| 1 | /*
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| 2 | * Copyright (c) 1985, 1993
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| 3 | * The Regents of the University of California. All rights reserved.
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| 4 | *
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| 5 | * Redistribution and use in source and binary forms, with or without
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| 6 | * modification, are permitted provided that the following conditions
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| 7 | * are met:
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| 8 | * 1. Redistributions of source code must retain the above copyright
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| 9 | * notice, this list of conditions and the following disclaimer.
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| 10 | * 2. Redistributions in binary form must reproduce the above copyright
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| 11 | * notice, this list of conditions and the following disclaimer in the
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| 12 | * documentation and/or other materials provided with the distribution.
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| 13 | * 3. All advertising materials mentioning features or use of this software
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| 14 | * must display the following acknowledgement:
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| 15 | * This product includes software developed by the University of
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| 16 | * California, Berkeley and its contributors.
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| 17 | * 4. Neither the name of the University nor the names of its contributors
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| 18 | * may be used to endorse or promote products derived from this software
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| 19 | * without specific prior written permission.
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| 20 | *
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| 21 | * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
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| 22 | * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
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| 23 | * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
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| 24 | * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
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| 25 | * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
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| 26 | * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
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| 27 | * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
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| 28 | * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
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| 29 | * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
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| 30 | * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
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| 31 | * SUCH DAMAGE.
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| 32 | */
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| 33 |
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| 34 | #ifndef lint
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| 35 | static char sccsid[] = "@(#)exp.c 8.1 (Berkeley) 6/4/93";
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| 36 | #endif /* not lint */
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| 37 | #include <sys/cdefs.h>
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| 38 | __FBSDID("$FreeBSD: src/lib/msun/bsdsrc/b_exp.c,v 1.7 2004/12/16 20:40:37 das Exp $");
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| 39 |
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| 40 |
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| 41 | /* EXP(X)
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| 42 | * RETURN THE EXPONENTIAL OF X
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| 43 | * DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS)
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| 44 | * CODED IN C BY K.C. NG, 1/19/85;
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| 45 | * REVISED BY K.C. NG on 2/6/85, 2/15/85, 3/7/85, 3/24/85, 4/16/85, 6/14/86.
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| 46 | *
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| 47 | * Required system supported functions:
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| 48 | * scalb(x,n)
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| 49 | * copysign(x,y)
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| 50 | * finite(x)
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| 51 | *
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| 52 | * Method:
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| 53 | * 1. Argument Reduction: given the input x, find r and integer k such
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| 54 | * that
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| 55 | * x = k*ln2 + r, |r| <= 0.5*ln2 .
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| 56 | * r will be represented as r := z+c for better accuracy.
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| 57 | *
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| 58 | * 2. Compute exp(r) by
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| 59 | *
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| 60 | * exp(r) = 1 + r + r*R1/(2-R1),
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| 61 | * where
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| 62 | * R1 = x - x^2*(p1+x^2*(p2+x^2*(p3+x^2*(p4+p5*x^2)))).
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| 63 | *
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| 64 | * 3. exp(x) = 2^k * exp(r) .
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| 65 | *
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| 66 | * Special cases:
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| 67 | * exp(INF) is INF, exp(NaN) is NaN;
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| 68 | * exp(-INF)= 0;
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| 69 | * for finite argument, only exp(0)=1 is exact.
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| 70 | *
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| 71 | * Accuracy:
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| 72 | * exp(x) returns the exponential of x nearly rounded. In a test run
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| 73 | * with 1,156,000 random arguments on a VAX, the maximum observed
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| 74 | * error was 0.869 ulps (units in the last place).
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| 75 | */
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| 76 |
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| 77 | #include "mathimpl.h"
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| 78 |
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| 79 | const static double p1 = 0x1.555555555553ep-3;
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| 80 | const static double p2 = -0x1.6c16c16bebd93p-9;
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| 81 | const static double p3 = 0x1.1566aaf25de2cp-14;
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| 82 | const static double p4 = -0x1.bbd41c5d26bf1p-20;
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| 83 | const static double p5 = 0x1.6376972bea4d0p-25;
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| 84 | const static double ln2hi = 0x1.62e42fee00000p-1;
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| 85 | const static double ln2lo = 0x1.a39ef35793c76p-33;
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| 86 | const static double lnhuge = 0x1.6602b15b7ecf2p9;
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| 87 | const static double lntiny = -0x1.77af8ebeae354p9;
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| 88 | const static double invln2 = 0x1.71547652b82fep0;
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| 89 |
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| 90 | #if 0
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| 91 | double exp(x)
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| 92 | double x;
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| 93 | {
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| 94 | double z,hi,lo,c;
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| 95 | int k;
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| 96 |
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| 97 | #if !defined(vax)&&!defined(tahoe)
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| 98 | if(x!=x) return(x); /* x is NaN */
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| 99 | #endif /* !defined(vax)&&!defined(tahoe) */
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| 100 | if( x <= lnhuge ) {
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| 101 | if( x >= lntiny ) {
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| 102 |
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| 103 | /* argument reduction : x --> x - k*ln2 */
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| 104 |
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| 105 | k=invln2*x+copysign(0.5,x); /* k=NINT(x/ln2) */
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| 106 |
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| 107 | /* express x-k*ln2 as hi-lo and let x=hi-lo rounded */
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| 108 |
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| 109 | hi=x-k*ln2hi;
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| 110 | x=hi-(lo=k*ln2lo);
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| 111 |
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| 112 | /* return 2^k*[1+x+x*c/(2+c)] */
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| 113 | z=x*x;
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| 114 | c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5))));
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| 115 | return scalb(1.0+(hi-(lo-(x*c)/(2.0-c))),k);
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| 116 |
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| 117 | }
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| 118 | /* end of x > lntiny */
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| 119 |
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| 120 | else
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| 121 | /* exp(-big#) underflows to zero */
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| 122 | if(finite(x)) return(scalb(1.0,-5000));
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| 123 |
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| 124 | /* exp(-INF) is zero */
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| 125 | else return(0.0);
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| 126 | }
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| 127 | /* end of x < lnhuge */
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| 128 |
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| 129 | else
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| 130 | /* exp(INF) is INF, exp(+big#) overflows to INF */
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| 131 | return( finite(x) ? scalb(1.0,5000) : x);
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| 132 | }
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| 133 | #endif
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| 134 |
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| 135 | /* returns exp(r = x + c) for |c| < |x| with no overlap. */
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| 136 |
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| 137 | double __exp__D(x, c)
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| 138 | double x, c;
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| 139 | {
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| 140 | double z,hi,lo;
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| 141 | int k;
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| 142 |
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| 143 | if (x != x) /* x is NaN */
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| 144 | return(x);
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| 145 | if ( x <= lnhuge ) {
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| 146 | if ( x >= lntiny ) {
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| 147 |
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| 148 | /* argument reduction : x --> x - k*ln2 */
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| 149 | z = invln2*x;
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| 150 | k = z + copysign(.5, x);
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| 151 |
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| 152 | /* express (x+c)-k*ln2 as hi-lo and let x=hi-lo rounded */
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| 153 |
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| 154 | hi=(x-k*ln2hi); /* Exact. */
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| 155 | x= hi - (lo = k*ln2lo-c);
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| 156 | /* return 2^k*[1+x+x*c/(2+c)] */
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| 157 | z=x*x;
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| 158 | c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5))));
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| 159 | c = (x*c)/(2.0-c);
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| 160 |
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| 161 | return scalb(1.+(hi-(lo - c)), k);
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| 162 | }
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| 163 | /* end of x > lntiny */
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| 164 |
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| 165 | else
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| 166 | /* exp(-big#) underflows to zero */
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| 167 | if(finite(x)) return(scalb(1.0,-5000));
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| 168 |
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| 169 | /* exp(-INF) is zero */
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| 170 | else return(0.0);
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| 171 | }
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| 172 | /* end of x < lnhuge */
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| 173 |
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| 174 | else
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| 175 | /* exp(INF) is INF, exp(+big#) overflows to INF */
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| 176 | return( finite(x) ? scalb(1.0,5000) : x);
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| 177 | }
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