e_pow.c revision 141296
10SN/A/* @(#)e_pow.c 1.5 04/04/22 SMI */ 217247Swetmore/* 30SN/A * ==================================================== 40SN/A * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved. 50SN/A * 60SN/A * Permission to use, copy, modify, and distribute this 72362SN/A * software is freely granted, provided that this notice 80SN/A * is preserved. 92362SN/A * ==================================================== 100SN/A */ 110SN/A 120SN/A#ifndef lint 130SN/Astatic char rcsid[] = "$FreeBSD: head/lib/msun/src/e_pow.c 141296 2005-02-04 18:26:06Z das $"; 140SN/A#endif 150SN/A 160SN/A/* __ieee754_pow(x,y) return x**y 170SN/A * 180SN/A * n 190SN/A * Method: Let x = 2 * (1+f) 200SN/A * 1. Compute and return log2(x) in two pieces: 212362SN/A * log2(x) = w1 + w2, 222362SN/A * where w1 has 53-24 = 29 bit trailing zeros. 232362SN/A * 2. Perform y*log2(x) = n+y' by simulating muti-precision 240SN/A * arithmetic, where |y'|<=0.5. 250SN/A * 3. Return x**y = 2**n*exp(y'*log2) 260SN/A * 270SN/A * Special cases: 280SN/A * 1. (anything) ** 0 is 1 290SN/A * 2. (anything) ** 1 is itself 3015973Swetmore * 3. (anything) ** NAN is NAN 310SN/A * 4. NAN ** (anything except 0) is NAN 320SN/A * 5. +-(|x| > 1) ** +INF is +INF 330SN/A * 6. +-(|x| > 1) ** -INF is +0 340SN/A * 7. +-(|x| < 1) ** +INF is +0 350SN/A * 8. +-(|x| < 1) ** -INF is +INF 360SN/A * 9. +-1 ** +-INF is NAN 370SN/A * 10. +0 ** (+anything except 0, NAN) is +0 380SN/A * 11. -0 ** (+anything except 0, NAN, odd integer) is +0 390SN/A * 12. +0 ** (-anything except 0, NAN) is +INF 400SN/A * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF 410SN/A * 14. -0 ** (odd integer) = -( +0 ** (odd integer) ) 420SN/A * 15. +INF ** (+anything except 0,NAN) is +INF 430SN/A * 16. +INF ** (-anything except 0,NAN) is +0 440SN/A * 17. -INF ** (anything) = -0 ** (-anything) 450SN/A * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer) 460SN/A * 19. (-anything except 0 and inf) ** (non-integer) is NAN 470SN/A * 480SN/A * Accuracy: 490SN/A * pow(x,y) returns x**y nearly rounded. In particular 500SN/A * pow(integer,integer) 510SN/A * always returns the correct integer provided it is 520SN/A * representable. 530SN/A * 540SN/A * Constants : 550SN/A * The hexadecimal values are the intended ones for the following 560SN/A * constants. The decimal values may be used, provided that the 576241SN/A * compiler will convert from decimal to binary accurately enough 586241SN/A * to produce the hexadecimal values shown. 590SN/A */ 606241SN/A 616241SN/A#include "math.h" 626241SN/A#include "math_private.h" 636241SN/A 640SN/Astatic const double 6512745Smartinbp[] = {1.0, 1.5,}, 660SN/Adp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */ 6711809Sdarcydp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */ 686128SN/Azero = 0.0, 690SN/Aone = 1.0, 700SN/Atwo = 2.0, 710SN/Atwo53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */ 720SN/Ahuge = 1.0e300, 730SN/Atiny = 1.0e-300, 740SN/A /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */ 750SN/AL1 = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */ 760SN/AL2 = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */ 770SN/AL3 = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */ 780SN/AL4 = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */ 790SN/AL5 = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */ 800SN/AL6 = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */ 810SN/AP1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ 820SN/AP2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */ 830SN/AP3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ 840SN/AP4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ 850SN/AP5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */ 860SN/Alg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */ 870SN/Alg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */ 880SN/Alg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */ 890SN/Aovt = 8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */ 900SN/Acp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */ 910SN/Acp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */ 920SN/Acp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/ 930SN/Aivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */ 940SN/Aivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/ 950SN/Aivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/ 960SN/A 970SN/Adouble 980SN/A__ieee754_pow(double x, double y) 990SN/A{ 1000SN/A double z,ax,z_h,z_l,p_h,p_l; 1010SN/A double y1,t1,t2,r,s,t,u,v,w; 1020SN/A int32_t i,j,k,yisint,n; 1030SN/A int32_t hx,hy,ix,iy; 1040SN/A u_int32_t lx,ly; 1050SN/A 1060SN/A EXTRACT_WORDS(hx,lx,x); 1070SN/A EXTRACT_WORDS(hy,ly,y); 1080SN/A ix = hx&0x7fffffff; iy = hy&0x7fffffff; 1090SN/A 1100SN/A /* y==zero: x**0 = 1 */ 1110SN/A if((iy|ly)==0) return one; 1120SN/A 1130SN/A /* +-NaN return x+y */ 1140SN/A if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) || 1150SN/A iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0))) 1160SN/A return x+y; 1170SN/A 1180SN/A /* determine if y is an odd int when x < 0 1190SN/A * yisint = 0 ... y is not an integer 12012891Sascarpino * yisint = 1 ... y is an odd int 12112891Sascarpino * yisint = 2 ... y is an even int 12214161Sascarpino */ 12314161Sascarpino yisint = 0; 12412891Sascarpino if(hx<0) { 12512891Sascarpino if(iy>=0x43400000) yisint = 2; /* even integer y */ 12612891Sascarpino else if(iy>=0x3ff00000) { 12712891Sascarpino k = (iy>>20)-0x3ff; /* exponent */ 1280SN/A if(k>20) { 1290SN/A j = ly>>(52-k); 13017247Swetmore if((j<<(52-k))==ly) yisint = 2-(j&1); 13117247Swetmore } else if(ly==0) { 1320SN/A j = iy>>(20-k); 1330SN/A if((j<<(20-k))==iy) yisint = 2-(j&1); 13415973Swetmore } 1350SN/A } 13615973Swetmore } 13715973Swetmore 13815973Swetmore /* special value of y */ 13915973Swetmore if(ly==0) { 14015973Swetmore if (iy==0x7ff00000) { /* y is +-inf */ 1410SN/A if(((ix-0x3ff00000)|lx)==0) 1420SN/A return y - y; /* inf**+-1 is NaN */ 1430SN/A else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */ 1440SN/A return (hy>=0)? y: zero; 1450SN/A else /* (|x|<1)**-,+inf = inf,0 */ 14615973Swetmore return (hy<0)?-y: zero; 1470SN/A } 1480SN/A if(iy==0x3ff00000) { /* y is +-1 */ 1490SN/A if(hy<0) return one/x; else return x; 1500SN/A } 1510SN/A if(hy==0x40000000) return x*x; /* y is 2 */ 1520SN/A if(hy==0x3fe00000) { /* y is 0.5 */ 1530SN/A if(hx>=0) /* x >= +0 */ 1540SN/A return sqrt(x); 1550SN/A } 1560SN/A } 1570SN/A 1580SN/A ax = fabs(x); 1590SN/A /* special value of x */ 1600SN/A if(lx==0) { 1610SN/A if(ix==0x7ff00000||ix==0||ix==0x3ff00000){ 1620SN/A z = ax; /*x is +-0,+-inf,+-1*/ 1630SN/A if(hy<0) z = one/z; /* z = (1/|x|) */ 1640SN/A if(hx<0) { 1650SN/A if(((ix-0x3ff00000)|yisint)==0) { 1660SN/A z = (z-z)/(z-z); /* (-1)**non-int is NaN */ 1670SN/A } else if(yisint==1) 16817247Swetmore z = -z; /* (x<0)**odd = -(|x|**odd) */ 16917247Swetmore } 1700SN/A return z; 1710SN/A } 1720SN/A } 1730SN/A 17415973Swetmore /* CYGNUS LOCAL + fdlibm-5.3 fix: This used to be 1750SN/A n = (hx>>31)+1; 17615973Swetmore but ANSI C says a right shift of a signed negative quantity is 17715973Swetmore implementation defined. */ 17815973Swetmore n = ((u_int32_t)hx>>31)-1; 17915973Swetmore 18015973Swetmore /* (x<0)**(non-int) is NaN */ 18115973Swetmore if((n|yisint)==0) return (x-x)/(x-x); 1820SN/A 1830SN/A s = one; /* s (sign of result -ve**odd) = -1 else = 1 */ 18415973Swetmore if((n|(yisint-1))==0) s = -one;/* (-ve)**(odd int) */ 1850SN/A 18615973Swetmore /* |y| is huge */ 1870SN/A if(iy>0x41e00000) { /* if |y| > 2**31 */ 1880SN/A if(iy>0x43f00000){ /* if |y| > 2**64, must o/uflow */ 1890SN/A if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny; 1900SN/A if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny; 1910SN/A } 1920SN/A /* over/underflow if x is not close to one */ 19315973Swetmore if(ix<0x3fefffff) return (hy<0)? s*huge*huge:s*tiny*tiny; 1940SN/A if(ix>0x3ff00000) return (hy>0)? s*huge*huge:s*tiny*tiny; 1950SN/A /* now |1-x| is tiny <= 2**-20, suffice to compute 1960SN/A log(x) by x-x^2/2+x^3/3-x^4/4 */ 1970SN/A t = ax-one; /* t has 20 trailing zeros */ 1980SN/A w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25)); 1990SN/A u = ivln2_h*t; /* ivln2_h has 21 sig. bits */ 2000SN/A v = t*ivln2_l-w*ivln2; 2010SN/A t1 = u+v; 2020SN/A SET_LOW_WORD(t1,0); 2030SN/A t2 = v-(t1-u); 2040SN/A } else { 2050SN/A double ss,s2,s_h,s_l,t_h,t_l; 2060SN/A n = 0; 2070SN/A /* take care subnormal number */ 2080SN/A if(ix<0x00100000) 2090SN/A {ax *= two53; n -= 53; GET_HIGH_WORD(ix,ax); } 2100SN/A n += ((ix)>>20)-0x3ff; 2110SN/A j = ix&0x000fffff; 21217247Swetmore /* determine interval */ 21317247Swetmore ix = j|0x3ff00000; /* normalize ix */ 2140SN/A if(j<=0x3988E) k=0; /* |x|<sqrt(3/2) */ 2150SN/A else if(j<0xBB67A) k=1; /* |x|<sqrt(3) */ 2160SN/A else {k=0;n+=1;ix -= 0x00100000;} 2170SN/A SET_HIGH_WORD(ax,ix); 21815973Swetmore 21915973Swetmore /* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */ 22015973Swetmore u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */ 2210SN/A v = one/(ax+bp[k]); 22215973Swetmore ss = u*v; 22315973Swetmore s_h = ss; 22415973Swetmore SET_LOW_WORD(s_h,0); 2250SN/A /* t_h=ax+bp[k] High */ 22615973Swetmore t_h = zero; 2270SN/A SET_HIGH_WORD(t_h,((ix>>1)|0x20000000)+0x00080000+(k<<18)); 2280SN/A t_l = ax - (t_h-bp[k]); 2290SN/A s_l = v*((u-s_h*t_h)-s_h*t_l); 2300SN/A /* compute log(ax) */ 2310SN/A s2 = ss*ss; 23215973Swetmore r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6))))); 2330SN/A r += s_l*(s_h+ss); 2340SN/A s2 = s_h*s_h; 2350SN/A t_h = 3.0+s2+r; 2360SN/A SET_LOW_WORD(t_h,0); 2370SN/A t_l = r-((t_h-3.0)-s2); 2380SN/A /* u+v = ss*(1+...) */ 2390SN/A u = s_h*t_h; 2400SN/A v = s_l*t_h+t_l*ss; 2410SN/A /* 2/(3log2)*(ss+...) */ 2420SN/A p_h = u+v; 2430SN/A SET_LOW_WORD(p_h,0); 2440SN/A p_l = v-(p_h-u); 2450SN/A z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */ 2460SN/A z_l = cp_l*p_h+p_l*cp+dp_l[k]; 2470SN/A /* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */ 2480SN/A t = (double)n; 2490SN/A t1 = (((z_h+z_l)+dp_h[k])+t); 2500SN/A SET_LOW_WORD(t1,0); 2510SN/A t2 = z_l-(((t1-t)-dp_h[k])-z_h); 2520SN/A } 2530SN/A 2540SN/A /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */ 2550SN/A y1 = y; 2560SN/A SET_LOW_WORD(y1,0); 2570SN/A p_l = (y-y1)*t1+y*t2; 2580SN/A p_h = y1*t1; 2590SN/A z = p_l+p_h; 2600SN/A EXTRACT_WORDS(j,i,z); 2610SN/A if (j>=0x40900000) { /* z >= 1024 */ 2620SN/A if(((j-0x40900000)|i)!=0) /* if z > 1024 */ 2630SN/A return s*huge*huge; /* overflow */ 2640SN/A else { 2650SN/A if(p_l+ovt>z-p_h) return s*huge*huge; /* overflow */ 2660SN/A } 2670SN/A } else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */ 2680SN/A if(((j-0xc090cc00)|i)!=0) /* z < -1075 */ 2690SN/A return s*tiny*tiny; /* underflow */ 2700SN/A else { 2710SN/A if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */ 2720SN/A } 2730SN/A } 2740SN/A /* 2750SN/A * compute 2**(p_h+p_l) 2760SN/A */ 2770SN/A i = j&0x7fffffff; 2780SN/A k = (i>>20)-0x3ff; 2790SN/A n = 0; 2800SN/A if(i>0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */ 2810SN/A n = j+(0x00100000>>(k+1)); 2820SN/A k = ((n&0x7fffffff)>>20)-0x3ff; /* new k for n */ 2830SN/A t = zero; 2840SN/A SET_HIGH_WORD(t,n&~(0x000fffff>>k)); 2850SN/A n = ((n&0x000fffff)|0x00100000)>>(20-k); 2860SN/A if(j<0) n = -n; 2870SN/A p_h -= t; 2880SN/A } 2890SN/A t = p_l+p_h; 2900SN/A SET_LOW_WORD(t,0); 2910SN/A u = t*lg2_h; 2920SN/A v = (p_l-(t-p_h))*lg2+t*lg2_l; 2930SN/A z = u+v; 2940SN/A w = v-(z-u); 2950SN/A t = z*z; 2960SN/A t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); 2970SN/A r = (z*t1)/(t1-two)-(w+z*w); 2980SN/A z = one-(r-z); 2990SN/A GET_HIGH_WORD(j,z); 3000SN/A j += (n<<20); 3010SN/A if((j>>20)<=0) z = scalbn(z,n); /* subnormal output */ 3020SN/A else SET_HIGH_WORD(z,j); 3030SN/A return s*z; 3040SN/A} 3050SN/A