1 /* $NetBSD: n_expm1.c,v 1.1 1995/10/10 23:36:46 ragge 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 static char sccsid[] = "@(#)expm1.c 8.1 (Berkeley) 6/4/93"; 37 #endif /* not lint */ 38 39 /* EXPM1(X) 40 * RETURN THE EXPONENTIAL OF X MINUS ONE 41 * DOUBLE PRECISION (IEEE 53 BITS, VAX D FORMAT 56 BITS) 42 * CODED IN C BY K.C. NG, 1/19/85; 43 * REVISED BY K.C. NG on 2/6/85, 3/7/85, 3/21/85, 4/16/85. 44 * 45 * Required system supported functions: 46 * scalb(x,n) 47 * copysign(x,y) 48 * finite(x) 49 * 50 * Kernel function: 51 * exp__E(x,c) 52 * 53 * Method: 54 * 1. Argument Reduction: given the input x, find r and integer k such 55 * that 56 * x = k*ln2 + r, |r| <= 0.5*ln2 . 57 * r will be represented as r := z+c for better accuracy. 58 * 59 * 2. Compute EXPM1(r)=exp(r)-1 by 60 * 61 * EXPM1(r=z+c) := z + exp__E(z,c) 62 * 63 * 3. EXPM1(x) = 2^k * ( EXPM1(r) + 1-2^-k ). 64 * 65 * Remarks: 66 * 1. When k=1 and z < -0.25, we use the following formula for 67 * better accuracy: 68 * EXPM1(x) = 2 * ( (z+0.5) + exp__E(z,c) ) 69 * 2. To avoid rounding error in 1-2^-k where k is large, we use 70 * EXPM1(x) = 2^k * { [z+(exp__E(z,c)-2^-k )] + 1 } 71 * when k>56. 72 * 73 * Special cases: 74 * EXPM1(INF) is INF, EXPM1(NaN) is NaN; 75 * EXPM1(-INF)= -1; 76 * for finite argument, only EXPM1(0)=0 is exact. 77 * 78 * Accuracy: 79 * EXPM1(x) returns the exact (exp(x)-1) nearly rounded. In a test run with 80 * 1,166,000 random arguments on a VAX, the maximum observed error was 81 * .872 ulps (units of the last place). 82 * 83 * Constants: 84 * The hexadecimal values are the intended ones for the following constants. 85 * The decimal values may be used, provided that the compiler will convert 86 * from decimal to binary accurately enough to produce the hexadecimal values 87 * shown. 88 */ 89 90 #include "mathimpl.h" 91 92 vc(ln2hi, 6.9314718055829871446E-1 ,7217,4031,0000,f7d0, 0, .B17217F7D00000) 93 vc(ln2lo, 1.6465949582897081279E-12 ,bcd5,2ce7,d9cc,e4f1, -39, .E7BCD5E4F1D9CC) 94 vc(lnhuge, 9.4961163736712506989E1 ,ec1d,43bd,9010,a73e, 7, .BDEC1DA73E9010) 95 vc(invln2, 1.4426950408889634148E0 ,aa3b,40b8,17f1,295c, 1, .B8AA3B295C17F1) 96 97 ic(ln2hi, 6.9314718036912381649E-1, -1, 1.62E42FEE00000) 98 ic(ln2lo, 1.9082149292705877000E-10, -33, 1.A39EF35793C76) 99 ic(lnhuge, 7.1602103751842355450E2, 9, 1.6602B15B7ECF2) 100 ic(invln2, 1.4426950408889633870E0, 0, 1.71547652B82FE) 101 102 #ifdef vccast 103 #define ln2hi vccast(ln2hi) 104 #define ln2lo vccast(ln2lo) 105 #define lnhuge vccast(lnhuge) 106 #define invln2 vccast(invln2) 107 #endif 108 109 double expm1(x) 110 double x; 111 { 112 const static double one=1.0, half=1.0/2.0; 113 double z,hi,lo,c; 114 int k; 115 #if defined(vax)||defined(tahoe) 116 static prec=56; 117 #else /* defined(vax)||defined(tahoe) */ 118 static prec=53; 119 #endif /* defined(vax)||defined(tahoe) */ 120 121 #if !defined(vax)&&!defined(tahoe) 122 if(x!=x) return(x); /* x is NaN */ 123 #endif /* !defined(vax)&&!defined(tahoe) */ 124 125 if( x <= lnhuge ) { 126 if( x >= -40.0 ) { 127 128 /* argument reduction : x - k*ln2 */ 129 k= invln2 *x+copysign(0.5,x); /* k=NINT(x/ln2) */ 130 hi=x-k*ln2hi ; 131 z=hi-(lo=k*ln2lo); 132 c=(hi-z)-lo; 133 134 if(k==0) return(z+__exp__E(z,c)); 135 if(k==1) 136 if(z< -0.25) 137 {x=z+half;x +=__exp__E(z,c); return(x+x);} 138 else 139 {z+=__exp__E(z,c); x=half+z; return(x+x);} 140 /* end of k=1 */ 141 142 else { 143 if(k<=prec) 144 { x=one-scalb(one,-k); z += __exp__E(z,c);} 145 else if(k<100) 146 { x = __exp__E(z,c)-scalb(one,-k); x+=z; z=one;} 147 else 148 { x = __exp__E(z,c)+z; z=one;} 149 150 return (scalb(x+z,k)); 151 } 152 } 153 /* end of x > lnunfl */ 154 155 else 156 /* expm1(-big#) rounded to -1 (inexact) */ 157 if(finite(x)) 158 { ln2hi+ln2lo; return(-one);} 159 160 /* expm1(-INF) is -1 */ 161 else return(-one); 162 } 163 /* end of x < lnhuge */ 164 165 else 166 /* expm1(INF) is INF, expm1(+big#) overflows to INF */ 167 return( finite(x) ? scalb(one,5000) : x); 168 } 169