215 lines
6.5 KiB
C
215 lines
6.5 KiB
C
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#ifndef lint
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static char sccsid[] = "@(#)expm1.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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/* EXPM1(X)
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* RETURN EXP(X) - 1
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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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* finite(x)
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*
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* Method:
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* 1. Argument Reduction: given the input x, find r and integer k such
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* that
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* x = k*ln2 + r, |r| <= 0.5*ln2 .
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* r will be represented as r := z+c for better accuracy.
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*
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* 2. Compute EXPM1(r)=exp(r)-1 by
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*
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* EXPM1(r=z+c) := z + exp__E(z,c)
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*
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* 3. EXPM1(x) = 2^k * ( EXPM1(r) + 1-2^-k ).
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*
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* Remarks:
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* 1. When k=1 and z < -0.25, we use the following formula for
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* better accuracy:
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* EXPM1(x) = 2 * ( (z+0.5) + exp__E(z,c) )
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* 2. To avoid rounding error in 1-2^-k where k is large, we use
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* EXPM1(x) = 2^k * { [z+(exp__E(z,c)-2^-k )] + 1 }
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* when k>56.
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*
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* Special cases:
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* EXPM1(INF) is INF, EXPM1(NaN) is NaN;
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* EXPM1(-INF)= -1;
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* for finite argument, only EXPM1(0)=0 is exact.
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*
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* Accuracy:
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* EXPM1(x) returns the exact (exp(x)-1) nearly rounded. In a test run with
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* 1,166,000 random arguments on a VAX, the maximum observed error was
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* .872 ulps (units of 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 expm1(x)
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double x;
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{
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double static one=1.0, half=1.0/2.0;
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double scalbn(),copysign(),exp__E(),z,hi,lo,c;
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int k,finite();
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#ifdef VAX
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static prec=56;
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#else /* IEEE double */
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static prec=53;
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#endif
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if(!finite(x)) {if(x!=x||x>0.0) return x+x; else return -1.0;}
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if(signbit(x)) {
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if (x > -0.346573590279972643113) { /* |x| < (ln2)/2 ? */
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if(x > -1e-17) {
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dummy(x+fmax); /* inexact unless x=0 */
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return x;
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} else return x+exp__E(x,0.0);
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} else if ( x > -0.693147180559945286227) {
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hi = x + ln2hi;
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z = hi + ln2lo;
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c = (hi-z)+ln2lo;
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return 0.5*(z+exp__E(z,c))-0.5;
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} else if ( x > -40.0) return exp(x)-1.0;
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else { dummy(1e-300+x); return fmin-1.0;}
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} else {
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if (x < 0.346573590279972643113) { /* x < (ln2)/2 ? */
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if(x < 1e-17) {
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dummy(x-fmax); /* inexact unless x=0 */
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return x;
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} else return x+exp__E(x,0.0);
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} else if ( x < 1.03972077083991792934) {
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hi = x - ln2hi;
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z = hi - ln2lo;
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c = (hi-z)-ln2lo;
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if (x < 0.443147180559945286227) {
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x = z+0.5; x += exp__E(z,c);
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} else {
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z += exp__E(z,c); x = 0.5 + z;
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}
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return x+x;
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} else if ( x < 70.0) {
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k= invln2 *x+copysign(0.5,x); /* k=NINT(x/ln2) */
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hi=x-k*ln2hi ;
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z=hi-(lo=k*ln2lo);
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c=(hi-z)-lo;
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if(k<=prec) { x=one-scalbn(one,-k); z += exp__E(z,c);}
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else { x = exp__E(z,c)-scalbn(one,-k); x+=z; z=one;}
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return scalbn(x+z,k);
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} else if (x <= lnovft) return exp(x)-1.0;
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else return exp(x);
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}
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}
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/* exp__E(x,c)
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* ASSUMPTION: c << x SO THAT fl(x+c)=x.
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* (c is the correction term for x)
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* exp__E RETURNS
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*
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* / exp(x+c) - 1 - x , 1E-19 < |x| < .3465736
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* exp__E(x,c) = |
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* \ 0 , |x| < 1E-19.
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*
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* DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS)
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* KERNEL FUNCTION OF EXP, EXPM1, POW FUNCTIONS
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* CODED IN C BY K.C. NG, 1/31/85;
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* REVISED BY K.C. NG on 3/16/85, 4/16/85.
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*
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* Required system supported function:
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* copysign(x,y)
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*
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* Method:
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* 1. Rational approximation. Let r=x+c.
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* Based on
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* 2 * sinh(r/2)
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* exp(r) - 1 = ---------------------- ,
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* cosh(r/2) - sinh(r/2)
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* exp__E(r) is computed using
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* x*x (x/2)*W - ( Q - ( 2*P + x*P ) )
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* --- + (c + x*[---------------------------------- + c ])
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* 2 1 - W
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* where P := p1*x^2 + p2*x^4,
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* Q := q1*x^2 + q2*x^4 (for 56 bits precision, add q3*x^6)
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* W := x/2-(Q-x*P),
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*
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* (See the listing below for the values of p1,p2,q1,q2,q3. The poly-
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* nomials P and Q may be regarded as the approximations to sinh
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* and cosh :
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* sinh(r/2) = r/2 + r * P , cosh(r/2) = 1 + Q . )
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*
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* The coefficients were obtained by a special Remez algorithm.
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*
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* Approximation error:
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*
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* | exp(x) - 1 | 2**(-57), (IEEE double)
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* | ------------ - (exp__E(x,0)+x)/x | <=
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* | x | 2**(-69). (VAX D)
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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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#ifdef VAX /* VAX D format */
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/* static double */
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/* p1 = 1.5150724356786683059E-2 , Hex 2^ -6 * .F83ABE67E1066A */
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/* p2 = 6.3112487873718332688E-5 , Hex 2^-13 * .845B4248CD0173 */
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/* q1 = 1.1363478204690669916E-1 , Hex 2^ -3 * .E8B95A44A2EC45 */
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/* q2 = 1.2624568129896839182E-3 , Hex 2^ -9 * .A5790572E4F5E7 */
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/* q3 = 1.5021856115869022674E-6 ; Hex 2^-19 * .C99EB4604AC395 */
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static long p1x[] = { 0x3abe3d78, 0x066a67e1};
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static long p2x[] = { 0x5b423984, 0x017348cd};
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static long q1x[] = { 0xb95a3ee8, 0xec4544a2};
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static long q2x[] = { 0x79053ba5, 0xf5e772e4};
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static long q3x[] = { 0x9eb436c9, 0xc395604a};
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#define p1 (*(double*)p1x)
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#define p2 (*(double*)p2x)
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#define q1 (*(double*)q1x)
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#define q2 (*(double*)q2x)
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#define q3 (*(double*)q3x)
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#else /* IEEE double */
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static double
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p1 = 1.3887401997267371720E-2 , /*Hex 2^ -7 * 1.C70FF8B3CC2CF */
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p2 = 3.3044019718331897649E-5 , /*Hex 2^-15 * 1.15317DF4526C4 */
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q1 = 1.1110813732786649355E-1 , /*Hex 2^ -4 * 1.C719538248597 */
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q2 = 9.9176615021572857300E-4 ; /*Hex 2^-10 * 1.03FC4CB8C98E8 */
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#endif
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static double exp__E(x,c)
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double x,c;
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{
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double static zero=0.0, one=1.0, half=1.0/2.0;
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double copysign(),z,p,q,xp,xh,w;
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z = x*x ;
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p = z*( p1 +z* p2 );
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#ifdef VAX
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q = z*( q1 +z*( q2 +z* q3 ));
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#else /* IEEE double */
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q = z*( q1 +z* q2 );
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#endif
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xp= x*p ;
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xh= x*half ;
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w = xh-(q-xp) ;
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p = p+p;
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c += x*((xh*w-(q-(p+xp)))/(one-w)+c);
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return(z*half+c);
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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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