musl/src/math/atan.c

/* origin: FreeBSD /usr/src/lib/msun/src/s_atan.c */
/*
 * ====================================================
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 *
 * Developed at SunPro, a Sun Microsystems, Inc. business.
 * Permission to use, copy, modify, and distribute this
 * software is freely granted, provided that this notice
 * is preserved.
 * ====================================================
 */
/* atan(x)
 * Method
 *   1. Reduce x to positive by atan(x) = -atan(-x).
 *   2. According to the integer k=4t+0.25 chopped, t=x, the argument
 *      is further reduced to one of the following intervals and the
 *      arctangent of t is evaluated by the corresponding formula:
 *
 *      [0,7/16]      atan(x) = t-t^3*(a1+t^2*(a2+...(a10+t^2*a11)...)
 *      [7/16,11/16]  atan(x) = atan(1/2) + atan( (t-0.5)/(1+t/2) )
 *      [11/16.19/16] atan(x) = atan( 1 ) + atan( (t-1)/(1+t) )
 *      [19/16,39/16] atan(x) = atan(3/2) + atan( (t-1.5)/(1+1.5t) )
 *      [39/16,INF]   atan(x) = atan(INF) + atan( -1/t )
 *
 * Constants:
 * The hexadecimal values are the intended ones for the following
 * constants. The decimal values may be used, provided that the
 * compiler will convert from decimal to binary accurately enough
 * to produce the hexadecimal values shown.
 */


#include "libm.h"

static const double atanhi[] = {
  4.63647609000806093515e-01, /* atan(0.5)hi 0x3FDDAC67, 0x0561BB4F */
  7.85398163397448278999e-01, /* atan(1.0)hi 0x3FE921FB, 0x54442D18 */
  9.82793723247329054082e-01, /* atan(1.5)hi 0x3FEF730B, 0xD281F69B */
  1.57079632679489655800e+00, /* atan(inf)hi 0x3FF921FB, 0x54442D18 */
};

static const double atanlo[] = {
  2.26987774529616870924e-17, /* atan(0.5)lo 0x3C7A2B7F, 0x222F65E2 */
  3.06161699786838301793e-17, /* atan(1.0)lo 0x3C81A626, 0x33145C07 */
  1.39033110312309984516e-17, /* atan(1.5)lo 0x3C700788, 0x7AF0CBBD */
  6.12323399573676603587e-17, /* atan(inf)lo 0x3C91A626, 0x33145C07 */
};

static const double aT[] = {
  3.33333333333329318027e-01, /* 0x3FD55555, 0x5555550D */
 -1.99999999998764832476e-01, /* 0xBFC99999, 0x9998EBC4 */
  1.42857142725034663711e-01, /* 0x3FC24924, 0x920083FF */
 -1.11111104054623557880e-01, /* 0xBFBC71C6, 0xFE231671 */
  9.09088713343650656196e-02, /* 0x3FB745CD, 0xC54C206E */
 -7.69187620504482999495e-02, /* 0xBFB3B0F2, 0xAF749A6D */
  6.66107313738753120669e-02, /* 0x3FB10D66, 0xA0D03D51 */
 -5.83357013379057348645e-02, /* 0xBFADDE2D, 0x52DEFD9A */
  4.97687799461593236017e-02, /* 0x3FA97B4B, 0x24760DEB */
 -3.65315727442169155270e-02, /* 0xBFA2B444, 0x2C6A6C2F */
  1.62858201153657823623e-02, /* 0x3F90AD3A, 0xE322DA11 */
};

static const double huge = 1.0e300;

double atan(double x)
{
	double w,s1,s2,z;
	int32_t ix,hx,id;

	GET_HIGH_WORD(hx, x);
	ix = hx & 0x7fffffff;
	if (ix >= 0x44100000) {   /* if |x| >= 2^66 */
		uint32_t low;

		GET_LOW_WORD(low, x);
		if (ix > 0x7ff00000 ||
		    (ix == 0x7ff00000 && low != 0))  /* NaN */
			return x+x;
		if (hx > 0)
			return  atanhi[3] + *(volatile double *)&atanlo[3];
		else
			return -atanhi[3] - *(volatile double *)&atanlo[3];
	}
	if (ix < 0x3fdc0000) {    /* |x| < 0.4375 */
		if (ix < 0x3e400000) {  /* |x| < 2^-27 */
			/* raise inexact */
			if (huge+x > 1.0)
				return x;
		}
		id = -1;
	} else {
		x = fabs(x);
		if (ix < 0x3ff30000) {  /* |x| < 1.1875 */
			if (ix < 0x3fe60000) {  /*  7/16 <= |x| < 11/16 */
				id = 0;
				x = (2.0*x-1.0)/(2.0+x);
			} else {                /* 11/16 <= |x| < 19/16 */
				id = 1;
				x = (x-1.0)/(x+1.0);
			}
		} else {
			if (ix < 0x40038000) {  /* |x| < 2.4375 */
				id = 2;
				x = (x-1.5)/(1.0+1.5*x);
			} else {                /* 2.4375 <= |x| < 2^66 */
				id = 3;
				x = -1.0/x;
			}
		}
	}
	/* end of argument reduction */
	z = x*x;
	w = z*z;
	/* break sum from i=0 to 10 aT[i]z**(i+1) into odd and even poly */
	s1 = z*(aT[0]+w*(aT[2]+w*(aT[4]+w*(aT[6]+w*(aT[8]+w*aT[10])))));
	s2 = w*(aT[1]+w*(aT[3]+w*(aT[5]+w*(aT[7]+w*aT[9]))));
	if (id < 0)
		return x - x*(s1+s2);
	z = atanhi[id] - (x*(s1+s2) - atanlo[id] - x);
	return hx < 0 ? -z : z;
}