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 #ifndef lint 35 static char sccsid[] = "@(#)exp.c 8.1 (Berkeley) 6/4/93"; 36 #endif /* not lint */ 37 #include <sys/cdefs.h> 38 __FBSDID("$FreeBSD$"); 39 40 41 /* EXP(X) 42 * RETURN THE EXPONENTIAL OF X 43 * DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS) 44 * CODED IN C BY K.C. NG, 1/19/85; 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. 46 * 47 * Required system supported functions: 48 * scalb(x,n) 49 * copysign(x,y) 50 * finite(x) 51 * 52 * Method: 53 * 1. Argument Reduction: given the input x, find r and integer k such 54 * that 55 * x = k*ln2 + r, |r| <= 0.5*ln2 . 56 * r will be represented as r := z+c for better accuracy. 57 * 58 * 2. Compute exp(r) by 59 * 60 * exp(r) = 1 + r + r*R1/(2-R1), 61 * where 62 * R1 = x - x^2*(p1+x^2*(p2+x^2*(p3+x^2*(p4+p5*x^2)))). 63 * 64 * 3. exp(x) = 2^k * exp(r) . 65 * 66 * Special cases: 67 * exp(INF) is INF, exp(NaN) is NaN; 68 * exp(-INF)= 0; 69 * for finite argument, only exp(0)=1 is exact. 70 * 71 * Accuracy: 72 * exp(x) returns the exponential of x nearly rounded. In a test run 73 * with 1,156,000 random arguments on a VAX, the maximum observed 74 * error was 0.869 ulps (units in the last place). 75 */ 76 77 #include "mathimpl.h" 78 79 const static double p1 = 0x1.555555555553ep-3; 80 const static double p2 = -0x1.6c16c16bebd93p-9; 81 const static double p3 = 0x1.1566aaf25de2cp-14; 82 const static double p4 = -0x1.bbd41c5d26bf1p-20; 83 const static double p5 = 0x1.6376972bea4d0p-25; 84 const static double ln2hi = 0x1.62e42fee00000p-1; 85 const static double ln2lo = 0x1.a39ef35793c76p-33; 86 const static double lnhuge = 0x1.6602b15b7ecf2p9; 87 const static double lntiny = -0x1.77af8ebeae354p9; 88 const static double invln2 = 0x1.71547652b82fep0; 89 90 #if 0 91 double exp(x) 92 double x; 93 { 94 double z,hi,lo,c; 95 int k; 96 97 #if !defined(vax)&&!defined(tahoe) 98 if(x!=x) return(x); /* x is NaN */ 99 #endif /* !defined(vax)&&!defined(tahoe) */ 100 if( x <= lnhuge ) { 101 if( x >= lntiny ) { 102 103 /* argument reduction : x --> x - k*ln2 */ 104 105 k=invln2*x+copysign(0.5,x); /* k=NINT(x/ln2) */ 106 107 /* express x-k*ln2 as hi-lo and let x=hi-lo rounded */ 108 109 hi=x-k*ln2hi; 110 x=hi-(lo=k*ln2lo); 111 112 /* return 2^k*[1+x+x*c/(2+c)] */ 113 z=x*x; 114 c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5)))); 115 return scalb(1.0+(hi-(lo-(x*c)/(2.0-c))),k); 116 117 } 118 /* end of x > lntiny */ 119 120 else 121 /* exp(-big#) underflows to zero */ 122 if(finite(x)) return(scalb(1.0,-5000)); 123 124 /* exp(-INF) is zero */ 125 else return(0.0); 126 } 127 /* end of x < lnhuge */ 128 129 else 130 /* exp(INF) is INF, exp(+big#) overflows to INF */ 131 return( finite(x) ? scalb(1.0,5000) : x); 132 } 133 #endif 134 135 /* returns exp(r = x + c) for |c| < |x| with no overlap. */ 136 137 double __exp__D(x, c) 138 double x, c; 139 { 140 double z,hi,lo; 141 int k; 142 143 if (x != x) /* x is NaN */ 144 return(x); 145 if ( x <= lnhuge ) { 146 if ( x >= lntiny ) { 147 148 /* argument reduction : x --> x - k*ln2 */ 149 z = invln2*x; 150 k = z + copysign(.5, x); 151 152 /* express (x+c)-k*ln2 as hi-lo and let x=hi-lo rounded */ 153 154 hi=(x-k*ln2hi); /* Exact. */ 155 x= hi - (lo = k*ln2lo-c); 156 /* return 2^k*[1+x+x*c/(2+c)] */ 157 z=x*x; 158 c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5)))); 159 c = (x*c)/(2.0-c); 160 161 return scalb(1.+(hi-(lo - c)), k); 162 } 163 /* end of x > lntiny */ 164 165 else 166 /* exp(-big#) underflows to zero */ 167 if(finite(x)) return(scalb(1.0,-5000)); 168 169 /* exp(-INF) is zero */ 170 else return(0.0); 171 } 172 /* end of x < lnhuge */ 173 174 else 175 /* exp(INF) is INF, exp(+big#) overflows to INF */ 176 return( finite(x) ? scalb(1.0,5000) : x); 177 } 178