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Diffstat (limited to 'sysdeps/ieee754/dbl-64/e_pow.c')
| -rw-r--r-- | sysdeps/ieee754/dbl-64/e_pow.c | 352 |
1 files changed, 352 insertions, 0 deletions
diff --git a/sysdeps/ieee754/dbl-64/e_pow.c b/sysdeps/ieee754/dbl-64/e_pow.c new file mode 100644 index 0000000000..1e1496f00d --- /dev/null +++ b/sysdeps/ieee754/dbl-64/e_pow.c @@ -0,0 +1,352 @@ +/* @(#)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. + * ==================================================== + */ +/* Modified by Naohiko Shimizu/Tokai University, Japan 1997/08/25, + for performance improvement on pipelined processors. +*/ + +#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.h" +#include "math_private.h" +#define zero C[0] +#define one C[1] +#define two C[2] +#define two53 C[3] +#define huge C[4] +#define tiny C[5] +#define L1 C[6] +#define L2 C[7] +#define L3 C[8] +#define L4 C[9] +#define L5 C[10] +#define L6 C[11] +#define P1 C[12] +#define P2 C[13] +#define P3 C[14] +#define P4 C[15] +#define P5 C[16] +#define lg2 C[17] +#define lg2_h C[18] +#define lg2_l C[19] +#define ovt C[20] +#define cp C[21] +#define cp_h C[22] +#define cp_l C[23] +#define ivln2 C[24] +#define ivln2_h C[25] +#define ivln2_l C[26] + +#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 */ +C[] = { +0.0, +1.0, +2.0, +9007199254740992.0 , +1.0e300, +1.0e-300, +5.99999999999994648725e-01 , +4.28571428578550184252e-01 , +3.33333329818377432918e-01 , +2.72728123808534006489e-01 , +2.30660745775561754067e-01 , +2.06975017800338417784e-01 , +1.66666666666666019037e-01 , +-2.77777777770155933842e-03 , +6.61375632143793436117e-05 , +-1.65339022054652515390e-06 , +4.13813679705723846039e-08 , +6.93147180559945286227e-01 , +6.93147182464599609375e-01 , +-1.90465429995776804525e-09 , +8.0085662595372944372e-0017 , +9.61796693925975554329e-01 , +9.61796700954437255859e-01 , +-7.02846165095275826516e-09 , +1.44269504088896338700e+00 , +1.44269502162933349609e+00 , +1.92596299112661746887e-08 }; + +#ifdef __STDC__ + double __ieee754_pow(double x, double y) +#else + double __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, t12,t14,r_1,r_2,r_3; + 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 C[1]; + + /* +-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((u_int32_t)(j<<(52-k))==ly) yisint = 2-(j&1); + } else if(ly==0) { + j = iy>>(20-k); + if((int32_t)(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: C[0]; + else /* (|x|<1)**-,+inf = inf,0 */ + return (hy<0)?-y: C[0]; + } + if(iy==0x3ff00000) { /* y is +-1 */ + if(hy<0) return C[1]/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 = C[1]/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)? C[4]*C[4]:C[5]*C[5]; + if(ix>=0x3ff00000) return (hy>0)? C[4]*C[4]:C[5]*C[5]; + } + /* over/underflow if x is not close to one */ + if(ix<0x3fefffff) return (hy<0)? C[4]*C[4]:C[5]*C[5]; + if(ix>0x3ff00000) return (hy>0)? C[4]*C[4]:C[5]*C[5]; + /* 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 = C[25]*t; /* ivln2_h has 21 sig. bits */ + v = t*C[26]-w*C[24]; + t1 = u+v; + SET_LOW_WORD(t1,0); + t2 = v-(t1-u); + } else { + double s2,s_h,s_l,t_h,t_l,s22,s24,s26,r1,r2,r3; + n = 0; + /* take care subnormal number */ + if(ix<0x00100000) + {ax *= C[3]; 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|<sqrt(3/2) */ + else if(j<0xBB67A) k=1; /* |x|<sqrt(3) */ + else {k=0;n+=1;ix -= 0x00100000;} + SET_HIGH_WORD(ax,ix); + + /* compute s = 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 = C[1]/(ax+bp[k]); + s = u*v; + s_h = s; + SET_LOW_WORD(s_h,0); + /* t_h=ax+bp[k] High */ + t_h = C[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 = s*s; +#ifdef DO_NOT_USE_THIS + r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6))))); +#else + r1 = C[10]+s2*C[11]; s22=s2*s2; + r2 = C[8]+s2*C[9]; s24=s22*s22; + r3 = C[6]+s2*C[7]; s26=s24*s22; + r = r3*s22 + r2*s24 + r1*s26; +#endif + 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 = C[22]*p_h; /* cp_h+cp_l = 2/(3*log2) */ + z_l = C[23]*p_h+p_l*C[21]+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 = C[1]; /* s (sign of result -ve**odd) = -1 else = 1 */ + if(((((u_int32_t)hx>>31)-1)|(yisint-1))==0) + s = -C[1];/* (-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*C[4]*C[4]; /* overflow */ + else { + if(p_l+C[20]>z-p_h) return s*C[4]*C[4]; /* overflow */ + } + } else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */ + if(((j-0xc090cc00)|i)!=0) /* z < -1075 */ + return s*C[5]*C[5]; /* underflow */ + else { + if(p_l<=z-p_h) return s*C[5]*C[5]; /* 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 = C[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*C[18]; + v = (p_l-(t-p_h))*C[17]+t*C[19]; + z = u+v; + w = v-(z-u); + t = z*z; +#ifdef DO_NOT_USE_THIS + t1 = z - t*(C[12]+t*(C[13]+t*(C[14]+t*(C[15]+t*C[16])))); +#else + r_1 = C[15]+t*C[16]; t12 = t*t; + r_2 = C[13]+t*C[14]; t14 = t12*t12; + r_3 = t*C[12]; + t1 = z - r_3 - t12*r_2 - t14*r_1; +#endif + r = (z*t1)/(t1-C[2])-(w+z*w); + z = C[1]-(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; +} |