1 /* $OpenBSD: b_exp__D.c,v 1.5 2009/10/27 23:59:29 deraadt Exp $ */ 2 /* 3 * Copyright (c) 1985, 1993 4 * The Regents of the University of California. All rights reserved. 5 * 6 * Redistribution and use in source and binary forms, with or without 7 * modification, are permitted provided that the following conditions 8 * are met: 9 * 1. Redistributions of source code must retain the above copyright 10 * notice, this list of conditions and the following disclaimer. 11 * 2. Redistributions in binary form must reproduce the above copyright 12 * notice, this list of conditions and the following disclaimer in the 13 * documentation and/or other materials provided with the distribution. 14 * 3. Neither the name of the University nor the names of its contributors 15 * may be used to endorse or promote products derived from this software 16 * without specific prior written permission. 17 * 18 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND 19 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 20 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 21 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE 22 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 23 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 24 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 25 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 26 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY 27 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF 28 * SUCH DAMAGE. 29 */ 30 31 /* EXP(X) 32 * RETURN THE EXPONENTIAL OF X 33 * DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS) 34 * CODED IN C BY K.C. NG, 1/19/85; 35 * REVISED BY K.C. NG on 2/6/85, 2/15/85, 3/7/85, 3/24/85, 4/16/85, 6/14/86. 36 * 37 * Required system supported functions: 38 * scalb(x,n) 39 * copysign(x,y) 40 * finite(x) 41 * 42 * Method: 43 * 1. Argument Reduction: given the input x, find r and integer k such 44 * that 45 * x = k*ln2 + r, |r| <= 0.5*ln2 . 46 * r will be represented as r := z+c for better accuracy. 47 * 48 * 2. Compute exp(r) by 49 * 50 * exp(r) = 1 + r + r*R1/(2-R1), 51 * where 52 * R1 = x - x^2*(p1+x^2*(p2+x^2*(p3+x^2*(p4+p5*x^2)))). 53 * 54 * 3. exp(x) = 2^k * exp(r) . 55 * 56 * Special cases: 57 * exp(INF) is INF, exp(NaN) is NaN; 58 * exp(-INF)= 0; 59 * for finite argument, only exp(0)=1 is exact. 60 * 61 * Accuracy: 62 * exp(x) returns the exponential of x nearly rounded. In a test run 63 * with 1,156,000 random arguments on a VAX, the maximum observed 64 * error was 0.869 ulps (units in the last place). 65 */ 66 67 #include "math.h" 68 #include "math_private.h" 69 70 static const double p1 = 0x1.555555555553ep-3; 71 static const double p2 = -0x1.6c16c16bebd93p-9; 72 static const double p3 = 0x1.1566aaf25de2cp-14; 73 static const double p4 = -0x1.bbd41c5d26bf1p-20; 74 static const double p5 = 0x1.6376972bea4d0p-25; 75 static const double ln2hi = 0x1.62e42fee00000p-1; 76 static const double ln2lo = 0x1.a39ef35793c76p-33; 77 static const double lnhuge = 0x1.6602b15b7ecf2p9; 78 static const double lntiny = -0x1.77af8ebeae354p9; 79 static const double invln2 = 0x1.71547652b82fep0; 80 81 /* returns exp(r = x + c) for |c| < |x| with no overlap. */ 82 83 double 84 __exp__D(double x, double c) 85 { 86 double z, hi, lo; 87 int k; 88 89 if (isnan(x)) /* x is NaN */ 90 return(x); 91 if ( x <= lnhuge ) { 92 if ( x >= lntiny ) { 93 94 /* argument reduction : x --> x - k*ln2 */ 95 z = invln2*x; 96 k = z + copysign(.5, x); 97 98 /* express (x+c)-k*ln2 as hi-lo and let x=hi-lo rounded */ 99 100 hi=(x-k*ln2hi); /* Exact. */ 101 x= hi - (lo = k*ln2lo-c); 102 /* return 2^k*[1+x+x*c/(2+c)] */ 103 z=x*x; 104 c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5)))); 105 c = (x*c)/(2.0-c); 106 107 return scalb(1.+(hi-(lo - c)), k); 108 } 109 /* end of x > lntiny */ 110 111 else 112 /* exp(-big#) underflows to zero */ 113 if(finite(x)) return(scalb(1.0,-5000)); 114 115 /* exp(-INF) is zero */ 116 else return(0.0); 117 } 118 /* end of x < lnhuge */ 119 120 else 121 /* exp(INF) is INF, exp(+big#) overflows to INF */ 122 return( finite(x) ? scalb(1.0,5000) : x); 123 } 124