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SDL/src/libm/e_pow.c

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Windows: Always set the system timer resolution to 1ms by default. An existing hint lets apps that don't need the timer resolution changed avoid this, to save battery, etc, but this fixes several problems in timing, audio callbacks not firing fast enough, etc. Fixes Bugzilla #2944.
2015-04-20 16:22:44 +00:00
/* @(#)e_pow.c 5.1 93/09/24 */
/*
* ====================================================
* 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.
* ====================================================
*/
#if defined(LIBM_SCCS) && !defined(lint)
static char rcsid[] = "$NetBSD: e_pow.c,v 1.9 1995/05/12 04:57:32 jtc Exp $";
#endif
/* __ieee754_pow(x,y) return x**y
*
* n
* Method: Let x = 2 * (1+f)
* 1. Compute and return log2(x) in two pieces:
* log2(x) = w1 + w2,
* where w1 has 53-24 = 29 bit trailing zeros.
* 2. Perform y*log2(x) = n+y' by simulating muti-precision
* arithmetic, where |y'|<=0.5.
* 3. Return x**y = 2**n*exp(y'*log2)
*
* Special cases:
* 1. (anything) ** 0 is 1
* 2. (anything) ** 1 is itself
* 3. (anything) ** NAN is NAN
* 4. NAN ** (anything except 0) is NAN
* 5. +-(|x| > 1) ** +INF is +INF
* 6. +-(|x| > 1) ** -INF is +0
* 7. +-(|x| < 1) ** +INF is +0
* 8. +-(|x| < 1) ** -INF is +INF
* 9. +-1 ** +-INF is NAN
* 10. +0 ** (+anything except 0, NAN) is +0
* 11. -0 ** (+anything except 0, NAN, odd integer) is +0
* 12. +0 ** (-anything except 0, NAN) is +INF
* 13. -0 ** (-anything except 0, NAN, odd integer) is +INF
* 14. -0 ** (odd integer) = -( +0 ** (odd integer) )
* 15. +INF ** (+anything except 0,NAN) is +INF
* 16. +INF ** (-anything except 0,NAN) is +0
* 17. -INF ** (anything) = -0 ** (-anything)
* 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
* 19. (-anything except 0 and inf) ** (non-integer) is NAN
*
* Accuracy:
* pow(x,y) returns x**y nearly rounded. In particular
* pow(integer,integer)
* always returns the correct integer provided it is
* representable.
*
* Constants :
* The hexadecimal values are the intended ones for the following
* constants. The decimal values may be used, provided that the
* compiler will convert from decimal to binary accurately enough
* to produce the hexadecimal values shown.
*/
#include "math_libm.h"
#include "math_private.h"
libm_hidden_proto(scalbn)
libm_hidden_proto(fabs)
#ifdef __STDC__
static const double
#else
static double
#endif
bp[] = { 1.0, 1.5, }, dp_h[] = {
0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
dp_l[] = {
0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
zero = 0.0, one = 1.0, two = 2.0, two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */
huge_val = 1.0e300, tiny = 1.0e-300,
/* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
L1 = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */
L2 = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */
L3 = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */
L4 = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */
L5 = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */
L6 = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */
P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */
lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */
ovt = 8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */
cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */
cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h */
ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2 */
ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail */
#ifdef __STDC__
double attribute_hidden __ieee754_pow(double x, double y)
#else
double attribute_hidden __ieee754_pow(x, y)
double x, y;
#endif
{
double z, ax, z_h, z_l, p_h, p_l;
double y1, t1, t2, r, s, t, u, v, w;
int32_t i, j, k, yisint, n;
int32_t hx, hy, ix, iy;
u_int32_t lx, ly;
EXTRACT_WORDS(hx, lx, x);
EXTRACT_WORDS(hy, ly, y);
ix = hx & 0x7fffffff;
iy = hy & 0x7fffffff;
/* y==zero: x**0 = 1 */
if ((iy | ly) == 0)
return one;
/* +-NaN return x+y */
if (ix > 0x7ff00000 || ((ix == 0x7ff00000) && (lx != 0)) ||
iy > 0x7ff00000 || ((iy == 0x7ff00000) && (ly != 0)))
return x + y;
/* determine if y is an odd int when x < 0
* yisint = 0 ... y is not an integer
* yisint = 1 ... y is an odd int
* yisint = 2 ... y is an even int
*/
yisint = 0;
if (hx < 0) {
if (iy >= 0x43400000)
yisint = 2; /* even integer y */
else if (iy >= 0x3ff00000) {
k = (iy >> 20) - 0x3ff; /* exponent */
if (k > 20) {
j = ly >> (52 - k);
if ((j << (52 - k)) == ly)
yisint = 2 - (j & 1);
} else if (ly == 0) {
j = iy >> (20 - k);
if ((j << (20 - k)) == iy)
yisint = 2 - (j & 1);
}
}
}
/* special value of y */
if (ly == 0) {
if (iy == 0x7ff00000) { /* y is +-inf */
if (((ix - 0x3ff00000) | lx) == 0)
return y - y; /* inf**+-1 is NaN */
else if (ix >= 0x3ff00000) /* (|x|>1)**+-inf = inf,0 */
return (hy >= 0) ? y : zero;
else /* (|x|<1)**-,+inf = inf,0 */
return (hy < 0) ? -y : zero;
}
if (iy == 0x3ff00000) { /* y is +-1 */
if (hy < 0)
return one / x;
else
return x;
}
if (hy == 0x40000000)
return x * x; /* y is 2 */
if (hy == 0x3fe00000) { /* y is 0.5 */
if (hx >= 0) /* x >= +0 */
return __ieee754_sqrt(x);
}
}
ax = fabs(x);
/* special value of x */
if (lx == 0) {
if (ix == 0x7ff00000 || ix == 0 || ix == 0x3ff00000) {
z = ax; /* x is +-0,+-inf,+-1 */
if (hy < 0)
z = one / z; /* z = (1/|x|) */
if (hx < 0) {
if (((ix - 0x3ff00000) | yisint) == 0) {
z = (z - z) / (z - z); /* (-1)**non-int is NaN */
} else if (yisint == 1)
z = -z; /* (x<0)**odd = -(|x|**odd) */
}
return z;
}
}
/* (x<0)**(non-int) is NaN */
if (((((u_int32_t) hx >> 31) - 1) | yisint) == 0)
return (x - x) / (x - x);
/* |y| is huge */
if (iy > 0x41e00000) { /* if |y| > 2**31 */
if (iy > 0x43f00000) { /* if |y| > 2**64, must o/uflow */
if (ix <= 0x3fefffff)
return (hy < 0) ? huge_val * huge_val : tiny * tiny;
if (ix >= 0x3ff00000)
return (hy > 0) ? huge_val * huge_val : tiny * tiny;
}
/* over/underflow if x is not close to one */
if (ix < 0x3fefffff)
return (hy < 0) ? huge_val * huge_val : tiny * tiny;
if (ix > 0x3ff00000)
return (hy > 0) ? huge_val * huge_val : tiny * tiny;
/* now |1-x| is tiny <= 2**-20, suffice to compute
log(x) by x-x^2/2+x^3/3-x^4/4 */
t = x - 1; /* t has 20 trailing zeros */
w = (t * t) * (0.5 - t * (0.3333333333333333333333 - t * 0.25));
u = ivln2_h * t; /* ivln2_h has 21 sig. bits */
v = t * ivln2_l - w * ivln2;
t1 = u + v;
SET_LOW_WORD(t1, 0);
t2 = v - (t1 - u);
} else {
double s2, s_h, s_l, t_h, t_l;
n = 0;
/* take care subnormal number */
if (ix < 0x00100000) {
ax *= two53;
n -= 53;
GET_HIGH_WORD(ix, ax);
}
n += ((ix) >> 20) - 0x3ff;
j = ix & 0x000fffff;
/* determine interval */
ix = j | 0x3ff00000; /* normalize ix */
if (j <= 0x3988E)
k = 0; /* |x|<sqrt(3/2) */
else if (j < 0xBB67A)
k = 1; /* |x|<sqrt(3) */
else {
k = 0;
n += 1;
ix -= 0x00100000;
}
SET_HIGH_WORD(ax, ix);
/* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
u = ax - bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
v = one / (ax + bp[k]);
s = u * v;
s_h = s;
SET_LOW_WORD(s_h, 0);
/* t_h=ax+bp[k] High */
t_h = zero;
SET_HIGH_WORD(t_h,
((ix >> 1) | 0x20000000) + 0x00080000 + (k << 18));
t_l = ax - (t_h - bp[k]);
s_l = v * ((u - s_h * t_h) - s_h * t_l);
/* compute log(ax) */
s2 = s * s;
r = s2 * s2 * (L1 +
s2 * (L2 +
s2 * (L3 +
s2 * (L4 + s2 * (L5 + s2 * L6)))));
r += s_l * (s_h + s);
s2 = s_h * s_h;
t_h = 3.0 + s2 + r;
SET_LOW_WORD(t_h, 0);
t_l = r - ((t_h - 3.0) - s2);
/* u+v = s*(1+...) */
u = s_h * t_h;
v = s_l * t_h + t_l * s;
/* 2/(3log2)*(s+...) */
p_h = u + v;
SET_LOW_WORD(p_h, 0);
p_l = v - (p_h - u);
z_h = cp_h * p_h; /* cp_h+cp_l = 2/(3*log2) */
z_l = cp_l * p_h + p_l * cp + dp_l[k];
/* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
t = (double) n;
t1 = (((z_h + z_l) + dp_h[k]) + t);
SET_LOW_WORD(t1, 0);
t2 = z_l - (((t1 - t) - dp_h[k]) - z_h);
}
s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
if (((((u_int32_t) hx >> 31) - 1) | (yisint - 1)) == 0)
s = -one; /* (-ve)**(odd int) */
/* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
y1 = y;
SET_LOW_WORD(y1, 0);
p_l = (y - y1) * t1 + y * t2;
p_h = y1 * t1;
z = p_l + p_h;
EXTRACT_WORDS(j, i, z);
if (j >= 0x40900000) { /* z >= 1024 */
if (((j - 0x40900000) | i) != 0) /* if z > 1024 */
return s * huge_val * huge_val; /* overflow */
else {
if (p_l + ovt > z - p_h)
return s * huge_val * huge_val; /* overflow */
}
} else if ((j & 0x7fffffff) >= 0x4090cc00) { /* z <= -1075 */
if (((j - 0xc090cc00) | i) != 0) /* z < -1075 */
return s * tiny * tiny; /* underflow */
else {
if (p_l <= z - p_h)
return s * tiny * tiny; /* underflow */
}
}
/*
* compute 2**(p_h+p_l)
*/
i = j & 0x7fffffff;
k = (i >> 20) - 0x3ff;
n = 0;
if (i > 0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */
n = j + (0x00100000 >> (k + 1));
k = ((n & 0x7fffffff) >> 20) - 0x3ff; /* new k for n */
t = zero;
SET_HIGH_WORD(t, n & ~(0x000fffff >> k));
n = ((n & 0x000fffff) | 0x00100000) >> (20 - k);
if (j < 0)
n = -n;
p_h -= t;
}
t = p_l + p_h;
SET_LOW_WORD(t, 0);
u = t * lg2_h;
v = (p_l - (t - p_h)) * lg2 + t * lg2_l;
z = u + v;
w = v - (z - u);
t = z * z;
t1 = z - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5))));
r = (z * t1) / (t1 - two) - (w + z * w);
z = one - (r - z);
GET_HIGH_WORD(j, z);
j += (n << 20);
if ((j >> 20) <= 0)
z = scalbn(z, n); /* subnormal output */
else
SET_HIGH_WORD(z, j);
return s * z;
}