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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 #include <sys/cdefs.h>
     15 __FBSDID("$FreeBSD$");
     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 <float.h>
     69 
     70 #include "math.h"
     71 #include "math_private.h"
     72 
     73 static const double
     74 ln2_hi  =  6.93147180369123816490e-01,	/* 3fe62e42 fee00000 */
     75 ln2_lo  =  1.90821492927058770002e-10,	/* 3dea39ef 35793c76 */
     76 two54   =  1.80143985094819840000e+16,  /* 43500000 00000000 */
     77 Lg1 = 6.666666666666735130e-01,  /* 3FE55555 55555593 */
     78 Lg2 = 3.999999999940941908e-01,  /* 3FD99999 9997FA04 */
     79 Lg3 = 2.857142874366239149e-01,  /* 3FD24924 94229359 */
     80 Lg4 = 2.222219843214978396e-01,  /* 3FCC71C5 1D8E78AF */
     81 Lg5 = 1.818357216161805012e-01,  /* 3FC74664 96CB03DE */
     82 Lg6 = 1.531383769920937332e-01,  /* 3FC39A09 D078C69F */
     83 Lg7 = 1.479819860511658591e-01;  /* 3FC2F112 DF3E5244 */
     84 
     85 static const double zero   =  0.0;
     86 static volatile double vzero = 0.0;
     87 
     88 double
     89 __ieee754_log(double x)
     90 {
     91 	double hfsq,f,s,z,R,w,t1,t2,dk;
     92 	int32_t k,hx,i,j;
     93 	u_int32_t lx;
     94 
     95 	EXTRACT_WORDS(hx,lx,x);
     96 
     97 	k=0;
     98 	if (hx < 0x00100000) {			/* x < 2**-1022  */
     99 	    if (((hx&0x7fffffff)|lx)==0)
    100 		return -two54/vzero;		/* log(+-0)=-inf */
    101 	    if (hx<0) return (x-x)/zero;	/* log(-#) = NaN */
    102 	    k -= 54; x *= two54; /* subnormal number, scale up x */
    103 	    GET_HIGH_WORD(hx,x);
    104 	}
    105 	if (hx >= 0x7ff00000) return x+x;
    106 	k += (hx>>20)-1023;
    107 	hx &= 0x000fffff;
    108 	i = (hx+0x95f64)&0x100000;
    109 	SET_HIGH_WORD(x,hx|(i^0x3ff00000));	/* normalize x or x/2 */
    110 	k += (i>>20);
    111 	f = x-1.0;
    112 	if((0x000fffff&(2+hx))<3) {	/* -2**-20 <= f < 2**-20 */
    113 	    if(f==zero) {
    114 		if(k==0) {
    115 		    return zero;
    116 		} else {
    117 		    dk=(double)k;
    118 		    return dk*ln2_hi+dk*ln2_lo;
    119 		}
    120 	    }
    121 	    R = f*f*(0.5-0.33333333333333333*f);
    122 	    if(k==0) return f-R; else {dk=(double)k;
    123 	    	     return dk*ln2_hi-((R-dk*ln2_lo)-f);}
    124 	}
    125  	s = f/(2.0+f);
    126 	dk = (double)k;
    127 	z = s*s;
    128 	i = hx-0x6147a;
    129 	w = z*z;
    130 	j = 0x6b851-hx;
    131 	t1= w*(Lg2+w*(Lg4+w*Lg6));
    132 	t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
    133 	i |= j;
    134 	R = t2+t1;
    135 	if(i>0) {
    136 	    hfsq=0.5*f*f;
    137 	    if(k==0) return f-(hfsq-s*(hfsq+R)); else
    138 		     return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f);
    139 	} else {
    140 	    if(k==0) return f-s*(f-R); else
    141 		     return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f);
    142 	}
    143 }
    144 
    145 #if (LDBL_MANT_DIG == 53)
    146 __weak_reference(log, logl);
    147 #endif
    148