b_exp.c revision 84210
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#include <sys/cdefs.h> 35__FBSDID("$FreeBSD: head/lib/msun/bsdsrc/b_exp.c 84210 2001-09-30 21:53:45Z dillon $"); 36 37#ifndef lint 38static char sccsid[] = "@(#)exp.c 8.1 (Berkeley) 6/4/93"; 39#endif /* not lint */ 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 * Constants: 77 * The hexadecimal values are the intended ones for the following constants. 78 * The decimal values may be used, provided that the compiler will convert 79 * from decimal to binary accurately enough to produce the hexadecimal values 80 * shown. 81 */ 82 83#include "mathimpl.h" 84 85vc(ln2hi, 6.9314718055829871446E-1 ,7217,4031,0000,f7d0, 0, .B17217F7D00000) 86vc(ln2lo, 1.6465949582897081279E-12 ,bcd5,2ce7,d9cc,e4f1, -39, .E7BCD5E4F1D9CC) 87vc(lnhuge, 9.4961163736712506989E1 ,ec1d,43bd,9010,a73e, 7, .BDEC1DA73E9010) 88vc(lntiny,-9.5654310917272452386E1 ,4f01,c3bf,33af,d72e, 7,-.BF4F01D72E33AF) 89vc(invln2, 1.4426950408889634148E0 ,aa3b,40b8,17f1,295c, 1, .B8AA3B295C17F1) 90vc(p1, 1.6666666666666602251E-1 ,aaaa,3f2a,a9f1,aaaa, -2, .AAAAAAAAAAA9F1) 91vc(p2, -2.7777777777015591216E-3 ,0b60,bc36,ec94,b5f5, -8,-.B60B60B5F5EC94) 92vc(p3, 6.6137563214379341918E-5 ,b355,398a,f15f,792e, -13, .8AB355792EF15F) 93vc(p4, -1.6533902205465250480E-6 ,ea0e,b6dd,5f84,2e93, -19,-.DDEA0E2E935F84) 94vc(p5, 4.1381367970572387085E-8 ,bb4b,3431,2683,95f5, -24, .B1BB4B95F52683) 95 96#ifdef vccast 97#define ln2hi vccast(ln2hi) 98#define ln2lo vccast(ln2lo) 99#define lnhuge vccast(lnhuge) 100#define lntiny vccast(lntiny) 101#define invln2 vccast(invln2) 102#define p1 vccast(p1) 103#define p2 vccast(p2) 104#define p3 vccast(p3) 105#define p4 vccast(p4) 106#define p5 vccast(p5) 107#endif 108 109ic(p1, 1.6666666666666601904E-1, -3, 1.555555555553E) 110ic(p2, -2.7777777777015593384E-3, -9, -1.6C16C16BEBD93) 111ic(p3, 6.6137563214379343612E-5, -14, 1.1566AAF25DE2C) 112ic(p4, -1.6533902205465251539E-6, -20, -1.BBD41C5D26BF1) 113ic(p5, 4.1381367970572384604E-8, -25, 1.6376972BEA4D0) 114ic(ln2hi, 6.9314718036912381649E-1, -1, 1.62E42FEE00000) 115ic(ln2lo, 1.9082149292705877000E-10,-33, 1.A39EF35793C76) 116ic(lnhuge, 7.1602103751842355450E2, 9, 1.6602B15B7ECF2) 117ic(lntiny,-7.5137154372698068983E2, 9, -1.77AF8EBEAE354) 118ic(invln2, 1.4426950408889633870E0, 0, 1.71547652B82FE) 119 120double exp(x) 121double x; 122{ 123 double z,hi,lo,c; 124 int k; 125 126#if !defined(vax)&&!defined(tahoe) 127 if(x!=x) return(x); /* x is NaN */ 128#endif /* !defined(vax)&&!defined(tahoe) */ 129 if( x <= lnhuge ) { 130 if( x >= lntiny ) { 131 132 /* argument reduction : x --> x - k*ln2 */ 133 134 k=invln2*x+copysign(0.5,x); /* k=NINT(x/ln2) */ 135 136 /* express x-k*ln2 as hi-lo and let x=hi-lo rounded */ 137 138 hi=x-k*ln2hi; 139 x=hi-(lo=k*ln2lo); 140 141 /* return 2^k*[1+x+x*c/(2+c)] */ 142 z=x*x; 143 c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5)))); 144 return scalb(1.0+(hi-(lo-(x*c)/(2.0-c))),k); 145 146 } 147 /* end of x > lntiny */ 148 149 else 150 /* exp(-big#) underflows to zero */ 151 if(finite(x)) return(scalb(1.0,-5000)); 152 153 /* exp(-INF) is zero */ 154 else return(0.0); 155 } 156 /* end of x < lnhuge */ 157 158 else 159 /* exp(INF) is INF, exp(+big#) overflows to INF */ 160 return( finite(x) ? scalb(1.0,5000) : x); 161} 162 163/* returns exp(r = x + c) for |c| < |x| with no overlap. */ 164 165double __exp__D(x, c) 166double x, c; 167{ 168 double z,hi,lo, t; 169 int k; 170 171#if !defined(vax)&&!defined(tahoe) 172 if (x!=x) return(x); /* x is NaN */ 173#endif /* !defined(vax)&&!defined(tahoe) */ 174 if ( x <= lnhuge ) { 175 if ( x >= lntiny ) { 176 177 /* argument reduction : x --> x - k*ln2 */ 178 z = invln2*x; 179 k = z + copysign(.5, x); 180 181 /* express (x+c)-k*ln2 as hi-lo and let x=hi-lo rounded */ 182 183 hi=(x-k*ln2hi); /* Exact. */ 184 x= hi - (lo = k*ln2lo-c); 185 /* return 2^k*[1+x+x*c/(2+c)] */ 186 z=x*x; 187 c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5)))); 188 c = (x*c)/(2.0-c); 189 190 return scalb(1.+(hi-(lo - c)), k); 191 } 192 /* end of x > lntiny */ 193 194 else 195 /* exp(-big#) underflows to zero */ 196 if(finite(x)) return(scalb(1.0,-5000)); 197 198 /* exp(-INF) is zero */ 199 else return(0.0); 200 } 201 /* end of x < lnhuge */ 202 203 else 204 /* exp(INF) is INF, exp(+big#) overflows to INF */ 205 return( finite(x) ? scalb(1.0,5000) : x); 206} 207