tor-browser

The Tor Browser
git clone https://git.dasho.dev/tor-browser.git
Log | Files | Refs | README | LICENSE

e_log.cpp (4452B)

      1 /* @(#)e_log.c 1.3 95/01/18 */
      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 //#include <sys/cdefs.h>
     14 //__FBSDID("$FreeBSD$");
     15 
     16 /* __ieee754_log(x)
     17 * Return the logrithm of x
     18 *
     19 * Method :                  
     20 *   1. Argument Reduction: find k and f such that 
     21 *			x = 2^k * (1+f), 
     22 *	   where  sqrt(2)/2 < 1+f < sqrt(2) .
     23 *
     24 *   2. Approximation of log(1+f).
     25 *	Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
     26 *		 = 2s + 2/3 s**3 + 2/5 s**5 + .....,
     27 *	     	 = 2s + s*R
     28 *      We use a special Reme algorithm on [0,0.1716] to generate 
     29 * 	a polynomial of degree 14 to approximate R The maximum error 
     30 *	of this polynomial approximation is bounded by 2**-58.45. In
     31 *	other words,
     32 *		        2      4      6      8      10      12      14
     33 *	    R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s  +Lg6*s  +Lg7*s
     34 *  	(the values of Lg1 to Lg7 are listed in the program)
     35 *	and
     36 *	    |      2          14          |     -58.45
     37 *	    | Lg1*s +...+Lg7*s    -  R(z) | <= 2 
     38 *	    |                             |
     39 *	Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
     40 *	In order to guarantee error in log below 1ulp, we compute log
     41 *	by
     42 *		log(1+f) = f - s*(f - R)	(if f is not too large)
     43 *		log(1+f) = f - (hfsq - s*(hfsq+R)).	(better accuracy)
     44 *	
     45 *	3. Finally,  log(x) = k*ln2 + log(1+f).  
     46 *			    = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
     47 *	   Here ln2 is split into two floating point number: 
     48 *			ln2_hi + ln2_lo,
     49 *	   where n*ln2_hi is always exact for |n| < 2000.
     50 *
     51 * Special cases:
     52 *	log(x) is NaN with signal if x < 0 (including -INF) ; 
     53 *	log(+INF) is +INF; log(0) is -INF with signal;
     54 *	log(NaN) is that NaN with no signal.
     55 *
     56 * Accuracy:
     57 *	according to an error analysis, the error is always less than
     58 *	1 ulp (unit in the last place).
     59 *
     60 * Constants:
     61 * The hexadecimal values are the intended ones for the following 
     62 * constants. The decimal values may be used, provided that the 
     63 * compiler will convert from decimal to binary accurately enough 
     64 * to produce the hexadecimal values shown.
     65 */
     66 
     67 #include <float.h>
     68 
     69 #include "math_private.h"
     70 
     71 static const double
     72 ln2_hi  =  6.93147180369123816490e-01,	/* 3fe62e42 fee00000 */
     73 ln2_lo  =  1.90821492927058770002e-10,	/* 3dea39ef 35793c76 */
     74 two54   =  1.80143985094819840000e+16,  /* 43500000 00000000 */
     75 Lg1 = 6.666666666666735130e-01,  /* 3FE55555 55555593 */
     76 Lg2 = 3.999999999940941908e-01,  /* 3FD99999 9997FA04 */
     77 Lg3 = 2.857142874366239149e-01,  /* 3FD24924 94229359 */
     78 Lg4 = 2.222219843214978396e-01,  /* 3FCC71C5 1D8E78AF */
     79 Lg5 = 1.818357216161805012e-01,  /* 3FC74664 96CB03DE */
     80 Lg6 = 1.531383769920937332e-01,  /* 3FC39A09 D078C69F */
     81 Lg7 = 1.479819860511658591e-01;  /* 3FC2F112 DF3E5244 */
     82 
     83 static const double zero   =  0.0;
     84 static volatile double vzero = 0.0;
     85 
     86 double
     87 __ieee754_log(double x)
     88 {
     89 double hfsq,f,s,z,R,w,t1,t2,dk;
     90 int32_t k,hx,i,j;
     91 u_int32_t lx;
     92 
     93 EXTRACT_WORDS(hx,lx,x);
     94 
     95 k=0;
     96 if (hx < 0x00100000) {			/* x < 2**-1022  */
     97     if (((hx&0x7fffffff)|lx)==0) 
     98 	return -two54/vzero;		/* log(+-0)=-inf */
     99     if (hx<0) return (x-x)/zero;	/* log(-#) = NaN */
    100     k -= 54; x *= two54; /* subnormal number, scale up x */
    101     GET_HIGH_WORD(hx,x);
    102 } 
    103 if (hx >= 0x7ff00000) return x+x;
    104 k += (hx>>20)-1023;
    105 hx &= 0x000fffff;
    106 i = (hx+0x95f64)&0x100000;
    107 SET_HIGH_WORD(x,hx|(i^0x3ff00000));	/* normalize x or x/2 */
    108 k += (i>>20);
    109 f = x-1.0;
    110 if((0x000fffff&(2+hx))<3) {	/* -2**-20 <= f < 2**-20 */
    111     if(f==zero) {
    112 	if(k==0) {
    113 	    return zero;
    114 	} else {
    115 	    dk=(double)k;
    116 	    return dk*ln2_hi+dk*ln2_lo;
    117 	}
    118     }
    119     R = f*f*(0.5-0.33333333333333333*f);
    120     if(k==0) return f-R; else {dk=(double)k;
    121     	     return dk*ln2_hi-((R-dk*ln2_lo)-f);}
    122 }
    123 	s = f/(2.0+f); 
    124 dk = (double)k;
    125 z = s*s;
    126 i = hx-0x6147a;
    127 w = z*z;
    128 j = 0x6b851-hx;
    129 t1= w*(Lg2+w*(Lg4+w*Lg6)); 
    130 t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7))); 
    131 i |= j;
    132 R = t2+t1;
    133 if(i>0) {
    134     hfsq=0.5*f*f;
    135     if(k==0) return f-(hfsq-s*(hfsq+R)); else
    136 	     return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f);
    137 } else {
    138     if(k==0) return f-s*(f-R); else
    139 	     return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f);
    140 }
    141 }