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 "@(#)log1p.c 1.3 (Berkeley) 8/21/85; 5.1 (ucb.elefunt) 11/30/87"; 17 #endif /* not lint */ 18 19 /* LOG1P(x) 20 * RETURN THE LOGARITHM OF 1+x 21 * DOUBLE PRECISION (VAX D FORMAT 56 bits, IEEE DOUBLE 53 BITS) 22 * CODED IN C BY K.C. NG, 1/19/85; 23 * REVISED BY K.C. NG on 2/6/85, 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 * 1+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(1+x) = k*ln2 + log(1+f). 50 * 51 * Remarks 1. In step 3 n*ln2 will be stored in two floating point numbers 52 * n*ln2hi + n*ln2lo, where ln2hi is chosen such that the last 53 * 20 bits (for VAX D format), or the last 21 bits ( for IEEE 54 * double) is 0. This ensures n*ln2hi is exactly representable. 55 * 2. In step 1, f may not be representable. A correction term c 56 * for f is computed. It follows that the correction term for 57 * f - t (the leading term of log(1+f) in step 2) is c-c*x. We 58 * add this correction term to n*ln2lo to attenuate the error. 59 * 60 * 61 * Special cases: 62 * log1p(x) is NaN with signal if x < -1; log1p(NaN) is NaN with no signal; 63 * log1p(INF) is +INF; log1p(-1) is -INF with signal; 64 * only log1p(0)=0 is exact for finite argument. 65 * 66 * Accuracy: 67 * log1p(x) returns the exact log(1+x) nearly rounded. In a test run 68 * with 1,536,000 random arguments on a VAX, the maximum observed 69 * error was .846 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 log1p(x) 103 double x; 104 { 105 static double zero=0.0, negone= -1.0, one=1.0, 106 half=1.0/2.0, small=1.0E-20; /* 1+small == 1 */ 107 double logb(),copysign(),scalb(),log__L(),z,s,t,c; 108 int k,finite(); 109 110 #if !defined(vax)&&!defined(tahoe) 111 if(x!=x) return(x); /* x is NaN */ 112 #endif /* !defined(vax)&&!defined(tahoe) */ 113 114 if(finite(x)) { 115 if( x > negone ) { 116 117 /* argument reduction */ 118 if(copysign(x,one)<small) return(x); 119 k=logb(one+x); z=scalb(x,-k); t=scalb(one,-k); 120 if(z+t >= sqrt2 ) 121 { k += 1 ; z *= half; t *= half; } 122 t += negone; x = z + t; 123 c = (t-x)+z ; /* correction term for x */ 124 125 /* compute log(1+x) */ 126 s = x/(2+x); t = x*x*half; 127 c += (k*ln2lo-c*x); 128 z = c+s*(t+log__L(s*s)); 129 x += (z - t) ; 130 131 return(k*ln2hi+x); 132 } 133 /* end of if (x > negone) */ 134 135 else { 136 #if defined(vax)||defined(tahoe) 137 extern double infnan(); 138 if ( x == negone ) 139 return (infnan(-ERANGE)); /* -INF */ 140 else 141 return (infnan(EDOM)); /* NaN */ 142 #else /* defined(vax)||defined(tahoe) */ 143 /* x = -1, return -INF with signal */ 144 if ( x == negone ) return( negone/zero ); 145 146 /* negative argument for log, return NaN with signal */ 147 else return ( zero / zero ); 148 #endif /* defined(vax)||defined(tahoe) */ 149 } 150 } 151 /* end of if (finite(x)) */ 152 153 /* log(-INF) is NaN */ 154 else if(x<0) 155 return(zero/zero); 156 157 /* log(+INF) is INF */ 158 else return(x); 159 } 160