1 2 /* @(#)e_log10.c 1.3 95/01/18 */ 3 /* 4 * ==================================================== 5 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 6 * 7 * Developed at SunSoft, a Sun Microsystems, Inc. business. 8 * Permission to use, copy, modify, and distribute this 9 * software is freely granted, provided that this notice 10 * is preserved. 11 * ==================================================== 12 */ 13 14 #include "cdefs-compat.h" 15 //__FBSDID("$FreeBSD: src/lib/msun/src/e_log2.c,v 1.4 2011/10/15 05:23:28 das Exp $"); 16 17 /* 18 * Return the base 2 logarithm of x. See e_log.c and k_log.h for most 19 * comments. 20 * 21 * This reduces x to {k, 1+f} exactly as in e_log.c, then calls the kernel, 22 * then does the combining and scaling steps 23 * log2(x) = (f - 0.5*f*f + k_log1p(f)) / ln2 + k 24 * in not-quite-routine extra precision. 25 */ 26 27 #include <openlibm_math.h> 28 29 #include "math_private.h" 30 #include "k_log.h" 31 32 static const double 33 two54 = 1.80143985094819840000e+16, /* 0x43500000, 0x00000000 */ 34 ivln2hi = 1.44269504072144627571e+00, /* 0x3ff71547, 0x65200000 */ 35 ivln2lo = 1.67517131648865118353e-10; /* 0x3de705fc, 0x2eefa200 */ 36 37 static const double zero = 0.0; 38 39 OLM_DLLEXPORT double 40 __ieee754_log2(double x) 41 { 42 double f,hfsq,hi,lo,r,val_hi,val_lo,w,y; 43 int32_t i,k,hx; 44 u_int32_t lx; 45 46 EXTRACT_WORDS(hx,lx,x); 47 48 k=0; 49 if (hx < 0x00100000) { /* x < 2**-1022 */ 50 if (((hx&0x7fffffff)|lx)==0) 51 return -two54/zero; /* log(+-0)=-inf */ 52 if (hx<0) return (x-x)/zero; /* log(-#) = NaN */ 53 k -= 54; x *= two54; /* subnormal number, scale up x */ 54 GET_HIGH_WORD(hx,x); 55 } 56 if (hx >= 0x7ff00000) return x+x; 57 if (hx == 0x3ff00000 && lx == 0) 58 return zero; /* log(1) = +0 */ 59 k += (hx>>20)-1023; 60 hx &= 0x000fffff; 61 i = (hx+0x95f64)&0x100000; 62 SET_HIGH_WORD(x,hx|(i^0x3ff00000)); /* normalize x or x/2 */ 63 k += (i>>20); 64 y = (double)k; 65 f = x - 1.0; 66 hfsq = 0.5*f*f; 67 r = k_log1p(f); 68 69 /* 70 * f-hfsq must (for args near 1) be evaluated in extra precision 71 * to avoid a large cancellation when x is near sqrt(2) or 1/sqrt(2). 72 * This is fairly efficient since f-hfsq only depends on f, so can 73 * be evaluated in parallel with R. Not combining hfsq with R also 74 * keeps R small (though not as small as a true `lo' term would be), 75 * so that extra precision is not needed for terms involving R. 76 * 77 * Compiler bugs involving extra precision used to break Dekker's 78 * theorem for spitting f-hfsq as hi+lo, unless double_t was used 79 * or the multi-precision calculations were avoided when double_t 80 * has extra precision. These problems are now automatically 81 * avoided as a side effect of the optimization of combining the 82 * Dekker splitting step with the clear-low-bits step. 83 * 84 * y must (for args near sqrt(2) and 1/sqrt(2)) be added in extra 85 * precision to avoid a very large cancellation when x is very near 86 * these values. Unlike the above cancellations, this problem is 87 * specific to base 2. It is strange that adding +-1 is so much 88 * harder than adding +-ln2 or +-log10_2. 89 * 90 * This uses Dekker's theorem to normalize y+val_hi, so the 91 * compiler bugs are back in some configurations, sigh. And I 92 * don't want to used double_t to avoid them, since that gives a 93 * pessimization and the support for avoiding the pessimization 94 * is not yet available. 95 * 96 * The multi-precision calculations for the multiplications are 97 * routine. 98 */ 99 hi = f - hfsq; 100 SET_LOW_WORD(hi,0); 101 lo = (f - hi) - hfsq + r; 102 val_hi = hi*ivln2hi; 103 val_lo = (lo+hi)*ivln2lo + lo*ivln2hi; 104 105 /* spadd(val_hi, val_lo, y), except for not using double_t: */ 106 w = y + val_hi; 107 val_lo += (y - w) + val_hi; 108 val_hi = w; 109 110 return val_lo + val_hi; 111 } 112