4.4BSD/usr/src/old/libm/libm/log.c
/*
* Copyright (c) 1985 Regents of the University of California.
*
* Use and reproduction of this software are granted in accordance with
* the terms and conditions specified in the Berkeley Software License
* Agreement (in particular, this entails acknowledgement of the programs'
* source, and inclusion of this notice) with the additional understanding
* that all recipients should regard themselves as participants in an
* ongoing research project and hence should feel obligated to report
* their experiences (good or bad) with these elementary function codes,
* using "sendbug 4bsd-bugs@BERKELEY", to the authors.
*/
#ifndef lint
static char sccsid[] = "@(#)log.c 4.5 (Berkeley) 8/21/85";
#endif not lint
/* LOG(X)
* RETURN THE LOGARITHM OF x
* DOUBLE PRECISION (VAX D FORMAT 56 bits or IEEE DOUBLE 53 BITS)
* CODED IN C BY K.C. NG, 1/19/85;
* REVISED BY K.C. NG on 2/7/85, 3/7/85, 3/24/85, 4/16/85.
*
* Required system supported functions:
* scalb(x,n)
* copysign(x,y)
* logb(x)
* finite(x)
*
* Required kernel function:
* log__L(z)
*
* Method :
* 1. Argument Reduction: find k and f such that
* x = 2^k * (1+f),
* where sqrt(2)/2 < 1+f < sqrt(2) .
*
* 2. Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
* = 2s + 2/3 s**3 + 2/5 s**5 + .....,
* log(1+f) is computed by
*
* log(1+f) = 2s + s*log__L(s*s)
* where
* log__L(z) = z*(L1 + z*(L2 + z*(... (L6 + z*L7)...)))
*
* See log__L() for the values of the coefficients.
*
* 3. Finally, log(x) = k*ln2 + log(1+f). (Here n*ln2 will be stored
* in two floating point number: n*ln2hi + n*ln2lo, n*ln2hi is exact
* since the last 20 bits of ln2hi is 0.)
*
* Special cases:
* log(x) is NaN with signal if x < 0 (including -INF) ;
* log(+INF) is +INF; log(0) is -INF with signal;
* log(NaN) is that NaN with no signal.
*
* Accuracy:
* log(x) returns the exact log(x) nearly rounded. In a test run with
* 1,536,000 random arguments on a VAX, the maximum observed error was
* .826 ulps (units in the last place).
*
* Constants:
* The hexadecimal values are the intended ones for the following constants.
* The decimal values may be used, provided that the compiler will convert
* from decimal to binary accurately enough to produce the hexadecimal values
* shown.
*/
#ifdef VAX /* VAX D format */
#include <errno.h>
/* double static */
/* ln2hi = 6.9314718055829871446E-1 , Hex 2^ 0 * .B17217F7D00000 */
/* ln2lo = 1.6465949582897081279E-12 , Hex 2^-39 * .E7BCD5E4F1D9CC */
/* sqrt2 = 1.4142135623730950622E0 ; Hex 2^ 1 * .B504F333F9DE65 */
static long ln2hix[] = { 0x72174031, 0x0000f7d0};
static long ln2lox[] = { 0xbcd52ce7, 0xd9cce4f1};
static long sqrt2x[] = { 0x04f340b5, 0xde6533f9};
#define ln2hi (*(double*)ln2hix)
#define ln2lo (*(double*)ln2lox)
#define sqrt2 (*(double*)sqrt2x)
#else /* IEEE double */
double static
ln2hi = 6.9314718036912381649E-1 , /*Hex 2^ -1 * 1.62E42FEE00000 */
ln2lo = 1.9082149292705877000E-10 , /*Hex 2^-33 * 1.A39EF35793C76 */
sqrt2 = 1.4142135623730951455E0 ; /*Hex 2^ 0 * 1.6A09E667F3BCD */
#endif
double log(x)
double x;
{
static double zero=0.0, negone= -1.0, half=1.0/2.0;
double logb(),scalb(),copysign(),log__L(),s,z,t;
int k,n,finite();
#ifndef VAX
if(x!=x) return(x); /* x is NaN */
#endif
if(finite(x)) {
if( x > zero ) {
/* argument reduction */
k=logb(x); x=scalb(x,-k);
if(k == -1022) /* subnormal no. */
{n=logb(x); x=scalb(x,-n); k+=n;}
if(x >= sqrt2 ) {k += 1; x *= half;}
x += negone ;
/* compute log(1+x) */
s=x/(2+x); t=x*x*half;
z=k*ln2lo+s*(t+log__L(s*s));
x += (z - t) ;
return(k*ln2hi+x);
}
/* end of if (x > zero) */
else {
#ifdef VAX
extern double infnan();
if ( x == zero )
return (infnan(-ERANGE)); /* -INF */
else
return (infnan(EDOM)); /* NaN */
#else /* IEEE double */
/* zero argument, return -INF with signal */
if ( x == zero )
return( negone/zero );
/* negative argument, return NaN with signal */
else
return ( zero / zero );
#endif
}
}
/* end of if (finite(x)) */
/* NOT REACHED ifdef VAX */
/* log(-INF) is NaN with signal */
else if (x<0)
return(zero/zero);
/* log(+INF) is +INF */
else return(x);
}