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