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. Neither the name of the University nor the names of its contributors 14 * may be used to endorse or promote products derived from this software 15 * without specific prior written permission. 16 * 17 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND 18 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 19 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 20 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE 21 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 22 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 23 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 24 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 25 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY 26 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF 27 * SUCH DAMAGE. 28 */ 29 30 /* @(#)exp.c 8.1 (Berkeley) 6/4/93 */ 31 #include "cdefs-compat.h" 32 //__FBSDID("$FreeBSD: src/lib/msun/bsdsrc/b_exp.c,v 1.9 2011/10/16 05:37:20 das Exp $"); 33 34 #include <openlibm_math.h> 35 36 /* EXP(X) 37 * RETURN THE EXPONENTIAL OF X 38 * DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS) 39 * CODED IN C BY K.C. NG, 1/19/85; 40 * REVISED BY K.C. NG on 2/6/85, 2/15/85, 3/7/85, 3/24/85, 4/16/85, 6/14/86. 41 * 42 * Required system supported functions: 43 * scalb(x,n) 44 * copysign(x,y) 45 * finite(x) 46 * 47 * Method: 48 * 1. Argument Reduction: given the input x, find r and integer k such 49 * that 50 * x = k*ln2 + r, |r| <= 0.5*ln2 . 51 * r will be represented as r := z+c for better accuracy. 52 * 53 * 2. Compute exp(r) by 54 * 55 * exp(r) = 1 + r + r*R1/(2-R1), 56 * where 57 * R1 = x - x^2*(p1+x^2*(p2+x^2*(p3+x^2*(p4+p5*x^2)))). 58 * 59 * 3. exp(x) = 2^k * exp(r) . 60 * 61 * Special cases: 62 * exp(INF) is INF, exp(NaN) is NaN; 63 * exp(-INF)= 0; 64 * for finite argument, only exp(0)=1 is exact. 65 * 66 * Accuracy: 67 * exp(x) returns the exponential of x nearly rounded. In a test run 68 * with 1,156,000 random arguments on a VAX, the maximum observed 69 * error was 0.869 ulps (units in the last place). 70 */ 71 72 #include "mathimpl.h" 73 74 static const double p1 = 0x1.555555555553ep-3; 75 static const double p2 = -0x1.6c16c16bebd93p-9; 76 static const double p3 = 0x1.1566aaf25de2cp-14; 77 static const double p4 = -0x1.bbd41c5d26bf1p-20; 78 static const double p5 = 0x1.6376972bea4d0p-25; 79 static const double ln2hi = 0x1.62e42fee00000p-1; 80 static const double ln2lo = 0x1.a39ef35793c76p-33; 81 static const double lnhuge = 0x1.6602b15b7ecf2p9; 82 static const double lntiny = -0x1.77af8ebeae354p9; 83 static const double invln2 = 0x1.71547652b82fep0; 84 85 #if 0 86 OLM_DLLEXPORT double exp(x) 87 double x; 88 { 89 double z,hi,lo,c; 90 int k; 91 92 #if !defined(vax)&&!defined(tahoe) 93 if(x!=x) return(x); /* x is NaN */ 94 #endif /* !defined(vax)&&!defined(tahoe) */ 95 if( x <= lnhuge ) { 96 if( x >= lntiny ) { 97 98 /* argument reduction : x --> x - k*ln2 */ 99 100 k=invln2*x+copysign(0.5,x); /* k=NINT(x/ln2) */ 101 102 /* express x-k*ln2 as hi-lo and let x=hi-lo rounded */ 103 104 hi=x-k*ln2hi; 105 x=hi-(lo=k*ln2lo); 106 107 /* return 2^k*[1+x+x*c/(2+c)] */ 108 z=x*x; 109 c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5)))); 110 return scalb(1.0+(hi-(lo-(x*c)/(2.0-c))),k); 111 112 } 113 /* end of x > lntiny */ 114 115 else 116 /* exp(-big#) underflows to zero */ 117 if(finite(x)) return(scalb(1.0,-5000)); 118 119 /* exp(-INF) is zero */ 120 else return(0.0); 121 } 122 /* end of x < lnhuge */ 123 124 else 125 /* exp(INF) is INF, exp(+big#) overflows to INF */ 126 return( finite(x) ? scalb(1.0,5000) : x); 127 } 128 #endif 129 130 /* returns exp(r = x + c) for |c| < |x| with no overlap. */ 131 132 double __exp__D(x, c) 133 double x, c; 134 { 135 double z,hi,lo; 136 int k; 137 138 if (x != x) /* x is NaN */ 139 return(x); 140 if ( x <= lnhuge ) { 141 if ( x >= lntiny ) { 142 143 /* argument reduction : x --> x - k*ln2 */ 144 z = invln2*x; 145 k = z + copysign(.5, x); 146 147 /* express (x+c)-k*ln2 as hi-lo and let x=hi-lo rounded */ 148 149 hi=(x-k*ln2hi); /* Exact. */ 150 x= hi - (lo = k*ln2lo-c); 151 /* return 2^k*[1+x+x*c/(2+c)] */ 152 z=x*x; 153 c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5)))); 154 c = (x*c)/(2.0-c); 155 156 return scalbn(1.+(hi-(lo - c)), k); 157 } 158 /* end of x > lntiny */ 159 160 else 161 /* exp(-big#) underflows to zero */ 162 if(isfinite(x)) return(scalbn(1.0,-5000)); 163 164 /* exp(-INF) is zero */ 165 else return(0.0); 166 } 167 /* end of x < lnhuge */ 168 169 else 170 /* exp(INF) is INF, exp(+big#) overflows to INF */ 171 return( isfinite(x) ? scalbn(1.0,5000) : x); 172 } 173