1141296Sdas 2141296Sdas/* @(#)e_log10.c 1.3 95/01/18 */ 32116Sjkh/* 42116Sjkh * ==================================================== 52116Sjkh * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 62116Sjkh * 7141296Sdas * Developed at SunSoft, a Sun Microsystems, Inc. business. 82116Sjkh * Permission to use, copy, modify, and distribute this 9141296Sdas * software is freely granted, provided that this notice 102116Sjkh * is preserved. 112116Sjkh * ==================================================== 122116Sjkh */ 132116Sjkh 14176451Sdas#include <sys/cdefs.h> 15176451Sdas__FBSDID("$FreeBSD$"); 162116Sjkh 17216247Sdas/* 18226375Sdas * Return the base 2 logarithm of x. See e_log.c and k_log.h for most 19226375Sdas * comments. 20226376Sdas * 21226376Sdas * This reduces x to {k, 1+f} exactly as in e_log.c, then calls the kernel, 22226376Sdas * then does the combining and scaling steps 23226376Sdas * log2(x) = (f - 0.5*f*f + k_log1p(f)) / ln2 + k 24226376Sdas * in not-quite-routine extra precision. 252116Sjkh */ 262116Sjkh 27251404Sdas#include <float.h> 28251404Sdas 292116Sjkh#include "math.h" 302116Sjkh#include "math_private.h" 31216211Sdas#include "k_log.h" 322116Sjkh 332116Sjkhstatic const double 342116Sjkhtwo54 = 1.80143985094819840000e+16, /* 0x43500000, 0x00000000 */ 35216247Sdasivln2hi = 1.44269504072144627571e+00, /* 0x3ff71547, 0x65200000 */ 36216247Sdasivln2lo = 1.67517131648865118353e-10; /* 0x3de705fc, 0x2eefa200 */ 372116Sjkh 382116Sjkhstatic const double zero = 0.0; 39251024Sdasstatic volatile double vzero = 0.0; 402116Sjkh 4197413Salfreddouble 42216211Sdas__ieee754_log2(double x) 432116Sjkh{ 44226376Sdas double f,hfsq,hi,lo,r,val_hi,val_lo,w,y; 452116Sjkh int32_t i,k,hx; 462116Sjkh u_int32_t lx; 472116Sjkh 482116Sjkh EXTRACT_WORDS(hx,lx,x); 492116Sjkh 50226375Sdas k=0; 51226375Sdas if (hx < 0x00100000) { /* x < 2**-1022 */ 52226375Sdas if (((hx&0x7fffffff)|lx)==0) 53251024Sdas return -two54/vzero; /* log(+-0)=-inf */ 54226375Sdas if (hx<0) return (x-x)/zero; /* log(-#) = NaN */ 55226375Sdas k -= 54; x *= two54; /* subnormal number, scale up x */ 562116Sjkh GET_HIGH_WORD(hx,x); 57226375Sdas } 582116Sjkh if (hx >= 0x7ff00000) return x+x; 59226376Sdas if (hx == 0x3ff00000 && lx == 0) 60226376Sdas return zero; /* log(1) = +0 */ 612116Sjkh k += (hx>>20)-1023; 62216211Sdas hx &= 0x000fffff; 63216211Sdas i = (hx+0x95f64)&0x100000; 64216211Sdas SET_HIGH_WORD(x,hx|(i^0x3ff00000)); /* normalize x or x/2 */ 65216211Sdas k += (i>>20); 66226376Sdas y = (double)k; 67226376Sdas f = x - 1.0; 68226376Sdas hfsq = 0.5*f*f; 69226376Sdas r = k_log1p(f); 70226376Sdas 71226376Sdas /* 72226376Sdas * f-hfsq must (for args near 1) be evaluated in extra precision 73226376Sdas * to avoid a large cancellation when x is near sqrt(2) or 1/sqrt(2). 74226376Sdas * This is fairly efficient since f-hfsq only depends on f, so can 75226376Sdas * be evaluated in parallel with R. Not combining hfsq with R also 76226376Sdas * keeps R small (though not as small as a true `lo' term would be), 77226376Sdas * so that extra precision is not needed for terms involving R. 78226376Sdas * 79226376Sdas * Compiler bugs involving extra precision used to break Dekker's 80226376Sdas * theorem for spitting f-hfsq as hi+lo, unless double_t was used 81226376Sdas * or the multi-precision calculations were avoided when double_t 82226376Sdas * has extra precision. These problems are now automatically 83226376Sdas * avoided as a side effect of the optimization of combining the 84226376Sdas * Dekker splitting step with the clear-low-bits step. 85226376Sdas * 86226376Sdas * y must (for args near sqrt(2) and 1/sqrt(2)) be added in extra 87226376Sdas * precision to avoid a very large cancellation when x is very near 88226376Sdas * these values. Unlike the above cancellations, this problem is 89226376Sdas * specific to base 2. It is strange that adding +-1 is so much 90226376Sdas * harder than adding +-ln2 or +-log10_2. 91226376Sdas * 92226376Sdas * This uses Dekker's theorem to normalize y+val_hi, so the 93226376Sdas * compiler bugs are back in some configurations, sigh. And I 94226376Sdas * don't want to used double_t to avoid them, since that gives a 95226376Sdas * pessimization and the support for avoiding the pessimization 96226376Sdas * is not yet available. 97226376Sdas * 98226376Sdas * The multi-precision calculations for the multiplications are 99226376Sdas * routine. 100226376Sdas */ 101226376Sdas hi = f - hfsq; 102216211Sdas SET_LOW_WORD(hi,0); 103226376Sdas lo = (f - hi) - hfsq + r; 104226376Sdas val_hi = hi*ivln2hi; 105226376Sdas val_lo = (lo+hi)*ivln2lo + lo*ivln2hi; 106226376Sdas 107226376Sdas /* spadd(val_hi, val_lo, y), except for not using double_t: */ 108226376Sdas w = y + val_hi; 109226376Sdas val_lo += (y - w) + val_hi; 110226376Sdas val_hi = w; 111226376Sdas 112226376Sdas return val_lo + val_hi; 1132116Sjkh} 114251292Sdas 115251292Sdas#if (LDBL_MANT_DIG == 53) 116251292Sdas__weak_reference(log2, log2l); 117251292Sdas#endif 118