/* * 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[] = "@(#)expm1.c 1.2 (Berkeley) 8/21/85"; #endif not lint /* EXPM1(X) * RETURN THE EXPONENTIAL OF X MINUS ONE * DOUBLE PRECISION (IEEE 53 BITS, VAX D FORMAT 56 BITS) * CODED IN C BY K.C. NG, 1/19/85; * REVISED BY K.C. NG on 2/6/85, 3/7/85, 3/21/85, 4/16/85. * * Required system supported functions: * scalb(x,n) * copysign(x,y) * finite(x) * * Kernel function: * exp__E(x,c) * * Method: * 1. Argument Reduction: given the input x, find r and integer k such * that * x = k*ln2 + r, |r| <= 0.5*ln2 . * r will be represented as r := z+c for better accuracy. * * 2. Compute EXPM1(r)=exp(r)-1 by * * EXPM1(r=z+c) := z + exp__E(z,c) * * 3. EXPM1(x) = 2^k * ( EXPM1(r) + 1-2^-k ). * * Remarks: * 1. When k=1 and z < -0.25, we use the following formula for * better accuracy: * EXPM1(x) = 2 * ( (z+0.5) + exp__E(z,c) ) * 2. To avoid rounding error in 1-2^-k where k is large, we use * EXPM1(x) = 2^k * { [z+(exp__E(z,c)-2^-k )] + 1 } * when k>56. * * Special cases: * EXPM1(INF) is INF, EXPM1(NaN) is NaN; * EXPM1(-INF)= -1; * for finite argument, only EXPM1(0)=0 is exact. * * Accuracy: * EXPM1(x) returns the exact (exp(x)-1) nearly rounded. In a test run with * 1,166,000 random arguments on a VAX, the maximum observed error was * .872 ulps (units of 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 */ /* double static */ /* ln2hi = 6.9314718055829871446E-1 , Hex 2^ 0 * .B17217F7D00000 */ /* ln2lo = 1.6465949582897081279E-12 , Hex 2^-39 * .E7BCD5E4F1D9CC */ /* lnhuge = 9.4961163736712506989E1 , Hex 2^ 7 * .BDEC1DA73E9010 */ /* invln2 = 1.4426950408889634148E0 ; Hex 2^ 1 * .B8AA3B295C17F1 */ static long ln2hix[] = { 0x72174031, 0x0000f7d0}; static long ln2lox[] = { 0xbcd52ce7, 0xd9cce4f1}; static long lnhugex[] = { 0xec1d43bd, 0x9010a73e}; static long invln2x[] = { 0xaa3b40b8, 0x17f1295c}; #define ln2hi (*(double*)ln2hix) #define ln2lo (*(double*)ln2lox) #define lnhuge (*(double*)lnhugex) #define invln2 (*(double*)invln2x) #else /* IEEE double */ double static ln2hi = 6.9314718036912381649E-1 , /*Hex 2^ -1 * 1.62E42FEE00000 */ ln2lo = 1.9082149292705877000E-10 , /*Hex 2^-33 * 1.A39EF35793C76 */ lnhuge = 7.1602103751842355450E2 , /*Hex 2^ 9 * 1.6602B15B7ECF2 */ invln2 = 1.4426950408889633870E0 ; /*Hex 2^ 0 * 1.71547652B82FE */ #endif double expm1(x) double x; { double static one=1.0, half=1.0/2.0; double scalb(), copysign(), exp__E(), z,hi,lo,c; int k,finite(); #ifdef VAX static prec=56; #else /* IEEE double */ static prec=53; #endif #ifndef VAX if(x!=x) return(x); /* x is NaN */ #endif if( x <= lnhuge ) { if( x >= -40.0 ) { /* argument reduction : x - k*ln2 */ k= invln2 *x+copysign(0.5,x); /* k=NINT(x/ln2) */ hi=x-k*ln2hi ; z=hi-(lo=k*ln2lo); c=(hi-z)-lo; if(k==0) return(z+exp__E(z,c)); if(k==1) if(z< -0.25) {x=z+half;x +=exp__E(z,c); return(x+x);} else {z+=exp__E(z,c); x=half+z; return(x+x);} /* end of k=1 */ else { if(k<=prec) { x=one-scalb(one,-k); z += exp__E(z,c);} else if(k<100) { x = exp__E(z,c)-scalb(one,-k); x+=z; z=one;} else { x = exp__E(z,c)+z; z=one;} return (scalb(x+z,k)); } } /* end of x > lnunfl */ else /* expm1(-big#) rounded to -1 (inexact) */ if(finite(x)) { ln2hi+ln2lo; return(-one);} /* expm1(-INF) is -1 */ else return(-one); } /* end of x < lnhuge */ else /* expm1(INF) is INF, expm1(+big#) overflows to INF */ return( finite(x) ? scalb(one,5000) : x); }