/* * 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 atanl = __atanl /* * atanl(x) * Table look-up algorithm * By K.C. Ng, March 9, 1989 * * Algorithm. * * The algorithm is based on atan(x)=atan(y)+atan((x-y)/(1+x*y)). * We use poly1(x) to approximate atan(x) for x in [0,1/8] with * error (relative) * |(atan(x)-poly1(x))/x|<= 2^-115.94 long double * |(atan(x)-poly1(x))/x|<= 2^-58.85 double * |(atan(x)-poly1(x))/x|<= 2^-25.53 float * and use poly2(x) to approximate atan(x) for x in [0,1/65] with * error (absolute) * |atan(x)-poly2(x)|<= 2^-122.15 long double * |atan(x)-poly2(x)|<= 2^-64.79 double * |atan(x)-poly2(x)|<= 2^-35.36 float * Here poly1 and poly2 are odd polynomial with the following form: * x + x^3*(a1+x^2*(a2+...)) * * (0). Purge off Inf and NaN and 0 * (1). Reduce x to positive by atan(x) = -atan(-x). * (2). For x <= 1/8, use * (2.1) if x < 2^(-prec/2-2), atan(x) = x with inexact * (2.2) Otherwise * atan(x) = poly1(x) * (3). For x >= 8 then * (3.1) if x >= 2^(prec+2), atan(x) = atan(inf) - pio2lo * (3.2) if x >= 2^(prec/3+2), atan(x) = atan(inf) - 1/x * (3.3) if x > 65, atan(x) = atan(inf) - poly2(1/x) * (3.4) Otherwise, atan(x) = atan(inf) - poly1(1/x) * * (4). Now x is in (0.125, 8) * Find y that match x to 4.5 bit after binary (easy). * If iy is the high word of y, then * single : j = (iy - 0x3e000000) >> 19 * double : j = (iy - 0x3fc00000) >> 16 * quad : j = (iy - 0x3ffc0000) >> 12 * * Let s = (x-y)/(1+x*y). Then * atan(x) = atan(y) + poly1(s) * = _TBL_atanl_hi[j] + (_TBL_atanl_lo[j] + poly2(s) ) * * Note. |s| <= 1.5384615385e-02 = 1/65. Maxium occurs at x = 1.03125 * */ #include "libm.h" extern const long double _TBL_atanl_hi[], _TBL_atanl_lo[]; static const long double one = 1.0L, p1 = -3.333333333333333333333333333331344526118e-0001L, p2 = 1.999999999999999999999999989931277668570e-0001L, p3 = -1.428571428571428571428553606221309530901e-0001L, p4 = 1.111111111111111111095219842737139747418e-0001L, p5 = -9.090909090909090825503603835248061123323e-0002L, p6 = 7.692307692307664052130743214708925258904e-0002L, p7 = -6.666666666660213835187713228363717388266e-0002L, p8 = 5.882352940152439399097283359608661949504e-0002L, p9 = -5.263157780447533993046614040509529668487e-0002L, p10 = 4.761895816878184933175855990886788439447e-0002L, p11 = -4.347345005832274022681019724553538135922e-0002L, p12 = 3.983031914579635037502589204647752042736e-0002L, p13 = -3.348206704469830575196657749413894897554e-0002L, q1 = -3.333333333333333333333333333195273650186e-0001L, q2 = 1.999999999999999999999988146114392615808e-0001L, q3 = -1.428571428571428571057630319435467111253e-0001L, q4 = 1.111111111111105373263048208994541544098e-0001L, q5 = -9.090909090421834209167373258681021816441e-0002L, q6 = 7.692305377813692706850171767150701644539e-0002L, q7 = -6.660896644393861499914731734305717901330e-0002L, pio2hi = 1.570796326794896619231321691639751398740e+0000L, pio2lo = 4.335905065061890512398522013021675984381e-0035L; #define i0 0 #define i1 3 long double atanl(long double x) { long double y, z, r, p, s; int *px = (int *) &x, *py = (int *) &y; int ix, iy, sign, j; ix = px[i0]; sign = ix & 0x80000000; ix ^= sign; /* for |x| < 1/8 */ if (ix < 0x3ffc0000) { if (ix < 0x3feb0000) { /* when |x| < 2**(-prec/6-2) */ if (ix < 0x3fc50000) { /* if |x| < 2**(-prec/2-2) */ s = one; *(3 - i0 + (int *) &s) = -1; /* s = 1-ulp */ *(1 + (int *) &s) = -1; *(2 + (int *) &s) = -1; *(i0 + (int *) &s) -= 1; if ((int) (s * x) < 1) return (x); /* raise inexact */ } z = x * x; if (ix < 0x3fe20000) { /* if |x| < 2**(-prec/4-1) */ return (x + (x * z) * p1); } else { /* if |x| < 2**(-prec/6-2) */ return (x + (x * z) * (p1 + z * p2)); } } z = x * x; return (x + (x * z) * (p1 + z * (p2 + z * (p3 + z * (p4 + z * (p5 + z * (p6 + z * (p7 + z * (p8 + z * (p9 + z * (p10 + z * (p11 + z * (p12 + z * p13))))))))))))); } /* for |x| >= 8.0 */ if (ix >= 0x40020000) { px[i0] = ix; if (ix < 0x40050400) { /* x < 65 */ r = one / x; z = r * r; /* * poly1 */ y = r * (one + z * (p1 + z * (p2 + z * (p3 + z * (p4 + z * (p5 + z * (p6 + z * (p7 + z * (p8 + z * (p9 + z * (p10 + z * (p11 + z * (p12 + z * p13))))))))))))); y -= pio2lo; } else if (ix < 0x40260000) { /* x < 2**(prec/3+2) */ r = one / x; z = r * r; /* * poly2 */ y = r * (one + z * (q1 + z * (q2 + z * (q3 + z * (q4 + z * (q5 + z * (q6 + z * q7))))))); y -= pio2lo; } else if (ix < 0x40720000) { /* x < 2**(prec+2) */ y = one / x - pio2lo; } else if (ix < 0x7fff0000) { /* x < inf */ y = -pio2lo; } else { /* x is inf or NaN */ if (((ix - 0x7fff0000) | px[1] | px[2] | px[i1]) != 0) return (x - x); y = -pio2lo; } if (sign == 0) return (pio2hi - y); else return (y - pio2hi); } /* now x is between 1/8 and 8 */ px[i0] = ix; iy = (ix + 0x00000800) & 0x7ffff000; py[i0] = iy; py[1] = py[2] = py[i1] = 0; j = (iy - 0x3ffc0000) >> 12; if (sign == 0) s = (x - y) / (one + x * y); else s = (y - x) / (one + x * y); z = s * s; if (ix == iy) p = s * (one + z * (q1 + z * (q2 + z * (q3 + z * q4)))); else p = s * (one + z * (q1 + z * (q2 + z * (q3 + z * (q4 + z * (q5 + z * (q6 + z * q7))))))); if (sign == 0) { r = p + _TBL_atanl_lo[j]; return (r + _TBL_atanl_hi[j]); } else { r = p - _TBL_atanl_lo[j]; return (r - _TBL_atanl_hi[j]); } }