e_pow.c
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上传日期：2007-01-17
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- /* @(#)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.
- * ====================================================
- */
- /* __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 "fdlibm.h"
- #ifndef _DOUBLE_IS_32BITS
- #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 */
- zero = 0.0,
- one = 1.0,
- two = 2.0,
- 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-0017, /* -(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*/
- #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;
- __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;
- /* y==zero: x**0 = 1 */
- if((iy|ly)==0) return one;
- /* +-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((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)
- return y - y; /* inf**+-1 is NaN */
- else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */
- return (hy>=0)? y: zero;
- else /* (|x|<1)**-,+inf = inf,0 */
- return (hy<0)?-y: zero;
- }
- if(iy==0x3ff00000) { /* y is +-1 */
- if(hy<0) return one/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 = one/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 */
- /* CYGNUS LOCAL: This used to be
- if((((hx>>31)+1)|yisint)==0) return (x-x)/(x-x);
- but ANSI C says a right shift of a signed negative quantity is
- implementation defined. */
- if(((((__uint32_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)? 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)? huge*huge:tiny*tiny;
- if(ix>0x3ff00000) return (hy>0)? huge*huge: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 = x-1; /* 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 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) 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 = one/(ax+bp[k]);
- s = u*v;
- s_h = s;
- SET_LOW_WORD(s_h,0);
- /* t_h=ax+bp[k] High */
- t_h = zero;
- 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;
- r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
- 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 = 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) = (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 = one; /* s (sign of result -ve**odd) = -1 else = 1 */
- if(((((__uint32_t)hx>>31)-1)|(yisint-1))==0)
- s = -one;/* (-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*huge*huge; /* overflow */
- else {
- if(p_l+ovt>z-p_h) return s*huge*huge; /* overflow */
- }
- } else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */
- if(((j-0xc090cc00)|i)!=0) /* z < -1075 */
- return s*tiny*tiny; /* underflow */
- else {
- 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 = zero;
- 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-two)-(w+z*w);
- z = one-(r-z);
- GET_HIGH_WORD(j,z);
- j += (n<<20);
- if((j>>20)<=0) z = scalbn(z,(int)n); /* subnormal output */
- else SET_HIGH_WORD(z,j);
- return s*z;
- }
- #endif /* defined(_DOUBLE_IS_32BITS) */
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