/*							log2l.c * *	Base 2 logarithm, long double precision * * * * SYNOPSIS: * * long double x, y, log2l(); * * y = log2l( x ); * * * * DESCRIPTION: * * Returns the base 2 logarithm of x. * * The argument is separated into its exponent and fractional * parts.  If the exponent is between -1 and +1, the (natural) * logarithm of the fraction is approximated by * *     log(1+x) = x - 0.5 x**2 + x**3 P(x)/Q(x). * * Otherwise, setting  z = 2(x-1)/x+1), *  *     log(x) = z + z**3 P(z)/Q(z). * * * * ACCURACY: * *                      Relative error: * arithmetic   domain     # trials      peak         rms *    IEEE      0.5, 2.0     100,000    1.3e-34     4.5e-35 *    IEEE     exp(+-10000)  100,000    9.6e-35     4.0e-35 * * In the tests over the interval exp(+-10000), the logarithms * of the random arguments were uniformly distributed over * [-10000, +10000]. * * ERROR MESSAGES: * * log singularity:  x = 0; returns MINLOG * log domain:       x < 0; returns MINLOG *//*Cephes Math Library Release 2.2:  January, 1991Copyright 1984, 1991 by Stephen L. MoshierDirect inquiries to 30 Frost Street, Cambridge, MA 02140*/#include "mconf.h"static char fname[] = {"log2l"};/* Coefficients for ln(1+x) = x - x**2/2 + x**3 P(x)/Q(x) * 1/sqrt(2) <= x < sqrt(2) * Theoretical peak relative error = 5.3e-37, * relative peak error spread = 2.3e-14 */static long double P[13] = { 1.538612243596254322971797716843006400388E-6L, 4.998469661968096229986658302195402690910E-1L, 2.321125933898420063925789532045674660756E1L, 4.114517881637811823002128927449878962058E2L, 3.824952356185897735160588078446136783779E3L, 2.128857716871515081352991964243375186031E4L, 7.594356839258970405033155585486712125861E4L, 1.797628303815655343403735250238293741397E5L, 2.854829159639697837788887080758954924001E5L, 3.007007295140399532324943111654767187848E5L, 2.014652742082537582487669938141683759923E5L, 7.771154681358524243729929227226708890930E4L, 1.313572404063446165910279910527789794488E4L};static long double Q[12] = {/* 1.000000000000000000000000000000000000000E0L, */ 4.839208193348159620282142911143429644326E1L, 9.104928120962988414618126155557301584078E2L, 9.147150349299596453976674231612674085381E3L, 5.605842085972455027590989944010492125825E4L, 2.248234257620569139969141618556349415120E5L, 6.132189329546557743179177159925690841200E5L, 1.158019977462989115839826904108208787040E6L, 1.514882452993549494932585972882995548426E6L, 1.347518538384329112529391120390701166528E6L, 7.777690340007566932935753241556479363645E5L, 2.626900195321832660448791748036714883242E5L, 3.940717212190338497730839731583397586124E4L};/* Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2), * where z = 2(x-1)/(x+1) * 1/sqrt(2) <= x < sqrt(2) * Theoretical peak relative error = 1.1e-35, * relative peak error spread 1.1e-9 */static long double R[6] = {-8.828896441624934385266096344596648080902E-1L, 8.057002716646055371965756206836056074715E1L,-2.024301798136027039250415126250455056397E3L, 2.048819892795278657810231591630928516206E4L,-8.977257995689735303686582344659576526998E4L, 1.418134209872192732479751274970992665513E5L};static long double S[6] = {/* 1.000000000000000000000000000000000000000E0L, */-1.186359407982897997337150403816839480438E2L, 3.998526750980007367835804959888064681098E3L,-5.748542087379434595104154610899551484314E4L, 4.001557694070773974936904547424676279307E5L,-1.332535117259762928288745111081235577029E6L, 1.701761051846631278975701529965589676574E6L};/* log2(e) - 1 */#define LOG2EA 4.4269504088896340735992468100189213742664595E-1L#define SQRTH 7.071067811865475244008443621048490392848359E-1Lextern long double MINLOGL;long double frexpl(), ldexpl(), polevll(), p1evll();long double log2l(x)long double x;{VOLATILE long double z;long double y;int e;/* Test for domain */if( x <= 0.0L )	{	if( x == 0.0L )		mtherr( fname, SING );	else		mtherr( fname, DOMAIN );	return( -16384.0L );	}/* separate mantissa from exponent *//* Note, frexp is used so that denormal numbers * will be handled properly. */x = frexpl( x, &e );/* logarithm using log(x) = z + z**3 P(z)/Q(z), * where z = 2(x-1)/x+1) */if( (e > 2) || (e < -2) ){if( x < SQRTH )	{ /* 2( 2x-1 )/( 2x+1 ) */	e -= 1;	z = x - 0.5L;	y = 0.5L * z + 0.5L;	}	else	{ /*  2 (x-1)/(x+1)   */	z = x - 0.5L;	z -= 0.5L;	y = 0.5L * x  + 0.5L;	}x = z / y;z = x*x;y = x * ( z * polevll( z, R, 5 ) / p1evll( z, S, 6 ) );goto done;}/* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */if( x < SQRTH )	{	e -= 1;	x = ldexpl( x, 1 ) - 1.0L; /*  2x - 1  */	}	else	{	x = x - 1.0L;	}z = x*x;y = x * ( z * polevll( x, P, 12 ) / p1evll( x, Q, 12 ) );y = y - ldexpl( z, -1 );   /* -0.5x^2 + ... */done:/* Multiply log of fraction by log2(e) * and base 2 exponent by 1 * * ***CAUTION*** * * This sequence of operations is critical and it may * be horribly defeated by some compiler optimizers. */z = y * LOG2EA;z += x * LOG2EA;z += y;z += x;z += e;return( z );}