1 /* $NetBSD: n_expm1.c,v 1.5 2002/06/15 00:10:17 matt 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. All advertising materials mentioning features or use of this software 15 * must display the following acknowledgement: 16 * This product includes software developed by the University of 17 * California, Berkeley and its contributors. 18 * 4. Neither the name of the University nor the names of its contributors 19 * may be used to endorse or promote products derived from this software 20 * without specific prior written permission. 21 * 22 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND 23 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 24 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 25 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE 26 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 27 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 28 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 29 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 30 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY 31 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF 32 * SUCH DAMAGE. 33 */ 34 35 #ifndef lint 36 #if 0 37 static char sccsid[] = "@(#)expm1.c 8.1 (Berkeley) 6/4/93"; 38 #endif 39 #endif /* not lint */ 40 41 /* EXPM1(X) 42 * RETURN THE EXPONENTIAL OF X MINUS ONE 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, 3/7/85, 3/21/85, 4/16/85. 46 * 47 * Required system supported functions: 48 * scalb(x,n) 49 * copysign(x,y) 50 * finite(x) 51 * 52 * Kernel function: 53 * exp__E(x,c) 54 * 55 * Method: 56 * 1. Argument Reduction: given the input x, find r and integer k such 57 * that 58 * x = k*ln2 + r, |r| <= 0.5*ln2 . 59 * r will be represented as r := z+c for better accuracy. 60 * 61 * 2. Compute EXPM1(r)=exp(r)-1 by 62 * 63 * EXPM1(r=z+c) := z + exp__E(z,c) 64 * 65 * 3. EXPM1(x) = 2^k * ( EXPM1(r) + 1-2^-k ). 66 * 67 * Remarks: 68 * 1. When k=1 and z < -0.25, we use the following formula for 69 * better accuracy: 70 * EXPM1(x) = 2 * ( (z+0.5) + exp__E(z,c) ) 71 * 2. To avoid rounding error in 1-2^-k where k is large, we use 72 * EXPM1(x) = 2^k * { [z+(exp__E(z,c)-2^-k )] + 1 } 73 * when k>56. 74 * 75 * Special cases: 76 * EXPM1(INF) is INF, EXPM1(NaN) is NaN; 77 * EXPM1(-INF)= -1; 78 * for finite argument, only EXPM1(0)=0 is exact. 79 * 80 * Accuracy: 81 * EXPM1(x) returns the exact (exp(x)-1) nearly rounded. In a test run with 82 * 1,166,000 random arguments on a VAX, the maximum observed error was 83 * .872 ulps (units of the last place). 84 * 85 * Constants: 86 * The hexadecimal values are the intended ones for the following constants. 87 * The decimal values may be used, provided that the compiler will convert 88 * from decimal to binary accurately enough to produce the hexadecimal values 89 * shown. 90 */ 91 92 #define _LIBM_STATIC 93 #include "mathimpl.h" 94 95 vc(ln2hi, 6.9314718055829871446E-1 ,7217,4031,0000,f7d0, 0, .B17217F7D00000) 96 vc(ln2lo, 1.6465949582897081279E-12 ,bcd5,2ce7,d9cc,e4f1, -39, .E7BCD5E4F1D9CC) 97 vc(lnhuge, 9.4961163736712506989E1 ,ec1d,43bd,9010,a73e, 7, .BDEC1DA73E9010) 98 vc(invln2, 1.4426950408889634148E0 ,aa3b,40b8,17f1,295c, 1, .B8AA3B295C17F1) 99 100 ic(ln2hi, 6.9314718036912381649E-1, -1, 1.62E42FEE00000) 101 ic(ln2lo, 1.9082149292705877000E-10, -33, 1.A39EF35793C76) 102 ic(lnhuge, 7.1602103751842355450E2, 9, 1.6602B15B7ECF2) 103 ic(invln2, 1.4426950408889633870E0, 0, 1.71547652B82FE) 104 105 #ifdef vccast 106 #define ln2hi vccast(ln2hi) 107 #define ln2lo vccast(ln2lo) 108 #define lnhuge vccast(lnhuge) 109 #define invln2 vccast(invln2) 110 #endif 111 112 #if defined(__vax__)||defined(tahoe) 113 #define PREC 56 114 #else /* defined(__vax__)||defined(tahoe) */ 115 #define PREC 53 116 #endif /* defined(__vax__)||defined(tahoe) */ 117 118 double 119 expm1(double x) 120 { 121 const static double one=1.0, half=1.0/2.0; 122 double z,hi,lo,c; 123 int k; 124 125 #if !defined(__vax__)&&!defined(tahoe) 126 if(x!=x) return(x); /* x is NaN */ 127 #endif /* !defined(__vax__)&&!defined(tahoe) */ 128 129 if( x <= lnhuge ) { 130 if( x >= -40.0 ) { 131 132 /* argument reduction : x - k*ln2 */ 133 k= invln2 *x+copysign(0.5,x); /* k=NINT(x/ln2) */ 134 hi=x-k*ln2hi ; 135 z=hi-(lo=k*ln2lo); 136 c=(hi-z)-lo; 137 138 if(k==0) return(z+__exp__E(z,c)); 139 if(k==1) 140 if(z< -0.25) 141 {x=z+half;x +=__exp__E(z,c); return(x+x);} 142 else 143 {z+=__exp__E(z,c); x=half+z; return(x+x);} 144 /* end of k=1 */ 145 146 else { 147 if(k<=PREC) 148 { x=one-scalb(one,-k); z += __exp__E(z,c);} 149 else if(k<100) 150 { x = __exp__E(z,c)-scalb(one,-k); x+=z; z=one;} 151 else 152 { x = __exp__E(z,c)+z; z=one;} 153 154 return (scalb(x+z,k)); 155 } 156 } 157 /* end of x > lnunfl */ 158 159 else 160 /* expm1(-big#) rounded to -1 (inexact) */ 161 if(finite(x)) 162 { c=ln2hi+ln2lo; return(-one);} /* ??? -ragge */ 163 164 /* expm1(-INF) is -1 */ 165 else return(-one); 166 } 167 /* end of x < lnhuge */ 168 169 else 170 /* expm1(INF) is INF, expm1(+big#) overflows to INF */ 171 return( finite(x) ? scalb(one,5000) : x); 172 } 173