1a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes 2a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes/* @(#)e_log10.c 1.3 95/01/18 */ 3a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes/* 4a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes * ==================================================== 5a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 6a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes * 7a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes * Developed at SunSoft, a Sun Microsystems, Inc. business. 8a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes * Permission to use, copy, modify, and distribute this 9a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes * software is freely granted, provided that this notice 10a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes * is preserved. 11a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes * ==================================================== 12a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes */ 13a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes 14a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes#include <sys/cdefs.h> 15a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes__FBSDID("$FreeBSD$"); 16a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes 17a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes/* 18a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes * Return the base 2 logarithm of x. See e_log.c and k_log.h for most 19a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes * comments. 20a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes * 21a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes * This reduces x to {k, 1+f} exactly as in e_log.c, then calls the kernel, 22a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes * then does the combining and scaling steps 23a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes * log2(x) = (f - 0.5*f*f + k_log1p(f)) / ln2 + k 24a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes * in not-quite-routine extra precision. 25a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes */ 26a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes 2778419467a2f88744ae2445fca5eb442877ebb1b0Elliott Hughes#include <float.h> 2878419467a2f88744ae2445fca5eb442877ebb1b0Elliott Hughes 29a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes#include "math.h" 30a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes#include "math_private.h" 31a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes#include "k_log.h" 32a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes 33a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughesstatic const double 34a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughestwo54 = 1.80143985094819840000e+16, /* 0x43500000, 0x00000000 */ 35a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughesivln2hi = 1.44269504072144627571e+00, /* 0x3ff71547, 0x65200000 */ 36a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughesivln2lo = 1.67517131648865118353e-10; /* 0x3de705fc, 0x2eefa200 */ 37a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes 38a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughesstatic const double zero = 0.0; 3978419467a2f88744ae2445fca5eb442877ebb1b0Elliott Hughesstatic volatile double vzero = 0.0; 40a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes 41a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughesdouble 42a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes__ieee754_log2(double x) 43a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes{ 44a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes double f,hfsq,hi,lo,r,val_hi,val_lo,w,y; 45a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes int32_t i,k,hx; 46a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes u_int32_t lx; 47a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes 48a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes EXTRACT_WORDS(hx,lx,x); 49a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes 50a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes k=0; 51a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes if (hx < 0x00100000) { /* x < 2**-1022 */ 52a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes if (((hx&0x7fffffff)|lx)==0) 5378419467a2f88744ae2445fca5eb442877ebb1b0Elliott Hughes return -two54/vzero; /* log(+-0)=-inf */ 54a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes if (hx<0) return (x-x)/zero; /* log(-#) = NaN */ 55a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes k -= 54; x *= two54; /* subnormal number, scale up x */ 56a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes GET_HIGH_WORD(hx,x); 57a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes } 58a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes if (hx >= 0x7ff00000) return x+x; 59a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes if (hx == 0x3ff00000 && lx == 0) 60a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes return zero; /* log(1) = +0 */ 61a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes k += (hx>>20)-1023; 62a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes hx &= 0x000fffff; 63a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes i = (hx+0x95f64)&0x100000; 64a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes SET_HIGH_WORD(x,hx|(i^0x3ff00000)); /* normalize x or x/2 */ 65a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes k += (i>>20); 66a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes y = (double)k; 67a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes f = x - 1.0; 68a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes hfsq = 0.5*f*f; 69a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes r = k_log1p(f); 70a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes 71a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes /* 72a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes * f-hfsq must (for args near 1) be evaluated in extra precision 73a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes * to avoid a large cancellation when x is near sqrt(2) or 1/sqrt(2). 74a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes * This is fairly efficient since f-hfsq only depends on f, so can 75a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes * be evaluated in parallel with R. Not combining hfsq with R also 76a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes * keeps R small (though not as small as a true `lo' term would be), 77a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes * so that extra precision is not needed for terms involving R. 78a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes * 79a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes * Compiler bugs involving extra precision used to break Dekker's 80a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes * theorem for spitting f-hfsq as hi+lo, unless double_t was used 81a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes * or the multi-precision calculations were avoided when double_t 82a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes * has extra precision. These problems are now automatically 83a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes * avoided as a side effect of the optimization of combining the 84a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes * Dekker splitting step with the clear-low-bits step. 85a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes * 86a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes * y must (for args near sqrt(2) and 1/sqrt(2)) be added in extra 87a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes * precision to avoid a very large cancellation when x is very near 88a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes * these values. Unlike the above cancellations, this problem is 89a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes * specific to base 2. It is strange that adding +-1 is so much 90a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes * harder than adding +-ln2 or +-log10_2. 91a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes * 92a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes * This uses Dekker's theorem to normalize y+val_hi, so the 93a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes * compiler bugs are back in some configurations, sigh. And I 94a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes * don't want to used double_t to avoid them, since that gives a 95a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes * pessimization and the support for avoiding the pessimization 96a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes * is not yet available. 97a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes * 98a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes * The multi-precision calculations for the multiplications are 99a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes * routine. 100a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes */ 101a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes hi = f - hfsq; 102a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes SET_LOW_WORD(hi,0); 103a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes lo = (f - hi) - hfsq + r; 104a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes val_hi = hi*ivln2hi; 105a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes val_lo = (lo+hi)*ivln2lo + lo*ivln2hi; 106a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes 107a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes /* spadd(val_hi, val_lo, y), except for not using double_t: */ 108a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes w = y + val_hi; 109a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes val_lo += (y - w) + val_hi; 110a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes val_hi = w; 111a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes 112a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes return val_lo + val_hi; 113a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes} 11478419467a2f88744ae2445fca5eb442877ebb1b0Elliott Hughes 11578419467a2f88744ae2445fca5eb442877ebb1b0Elliott Hughes#if (LDBL_MANT_DIG == 53) 11678419467a2f88744ae2445fca5eb442877ebb1b0Elliott Hughes__weak_reference(log2, log2l); 11778419467a2f88744ae2445fca5eb442877ebb1b0Elliott Hughes#endif 118