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 4.4 (Berkeley) 06/03/85"; 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 extern errno; 72 static long NaN_[] = {0x8000, 0x0}; 73 #define NaN (*(double *) NaN_) 74 75 /* double static */ 76 /* ln2hi = 6.9314718055829871446E-1 , Hex 2^ 0 * .B17217F7D00000 */ 77 /* ln2lo = 1.6465949582897081279E-12 , Hex 2^-39 * .E7BCD5E4F1D9CC */ 78 /* sqrt2 = 1.4142135623730950622E0 ; Hex 2^ 1 * .B504F333F9DE65 */ 79 static long ln2hix[] = { 0x72174031, 0x0000f7d0}; 80 static long ln2lox[] = { 0xbcd52ce7, 0xd9cce4f1}; 81 static long sqrt2x[] = { 0x04f340b5, 0xde6533f9}; 82 #define ln2hi (*(double*)ln2hix) 83 #define ln2lo (*(double*)ln2lox) 84 #define sqrt2 (*(double*)sqrt2x) 85 #else /* IEEE double format */ 86 double static 87 ln2hi = 6.9314718036912381649E-1 , /*Hex 2^ -1 * 1.62E42FEE00000 */ 88 ln2lo = 1.9082149292705877000E-10 , /*Hex 2^-33 * 1.A39EF35793C76 */ 89 sqrt2 = 1.4142135623730951455E0 ; /*Hex 2^ 0 * 1.6A09E667F3BCD */ 90 #endif 91 92 double log(x) 93 double x; 94 { 95 static double zero=0.0, negone= -1.0, half=1.0/2.0; 96 double logb(),scalb(),copysign(),log__L(),s,z,t; 97 int k,n,finite(); 98 99 if(x!=x) return(x); 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 errno = EDOM; 122 return (NaN); 123 #else 124 /* zero argument, return -INF with signal */ 125 if ( x == zero ) 126 return( negone/zero ); 127 128 /* negative argument, return NAN with signal */ 129 else 130 return ( zero / zero ); 131 #endif 132 } 133 } 134 /* end of if (finite(x)) */ 135 /* NOT REACHED ifdef VAX */ 136 137 /* log(-INF) is NAN with signal */ 138 else if (x<0) 139 return(zero/zero); 140 141 /* log(+INF) is +INF */ 142 else return(x); 143 144 } 145