1 /* 2 * Copyright (c) 1985, 1993 3 * The Regents of the University of California. All rights reserved. 4 * 5 * %sccs.include.redist.c% 6 */ 7 8 #ifndef lint 9 static char sccsid[] = "@(#)log1p.c 8.1 (Berkeley) 06/04/93"; 10 #endif /* not lint */ 11 12 /* LOG1P(x) 13 * RETURN THE LOGARITHM OF 1+x 14 * DOUBLE PRECISION (VAX D FORMAT 56 bits, IEEE DOUBLE 53 BITS) 15 * CODED IN C BY K.C. NG, 1/19/85; 16 * REVISED BY K.C. NG on 2/6/85, 3/7/85, 3/24/85, 4/16/85. 17 * 18 * Required system supported functions: 19 * scalb(x,n) 20 * copysign(x,y) 21 * logb(x) 22 * finite(x) 23 * 24 * Required kernel function: 25 * log__L(z) 26 * 27 * Method : 28 * 1. Argument Reduction: find k and f such that 29 * 1+x = 2^k * (1+f), 30 * where sqrt(2)/2 < 1+f < sqrt(2) . 31 * 32 * 2. Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) 33 * = 2s + 2/3 s**3 + 2/5 s**5 + ....., 34 * log(1+f) is computed by 35 * 36 * log(1+f) = 2s + s*log__L(s*s) 37 * where 38 * log__L(z) = z*(L1 + z*(L2 + z*(... (L6 + z*L7)...))) 39 * 40 * See log__L() for the values of the coefficients. 41 * 42 * 3. Finally, log(1+x) = k*ln2 + log(1+f). 43 * 44 * Remarks 1. In step 3 n*ln2 will be stored in two floating point numbers 45 * n*ln2hi + n*ln2lo, where ln2hi is chosen such that the last 46 * 20 bits (for VAX D format), or the last 21 bits ( for IEEE 47 * double) is 0. This ensures n*ln2hi is exactly representable. 48 * 2. In step 1, f may not be representable. A correction term c 49 * for f is computed. It follows that the correction term for 50 * f - t (the leading term of log(1+f) in step 2) is c-c*x. We 51 * add this correction term to n*ln2lo to attenuate the error. 52 * 53 * 54 * Special cases: 55 * log1p(x) is NaN with signal if x < -1; log1p(NaN) is NaN with no signal; 56 * log1p(INF) is +INF; log1p(-1) is -INF with signal; 57 * only log1p(0)=0 is exact for finite argument. 58 * 59 * Accuracy: 60 * log1p(x) returns the exact log(1+x) nearly rounded. In a test run 61 * with 1,536,000 random arguments on a VAX, the maximum observed 62 * error was .846 ulps (units in the last place). 63 * 64 * Constants: 65 * The hexadecimal values are the intended ones for the following constants. 66 * The decimal values may be used, provided that the compiler will convert 67 * from decimal to binary accurately enough to produce the hexadecimal values 68 * shown. 69 */ 70 71 #include <errno.h> 72 #include "mathimpl.h" 73 74 vc(ln2hi, 6.9314718055829871446E-1 ,7217,4031,0000,f7d0, 0, .B17217F7D00000) 75 vc(ln2lo, 1.6465949582897081279E-12 ,bcd5,2ce7,d9cc,e4f1, -39, .E7BCD5E4F1D9CC) 76 vc(sqrt2, 1.4142135623730950622E0 ,04f3,40b5,de65,33f9, 1, .B504F333F9DE65) 77 78 ic(ln2hi, 6.9314718036912381649E-1, -1, 1.62E42FEE00000) 79 ic(ln2lo, 1.9082149292705877000E-10, -33, 1.A39EF35793C76) 80 ic(sqrt2, 1.4142135623730951455E0, 0, 1.6A09E667F3BCD) 81 82 #ifdef vccast 83 #define ln2hi vccast(ln2hi) 84 #define ln2lo vccast(ln2lo) 85 #define sqrt2 vccast(sqrt2) 86 #endif 87 88 double log1p(x) 89 double x; 90 { 91 const static double zero=0.0, negone= -1.0, one=1.0, 92 half=1.0/2.0, small=1.0E-20; /* 1+small == 1 */ 93 double z,s,t,c; 94 int k; 95 96 #if !defined(vax)&&!defined(tahoe) 97 if(x!=x) return(x); /* x is NaN */ 98 #endif /* !defined(vax)&&!defined(tahoe) */ 99 100 if(finite(x)) { 101 if( x > negone ) { 102 103 /* argument reduction */ 104 if(copysign(x,one)<small) return(x); 105 k=logb(one+x); z=scalb(x,-k); t=scalb(one,-k); 106 if(z+t >= sqrt2 ) 107 { k += 1 ; z *= half; t *= half; } 108 t += negone; x = z + t; 109 c = (t-x)+z ; /* correction term for x */ 110 111 /* compute log(1+x) */ 112 s = x/(2+x); t = x*x*half; 113 c += (k*ln2lo-c*x); 114 z = c+s*(t+__log__L(s*s)); 115 x += (z - t) ; 116 117 return(k*ln2hi+x); 118 } 119 /* end of if (x > negone) */ 120 121 else { 122 #if defined(vax)||defined(tahoe) 123 if ( x == negone ) 124 return (infnan(-ERANGE)); /* -INF */ 125 else 126 return (infnan(EDOM)); /* NaN */ 127 #else /* defined(vax)||defined(tahoe) */ 128 /* x = -1, return -INF with signal */ 129 if ( x == negone ) return( negone/zero ); 130 131 /* negative argument for log, return NaN with signal */ 132 else return ( zero / zero ); 133 #endif /* defined(vax)||defined(tahoe) */ 134 } 135 } 136 /* end of if (finite(x)) */ 137 138 /* log(-INF) is NaN */ 139 else if(x<0) 140 return(zero/zero); 141 142 /* log(+INF) is INF */ 143 else return(x); 144 } 145