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 /* @(#)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|> 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; }