164 lines
5.2 KiB
C
164 lines
5.2 KiB
C
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#ifndef lint
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static char sccsid[] = "@(#)log1p.c 1.1 94/10/31 SMI";
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#endif
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/*
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* Copyright (c) 1987 by Sun Microsystems, Inc.
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*/
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/* LOG1P(x)
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* RETURN THE LOGARITHM OF 1+x
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* IEEE DOUBLE PRECISION
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* CODE BASED ON 4.3BSD, MODIFIED BY K.C. NG, 6/29/87.
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*
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* Required system supported functions:
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* scalbn(x,n)
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* copysign(x,y)
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* ilogb(x)
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* finite(x)
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*
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* Required kernel function:
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* log__L(z)
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*
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* Method :
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* 1. Argument Reduction: find k and f such that
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* 1+x = 2^k * (1+f),
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* where sqrt(2)/2 < 1+f < sqrt(2) .
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*
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* 2. Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
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* = 2s + 2/3 s**3 + 2/5 s**5 + .....,
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* log(1+f) is computed by
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*
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* log(1+f) = 2s + s*log__L(s*s)
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* where
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* log__L(z) = z*(L1 + z*(L2 + z*(... (L6 + z*L7)...)))
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*
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* See log__L() for the values of the coefficients.
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*
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* 3. Finally, log(1+x) = k*ln2 + log(1+f).
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*
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* Remarks 1. In step 3 n*ln2 will be stored in two floating point numbers
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* n*ln2hi + n*ln2lo, where ln2hi is chosen such that the last
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* 20 bits (for VAX D format), or the last 21 bits ( for IEEE
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* double) is 0. This ensures n*ln2hi is exactly representable.
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* 2. In step 1, f may not be representable. A correction term c
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* for f is computed. It follows that the correction term for
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* f - t (the leading term of log(1+f) in step 2) is c-c*x. We
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* add this correction term to n*ln2lo to attenuate the error.
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*
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*
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* Special cases:
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* log1p(x) is NaN with signal if x < -1; log1p(NaN) is NaN with no signal;
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* log1p(INF) is +INF; log1p(-1) is -INF with signal;
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* only log1p(0)=0 is exact for finite argument.
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*
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* Accuracy:
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* log1p(x) returns the exact log(1+x) nearly rounded. In a test run
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* with 1,536,000 random arguments on a VAX, the maximum observed
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* error was .846 ulps (units in the last place).
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*
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* Constants:
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* The hexadecimal values are the intended ones for the following constants.
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* The decimal values may be used, provided that the compiler will convert
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* from decimal to binary accurately enough to produce the hexadecimal values
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* shown.
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*/
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#include <math.h>
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#include "libm.h"
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double log1p(x)
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double x;
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{
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static double zero=0.0, negone= -1.0, one=1.0,
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half=1.0/2.0, small=1.0E-20; /* 1+small == 1 */
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double log__L(),z,s,t,c;
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int ilogb();
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int k,finite();
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if(!finite(x)) return Inf+x; /* x is +-INF or NaN */
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else { if( x > negone ) {
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/* argument reduction */
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if(fabs(x)<small) {
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dummy(fmax-fabs(x)); /* raise inexact if x is not zero */
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return(x);
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}
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if (x>1e300) k=ilogb(x); else k=ilogb(one+x);
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z=scalbn(x,-k); t=scalbn(one,-k);
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if(z+t >= sqrt2 )
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{ k += 1 ; z *= half; t *= half; }
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t += negone; x = z + t;
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c = (t-x)+z ; /* correction term for x */
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/* compute log(1+x) */
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s = x/(2+x); t = x*x*half;
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c += (k*ln2lo-c*x);
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z = c+s*(t+log__L(s*s));
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x += (z - t) ;
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return(k*ln2hi+x);
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}
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/* end of if (x > negone) */
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else {
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if ( x == negone ) return( negone/zero );
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else return ( zero / zero );
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}
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}
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}
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#ifdef VAX /* VAX D format (56 bits) */
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/* static double */
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/* L1 = 6.6666666666666703212E-1 , Hex 2^ 0 * .AAAAAAAAAAAAC5 */
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/* L2 = 3.9999999999970461961E-1 , Hex 2^ -1 * .CCCCCCCCCC2684 */
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/* L3 = 2.8571428579395698188E-1 , Hex 2^ -1 * .92492492F85782 */
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/* L4 = 2.2222221233634724402E-1 , Hex 2^ -2 * .E38E3839B7AF2C */
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/* L5 = 1.8181879517064680057E-1 , Hex 2^ -2 * .BA2EB4CC39655E */
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/* L6 = 1.5382888777946145467E-1 , Hex 2^ -2 * .9D8551E8C5781D */
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/* L7 = 1.3338356561139403517E-1 , Hex 2^ -2 * .8895B3907FCD92 */
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/* L8 = 1.2500000000000000000E-1 , Hex 2^ -2 * .80000000000000 */
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static long L1x[] = { 0xaaaa402a, 0xaac5aaaa};
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static long L2x[] = { 0xcccc3fcc, 0x2684cccc};
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static long L3x[] = { 0x49243f92, 0x578292f8};
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static long L4x[] = { 0x8e383f63, 0xaf2c39b7};
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static long L5x[] = { 0x2eb43f3a, 0x655ecc39};
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static long L6x[] = { 0x85513f1d, 0x781de8c5};
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static long L7x[] = { 0x95b33f08, 0xcd92907f};
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static long L8x[] = { 0x00003f00, 0x00000000};
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#define L1 (*(double*)L1x)
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#define L2 (*(double*)L2x)
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#define L3 (*(double*)L3x)
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#define L4 (*(double*)L4x)
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#define L5 (*(double*)L5x)
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#define L6 (*(double*)L6x)
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#define L7 (*(double*)L7x)
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#define L8 (*(double*)L8x)
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#else /* IEEE double */
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static double
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L1 = 6.6666666666667340202E-1 , /*Hex 2^ -1 * 1.5555555555592 */
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L2 = 3.9999999999416702146E-1 , /*Hex 2^ -2 * 1.999999997FF24 */
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L3 = 2.8571428742008753154E-1 , /*Hex 2^ -2 * 1.24924941E07B4 */
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L4 = 2.2222198607186277597E-1 , /*Hex 2^ -3 * 1.C71C52150BEA6 */
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L5 = 1.8183562745289935658E-1 , /*Hex 2^ -3 * 1.74663CC94342F */
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L6 = 1.5314087275331442206E-1 , /*Hex 2^ -3 * 1.39A1EC014045B */
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L7 = 1.4795612545334174692E-1 ; /*Hex 2^ -3 * 1.2F039F0085122 */
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#endif
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static double log__L(z)
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double z;
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{
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#ifdef VAX
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return(z*(L1+z*(L2+z*(L3+z*(L4+z*(L5+z*(L6+z*(L7+z*L8))))))));
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#else /* IEEE double */
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return(z*(L1+z*(L2+z*(L3+z*(L4+z*(L5+z*(L6+z*L7)))))));
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#endif
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}
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static dummy(x)
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double x;
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{
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return 1;
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}
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