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[] = "@(#)log1p.c 1.3 (Berkeley) 08/21/85"; 16 #endif not lint 17 18 /* LOG1P(x) 19 * RETURN THE LOGARITHM OF 1+x 20 * DOUBLE PRECISION (VAX D FORMAT 56 bits, IEEE DOUBLE 53 BITS) 21 * CODED IN C BY K.C. NG, 1/19/85; 22 * REVISED BY K.C. NG on 2/6/85, 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 * 1+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(1+x) = k*ln2 + log(1+f). 49 * 50 * Remarks 1. In step 3 n*ln2 will be stored in two floating point numbers 51 * n*ln2hi + n*ln2lo, where ln2hi is chosen such that the last 52 * 20 bits (for VAX D format), or the last 21 bits ( for IEEE 53 * double) is 0. This ensures n*ln2hi is exactly representable. 54 * 2. In step 1, f may not be representable. A correction term c 55 * for f is computed. It follows that the correction term for 56 * f - t (the leading term of log(1+f) in step 2) is c-c*x. We 57 * add this correction term to n*ln2lo to attenuate the error. 58 * 59 * 60 * Special cases: 61 * log1p(x) is NaN with signal if x < -1; log1p(NaN) is NaN with no signal; 62 * log1p(INF) is +INF; log1p(-1) is -INF with signal; 63 * only log1p(0)=0 is exact for finite argument. 64 * 65 * Accuracy: 66 * log1p(x) returns the exact log(1+x) nearly rounded. In a test run 67 * with 1,536,000 random arguments on a VAX, the maximum observed 68 * error was .846 ulps (units in the last place). 69 * 70 * Constants: 71 * The hexadecimal values are the intended ones for the following constants. 72 * The decimal values may be used, provided that the compiler will convert 73 * from decimal to binary accurately enough to produce the hexadecimal values 74 * shown. 75 */ 76 77 #ifdef VAX /* VAX D format */ 78 #include <errno.h> 79 80 /* double static */ 81 /* ln2hi = 6.9314718055829871446E-1 , Hex 2^ 0 * .B17217F7D00000 */ 82 /* ln2lo = 1.6465949582897081279E-12 , Hex 2^-39 * .E7BCD5E4F1D9CC */ 83 /* sqrt2 = 1.4142135623730950622E0 ; Hex 2^ 1 * .B504F333F9DE65 */ 84 static long ln2hix[] = { 0x72174031, 0x0000f7d0}; 85 static long ln2lox[] = { 0xbcd52ce7, 0xd9cce4f1}; 86 static long sqrt2x[] = { 0x04f340b5, 0xde6533f9}; 87 #define ln2hi (*(double*)ln2hix) 88 #define ln2lo (*(double*)ln2lox) 89 #define sqrt2 (*(double*)sqrt2x) 90 #else /* IEEE double */ 91 double static 92 ln2hi = 6.9314718036912381649E-1 , /*Hex 2^ -1 * 1.62E42FEE00000 */ 93 ln2lo = 1.9082149292705877000E-10 , /*Hex 2^-33 * 1.A39EF35793C76 */ 94 sqrt2 = 1.4142135623730951455E0 ; /*Hex 2^ 0 * 1.6A09E667F3BCD */ 95 #endif 96 97 double log1p(x) 98 double x; 99 { 100 static double zero=0.0, negone= -1.0, one=1.0, 101 half=1.0/2.0, small=1.0E-20; /* 1+small == 1 */ 102 double logb(),copysign(),scalb(),log__L(),z,s,t,c; 103 int k,finite(); 104 105 #ifndef VAX 106 if(x!=x) return(x); /* x is NaN */ 107 #endif 108 109 if(finite(x)) { 110 if( x > negone ) { 111 112 /* argument reduction */ 113 if(copysign(x,one)<small) return(x); 114 k=logb(one+x); z=scalb(x,-k); t=scalb(one,-k); 115 if(z+t >= sqrt2 ) 116 { k += 1 ; z *= half; t *= half; } 117 t += negone; x = z + t; 118 c = (t-x)+z ; /* correction term for x */ 119 120 /* compute log(1+x) */ 121 s = x/(2+x); t = x*x*half; 122 c += (k*ln2lo-c*x); 123 z = c+s*(t+log__L(s*s)); 124 x += (z - t) ; 125 126 return(k*ln2hi+x); 127 } 128 /* end of if (x > negone) */ 129 130 else { 131 #ifdef VAX 132 extern double infnan(); 133 if ( x == negone ) 134 return (infnan(-ERANGE)); /* -INF */ 135 else 136 return (infnan(EDOM)); /* NaN */ 137 #else /* IEEE double */ 138 /* x = -1, return -INF with signal */ 139 if ( x == negone ) return( negone/zero ); 140 141 /* negative argument for log, return NaN with signal */ 142 else return ( zero / zero ); 143 #endif 144 } 145 } 146 /* end of if (finite(x)) */ 147 148 /* log(-INF) is NaN */ 149 else if(x<0) 150 return(zero/zero); 151 152 /* log(+INF) is INF */ 153 else return(x); 154 } 155