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[] = "@(#)log.c 4.5 (Berkeley) 08/21/85"; 16 #endif not lint 17 18 /* LOG(X) 19 * RETURN THE LOGARITHM OF x 20 * DOUBLE PRECISION (VAX D FORMAT 56 bits or IEEE DOUBLE 53 BITS) 21 * CODED IN C BY K.C. NG, 1/19/85; 22 * REVISED BY K.C. NG on 2/7/85, 3/7/85, 3/24/85, 4/16/85. 23 * 24 * Required system supported functions: 25 * scalb(x,n) 26 * copysign(x,y) 27 * logb(x) 28 * finite(x) 29 * 30 * Required kernel function: 31 * log__L(z) 32 * 33 * Method : 34 * 1. Argument Reduction: find k and f such that 35 * x = 2^k * (1+f), 36 * where sqrt(2)/2 < 1+f < sqrt(2) . 37 * 38 * 2. Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) 39 * = 2s + 2/3 s**3 + 2/5 s**5 + ....., 40 * log(1+f) is computed by 41 * 42 * log(1+f) = 2s + s*log__L(s*s) 43 * where 44 * log__L(z) = z*(L1 + z*(L2 + z*(... (L6 + z*L7)...))) 45 * 46 * See log__L() for the values of the coefficients. 47 * 48 * 3. Finally, log(x) = k*ln2 + log(1+f). (Here n*ln2 will be stored 49 * in two floating point number: n*ln2hi + n*ln2lo, n*ln2hi is exact 50 * since the last 20 bits of ln2hi is 0.) 51 * 52 * Special cases: 53 * log(x) is NaN with signal if x < 0 (including -INF) ; 54 * log(+INF) is +INF; log(0) is -INF with signal; 55 * log(NaN) is that NaN with no signal. 56 * 57 * Accuracy: 58 * log(x) returns the exact log(x) nearly rounded. In a test run with 59 * 1,536,000 random arguments on a VAX, the maximum observed error was 60 * .826 ulps (units in the last place). 61 * 62 * Constants: 63 * The hexadecimal values are the intended ones for the following constants. 64 * The decimal values may be used, provided that the compiler will convert 65 * from decimal to binary accurately enough to produce the hexadecimal values 66 * shown. 67 */ 68 69 #ifdef VAX /* VAX D format */ 70 #include <errno.h> 71 72 /* double static */ 73 /* ln2hi = 6.9314718055829871446E-1 , Hex 2^ 0 * .B17217F7D00000 */ 74 /* ln2lo = 1.6465949582897081279E-12 , Hex 2^-39 * .E7BCD5E4F1D9CC */ 75 /* sqrt2 = 1.4142135623730950622E0 ; Hex 2^ 1 * .B504F333F9DE65 */ 76 static long ln2hix[] = { 0x72174031, 0x0000f7d0}; 77 static long ln2lox[] = { 0xbcd52ce7, 0xd9cce4f1}; 78 static long sqrt2x[] = { 0x04f340b5, 0xde6533f9}; 79 #define ln2hi (*(double*)ln2hix) 80 #define ln2lo (*(double*)ln2lox) 81 #define sqrt2 (*(double*)sqrt2x) 82 #else /* IEEE double */ 83 double static 84 ln2hi = 6.9314718036912381649E-1 , /*Hex 2^ -1 * 1.62E42FEE00000 */ 85 ln2lo = 1.9082149292705877000E-10 , /*Hex 2^-33 * 1.A39EF35793C76 */ 86 sqrt2 = 1.4142135623730951455E0 ; /*Hex 2^ 0 * 1.6A09E667F3BCD */ 87 #endif 88 89 double log(x) 90 double x; 91 { 92 static double zero=0.0, negone= -1.0, half=1.0/2.0; 93 double logb(),scalb(),copysign(),log__L(),s,z,t; 94 int k,n,finite(); 95 96 #ifndef VAX 97 if(x!=x) return(x); /* x is NaN */ 98 #endif 99 if(finite(x)) { 100 if( x > zero ) { 101 102 /* argument reduction */ 103 k=logb(x); x=scalb(x,-k); 104 if(k == -1022) /* subnormal no. */ 105 {n=logb(x); x=scalb(x,-n); k+=n;} 106 if(x >= sqrt2 ) {k += 1; x *= half;} 107 x += negone ; 108 109 /* compute log(1+x) */ 110 s=x/(2+x); t=x*x*half; 111 z=k*ln2lo+s*(t+log__L(s*s)); 112 x += (z - t) ; 113 114 return(k*ln2hi+x); 115 } 116 /* end of if (x > zero) */ 117 118 else { 119 #ifdef VAX 120 extern double infnan(); 121 if ( x == zero ) 122 return (infnan(-ERANGE)); /* -INF */ 123 else 124 return (infnan(EDOM)); /* NaN */ 125 #else /* IEEE double */ 126 /* zero argument, return -INF with signal */ 127 if ( x == zero ) 128 return( negone/zero ); 129 130 /* negative argument, return NaN with signal */ 131 else 132 return ( zero / zero ); 133 #endif 134 } 135 } 136 /* end of if (finite(x)) */ 137 /* NOT REACHED ifdef VAX */ 138 139 /* log(-INF) is NaN with signal */ 140 else if (x<0) 141 return(zero/zero); 142 143 /* log(+INF) is +INF */ 144 else return(x); 145 146 } 147