/* log2.c * * Base 2 logarithm * * * * SYNOPSIS: * * double x, y, log2(); * * y = log2( 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 base e * 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 2.0e-16 5.5e-17 * IEEE exp(+-700) 40000 1.3e-16 4.6e-17 * * In the tests over the interval [exp(+-700)], the logarithms * of the random arguments were uniformly distributed. * * ERROR MESSAGES: * * log2 singularity: x = 0; returns -INFINITY * log2 domain: x < 0; returns NAN */ /* Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1995, 2000 by Stephen L. Moshier */ #include static char fname[] = {"log2"}; /* Coefficients for log(1+x) = x - x**2/2 + x**3 P(x)/Q(x) * 1/sqrt(2) <= x < sqrt(2) */ #ifdef UNK static double P[] = { 1.01875663804580931796E-4, 4.97494994976747001425E-1, 4.70579119878881725854E0, 1.44989225341610930846E1, 1.79368678507819816313E1, 7.70838733755885391666E0, }; static double Q[] = { /* 1.00000000000000000000E0, */ 1.12873587189167450590E1, 4.52279145837532221105E1, 8.29875266912776603211E1, 7.11544750618563894466E1, 2.31251620126765340583E1, }; #define LOG2EA 0.44269504088896340735992 #endif #ifdef DEC static unsigned short P[] = { 0037777,0127270,0162547,0057274, 0041001,0054665,0164317,0005341, 0041451,0034104,0031640,0105773, 0041677,0011276,0123617,0160135, 0041701,0126603,0053215,0117250, 0041420,0115777,0135206,0030232, }; static unsigned short Q[] = { /*0040200,0000000,0000000,0000000,*/ 0041220,0144332,0045272,0174241, 0041742,0164566,0035720,0130431, 0042246,0126327,0166065,0116357, 0042372,0033420,0157525,0124560, 0042271,0167002,0066537,0172303, 0041730,0164777,0113711,0044407, }; static unsigned short L[5] = {0037742,0124354,0122560,0057703}; #define LOG2EA (*(double *)(&L[0])) #endif #ifdef IBMPC static unsigned short P[] = { 0x1bb0,0x93c3,0xb4c2,0x3f1a, 0x52f2,0x3f56,0xd6f5,0x3fdf, 0x6911,0xed92,0xd2ba,0x4012, 0xeb2e,0xc63e,0xff72,0x402c, 0xc84d,0x924b,0xefd6,0x4031, 0xdcf8,0x7d7e,0xd563,0x401e, }; static unsigned short Q[] = { /*0x0000,0x0000,0x0000,0x3ff0,*/ 0xef8e,0xae97,0x9320,0x4026, 0xc033,0x4e19,0x9d2c,0x4046, 0xbdbd,0xa326,0xbf33,0x4054, 0xae21,0xeb5e,0xc9e2,0x4051, 0x25b2,0x9e1f,0x200a,0x4037, }; static unsigned short L[5] = {0x0bf8,0x94ae,0x551d,0x3fdc}; #define LOG2EA (*(double *)(&L[0])) #endif #ifdef MIEEE static unsigned short P[] = { 0x3f1a,0xb4c2,0x93c3,0x1bb0, 0x3fdf,0xd6f5,0x3f56,0x52f2, 0x4012,0xd2ba,0xed92,0x6911, 0x402c,0xff72,0xc63e,0xeb2e, 0x4031,0xefd6,0x924b,0xc84d, 0x401e,0xd563,0x7d7e,0xdcf8, }; static unsigned short Q[] = { /*0x3ff0,0x0000,0x0000,0x0000,*/ 0x4026,0x9320,0xae97,0xef8e, 0x4046,0x9d2c,0x4e19,0xc033, 0x4054,0xbf33,0xa326,0xbdbd, 0x4051,0xc9e2,0xeb5e,0xae21, 0x4037,0x200a,0x9e1f,0x25b2, }; static unsigned short L[5] = {0x3fdc,0x551d,0x94ae,0x0bf8}; #define LOG2EA (*(double *)(&L[0])) #endif /* Coefficients for log(x) = z + z**3 P(z)/Q(z), * where z = 2(x-1)/(x+1) * 1/sqrt(2) <= x < sqrt(2) */ #ifdef UNK static double R[3] = { -7.89580278884799154124E-1, 1.63866645699558079767E1, -6.41409952958715622951E1, }; static double S[3] = { /* 1.00000000000000000000E0,*/ -3.56722798256324312549E1, 3.12093766372244180303E2, -7.69691943550460008604E2, }; /* log2(e) - 1 */ #define LOG2EA 0.44269504088896340735992 #endif #ifdef DEC static unsigned short R[12] = { 0140112,0020756,0161540,0072035, 0041203,0013743,0114023,0155527, 0141600,0044060,0104421,0050400, }; static unsigned short S[12] = { /*0040200,0000000,0000000,0000000,*/ 0141416,0130152,0017543,0064122, 0042234,0006000,0104527,0020155, 0142500,0066110,0146631,0174731, }; /* log2(e) - 1 */ #define LOG2EA 0.44269504088896340735992L #endif #ifdef IBMPC static unsigned short R[12] = { 0x0e84,0xdc6c,0x443d,0xbfe9, 0x7b6b,0x7302,0x62fc,0x4030, 0x2a20,0x1122,0x0906,0xc050, }; static unsigned short S[12] = { /*0x0000,0x0000,0x0000,0x3ff0,*/ 0x6d0a,0x43ec,0xd60d,0xc041, 0xe40e,0x112a,0x8180,0x4073, 0x3f3b,0x19b3,0x0d89,0xc088, }; #endif #ifdef MIEEE static unsigned short R[12] = { 0xbfe9,0x443d,0xdc6c,0x0e84, 0x4030,0x62fc,0x7302,0x7b6b, 0xc050,0x0906,0x1122,0x2a20, }; static unsigned short S[12] = { /*0x3ff0,0x0000,0x0000,0x0000,*/ 0xc041,0xd60d,0x43ec,0x6d0a, 0x4073,0x8180,0x112a,0xe40e, 0xc088,0x0d89,0x19b3,0x3f3b, }; #endif #ifdef ANSIPROT extern double frexp ( double, int * ); extern double ldexp ( double, int ); extern double polevl ( double, void *, int ); extern double p1evl ( double, void *, int ); extern int isnan ( double ); extern int isfinite ( double ); #else double frexp(), ldexp(), polevl(), p1evl(); int isnan(), isfinite(); #endif #define SQRTH 0.70710678118654752440 extern double LOGE2, INFINITY, NAN; double log2(x) double x; { int e; double y; VOLATILE double z; #ifdef DEC short *q; #endif #ifdef NANS if( isnan(x) ) return(x); #endif #ifdef INFINITIES if( x == INFINITY ) return(x); #endif /* Test for domain */ if( x <= 0.0 ) { if( x == 0.0 ) { mtherr( fname, SING ); return( -INFINITY ); } else { mtherr( fname, DOMAIN ); return( NAN ); } } /* separate mantissa from exponent */ #ifdef DEC q = (short *)&x; e = *q; /* short containing exponent */ e = ((e >> 7) & 0377) - 0200; /* the exponent */ *q &= 0177; /* strip exponent from x */ *q |= 040000; /* x now between 0.5 and 1 */ #endif /* Note, frexp is used so that denormal numbers * will be handled properly. */ #ifdef IBMPC x = frexp( x, &e ); /* q = (short *)&x; q += 3; e = *q; e = ((e >> 4) & 0x0fff) - 0x3fe; *q &= 0x0f; *q |= 0x3fe0; */ #endif /* Equivalent C language standard library function: */ #ifdef UNK x = frexp( x, &e ); #endif #ifdef MIEEE x = frexp( x, &e ); #endif /* 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 - 0.5; y = 0.5 * z + 0.5; } else { /* 2 (x-1)/(x+1) */ z = x - 0.5; z -= 0.5; y = 0.5 * x + 0.5; } x = z / y; z = x*x; y = x * ( z * polevl( z, R, 2 ) / p1evl( z, S, 3 ) ); goto ldone; } /* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */ if( x < SQRTH ) { e -= 1; x = ldexp( x, 1 ) - 1.0; /* 2x - 1 */ } else { x = x - 1.0; } z = x*x; #if DEC y = x * ( z * polevl( x, P, 5 ) / p1evl( x, Q, 6 ) ) - ldexp( z, -1 ); #else y = x * ( z * polevl( x, P, 5 ) / p1evl( x, Q, 5 ) ) - ldexp( z, -1 ); #endif ldone: /* 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 ); }