1*24603Szliu /* 2*24603Szliu * Copyright (c) 1985 Regents of the University of California. 3*24603Szliu * 4*24603Szliu * Use and reproduction of this software are granted in accordance with 5*24603Szliu * the terms and conditions specified in the Berkeley Software License 6*24603Szliu * Agreement (in particular, this entails acknowledgement of the programs' 7*24603Szliu * source, and inclusion of this notice) with the additional understanding 8*24603Szliu * that all recipients should regard themselves as participants in an 9*24603Szliu * ongoing research project and hence should feel obligated to report 10*24603Szliu * their experiences (good or bad) with these elementary function codes, 11*24603Szliu * using "sendbug 4bsd-bugs@BERKELEY", to the authors. 12*24603Szliu */ 13*24603Szliu 14*24603Szliu #ifndef lint 15*24603Szliu static char sccsid[] = "@(#)log1p.c 1.1 (ELEFUNT) 09/06/85"; 16*24603Szliu #endif not lint 17*24603Szliu 18*24603Szliu /* LOG1P(x) 19*24603Szliu * RETURN THE LOGARITHM OF 1+x 20*24603Szliu * DOUBLE PRECISION (VAX D FORMAT 56 bits, IEEE DOUBLE 53 BITS) 21*24603Szliu * CODED IN C BY K.C. NG, 1/19/85; 22*24603Szliu * REVISED BY K.C. NG on 2/6/85, 3/7/85, 3/24/85, 4/16/85. 23*24603Szliu * 24*24603Szliu * Required system supported functions: 25*24603Szliu * scalb(x,n) 26*24603Szliu * copysign(x,y) 27*24603Szliu * logb(x) 28*24603Szliu * finite(x) 29*24603Szliu * 30*24603Szliu * Required kernel function: 31*24603Szliu * log__L(z) 32*24603Szliu * 33*24603Szliu * Method : 34*24603Szliu * 1. Argument Reduction: find k and f such that 35*24603Szliu * 1+x = 2^k * (1+f), 36*24603Szliu * where sqrt(2)/2 < 1+f < sqrt(2) . 37*24603Szliu * 38*24603Szliu * 2. Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) 39*24603Szliu * = 2s + 2/3 s**3 + 2/5 s**5 + ....., 40*24603Szliu * log(1+f) is computed by 41*24603Szliu * 42*24603Szliu * log(1+f) = 2s + s*log__L(s*s) 43*24603Szliu * where 44*24603Szliu * log__L(z) = z*(L1 + z*(L2 + z*(... (L6 + z*L7)...))) 45*24603Szliu * 46*24603Szliu * See log__L() for the values of the coefficients. 47*24603Szliu * 48*24603Szliu * 3. Finally, log(1+x) = k*ln2 + log(1+f). 49*24603Szliu * 50*24603Szliu * Remarks 1. In step 3 n*ln2 will be stored in two floating point numbers 51*24603Szliu * n*ln2hi + n*ln2lo, where ln2hi is chosen such that the last 52*24603Szliu * 20 bits (for VAX D format), or the last 21 bits ( for IEEE 53*24603Szliu * double) is 0. This ensures n*ln2hi is exactly representable. 54*24603Szliu * 2. In step 1, f may not be representable. A correction term c 55*24603Szliu * for f is computed. It follows that the correction term for 56*24603Szliu * f - t (the leading term of log(1+f) in step 2) is c-c*x. We 57*24603Szliu * add this correction term to n*ln2lo to attenuate the error. 58*24603Szliu * 59*24603Szliu * 60*24603Szliu * Special cases: 61*24603Szliu * log1p(x) is NaN with signal if x < -1; log1p(NaN) is NaN with no signal; 62*24603Szliu * log1p(INF) is +INF; log1p(-1) is -INF with signal; 63*24603Szliu * only log1p(0)=0 is exact for finite argument. 64*24603Szliu * 65*24603Szliu * Accuracy: 66*24603Szliu * log1p(x) returns the exact log(1+x) nearly rounded. In a test run 67*24603Szliu * with 1,536,000 random arguments on a VAX, the maximum observed 68*24603Szliu * error was .846 ulps (units in the last place). 69*24603Szliu * 70*24603Szliu * Constants: 71*24603Szliu * The hexadecimal values are the intended ones for the following constants. 72*24603Szliu * The decimal values may be used, provided that the compiler will convert 73*24603Szliu * from decimal to binary accurately enough to produce the hexadecimal values 74*24603Szliu * shown. 75*24603Szliu */ 76*24603Szliu 77*24603Szliu #ifdef VAX /* VAX D format */ 78*24603Szliu #include <errno.h> 79*24603Szliu 80*24603Szliu /* double static */ 81*24603Szliu /* ln2hi = 6.9314718055829871446E-1 , Hex 2^ 0 * .B17217F7D00000 */ 82*24603Szliu /* ln2lo = 1.6465949582897081279E-12 , Hex 2^-39 * .E7BCD5E4F1D9CC */ 83*24603Szliu /* sqrt2 = 1.4142135623730950622E0 ; Hex 2^ 1 * .B504F333F9DE65 */ 84*24603Szliu static long ln2hix[] = { 0x72174031, 0x0000f7d0}; 85*24603Szliu static long ln2lox[] = { 0xbcd52ce7, 0xd9cce4f1}; 86*24603Szliu static long sqrt2x[] = { 0x04f340b5, 0xde6533f9}; 87*24603Szliu #define ln2hi (*(double*)ln2hix) 88*24603Szliu #define ln2lo (*(double*)ln2lox) 89*24603Szliu #define sqrt2 (*(double*)sqrt2x) 90*24603Szliu #else /* IEEE double */ 91*24603Szliu double static 92*24603Szliu ln2hi = 6.9314718036912381649E-1 , /*Hex 2^ -1 * 1.62E42FEE00000 */ 93*24603Szliu ln2lo = 1.9082149292705877000E-10 , /*Hex 2^-33 * 1.A39EF35793C76 */ 94*24603Szliu sqrt2 = 1.4142135623730951455E0 ; /*Hex 2^ 0 * 1.6A09E667F3BCD */ 95*24603Szliu #endif 96*24603Szliu 97*24603Szliu double log1p(x) 98*24603Szliu double x; 99*24603Szliu { 100*24603Szliu static double zero=0.0, negone= -1.0, one=1.0, 101*24603Szliu half=1.0/2.0, small=1.0E-20; /* 1+small == 1 */ 102*24603Szliu double logb(),copysign(),scalb(),log__L(),z,s,t,c; 103*24603Szliu int k,finite(); 104*24603Szliu 105*24603Szliu #ifndef VAX 106*24603Szliu if(x!=x) return(x); /* x is NaN */ 107*24603Szliu #endif 108*24603Szliu 109*24603Szliu if(finite(x)) { 110*24603Szliu if( x > negone ) { 111*24603Szliu 112*24603Szliu /* argument reduction */ 113*24603Szliu if(copysign(x,one)<small) return(x); 114*24603Szliu k=logb(one+x); z=scalb(x,-k); t=scalb(one,-k); 115*24603Szliu if(z+t >= sqrt2 ) 116*24603Szliu { k += 1 ; z *= half; t *= half; } 117*24603Szliu t += negone; x = z + t; 118*24603Szliu c = (t-x)+z ; /* correction term for x */ 119*24603Szliu 120*24603Szliu /* compute log(1+x) */ 121*24603Szliu s = x/(2+x); t = x*x*half; 122*24603Szliu c += (k*ln2lo-c*x); 123*24603Szliu z = c+s*(t+log__L(s*s)); 124*24603Szliu x += (z - t) ; 125*24603Szliu 126*24603Szliu return(k*ln2hi+x); 127*24603Szliu } 128*24603Szliu /* end of if (x > negone) */ 129*24603Szliu 130*24603Szliu else { 131*24603Szliu #ifdef VAX 132*24603Szliu extern double infnan(); 133*24603Szliu if ( x == negone ) 134*24603Szliu return (infnan(-ERANGE)); /* -INF */ 135*24603Szliu else 136*24603Szliu return (infnan(EDOM)); /* NaN */ 137*24603Szliu #else /* IEEE double */ 138*24603Szliu /* x = -1, return -INF with signal */ 139*24603Szliu if ( x == negone ) return( negone/zero ); 140*24603Szliu 141*24603Szliu /* negative argument for log, return NaN with signal */ 142*24603Szliu else return ( zero / zero ); 143*24603Szliu #endif 144*24603Szliu } 145*24603Szliu } 146*24603Szliu /* end of if (finite(x)) */ 147*24603Szliu 148*24603Szliu /* log(-INF) is NaN */ 149*24603Szliu else if(x<0) 150*24603Szliu return(zero/zero); 151*24603Szliu 152*24603Szliu /* log(+INF) is INF */ 153*24603Szliu else return(x); 154*24603Szliu } 155