atan.c File Reference

Inverse tangent function. More...

#include "prim.h"

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Functions
double	atan (double x)
	Inverse tangent function. More...

Variables
static const double	atanhi []
static const double	atanlo []
static const double	aT []
static const double	one = 1.0
static const double	huge = 1.0e300

Detailed Description

Inverse tangent function.

Definition in file atan.c.

Function Documentation

atan()

double atan

(

double

x

)

Inverse tangent function.

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.

Definition at line 103 of file atan.c.

References aT, atanhi, atanlo, fabs(), huge, and one.

Referenced by acot(), acvs(), aexs(), ahv(), RealNumber::ArcTangent(), atan2(), and aver().

{
    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)))
            return NAN;
        if (hx > 0)
            return atanhi[3] + atanlo[3];
        else
            return -atanhi[3] - atanlo[3];
    }
    if (ix < 0x3fdc0000)
    { /* |x| < 0.4375 */
        if (ix < 0x3e200000)
        { /* |x| < 2^-29 */
            if (huge + x > one)
                return x; /* raise inexact */
        }
        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 - one) / (2.0 + x);
            }
            else
            { /* 11/16<=|x|< 19/16 */
                id = 1;
                x = (x - one) / (x + one);
            }
        }
        else
        {
            if (ix < 0x40038000)
            { /* |x| < 2.4375 */
                id = 2;
                x = (x - 1.5) / (one + 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;
}

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Variable Documentation

aT

const double aT[]

static

Initial value:

= {
    3.33333333333329318027e-01,  
    -1.99999999998764832476e-01, 
    1.42857142725034663711e-01,  
    -1.11111104054623557880e-01, 
    9.09088713343650656196e-02,  
    -7.69187620504482999495e-02, 
    6.66107313738753120669e-02,  
    -5.83357013379057348645e-02, 
    4.97687799461593236017e-02,  
    -3.65315727442169155270e-02, 
    1.62858201153657823623e-02,  
}

Definition at line 62 of file atan.c.

Referenced by atan().

atanhi

const double atanhi[]

static

Initial value:

= {
    4.63647609000806093515e-01, 
    7.85398163397448278999e-01, 
    9.82793723247329054082e-01, 
    1.57079632679489655800e+00, 
}

Definition at line 48 of file atan.c.

Referenced by atan().

atanlo

const double atanlo[]

static

Initial value:

= {
    2.26987774529616870924e-17, 
    3.06161699786838301793e-17, 
    1.39033110312309984516e-17, 
    6.12323399573676603587e-17, 
}

Definition at line 55 of file atan.c.

Referenced by atan().

huge

const double huge = 1.0e300

static

Definition at line 78 of file atan.c.

Referenced by atan().

one

const double one = 1.0

static

Definition at line 77 of file atan.c.

Referenced by atan().