1 /* 2 * Copyright (c) 1985 Regents of the University of California. 3 * 4 * Use and reproduction of this software are granted in accordance with 5 * the terms and conditions specified in the Berkeley Software License 6 * Agreement (in particular, this entails acknowledgement of the programs' 7 * source, and inclusion of this notice) with the additional understanding 8 * that all recipients should regard themselves as participants in an 9 * ongoing research project and hence should feel obligated to report 10 * their experiences (good or bad) with these elementary function codes, 11 * using "sendbug 4bsd-bugs@BERKELEY", to the authors. 12 */ 13 14 #ifndef lint 15 static char sccsid[] = 16 "@(#)log.c 4.5 (Berkeley) 8/21/85; 1.3 (ucb.elefunt) 03/16/86"; 17 #endif not lint 18 19 /* LOG(X) 20 * RETURN THE LOGARITHM OF x 21 * DOUBLE PRECISION (VAX D FORMAT 56 bits or IEEE DOUBLE 53 BITS) 22 * CODED IN C BY K.C. NG, 1/19/85; 23 * REVISED BY K.C. NG on 2/7/85, 3/7/85, 3/24/85, 4/16/85. 24 * 25 * Required system supported functions: 26 * scalb(x,n) 27 * copysign(x,y) 28 * logb(x) 29 * finite(x) 30 * 31 * Required kernel function: 32 * log__L(z) 33 * 34 * Method : 35 * 1. Argument Reduction: find k and f such that 36 * x = 2^k * (1+f), 37 * where sqrt(2)/2 < 1+f < sqrt(2) . 38 * 39 * 2. Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) 40 * = 2s + 2/3 s**3 + 2/5 s**5 + ....., 41 * log(1+f) is computed by 42 * 43 * log(1+f) = 2s + s*log__L(s*s) 44 * where 45 * log__L(z) = z*(L1 + z*(L2 + z*(... (L6 + z*L7)...))) 46 * 47 * See log__L() for the values of the coefficients. 48 * 49 * 3. Finally, log(x) = k*ln2 + log(1+f). (Here n*ln2 will be stored 50 * in two floating point number: n*ln2hi + n*ln2lo, n*ln2hi is exact 51 * since the last 20 bits of ln2hi is 0.) 52 * 53 * Special cases: 54 * log(x) is NaN with signal if x < 0 (including -INF) ; 55 * log(+INF) is +INF; log(0) is -INF with signal; 56 * log(NaN) is that NaN with no signal. 57 * 58 * Accuracy: 59 * log(x) returns the exact log(x) nearly rounded. In a test run with 60 * 1,536,000 random arguments on a VAX, the maximum observed error was 61 * .826 ulps (units in the last place). 62 * 63 * Constants: 64 * The hexadecimal values are the intended ones for the following constants. 65 * The decimal values may be used, provided that the compiler will convert 66 * from decimal to binary accurately enough to produce the hexadecimal values 67 * shown. 68 */ 69 70 #ifdef VAX /* VAX D format */ 71 #include <errno.h> 72 73 /* static double */ 74 /* ln2hi = 6.9314718055829871446E-1 , Hex 2^ 0 * .B17217F7D00000 */ 75 /* ln2lo = 1.6465949582897081279E-12 , Hex 2^-39 * .E7BCD5E4F1D9CC */ 76 /* sqrt2 = 1.4142135623730950622E0 ; Hex 2^ 1 * .B504F333F9DE65 */ 77 static long ln2hix[] = { 0x72174031, 0x0000f7d0}; 78 static long ln2lox[] = { 0xbcd52ce7, 0xd9cce4f1}; 79 static long sqrt2x[] = { 0x04f340b5, 0xde6533f9}; 80 #define ln2hi (*(double*)ln2hix) 81 #define ln2lo (*(double*)ln2lox) 82 #define sqrt2 (*(double*)sqrt2x) 83 #else /* IEEE double */ 84 static double 85 ln2hi = 6.9314718036912381649E-1 , /*Hex 2^ -1 * 1.62E42FEE00000 */ 86 ln2lo = 1.9082149292705877000E-10 , /*Hex 2^-33 * 1.A39EF35793C76 */ 87 sqrt2 = 1.4142135623730951455E0 ; /*Hex 2^ 0 * 1.6A09E667F3BCD */ 88 #endif 89 90 double log(x) 91 double x; 92 { 93 static double zero=0.0, negone= -1.0, half=1.0/2.0; 94 double logb(),scalb(),copysign(),log__L(),s,z,t; 95 int k,n,finite(); 96 97 #ifndef VAX 98 if(x!=x) return(x); /* x is NaN */ 99 #endif 100 if(finite(x)) { 101 if( x > zero ) { 102 103 /* argument reduction */ 104 k=logb(x); x=scalb(x,-k); 105 if(k == -1022) /* subnormal no. */ 106 {n=logb(x); x=scalb(x,-n); k+=n;} 107 if(x >= sqrt2 ) {k += 1; x *= half;} 108 x += negone ; 109 110 /* compute log(1+x) */ 111 s=x/(2+x); t=x*x*half; 112 z=k*ln2lo+s*(t+log__L(s*s)); 113 x += (z - t) ; 114 115 return(k*ln2hi+x); 116 } 117 /* end of if (x > zero) */ 118 119 else { 120 #ifdef VAX 121 extern double infnan(); 122 if ( x == zero ) 123 return (infnan(-ERANGE)); /* -INF */ 124 else 125 return (infnan(EDOM)); /* NaN */ 126 #else /* IEEE double */ 127 /* zero argument, return -INF with signal */ 128 if ( x == zero ) 129 return( negone/zero ); 130 131 /* negative argument, return NaN with signal */ 132 else 133 return ( zero / zero ); 134 #endif 135 } 136 } 137 /* end of if (finite(x)) */ 138 /* NOT REACHED ifdef VAX */ 139 140 /* log(-INF) is NaN with signal */ 141 else if (x<0) 142 return(zero/zero); 143 144 /* log(+INF) is +INF */ 145 else return(x); 146 147 } 148