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math: lgamma cleanup (simpler sin(pi*x) for the negative case)
* simplify sin_pi(x) (don't care about inexact here, the result is inexact anyway, and x is not so small to underflow) * in lgammal add the previously removed special case for x==1 and x==2 (to fix the sign of zero in downward rounding mode) * only define lgammal on supported long double platforms * change tgamma so the generated code is a bit smaller
This commit is contained in:
Szabolcs Nagy
parent
326e5c2e27
commit
ebbaf2180e
@ -82,7 +82,6 @@
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#include "libc.h"
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static const double
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two52= 4.50359962737049600000e+15, /* 0x43300000, 0x00000000 */
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pi = 3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */
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a0 = 7.72156649015328655494e-02, /* 0x3FB3C467, 0xE37DB0C8 */
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a1 = 3.22467033424113591611e-01, /* 0x3FD4A34C, 0xC4A60FAD */
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@ -147,91 +146,62 @@ w4 = -5.95187557450339963135e-04, /* 0xBF4380CB, 0x8C0FE741 */
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w5 = 8.36339918996282139126e-04, /* 0x3F4B67BA, 0x4CDAD5D1 */
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w6 = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */
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/* sin(pi*x) assuming x > 2^-100, if sin(pi*x)==0 the sign is arbitrary */
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static double sin_pi(double x)
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{
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double y,z;
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int n,ix;
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int n;
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GET_HIGH_WORD(ix, x);
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ix &= 0x7fffffff;
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/* spurious inexact if odd int */
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x = 2.0*(x*0.5 - floor(x*0.5)); /* x mod 2.0 */
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if (ix < 0x3fd00000)
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return __sin(pi*x, 0.0, 0);
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n = (int)(x*4.0);
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n = (n+1)/2;
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x -= n*0.5f;
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x *= pi;
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y = -x; /* negative x is assumed */
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/*
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* argument reduction, make sure inexact flag not raised if input
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* is an integer
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*/
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z = floor(y);
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if (z != y) { /* inexact anyway */
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y *= 0.5;
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y = 2.0*(y - floor(y)); /* y = |x| mod 2.0 */
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n = (int)(y*4.0);
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} else {
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if (ix >= 0x43400000) {
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y = 0.0; /* y must be even */
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n = 0;
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} else {
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if (ix < 0x43300000)
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z = y + two52; /* exact */
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GET_LOW_WORD(n, z);
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n &= 1;
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y = n;
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n <<= 2;
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}
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}
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switch (n) {
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case 0: y = __sin(pi*y, 0.0, 0); break;
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case 1:
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case 2: y = __cos(pi*(0.5-y), 0.0); break;
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case 3:
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case 4: y = __sin(pi*(1.0-y), 0.0, 0); break;
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case 5:
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case 6: y = -__cos(pi*(y-1.5), 0.0); break;
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default: y = __sin(pi*(y-2.0), 0.0, 0); break;
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default: /* case 4: */
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case 0: return __sin(x, 0.0, 0);
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case 1: return __cos(x, 0.0);
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case 2: return __sin(-x, 0.0, 0);
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case 3: return -__cos(x, 0.0);
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}
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return -y;
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}
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double __lgamma_r(double x, int *signgamp)
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{
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double t,y,z,nadj,p,p1,p2,p3,q,r,w;
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int32_t hx;
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int i,lx,ix;
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EXTRACT_WORDS(hx, lx, x);
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union {double f; uint64_t i;} u = {x};
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double_t t,y,z,nadj,p,p1,p2,p3,q,r,w;
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uint32_t ix;
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int sign,i;
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/* purge off +-inf, NaN, +-0, tiny and negative arguments */
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*signgamp = 1;
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ix = hx & 0x7fffffff;
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sign = u.i>>63;
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ix = u.i>>32 & 0x7fffffff;
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if (ix >= 0x7ff00000)
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return x*x;
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if ((ix|lx) == 0)
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return 1.0/0.0;
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if (ix < 0x3b900000) { /* |x|<2**-70, return -log(|x|) */
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if(hx < 0) {
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if (ix < (0x3ff-70)<<20) { /* |x|<2**-70, return -log(|x|) */
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if(sign) {
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x = -x;
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*signgamp = -1;
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return -log(-x);
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}
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return -log(x);
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}
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if (hx < 0) {
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if (ix >= 0x43300000) /* |x|>=2**52, must be -integer */
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return 1.0/0.0;
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if (sign) {
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x = -x;
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t = sin_pi(x);
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if (t == 0.0) /* -integer */
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return 1.0/0.0;
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nadj = log(pi/fabs(t*x));
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if (t < 0.0)
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return 1.0/(x-x);
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if (t > 0.0)
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*signgamp = -1;
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x = -x;
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else
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t = -t;
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nadj = log(pi/(t*x));
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}
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/* purge off 1 and 2 */
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if (((ix - 0x3ff00000)|lx) == 0 || ((ix - 0x40000000)|lx) == 0)
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if ((ix == 0x3ff00000 || ix == 0x40000000) && (uint32_t)u.i == 0)
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r = 0;
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/* for x < 2.0 */
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else if (ix < 0x40000000) {
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@ -306,7 +276,7 @@ double __lgamma_r(double x, int *signgamp)
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r = (x-0.5)*(t-1.0)+w;
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} else /* 2**58 <= x <= inf */
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r = x*(log(x)-1.0);
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if (hx < 0)
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if (sign)
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r = nadj - r;
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return r;
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}
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@ -17,7 +17,6 @@
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#include "libc.h"
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static const float
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two23= 8.3886080000e+06, /* 0x4b000000 */
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pi = 3.1415927410e+00, /* 0x40490fdb */
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a0 = 7.7215664089e-02, /* 0x3d9e233f */
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a1 = 3.2246702909e-01, /* 0x3ea51a66 */
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@ -82,87 +81,58 @@ w4 = -5.9518753551e-04, /* 0xba1c065c */
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w5 = 8.3633989561e-04, /* 0x3a5b3dd2 */
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w6 = -1.6309292987e-03; /* 0xbad5c4e8 */
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static float sin_pif(float x)
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/* sin(pi*x) assuming x > 2^-100, if sin(pi*x)==0 the sign is arbitrary */
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static float sin_pi(float x)
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{
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float y,z;
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int n,ix;
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double_t y;
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int n;
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GET_FLOAT_WORD(ix, x);
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ix &= 0x7fffffff;
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/* spurious inexact if odd int */
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x = 2*(x*0.5f - floorf(x*0.5f)); /* x mod 2.0 */
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if(ix < 0x3e800000)
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return __sindf(pi*x);
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y = -x; /* negative x is assumed */
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/*
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* argument reduction, make sure inexact flag not raised if input
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* is an integer
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*/
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z = floorf(y);
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if (z != y) { /* inexact anyway */
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y *= 0.5f;
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y = 2.0f*(y - floorf(y)); /* y = |x| mod 2.0 */
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n = (int)(y*4.0f);
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} else {
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if (ix >= 0x4b800000) {
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y = 0.0f; /* y must be even */
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n = 0;
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} else {
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if (ix < 0x4b000000)
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z = y + two23; /* exact */
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GET_FLOAT_WORD(n, z);
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n &= 1;
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y = n;
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n <<= 2;
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}
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}
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n = (int)(x*4);
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n = (n+1)/2;
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y = x - n*0.5f;
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y *= 3.14159265358979323846;
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switch (n) {
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case 0: y = __sindf(pi*y); break;
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case 1:
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case 2: y = __cosdf(pi*(0.5f - y)); break;
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case 3:
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case 4: y = __sindf(pi*(1.0f - y)); break;
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case 5:
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case 6: y = -__cosdf(pi*(y - 1.5f)); break;
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default: y = __sindf(pi*(y - 2.0f)); break;
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default: /* case 4: */
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case 0: return __sindf(y);
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case 1: return __cosdf(y);
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case 2: return __sindf(-y);
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case 3: return -__cosdf(y);
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}
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return -y;
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}
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float __lgammaf_r(float x, int *signgamp)
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{
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union {float f; uint32_t i;} u = {x};
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float t,y,z,nadj,p,p1,p2,p3,q,r,w;
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int32_t hx;
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int i,ix;
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GET_FLOAT_WORD(hx, x);
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uint32_t ix;
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int i,sign;
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/* purge off +-inf, NaN, +-0, tiny and negative arguments */
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*signgamp = 1;
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ix = hx & 0x7fffffff;
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sign = u.i>>31;
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ix = u.i & 0x7fffffff;
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if (ix >= 0x7f800000)
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return x*x;
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if (ix == 0)
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return 1.0f/0.0f;
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if (ix < 0x35000000) { /* |x| < 2**-21, return -log(|x|) */
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if (hx < 0) {
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if (sign) {
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*signgamp = -1;
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return -logf(-x);
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x = -x;
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}
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return -logf(x);
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}
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if (hx < 0) {
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if (ix >= 0x4b000000) /* |x| >= 2**23, must be -integer */
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return 1.0f/0.0f;
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t = sin_pif(x);
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if (t == 0.0f) /* -integer */
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return 1.0f/0.0f;
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nadj = logf(pi/fabsf(t*x));
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if (t < 0.0f)
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*signgamp = -1;
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if (sign) {
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x = -x;
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t = sin_pi(x);
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if (t == 0.0f) /* -integer */
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return 1.0f/(x-x);
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if (t > 0.0f)
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*signgamp = -1;
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else
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t = -t;
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nadj = logf(pi/(t*x));
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}
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/* purge off 1 and 2 */
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@ -241,7 +211,7 @@ float __lgammaf_r(float x, int *signgamp)
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r = (x-0.5f)*(t-1.0f)+w;
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} else /* 2**58 <= x <= inf */
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r = x*(logf(x)-1.0f);
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if (hx < 0)
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if (sign)
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r = nadj - r;
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return r;
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}
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@ -99,7 +99,6 @@ long double __lgammal_r(long double x, int *sg)
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#elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
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static const long double
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pi = 3.14159265358979323846264L,
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two63 = 9.223372036854775808e18L,
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/* lgam(1+x) = 0.5 x + x a(x)/b(x)
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-0.268402099609375 <= x <= 0
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@ -201,61 +200,27 @@ w5 = 8.412723297322498080632E-4L,
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w6 = -1.880801938119376907179E-3L,
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w7 = 4.885026142432270781165E-3L;
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/* sin(pi*x) assuming x > 2^-1000, if sin(pi*x)==0 the sign is arbitrary */
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static long double sin_pi(long double x)
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{
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union ldshape u = {x};
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uint32_t ix = (u.i.se & 0x7fffU)<<16 | u.i.m>>48;
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long double y, z;
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int n;
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if (ix < 0x3ffd8000) /* 0.25 */
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return sinl(pi * x);
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y = -x; /* x is assume negative */
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/* spurious inexact if odd int */
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x *= 0.5;
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x = 2.0*(x - floorl(x)); /* x mod 2.0 */
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/*
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* argument reduction, make sure inexact flag not raised if input
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* is an integer
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*/
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z = floorl(y);
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if (z != y) { /* inexact anyway */
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y *= 0.5;
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y = 2.0*(y - floorl(y));/* y = |x| mod 2.0 */
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n = (int) (y*4.0);
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} else {
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if (ix >= 0x403f8000) { /* 2^64 */
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y = 0.0; /* y must be even */
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n = 0;
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} else {
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if (ix < 0x403e8000) /* 2^63 */
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z = y + two63; /* exact */
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u.f = z;
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n = u.i.m & 1;
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y = n;
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n <<= 2;
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}
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}
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n = (int)(x*4.0);
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n = (n+1)/2;
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x -= n*0.5f;
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x *= pi;
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switch (n) {
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case 0:
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y = sinl(pi * y);
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break;
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case 1:
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case 2:
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y = cosl(pi * (0.5 - y));
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break;
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case 3:
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case 4:
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y = sinl(pi * (1.0 - y));
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break;
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case 5:
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case 6:
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y = -cosl(pi * (y - 1.5));
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break;
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default:
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y = sinl(pi * (y - 2.0));
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break;
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default: /* case 4: */
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case 0: return __sinl(x, 0.0, 0);
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case 1: return __cosl(x, 0.0);
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case 2: return __sinl(-x, 0.0, 0);
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case 3: return -__cosl(x, 0.0);
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}
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return -y;
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}
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long double __lgammal_r(long double x, int *sg) {
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@ -267,31 +232,32 @@ long double __lgammal_r(long double x, int *sg) {
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*sg = 1;
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/* purge off +-inf, NaN, +-0, and negative arguments */
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/* purge off +-inf, NaN, +-0, tiny and negative arguments */
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if (ix >= 0x7fff0000)
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return x * x;
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if (x == 0) {
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*sg -= 2*sign;
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return 1.0 / fabsl(x);
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}
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if (ix < 0x3fc08000) { /* |x|<2**-63, return -log(|x|) */
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if (sign) {
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*sg = -1;
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return -logl(-x);
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x = -x;
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}
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return -logl(x);
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}
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if (sign) {
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t = sin_pi (x);
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if (t == 0.0)
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return 1.0 / fabsl(t); /* -integer */
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nadj = logl(pi / fabsl(t * x));
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if (t < 0.0)
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*sg = -1;
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x = -x;
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t = sin_pi(x);
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if (t == 0.0)
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return 1.0 / (x-x); /* -integer */
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if (t > 0.0)
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*sg = -1;
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else
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t = -t;
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nadj = logl(pi / (t * x));
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}
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if (ix < 0x40008000) { /* x < 2.0 */
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/* purge off 1 and 2 (so the sign is ok with downward rounding) */
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if ((ix == 0x3fff8000 || ix == 0x40008000) && u.i.m == 0) {
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r = 0;
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} else if (ix < 0x40008000) { /* x < 2.0 */
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if (ix <= 0x3ffee666) { /* 8.99993896484375e-1 */
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/* lgamma(x) = lgamma(x+1) - log(x) */
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r = -logl(x);
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@ -376,6 +342,7 @@ long double __lgammal_r(long double x, int *sg) {
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}
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#endif
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#if (LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024) || (LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384)
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extern int __signgam;
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long double lgammal(long double x)
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@ -384,3 +351,4 @@ long double lgammal(long double x)
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}
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weak_alias(__lgammal_r, lgammal_r);
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#endif
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@ -26,7 +26,7 @@ most ideas and constants are from boost and python
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static const double pi = 3.141592653589793238462643383279502884;
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/* sin(pi x) with x > 0 && isnormal(x) assumption */
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/* sin(pi x) with x > 0x1p-100, if sin(pi*x)==0 the sign is arbitrary */
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static double sinpi(double x)
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{
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int n;
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@ -49,8 +49,7 @@ static double sinpi(double x)
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case 1:
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return __cos(x, 0);
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case 2:
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/* sin(0-x) and -sin(x) have different sign at 0 */
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return __sin(0-x, 0, 0);
|
||||
return __sin(-x, 0, 0);
|
||||
case 3:
|
||||
return -__cos(x, 0);
|
||||
}
|
||||
@ -108,35 +107,33 @@ static double S(double x)
|
||||
|
||||
double tgamma(double x)
|
||||
{
|
||||
double absx, y, dy, z, r;
|
||||
union {double f; uint64_t i;} u = {x};
|
||||
double absx, y;
|
||||
double_t dy, z, r;
|
||||
uint32_t ix = u.i>>32 & 0x7fffffff;
|
||||
int sign = u.i>>63;
|
||||
|
||||
/* special cases */
|
||||
if (!isfinite(x))
|
||||
if (ix >= 0x7ff00000)
|
||||
/* tgamma(nan)=nan, tgamma(inf)=inf, tgamma(-inf)=nan with invalid */
|
||||
return x + INFINITY;
|
||||
if (ix < (0x3ff-54)<<20)
|
||||
/* |x| < 2^-54: tgamma(x) ~ 1/x, +-0 raises div-by-zero */
|
||||
return 1/x;
|
||||
|
||||
/* integer arguments */
|
||||
/* raise inexact when non-integer */
|
||||
if (x == floor(x)) {
|
||||
if (x == 0)
|
||||
/* tgamma(+-0)=+-inf with divide-by-zero */
|
||||
return 1/x;
|
||||
if (x < 0)
|
||||
if (sign)
|
||||
return 0/0.0;
|
||||
if (x <= sizeof fact/sizeof *fact)
|
||||
return fact[(int)x - 1];
|
||||
}
|
||||
|
||||
absx = fabs(x);
|
||||
|
||||
/* x ~ 0: tgamma(x) ~ 1/x */
|
||||
if (absx < 0x1p-54)
|
||||
return 1/x;
|
||||
|
||||
/* x >= 172: tgamma(x)=inf with overflow */
|
||||
/* x =< -184: tgamma(x)=+-0 with underflow */
|
||||
if (absx >= 184) {
|
||||
if (x < 0) {
|
||||
if (ix >= 0x40670000) { /* |x| >= 184 */
|
||||
if (sign) {
|
||||
FORCE_EVAL((float)(0x1p-126/x));
|
||||
if (floor(x) * 0.5 == floor(x * 0.5))
|
||||
return 0;
|
||||
@ -146,6 +143,8 @@ double tgamma(double x)
|
||||
return x;
|
||||
}
|
||||
|
||||
absx = sign ? -x : x;
|
||||
|
||||
/* handle the error of x + g - 0.5 */
|
||||
y = absx + gmhalf;
|
||||
if (absx > gmhalf) {
|
||||
@ -160,20 +159,21 @@ double tgamma(double x)
|
||||
r = S(absx) * exp(-y);
|
||||
if (x < 0) {
|
||||
/* reflection formula for negative x */
|
||||
/* sinpi(absx) is not 0, integers are already handled */
|
||||
r = -pi / (sinpi(absx) * absx * r);
|
||||
dy = -dy;
|
||||
z = -z;
|
||||
}
|
||||
r += dy * (gmhalf+0.5) * r / y;
|
||||
z = pow(y, 0.5*z);
|
||||
r = r * z * z;
|
||||
return r;
|
||||
y = r * z * z;
|
||||
return y;
|
||||
}
|
||||
|
||||
#if 0
|
||||
double __lgamma_r(double x, int *sign)
|
||||
{
|
||||
double r, absx, z, zz, w;
|
||||
double r, absx;
|
||||
|
||||
*sign = 1;
|
||||
|
||||
|
||||
Loading…
Reference in New Issue
Block a user