/* * CDDL HEADER START * * The contents of this file are subject to the terms of the * Common Development and Distribution License (the "License"). * You may not use this file except in compliance with the License. * * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE * or http://www.opensolaris.org/os/licensing. * See the License for the specific language governing permissions * and limitations under the License. * * When distributing Covered Code, include this CDDL HEADER in each * file and include the License file at usr/src/OPENSOLARIS.LICENSE. * If applicable, add the following below this CDDL HEADER, with the * fields enclosed by brackets "[]" replaced with your own identifying * information: Portions Copyright [yyyy] [name of copyright owner] * * CDDL HEADER END */ /* * Copyright 2011 Nexenta Systems, Inc. All rights reserved. */ /* * Copyright 2006 Sun Microsystems, Inc. All rights reserved. * Use is subject to license terms. */ #pragma weak __atan = atan /* INDENT OFF */ /* * atan(x) * Accurate Table look-up algorithm with polynomial approximation in * partially product form. * * -- K.C. Ng, October 17, 2004 * * Algorithm * * (1). Purge off Inf and NaN and 0 * (2). Reduce x to positive by atan(x) = -atan(-x). * (3). For x <= 1/8 and let z = x*x, return * (2.1) if x < 2^(-prec/2), atan(x) = x with inexact flag raised * (2.2) if x < 2^(-prec/4-1), atan(x) = x+(x/3)(x*x) * (2.3) if x < 2^(-prec/6-2), atan(x) = x+(z-5/3)(z*x/5) * (2.4) Otherwise * atan(x) = poly1(x) = x + A * B, * where * A = (p1*x*z) * (p2+z(p3+z)) * B = (p4+z)+z*z) * (p5+z(p6+z)) * Note: (i) domain of poly1 is [0, 1/8], (ii) remez relative * approximation error of poly1 is bounded by * |(atan(x)-poly1(x))/x| <= 2^-57.61 * (4). For x >= 8 then * (3.1) if x >= 2^prec, atan(x) = atan(inf) - pio2lo * (3.2) if x >= 2^(prec/3), atan(x) = atan(inf) - 1/x * (3.3) if x <= 65, atan(x) = atan(inf) - poly1(1/x) * (3.4) otherwise atan(x) = atan(inf) - poly2(1/x) * where * poly2(r) = (q1*r) * (q2+z(q3+z)) * (q4+z), * its domain is [0, 0.0154]; and its remez absolute * approximation error is bounded by * |atan(x)-poly2(x)|<= 2^-59.45 * * (5). Now x is in (0.125, 8). * Recall identity * atan(x) = atan(y) + atan((x-y)/(1+x*y)). * Let j = (ix - 0x3fc00000) >> 16, 0 <= j < 96, where ix is the high * part of x in IEEE double format. Then * atan(x) = atan(y[j]) + poly2((x-y[j])/(1+x*y[j])) * where y[j] are carefully chosen so that it matches x to around 4.5 * bits and at the same time atan(y[j]) is very close to an IEEE double * floating point number. Calculation indicates that * max|(x-y[j])/(1+x*y[j])| < 0.0154 * j,x * * Accuracy: Maximum error observed is bounded by 0.6 ulp after testing * more than 10 million random arguments */ /* INDENT ON */ #include "libm.h" #include "libm_protos.h" extern const double _TBL_atan[]; static const double g[] = { /* one = */ 1.0, /* p1 = */ 8.02176624254765935351230154992663301527500152588e-0002, /* p2 = */ 1.27223421700559402580665846471674740314483642578e+0000, /* p3 = */ -1.20606901800503640842521235754247754812240600586e+0000, /* p4 = */ -2.36088967922325565496066701598465442657470703125e+0000, /* p5 = */ 1.38345799501389166152875986881554126739501953125e+0000, /* p6 = */ 1.06742368078953453469637224770849570631980895996e+0000, /* q1 = */ -1.42796626333911796935538518482644576579332351685e-0001, /* q2 = */ 3.51427110447873227059810477159863497078605962912e+0000, /* q3 = */ 5.92129112708164262457444237952586263418197631836e-0001, /* q4 = */ -1.99272234785683144409063061175402253866195678711e+0000, /* pio2hi */ 1.570796326794896558e+00, /* pio2lo */ 6.123233995736765886e-17, /* t1 = */ -0.333333333333333333333333333333333, /* t2 = */ 0.2, /* t3 = */ -1.666666666666666666666666666666666, }; #define one g[0] #define p1 g[1] #define p2 g[2] #define p3 g[3] #define p4 g[4] #define p5 g[5] #define p6 g[6] #define q1 g[7] #define q2 g[8] #define q3 g[9] #define q4 g[10] #define pio2hi g[11] #define pio2lo g[12] #define t1 g[13] #define t2 g[14] #define t3 g[15] double atan(double x) { double y, z, r, p, s; int ix, lx, hx, j; hx = ((int *) &x)[HIWORD]; lx = ((int *) &x)[LOWORD]; ix = hx & ~0x80000000; j = ix >> 20; /* for |x| < 1/8 */ if (j < 0x3fc) { if (j < 0x3f5) { /* when |x| < 2**(-prec/6-2) */ if (j < 0x3e3) { /* if |x| < 2**(-prec/2-2) */ return ((int) x == 0 ? x : one); } if (j < 0x3f1) { /* if |x| < 2**(-prec/4-1) */ return (x + (x * t1) * (x * x)); } else { /* if |x| < 2**(-prec/6-2) */ z = x * x; s = t2 * x; return (x + (t3 + z) * (s * z)); } } z = x * x; s = p1 * x; return (x + ((s * z) * (p2 + z * (p3 + z))) * (((p4 + z) + z * z) * (p5 + z * (p6 + z)))); } /* for |x| >= 8.0 */ if (j >= 0x402) { if (j < 0x436) { r = one / x; if (hx >= 0) { y = pio2hi; p = pio2lo; } else { y = -pio2hi; p = -pio2lo; } if (ix < 0x40504000) { /* x < 65 */ z = r * r; s = p1 * r; return (y + ((p - r) - ((s * z) * (p2 + z * (p3 + z))) * (((p4 + z) + z * z) * (p5 + z * (p6 + z))))); } else if (j < 0x412) { z = r * r; return (y + (p - ((q1 * r) * (q4 + z)) * (q2 + z * (q3 + z)))); } else return (y + (p - r)); } else { if (j >= 0x7ff) /* x is inf or NaN */ if (((ix - 0x7ff00000) | lx) != 0) #if defined(FPADD_TRAPS_INCOMPLETE_ON_NAN) return (ix >= 0x7ff80000 ? x : x - x); /* assumes sparc-like QNaN */ #else return (x - x); #endif y = -pio2lo; return (hx >= 0 ? pio2hi - y : y - pio2hi); } } else { /* now x is between 1/8 and 8 */ double *w, w0, w1, s, z; w = (double *) _TBL_atan + (((ix - 0x3fc00000) >> 16) << 1); w0 = (hx >= 0)? w[0] : -w[0]; s = (x - w0) / (one + x * w0); w1 = (hx >= 0)? w[1] : -w[1]; z = s * s; return (((q1 * s) * (q4 + z)) * (q2 + z * (q3 + z)) + w1); } }
