1994-05-27 05:00:24 +00:00
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/*
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* Copyright (c) 1985, 1993
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* The Regents of the University of California. All rights reserved.
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*
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* Redistribution and use in source and binary forms, with or without
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* modification, are permitted provided that the following conditions
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* are met:
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* 1. Redistributions of source code must retain the above copyright
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* notice, this list of conditions and the following disclaimer.
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* 2. Redistributions in binary form must reproduce the above copyright
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* notice, this list of conditions and the following disclaimer in the
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* documentation and/or other materials provided with the distribution.
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* 3. All advertising materials mentioning features or use of this software
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* must display the following acknowledgement:
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* This product includes software developed by the University of
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* California, Berkeley and its contributors.
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* 4. Neither the name of the University nor the names of its contributors
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* may be used to endorse or promote products derived from this software
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* without specific prior written permission.
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*
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* THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
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* ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
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* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
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* ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
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* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
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* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
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* OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
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* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
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* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
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* OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
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* SUCH DAMAGE.
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*/
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#ifndef lint
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static char sccsid[] = "@(#)exp__E.c 8.1 (Berkeley) 6/4/93";
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#endif /* not lint */
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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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1995-05-30 05:51:47 +00:00
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* exp__E(x,c) = |
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1994-05-27 05:00:24 +00:00
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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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1995-05-30 05:51:47 +00:00
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* copysign(x,y)
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1994-05-27 05:00:24 +00:00
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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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1995-05-30 05:51:47 +00:00
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* 2 * sinh(r/2)
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1994-05-27 05:00:24 +00:00
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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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1995-05-30 05:51:47 +00:00
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* | ------------ - (exp__E(x,0)+x)/x | <=
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1994-05-27 05:00:24 +00:00
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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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#include "mathimpl.h"
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vc(p1, 1.5150724356786683059E-2 ,3abe,3d78,066a,67e1, -6, .F83ABE67E1066A)
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vc(p2, 6.3112487873718332688E-5 ,5b42,3984,0173,48cd, -13, .845B4248CD0173)
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vc(q1, 1.1363478204690669916E-1 ,b95a,3ee8,ec45,44a2, -3, .E8B95A44A2EC45)
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vc(q2, 1.2624568129896839182E-3 ,7905,3ba5,f5e7,72e4, -9, .A5790572E4F5E7)
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vc(q3, 1.5021856115869022674E-6 ,9eb4,36c9,c395,604a, -19, .C99EB4604AC395)
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ic(p1, 1.3887401997267371720E-2, -7, 1.C70FF8B3CC2CF)
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ic(p2, 3.3044019718331897649E-5, -15, 1.15317DF4526C4)
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ic(q1, 1.1110813732786649355E-1, -4, 1.C719538248597)
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ic(q2, 9.9176615021572857300E-4, -10, 1.03FC4CB8C98E8)
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#ifdef vccast
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#define p1 vccast(p1)
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#define p2 vccast(p2)
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#define q1 vccast(q1)
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#define q2 vccast(q2)
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#define q3 vccast(q3)
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#endif
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double __exp__E(x,c)
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double x,c;
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{
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const static double zero=0.0, one=1.0, half=1.0/2.0, small=1.0E-19;
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double z,p,q,xp,xh,w;
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if(copysign(x,one)>small) {
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z = x*x ;
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p = z*( p1 +z* p2 );
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#if defined(vax)||defined(tahoe)
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q = z*( q1 +z*( q2 +z* q3 ));
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#else /* defined(vax)||defined(tahoe) */
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q = z*( q1 +z* q2 );
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#endif /* defined(vax)||defined(tahoe) */
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1995-05-30 05:51:47 +00:00
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xp= x*p ;
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1994-05-27 05:00:24 +00:00
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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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/* end of |x| > small */
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else {
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if(x!=zero) one+small; /* raise the inexact flag */
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return(copysign(zero,x));
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}
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}
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