| /* origin: FreeBSD /usr/src/lib/msun/src/e_pow.c */ |
| /* |
| * ==================================================== |
| * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved. |
| * |
| * Permission to use, copy, modify, and distribute this |
| * software is freely granted, provided that this notice |
| * is preserved. |
| * ==================================================== |
| */ |
| /* 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. 1 ** (anything) is 1 |
| * 3. (anything except 1) ** 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 1 |
| * 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, raise |
| * divbyzero |
| * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF, raise |
| * divbyzero |
| * 14. -0 ** (+odd integer) is -0 |
| * 15. -0 ** (-odd integer) is -INF, raise divbyzero |
| * 16. +INF ** (+anything except 0,NAN) is +INF |
| * 17. +INF ** (-anything except 0,NAN) is +0 |
| * 18. -INF ** (+odd integer) is -INF |
| * 19. -INF ** (anything) = -0 ** (-anything), (anything except odd |
| * integer) |
| * 20. (anything) ** 1 is (anything) |
| * 21. (anything) ** -1 is 1/(anything) |
| * 22. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer) |
| * 23. (-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 "libm.h" |
| |
| static const double bp[] = |
| { |
| 1.0, 1.5, |
| }, |
| dp_h[] = |
| { |
| 0.0, 5.84962487220764160156e-01, |
| }, /* 0x3FE2B803, 0x40000000 */ |
| dp_l[] = |
| { |
| 0.0, 1.35003920212974897128e-08, |
| }, /* 0x3E4CFDEB, 0x43CFD006 */ |
| two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */ |
| huge = 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-017, /* -(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*/ |
| |
| double pow(double x, double y) { |
| 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; |
| uint32_t lx, ly; |
| |
| EXTRACT_WORDS(hx, lx, x); |
| EXTRACT_WORDS(hy, ly, y); |
| ix = hx & 0x7fffffff; |
| iy = hy & 0x7fffffff; |
| |
| /* x**0 = 1, even if x is NaN */ |
| if ((iy | ly) == 0) |
| return 1.0; |
| /* 1**y = 1, even if y is NaN */ |
| if (hx == 0x3ff00000 && lx == 0) |
| return 1.0; |
| /* NaN if either arg is NaN */ |
| 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) /* (-1)**+-inf is 1 */ |
| return 1.0; |
| else if (ix >= 0x3ff00000) /* (|x|>1)**+-inf = inf,0 */ |
| return hy >= 0 ? y : 0.0; |
| else /* (|x|<1)**+-inf = 0,inf */ |
| return hy >= 0 ? 0.0 : -y; |
| } |
| if (iy == 0x3ff00000) { /* y is +-1 */ |
| if (hy >= 0) |
| return x; |
| y = 1 / x; |
| #if FLT_EVAL_METHOD != 0 |
| { |
| union { |
| double f; |
| uint64_t i; |
| } u = {y}; |
| uint64_t i = u.i & -1ULL / 2; |
| if (i >> 52 == 0 && (i & (i - 1))) |
| FORCE_EVAL((float)y); |
| } |
| #endif |
| return y; |
| } |
| if (hy == 0x40000000) /* y is 2 */ |
| return x * x; |
| if (hy == 0x3fe00000) { /* y is 0.5 */ |
| if (hx >= 0) /* x >= +0 */ |
| return sqrt(x); |
| } |
| } |
| |
| ax = fabs(x); |
| /* special value of x */ |
| if (lx == 0) { |
| if (ix == 0x7ff00000 || ix == 0 || |
| ix == 0x3ff00000) { /* x is +-0,+-inf,+-1 */ |
| z = ax; |
| if (hy < 0) /* z = (1/|x|) */ |
| z = 1.0 / z; |
| 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; |
| } |
| } |
| |
| s = 1.0; /* sign of result */ |
| if (hx < 0) { |
| if (yisint == 0) /* (x<0)**(non-int) is NaN */ |
| return (x - x) / (x - x); |
| if (yisint == 1) /* (x<0)**(odd int) */ |
| s = -1.0; |
| } |
| |
| /* |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 * huge : tiny * tiny; |
| if (ix >= 0x3ff00000) |
| return hy > 0 ? huge * huge : tiny * tiny; |
| } |
| /* over/underflow if x is not close to one */ |
| if (ix < 0x3fefffff) |
| return hy < 0 ? s * huge * huge : s * tiny * tiny; |
| if (ix > 0x3ff00000) |
| return hy > 0 ? s * huge * huge : s * 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 = ax - 1.0; /* 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 ss, 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) /* |x|<sqrt(3/2) */ |
| k = 0; |
| else if (j < 0xBB67A) /* |x|<sqrt(3) */ |
| k = 1; |
| else { |
| k = 0; |
| n += 1; |
| ix -= 0x00100000; |
| } |
| SET_HIGH_WORD(ax, ix); |
| |
| /* compute ss = 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 = 1.0 / (ax + bp[k]); |
| ss = u * v; |
| s_h = ss; |
| SET_LOW_WORD(s_h, 0); |
| /* t_h=ax+bp[k] High */ |
| t_h = 0.0; |
| 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 = ss * ss; |
| r = s2 * s2 * |
| (L1 + s2 * (L2 + s2 * (L3 + s2 * (L4 + s2 * (L5 + s2 * L6))))); |
| r += s_l * (s_h + ss); |
| 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 = ss*(1+...) */ |
| u = s_h * t_h; |
| v = s_l * t_h + t_l * ss; |
| /* 2/(3log2)*(ss+...) */ |
| 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) = (ss+..)*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); |
| } |
| |
| /* 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 * huge; /* overflow */ |
| if (p_l + ovt > z - p_h) |
| return s * huge * huge; /* overflow */ |
| } else if ((j & 0x7fffffff) >= 0x4090cc00) { |
| /* z <= -1075 */ // FIXME: instead of abs(j) use unsigned j |
| if (((j - 0xc090cc00) | i) != 0) /* z < -1075 */ |
| return s * tiny * tiny; /* underflow */ |
| 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 = 0.0; |
| 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 - 2.0) - (w + z * w); |
| z = 1.0 - (r - z); |
| GET_HIGH_WORD(j, z); |
| j += n << 20; |
| if ((j >> 20) <= 0) /* subnormal output */ |
| z = scalbn(z, n); |
| else |
| SET_HIGH_WORD(z, j); |
| return s * z; |
| } |
