/* 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 30000 9.8e-20 2.7e-20 * IEEE exp(+-10000) 70000 5.4e-20 2.3e-20 * * 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 -INFINITYL * log domain: x < 0; returns NANL */ /* Cephes Math Library Release 2.8: May, 1998 Copyright 1984, 1991, 1998 by Stephen L. Moshier */ #include /* 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 = 6.2e-22 */ #ifdef UNK static long double P[] = { 4.9962495940332550844739E-1L, 1.0767376367209449010438E1L, 7.7671073698359539859595E1L, 2.5620629828144409632571E2L, 4.2401812743503691187826E2L, 3.4258224542413922935104E2L, 1.0747524399916215149070E2L, }; static long double Q[] = { /* 1.0000000000000000000000E0,*/ 2.3479774160285863271658E1L, 1.9444210022760132894510E2L, 7.7952888181207260646090E2L, 1.6911722418503949084863E3L, 2.0307734695595183428202E3L, 1.2695660352705325274404E3L, 3.2242573199748645407652E2L, }; #endif #ifdef IBMPC static short P[] = { 0xfe72,0xce22,0xd7b9,0xffce,0x3ffd, XPD 0xb778,0x0e34,0x2c71,0xac47,0x4002, XPD 0xea8b,0xc751,0x96f8,0x9b57,0x4005, XPD 0xfeaf,0x6a02,0x67fb,0x801a,0x4007, XPD 0x6b5a,0xf252,0x51ff,0xd402,0x4007, XPD 0x39ce,0x9f76,0x8704,0xab4a,0x4007, XPD 0x1b39,0x740b,0x532e,0xd6f3,0x4005, XPD }; static short Q[] = { /*0x0000,0x0000,0x0000,0x8000,0x3fff,*/ 0x2f3a,0xbf26,0x93d5,0xbbd6,0x4003, XPD 0x13c8,0x031a,0x2d7b,0xc271,0x4006, XPD 0x449d,0x1993,0xd933,0xc2e1,0x4008, XPD 0x5b65,0x574e,0x8301,0xd365,0x4009, XPD 0xa65d,0x3bd2,0xc043,0xfdd8,0x4009, XPD 0x3b21,0xffea,0x1cf5,0x9eb2,0x4009, XPD 0x545c,0xd708,0x7e62,0xa136,0x4007, XPD }; #endif #ifdef MIEEE static long P[] = { 0x3ffd0000,0xffced7b9,0xce22fe72, 0x40020000,0xac472c71,0x0e34b778, 0x40050000,0x9b5796f8,0xc751ea8b, 0x40070000,0x801a67fb,0x6a02feaf, 0x40070000,0xd40251ff,0xf2526b5a, 0x40070000,0xab4a8704,0x9f7639ce, 0x40050000,0xd6f3532e,0x740b1b39, }; static long Q[] = { /*0x3fff0000,0x80000000,0x00000000,*/ 0x40030000,0xbbd693d5,0xbf262f3a, 0x40060000,0xc2712d7b,0x031a13c8, 0x40080000,0xc2e1d933,0x1993449d, 0x40090000,0xd3658301,0x574e5b65, 0x40090000,0xfdd8c043,0x3bd2a65d, 0x40090000,0x9eb21cf5,0xffea3b21, 0x40070000,0xa1367e62,0xd708545c, }; #endif /* 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 = 6.16e-22 */ #ifdef UNK static long double R[4] = { 1.9757429581415468984296E-3L, -7.1990767473014147232598E-1L, 1.0777257190312272158094E1L, -3.5717684488096787370998E1L, }; static long double S[4] = { /* 1.00000000000000000000E0L,*/ -2.6201045551331104417768E1L, 1.9361891836232102174846E2L, -4.2861221385716144629696E2L, }; /* log2(e) - 1 */ #define LOG2EA 4.4269504088896340735992e-1L #endif #ifdef IBMPC static short R[] = { 0x6ef4,0xf922,0x7763,0x817b,0x3ff6, XPD 0x15fd,0x1af9,0xde8f,0xb84b,0xbffe, XPD 0x8b96,0x4f8d,0xa53c,0xac6f,0x4002, XPD 0x8932,0xb4e3,0xe8ae,0x8ede,0xc004, XPD }; static short S[] = { /*0x0000,0x0000,0x0000,0x8000,0x3fff,*/ 0x7ce4,0x1fc9,0xbdc5,0xd19b,0xc003, XPD 0x0af3,0x0d10,0x716f,0xc19e,0x4006, XPD 0x4d7d,0x0f55,0x5d06,0xd64e,0xc007, XPD }; static short LG2EA[] = {0xc2ef,0x705f,0xeca5,0xe2a8,0x3ffd, XPD}; #define LOG2EA *(long double *)LG2EA #endif #ifdef MIEEE static long R[12] = { 0x3ff60000,0x817b7763,0xf9226ef4, 0xbffe0000,0xb84bde8f,0x1af915fd, 0x40020000,0xac6fa53c,0x4f8d8b96, 0xc0040000,0x8edee8ae,0xb4e38932, }; static long S[9] = { /*0x3fff0000,0x80000000,0x00000000,*/ 0xc0030000,0xd19bbdc5,0x1fc97ce4, 0x40060000,0xc19e716f,0x0d100af3, 0xc0070000,0xd64e5d06,0x0f554d7d, }; static long LG2EA[] = {0x3ffd0000,0xe2a8eca5,0x705fc2ef}; #define LOG2EA *(long double *)LG2EA #endif #define SQRTH 0.70710678118654752440L extern long double MINLOGL; #ifdef ANSIPROT extern long double frexpl ( long double, int * ); extern long double ldexpl ( long double, int ); extern long double polevll ( long double, void *, int ); extern long double p1evll ( long double, void *, int ); extern int isnanl ( long double ); #else long double frexpl(), ldexpl(), polevll(), p1evll(); extern int isnanl (); #endif #ifdef INFINITIES extern long double INFINITYL; #endif #ifdef NANS extern long double NANL; #endif long double log2l(x) long double x; { VOLATILE long double z; long double y; int e; #ifdef NANS if( isnanl(x) ) return(x); #endif #ifdef INFINITIES if( x == INFINITYL ) return(x); #endif /* Test for domain */ if( x <= 0.0L ) { if( x == 0.0L ) { #ifdef INFINITIES return( -INFINITYL ); #else mtherr( "log2l", SING ); return( -16384.0L ); #endif } else { #ifdef NANS return( NANL ); #else mtherr( "log2l", DOMAIN ); return( -16384.0L ); #endif } } /* 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, 3 ) / p1evll( z, S, 3 ) ); 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, 6 ) / p1evll( x, Q, 7 ) ); 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 ); }