/* 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 if ((ix|lx) != 0) /* (|x|<1)**+-inf = 0,inf if x!=0 */ 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; }