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| 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) && !defined(__le32__) && !defined(__le64__) |
| 116 | +#define __weak_reference(sym,alias) \ |
| 117 | + __asm__(".weak " #alias); \ |
| 118 | + __asm__(".equ " #alias ", " #sym) |
| 119 | +__weak_reference(log2, log2l); |
| 120 | +#endif |
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