1 /* 2 * Copyright (c) 1985 Regents of the University of California. 3 * All rights reserved. 4 * 5 * Redistribution and use in source and binary forms are permitted 6 * provided that the above copyright notice and this paragraph are 7 * duplicated in all such forms and that any documentation, 8 * advertising materials, and other materials related to such 9 * distribution and use acknowledge that the software was developed 10 * by the University of California, Berkeley. The name of the 11 * University may not be used to endorse or promote products derived 12 * from this software without specific prior written permission. 13 * THIS SOFTWARE IS PROVIDED ``AS IS'' AND WITHOUT ANY EXPRESS OR 14 * IMPLIED WARRANTIES, INCLUDING, WITHOUT LIMITATION, THE IMPLIED 15 * WARRANTIES OF MERCHANTIBILITY AND FITNESS FOR A PARTICULAR PURPOSE. 16 * 17 * All recipients should regard themselves as participants in an ongoing 18 * research project and hence should feel obligated to report their 19 * experiences (good or bad) with these elementary function codes, using 20 * the sendbug(8) program, to the authors. 21 */ 22 23 #ifndef lint 24 static char sccsid[] = "@(#)log.c 5.3 (Berkeley) 06/30/88"; 25 #endif /* not lint */ 26 27 /* LOG(X) 28 * RETURN THE LOGARITHM OF x 29 * DOUBLE PRECISION (VAX D FORMAT 56 bits or IEEE DOUBLE 53 BITS) 30 * CODED IN C BY K.C. NG, 1/19/85; 31 * REVISED BY K.C. NG on 2/7/85, 3/7/85, 3/24/85, 4/16/85. 32 * 33 * Required system supported functions: 34 * scalb(x,n) 35 * copysign(x,y) 36 * logb(x) 37 * finite(x) 38 * 39 * Required kernel function: 40 * log__L(z) 41 * 42 * Method : 43 * 1. Argument Reduction: find k and f such that 44 * x = 2^k * (1+f), 45 * where sqrt(2)/2 < 1+f < sqrt(2) . 46 * 47 * 2. Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) 48 * = 2s + 2/3 s**3 + 2/5 s**5 + ....., 49 * log(1+f) is computed by 50 * 51 * log(1+f) = 2s + s*log__L(s*s) 52 * where 53 * log__L(z) = z*(L1 + z*(L2 + z*(... (L6 + z*L7)...))) 54 * 55 * See log__L() for the values of the coefficients. 56 * 57 * 3. Finally, log(x) = k*ln2 + log(1+f). (Here n*ln2 will be stored 58 * in two floating point number: n*ln2hi + n*ln2lo, n*ln2hi is exact 59 * since the last 20 bits of ln2hi is 0.) 60 * 61 * Special cases: 62 * log(x) is NaN with signal if x < 0 (including -INF) ; 63 * log(+INF) is +INF; log(0) is -INF with signal; 64 * log(NaN) is that NaN with no signal. 65 * 66 * Accuracy: 67 * log(x) returns the exact log(x) nearly rounded. In a test run with 68 * 1,536,000 random arguments on a VAX, the maximum observed error was 69 * .826 ulps (units in the last place). 70 * 71 * Constants: 72 * The hexadecimal values are the intended ones for the following constants. 73 * The decimal values may be used, provided that the compiler will convert 74 * from decimal to binary accurately enough to produce the hexadecimal values 75 * shown. 76 */ 77 78 #if defined(vax)||defined(tahoe) /* VAX D format */ 79 #include <errno.h> 80 #ifdef vax 81 #define _0x(A,B) 0x/**/A/**/B 82 #else /* vax */ 83 #define _0x(A,B) 0x/**/B/**/A 84 #endif /* vax */ 85 /* static double */ 86 /* ln2hi = 6.9314718055829871446E-1 , Hex 2^ 0 * .B17217F7D00000 */ 87 /* ln2lo = 1.6465949582897081279E-12 , Hex 2^-39 * .E7BCD5E4F1D9CC */ 88 /* sqrt2 = 1.4142135623730950622E0 ; Hex 2^ 1 * .B504F333F9DE65 */ 89 static long ln2hix[] = { _0x(7217,4031), _0x(0000,f7d0)}; 90 static long ln2lox[] = { _0x(bcd5,2ce7), _0x(d9cc,e4f1)}; 91 static long sqrt2x[] = { _0x(04f3,40b5), _0x(de65,33f9)}; 92 #define ln2hi (*(double*)ln2hix) 93 #define ln2lo (*(double*)ln2lox) 94 #define sqrt2 (*(double*)sqrt2x) 95 #else /* defined(vax)||defined(tahoe) */ 96 static double 97 ln2hi = 6.9314718036912381649E-1 , /*Hex 2^ -1 * 1.62E42FEE00000 */ 98 ln2lo = 1.9082149292705877000E-10 , /*Hex 2^-33 * 1.A39EF35793C76 */ 99 sqrt2 = 1.4142135623730951455E0 ; /*Hex 2^ 0 * 1.6A09E667F3BCD */ 100 #endif /* defined(vax)||defined(tahoe) */ 101 102 double log(x) 103 double x; 104 { 105 static double zero=0.0, negone= -1.0, half=1.0/2.0; 106 double logb(),scalb(),copysign(),log__L(),s,z,t; 107 int k,n,finite(); 108 109 #if !defined(vax)&&!defined(tahoe) 110 if(x!=x) return(x); /* x is NaN */ 111 #endif /* !defined(vax)&&!defined(tahoe) */ 112 if(finite(x)) { 113 if( x > zero ) { 114 115 /* argument reduction */ 116 k=logb(x); x=scalb(x,-k); 117 if(k == -1022) /* subnormal no. */ 118 {n=logb(x); x=scalb(x,-n); k+=n;} 119 if(x >= sqrt2 ) {k += 1; x *= half;} 120 x += negone ; 121 122 /* compute log(1+x) */ 123 s=x/(2+x); t=x*x*half; 124 z=k*ln2lo+s*(t+log__L(s*s)); 125 x += (z - t) ; 126 127 return(k*ln2hi+x); 128 } 129 /* end of if (x > zero) */ 130 131 else { 132 #if defined(vax)||defined(tahoe) 133 extern double infnan(); 134 if ( x == zero ) 135 return (infnan(-ERANGE)); /* -INF */ 136 else 137 return (infnan(EDOM)); /* NaN */ 138 #else /* defined(vax)||defined(tahoe) */ 139 /* zero argument, return -INF with signal */ 140 if ( x == zero ) 141 return( negone/zero ); 142 143 /* negative argument, return NaN with signal */ 144 else 145 return ( zero / zero ); 146 #endif /* defined(vax)||defined(tahoe) */ 147 } 148 } 149 /* end of if (finite(x)) */ 150 /* NOTREACHED if defined(vax)||defined(tahoe) */ 151 152 /* log(-INF) is NaN with signal */ 153 else if (x<0) 154 return(zero/zero); 155 156 /* log(+INF) is +INF */ 157 else return(x); 158 159 } 160