diff options
Diffstat (limited to 'system')
-rw-r--r-- | system/lib/libc/musl/src/math/__cosdf.c | 35 | ||||
-rw-r--r-- | system/lib/libc/musl/src/math/__sindf.c | 36 | ||||
-rw-r--r-- | system/lib/libc/musl/src/math/lgamma.c | 9 | ||||
-rw-r--r-- | system/lib/libc/musl/src/math/lgamma_r.c | 314 | ||||
-rw-r--r-- | system/lib/libc/musl/src/math/lgammaf.c | 9 | ||||
-rw-r--r-- | system/lib/libc/musl/src/math/lgammaf_r.c | 249 | ||||
-rw-r--r-- | system/lib/libc/musl/src/math/lgammal.c | 386 | ||||
-rw-r--r-- | system/lib/libc/musl/src/math/signgam.c | 6 | ||||
-rw-r--r-- | system/lib/libc/musl/src/math/tgamma.c | 222 | ||||
-rw-r--r-- | system/lib/libc/musl/src/math/tgammaf.c | 6 | ||||
-rw-r--r-- | system/lib/libc/musl/src/math/tgammal.c | 275 | ||||
-rw-r--r-- | system/lib/libcextra.symbols | 16 |
12 files changed, 1563 insertions, 0 deletions
diff --git a/system/lib/libc/musl/src/math/__cosdf.c b/system/lib/libc/musl/src/math/__cosdf.c new file mode 100644 index 00000000..2124989b --- /dev/null +++ b/system/lib/libc/musl/src/math/__cosdf.c @@ -0,0 +1,35 @@ +/* origin: FreeBSD /usr/src/lib/msun/src/k_cosf.c */ +/* + * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com. + * Debugged and optimized by Bruce D. Evans. + */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +#include "libm.h" + +/* |cos(x) - c(x)| < 2**-34.1 (~[-5.37e-11, 5.295e-11]). */ +static const double +C0 = -0x1ffffffd0c5e81.0p-54, /* -0.499999997251031003120 */ +C1 = 0x155553e1053a42.0p-57, /* 0.0416666233237390631894 */ +C2 = -0x16c087e80f1e27.0p-62, /* -0.00138867637746099294692 */ +C3 = 0x199342e0ee5069.0p-68; /* 0.0000243904487962774090654 */ + +float __cosdf(double x) +{ + double_t r, w, z; + + /* Try to optimize for parallel evaluation as in __tandf.c. */ + z = x*x; + w = z*z; + r = C2+z*C3; + return ((1.0+z*C0) + w*C1) + (w*z)*r; +} diff --git a/system/lib/libc/musl/src/math/__sindf.c b/system/lib/libc/musl/src/math/__sindf.c new file mode 100644 index 00000000..8fec2a3f --- /dev/null +++ b/system/lib/libc/musl/src/math/__sindf.c @@ -0,0 +1,36 @@ +/* origin: FreeBSD /usr/src/lib/msun/src/k_sinf.c */ +/* + * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com. + * Optimized by Bruce D. Evans. + */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +#include "libm.h" + +/* |sin(x)/x - s(x)| < 2**-37.5 (~[-4.89e-12, 4.824e-12]). */ +static const double +S1 = -0x15555554cbac77.0p-55, /* -0.166666666416265235595 */ +S2 = 0x111110896efbb2.0p-59, /* 0.0083333293858894631756 */ +S3 = -0x1a00f9e2cae774.0p-65, /* -0.000198393348360966317347 */ +S4 = 0x16cd878c3b46a7.0p-71; /* 0.0000027183114939898219064 */ + +float __sindf(double x) +{ + double_t r, s, w, z; + + /* Try to optimize for parallel evaluation as in __tandf.c. */ + z = x*x; + w = z*z; + r = S3 + z*S4; + s = z*x; + return (x + s*(S1 + z*S2)) + s*w*r; +} diff --git a/system/lib/libc/musl/src/math/lgamma.c b/system/lib/libc/musl/src/math/lgamma.c new file mode 100644 index 00000000..e25ec8e6 --- /dev/null +++ b/system/lib/libc/musl/src/math/lgamma.c @@ -0,0 +1,9 @@ +#include <math.h> + +extern int __signgam; +double __lgamma_r(double, int *); + +double lgamma(double x) +{ + return __lgamma_r(x, &__signgam); +} diff --git a/system/lib/libc/musl/src/math/lgamma_r.c b/system/lib/libc/musl/src/math/lgamma_r.c new file mode 100644 index 00000000..82e296f5 --- /dev/null +++ b/system/lib/libc/musl/src/math/lgamma_r.c @@ -0,0 +1,314 @@ +/* origin: FreeBSD /usr/src/lib/msun/src/e_lgamma_r.c */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunSoft, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + * + */ +/* lgamma_r(x, signgamp) + * Reentrant version of the logarithm of the Gamma function + * with user provide pointer for the sign of Gamma(x). + * + * Method: + * 1. Argument Reduction for 0 < x <= 8 + * Since gamma(1+s)=s*gamma(s), for x in [0,8], we may + * reduce x to a number in [1.5,2.5] by + * lgamma(1+s) = log(s) + lgamma(s) + * for example, + * lgamma(7.3) = log(6.3) + lgamma(6.3) + * = log(6.3*5.3) + lgamma(5.3) + * = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3) + * 2. Polynomial approximation of lgamma around its + * minimun ymin=1.461632144968362245 to maintain monotonicity. + * On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use + * Let z = x-ymin; + * lgamma(x) = -1.214862905358496078218 + z^2*poly(z) + * where + * poly(z) is a 14 degree polynomial. + * 2. Rational approximation in the primary interval [2,3] + * We use the following approximation: + * s = x-2.0; + * lgamma(x) = 0.5*s + s*P(s)/Q(s) + * with accuracy + * |P/Q - (lgamma(x)-0.5s)| < 2**-61.71 + * Our algorithms are based on the following observation + * + * zeta(2)-1 2 zeta(3)-1 3 + * lgamma(2+s) = s*(1-Euler) + --------- * s - --------- * s + ... + * 2 3 + * + * where Euler = 0.5771... is the Euler constant, which is very + * close to 0.5. + * + * 3. For x>=8, we have + * lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+.... + * (better formula: + * lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...) + * Let z = 1/x, then we approximation + * f(z) = lgamma(x) - (x-0.5)(log(x)-1) + * by + * 3 5 11 + * w = w0 + w1*z + w2*z + w3*z + ... + w6*z + * where + * |w - f(z)| < 2**-58.74 + * + * 4. For negative x, since (G is gamma function) + * -x*G(-x)*G(x) = pi/sin(pi*x), + * we have + * G(x) = pi/(sin(pi*x)*(-x)*G(-x)) + * since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0 + * Hence, for x<0, signgam = sign(sin(pi*x)) and + * lgamma(x) = log(|Gamma(x)|) + * = log(pi/(|x*sin(pi*x)|)) - lgamma(-x); + * Note: one should avoid compute pi*(-x) directly in the + * computation of sin(pi*(-x)). + * + * 5. Special Cases + * lgamma(2+s) ~ s*(1-Euler) for tiny s + * lgamma(1) = lgamma(2) = 0 + * lgamma(x) ~ -log(|x|) for tiny x + * lgamma(0) = lgamma(neg.integer) = inf and raise divide-by-zero + * lgamma(inf) = inf + * lgamma(-inf) = inf (bug for bug compatible with C99!?) + * + */ + +#include "libm.h" +#include "libc.h" + +static const double +two52= 4.50359962737049600000e+15, /* 0x43300000, 0x00000000 */ +pi = 3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */ +a0 = 7.72156649015328655494e-02, /* 0x3FB3C467, 0xE37DB0C8 */ +a1 = 3.22467033424113591611e-01, /* 0x3FD4A34C, 0xC4A60FAD */ +a2 = 6.73523010531292681824e-02, /* 0x3FB13E00, 0x1A5562A7 */ +a3 = 2.05808084325167332806e-02, /* 0x3F951322, 0xAC92547B */ +a4 = 7.38555086081402883957e-03, /* 0x3F7E404F, 0xB68FEFE8 */ +a5 = 2.89051383673415629091e-03, /* 0x3F67ADD8, 0xCCB7926B */ +a6 = 1.19270763183362067845e-03, /* 0x3F538A94, 0x116F3F5D */ +a7 = 5.10069792153511336608e-04, /* 0x3F40B6C6, 0x89B99C00 */ +a8 = 2.20862790713908385557e-04, /* 0x3F2CF2EC, 0xED10E54D */ +a9 = 1.08011567247583939954e-04, /* 0x3F1C5088, 0x987DFB07 */ +a10 = 2.52144565451257326939e-05, /* 0x3EFA7074, 0x428CFA52 */ +a11 = 4.48640949618915160150e-05, /* 0x3F07858E, 0x90A45837 */ +tc = 1.46163214496836224576e+00, /* 0x3FF762D8, 0x6356BE3F */ +tf = -1.21486290535849611461e-01, /* 0xBFBF19B9, 0xBCC38A42 */ +/* tt = -(tail of tf) */ +tt = -3.63867699703950536541e-18, /* 0xBC50C7CA, 0xA48A971F */ +t0 = 4.83836122723810047042e-01, /* 0x3FDEF72B, 0xC8EE38A2 */ +t1 = -1.47587722994593911752e-01, /* 0xBFC2E427, 0x8DC6C509 */ +t2 = 6.46249402391333854778e-02, /* 0x3FB08B42, 0x94D5419B */ +t3 = -3.27885410759859649565e-02, /* 0xBFA0C9A8, 0xDF35B713 */ +t4 = 1.79706750811820387126e-02, /* 0x3F9266E7, 0x970AF9EC */ +t5 = -1.03142241298341437450e-02, /* 0xBF851F9F, 0xBA91EC6A */ +t6 = 6.10053870246291332635e-03, /* 0x3F78FCE0, 0xE370E344 */ +t7 = -3.68452016781138256760e-03, /* 0xBF6E2EFF, 0xB3E914D7 */ +t8 = 2.25964780900612472250e-03, /* 0x3F6282D3, 0x2E15C915 */ +t9 = -1.40346469989232843813e-03, /* 0xBF56FE8E, 0xBF2D1AF1 */ +t10 = 8.81081882437654011382e-04, /* 0x3F4CDF0C, 0xEF61A8E9 */ +t11 = -5.38595305356740546715e-04, /* 0xBF41A610, 0x9C73E0EC */ +t12 = 3.15632070903625950361e-04, /* 0x3F34AF6D, 0x6C0EBBF7 */ +t13 = -3.12754168375120860518e-04, /* 0xBF347F24, 0xECC38C38 */ +t14 = 3.35529192635519073543e-04, /* 0x3F35FD3E, 0xE8C2D3F4 */ +u0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */ +u1 = 6.32827064025093366517e-01, /* 0x3FE4401E, 0x8B005DFF */ +u2 = 1.45492250137234768737e+00, /* 0x3FF7475C, 0xD119BD6F */ +u3 = 9.77717527963372745603e-01, /* 0x3FEF4976, 0x44EA8450 */ +u4 = 2.28963728064692451092e-01, /* 0x3FCD4EAE, 0xF6010924 */ +u5 = 1.33810918536787660377e-02, /* 0x3F8B678B, 0xBF2BAB09 */ +v1 = 2.45597793713041134822e+00, /* 0x4003A5D7, 0xC2BD619C */ +v2 = 2.12848976379893395361e+00, /* 0x40010725, 0xA42B18F5 */ +v3 = 7.69285150456672783825e-01, /* 0x3FE89DFB, 0xE45050AF */ +v4 = 1.04222645593369134254e-01, /* 0x3FBAAE55, 0xD6537C88 */ +v5 = 3.21709242282423911810e-03, /* 0x3F6A5ABB, 0x57D0CF61 */ +s0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */ +s1 = 2.14982415960608852501e-01, /* 0x3FCB848B, 0x36E20878 */ +s2 = 3.25778796408930981787e-01, /* 0x3FD4D98F, 0x4F139F59 */ +s3 = 1.46350472652464452805e-01, /* 0x3FC2BB9C, 0xBEE5F2F7 */ +s4 = 2.66422703033638609560e-02, /* 0x3F9B481C, 0x7E939961 */ +s5 = 1.84028451407337715652e-03, /* 0x3F5E26B6, 0x7368F239 */ +s6 = 3.19475326584100867617e-05, /* 0x3F00BFEC, 0xDD17E945 */ +r1 = 1.39200533467621045958e+00, /* 0x3FF645A7, 0x62C4AB74 */ +r2 = 7.21935547567138069525e-01, /* 0x3FE71A18, 0x93D3DCDC */ +r3 = 1.71933865632803078993e-01, /* 0x3FC601ED, 0xCCFBDF27 */ +r4 = 1.86459191715652901344e-02, /* 0x3F9317EA, 0x742ED475 */ +r5 = 7.77942496381893596434e-04, /* 0x3F497DDA, 0xCA41A95B */ +r6 = 7.32668430744625636189e-06, /* 0x3EDEBAF7, 0xA5B38140 */ +w0 = 4.18938533204672725052e-01, /* 0x3FDACFE3, 0x90C97D69 */ +w1 = 8.33333333333329678849e-02, /* 0x3FB55555, 0x5555553B */ +w2 = -2.77777777728775536470e-03, /* 0xBF66C16C, 0x16B02E5C */ +w3 = 7.93650558643019558500e-04, /* 0x3F4A019F, 0x98CF38B6 */ +w4 = -5.95187557450339963135e-04, /* 0xBF4380CB, 0x8C0FE741 */ +w5 = 8.36339918996282139126e-04, /* 0x3F4B67BA, 0x4CDAD5D1 */ +w6 = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */ + +static double sin_pi(double x) +{ + double y,z; + int n,ix; + + GET_HIGH_WORD(ix, x); + ix &= 0x7fffffff; + + if (ix < 0x3fd00000) + return __sin(pi*x, 0.0, 0); + + y = -x; /* negative x is assumed */ + + /* + * argument reduction, make sure inexact flag not raised if input + * is an integer + */ + z = floor(y); + if (z != y) { /* inexact anyway */ + y *= 0.5; + y = 2.0*(y - floor(y)); /* y = |x| mod 2.0 */ + n = (int)(y*4.0); + } else { + if (ix >= 0x43400000) { + y = 0.0; /* y must be even */ + n = 0; + } else { + if (ix < 0x43300000) + z = y + two52; /* exact */ + GET_LOW_WORD(n, z); + n &= 1; + y = n; + n <<= 2; + } + } + switch (n) { + case 0: y = __sin(pi*y, 0.0, 0); break; + case 1: + case 2: y = __cos(pi*(0.5-y), 0.0); break; + case 3: + case 4: y = __sin(pi*(1.0-y), 0.0, 0); break; + case 5: + case 6: y = -__cos(pi*(y-1.5), 0.0); break; + default: y = __sin(pi*(y-2.0), 0.0, 0); break; + } + return -y; +} + + +double __lgamma_r(double x, int *signgamp) +{ + double t,y,z,nadj,p,p1,p2,p3,q,r,w; + int32_t hx; + int i,lx,ix; + + EXTRACT_WORDS(hx, lx, x); + + /* purge off +-inf, NaN, +-0, tiny and negative arguments */ + *signgamp = 1; + ix = hx & 0x7fffffff; + if (ix >= 0x7ff00000) + return x*x; + if ((ix|lx) == 0) + return 1.0/0.0; + if (ix < 0x3b900000) { /* |x|<2**-70, return -log(|x|) */ + if(hx < 0) { + *signgamp = -1; + return -log(-x); + } + return -log(x); + } + if (hx < 0) { + if (ix >= 0x43300000) /* |x|>=2**52, must be -integer */ + return 1.0/0.0; + t = sin_pi(x); + if (t == 0.0) /* -integer */ + return 1.0/0.0; + nadj = log(pi/fabs(t*x)); + if (t < 0.0) + *signgamp = -1; + x = -x; + } + + /* purge off 1 and 2 */ + if (((ix - 0x3ff00000)|lx) == 0 || ((ix - 0x40000000)|lx) == 0) + r = 0; + /* for x < 2.0 */ + else if (ix < 0x40000000) { + if (ix <= 0x3feccccc) { /* lgamma(x) = lgamma(x+1)-log(x) */ + r = -log(x); + if (ix >= 0x3FE76944) { + y = 1.0 - x; + i = 0; + } else if (ix >= 0x3FCDA661) { + y = x - (tc-1.0); + i = 1; + } else { + y = x; + i = 2; + } + } else { + r = 0.0; + if (ix >= 0x3FFBB4C3) { /* [1.7316,2] */ + y = 2.0 - x; + i = 0; + } else if(ix >= 0x3FF3B4C4) { /* [1.23,1.73] */ + y = x - tc; + i = 1; + } else { + y = x - 1.0; + i = 2; + } + } + switch (i) { + case 0: + z = y*y; + p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*a10)))); + p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*a11))))); + p = y*p1+p2; + r += (p-0.5*y); + break; + case 1: + z = y*y; + w = z*y; + p1 = t0+w*(t3+w*(t6+w*(t9 +w*t12))); /* parallel comp */ + p2 = t1+w*(t4+w*(t7+w*(t10+w*t13))); + p3 = t2+w*(t5+w*(t8+w*(t11+w*t14))); + p = z*p1-(tt-w*(p2+y*p3)); + r += tf + p; + break; + case 2: + p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*u5))))); + p2 = 1.0+y*(v1+y*(v2+y*(v3+y*(v4+y*v5)))); + r += -0.5*y + p1/p2; + } + } else if (ix < 0x40200000) { /* x < 8.0 */ + i = (int)x; + y = x - (double)i; + p = y*(s0+y*(s1+y*(s2+y*(s3+y*(s4+y*(s5+y*s6)))))); + q = 1.0+y*(r1+y*(r2+y*(r3+y*(r4+y*(r5+y*r6))))); + r = 0.5*y+p/q; + z = 1.0; /* lgamma(1+s) = log(s) + lgamma(s) */ + switch (i) { + case 7: z *= y + 6.0; /* FALLTHRU */ + case 6: z *= y + 5.0; /* FALLTHRU */ + case 5: z *= y + 4.0; /* FALLTHRU */ + case 4: z *= y + 3.0; /* FALLTHRU */ + case 3: z *= y + 2.0; /* FALLTHRU */ + r += log(z); + break; + } + } else if (ix < 0x43900000) { /* 8.0 <= x < 2**58 */ + t = log(x); + z = 1.0/x; + y = z*z; + w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*w6))))); + r = (x-0.5)*(t-1.0)+w; + } else /* 2**58 <= x <= inf */ + r = x*(log(x)-1.0); + if (hx < 0) + r = nadj - r; + return r; +} + +weak_alias(__lgamma_r, lgamma_r); diff --git a/system/lib/libc/musl/src/math/lgammaf.c b/system/lib/libc/musl/src/math/lgammaf.c new file mode 100644 index 00000000..badb6dfe --- /dev/null +++ b/system/lib/libc/musl/src/math/lgammaf.c @@ -0,0 +1,9 @@ +#include <math.h> + +extern int __signgam; +float __lgammaf_r(float, int *); + +float lgammaf(float x) +{ + return __lgammaf_r(x, &__signgam); +} diff --git a/system/lib/libc/musl/src/math/lgammaf_r.c b/system/lib/libc/musl/src/math/lgammaf_r.c new file mode 100644 index 00000000..dc65bace --- /dev/null +++ b/system/lib/libc/musl/src/math/lgammaf_r.c @@ -0,0 +1,249 @@ +/* origin: FreeBSD /usr/src/lib/msun/src/e_lgammaf_r.c */ +/* + * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com. + */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +#include "libm.h" +#include "libc.h" + +static const float +two23= 8.3886080000e+06, /* 0x4b000000 */ +pi = 3.1415927410e+00, /* 0x40490fdb */ +a0 = 7.7215664089e-02, /* 0x3d9e233f */ +a1 = 3.2246702909e-01, /* 0x3ea51a66 */ +a2 = 6.7352302372e-02, /* 0x3d89f001 */ +a3 = 2.0580807701e-02, /* 0x3ca89915 */ +a4 = 7.3855509982e-03, /* 0x3bf2027e */ +a5 = 2.8905137442e-03, /* 0x3b3d6ec6 */ +a6 = 1.1927076848e-03, /* 0x3a9c54a1 */ +a7 = 5.1006977446e-04, /* 0x3a05b634 */ +a8 = 2.2086278477e-04, /* 0x39679767 */ +a9 = 1.0801156895e-04, /* 0x38e28445 */ +a10 = 2.5214456400e-05, /* 0x37d383a2 */ +a11 = 4.4864096708e-05, /* 0x383c2c75 */ +tc = 1.4616321325e+00, /* 0x3fbb16c3 */ +tf = -1.2148628384e-01, /* 0xbdf8cdcd */ +/* tt = -(tail of tf) */ +tt = 6.6971006518e-09, /* 0x31e61c52 */ +t0 = 4.8383611441e-01, /* 0x3ef7b95e */ +t1 = -1.4758771658e-01, /* 0xbe17213c */ +t2 = 6.4624942839e-02, /* 0x3d845a15 */ +t3 = -3.2788541168e-02, /* 0xbd064d47 */ +t4 = 1.7970675603e-02, /* 0x3c93373d */ +t5 = -1.0314224288e-02, /* 0xbc28fcfe */ +t6 = 6.1005386524e-03, /* 0x3bc7e707 */ +t7 = -3.6845202558e-03, /* 0xbb7177fe */ +t8 = 2.2596477065e-03, /* 0x3b141699 */ +t9 = -1.4034647029e-03, /* 0xbab7f476 */ +t10 = 8.8108185446e-04, /* 0x3a66f867 */ +t11 = -5.3859531181e-04, /* 0xba0d3085 */ +t12 = 3.1563205994e-04, /* 0x39a57b6b */ +t13 = -3.1275415677e-04, /* 0xb9a3f927 */ +t14 = 3.3552918467e-04, /* 0x39afe9f7 */ +u0 = -7.7215664089e-02, /* 0xbd9e233f */ +u1 = 6.3282704353e-01, /* 0x3f2200f4 */ +u2 = 1.4549225569e+00, /* 0x3fba3ae7 */ +u3 = 9.7771751881e-01, /* 0x3f7a4bb2 */ +u4 = 2.2896373272e-01, /* 0x3e6a7578 */ +u5 = 1.3381091878e-02, /* 0x3c5b3c5e */ +v1 = 2.4559779167e+00, /* 0x401d2ebe */ +v2 = 2.1284897327e+00, /* 0x4008392d */ +v3 = 7.6928514242e-01, /* 0x3f44efdf */ +v4 = 1.0422264785e-01, /* 0x3dd572af */ +v5 = 3.2170924824e-03, /* 0x3b52d5db */ +s0 = -7.7215664089e-02, /* 0xbd9e233f */ +s1 = 2.1498242021e-01, /* 0x3e5c245a */ +s2 = 3.2577878237e-01, /* 0x3ea6cc7a */ +s3 = 1.4635047317e-01, /* 0x3e15dce6 */ +s4 = 2.6642270386e-02, /* 0x3cda40e4 */ +s5 = 1.8402845599e-03, /* 0x3af135b4 */ +s6 = 3.1947532989e-05, /* 0x3805ff67 */ +r1 = 1.3920053244e+00, /* 0x3fb22d3b */ +r2 = 7.2193557024e-01, /* 0x3f38d0c5 */ +r3 = 1.7193385959e-01, /* 0x3e300f6e */ +r4 = 1.8645919859e-02, /* 0x3c98bf54 */ +r5 = 7.7794247773e-04, /* 0x3a4beed6 */ +r6 = 7.3266842264e-06, /* 0x36f5d7bd */ +w0 = 4.1893854737e-01, /* 0x3ed67f1d */ +w1 = 8.3333335817e-02, /* 0x3daaaaab */ +w2 = -2.7777778450e-03, /* 0xbb360b61 */ +w3 = 7.9365057172e-04, /* 0x3a500cfd */ +w4 = -5.9518753551e-04, /* 0xba1c065c */ +w5 = 8.3633989561e-04, /* 0x3a5b3dd2 */ +w6 = -1.6309292987e-03; /* 0xbad5c4e8 */ + +static float sin_pif(float x) +{ + float y,z; + int n,ix; + + GET_FLOAT_WORD(ix, x); + ix &= 0x7fffffff; + + if(ix < 0x3e800000) + return __sindf(pi*x); + + y = -x; /* negative x is assumed */ + + /* + * argument reduction, make sure inexact flag not raised if input + * is an integer + */ + z = floorf(y); + if (z != y) { /* inexact anyway */ + y *= 0.5f; + y = 2.0f*(y - floorf(y)); /* y = |x| mod 2.0 */ + n = (int)(y*4.0f); + } else { + if (ix >= 0x4b800000) { + y = 0.0f; /* y must be even */ + n = 0; + } else { + if (ix < 0x4b000000) + z = y + two23; /* exact */ + GET_FLOAT_WORD(n, z); + n &= 1; + y = n; + n <<= 2; + } + } + switch (n) { + case 0: y = __sindf(pi*y); break; + case 1: + case 2: y = __cosdf(pi*(0.5f - y)); break; + case 3: + case 4: y = __sindf(pi*(1.0f - y)); break; + case 5: + case 6: y = -__cosdf(pi*(y - 1.5f)); break; + default: y = __sindf(pi*(y - 2.0f)); break; + } + return -y; +} + + +float __lgammaf_r(float x, int *signgamp) +{ + float t,y,z,nadj,p,p1,p2,p3,q,r,w; + int32_t hx; + int i,ix; + + GET_FLOAT_WORD(hx, x); + + /* purge off +-inf, NaN, +-0, tiny and negative arguments */ + *signgamp = 1; + ix = hx & 0x7fffffff; + if (ix >= 0x7f800000) + return x*x; + if (ix == 0) + return 1.0f/0.0f; + if (ix < 0x35000000) { /* |x| < 2**-21, return -log(|x|) */ + if (hx < 0) { + *signgamp = -1; + return -logf(-x); + } + return -logf(x); + } + if (hx < 0) { + if (ix >= 0x4b000000) /* |x| >= 2**23, must be -integer */ + return 1.0f/0.0f; + t = sin_pif(x); + if (t == 0.0f) /* -integer */ + return 1.0f/0.0f; + nadj = logf(pi/fabsf(t*x)); + if (t < 0.0f) + *signgamp = -1; + x = -x; + } + + /* purge off 1 and 2 */ + if (ix == 0x3f800000 || ix == 0x40000000) + r = 0; + /* for x < 2.0 */ + else if (ix < 0x40000000) { + if (ix <= 0x3f666666) { /* lgamma(x) = lgamma(x+1)-log(x) */ + r = -logf(x); + if (ix >= 0x3f3b4a20) { + y = 1.0f - x; + i = 0; + } else if (ix >= 0x3e6d3308) { + y = x - (tc-1.0f); + i = 1; + } else { + y = x; + i = 2; + } + } else { + r = 0.0f; + if (ix >= 0x3fdda618) { /* [1.7316,2] */ + y = 2.0f - x; + i = 0; + } else if (ix >= 0x3F9da620) { /* [1.23,1.73] */ + y = x - tc; + i = 1; + } else { + y = x - 1.0f; + i = 2; + } + } + switch(i) { + case 0: + z = y*y; + p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*a10)))); + p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*a11))))); + p = y*p1+p2; + r += p - 0.5f*y; + break; + case 1: + z = y*y; + w = z*y; + p1 = t0+w*(t3+w*(t6+w*(t9 +w*t12))); /* parallel comp */ + p2 = t1+w*(t4+w*(t7+w*(t10+w*t13))); + p3 = t2+w*(t5+w*(t8+w*(t11+w*t14))); + p = z*p1-(tt-w*(p2+y*p3)); + r += (tf + p); + break; + case 2: + p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*u5))))); + p2 = 1.0f+y*(v1+y*(v2+y*(v3+y*(v4+y*v5)))); + r += -0.5f*y + p1/p2; + } + } else if (ix < 0x41000000) { /* x < 8.0 */ + i = (int)x; + y = x - (float)i; + p = y*(s0+y*(s1+y*(s2+y*(s3+y*(s4+y*(s5+y*s6)))))); + q = 1.0f+y*(r1+y*(r2+y*(r3+y*(r4+y*(r5+y*r6))))); + r = 0.5f*y+p/q; + z = 1.0f; /* lgamma(1+s) = log(s) + lgamma(s) */ + switch (i) { + case 7: z *= y + 6.0f; /* FALLTHRU */ + case 6: z *= y + 5.0f; /* FALLTHRU */ + case 5: z *= y + 4.0f; /* FALLTHRU */ + case 4: z *= y + 3.0f; /* FALLTHRU */ + case 3: z *= y + 2.0f; /* FALLTHRU */ + r += logf(z); + break; + } + } else if (ix < 0x5c800000) { /* 8.0 <= x < 2**58 */ + t = logf(x); + z = 1.0f/x; + y = z*z; + w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*w6))))); + r = (x-0.5f)*(t-1.0f)+w; + } else /* 2**58 <= x <= inf */ + r = x*(logf(x)-1.0f); + if (hx < 0) + r = nadj - r; + return r; +} + +weak_alias(__lgammaf_r, lgammaf_r); diff --git a/system/lib/libc/musl/src/math/lgammal.c b/system/lib/libc/musl/src/math/lgammal.c new file mode 100644 index 00000000..cc4895eb --- /dev/null +++ b/system/lib/libc/musl/src/math/lgammal.c @@ -0,0 +1,386 @@ +/* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_lgammal.c */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ +/* + * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net> + * + * Permission to use, copy, modify, and distribute this software for any + * purpose with or without fee is hereby granted, provided that the above + * copyright notice and this permission notice appear in all copies. + * + * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES + * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF + * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR + * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES + * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN + * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF + * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE. + */ +/* lgammal(x) + * Reentrant version of the logarithm of the Gamma function + * with user provide pointer for the sign of Gamma(x). + * + * Method: + * 1. Argument Reduction for 0 < x <= 8 + * Since gamma(1+s)=s*gamma(s), for x in [0,8], we may + * reduce x to a number in [1.5,2.5] by + * lgamma(1+s) = log(s) + lgamma(s) + * for example, + * lgamma(7.3) = log(6.3) + lgamma(6.3) + * = log(6.3*5.3) + lgamma(5.3) + * = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3) + * 2. Polynomial approximation of lgamma around its + * minimun ymin=1.461632144968362245 to maintain monotonicity. + * On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use + * Let z = x-ymin; + * lgamma(x) = -1.214862905358496078218 + z^2*poly(z) + * 2. Rational approximation in the primary interval [2,3] + * We use the following approximation: + * s = x-2.0; + * lgamma(x) = 0.5*s + s*P(s)/Q(s) + * Our algorithms are based on the following observation + * + * zeta(2)-1 2 zeta(3)-1 3 + * lgamma(2+s) = s*(1-Euler) + --------- * s - --------- * s + ... + * 2 3 + * + * where Euler = 0.5771... is the Euler constant, which is very + * close to 0.5. + * + * 3. For x>=8, we have + * lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+.... + * (better formula: + * lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...) + * Let z = 1/x, then we approximation + * f(z) = lgamma(x) - (x-0.5)(log(x)-1) + * by + * 3 5 11 + * w = w0 + w1*z + w2*z + w3*z + ... + w6*z + * + * 4. For negative x, since (G is gamma function) + * -x*G(-x)*G(x) = pi/sin(pi*x), + * we have + * G(x) = pi/(sin(pi*x)*(-x)*G(-x)) + * since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0 + * Hence, for x<0, signgam = sign(sin(pi*x)) and + * lgamma(x) = log(|Gamma(x)|) + * = log(pi/(|x*sin(pi*x)|)) - lgamma(-x); + * Note: one should avoid compute pi*(-x) directly in the + * computation of sin(pi*(-x)). + * + * 5. Special Cases + * lgamma(2+s) ~ s*(1-Euler) for tiny s + * lgamma(1)=lgamma(2)=0 + * lgamma(x) ~ -log(x) for tiny x + * lgamma(0) = lgamma(inf) = inf + * lgamma(-integer) = +-inf + * + */ + +#define _GNU_SOURCE +#include "libm.h" +#include "libc.h" + +#if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024 +double __lgamma_r(double x, int *sg); + +long double __lgammal_r(long double x, int *sg) +{ + return __lgamma_r(x, sg); +} +#elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384 +static const long double +pi = 3.14159265358979323846264L, +two63 = 9.223372036854775808e18L, + +/* lgam(1+x) = 0.5 x + x a(x)/b(x) + -0.268402099609375 <= x <= 0 + peak relative error 6.6e-22 */ +a0 = -6.343246574721079391729402781192128239938E2L, +a1 = 1.856560238672465796768677717168371401378E3L, +a2 = 2.404733102163746263689288466865843408429E3L, +a3 = 8.804188795790383497379532868917517596322E2L, +a4 = 1.135361354097447729740103745999661157426E2L, +a5 = 3.766956539107615557608581581190400021285E0L, + +b0 = 8.214973713960928795704317259806842490498E3L, +b1 = 1.026343508841367384879065363925870888012E4L, +b2 = 4.553337477045763320522762343132210919277E3L, +b3 = 8.506975785032585797446253359230031874803E2L, +b4 = 6.042447899703295436820744186992189445813E1L, +/* b5 = 1.000000000000000000000000000000000000000E0 */ + + +tc = 1.4616321449683623412626595423257213284682E0L, +tf = -1.2148629053584961146050602565082954242826E-1, /* double precision */ +/* tt = (tail of tf), i.e. tf + tt has extended precision. */ +tt = 3.3649914684731379602768989080467587736363E-18L, +/* lgam ( 1.4616321449683623412626595423257213284682E0 ) = +-1.2148629053584960809551455717769158215135617312999903886372437313313530E-1 */ + +/* lgam (x + tc) = tf + tt + x g(x)/h(x) + -0.230003726999612341262659542325721328468 <= x + <= 0.2699962730003876587373404576742786715318 + peak relative error 2.1e-21 */ +g0 = 3.645529916721223331888305293534095553827E-18L, +g1 = 5.126654642791082497002594216163574795690E3L, +g2 = 8.828603575854624811911631336122070070327E3L, +g3 = 5.464186426932117031234820886525701595203E3L, +g4 = 1.455427403530884193180776558102868592293E3L, +g5 = 1.541735456969245924860307497029155838446E2L, +g6 = 4.335498275274822298341872707453445815118E0L, + +h0 = 1.059584930106085509696730443974495979641E4L, +h1 = 2.147921653490043010629481226937850618860E4L, +h2 = 1.643014770044524804175197151958100656728E4L, +h3 = 5.869021995186925517228323497501767586078E3L, +h4 = 9.764244777714344488787381271643502742293E2L, +h5 = 6.442485441570592541741092969581997002349E1L, +/* h6 = 1.000000000000000000000000000000000000000E0 */ + + +/* lgam (x+1) = -0.5 x + x u(x)/v(x) + -0.100006103515625 <= x <= 0.231639862060546875 + peak relative error 1.3e-21 */ +u0 = -8.886217500092090678492242071879342025627E1L, +u1 = 6.840109978129177639438792958320783599310E2L, +u2 = 2.042626104514127267855588786511809932433E3L, +u3 = 1.911723903442667422201651063009856064275E3L, +u4 = 7.447065275665887457628865263491667767695E2L, +u5 = 1.132256494121790736268471016493103952637E2L, +u6 = 4.484398885516614191003094714505960972894E0L, + +v0 = 1.150830924194461522996462401210374632929E3L, +v1 = 3.399692260848747447377972081399737098610E3L, +v2 = 3.786631705644460255229513563657226008015E3L, +v3 = 1.966450123004478374557778781564114347876E3L, +v4 = 4.741359068914069299837355438370682773122E2L, +v5 = 4.508989649747184050907206782117647852364E1L, +/* v6 = 1.000000000000000000000000000000000000000E0 */ + + +/* lgam (x+2) = .5 x + x s(x)/r(x) + 0 <= x <= 1 + peak relative error 7.2e-22 */ +s0 = 1.454726263410661942989109455292824853344E6L, +s1 = -3.901428390086348447890408306153378922752E6L, +s2 = -6.573568698209374121847873064292963089438E6L, +s3 = -3.319055881485044417245964508099095984643E6L, +s4 = -7.094891568758439227560184618114707107977E5L, +s5 = -6.263426646464505837422314539808112478303E4L, +s6 = -1.684926520999477529949915657519454051529E3L, + +r0 = -1.883978160734303518163008696712983134698E7L, +r1 = -2.815206082812062064902202753264922306830E7L, +r2 = -1.600245495251915899081846093343626358398E7L, +r3 = -4.310526301881305003489257052083370058799E6L, +r4 = -5.563807682263923279438235987186184968542E5L, +r5 = -3.027734654434169996032905158145259713083E4L, +r6 = -4.501995652861105629217250715790764371267E2L, +/* r6 = 1.000000000000000000000000000000000000000E0 */ + + +/* lgam(x) = ( x - 0.5 ) * log(x) - x + LS2PI + 1/x w(1/x^2) + x >= 8 + Peak relative error 1.51e-21 +w0 = LS2PI - 0.5 */ +w0 = 4.189385332046727417803e-1L, +w1 = 8.333333333333331447505E-2L, +w2 = -2.777777777750349603440E-3L, +w3 = 7.936507795855070755671E-4L, +w4 = -5.952345851765688514613E-4L, +w5 = 8.412723297322498080632E-4L, +w6 = -1.880801938119376907179E-3L, +w7 = 4.885026142432270781165E-3L; + +static long double sin_pi(long double x) +{ + union ldshape u = {x}; + uint32_t ix = (u.i.se & 0x7fffU)<<16 | u.i.m>>48; + long double y, z; + int n; + + if (ix < 0x3ffd8000) /* 0.25 */ + return sinl(pi * x); + y = -x; /* x is assume negative */ + + /* + * argument reduction, make sure inexact flag not raised if input + * is an integer + */ + z = floorl(y); + if (z != y) { /* inexact anyway */ + y *= 0.5; + y = 2.0*(y - floorl(y));/* y = |x| mod 2.0 */ + n = (int) (y*4.0); + } else { + if (ix >= 0x403f8000) { /* 2^64 */ + y = 0.0; /* y must be even */ + n = 0; + } else { + if (ix < 0x403e8000) /* 2^63 */ + z = y + two63; /* exact */ + u.f = z; + n = u.i.m & 1; + y = n; + n <<= 2; + } + } + + switch (n) { + case 0: + y = sinl(pi * y); + break; + case 1: + case 2: + y = cosl(pi * (0.5 - y)); + break; + case 3: + case 4: + y = sinl(pi * (1.0 - y)); + break; + case 5: + case 6: + y = -cosl(pi * (y - 1.5)); + break; + default: + y = sinl(pi * (y - 2.0)); + break; + } + return -y; +} + +long double __lgammal_r(long double x, int *sg) { + long double t, y, z, nadj, p, p1, p2, q, r, w; + union ldshape u = {x}; + uint32_t ix = (u.i.se & 0x7fffU)<<16 | u.i.m>>48; + int sign = u.i.se >> 15; + int i; + + *sg = 1; + + /* purge off +-inf, NaN, +-0, and negative arguments */ + if (ix >= 0x7fff0000) + return x * x; + if (x == 0) { + *sg -= 2*sign; + return 1.0 / fabsl(x); + } + if (ix < 0x3fc08000) { /* |x|<2**-6 |