| /* @(#)e_log.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. |
| * ==================================================== |
| */ |
| |
| #if defined(LIBM_SCCS) && !defined(lint) |
| static char rcsid[] = "$NetBSD: e_log.c,v 1.8 1995/05/10 20:45:49 jtc Exp $"; |
| #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" |
| |
| #ifdef __STDC__ |
| static const double |
| #else |
| static double |
| #endif |
| ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */ |
| ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */ |
| 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 */ |
| |
| #ifdef __STDC__ |
| double __ieee754_log(double x) |
| #else |
| double __ieee754_log(x) |
| double x; |
| #endif |
| { |
| 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) 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); |
| } |
| } |
