/* @(#)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;

}