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 <sys/cdefs.h> 15 __FBSDID("$FreeBSD$"); 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 <float.h> 28 29 #include "math.h" 30 #include "math_private.h" 31 #include "k_log.h" 32 33 static const double 34 two54 = 1.80143985094819840000e+16, /* 0x43500000, 0x00000000 */ 35 ivln2hi = 1.44269504072144627571e+00, /* 0x3ff71547, 0x65200000 */ 36 ivln2lo = 1.67517131648865118353e-10; /* 0x3de705fc, 0x2eefa200 */ 37 38 static const double zero = 0.0; 39 static volatile double vzero = 0.0; 40 41 double 42 __ieee754_log2(double x) 43 { 44 double f,hfsq,hi,lo,r,val_hi,val_lo,w,y; 45 int32_t i,k,hx; 46 u_int32_t lx; 47 48 EXTRACT_WORDS(hx,lx,x); 49 50 k=0; 51 if (hx < 0x00100000) { /* x < 2**-1022 */ 52 if (((hx&0x7fffffff)|lx)==0) 53 return -two54/vzero; /* log(+-0)=-inf */ 54 if (hx<0) return (x-x)/zero; /* log(-#) = NaN */ 55 k -= 54; x *= two54; /* subnormal number, scale up x */ 56 GET_HIGH_WORD(hx,x); 57 } 58 if (hx >= 0x7ff00000) return x+x; 59 if (hx == 0x3ff00000 && lx == 0) 60 return zero; /* log(1) = +0 */ 61 k += (hx>>20)-1023; 62 hx &= 0x000fffff; 63 i = (hx+0x95f64)&0x100000; 64 SET_HIGH_WORD(x,hx|(i^0x3ff00000)); /* normalize x or x/2 */ 65 k += (i>>20); 66 y = (double)k; 67 f = x - 1.0; 68 hfsq = 0.5*f*f; 69 r = k_log1p(f); 70 71 /* 72 * f-hfsq must (for args near 1) be evaluated in extra precision 73 * to avoid a large cancellation when x is near sqrt(2) or 1/sqrt(2). 74 * This is fairly efficient since f-hfsq only depends on f, so can 75 * be evaluated in parallel with R. Not combining hfsq with R also 76 * keeps R small (though not as small as a true `lo' term would be), 77 * so that extra precision is not needed for terms involving R. 78 * 79 * Compiler bugs involving extra precision used to break Dekker's 80 * theorem for spitting f-hfsq as hi+lo, unless double_t was used 81 * or the multi-precision calculations were avoided when double_t 82 * has extra precision. These problems are now automatically 83 * avoided as a side effect of the optimization of combining the 84 * Dekker splitting step with the clear-low-bits step. 85 * 86 * y must (for args near sqrt(2) and 1/sqrt(2)) be added in extra 87 * precision to avoid a very large cancellation when x is very near 88 * these values. Unlike the above cancellations, this problem is 89 * specific to base 2. It is strange that adding +-1 is so much 90 * harder than adding +-ln2 or +-log10_2. 91 * 92 * This uses Dekker's theorem to normalize y+val_hi, so the 93 * compiler bugs are back in some configurations, sigh. And I 94 * don't want to used double_t to avoid them, since that gives a 95 * pessimization and the support for avoiding the pessimization 96 * is not yet available. 97 * 98 * The multi-precision calculations for the multiplications are 99 * routine. 100 */ 101 hi = f - hfsq; 102 SET_LOW_WORD(hi,0); 103 lo = (f - hi) - hfsq + r; 104 val_hi = hi*ivln2hi; 105 val_lo = (lo+hi)*ivln2lo + lo*ivln2hi; 106 107 /* spadd(val_hi, val_lo, y), except for not using double_t: */ 108 w = y + val_hi; 109 val_lo += (y - w) + val_hi; 110 val_hi = w; 111 112 return val_lo + val_hi; 113 } 114 115 #if (LDBL_MANT_DIG == 53) 116 __weak_reference(log2, log2l); 117 #endif 118
