/** @file | |
Compute the 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_log.c 5.1 93/09/24 | |
NetBSD: e_log.c,v 1.12 2002/05/26 22:01:51 wiz Exp | |
**/ | |
#include <LibConfig.h> | |
#include <sys/EfiCdefs.h> | |
#if defined(_MSC_VER) /* Handle Microsoft VC++ compiler specifics. */ | |
// potential divide by 0 -- near line 118, (x-x)/zero is on purpose | |
#pragma warning ( disable : 4723 ) | |
#endif | |
/* __ieee754_log(x) | |
* Return the logrithm of x | |
* | |
* Method : | |
* 1. Argument Reduction: find k and f such that | |
* x = 2^k * (1+f), | |
* where sqrt(2)/2 < 1+f < sqrt(2) . | |
* | |
* 2. Approximation of log(1+f). | |
* Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) | |
* = 2s + 2/3 s**3 + 2/5 s**5 + ....., | |
* = 2s + s*R | |
* We use a special Reme algorithm on [0,0.1716] to generate | |
* a polynomial of degree 14 to approximate R The maximum error | |
* of this polynomial approximation is bounded by 2**-58.45. In | |
* other words, | |
* 2 4 6 8 10 12 14 | |
* R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s | |
* (the values of Lg1 to Lg7 are listed in the program) | |
* and | |
* | 2 14 | -58.45 | |
* | Lg1*s +...+Lg7*s - R(z) | <= 2 | |
* | | | |
* Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2. | |
* In order to guarantee error in log below 1ulp, we compute log | |
* by | |
* log(1+f) = f - s*(f - R) (if f is not too large) | |
* log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy) | |
* | |
* 3. Finally, log(x) = k*ln2 + log(1+f). | |
* = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo))) | |
* Here ln2 is split into two floating point number: | |
* ln2_hi + ln2_lo, | |
* where n*ln2_hi is always exact for |n| < 2000. | |
* | |
* Special cases: | |
* log(x) is NaN with signal if x < 0 (including -INF) ; | |
* log(+INF) is +INF; log(0) is -INF with signal; | |
* log(NaN) is that NaN with no signal. | |
* | |
* Accuracy: | |
* according to an error analysis, the error is always less than | |
* 1 ulp (unit in the last place). | |
* | |
* 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 | |
ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */ | |
ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */ | |
two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */ | |
Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */ | |
Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ | |
Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */ | |
Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ | |
Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ | |
Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ | |
Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ | |
static const double zero = 0.0; | |
double | |
__ieee754_log(double x) | |
{ | |
double hfsq,f,s,z,R,w,t1,t2,dk; | |
int32_t k,hx,i,j; | |
u_int32_t lx; | |
EXTRACT_WORDS(hx,lx,x); | |
k=0; | |
if (hx < 0x00100000) { /* x < 2**-1022 */ | |
if (((hx&0x7fffffff)|lx)==0) | |
return -two54/zero; /* log(+-0)=-inf */ | |
if (hx<0) { | |
errno = EDOM; | |
return (x-x)/zero; /* log(-#) = NaN */ | |
} | |
k -= 54; x *= two54; /* subnormal number, scale up x */ | |
GET_HIGH_WORD(hx,x); | |
} | |
if (hx >= 0x7ff00000) return x+x; | |
k += (hx>>20)-1023; | |
hx &= 0x000fffff; | |
i = (hx+0x95f64)&0x100000; | |
SET_HIGH_WORD(x,hx|(i^0x3ff00000)); /* normalize x or x/2 */ | |
k += (i>>20); | |
f = x-1.0; | |
if((0x000fffff&(2+hx))<3) { /* |f| < 2**-20 */ | |
if(f==zero) { if(k==0) return zero; else {dk=(double)k; | |
return dk*ln2_hi+dk*ln2_lo;} | |
} | |
R = f*f*(0.5-0.33333333333333333*f); | |
if(k==0) return f-R; else {dk=(double)k; | |
return dk*ln2_hi-((R-dk*ln2_lo)-f);} | |
} | |
s = f/(2.0+f); | |
dk = (double)k; | |
z = s*s; | |
i = hx-0x6147a; | |
w = z*z; | |
j = 0x6b851-hx; | |
t1= w*(Lg2+w*(Lg4+w*Lg6)); | |
t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7))); | |
i |= j; | |
R = t2+t1; | |
if(i>0) { | |
hfsq=0.5*f*f; | |
if(k==0) return f-(hfsq-s*(hfsq+R)); else | |
return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f); | |
} else { | |
if(k==0) return f-s*(f-R); else | |
return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f); | |
} | |
} |