/** @file | |
Compute the base 10 logrithm of x. | |
Copyright (c) 2010 - 2011, Intel Corporation. All rights reserved.<BR> | |
This program and the accompanying materials are licensed and made available under | |
the terms and conditions of the BSD License that accompanies this distribution. | |
The full text of the license may be found at | |
http://opensource.org/licenses/bsd-license. | |
THE PROGRAM IS DISTRIBUTED UNDER THE BSD LICENSE ON AN "AS IS" BASIS, | |
WITHOUT WARRANTIES OR REPRESENTATIONS OF ANY KIND, EITHER EXPRESS OR IMPLIED. | |
* ==================================================== | |
* 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. | |
* ==================================================== | |
e_pow.c 5.1 93/09/24 | |
NetBSD: e_pow.c,v 1.13 2004/06/30 18:43:15 drochner Exp | |
**/ | |
#include <LibConfig.h> | |
#include <sys/EfiCdefs.h> | |
#if defined(_MSC_VER) /* Handle Microsoft VC++ compiler specifics. */ | |
// C4723: potential divide by zero. | |
#pragma warning ( disable : 4723 ) | |
// C4756: overflow in constant arithmetic | |
#pragma warning ( disable : 4756 ) | |
#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 multi-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" | |
#include <errno.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 */ | |
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*/ | |
double | |
__ieee754_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; | |
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 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((u_int32_t)(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; | |
} | |
} | |
n = (hx>>31)+1; | |
/* (x<0)**(non-int) is NaN */ | |
if((n|yisint)==0) { | |
errno = EDOM; | |
return (x-x)/(x-x); | |
} | |
s = one; /* s (sign of result -ve**odd) = -1 else = 1 */ | |
if((n|(yisint-1))==0) s = -one;/* (-ve)**(odd int) */ | |
/* |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-one; /* 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) 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 ss = 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]); | |
ss = u*v; | |
s_h = ss; | |
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 = 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 */ | |
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,n); /* subnormal output */ | |
else SET_HIGH_WORD(z,j); | |
return s*z; | |
} |