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