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 ```/* 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) { ``` ``` uint32_t j2 = ly>>(52-k); ``` ``` if ((j2<<(52-k)) == ly) ``` ``` yisint = 2 - (j2&1); ``` ``` } else if (ly == 0) { ``` ``` uint32_t j2 = iy>>(20-k); ``` ``` if ((j2<<(20-k)) == (uint32_t)iy) ``` ``` yisint = 2 - (j2&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|>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; ``` ```} ```