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 5.4 (Berkeley) 09/22/88"; 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 #include <errno.h> 87 #include "mathimpl.h" 88 89 vc(ln2hi, 6.9314718055829871446E-1 ,7217,4031,0000,f7d0, 0, .B17217F7D00000) 90 vc(ln2lo, 1.6465949582897081279E-12 ,bcd5,2ce7,d9cc,e4f1, -39, .E7BCD5E4F1D9CC) 91 vc(sqrt2, 1.4142135623730950622E0 ,04f3,40b5,de65,33f9, 1, .B504F333F9DE65) 92 93 ic(ln2hi, 6.9314718036912381649E-1, -1, 1.62E42FEE00000) 94 ic(ln2lo, 1.9082149292705877000E-10, -33, 1.A39EF35793C76) 95 ic(sqrt2, 1.4142135623730951455E0, 0, 1.6A09E667F3BCD) 96 97 #ifdef vccast 98 #define ln2hi vccast(ln2hi) 99 #define ln2lo vccast(ln2lo) 100 #define sqrt2 vccast(sqrt2) 101 #endif 102 103 double log1p(x) 104 double x; 105 { 106 const static double zero=0.0, negone= -1.0, one=1.0, 107 half=1.0/2.0, small=1.0E-20; /* 1+small == 1 */ 108 double z,s,t,c; 109 int k; 110 111 #if !defined(vax)&&!defined(tahoe) 112 if(x!=x) return(x); /* x is NaN */ 113 #endif /* !defined(vax)&&!defined(tahoe) */ 114 115 if(finite(x)) { 116 if( x > negone ) { 117 118 /* argument reduction */ 119 if(copysign(x,one)<small) return(x); 120 k=logb(one+x); z=scalb(x,-k); t=scalb(one,-k); 121 if(z+t >= sqrt2 ) 122 { k += 1 ; z *= half; t *= half; } 123 t += negone; x = z + t; 124 c = (t-x)+z ; /* correction term for x */ 125 126 /* compute log(1+x) */ 127 s = x/(2+x); t = x*x*half; 128 c += (k*ln2lo-c*x); 129 z = c+s*(t+log__L(s*s)); 130 x += (z - t) ; 131 132 return(k*ln2hi+x); 133 } 134 /* end of if (x > negone) */ 135 136 else { 137 #if defined(vax)||defined(tahoe) 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