acos.c File Reference

Inverse cosine function. More...

#include "prim.h"

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Functions
double	acos (double x)
	Inverse cosine function. More...

Variables
static const double	one = 1.00000000000000000000e+00
static const double	pi = 3.14159265358979311600e+00
static const double	pio2_hi = 1.57079632679489655800e+00
static const double	pio2_lo = 6.12323399573676603587e-17
static const double	pS0 = 1.66666666666666657415e-01
static const double	pS1 = -3.25565818622400915405e-01
static const double	pS2 = 2.01212532134862925881e-01
static const double	pS3 = -4.00555345006794114027e-02
static const double	pS4 = 7.91534994289814532176e-04
static const double	pS5 = 3.47933107596021167570e-05
static const double	qS1 = -2.40339491173441421878e+00
static const double	qS2 = 2.02094576023350569471e+00
static const double	qS3 = -6.88283971605453293030e-01
static const double	qS4 = 7.70381505559019352791e-02

Detailed Description

Inverse cosine function.

Definition in file acos.c.

Function Documentation

acos()

double acos

(

double

x

)

Inverse cosine function.

Method
    acos(x)  = pi/2 - asin(x)
    acos(-x) = pi/2 + asin(x)

    For |x|<=0.5
    acos(x)  = pi/2 - (x + x*x^2*R(x^2))        (see asin.c)

    For x>0.5
    acos(x)  = pi/2 - (pi/2 - 2asin(sqrt((1-x)/2)))
             = 2asin(sqrt((1-x)/2))
             = 2s + 2s*z*R(z)   ...z=(1-x)/2, s=sqrt(z)
             = 2f + (2c + 2s*z*R(z))
    where f=hi part of s, and c = (z-f*f)/(s+f) is the correction term
    for f so that f+c ~ sqrt(z).

    For x<-0.5
    acos(x)  = pi - 2asin(sqrt((1-|x|)/2))
             = pi - 0.5*(s+s*z*R(z)), where z=(1-|x|)/2,s=sqrt(z)

Special cases
    if x is NaN, return NaN
    if |x|>1, return NaN

Definition at line 92 of file acos.c.

References one, pi, pio2_hi, pio2_lo, pS0, pS1, pS2, pS3, pS4, pS5, qS1, qS2, qS3, qS4, and sqrt().

Referenced by ahvc(), RealNumber::ArcCosine(), asec(), and avcs().

{
    double z, p, q, r, w, s, c, df;
    int32_t hx, ix;
    uint32_t lx;

    GET_HIGH_WORD(hx, x);
    ix = hx & 0x7FFFFFFF;

    // |x| >= 1
    if (ix >= 0x3FF00000)
    {
        GET_LOW_WORD(lx, x);

        // |x|==1
        if (((ix - 0x3FF00000) | lx) == 0)
        {
            if (hx > 0)
            {
                // acos(1) = 0
                return 0.0;
            }

            // acos(-1) = pi
            return pi + 2.0 * pio2_lo;
        }

        // acos(|x|>1) is NaN
        return NAN;
    }

    // |x| < 0.5
    if (ix < 0x3FE00000)
    {
        // if |x|<2**-57
        if (ix <= 0x3C600000)
        {
            return pio2_hi + pio2_lo;
        }
        z = x * x;
        p = z * (pS0 + z * (pS1 + z * (pS2 + z * (pS3 + z * (pS4 + z * pS5)))));
        q = one + z * (qS1 + z * (qS2 + z * (qS3 + z * qS4)));
        r = p / q;
        return pio2_hi - (x - (pio2_lo - x * r));
    }

    // x < -0.5
    if (hx < 0)
    {
        z = (one + x) * 0.5;
        p = z * (pS0 + z * (pS1 + z * (pS2 + z * (pS3 + z * (pS4 + z * pS5)))));
        q = one + z * (qS1 + z * (qS2 + z * (qS3 + z * qS4)));
        s = sqrt(z);
        r = p / q;
        w = r * s - pio2_lo;
        return pi - 2.0 * (s + w);
    }

    // x > 0.5
    z = (one - x) * 0.5;
    s = sqrt(z);
    df = s;
    SET_LOW_WORD(df, 0);
    c = (z - df * df) / (s + df);
    p = z * (pS0 + z * (pS1 + z * (pS2 + z * (pS3 + z * (pS4 + z * pS5)))));
    q = one + z * (qS1 + z * (qS2 + z * (qS3 + z * qS4)));
    r = p / q;
    w = r * s + c;
    return 2.0 * (df + w);
}

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Variable Documentation

one

const double one = 1.00000000000000000000e+00

static

Definition at line 49 of file acos.c.

Referenced by acos().

pi

const double pi = 3.14159265358979311600e+00

static

Definition at line 50 of file acos.c.

Referenced by acos().

pio2_hi

const double pio2_hi = 1.57079632679489655800e+00

static

Definition at line 51 of file acos.c.

Referenced by acos().

pio2_lo

const double pio2_lo = 6.12323399573676603587e-17

static

Definition at line 52 of file acos.c.

Referenced by acos().

pS0

const double pS0 = 1.66666666666666657415e-01

static

Definition at line 53 of file acos.c.

Referenced by acos().

pS1

const double pS1 = -3.25565818622400915405e-01

static

Definition at line 54 of file acos.c.

Referenced by acos().

pS2

const double pS2 = 2.01212532134862925881e-01

static

Definition at line 55 of file acos.c.

Referenced by acos().

pS3

const double pS3 = -4.00555345006794114027e-02

static

Definition at line 56 of file acos.c.

Referenced by acos().

pS4

const double pS4 = 7.91534994289814532176e-04

static

Definition at line 57 of file acos.c.

Referenced by acos().

pS5

const double pS5 = 3.47933107596021167570e-05

static

Definition at line 58 of file acos.c.

Referenced by acos().

qS1

const double qS1 = -2.40339491173441421878e+00

static

Definition at line 59 of file acos.c.

Referenced by acos().

qS2

const double qS2 = 2.02094576023350569471e+00

static

Definition at line 60 of file acos.c.

Referenced by acos().

qS3

const double qS3 = -6.88283971605453293030e-01

static

Definition at line 61 of file acos.c.

Referenced by acos().

qS4

const double qS4 = 7.70381505559019352791e-02

static

Definition at line 62 of file acos.c.

Referenced by acos().