vx32

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e_log.c (4400B)

      1 /* @(#)e_log.c 5.1 93/09/24 */
      2 /*
      3  * ====================================================
      4  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
      5  *
      6  * Developed at SunPro, 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 #ifndef lint
     14 static char rcsid[] = "$FreeBSD: src/lib/msun/src/e_log.c,v 1.9 2003/07/23 04:53:46 peter Exp $";
     15 #endif
     16 
     17 /* __ieee754_log(x)
     18  * Return the logrithm of x
     19  *
     20  * Method :
     21  *   1. Argument Reduction: find k and f such that
     22  *			x = 2^k * (1+f),
     23  *	   where  sqrt(2)/2 < 1+f < sqrt(2) .
     24  *
     25  *   2. Approximation of log(1+f).
     26  *	Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
     27  *		 = 2s + 2/3 s**3 + 2/5 s**5 + .....,
     28  *	     	 = 2s + s*R
     29  *      We use a special Reme algorithm on [0,0.1716] to generate
     30  * 	a polynomial of degree 14 to approximate R The maximum error
     31  *	of this polynomial approximation is bounded by 2**-58.45. In
     32  *	other words,
     33  *		        2      4      6      8      10      12      14
     34  *	    R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s  +Lg6*s  +Lg7*s
     35  *  	(the values of Lg1 to Lg7 are listed in the program)
     36  *	and
     37  *	    |      2          14          |     -58.45
     38  *	    | Lg1*s +...+Lg7*s    -  R(z) | <= 2
     39  *	    |                             |
     40  *	Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
     41  *	In order to guarantee error in log below 1ulp, we compute log
     42  *	by
     43  *		log(1+f) = f - s*(f - R)	(if f is not too large)
     44  *		log(1+f) = f - (hfsq - s*(hfsq+R)).	(better accuracy)
     45  *
     46  *	3. Finally,  log(x) = k*ln2 + log(1+f).
     47  *			    = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
     48  *	   Here ln2 is split into two floating point number:
     49  *			ln2_hi + ln2_lo,
     50  *	   where n*ln2_hi is always exact for |n| < 2000.
     51  *
     52  * Special cases:
     53  *	log(x) is NaN with signal if x < 0 (including -INF) ;
     54  *	log(+INF) is +INF; log(0) is -INF with signal;
     55  *	log(NaN) is that NaN with no signal.
     56  *
     57  * Accuracy:
     58  *	according to an error analysis, the error is always less than
     59  *	1 ulp (unit in the last place).
     60  *
     61  * Constants:
     62  * The hexadecimal values are the intended ones for the following
     63  * constants. The decimal values may be used, provided that the
     64  * compiler will convert from decimal to binary accurately enough
     65  * to produce the hexadecimal values shown.
     66  */
     67 
     68 #include "math.h"
     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 
     85 double
     86 __ieee754_log(double x)
     87 {
     88 	double hfsq,f,s,z,R,w,t1,t2,dk;
     89 	int32_t k,hx,i,j;
     90 	u_int32_t lx;
     91 
     92 	EXTRACT_WORDS(hx,lx,x);
     93 
     94 	k=0;
     95 	if (hx < 0x00100000) {			/* x < 2**-1022  */
     96 	    if (((hx&0x7fffffff)|lx)==0)
     97 		return -two54/zero;		/* log(+-0)=-inf */
     98 	    if (hx<0) return (x-x)/zero;	/* log(-#) = NaN */
     99 	    k -= 54; x *= two54; /* subnormal number, scale up x */
    100 	    GET_HIGH_WORD(hx,x);
    101 	}
    102 	if (hx >= 0x7ff00000) return x+x;
    103 	k += (hx>>20)-1023;
    104 	hx &= 0x000fffff;
    105 	i = (hx+0x95f64)&0x100000;
    106 	SET_HIGH_WORD(x,hx|(i^0x3ff00000));	/* normalize x or x/2 */
    107 	k += (i>>20);
    108 	f = x-1.0;
    109 	if((0x000fffff&(2+hx))<3) {	/* |f| < 2**-20 */
    110 	    if(f==zero) if(k==0) return zero;  else {dk=(double)k;
    111 				 return dk*ln2_hi+dk*ln2_lo;}
    112 	    R = f*f*(0.5-0.33333333333333333*f);
    113 	    if(k==0) return f-R; else {dk=(double)k;
    114 	    	     return dk*ln2_hi-((R-dk*ln2_lo)-f);}
    115 	}
    116  	s = f/(2.0+f);
    117 	dk = (double)k;
    118 	z = s*s;
    119 	i = hx-0x6147a;
    120 	w = z*z;
    121 	j = 0x6b851-hx;
    122 	t1= w*(Lg2+w*(Lg4+w*Lg6));
    123 	t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
    124 	i |= j;
    125 	R = t2+t1;
    126 	if(i>0) {
    127 	    hfsq=0.5*f*f;
    128 	    if(k==0) return f-(hfsq-s*(hfsq+R)); else
    129 		     return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f);
    130 	} else {
    131 	    if(k==0) return f-s*(f-R); else
    132 		     return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f);
    133 	}
    134 }