1 /* $NetBSD: n_log1p.c,v 1.1 1995/10/10 23:37:00 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[] = "@(#)log1p.c 8.1 (Berkeley) 6/4/93"; 37 #endif /* not lint */ 38 39 /* LOG1P(x) 40 * RETURN THE LOGARITHM OF 1+x 41 * DOUBLE PRECISION (VAX D FORMAT 56 bits, IEEE DOUBLE 53 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, 3/24/85, 4/16/85. 44 * 45 * Required system supported functions: 46 * scalb(x,n) 47 * copysign(x,y) 48 * logb(x) 49 * finite(x) 50 * 51 * Required kernel function: 52 * log__L(z) 53 * 54 * Method : 55 * 1. Argument Reduction: find k and f such that 56 * 1+x = 2^k * (1+f), 57 * where sqrt(2)/2 < 1+f < sqrt(2) . 58 * 59 * 2. Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) 60 * = 2s + 2/3 s**3 + 2/5 s**5 + ....., 61 * log(1+f) is computed by 62 * 63 * log(1+f) = 2s + s*log__L(s*s) 64 * where 65 * log__L(z) = z*(L1 + z*(L2 + z*(... (L6 + z*L7)...))) 66 * 67 * See log__L() for the values of the coefficients. 68 * 69 * 3. Finally, log(1+x) = k*ln2 + log(1+f). 70 * 71 * Remarks 1. In step 3 n*ln2 will be stored in two floating point numbers 72 * n*ln2hi + n*ln2lo, where ln2hi is chosen such that the last 73 * 20 bits (for VAX D format), or the last 21 bits ( for IEEE 74 * double) is 0. This ensures n*ln2hi is exactly representable. 75 * 2. In step 1, f may not be representable. A correction term c 76 * for f is computed. It follows that the correction term for 77 * f - t (the leading term of log(1+f) in step 2) is c-c*x. We 78 * add this correction term to n*ln2lo to attenuate the error. 79 * 80 * 81 * Special cases: 82 * log1p(x) is NaN with signal if x < -1; log1p(NaN) is NaN with no signal; 83 * log1p(INF) is +INF; log1p(-1) is -INF with signal; 84 * only log1p(0)=0 is exact for finite argument. 85 * 86 * Accuracy: 87 * log1p(x) returns the exact log(1+x) nearly rounded. In a test run 88 * with 1,536,000 random arguments on a VAX, the maximum observed 89 * error was .846 ulps (units in the last place). 90 * 91 * Constants: 92 * The hexadecimal values are the intended ones for the following constants. 93 * The decimal values may be used, provided that the compiler will convert 94 * from decimal to binary accurately enough to produce the hexadecimal values 95 * shown. 96 */ 97 98 #include <errno.h> 99 #include "mathimpl.h" 100 101 vc(ln2hi, 6.9314718055829871446E-1 ,7217,4031,0000,f7d0, 0, .B17217F7D00000) 102 vc(ln2lo, 1.6465949582897081279E-12 ,bcd5,2ce7,d9cc,e4f1, -39, .E7BCD5E4F1D9CC) 103 vc(sqrt2, 1.4142135623730950622E0 ,04f3,40b5,de65,33f9, 1, .B504F333F9DE65) 104 105 ic(ln2hi, 6.9314718036912381649E-1, -1, 1.62E42FEE00000) 106 ic(ln2lo, 1.9082149292705877000E-10, -33, 1.A39EF35793C76) 107 ic(sqrt2, 1.4142135623730951455E0, 0, 1.6A09E667F3BCD) 108 109 #ifdef vccast 110 #define ln2hi vccast(ln2hi) 111 #define ln2lo vccast(ln2lo) 112 #define sqrt2 vccast(sqrt2) 113 #endif 114 115 double log1p(x) 116 double x; 117 { 118 const static double zero=0.0, negone= -1.0, one=1.0, 119 half=1.0/2.0, small=1.0E-20; /* 1+small == 1 */ 120 double z,s,t,c; 121 int k; 122 123 #if !defined(vax)&&!defined(tahoe) 124 if(x!=x) return(x); /* x is NaN */ 125 #endif /* !defined(vax)&&!defined(tahoe) */ 126 127 if(finite(x)) { 128 if( x > negone ) { 129 130 /* argument reduction */ 131 if(copysign(x,one)<small) return(x); 132 k=logb(one+x); z=scalb(x,-k); t=scalb(one,-k); 133 if(z+t >= sqrt2 ) 134 { k += 1 ; z *= half; t *= half; } 135 t += negone; x = z + t; 136 c = (t-x)+z ; /* correction term for x */ 137 138 /* compute log(1+x) */ 139 s = x/(2+x); t = x*x*half; 140 c += (k*ln2lo-c*x); 141 z = c+s*(t+__log__L(s*s)); 142 x += (z - t) ; 143 144 return(k*ln2hi+x); 145 } 146 /* end of if (x > negone) */ 147 148 else { 149 #if defined(vax)||defined(tahoe) 150 if ( x == negone ) 151 return (infnan(-ERANGE)); /* -INF */ 152 else 153 return (infnan(EDOM)); /* NaN */ 154 #else /* defined(vax)||defined(tahoe) */ 155 /* x = -1, return -INF with signal */ 156 if ( x == negone ) return( negone/zero ); 157 158 /* negative argument for log, return NaN with signal */ 159 else return ( zero / zero ); 160 #endif /* defined(vax)||defined(tahoe) */ 161 } 162 } 163 /* end of if (finite(x)) */ 164 165 /* log(-INF) is NaN */ 166 else if(x<0) 167 return(zero/zero); 168 169 /* log(+INF) is INF */ 170 else return(x); 171 } 172