/* atan.c * * Inverse circular tangent * (arctangent) * * * * SYNOPSIS: * * double x, y, atan(); * * y = atan( x ); * * * * DESCRIPTION: * * Returns radian angle between -pi/2 and +pi/2 whose tangent * is x. * * Range reduction is from three intervals into the interval * from zero to 0.66. The approximant uses a rational * function of degree 4/5 of the form x + x**3 P(x)/Q(x). * * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * DEC -10, 10 50000 2.4e-17 8.3e-18 * IEEE -10, 10 10^6 1.8e-16 5.0e-17 * */ /* atan2() * * Quadrant correct inverse circular tangent * * * * SYNOPSIS: * * double x, y, z, atan2(); * * z = atan2( y, x ); * * * * DESCRIPTION: * * Returns radian angle whose tangent is y/x. * Define compile time symbol ANSIC = 1 for ANSI standard, * range -PI < z <= +PI, args (y,x); else ANSIC = 0 for range * 0 to 2PI, args (x,y). * * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * IEEE -10, 10 10^6 2.5e-16 6.9e-17 * See atan.c. * */ /* atan.c */ /* Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1995, 2000 by Stephen L. Moshier */ #include "mconf.h" /* arctan(x) = x + x^3 P(x^2)/Q(x^2) 0 <= x <= 0.66 Peak relative error = 2.6e-18 */ #ifdef UNK static double P[5] = { -8.750608600031904122785E-1, -1.615753718733365076637E1, -7.500855792314704667340E1, -1.228866684490136173410E2, -6.485021904942025371773E1, }; static double Q[5] = { /* 1.000000000000000000000E0, */ 2.485846490142306297962E1, 1.650270098316988542046E2, 4.328810604912902668951E2, 4.853903996359136964868E2, 1.945506571482613964425E2, }; /* tan( 3*pi/8 ) */ static double T3P8 = 2.41421356237309504880; #endif #ifdef DEC static short P[20] = { 0140140,0001775,0007671,0026242, 0141201,0041242,0155534,0001715, 0141626,0002141,0132100,0011625, 0141765,0142771,0064055,0150453, 0141601,0131517,0164507,0062164, }; static short Q[20] = { /* 0040200,0000000,0000000,0000000, */ 0041306,0157042,0154243,0000742, 0042045,0003352,0016707,0150452, 0042330,0070306,0113425,0170730, 0042362,0130770,0116602,0047520, 0042102,0106367,0156753,0013541, }; /* tan( 3*pi/8 ) = 2.41421356237309504880 */ static unsigned short T3P8A[] = {040432,0101171,0114774,0167462,}; #define T3P8 *(double *)T3P8A #endif #ifdef IBMPC static short P[20] = { 0x2594,0xa1f7,0x007f,0xbfec, 0x807a,0x5b6b,0x2854,0xc030, 0x0273,0x3688,0xc08c,0xc052, 0xba25,0x2d05,0xb8bf,0xc05e, 0xec8e,0xfd28,0x3669,0xc050, }; static short Q[20] = { /* 0x0000,0x0000,0x0000,0x3ff0, */ 0x603c,0x5b14,0xdbc4,0x4038, 0xfa25,0x43b8,0xa0dd,0x4064, 0xbe3b,0xd2e2,0x0e18,0x407b, 0x49ea,0x13b0,0x563f,0x407e, 0x62ec,0xfbbd,0x519e,0x4068, }; /* tan( 3*pi/8 ) = 2.41421356237309504880 */ static unsigned short T3P8A[] = {0x9de6,0x333f,0x504f,0x4003}; #define T3P8 *(double *)T3P8A #endif #ifdef MIEEE static short P[20] = { 0xbfec,0x007f,0xa1f7,0x2594, 0xc030,0x2854,0x5b6b,0x807a, 0xc052,0xc08c,0x3688,0x0273, 0xc05e,0xb8bf,0x2d05,0xba25, 0xc050,0x3669,0xfd28,0xec8e, }; static short Q[20] = { /* 0x3ff0,0x0000,0x0000,0x0000, */ 0x4038,0xdbc4,0x5b14,0x603c, 0x4064,0xa0dd,0x43b8,0xfa25, 0x407b,0x0e18,0xd2e2,0xbe3b, 0x407e,0x563f,0x13b0,0x49ea, 0x4068,0x519e,0xfbbd,0x62ec, }; /* tan( 3*pi/8 ) = 2.41421356237309504880 */ static unsigned short T3P8A[] = { 0x4003,0x504f,0x333f,0x9de6 }; #define T3P8 *(double *)T3P8A #endif #ifdef ANSIPROT extern double polevl ( double, void *, int ); extern double p1evl ( double, void *, int ); extern double atan ( double ); extern double fabs ( double ); extern int signbit ( double ); extern int isnan ( double ); #else double polevl(), p1evl(), atan(), fabs(); int signbit(), isnan(); #endif extern double PI, PIO2, PIO4, INFINITY, NEGZERO, MAXNUM; /* pi/2 = PIO2 + MOREBITS. */ #ifdef DEC #define MOREBITS 5.721188726109831840122E-18 #else #define MOREBITS 6.123233995736765886130E-17 #endif double atan(x) double x; { double y, z; short sign, flag; #ifdef MINUSZERO if( x == 0.0 ) return(x); #endif #ifdef INFINITIES if(x == INFINITY) return(PIO2); if(x == -INFINITY) return(-PIO2); #endif /* make argument positive and save the sign */ sign = 1; if( x < 0.0 ) { sign = -1; x = -x; } /* range reduction */ flag = 0; if( x > T3P8 ) { y = PIO2; flag = 1; x = -( 1.0/x ); } else if( x <= 0.66 ) { y = 0.0; } else { y = PIO4; flag = 2; x = (x-1.0)/(x+1.0); } z = x * x; z = z * polevl( z, P, 4 ) / p1evl( z, Q, 5 ); z = x * z + x; if( flag == 2 ) z += 0.5 * MOREBITS; else if( flag == 1 ) z += MOREBITS; y = y + z; if( sign < 0 ) y = -y; return(y); } /* atan2 */ #ifdef ANSIC double atan2( y, x ) #else double atan2( x, y ) #endif double x, y; { double z, w; short code; code = 0; #ifdef NANS if( isnan(x) ) return(x); if( isnan(y) ) return(y); #endif #ifdef MINUSZERO if( y == 0.0 ) { if( signbit(y) ) { if( x > 0.0 ) z = y; else if( x < 0.0 ) z = -PI; else { if( signbit(x) ) z = -PI; else z = y; } } else /* y is +0 */ { if( x == 0.0 ) { if( signbit(x) ) z = PI; else z = 0.0; } else if( x > 0.0 ) z = 0.0; else z = PI; } return z; } if( x == 0.0 ) { if( y > 0.0 ) z = PIO2; else z = -PIO2; return z; } #endif /* MINUSZERO */ #ifdef INFINITIES if( x == INFINITY ) { if( y == INFINITY ) z = 0.25 * PI; else if( y == -INFINITY ) z = -0.25 * PI; else if( y < 0.0 ) z = NEGZERO; else z = 0.0; return z; } if( x == -INFINITY ) { if( y == INFINITY ) z = 0.75 * PI; else if( y <= -INFINITY ) z = -0.75 * PI; else if( y >= 0.0 ) z = PI; else z = -PI; return z; } if( y == INFINITY ) return( PIO2 ); if( y == -INFINITY ) return( -PIO2 ); #endif if( x < 0.0 ) code = 2; if( y < 0.0 ) code |= 1; #ifdef INFINITIES if( x == 0.0 ) #else if( fabs(x) <= (fabs(y) / MAXNUM) ) #endif { if( code & 1 ) { #if ANSIC return( -PIO2 ); #else return( 3.0*PIO2 ); #endif } if( y == 0.0 ) return( 0.0 ); return( PIO2 ); } if( y == 0.0 ) { if( code & 2 ) return( PI ); return( 0.0 ); } switch( code ) { #if ANSIC default: case 0: case 1: w = 0.0; break; case 2: w = PI; break; case 3: w = -PI; break; #else default: case 0: w = 0.0; break; case 1: w = 2.0 * PI; break; case 2: case 3: w = PI; break; #endif } z = w + atan( y/x ); #ifdef MINUSZERO if( z == 0.0 && y < 0 ) z = NEGZERO; #endif return( z ); }