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 35 /* EXP(X) 36 * RETURN THE EXPONENTIAL OF X 37 * DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS) 38 * CODED IN C BY K.C. NG, 1/19/85; 39 * REVISED BY K.C. NG on 2/6/85, 2/15/85, 3/7/85, 3/24/85, 4/16/85, 6/14/86. 40 * 41 * Required system supported functions: 42 * scalb(x,n) 43 * copysign(x,y) 44 * finite(x) 45 * 46 * Method: 47 * 1. Argument Reduction: given the input x, find r and integer k such 48 * that 49 * x = k*ln2 + r, |r| <= 0.5*ln2 . 50 * r will be represented as r := z+c for better accuracy. 51 * 52 * 2. Compute exp(r) by 53 * 54 * exp(r) = 1 + r + r*R1/(2-R1), 55 * where 56 * R1 = x - x^2*(p1+x^2*(p2+x^2*(p3+x^2*(p4+p5*x^2)))). 57 * 58 * 3. exp(x) = 2^k * exp(r) . 59 * 60 * Special cases: 61 * exp(INF) is INF, exp(NaN) is NaN; 62 * exp(-INF)= 0; 63 * for finite argument, only exp(0)=1 is exact. 64 * 65 * Accuracy: 66 * exp(x) returns the exponential of x nearly rounded. In a test run 67 * with 1,156,000 random arguments on a VAX, the maximum observed 68 * error was 0.869 ulps (units in the last place). 69 */ 70 71 #include "mathimpl.h" 72 73 static const double p1 = 0x1.555555555553ep-3; 74 static const double p2 = -0x1.6c16c16bebd93p-9; 75 static const double p3 = 0x1.1566aaf25de2cp-14; 76 static const double p4 = -0x1.bbd41c5d26bf1p-20; 77 static const double p5 = 0x1.6376972bea4d0p-25; 78 static const double ln2hi = 0x1.62e42fee00000p-1; 79 static const double ln2lo = 0x1.a39ef35793c76p-33; 80 static const double lnhuge = 0x1.6602b15b7ecf2p9; 81 static const double lntiny = -0x1.77af8ebeae354p9; 82 static const double invln2 = 0x1.71547652b82fep0; 83 84 #if 0 85 DLLEXPORT double exp(x) 86 double x; 87 { 88 double z,hi,lo,c; 89 int k; 90 91 #if !defined(vax)&&!defined(tahoe) 92 if(x!=x) return(x); /* x is NaN */ 93 #endif /* !defined(vax)&&!defined(tahoe) */ 94 if( x <= lnhuge ) { 95 if( x >= lntiny ) { 96 97 /* argument reduction : x --> x - k*ln2 */ 98 99 k=invln2*x+copysign(0.5,x); /* k=NINT(x/ln2) */ 100 101 /* express x-k*ln2 as hi-lo and let x=hi-lo rounded */ 102 103 hi=x-k*ln2hi; 104 x=hi-(lo=k*ln2lo); 105 106 /* return 2^k*[1+x+x*c/(2+c)] */ 107 z=x*x; 108 c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5)))); 109 return scalb(1.0+(hi-(lo-(x*c)/(2.0-c))),k); 110 111 } 112 /* end of x > lntiny */ 113 114 else 115 /* exp(-big#) underflows to zero */ 116 if(finite(x)) return(scalb(1.0,-5000)); 117 118 /* exp(-INF) is zero */ 119 else return(0.0); 120 } 121 /* end of x < lnhuge */ 122 123 else 124 /* exp(INF) is INF, exp(+big#) overflows to INF */ 125 return( finite(x) ? scalb(1.0,5000) : x); 126 } 127 #endif 128 129 /* returns exp(r = x + c) for |c| < |x| with no overlap. */ 130 131 double __exp__D(x, c) 132 double x, c; 133 { 134 double z,hi,lo; 135 int k; 136 137 if (x != x) /* x is NaN */ 138 return(x); 139 if ( x <= lnhuge ) { 140 if ( x >= lntiny ) { 141 142 /* argument reduction : x --> x - k*ln2 */ 143 z = invln2*x; 144 k = z + copysign(.5, x); 145 146 /* express (x+c)-k*ln2 as hi-lo and let x=hi-lo rounded */ 147 148 hi=(x-k*ln2hi); /* Exact. */ 149 x= hi - (lo = k*ln2lo-c); 150 /* return 2^k*[1+x+x*c/(2+c)] */ 151 z=x*x; 152 c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5)))); 153 c = (x*c)/(2.0-c); 154 155 return scalbn(1.+(hi-(lo - c)), k); 156 } 157 /* end of x > lntiny */ 158 159 else 160 /* exp(-big#) underflows to zero */ 161 if(isfinite(x)) return(scalbn(1.0,-5000)); 162 163 /* exp(-INF) is zero */ 164 else return(0.0); 165 } 166 /* end of x < lnhuge */ 167 168 else 169 /* exp(INF) is INF, exp(+big#) overflows to INF */ 170 return( isfinite(x) ? scalbn(1.0,5000) : x); 171 } 172