| /* |
| * Copyright (c) 1985, 1993 |
| * The Regents of the University of California. All rights reserved. |
| * |
| * Redistribution and use in source and binary forms, with or without |
| * modification, are permitted provided that the following conditions |
| * are met: |
| * 1. Redistributions of source code must retain the above copyright |
| * notice, this list of conditions and the following disclaimer. |
| * 2. Redistributions in binary form must reproduce the above copyright |
| * notice, this list of conditions and the following disclaimer in the |
| * documentation and/or other materials provided with the distribution. |
| * 3. All advertising materials mentioning features or use of this software |
| * must display the following acknowledgement: |
| * This product includes software developed by the University of |
| * California, Berkeley and its contributors. |
| * 4. Neither the name of the University nor the names of its contributors |
| * may be used to endorse or promote products derived from this software |
| * without specific prior written permission. |
| * |
| * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND |
| * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE |
| * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE |
| * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE |
| * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL |
| * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS |
| * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) |
| * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT |
| * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY |
| * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF |
| * SUCH DAMAGE. |
| */ |
| |
| /* @(#)exp.c 8.1 (Berkeley) 6/4/93 */ |
| #include <sys/cdefs.h> |
| __FBSDID("$FreeBSD$"); |
| |
| |
| /* EXP(X) |
| * RETURN THE EXPONENTIAL OF X |
| * 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, 2/15/85, 3/7/85, 3/24/85, 4/16/85, 6/14/86. |
| * |
| * Required system supported functions: |
| * scalb(x,n) |
| * copysign(x,y) |
| * finite(x) |
| * |
| * 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 exp(r) by |
| * |
| * exp(r) = 1 + r + r*R1/(2-R1), |
| * where |
| * R1 = x - x^2*(p1+x^2*(p2+x^2*(p3+x^2*(p4+p5*x^2)))). |
| * |
| * 3. exp(x) = 2^k * exp(r) . |
| * |
| * Special cases: |
| * exp(INF) is INF, exp(NaN) is NaN; |
| * exp(-INF)= 0; |
| * for finite argument, only exp(0)=1 is exact. |
| * |
| * Accuracy: |
| * exp(x) returns the exponential of x nearly rounded. In a test run |
| * with 1,156,000 random arguments on a VAX, the maximum observed |
| * error was 0.869 ulps (units in the last place). |
| */ |
| |
| #include "mathimpl.h" |
| |
| static const double p1 = 0x1.555555555553ep-3; |
| static const double p2 = -0x1.6c16c16bebd93p-9; |
| static const double p3 = 0x1.1566aaf25de2cp-14; |
| static const double p4 = -0x1.bbd41c5d26bf1p-20; |
| static const double p5 = 0x1.6376972bea4d0p-25; |
| static const double ln2hi = 0x1.62e42fee00000p-1; |
| static const double ln2lo = 0x1.a39ef35793c76p-33; |
| static const double lnhuge = 0x1.6602b15b7ecf2p9; |
| static const double lntiny = -0x1.77af8ebeae354p9; |
| static const double invln2 = 0x1.71547652b82fep0; |
| |
| #if 0 |
| double exp(x) |
| double x; |
| { |
| double z,hi,lo,c; |
| int k; |
| |
| #if !defined(vax)&&!defined(tahoe) |
| if(x!=x) return(x); /* x is NaN */ |
| #endif /* !defined(vax)&&!defined(tahoe) */ |
| if( x <= lnhuge ) { |
| if( x >= lntiny ) { |
| |
| /* argument reduction : x --> x - k*ln2 */ |
| |
| k=invln2*x+copysign(0.5,x); /* k=NINT(x/ln2) */ |
| |
| /* express x-k*ln2 as hi-lo and let x=hi-lo rounded */ |
| |
| hi=x-k*ln2hi; |
| x=hi-(lo=k*ln2lo); |
| |
| /* return 2^k*[1+x+x*c/(2+c)] */ |
| z=x*x; |
| c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5)))); |
| return scalb(1.0+(hi-(lo-(x*c)/(2.0-c))),k); |
| |
| } |
| /* end of x > lntiny */ |
| |
| else |
| /* exp(-big#) underflows to zero */ |
| if(finite(x)) return(scalb(1.0,-5000)); |
| |
| /* exp(-INF) is zero */ |
| else return(0.0); |
| } |
| /* end of x < lnhuge */ |
| |
| else |
| /* exp(INF) is INF, exp(+big#) overflows to INF */ |
| return( finite(x) ? scalb(1.0,5000) : x); |
| } |
| #endif |
| |
| /* returns exp(r = x + c) for |c| < |x| with no overlap. */ |
| |
| double __exp__D(x, c) |
| double x, c; |
| { |
| double z,hi,lo; |
| int k; |
| |
| if (x != x) /* x is NaN */ |
| return(x); |
| if ( x <= lnhuge ) { |
| if ( x >= lntiny ) { |
| |
| /* argument reduction : x --> x - k*ln2 */ |
| z = invln2*x; |
| k = z + copysign(.5, x); |
| |
| /* express (x+c)-k*ln2 as hi-lo and let x=hi-lo rounded */ |
| |
| hi=(x-k*ln2hi); /* Exact. */ |
| x= hi - (lo = k*ln2lo-c); |
| /* return 2^k*[1+x+x*c/(2+c)] */ |
| z=x*x; |
| c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5)))); |
| c = (x*c)/(2.0-c); |
| |
| return scalb(1.+(hi-(lo - c)), k); |
| } |
| /* end of x > lntiny */ |
| |
| else |
| /* exp(-big#) underflows to zero */ |
| if(finite(x)) return(scalb(1.0,-5000)); |
| |
| /* exp(-INF) is zero */ |
| else return(0.0); |
| } |
| /* end of x < lnhuge */ |
| |
| else |
| /* exp(INF) is INF, exp(+big#) overflows to INF */ |
| return( finite(x) ? scalb(1.0,5000) : x); |
| } |
