/* @(#)e_log.c 1.3 95/01/18 */ /* * ==================================================== * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * * Developed at SunSoft, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== */ /* 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 #include "math_private.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 log(double x) { double hfsq,f,s,z,R,w,t1,t2,dk; int32_t k,hx,i,j; uint32_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); } }