1 /* $NetBSD: n_exp.c,v 1.9 2014/10/10 20:58:09 martin 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 #ifndef lint 32 #if 0 33 static char sccsid[] = "@(#)exp.c 8.1 (Berkeley) 6/4/93"; 34 #endif 35 #endif /* not lint */ 36 37 /* EXP(X) 38 * RETURN THE EXPONENTIAL OF X 39 * DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS) 40 * CODED IN C BY K.C. NG, 1/19/85; 41 * REVISED BY K.C. NG on 2/6/85, 2/15/85, 3/7/85, 3/24/85, 4/16/85, 6/14/86. 42 * 43 * Required system supported functions: 44 * scalb(x,n) 45 * copysign(x,y) 46 * finite(x) 47 * 48 * Method: 49 * 1. Argument Reduction: given the input x, find r and integer k such 50 * that 51 * x = k*ln2 + r, |r| <= 0.5*ln2 . 52 * r will be represented as r := z+c for better accuracy. 53 * 54 * 2. Compute exp(r) by 55 * 56 * exp(r) = 1 + r + r*R1/(2-R1), 57 * where 58 * R1 = x - x^2*(p1+x^2*(p2+x^2*(p3+x^2*(p4+p5*x^2)))). 59 * 60 * 3. exp(x) = 2^k * exp(r) . 61 * 62 * Special cases: 63 * exp(INF) is INF, exp(NaN) is NaN; 64 * exp(-INF)= 0; 65 * for finite argument, only exp(0)=1 is exact. 66 * 67 * Accuracy: 68 * exp(x) returns the exponential of x nearly rounded. In a test run 69 * with 1,156,000 random arguments on a VAX, the maximum observed 70 * error was 0.869 ulps (units in the last place). 71 * 72 * Constants: 73 * The hexadecimal values are the intended ones for the following constants. 74 * The decimal values may be used, provided that the compiler will convert 75 * from decimal to binary accurately enough to produce the hexadecimal values 76 * shown. 77 */ 78 79 #define _LIBM_STATIC 80 #include "../src/namespace.h" 81 #include "mathimpl.h" 82 83 #ifdef __weak_alias 84 __weak_alias(exp, _exp); 85 __weak_alias(_expl, _exp); 86 __weak_alias(expf, _expf); 87 #endif 88 89 vc(ln2hi, 6.9314718055829871446E-1 ,7217,4031,0000,f7d0, 0, .B17217F7D00000) 90 vc(ln2lo, 1.6465949582897081279E-12 ,bcd5,2ce7,d9cc,e4f1, -39, .E7BCD5E4F1D9CC) 91 vc(lnhuge, 9.4961163736712506989E1 ,ec1d,43bd,9010,a73e, 7, .BDEC1DA73E9010) 92 vc(lntiny,-9.5654310917272452386E1 ,4f01,c3bf,33af,d72e, 7,-.BF4F01D72E33AF) 93 vc(invln2, 1.4426950408889634148E0 ,aa3b,40b8,17f1,295c, 1, .B8AA3B295C17F1) 94 vc(p1, 1.6666666666666602251E-1 ,aaaa,3f2a,a9f1,aaaa, -2, .AAAAAAAAAAA9F1) 95 vc(p2, -2.7777777777015591216E-3 ,0b60,bc36,ec94,b5f5, -8,-.B60B60B5F5EC94) 96 vc(p3, 6.6137563214379341918E-5 ,b355,398a,f15f,792e, -13, .8AB355792EF15F) 97 vc(p4, -1.6533902205465250480E-6 ,ea0e,b6dd,5f84,2e93, -19,-.DDEA0E2E935F84) 98 vc(p5, 4.1381367970572387085E-8 ,bb4b,3431,2683,95f5, -24, .B1BB4B95F52683) 99 100 #ifdef vccast 101 #define ln2hi vccast(ln2hi) 102 #define ln2lo vccast(ln2lo) 103 #define lnhuge vccast(lnhuge) 104 #define lntiny vccast(lntiny) 105 #define invln2 vccast(invln2) 106 #define p1 vccast(p1) 107 #define p2 vccast(p2) 108 #define p3 vccast(p3) 109 #define p4 vccast(p4) 110 #define p5 vccast(p5) 111 #endif 112 113 ic(p1, 1.6666666666666601904E-1, -3, 1.555555555553E) 114 ic(p2, -2.7777777777015593384E-3, -9, -1.6C16C16BEBD93) 115 ic(p3, 6.6137563214379343612E-5, -14, 1.1566AAF25DE2C) 116 ic(p4, -1.6533902205465251539E-6, -20, -1.BBD41C5D26BF1) 117 ic(p5, 4.1381367970572384604E-8, -25, 1.6376972BEA4D0) 118 ic(ln2hi, 6.9314718036912381649E-1, -1, 1.62E42FEE00000) 119 ic(ln2lo, 1.9082149292705877000E-10,-33, 1.A39EF35793C76) 120 ic(lnhuge, 7.1602103751842355450E2, 9, 1.6602B15B7ECF2) 121 ic(lntiny,-7.5137154372698068983E2, 9, -1.77AF8EBEAE354) 122 ic(invln2, 1.4426950408889633870E0, 0, 1.71547652B82FE) 123 124 double 125 exp(double x) 126 { 127 double z,hi,lo,c; 128 int k; 129 130 #if !defined(__vax__)&&!defined(tahoe) 131 if(x!=x) return(x); /* x is NaN */ 132 #endif /* !defined(__vax__)&&!defined(tahoe) */ 133 if( x <= lnhuge ) { 134 if( x >= lntiny ) { 135 136 /* argument reduction : x --> x - k*ln2 */ 137 138 k=invln2*x+copysign(0.5,x); /* k=NINT(x/ln2) */ 139 140 /* express x-k*ln2 as hi-lo and let x=hi-lo rounded */ 141 142 hi=x-k*ln2hi; 143 x=hi-(lo=k*ln2lo); 144 145 /* return 2^k*[1+x+x*c/(2+c)] */ 146 z=x*x; 147 c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5)))); 148 return scalb(1.0+(hi-(lo-(x*c)/(2.0-c))),k); 149 150 } 151 /* end of x > lntiny */ 152 153 else 154 /* exp(-big#) underflows to zero */ 155 if(finite(x)) return(scalb(1.0,-5000)); 156 157 /* exp(-INF) is zero */ 158 else return(0.0); 159 } 160 /* end of x < lnhuge */ 161 162 else 163 /* exp(INF) is INF, exp(+big#) overflows to INF */ 164 return( finite(x) ? scalb(1.0,5000) : x); 165 } 166 167 float 168 expf(float x) 169 { 170 return(exp((double)x)); 171 } 172 173 /* returns exp(r = x + c) for |c| < |x| with no overlap. */ 174 175 double 176 __exp__D(double x, double c) 177 { 178 double z,hi,lo; 179 int k; 180 181 #if !defined(__vax__)&&!defined(tahoe) 182 if (x!=x) return(x); /* x is NaN */ 183 #endif /* !defined(__vax__)&&!defined(tahoe) */ 184 if ( x <= lnhuge ) { 185 if ( x >= lntiny ) { 186 187 /* argument reduction : x --> x - k*ln2 */ 188 z = invln2*x; 189 k = z + copysign(.5, x); 190 191 /* express (x+c)-k*ln2 as hi-lo and let x=hi-lo rounded */ 192 193 hi=(x-k*ln2hi); /* Exact. */ 194 x= hi - (lo = k*ln2lo-c); 195 /* return 2^k*[1+x+x*c/(2+c)] */ 196 z=x*x; 197 c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5)))); 198 c = (x*c)/(2.0-c); 199 200 return scalb(1.+(hi-(lo - c)), k); 201 } 202 /* end of x > lntiny */ 203 204 else 205 /* exp(-big#) underflows to zero */ 206 if(finite(x)) return(scalb(1.0,-5000)); 207 208 /* exp(-INF) is zero */ 209 else return(0.0); 210 } 211 /* end of x < lnhuge */ 212 213 else 214 /* exp(INF) is INF, exp(+big#) overflows to INF */ 215 return( finite(x) ? scalb(1.0,5000) : x); 216 } 217
Indexes created Sat Sep 20 22:09:52 GMT 2025