/**
* Convert radians to degrees, with error of less than 0.5 ULP
* @param x angle in radians
* @return x converted into degrees
*/
public static double toDegrees(double x)
{
if (Double.IsInfinity(x) || x == 0.0)
{ // Matches +/- 0.0; return correct sign
return(x);
}
// These are 180/PI split into high and low order bits
double facta = 57.2957763671875;
double factb = 3.145894820876798E-6;
double xa = BitOps.HighPart(x);
double xb = x - xa;
return(xb * factb + xb * facta + xa * factb + xa * facta);
}
/**
* Convert degrees to radians, with error of less than 0.5 ULP
* @param x angle in degrees
* @return x converted into radians
*/
public static double ToRadians(double x)
{
if (Double.IsInfinity(x) || x == 0.0)
{ // Matches +/- 0.0; return correct sign
return(x);
}
// These are PI/180 split into high and low order bits
double facta = 0.01745329052209854;
double factb = 1.997844754509471E-9;
double xa = BitOps.HighPart(x);
double xb = x - xa;
double result = xb * factb + xb * facta + xa * factb + xa * facta;
if (result == 0)
{
result = result * x; // ensure correct sign if calculation underflows
}
return(result);
}
/**
* Two arguments arctangent function
* @param y ordinate
* @param x abscissa
* @return phase angle of point (x,y) between {@code -PI} and {@code PI}
*/
public static double Arc(double y, double x)
{
if (y == 0)
{
double result = x * y;
double invx = 1d / x;
double invy = 1d / y;
if (invx == 0)
{ // X is infinite
if (x > 0)
{
return(y); // return +/- 0.0
}
else
{
return(BitOps.CopySign(Math.PI, y));
}
}
if (x < 0 || invx < 0)
{
if (y < 0 || invy < 0)
{
return(-Math.PI);
}
else
{
return(Math.PI);
}
}
else
{
return(result);
}
}
// y cannot now be zero
if (y == Double.PositiveInfinity)
{
if (x == Double.PositiveInfinity)
{
return(Math.PI * F_1_4);
}
if (x == Double.PositiveInfinity)
{
return(Math.PI * F_3_4);
}
return(Math.PI * F_1_2);
}
if (y == Double.NegativeInfinity)
{
if (x == Double.PositiveInfinity)
{
return(-Math.PI * F_1_4);
}
if (x == Double.NegativeInfinity)
{
return(-Math.PI * F_3_4);
}
return(-Math.PI * F_1_2);
}
if (x == Double.PositiveInfinity)
{
if (y > 0 || 1 / y > 0)
{
return(0d);
}
if (y < 0 || 1 / y < 0)
{
return(-0d);
}
}
if (x == Double.NegativeInfinity)
{
if (y > 0.0 || 1 / y > 0.0)
{
return(Math.PI);
}
if (y < 0 || 1 / y < 0)
{
return(-Math.PI);
}
}
// Neither y nor x can be infinite or NAN here
if (x == 0)
{
if (y > 0 || 1 / y > 0)
{
return(Math.PI * F_1_2);
}
if (y < 0 || 1 / y < 0)
{
return(-Math.PI * F_1_2);
}
}
// Compute ratio r = y/x
double r = y / x;
if (Double.IsInfinity(r))
{ // bypass calculations that can create NaN
return(Arc(r, 0, x < 0));
}
double ra = BitOps.HighPart(r);
double rb = r - ra;
// Split x
double xa = BitOps.HighPart(x);
double xb = x - xa;
rb += (y - ra * xa - ra * xb - rb * xa - rb * xb) / x;
double temp = ra + rb;
rb = -(temp - ra - rb);
ra = temp;
if (ra == 0)
{ // Fix up the sign so atan works correctly
ra = BitOps.CopySign(0d, y);
}
// Call atan
return(Arc(ra, rb, x < 0));
}
/** Internal helper function to compute arctangent.
* @param xa number from which arctangent is requested
* @param xb extra bits for x (may be 0.0)
* @param leftPlane if true, result angle must be put in the left half plane
* @return atan(xa + xb) (or angle shifted by {@code PI} if leftPlane is true)
*/
private static double Arc(double xa, double xb, bool leftPlane)
{
bool negate = false;
int idx;
if (xa == 0.0)
{ // Matches +/- 0.0; return correct sign
return(leftPlane ? BitOps.CopySign(Math.PI, xa) : xa);
}
if (xa < 0)
{
// negative
xa = -xa;
xb = -xb;
negate = true;
}
if (xa > 1.633123935319537E16)
{ // Very large input
return((negate ^ leftPlane) ? (-Math.PI * F_1_2) : (Math.PI * F_1_2));
}
/* Estimate the closest tabulated arctan value, compute eps = xa-tangentTable */
if (xa < 1)
{
idx = (int)(((-1.7168146928204136 * xa * xa + 8.0) * xa) + 0.5);
}
else
{
double oneOverXa = 1 / xa;
idx = (int)(-((-1.7168146928204136 * oneOverXa * oneOverXa + 8.0) * oneOverXa) + 13.07);
}
double epsA = xa - TANGENT_TABLE_A[idx];
double epsB = -(epsA - xa + TANGENT_TABLE_A[idx]);
epsB += xb - TANGENT_TABLE_B[idx];
double temp = epsA + epsB;
epsB = -(temp - epsA - epsB);
epsA = temp;
/* Compute eps = eps / (1.0 + xa*tangent) */
temp = xa * BitOps.HEX_40000000;
double ya = xa + temp - temp;
double yb = xb + xa - ya;
xa = ya;
xb += yb;
//if (idx > 8 || idx == 0)
if (idx == 0)
{
/* If the slope of the arctan is gentle enough (< 0.45), this approximation will suffice */
//double denom = 1.0 / (1.0 + xa*tangentTableA[idx] + xb*tangentTableA[idx] + xa*tangentTableB[idx] + xb*tangentTableB[idx]);
double denom = 1d / (1d + (xa + xb) * (TANGENT_TABLE_A[idx] + TANGENT_TABLE_B[idx]));
//double denom = 1.0 / (1.0 + xa*tangentTableA[idx]);
ya = epsA * denom;
yb = epsB * denom;
}
else
{
double temp2 = xa * TANGENT_TABLE_A[idx];
double za = 1d + temp2;
double zb = -(za - 1d - temp2);
temp2 = xb * TANGENT_TABLE_A[idx] + xa * TANGENT_TABLE_B[idx];
temp = za + temp2;
zb += -(temp - za - temp2);
za = temp;
zb += xb * TANGENT_TABLE_B[idx];
ya = epsA / za;
temp = ya * BitOps.HEX_40000000;
double yaa = (ya + temp) - temp;
double yab = ya - yaa;
temp = za * BitOps.HEX_40000000;
double zaa = (za + temp) - temp;
double zab = za - zaa;
/* Correct for rounding in division */
yb = (epsA - yaa * zaa - yaa * zab - yab * zaa - yab * zab) / za;
yb += -epsA * zb / za / za;
yb += epsB / za;
}
epsA = ya;
epsB = yb;
/* Evaluate polynomial */
double epsA2 = epsA * epsA;
/*
* yb = -0.09001346640161823;
* yb = yb * epsA2 + 0.11110718400605211;
* yb = yb * epsA2 + -0.1428571349122913;
* yb = yb * epsA2 + 0.19999999999273194;
* yb = yb * epsA2 + -0.33333333333333093;
* yb = yb * epsA2 * epsA;
*/
yb = 0.07490822288864472;
yb = yb * epsA2 + -0.09088450866185192;
yb = yb * epsA2 + 0.11111095942313305;
yb = yb * epsA2 + -0.1428571423679182;
yb = yb * epsA2 + 0.19999999999923582;
yb = yb * epsA2 + -0.33333333333333287;
yb = yb * epsA2 * epsA;
ya = epsA;
temp = ya + yb;
yb = -(temp - ya - yb);
ya = temp;
/* Add in effect of epsB. atan'(x) = 1/(1+x^2) */
yb += epsB / (1d + epsA * epsA);
//result = yb + eighths[idx] + ya;
double za = BitOps.EIGHTHS[idx] + ya;
double zb = -(za - BitOps.EIGHTHS[idx] - ya);
temp = za + yb;
zb += -(temp - za - yb);
za = temp;
double result = za + zb;
double resultb = -(result - za - zb);
if (leftPlane)
{
// Result is in the left plane
double pia = 1.5707963267948966 * 2;
double pib = 6.123233995736766E-17 * 2;
za = pia - result;
zb = -(za - pia + result);
zb += pib - resultb;
result = za + zb;
resultb = -(result - za - zb);
}
if (negate ^ leftPlane)
{
result = -result;
}
return(result);
}
/** Compute the arc cosine of a number.
* @param x number on which evaluation is done
* @return arc cosine of x
*/
public static double Arc(double x)
{
if (x > 1.0 || x < -1.0)
{
return(Double.NaN);
}
if (x == -1.0)
{
return(Math.PI);
}
if (x == 1.0)
{
return(0.0);
}
if (x == 0)
{
return(Math.PI / 2.0);
}
/* Compute acos(x) = atan(sqrt(1-x*x)/x) */
/* Split x */
double temp = x * BitOps.HEX_40000000;
double xa = x + temp - temp;
double xb = x - xa;
/* Square it */
double ya = xa * xa;
double yb = xa * xb * 2.0 + xb * xb;
/* Subtract from 1 */
ya = -ya;
yb = -yb;
double za = 1.0 + ya;
double zb = -(za - 1.0 - ya);
temp = za + yb;
zb += -(temp - za - yb);
za = temp;
/* Square root */
double y = Math.Sqrt(za);
temp = y * BitOps.HEX_40000000;
ya = y + temp - temp;
yb = y - ya;
/* Extend precision of sqrt */
yb += (za - ya * ya - 2 * ya * yb - yb * yb) / (2.0 * y);
/* Contribution of zb to sqrt */
yb += zb / (2.0 * y);
y = ya + yb;
yb = -(y - ya - yb);
// Compute ratio r = y/x
double r = y / x;
// Did r overflow?
if (double.IsInfinity(r))
{ // x is effectively zero
return(Math.PI / 2); // so return the appropriate value
}
double ra = BitOps.HighPart(r);
double rb = r - ra;
rb += (y - ra * xa - ra * xb - rb * xa - rb * xb) / x; // Correct for rounding in division
rb += yb / x; // Add in effect additional bits of sqrt.
temp = ra + rb;
rb = -(temp - ra - rb);
ra = temp;
return(atan(ra, rb, x < 0));
}