1 /* 2 * Copyright (c) 1985 Regents of the University of California. 3 * All rights reserved. 4 * 5 * Redistribution and use in source and binary forms are permitted 6 * provided that the above copyright notice and this paragraph are 7 * duplicated in all such forms and that any documentation, 8 * advertising materials, and other materials related to such 9 * distribution and use acknowledge that the software was developed 10 * by the University of California, Berkeley. The name of the 11 * University may not be used to endorse or promote products derived 12 * from this software without specific prior written permission. 13 * THIS SOFTWARE IS PROVIDED ``AS IS'' AND WITHOUT ANY EXPRESS OR 14 * IMPLIED WARRANTIES, INCLUDING, WITHOUT LIMITATION, THE IMPLIED 15 * WARRANTIES OF MERCHANTIBILITY AND FITNESS FOR A PARTICULAR PURPOSE. 16 * 17 * All recipients should regard themselves as participants in an ongoing 18 * research project and hence should feel obligated to report their 19 * experiences (good or bad) with these elementary function codes, using 20 * the sendbug(8) program, to the authors. 21 */ 22 23 #ifndef lint 24 static char sccsid[] = "@(#)log1p.c 5.3 (Berkeley) 06/30/88"; 25 #endif /* not lint */ 26 27 /* LOG1P(x) 28 * RETURN THE LOGARITHM OF 1+x 29 * DOUBLE PRECISION (VAX D FORMAT 56 bits, IEEE DOUBLE 53 BITS) 30 * CODED IN C BY K.C. NG, 1/19/85; 31 * REVISED BY K.C. NG on 2/6/85, 3/7/85, 3/24/85, 4/16/85. 32 * 33 * Required system supported functions: 34 * scalb(x,n) 35 * copysign(x,y) 36 * logb(x) 37 * finite(x) 38 * 39 * Required kernel function: 40 * log__L(z) 41 * 42 * Method : 43 * 1. Argument Reduction: find k and f such that 44 * 1+x = 2^k * (1+f), 45 * where sqrt(2)/2 < 1+f < sqrt(2) . 46 * 47 * 2. Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) 48 * = 2s + 2/3 s**3 + 2/5 s**5 + ....., 49 * log(1+f) is computed by 50 * 51 * log(1+f) = 2s + s*log__L(s*s) 52 * where 53 * log__L(z) = z*(L1 + z*(L2 + z*(... (L6 + z*L7)...))) 54 * 55 * See log__L() for the values of the coefficients. 56 * 57 * 3. Finally, log(1+x) = k*ln2 + log(1+f). 58 * 59 * Remarks 1. In step 3 n*ln2 will be stored in two floating point numbers 60 * n*ln2hi + n*ln2lo, where ln2hi is chosen such that the last 61 * 20 bits (for VAX D format), or the last 21 bits ( for IEEE 62 * double) is 0. This ensures n*ln2hi is exactly representable. 63 * 2. In step 1, f may not be representable. A correction term c 64 * for f is computed. It follows that the correction term for 65 * f - t (the leading term of log(1+f) in step 2) is c-c*x. We 66 * add this correction term to n*ln2lo to attenuate the error. 67 * 68 * 69 * Special cases: 70 * log1p(x) is NaN with signal if x < -1; log1p(NaN) is NaN with no signal; 71 * log1p(INF) is +INF; log1p(-1) is -INF with signal; 72 * only log1p(0)=0 is exact for finite argument. 73 * 74 * Accuracy: 75 * log1p(x) returns the exact log(1+x) nearly rounded. In a test run 76 * with 1,536,000 random arguments on a VAX, the maximum observed 77 * error was .846 ulps (units in the last place). 78 * 79 * Constants: 80 * The hexadecimal values are the intended ones for the following constants. 81 * The decimal values may be used, provided that the compiler will convert 82 * from decimal to binary accurately enough to produce the hexadecimal values 83 * shown. 84 */ 85 86 #if defined(vax)||defined(tahoe) /* VAX D format */ 87 #include <errno.h> 88 #ifdef vax 89 #define _0x(A,B) 0x/**/A/**/B 90 #else /* vax */ 91 #define _0x(A,B) 0x/**/B/**/A 92 #endif /* vax */ 93 /* static double */ 94 /* ln2hi = 6.9314718055829871446E-1 , Hex 2^ 0 * .B17217F7D00000 */ 95 /* ln2lo = 1.6465949582897081279E-12 , Hex 2^-39 * .E7BCD5E4F1D9CC */ 96 /* sqrt2 = 1.4142135623730950622E0 ; Hex 2^ 1 * .B504F333F9DE65 */ 97 static long ln2hix[] = { _0x(7217,4031), _0x(0000,f7d0)}; 98 static long ln2lox[] = { _0x(bcd5,2ce7), _0x(d9cc,e4f1)}; 99 static long sqrt2x[] = { _0x(04f3,40b5), _0x(de65,33f9)}; 100 #define ln2hi (*(double*)ln2hix) 101 #define ln2lo (*(double*)ln2lox) 102 #define sqrt2 (*(double*)sqrt2x) 103 #else /* defined(vax)||defined(tahoe) */ 104 static double 105 ln2hi = 6.9314718036912381649E-1 , /*Hex 2^ -1 * 1.62E42FEE00000 */ 106 ln2lo = 1.9082149292705877000E-10 , /*Hex 2^-33 * 1.A39EF35793C76 */ 107 sqrt2 = 1.4142135623730951455E0 ; /*Hex 2^ 0 * 1.6A09E667F3BCD */ 108 #endif /* defined(vax)||defined(tahoe) */ 109 110 double log1p(x) 111 double x; 112 { 113 static double zero=0.0, negone= -1.0, one=1.0, 114 half=1.0/2.0, small=1.0E-20; /* 1+small == 1 */ 115 double logb(),copysign(),scalb(),log__L(),z,s,t,c; 116 int k,finite(); 117 118 #if !defined(vax)&&!defined(tahoe) 119 if(x!=x) return(x); /* x is NaN */ 120 #endif /* !defined(vax)&&!defined(tahoe) */ 121 122 if(finite(x)) { 123 if( x > negone ) { 124 125 /* argument reduction */ 126 if(copysign(x,one)<small) return(x); 127 k=logb(one+x); z=scalb(x,-k); t=scalb(one,-k); 128 if(z+t >= sqrt2 ) 129 { k += 1 ; z *= half; t *= half; } 130 t += negone; x = z + t; 131 c = (t-x)+z ; /* correction term for x */ 132 133 /* compute log(1+x) */ 134 s = x/(2+x); t = x*x*half; 135 c += (k*ln2lo-c*x); 136 z = c+s*(t+log__L(s*s)); 137 x += (z - t) ; 138 139 return(k*ln2hi+x); 140 } 141 /* end of if (x > negone) */ 142 143 else { 144 #if defined(vax)||defined(tahoe) 145 extern double infnan(); 146 if ( x == negone ) 147 return (infnan(-ERANGE)); /* -INF */ 148 else 149 return (infnan(EDOM)); /* NaN */ 150 #else /* defined(vax)||defined(tahoe) */ 151 /* x = -1, return -INF with signal */ 152 if ( x == negone ) return( negone/zero ); 153 154 /* negative argument for log, return NaN with signal */ 155 else return ( zero / zero ); 156 #endif /* defined(vax)||defined(tahoe) */ 157 } 158 } 159 /* end of if (finite(x)) */ 160 161 /* log(-INF) is NaN */ 162 else if(x<0) 163 return(zero/zero); 164 165 /* log(+INF) is INF */ 166 else return(x); 167 } 168