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      1 
      2 /* @(#)e_log.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 #ifndef lint
     15 static char rcsid[] = "$FreeBSD: src/lib/msun/src/e_log.c,v 1.10 2005/02/04 18:26:06 das Exp $";
     16 #endif
     17 
     18 /* __ieee754_log(x)
     19  * Return the logrithm of x
     20  *
     21  * Method :
     22  *   1. Argument Reduction: find k and f such that
     23  *			x = 2^k * (1+f),
     24  *	   where  sqrt(2)/2 < 1+f < sqrt(2) .
     25  *
     26  *   2. Approximation of log(1+f).
     27  *	Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
     28  *		 = 2s + 2/3 s**3 + 2/5 s**5 + .....,
     29  *	     	 = 2s + s*R
     30  *      We use a special Reme algorithm on [0,0.1716] to generate
     31  * 	a polynomial of degree 14 to approximate R The maximum error
     32  *	of this polynomial approximation is bounded by 2**-58.45. In
     33  *	other words,
     34  *		        2      4      6      8      10      12      14
     35  *	    R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s  +Lg6*s  +Lg7*s
     36  *  	(the values of Lg1 to Lg7 are listed in the program)
     37  *	and
     38  *	    |      2          14          |     -58.45
     39  *	    | Lg1*s +...+Lg7*s    -  R(z) | <= 2
     40  *	    |                             |
     41  *	Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
     42  *	In order to guarantee error in log below 1ulp, we compute log
     43  *	by
     44  *		log(1+f) = f - s*(f - R)	(if f is not too large)
     45  *		log(1+f) = f - (hfsq - s*(hfsq+R)).	(better accuracy)
     46  *
     47  *	3. Finally,  log(x) = k*ln2 + log(1+f).
     48  *			    = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
     49  *	   Here ln2 is split into two floating point number:
     50  *			ln2_hi + ln2_lo,
     51  *	   where n*ln2_hi is always exact for |n| < 2000.
     52  *
     53  * Special cases:
     54  *	log(x) is NaN with signal if x < 0 (including -INF) ;
     55  *	log(+INF) is +INF; log(0) is -INF with signal;
     56  *	log(NaN) is that NaN with no signal.
     57  *
     58  * Accuracy:
     59  *	according to an error analysis, the error is always less than
     60  *	1 ulp (unit in the last place).
     61  *
     62  * Constants:
     63  * The hexadecimal values are the intended ones for the following
     64  * constants. The decimal values may be used, provided that the
     65  * compiler will convert from decimal to binary accurately enough
     66  * to produce the hexadecimal values shown.
     67  */
     68 
     69 #include "math.h"
     70 #include "math_private.h"
     71 
     72 static const double
     73 ln2_hi  =  6.93147180369123816490e-01,	/* 3fe62e42 fee00000 */
     74 ln2_lo  =  1.90821492927058770002e-10,	/* 3dea39ef 35793c76 */
     75 two54   =  1.80143985094819840000e+16,  /* 43500000 00000000 */
     76 Lg1 = 6.666666666666735130e-01,  /* 3FE55555 55555593 */
     77 Lg2 = 3.999999999940941908e-01,  /* 3FD99999 9997FA04 */
     78 Lg3 = 2.857142874366239149e-01,  /* 3FD24924 94229359 */
     79 Lg4 = 2.222219843214978396e-01,  /* 3FCC71C5 1D8E78AF */
     80 Lg5 = 1.818357216161805012e-01,  /* 3FC74664 96CB03DE */
     81 Lg6 = 1.531383769920937332e-01,  /* 3FC39A09 D078C69F */
     82 Lg7 = 1.479819860511658591e-01;  /* 3FC2F112 DF3E5244 */
     83 
     84 static const double zero   =  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/zero;		/* 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) {	/* |f| < 2**-20 */
    111 	    if(f==zero) if(k==0) return zero;  else {dk=(double)k;
    112 				 return dk*ln2_hi+dk*ln2_lo;}
    113 	    R = f*f*(0.5-0.33333333333333333*f);
    114 	    if(k==0) return f-R; else {dk=(double)k;
    115 	    	     return dk*ln2_hi-((R-dk*ln2_lo)-f);}
    116 	}
    117  	s = f/(2.0+f);
    118 	dk = (double)k;
    119 	z = s*s;
    120 	i = hx-0x6147a;
    121 	w = z*z;
    122 	j = 0x6b851-hx;
    123 	t1= w*(Lg2+w*(Lg4+w*Lg6));
    124 	t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
    125 	i |= j;
    126 	R = t2+t1;
    127 	if(i>0) {
    128 	    hfsq=0.5*f*f;
    129 	    if(k==0) return f-(hfsq-s*(hfsq+R)); else
    130 		     return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f);
    131 	} else {
    132 	    if(k==0) return f-s*(f-R); else
    133 		     return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f);
    134 	}
    135 }
    136