1 /* 2 * Copyright (c) 1985 Regents of the University of California. 3 * All rights reserved. 4 * 5 * %sccs.include.redist.c% 6 */ 7 8 #ifndef lint 9 static char sccsid[] = "@(#)expm1.c 5.6 (Berkeley) 10/09/90"; 10 #endif /* not lint */ 11 12 /* EXPM1(X) 13 * RETURN THE EXPONENTIAL OF X MINUS ONE 14 * DOUBLE PRECISION (IEEE 53 BITS, VAX D FORMAT 56 BITS) 15 * CODED IN C BY K.C. NG, 1/19/85; 16 * REVISED BY K.C. NG on 2/6/85, 3/7/85, 3/21/85, 4/16/85. 17 * 18 * Required system supported functions: 19 * scalb(x,n) 20 * copysign(x,y) 21 * finite(x) 22 * 23 * Kernel function: 24 * exp__E(x,c) 25 * 26 * Method: 27 * 1. Argument Reduction: given the input x, find r and integer k such 28 * that 29 * x = k*ln2 + r, |r| <= 0.5*ln2 . 30 * r will be represented as r := z+c for better accuracy. 31 * 32 * 2. Compute EXPM1(r)=exp(r)-1 by 33 * 34 * EXPM1(r=z+c) := z + exp__E(z,c) 35 * 36 * 3. EXPM1(x) = 2^k * ( EXPM1(r) + 1-2^-k ). 37 * 38 * Remarks: 39 * 1. When k=1 and z < -0.25, we use the following formula for 40 * better accuracy: 41 * EXPM1(x) = 2 * ( (z+0.5) + exp__E(z,c) ) 42 * 2. To avoid rounding error in 1-2^-k where k is large, we use 43 * EXPM1(x) = 2^k * { [z+(exp__E(z,c)-2^-k )] + 1 } 44 * when k>56. 45 * 46 * Special cases: 47 * EXPM1(INF) is INF, EXPM1(NaN) is NaN; 48 * EXPM1(-INF)= -1; 49 * for finite argument, only EXPM1(0)=0 is exact. 50 * 51 * Accuracy: 52 * EXPM1(x) returns the exact (exp(x)-1) nearly rounded. In a test run with 53 * 1,166,000 random arguments on a VAX, the maximum observed error was 54 * .872 ulps (units of the last place). 55 * 56 * Constants: 57 * The hexadecimal values are the intended ones for the following constants. 58 * The decimal values may be used, provided that the compiler will convert 59 * from decimal to binary accurately enough to produce the hexadecimal values 60 * shown. 61 */ 62 63 #include "mathimpl.h" 64 65 vc(ln2hi, 6.9314718055829871446E-1 ,7217,4031,0000,f7d0, 0, .B17217F7D00000) 66 vc(ln2lo, 1.6465949582897081279E-12 ,bcd5,2ce7,d9cc,e4f1, -39, .E7BCD5E4F1D9CC) 67 vc(lnhuge, 9.4961163736712506989E1 ,ec1d,43bd,9010,a73e, 7, .BDEC1DA73E9010) 68 vc(invln2, 1.4426950408889634148E0 ,aa3b,40b8,17f1,295c, 1, .B8AA3B295C17F1) 69 70 ic(ln2hi, 6.9314718036912381649E-1, -1, 1.62E42FEE00000) 71 ic(ln2lo, 1.9082149292705877000E-10, -33, 1.A39EF35793C76) 72 ic(lnhuge, 7.1602103751842355450E2, 9, 1.6602B15B7ECF2) 73 ic(invln2, 1.4426950408889633870E0, 0, 1.71547652B82FE) 74 75 #ifdef vccast 76 #define ln2hi vccast(ln2hi) 77 #define ln2lo vccast(ln2lo) 78 #define lnhuge vccast(lnhuge) 79 #define invln2 vccast(invln2) 80 #endif 81 82 double expm1(x) 83 double x; 84 { 85 const static double one=1.0, half=1.0/2.0; 86 double z,hi,lo,c; 87 int k; 88 #if defined(vax)||defined(tahoe) 89 static prec=56; 90 #else /* defined(vax)||defined(tahoe) */ 91 static prec=53; 92 #endif /* defined(vax)||defined(tahoe) */ 93 94 #if !defined(vax)&&!defined(tahoe) 95 if(x!=x) return(x); /* x is NaN */ 96 #endif /* !defined(vax)&&!defined(tahoe) */ 97 98 if( x <= lnhuge ) { 99 if( x >= -40.0 ) { 100 101 /* argument reduction : x - k*ln2 */ 102 k= invln2 *x+copysign(0.5,x); /* k=NINT(x/ln2) */ 103 hi=x-k*ln2hi ; 104 z=hi-(lo=k*ln2lo); 105 c=(hi-z)-lo; 106 107 if(k==0) return(z+exp__E(z,c)); 108 if(k==1) 109 if(z< -0.25) 110 {x=z+half;x +=exp__E(z,c); return(x+x);} 111 else 112 {z+=exp__E(z,c); x=half+z; return(x+x);} 113 /* end of k=1 */ 114 115 else { 116 if(k<=prec) 117 { x=one-scalb(one,-k); z += exp__E(z,c);} 118 else if(k<100) 119 { x = exp__E(z,c)-scalb(one,-k); x+=z; z=one;} 120 else 121 { x = exp__E(z,c)+z; z=one;} 122 123 return (scalb(x+z,k)); 124 } 125 } 126 /* end of x > lnunfl */ 127 128 else 129 /* expm1(-big#) rounded to -1 (inexact) */ 130 if(finite(x)) 131 { ln2hi+ln2lo; return(-one);} 132 133 /* expm1(-INF) is -1 */ 134 else return(-one); 135 } 136 /* end of x < lnhuge */ 137 138 else 139 /* expm1(INF) is INF, expm1(+big#) overflows to INF */ 140 return( finite(x) ? scalb(one,5000) : x); 141 } 142