1 2 /* 3 * ==================================================== 4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 5 * 6 * Developed at SunSoft, a Sun Microsystems, Inc. business. 7 * Permission to use, copy, modify, and distribute this 8 * software is freely granted, provided that this notice 9 * is preserved. 10 * ==================================================== 11 */ 12 13 /* 14 * Return the base 2 logarithm of x. See e_log.c and k_log.h for most 15 * comments. 16 * 17 * This reduces x to {k, 1+f} exactly as in e_log.c, then calls the kernel, 18 * then does the combining and scaling steps 19 * log2(x) = (f - 0.5*f*f + k_log1p(f)) / ln2 + k 20 * in not-quite-routine extra precision. 21 */ 22 23 #include <float.h> 24 25 #include "math.h" 26 #include "math_private.h" 27 #include "k_log.h" 28 29 static const double 30 two54 = 1.80143985094819840000e+16, /* 0x43500000, 0x00000000 */ 31 ivln2hi = 1.44269504072144627571e+00, /* 0x3ff71547, 0x65200000 */ 32 ivln2lo = 1.67517131648865118353e-10; /* 0x3de705fc, 0x2eefa200 */ 33 34 static const double zero = 0.0; 35 static volatile double vzero = 0.0; 36 37 double 38 log2(double x) 39 { 40 double f,hfsq,hi,lo,r,val_hi,val_lo,w,y; 41 int32_t i,k,hx; 42 u_int32_t lx; 43 44 EXTRACT_WORDS(hx,lx,x); 45 46 k=0; 47 if (hx < 0x00100000) { /* x < 2**-1022 */ 48 if (((hx&0x7fffffff)|lx)==0) 49 return -two54/vzero; /* log(+-0)=-inf */ 50 if (hx<0) return (x-x)/zero; /* log(-#) = NaN */ 51 k -= 54; x *= two54; /* subnormal number, scale up x */ 52 GET_HIGH_WORD(hx,x); 53 } 54 if (hx >= 0x7ff00000) return x+x; 55 if (hx == 0x3ff00000 && lx == 0) 56 return zero; /* log(1) = +0 */ 57 k += (hx>>20)-1023; 58 hx &= 0x000fffff; 59 i = (hx+0x95f64)&0x100000; 60 SET_HIGH_WORD(x,hx|(i^0x3ff00000)); /* normalize x or x/2 */ 61 k += (i>>20); 62 y = (double)k; 63 f = x - 1.0; 64 hfsq = 0.5*f*f; 65 r = k_log1p(f); 66 67 /* 68 * f-hfsq must (for args near 1) be evaluated in extra precision 69 * to avoid a large cancellation when x is near sqrt(2) or 1/sqrt(2). 70 * This is fairly efficient since f-hfsq only depends on f, so can 71 * be evaluated in parallel with R. Not combining hfsq with R also 72 * keeps R small (though not as small as a true `lo' term would be), 73 * so that extra precision is not needed for terms involving R. 74 * 75 * Compiler bugs involving extra precision used to break Dekker's 76 * theorem for spitting f-hfsq as hi+lo, unless double_t was used 77 * or the multi-precision calculations were avoided when double_t 78 * has extra precision. These problems are now automatically 79 * avoided as a side effect of the optimization of combining the 80 * Dekker splitting step with the clear-low-bits step. 81 * 82 * y must (for args near sqrt(2) and 1/sqrt(2)) be added in extra 83 * precision to avoid a very large cancellation when x is very near 84 * these values. Unlike the above cancellations, this problem is 85 * specific to base 2. It is strange that adding +-1 is so much 86 * harder than adding +-ln2 or +-log10_2. 87 * 88 * This uses Dekker's theorem to normalize y+val_hi, so the 89 * compiler bugs are back in some configurations, sigh. And I 90 * don't want to used double_t to avoid them, since that gives a 91 * pessimization and the support for avoiding the pessimization 92 * is not yet available. 93 * 94 * The multi-precision calculations for the multiplications are 95 * routine. 96 */ 97 hi = f - hfsq; 98 SET_LOW_WORD(hi,0); 99 lo = (f - hi) - hfsq + r; 100 val_hi = hi*ivln2hi; 101 val_lo = (lo+hi)*ivln2lo + lo*ivln2hi; 102 103 /* spadd(val_hi, val_lo, y), except for not using double_t: */ 104 w = y + val_hi; 105 val_lo += (y - w) + val_hi; 106 val_hi = w; 107 108 return val_lo + val_hi; 109 } 110 111 #if (LDBL_MANT_DIG == 53) 112 __weak_reference(log2, log2l); 113 #endif 114