z))+1 bits, which is | |
| * conveniently the value returned by bit_length(z). The | |
| * product x*y will require at most | |
| * bit_length(x) + bit_length(y) bits to store, based | |
| * on the idea that lg product = lg x + lg y. | |
| * | |
| * We know that stop - 2 is the largest number to be multiplied. From | |
| * there, we have: bit_length(answer) <= num_operands * | |
| * bit_length(stop - 2) | |
| */ | |
| num_operands = (stop - start) / 2; | |
| /* The "num_operands <= 8 * SIZEOF_LONG" check guards against the | |
| * unlikely case of an overflow in num_operands * max_bits. */ | |
| if (num_operands <= 8 * SIZEOF_LONG && | |
| num_operands * max_bits <= 8 * SIZEOF_LONG) { | |
| unsigned long j, total; | |
| for (total = start, j = start + 2; j < stop; j += 2) | |
| total *= j; | |
| return PyLong_FromUnsignedLong(total); | |
| } | |
| /* find midpoint of range(start, stop), rounded up to next odd number. */ | |
| midpoint = (start + num_operands) | 1; | |
| left = factorial_partial_product(start, midpoint, | |
| bit_length(midpoint - 2)); | |
| if (left == NULL) | |
| goto error; | |
| right = factorial_partial_product(midpoint, stop, max_bits); | |
| if (right == NULL) | |
| goto error; | |
| result = PyNumber_Multiply(left, right); | |
| error: | |
| Py_XDECREF(left); | |
| Py_XDECREF(right); | |
| return result; | |
| } | |
| /* factorial_odd_part: compute the odd part of factorial(n). */ | |
| static PyObject * | |
| factorial_odd_part(unsigned long n) | |
| { | |
| long i; | |
| unsigned long v, lower, upper; | |
| PyObject *partial, *tmp, *inner, *outer; | |
| inner = PyLong_FromLong(1); | |
| if (inner == NULL) | |
| return NULL; | |
| outer = inner; | |
| Py_INCREF(outer); | |
| upper = 3; | |
| for (i = bit_length(n) - 2; i >= 0; i--) { | |
| v = n >> i; | |
| if (v <= 2) | |
| continue; | |
| lower = upper; | |
| /* (v + 1) | 1 = least odd integer strictly larger than n / 2**i */ | |
| upper = (v + 1) | 1; | |
| /* Here inner is the product of all odd integers j in the range (0, | |
| n/2**(i+1)]. The factorial_partial_product call below gives the | |
| product of all odd integers j in the range (n/2**(i+1), n/2**i]. */ | |
| partial = factorial_partial_product(lower, upper, bit_length(upper-2)); | |
| /* inner *= partial */ | |
| if (partial == NULL) | |
| goto error; | |
| tmp = PyNumber_Multiply(inner, partial); | |
| Py_DECREF(partial); | |
| if (tmp == NULL) | |
| goto error; | |
| Py_DECREF(inner); | |
| inner = tmp; | |
| /* Now inner is the product of all odd integers j in the range (0, | |
| n/2**i], giving the inner product in the formula above. */ | |
| /* outer *= inner; */ | |
| tmp = PyNumber_Multiply(outer, inner); | |
| if (tmp == NULL) | |
| goto error; | |
| Py_DECREF(outer); | |
| outer = tmp; | |
| } | |
| Py_DECREF(inner); | |
| return outer; | |
| error: | |
| Py_DECREF(outer); | |
| Py_DECREF(inner); | |
| return NULL; | |
| } | |
| /* Lookup table for small factorial values */ | |
| static const unsigned long SmallFactorials[] = { | |
| 1, 1, 2, 6, 24, 120, 720, 5040, 40320, | |
| 362880, 3628800, 39916800, 479001600, | |
| #if SIZEOF_LONG >= 8 | |
| 6227020800, 87178291200, 1307674368000, | |
| 20922789888000, 355687428096000, 6402373705728000, | |
| 121645100408832000, 2432902008176640000 | |
| #endif | |
| }; | |
| /*[clinic input] | |
| math.factorial | |
| x as arg: object | |
| / | |
| Find x!. | |
| Raise a ValueError if x is negative or non-integral. | |
| [clinic start generated code]*/ | |
| static PyObject * | |
| math_factorial(PyObject *module, PyObject *arg) | |
| /*[clinic end generated code: output=6686f26fae00e9ca input=6d1c8105c0d91fb4]*/ | |
| { | |
| long x, two_valuation; | |
| int overflow; | |
| PyObject *result, *odd_part, *pyint_form; | |
| if (PyFloat_Check(arg)) { | |
| if (PyErr_WarnEx(PyExc_DeprecationWarning, | |
| "Using factorial() with floats is deprecated", | |
| 1) < 0) | |
| { | |
| return NULL; | |
| } | |
| PyObject *lx; | |
| double dx = PyFloat_AS_DOUBLE((PyFloatObject *)arg); | |
| if (!(Py_IS_FINITE(dx) && dx == floor(dx))) { | |
| PyErr_SetString(PyExc_ValueError, | |
| "factorial() only accepts integral values"); | |
| return NULL; | |
| } | |
| lx = PyLong_FromDouble(dx); | |
| if (lx == NULL) | |
| return NULL; | |
| x = PyLong_AsLongAndOverflow(lx, &overflow); | |
| Py_DECREF(lx); | |
| } | |
| else { | |
| pyint_form = PyNumber_Index(arg); | |
| if (pyint_form == NULL) { | |
| return NULL; | |
| } | |
| x = PyLong_AsLongAndOverflow(pyint_form, &overflow); | |
| Py_DECREF(pyint_form); | |
| } | |
| if (x == -1 && PyErr_Occurred()) { | |
| return NULL; | |
| } | |
| else if (overflow == 1) { | |
| PyErr_Format(PyExc_OverflowError, | |
| "factorial() argument should not exceed %ld", | |
| LONG_MAX); | |
| return NULL; | |
| } | |
| else if (overflow == -1 || x < 0) { | |
| PyErr_SetString(PyExc_ValueError, | |
| "factorial() not defined for negative values"); | |
| return NULL; | |
| } | |
| /* use lookup table if x is small */ | |
| if (x < (long)Py_ARRAY_LENGTH(SmallFactorials)) | |
| return PyLong_FromUnsignedLong(SmallFactorials[x]); | |
| /* else express in the form odd_part * 2**two_valuation, and compute as | |
| odd_part << two_valuation. */ | |
| odd_part = factorial_odd_part(x); | |
| if (odd_part == NULL) | |
| return NULL; | |
| two_valuation = x - count_set_bits(x); | |
| result = _PyLong_Lshift(odd_part, two_valuation); | |
| Py_DECREF(odd_part); | |
| return result; | |
| } | |
| /*[clinic input] | |
| math.trunc | |
| x: object | |
| / | |
| Truncates the Real x to the nearest Integral toward 0. | |
| Uses the __trunc__ magic method. | |
| [clinic start generated code]*/ | |
| static PyObject * | |
| math_trunc(PyObject *module, PyObject *x) | |
| /*[clinic end generated code: output=34b9697b707e1031 input=2168b34e0a09134d]*/ | |
| { | |
| _Py_IDENTIFIER(__trunc__); | |
| PyObject *trunc, *result; | |
| if (Py_TYPE(x)->tp_dict == NULL) { | |
| if (PyType_Ready(Py_TYPE(x)) < 0) | |
| return NULL; | |
| } | |
| trunc = _PyObject_LookupSpecial(x, &PyId___trunc__); | |
| if (trunc == NULL) { | |
| if (!PyErr_Occurred()) | |
| PyErr_Format(PyExc_TypeError, | |
| "type %.100s doesn't define __trunc__ method", | |
| Py_TYPE(x)->tp_name); | |
| return NULL; | |
| } | |
| result = _PyObject_CallNoArg(trunc); | |
| Py_DECREF(trunc); | |
| return result; | |
| } | |
| /*[clinic input] | |
| math.frexp | |
| x: double | |
| / | |
| Return the mantissa and exponent of x, as pair (m, e). | |
| m is a float and e is an int, such that x = m * 2.**e. | |
| If x is 0, m and e are both 0. Else 0.5 <= abs(m) < 1.0. | |
| [clinic start generated code]*/ | |
| static PyObject * | |
| math_frexp_impl(PyObject *module, double x) | |
| /*[clinic end generated code: output=03e30d252a15ad4a input=96251c9e208bc6e9]*/ | |
| { | |
| int i; | |
| /* deal with special cases directly, to sidestep platform | |
| differences */ | |
| if (Py_IS_NAN(x) || Py_IS_INFINITY(x) || !x) { | |
| i = 0; | |
| } | |
| else { | |
| PyFPE_START_PROTECT("in math_frexp", return 0); | |
| x = frexp(x, &i); | |
| PyFPE_END_PROTECT(x); | |
| } | |
| return Py_BuildValue("(di)", x, i); | |
| } | |
| /*[clinic input] | |
| math.ldexp | |
| x: double | |
| i: object | |
| / | |
| Return x * (2**i). | |
| This is essentially the inverse of frexp(). | |
| [clinic start generated code]*/ | |
| static PyObject * | |
| math_ldexp_impl(PyObject *module, double x, PyObject *i) | |
| /*[clinic end generated code: output=b6892f3c2df9cc6a input=17d5970c1a40a8c1]*/ | |
| { | |
| double r; | |
| long exp; | |
| int overflow; | |
| if (PyLong_Check(i)) { | |
| /* on overflow, replace exponent with either LONG_MAX | |
| or LONG_MIN, depending on the sign. */ | |
| exp = PyLong_AsLongAndOverflow(i, &overflow); | |
| if (exp == -1 && PyErr_Occurred()) | |
| return NULL; | |
| if (overflow) | |
| exp = overflow < 0 ? LONG_MIN : LONG_MAX; | |
| } | |
| else { | |
| PyErr_SetString(PyExc_TypeError, | |
| "Expected an int as second argument to ldexp."); | |
| return NULL; | |
| } | |
| if (x == 0. || !Py_IS_FINITE(x)) { | |
| /* NaNs, zeros and infinities are returned unchanged */ | |
| r = x; | |
| errno = 0; | |
| } else if (exp > INT_MAX) { | |
| /* overflow */ | |
| r = copysign(Py_HUGE_VAL, x); | |
| errno = ERANGE; | |
| } else if (exp < INT_MIN) { | |
| /* underflow to +-0 */ | |
| r = copysign(0., x); | |
| errno = 0; | |
| } else { | |
| errno = 0; | |
| PyFPE_START_PROTECT("in math_ldexp", return 0); | |
| r = ldexp(x, (int)exp); | |
| PyFPE_END_PROTECT(r); | |
| if (Py_IS_INFINITY(r)) | |
| errno = ERANGE; | |
| } | |
| if (errno && is_error(r)) | |
| return NULL; | |
| return PyFloat_FromDouble(r); | |
| } | |
| /*[clinic input] | |
| math.modf | |
| x: double | |
| / | |
| Return the fractional and integer parts of x. | |
| Both results carry the sign of x and are floats. | |
| [clinic start generated code]*/ | |
| static PyObject * | |
| math_modf_impl(PyObject *module, double x) | |
| /*[clinic end generated code: output=90cee0260014c3c0 input=b4cfb6786afd9035]*/ | |
| { | |
| double y; | |
| /* some platforms don't do the right thing for NaNs and | |
| infinities, so we take care of special cases directly. */ | |
| if (!Py_IS_FINITE(x)) { | |
| if (Py_IS_INFINITY(x)) | |
| return Py_BuildValue("(dd)", copysign(0., x), x); | |
| else if (Py_IS_NAN(x)) | |
| return Py_BuildValue("(dd)", x, x); | |
| } | |
| errno = 0; | |
| PyFPE_START_PROTECT("in math_modf", return 0); | |
| x = modf(x, &y); | |
| PyFPE_END_PROTECT(x); | |
| return Py_BuildValue("(dd)", x, y); | |
| } | |
| /* A decent logarithm is easy to compute even for huge ints, but libm can't | |
| do that by itself -- loghelper can. func is log or log10, and name is | |
| "log" or "log10". Note that overflow of the result isn't possible: an int | |
| can contain no more than INT_MAX * SHIFT bits, so has value certainly less | |
| than 2**(2**64 * 2**16) == 2**2**80, and log2 of that is 2**80, which is | |
| small enough to fit in an IEEE single. log and log10 are even smaller. | |
| However, intermediate overflow is possible for an int if the number of bits | |
| in that int is larger than PY_SSIZE_T_MAX. */ | |
| static PyObject* | |
| loghelper(PyObject* arg, double (*func)(double), const char *funcname) | |
| { | |
| /* If it is int, do it ourselves. */ | |
| if (PyLong_Check(arg)) { | |
| double x, result; | |
| Py_ssize_t e; | |
| /* Negative or zero inputs give a ValueError. */ | |
| if (Py_SIZE(arg) <= 0) { | |
| 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); | |
| } | |
| /*[clinic input] | |
| math.log | |
| x: object | |
| [ | |
| base: object(c_default="NULL") = math.e | |
| ] | |
| / | |
| Return the logarithm of x to the given base. | |
| If the base not specified, returns the natural logarithm (base e) of x. | |
| [clinic start generated code]*/ | |
| static PyObject * | |
| math_log_impl(PyObject *module, PyObject *x, int group_right_1, | |
| PyObject *base) | |
| /*[clinic end generated code: output=7b5a39e526b73fc9 input=0f62d5726cbfebbd]*/ | |
| { | |
| PyObject *num, *den; | |
| PyObject *ans; | |
| num = loghelper(x, 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_TrueDivide(num, den); | |
| Py_DECREF(num); | |
| Py_DECREF(den); | |
| return ans; | |
| } | |
| /*[clinic input] | |
| math.log2 | |
| x: object | |
| / | |
| Return the base 2 logarithm of x. | |
| [clinic start generated code]*/ | |
| static PyObject * | |
| math_log2(PyObject *module, PyObject *x) | |
| /*[clinic end generated code: output=5425899a4d5d6acb input=08321262bae4f39b]*/ | |
| { | |
| return loghelper(x, m_log2, "log2"); | |
| } | |
| /*[clinic input] | |
| math.log10 | |
| x: object | |
| / | |
| Return the base 10 logarithm of x. | |
| [clinic start generated code]*/ | |
| static PyObject * | |
| math_log10(PyObject *module, PyObject *x) | |
| /*[clinic end generated code: output=be72a64617df9c6f input=b2469d02c6469e53]*/ | |
| { | |
| return loghelper(x, m_log10, "log10"); | |
| } | |
| /*[clinic input] | |
| math.fmod | |
| x: double | |
| y: double | |
| / | |
| Return fmod(x, y), according to platform C. | |
| x % y may differ. | |
| [clinic start generated code]*/ | |
| static PyObject * | |
| math_fmod_impl(PyObject *module, double x, double y) | |
| /*[clinic end generated code: output=7559d794343a27b5 input=4f84caa8cfc26a03]*/ | |
| { | |
| double r; | |
| /* fmod(x, +/-Inf) returns x for finite x. */ | |
| if (Py_IS_INFINITY(y) && Py_IS_FINITE(x)) | |
| return PyFloat_FromDouble(x); | |
| errno = 0; | |
| 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); | |
| } | |
| /* | |
| Given an *n* length *vec* of values and a value *max*, compute: | |
| max * sqrt(sum((x / max) ** 2 for x in vec)) | |
| The value of the *max* variable must be non-negative and | |
| equal to the absolute value of the largest magnitude | |
| entry in the vector. If n==0, then *max* should be 0.0. | |
| If an infinity is present in the vec, *max* should be INF. | |
| The *found_nan* variable indicates whether some member of | |
| the *vec* is a NaN. | |
| To improve accuracy and to increase the number of cases where | |
| vector_norm() is commutative, we use a variant of Neumaier | |
| summation specialized to exploit that we always know that | |
| |csum| >= |x|. | |
| The *csum* variable tracks the cumulative sum and *frac* tracks | |
| the cumulative fractional errors at each step. Since this | |
| variant assumes that |csum| >= |x| at each step, we establish | |
| the precondition by starting the accumulation from 1.0 which | |
| represents the largest possible value of (x/max)**2. | |
| After the loop is finished, the initial 1.0 is subtracted out | |
| for a net zero effect on the final sum. Since *csum* will be | |
| greater than 1.0, the subtraction of 1.0 will not cause | |
| fractional digits to be dropped from *csum*. | |
| */ | |
| static inline double | |
| vector_norm(Py_ssize_t n, double *vec, double max, int found_nan) | |
| { | |
| double x, csum = 1.0, oldcsum, frac = 0.0; | |
| Py_ssize_t i; | |
| if (Py_IS_INFINITY(max)) { | |
| return max; | |
| } | |
| if (found_nan) { | |
| return Py_NAN; | |
| } | |
| if (max == 0.0 || n <= 1) { | |
| return max; | |
| } | |
| for (i=0 ; i < n ; i++) { | |
| x = vec[i]; | |
| assert(Py_IS_FINITE(x) && fabs(x) <= max); | |
| x /= max; | |
| x = x*x; | |
| oldcsum = csum; | |
| csum += x; | |
| assert(csum >= x); | |
| frac += (oldcsum - csum) + x; | |
| } | |
| return max * sqrt(csum - 1.0 + frac); | |
| } | |
| #define NUM_STACK_ELEMS 16 | |
| /*[clinic input] | |
| math.dist | |
| p: object | |
| q: object | |
| / | |
| Return the Euclidean distance between two points p and q. | |
| The points should be specified as sequences (or iterables) of | |
| coordinates. Both inputs must have the same dimension. | |
| Roughly equivalent to: | |
| sqrt(sum((px - qx) ** 2.0 for px, qx in zip(p, q))) | |
| [clinic start generated code]*/ | |
| static PyObject * | |
| math_dist_impl(PyObject *module, PyObject *p, PyObject *q) | |
| /*[clinic end generated code: output=56bd9538d06bbcfe input=74e85e1b6092e68e]*/ | |
| { | |
| PyObject *item; | |
| double max = 0.0; | |
| double x, px, qx, result; | |
| Py_ssize_t i, m, n; | |
| int found_nan = 0, p_allocated = 0, q_allocated = 0; | |
| double diffs_on_stack[NUM_STACK_ELEMS]; | |
| double *diffs = diffs_on_stack; | |
| if (!PyTuple_Check(p)) { | |
| p = PySequence_Tuple(p); | |
| if (p == NULL) { | |
| return NULL; | |
| } | |
| p_allocated = 1; | |
| } | |
| if (!PyTuple_Check(q)) { | |
| q = PySequence_Tuple(q); | |
| if (q == NULL) { | |
| if (p_allocated) { | |
| Py_DECREF(p); | |
| } | |
| return NULL; | |
| } | |
| q_allocated = 1; | |
| } | |
| m = PyTuple_GET_SIZE(p); | |
| n = PyTuple_GET_SIZE(q); | |
| if (m != n) { | |
| PyErr_SetString(PyExc_ValueError, | |
| "both points must have the same number of dimensions"); | |
| return NULL; | |
| } | |
| if (n > NUM_STACK_ELEMS) { | |
| diffs = (double *) PyObject_Malloc(n * sizeof(double)); | |
| if (diffs == NULL) { | |
| return PyErr_NoMemory(); | |
| } | |
| } | |
| for (i=0 ; i<n ; i++) { | |
| item = PyTuple_GET_ITEM(p, i); | |
| ASSIGN_DOUBLE(px, item, error_exit); | |
| item = PyTuple_GET_ITEM(q, i); | |
| ASSIGN_DOUBLE(qx, item, error_exit); | |
| x = fabs(px - qx); | |
| diffs[i] = x; | |
| found_nan |= Py_IS_NAN(x); | |
| if (x > max) { | |
| max = x; | |
| } | |
| } | |
| result = vector_norm(n, diffs, max, found_nan); | |
| if (diffs != diffs_on_stack) { | |
| PyObject_Free(diffs); | |
| } | |
| if (p_allocated) { | |
| Py_DECREF(p); | |
| } | |
| if (q_allocated) { | |
| Py_DECREF(q); | |
| } | |
| return PyFloat_FromDouble(result); | |
| error_exit: | |
| if (diffs != diffs_on_stack) { | |
| PyObject_Free(diffs); | |
| } | |
| if (p_allocated) { | |
| Py_DECREF(p); | |
| } | |
| if (q_allocated) { | |
| Py_DECREF(q); | |
| } | |
| return NULL; | |
| } | |
| /* AC: cannot convert yet, waiting for *args support */ | |
| static PyObject * | |
| math_hypot(PyObject *self, PyObject *const *args, Py_ssize_t nargs) | |
| { | |
| Py_ssize_t i; | |
| PyObject *item; | |
| double max = 0.0; | |
| double x, result; | |
| int found_nan = 0; | |
| double coord_on_stack[NUM_STACK_ELEMS]; | |
| double *coordinates = coord_on_stack; | |
| if (nargs > NUM_STACK_ELEMS) { | |
| coordinates = (double *) PyObject_Malloc(nargs * sizeof(double)); | |
| if (coordinates == NULL) { | |
| return PyErr_NoMemory(); | |
| } | |
| } | |
| for (i = 0; i < nargs; i++) { | |
| item = args[i]; | |
| ASSIGN_DOUBLE(x, item, error_exit); | |
| x = fabs(x); | |
| coordinates[i] = x; | |
| found_nan |= Py_IS_NAN(x); | |
| if (x > max) { | |
| max = x; | |
| } | |
| } | |
| result = vector_norm(nargs, coordinates, max, found_nan); | |
| if (coordinates != coord_on_stack) { | |
| PyObject_Free(coordinates); | |
| } | |
| return PyFloat_FromDouble(result); | |
| error_exit: | |
| if (coordinates != coord_on_stack) { | |
| PyObject_Free(coordinates); | |
| } | |
| return NULL; | |
| } | |
| #undef NUM_STACK_ELEMS | |
| PyDoc_STRVAR(math_hypot_doc, | |
| "hypot(*coordinates) -> value\n\n\ | |
| Multidimensional Euclidean distance from the origin to a point.\n\ | |
| \n\ | |
| Roughly equivalent to:\n\ | |
| sqrt(sum(x**2 for x in coordinates))\n\ | |
| \n\ | |
| For a two dimensional point (x, y), gives the hypotenuse\n\ | |
| using the Pythagorean theorem: sqrt(x*x + y*y).\n\ | |
| \n\ | |
| For example, the hypotenuse of a 3/4/5 right triangle is:\n\ | |
| \n\ | |
| >>> hypot(3.0, 4.0)\n\ | |
| 5.0\n\ | |
| "); | |
| /* pow can't use math_2, but needs its own wrapper: the problem is | |
| that an infinite result can arise either as a result of overflow | |
| (in which case OverflowError should be raised) or as a result of | |
| e.g. 0.**-5. (for which ValueError needs to be raised.) | |
| */ | |
| /*[clinic input] | |
| math.pow | |
| x: double | |
| y: double | |
| / | |
| Return x**y (x to the power of y). | |
| [clinic start generated code]*/ | |
| static PyObject * | |
| math_pow_impl(PyObject *module, double x, double y) | |
| /*[clinic end generated code: output=fff93e65abccd6b0 input=c26f1f6075088bfd]*/ | |
| { | |
| double r; | |
| int odd_y; | |
| /* deal directly with IEEE specials, to cope with problems on various | |
| platforms whose semantics don't exactly match C99 */ | |
| r = 0.; /* silence compiler warning */ | |
| if (!Py_IS_FINITE(x) || !Py_IS_FINITE(y)) { | |
| errno = 0; | |
| if (Py_IS_NAN(x)) | |
| r = y == 0. ? 1. : x; /* NaN**0 = 1 */ | |
| else if (Py_IS_NAN(y)) | |
| r = x == 1. ? 1. : y; /* 1**NaN = 1 */ | |
| else if (Py_IS_INFINITY(x)) { | |
| odd_y = Py_IS_FINITE(y) && fmod(fabs(y), 2.0) == 1.0; | |
| if (y > 0.) | |
| r = odd_y ? x : fabs(x); | |
| else if (y == 0.) | |
| r = 1.; | |
| else /* y < 0. */ | |
| r = odd_y ? copysign(0., x) : 0.; | |
| } | |
| else if (Py_IS_INFINITY(y)) { | |
| if (fabs(x) == 1.0) | |
| r = 1.; | |
| else if (y > 0. && fabs(x) > 1.0) | |
| r = y; | |
| else if (y < 0. && fabs(x) < 1.0) { | |
| r = -y; /* result is +inf */ | |
| 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); | |
| } | |
| static const double degToRad = Py_MATH_PI / 180.0; | |
| static const double radToDeg = 180.0 / Py_MATH_PI; | |
| /*[clinic input] | |
| math.degrees | |
| x: double | |
| / | |
| Convert angle x from radians to degrees. | |
| [clinic start generated code]*/ | |
| static PyObject * | |
| math_degrees_impl(PyObject *module, double x) | |
| /*[clinic end generated code: output=7fea78b294acd12f input=81e016555d6e3660]*/ | |
| { | |
| return PyFloat_FromDouble(x * radToDeg); | |
| } | |
| /*[clinic input] | |
| math.radians | |
| x: double | |
| / | |
| Convert angle x from degrees to radians. | |
| [clinic start generated code]*/ | |
| static PyObject * | |
| math_radians_impl(PyObject *module, double x) | |
| /*[clinic end generated code: output=34daa47caf9b1590 input=91626fc489fe3d63]*/ | |
| { | |
| return PyFloat_FromDouble(x * degToRad); | |
| } | |
| /*[clinic input] | |
| math.isfinite | |
| x: double | |
| / | |
| Return True if x is neither an infinity nor a NaN, and False otherwise. | |
| [clinic start generated code]*/ | |
| static PyObject * | |
| math_isfinite_impl(PyObject *module, double x) | |
| /*[clinic end generated code: output=8ba1f396440c9901 input=46967d254812e54a]*/ | |
| { | |
| return PyBool_FromLong((long)Py_IS_FINITE(x)); | |
| } | |
| /*[clinic input] | |
| math.isnan | |
| x: double | |
| / | |
| Return True if x is a NaN (not a number), and False otherwise. | |
| [clinic start generated code]*/ | |
| static PyObject * | |
| math_isnan_impl(PyObject *module, double x) | |
| /*[clinic end generated code: output=f537b4d6df878c3e input=935891e66083f46a]*/ | |
| { | |
| return PyBool_FromLong((long)Py_IS_NAN(x)); | |
| } | |
| /*[clinic input] | |
| math.isinf | |
| x: double | |
| / | |
| Return True if x is a positive or negative infinity, and False otherwise. | |
| [clinic start generated code]*/ | |
| static PyObject * | |
| math_isinf_impl(PyObject *module, double x) | |
| /*[clinic end generated code: output=9f00cbec4de7b06b input=32630e4212cf961f]*/ | |
| { | |
| return PyBool_FromLong((long)Py_IS_INFINITY(x)); | |
| } | |
| /*[clinic input] | |
| math.isclose -> bool | |
| a: double | |
| b: double | |
| * | |
| rel_tol: double = 1e-09 | |
| maximum difference for being considered "close", relative to the | |
| magnitude of the input values | |
| abs_tol: double = 0.0 | |
| maximum difference for being considered "close", regardless of the | |
| magnitude of the input values | |
| Determine whether two floating point numbers are close in value. | |
| Return True if a is close in value to b, and False otherwise. | |
| For the values to be considered close, the difference between them | |
| must be smaller than at least one of the tolerances. | |
| -inf, inf and NaN behave similarly to the IEEE 754 Standard. That | |
| is, NaN is not close to anything, even itself. inf and -inf are | |
| only close to themselves. | |
| [clinic start generated code]*/ | |
| static int | |
| math_isclose_impl(PyObject *module, double a, double b, double rel_tol, | |
| double abs_tol) | |
| /*[clinic end generated code: output=b73070207511952d input=f28671871ea5bfba]*/ | |
| { | |
| double diff = 0.0; | |
| /* sanity check on the inputs */ | |
| if (rel_tol < 0.0 || abs_tol < 0.0 ) { | |
| PyErr_SetString(PyExc_ValueError, | |
| "tolerances must be non-negative"); | |
| return -1; | |
| } | |
| if ( a == b ) { | |
| /* short circuit exact equality -- needed to catch two infinities of | |
| the same sign. And perhaps speeds things up a bit sometimes. | |
| */ | |
| return 1; | |
| } | |
| /* This catches the case of two infinities of opposite sign, or | |
| one infinity and one finite number. Two infinities of opposite | |
| sign would otherwise have an infinite relative tolerance. | |
| Two infinities of the same sign are caught by the equality check | |
| above. | |
| */ | |
| if (Py_IS_INFINITY(a) || Py_IS_INFINITY(b)) { | |
| return 0; | |
| } | |
| /* now do the regular computation | |
| this is essentially the "weak" test from the Boost library | |
| */ | |
| diff = fabs(b - a); | |
| return (((diff <= fabs(rel_tol * b)) || | |
| (diff <= fabs(rel_tol * a))) || | |
| (diff <= abs_tol)); | |
| } | |
| static inline int | |
| _check_long_mult_overflow(long a, long b) { | |
| /* From Python2's int_mul code: | |
| Integer overflow checking for * is painful: Python tried a couple ways, but | |
| they didn't work on all platforms, or failed in endcases (a product of | |
| -sys.maxint-1 has been a particular pain). | |
| Here's another way: | |
| The native long product x*y is either exactly right or *way* off, being | |
| just the last n bits of the true product, where n is the number of bits | |
| in a long (the delivered product is the true product plus i*2**n for | |
| some integer i). | |
| The native double product (double)x * (double)y is subject to three | |
| rounding errors: on a sizeof(long)==8 box, each cast to double can lose | |
| info, and even on a sizeof(long)==4 box, the multiplication can lose info. | |
| But, unlike the native long product, it's not in *range* trouble: even | |
| if sizeof(long)==32 (256-bit longs), the product easily fits in the | |
| dynamic range of a double. So the leading 50 (or so) bits of the double | |
| product are correct. | |
| We check these two ways against each other, and declare victory if they're | |
| approximately the same. Else, because the native long product is the only | |
| one that can lose catastrophic amounts of information, it's the native long | |
| product that must have overflowed. | |
| */ | |
| long longprod = (long)((unsigned long)a * b); | |
| double doubleprod = (double)a * (double)b; | |
| double doubled_longprod = (double)longprod; | |
| if (doubled_longprod == doubleprod) { | |
| return 0; | |
| } | |
| const double diff = doubled_longprod - doubleprod; | |
| const double absdiff = diff >= 0.0 ? diff : -diff; | |
| const double absprod = doubleprod >= 0.0 ? doubleprod : -doubleprod; | |
| if (32.0 * absdiff <= absprod) { | |
| return 0; | |
| } | |
| return 1; | |
| } | |
| /*[clinic input] | |
| math.prod | |
| iterable: object | |
| / | |
| * | |
| start: object(c_default="NULL") = 1 | |
| Calculate the product of all the elements in the input iterable. | |
| The default start value for the product is 1. | |
| When the iterable is empty, return the start value. This function is | |
| intended specifically for use with numeric values and may reject | |
| non-numeric types. | |
| [clinic start generated code]*/ | |
| static PyObject * | |
| math_prod_impl(PyObject *module, PyObject *iterable, PyObject *start) | |
| /*[clinic end generated code: output=36153bedac74a198 input=4c5ab0682782ed54]*/ | |
| { | |
| PyObject *result = start; | |
| PyObject *temp, *item, *iter; | |
| iter = PyObject_GetIter(iterable); | |
| if (iter == NULL) { | |
| return NULL; | |
| } | |
| if (result == NULL) { | |
| result = PyLong_FromLong(1); | |
| if (result == NULL) { | |
| Py_DECREF(iter); | |
| return NULL; | |
| } | |
| } else { | |
| Py_INCREF(result); | |
| } | |
| #ifndef SLOW_PROD | |
| /* Fast paths for integers keeping temporary products in C. | |
| * Assumes all inputs are the same type. | |
| * If the assumption fails, default to use PyObjects instead. | |
| */ | |
| if (PyLong_CheckExact(result)) { | |
| int overflow; | |
| long i_result = PyLong_AsLongAndOverflow(result, &overflow); | |
| /* If this already overflowed, don't even enter the loop. */ | |
| if (overflow == 0) { | |
| Py_DECREF(result); | |
| result = NULL; | |
| } | |
| /* Loop over all the items in the iterable until we finish, we overflow | |
| * or we found a non integer element */ | |
| while(result == NULL) { | |
| item = PyIter_Next(iter); | |
| if (item == NULL) { | |
| Py_DECREF(iter); | |
| if (PyErr_Occurred()) { | |
| return NULL; | |
| } | |
| return PyLong_FromLong(i_result); | |
| } | |
| if (PyLong_CheckExact(item)) { | |
| long b = PyLong_AsLongAndOverflow(item, &overflow); | |
| if (overflow == 0 && !_check_long_mult_overflow(i_result, b)) { | |
| long x = i_result * b; | |
| i_result = x; | |
| Py_DECREF(item); | |
| continue; | |
| } | |
| } | |
| /* Either overflowed or is not an int. | |
| * Restore real objects and process normally */ | |
| result = PyLong_FromLong(i_result); | |
| if (result == NULL) { | |
| Py_DECREF(item); | |
| Py_DECREF(iter); | |
| return NULL; | |
| } | |
| temp = PyNumber_Multiply(result, item); | |
| Py_DECREF(result); | |
| Py_DECREF(item); | |
| result = temp; | |
| if (result == NULL) { | |
| Py_DECREF(iter); | |
| return NULL; | |
| } | |
| } | |
| } | |
| /* Fast paths for floats keeping temporary products in C. | |
| * Assumes all inputs are the same type. | |
| * If the assumption fails, default to use PyObjects instead. | |
| */ | |
| if (PyFloat_CheckExact(result)) { | |
| double f_result = PyFloat_AS_DOUBLE(result); | |
| Py_DECREF(result); | |
| result = NULL; | |
| while(result == NULL) { | |
| item = PyIter_Next(iter); | |
| if (item == NULL) { | |
| Py_DECREF(iter); | |
| if (PyErr_Occurred()) { | |
| return NULL; | |
| } | |
| return PyFloat_FromDouble(f_result); | |
| } | |
| if (PyFloat_CheckExact(item)) { | |
| f_result *= PyFloat_AS_DOUBLE(item); | |
| Py_DECREF(item); | |
| continue; | |
| } | |
| if (PyLong_CheckExact(item)) { | |
| long value; | |
| int overflow; | |
| value = PyLong_AsLongAndOverflow(item, &overflow); | |
| if (!overflow) { | |
| f_result *= (double)value; | |
| Py_DECREF(item); | |
| continue; | |
| } | |
| } | |
| result = PyFloat_FromDouble(f_result); | |
| if (result == NULL) { | |
| Py_DECREF(item); | |
| Py_DECREF(iter); | |
| return NULL; | |
| } | |
| temp = PyNumber_Multiply(result, item); | |
| Py_DECREF(result); | |
| Py_DECREF(item); | |
| result = temp; | |
| if (result == NULL) { | |
| Py_DECREF(iter); | |
| return NULL; | |
| } | |
| } | |
| } | |
| #endif | |
| /* Consume rest of the iterable (if any) that could not be handled | |
| * by specialized functions above.*/ | |
| for(;;) { | |
| item = PyIter_Next(iter); | |
| if (item == NULL) { | |
| /* error, or end-of-sequence */ | |
| if (PyErr_Occurred()) { | |
| Py_DECREF(result); | |
| result = NULL; | |
| } | |
| break; | |
| } | |
| temp = PyNumber_Multiply(result, item); | |
| Py_DECREF(result); | |
| Py_DECREF(item); | |
| result = temp; | |
| if (result == NULL) | |
| break; | |
| } | |
| Py_DECREF(iter); | |
| return result; | |
| } | |
| /*[clinic input] | |
| math.perm | |
| n: object | |
| k: object = None | |
| / | |
| Number of ways to choose k items from n items without repetition and with order. | |
| Evaluates to n! / (n - k)! when k <= n and evaluates | |
| to zero when k > n. | |
| If k is not specified or is None, then k defaults to n | |
| and the function returns n!. | |
| Raises TypeError if either of the arguments are not integers. | |
| Raises ValueError if either of the arguments are negative. | |
| [clinic start generated code]*/ | |
| static PyObject * | |
| math_perm_impl(PyObject *module, PyObject *n, PyObject *k) | |
| /*[clinic end generated code: output=e021a25469653e23 input=5311c5a00f359b53]*/ | |
| { | |
| PyObject *result = NULL, *factor = NULL; | |
| int overflow, cmp; | |
| long long i, factors; | |
| if (k == Py_None) { | |
| return math_factorial(module, n); | |
| } | |
| n = PyNumber_Index(n); | |
| if (n == NULL) { | |
| return NULL; | |
| } | |
| if (!PyLong_CheckExact(n)) { | |
| Py_SETREF(n, _PyLong_Copy((PyLongObject *)n)); | |
| if (n == NULL) { | |
| return NULL; | |
| } | |
| } | |
| k = PyNumber_Index(k); | |
| if (k == NULL) { | |
| Py_DECREF(n); | |
| return NULL; | |
| } | |
| if (!PyLong_CheckExact(k)) { | |
| Py_SETREF(k, _PyLong_Copy((PyLongObject *)k)); | |
| if (k == NULL) { | |
| Py_DECREF(n); | |
| return NULL; | |
| } | |
| } | |
| if (Py_SIZE(n) < 0) { | |
| PyErr_SetString(PyExc_ValueError, | |
| "n must be a non-negative integer"); | |
| goto error; | |
| } | |
| if (Py_SIZE(k) < 0) { | |
| PyErr_SetString(PyExc_ValueError, | |
| "k must be a non-negative integer"); | |
| goto error; | |
| } | |
| cmp = PyObject_RichCompareBool(n, k, Py_LT); | |
| if (cmp != 0) { | |
| if (cmp > 0) { | |
| result = PyLong_FromLong(0); | |
| goto done; | |
| } | |
| goto error; | |
| } | |
| factors = PyLong_AsLongLongAndOverflow(k, &overflow); | |
| if (overflow > 0) { | |
| PyErr_Format(PyExc_OverflowError, | |
| "k must not exceed %lld", | |
| LLONG_MAX); | |
| goto error; | |
| } | |
| else if (factors == -1) { | |
| /* k is nonnegative, so a return value of -1 can only indicate error */ | |
| goto error; | |
| } | |
| if (factors == 0) { | |
| result = PyLong_FromLong(1); | |
| goto done; | |
| } | |
| result = n; | |
| Py_INCREF(result); | |
| if (factors == 1) { | |
| goto done; | |
| } | |
| factor = n; | |
| Py_INCREF(factor); | |
| for (i = 1; i < factors; ++i) { | |
| Py_SETREF(factor, PyNumber_Subtract(factor, _PyLong_One)); | |
| if (factor == NULL) { | |
| goto error; | |
| } | |
| Py_SETREF(result, PyNumber_Multiply(result, factor)); | |
| if (result == NULL) { | |
| goto error; | |
| } | |
| } | |
| Py_DECREF(factor); | |
| done: | |
| Py_DECREF(n); | |
| Py_DECREF(k); | |
| return result; | |
| error: | |
| Py_XDECREF(factor); | |
| Py_XDECREF(result); | |
| Py_DECREF(n); | |
| Py_DECREF(k); | |
| return NULL; | |
| } | |
| /*[clinic input] | |
| math.comb | |
| n: object | |
| k: object | |
| / | |
| Number of ways to choose k items from n items without repetition and without order. | |
| Evaluates to n! / (k! * (n - k)!) when k <= n and evaluates | |
| to zero when k > n. | |
| Also called the binomial coefficient because it is equivalent | |
| to the coefficient of k-th term in polynomial expansion of the | |
| expression (1 + x)**n. | |
| Raises TypeError if either of the arguments are not integers. | |
| Raises ValueError if either of the arguments are negative. | |
| [clinic start generated code]*/ | |
| static PyObject * | |
| math_comb_impl(PyObject *module, PyObject *n, PyObject *k) | |
| /*[clinic end generated code: output=bd2cec8d854f3493 input=9a05315af2518709]*/ | |
| { | |
| PyObject *result = NULL, *factor = NULL, *temp; | |
| int overflow, cmp; | |
| long long i, factors; | |
| n = PyNumber_Index(n); | |
| if (n == NULL) { | |
| return NULL; | |
| } | |
| if (!PyLong_CheckExact(n)) { | |
| Py_SETREF(n, _PyLong_Copy((PyLongObject *)n)); | |
| if (n == NULL) { | |
| return NULL; | |
| } | |
| } | |
| k = PyNumber_Index(k); | |
| if (k == NULL) { | |
| Py_DECREF(n); | |
| return NULL; | |
| } | |
| if (!PyLong_CheckExact(k)) { | |
| Py_SETREF(k, _PyLong_Copy((PyLongObject *)k)); | |
| if (k == NULL) { | |
| Py_DECREF(n); | |
| return NULL; | |
| } | |
| } | |
| if (Py_SIZE(n) < 0) { | |
| PyErr_SetString(PyExc_ValueError, | |
| "n must be a non-negative integer"); | |
| goto error; | |
| } | |
| if (Py_SIZE(k) < 0) { | |
| PyErr_SetString(PyExc_ValueError, | |
| "k must be a non-negative integer"); | |
| goto error; | |
| } | |
| /* k = min(k, n - k) */ | |
| temp = PyNumber_Subtract(n, k); | |
| if (temp == NULL) { | |
| goto error; | |
| } | |
| if (Py_SIZE(temp) < 0) { | |
| Py_DECREF(temp); | |
| result = PyLong_FromLong(0); | |
| goto done; | |
| } | |
| cmp = PyObject_RichCompareBool(temp, k, Py_LT); | |
| if (cmp > 0) { | |
| Py_SETREF(k, temp); | |
| } | |
| else { | |
| Py_DECREF(temp); | |
| if (cmp < 0) { | |
| goto error; | |
| } | |
| } | |
| factors = PyLong_AsLongLongAndOverflow(k, &overflow); | |
| if (overflow > 0) { | |
| PyErr_Format(PyExc_OverflowError, | |
| "min(n - k, k) must not exceed %lld", | |
| LLONG_MAX); | |
| goto error; | |
| } | |
| if (factors == -1) { | |
| /* k is nonnegative, so a return value of -1 can only indicate error */ | |
| goto error; | |
| } | |
| if (factors == 0) { | |
| result = PyLong_FromLong(1); | |
| goto done; | |
| } | |
| result = n; | |
| Py_INCREF(result); | |
| if (factors == 1) { | |
| goto done; | |
| } | |
| factor = n; | |
| Py_INCREF(factor); | |
| for (i = 1; i < factors; ++i) { | |
| Py_SETREF(factor, PyNumber_Subtract(factor, _PyLong_One)); | |
| if (factor == NULL) { | |
| goto error; | |
| } | |
| Py_SETREF(result, PyNumber_Multiply(result, factor)); | |
| if (result == NULL) { | |
| goto error; | |
| } | |
| temp = PyLong_FromUnsignedLongLong((unsigned long long)i + 1); | |
| if (temp == NULL) { | |
| goto error; | |
| } | |
| Py_SETREF(result, PyNumber_FloorDivide(result, temp)); | |
| Py_DECREF(temp); | |
| if (result == NULL) { | |
| goto error; | |
| } | |
| } | |
| Py_DECREF(factor); | |
| done: | |
| Py_DECREF(n); | |
| Py_DECREF(k); | |
| return result; | |
| error: | |
| Py_XDECREF(factor); | |
| Py_XDECREF(result); | |
| Py_DECREF(n); | |
| Py_DECREF(k); | |
| return NULL; | |
| } | |
| static PyMethodDef math_methods[] = { | |
| {"acos", math_acos, METH_O, math_acos_doc}, | |
| {"acosh", math_acosh, METH_O, math_acosh_doc}, | |
| {"asin", math_asin, METH_O, math_asin_doc}, | |
| {"asinh", math_asinh, METH_O, math_asinh_doc}, | |
| {"atan", math_atan, METH_O, math_atan_doc}, | |
| {"atan2", (PyCFunction)(void(*)(void))math_atan2, METH_FASTCALL, math_atan2_doc}, | |
| {"atanh", math_atanh, METH_O, math_atanh_doc}, | |
| MATH_CEIL_METHODDEF | |
| {"copysign", (PyCFunction)(void(*)(void))math_copysign, METH_FASTCALL, math_copysign_doc}, | |
| {"cos", math_cos, METH_O, math_cos_doc}, | |
| {"cosh", math_cosh, METH_O, math_cosh_doc}, | |
| MATH_DEGREES_METHODDEF | |
| MATH_DIST_METHODDEF | |
| {"erf", math_erf, METH_O, math_erf_doc}, | |
| {"erfc", math_erfc, METH_O, math_erfc_doc}, | |
| {"exp", math_exp, METH_O, math_exp_doc}, | |
| {"expm1", math_expm1, METH_O, math_expm1_doc}, | |
| {"fabs", math_fabs, METH_O, math_fabs_doc}, | |
| MATH_FACTORIAL_METHODDEF | |
| MATH_FLOOR_METHODDEF | |
| MATH_FMOD_METHODDEF | |
| MATH_FREXP_METHODDEF | |
| MATH_FSUM_METHODDEF | |
| {"gamma", math_gamma, METH_O, math_gamma_doc}, | |
| MATH_GCD_METHODDEF | |
| {"hypot", (PyCFunction)(void(*)(void))math_hypot, METH_FASTCALL, math_hypot_doc}, | |
| MATH_ISCLOSE_METHODDEF | |
| MATH_ISFINITE_METHODDEF | |
| MATH_ISINF_METHODDEF | |
| MATH_ISNAN_METHODDEF | |
| MATH_ISQRT_METHODDEF | |
| MATH_LDEXP_METHODDEF | |
| {"lgamma", math_lgamma, METH_O, math_lgamma_doc}, | |
| MATH_LOG_METHODDEF | |
| {"log1p", math_log1p, METH_O, math_log1p_doc}, | |
| MATH_LOG10_METHODDEF | |
| MATH_LOG2_METHODDEF | |
| MATH_MODF_METHODDEF | |
| MATH_POW_METHODDEF | |
| MATH_RADIANS_METHODDEF | |
| {"remainder", (PyCFunction)(void(*)(void))math_remainder, METH_FASTCALL, math_remainder_doc}, | |
| {"sin", math_sin, METH_O, math_sin_doc}, | |
| {"sinh", math_sinh, METH_O, math_sinh_doc}, | |
| {"sqrt", math_sqrt, METH_O, math_sqrt_doc}, | |
| {"tan", math_tan, METH_O, math_tan_doc}, | |
| {"tanh", math_tanh, METH_O, math_tanh_doc}, | |
| MATH_TRUNC_METHODDEF | |
| MATH_PROD_METHODDEF | |
| MATH_PERM_METHODDEF | |
| MATH_COMB_METHODDEF | |
| {NULL, NULL} /* sentinel */ | |
| }; | |
| PyDoc_STRVAR(module_doc, | |
| "This module provides access to the mathematical functions\n" | |
| "defined by the C standard."); | |
| static struct PyModuleDef mathmodule = { | |
| PyModuleDef_HEAD_INIT, | |
| "math", | |
| module_doc, | |
| -1, | |
| math_methods, | |
| NULL, | |
| NULL, | |
| NULL, | |
| NULL | |
| }; | |
| PyMODINIT_FUNC | |
| PyInit_math(void) | |
| { | |
| PyObject *m; | |
| m = PyModule_Create(&mathmodule); | |
| if (m == NULL) | |
| goto finally; | |
| PyModule_AddObject(m, "pi", PyFloat_FromDouble(Py_MATH_PI)); | |
| PyModule_AddObject(m, "e", PyFloat_FromDouble(Py_MATH_E)); | |
| PyModule_AddObject(m, "tau", PyFloat_FromDouble(Py_MATH_TAU)); /* 2pi */ | |
| PyModule_AddObject(m, "inf", PyFloat_FromDouble(m_inf())); | |
| #if !defined(PY_NO_SHORT_FLOAT_REPR) || defined(Py_NAN) | |
| PyModule_AddObject(m, "nan", PyFloat_FromDouble(m_nan())); | |
| #endif | |
| finally: | |
| return m; | |
| } |