Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2

Percentage Accurate: 89.0% → 99.5%

Time: 13.5s

Alternatives: 15

Speedup: 0.8×

Specification

Math

\[\begin{array}{l} \\ \frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))

C

double code(double x, double y, double z) {
	return (1.0 / x) / (y * (1.0 + (z * z)));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (1.0d0 / x) / (y * (1.0d0 + (z * z)))
end function

Java

public static double code(double x, double y, double z) {
	return (1.0 / x) / (y * (1.0 + (z * z)));
}

Python

def code(x, y, z):
	return (1.0 / x) / (y * (1.0 + (z * z)))

Julia

function code(x, y, z)
	return Float64(Float64(1.0 / x) / Float64(y * Float64(1.0 + Float64(z * z))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (1.0 / x) / (y * (1.0 + (z * z)));
end

Wolfram

code[x_, y_, z_] := N[(N[(1.0 / x), $MachinePrecision] / N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 89.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))

C

double code(double x, double y, double z) {
	return (1.0 / x) / (y * (1.0 + (z * z)));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (1.0d0 / x) / (y * (1.0d0 + (z * z)))
end function

Java

public static double code(double x, double y, double z) {
	return (1.0 / x) / (y * (1.0 + (z * z)));
}

Python

def code(x, y, z):
	return (1.0 / x) / (y * (1.0 + (z * z)))

Julia

function code(x, y, z)
	return Float64(Float64(1.0 / x) / Float64(y * Float64(1.0 + Float64(z * z))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (1.0 / x) / (y * (1.0 + (z * z)));
end

Wolfram

code[x_, y_, z_] := N[(N[(1.0 / x), $MachinePrecision] / N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 4.6 \cdot 10^{+228}:\\ \;\;\;\;\frac{\frac{\frac{1}{x_m \cdot \mathsf{hypot}\left(1, z_m\right)}}{\mathsf{hypot}\left(1, z_m\right)}}{y_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x_m}^{-1}}{y_m \cdot \mathsf{hypot}\left(1, z_m\right)}}{\mathsf{hypot}\left(1, z_m\right)}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z_m 4.6e+228)
     (/ (/ (/ 1.0 (* x_m (hypot 1.0 z_m))) (hypot 1.0 z_m)) y_m)
     (/ (/ (pow x_m -1.0) (* y_m (hypot 1.0 z_m))) (hypot 1.0 z_m))))))

C

z_m = fabs(z);
x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 4.6e+228) {
		tmp = ((1.0 / (x_m * hypot(1.0, z_m))) / hypot(1.0, z_m)) / y_m;
	} else {
		tmp = (pow(x_m, -1.0) / (y_m * hypot(1.0, z_m))) / hypot(1.0, z_m);
	}
	return y_s * (x_s * tmp);
}

Java

z_m = Math.abs(z);
x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 4.6e+228) {
		tmp = ((1.0 / (x_m * Math.hypot(1.0, z_m))) / Math.hypot(1.0, z_m)) / y_m;
	} else {
		tmp = (Math.pow(x_m, -1.0) / (y_m * Math.hypot(1.0, z_m))) / Math.hypot(1.0, z_m);
	}
	return y_s * (x_s * tmp);
}

Python

z_m = math.fabs(z)
x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x_m, y_m, z_m] = sort([x_m, y_m, z_m])
def code(y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if z_m <= 4.6e+228:
		tmp = ((1.0 / (x_m * math.hypot(1.0, z_m))) / math.hypot(1.0, z_m)) / y_m
	else:
		tmp = (math.pow(x_m, -1.0) / (y_m * math.hypot(1.0, z_m))) / math.hypot(1.0, z_m)
	return y_s * (x_s * tmp)

Julia

z_m = abs(z)
x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (z_m <= 4.6e+228)
		tmp = Float64(Float64(Float64(1.0 / Float64(x_m * hypot(1.0, z_m))) / hypot(1.0, z_m)) / y_m);
	else
		tmp = Float64(Float64((x_m ^ -1.0) / Float64(y_m * hypot(1.0, z_m))) / hypot(1.0, z_m));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

z_m = abs(z);
x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
x_m, y_m, z_m = num2cell(sort([x_m, y_m, z_m])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (z_m <= 4.6e+228)
		tmp = ((1.0 / (x_m * hypot(1.0, z_m))) / hypot(1.0, z_m)) / y_m;
	else
		tmp = ((x_m ^ -1.0) / (y_m * hypot(1.0, z_m))) / hypot(1.0, z_m);
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 4.6e+228], N[(N[(N[(1.0 / N[(x$95$m * N[Sqrt[1.0 ^ 2 + z$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z$95$m ^ 2], $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(N[Power[x$95$m, -1.0], $MachinePrecision] / N[(y$95$m * N[Sqrt[1.0 ^ 2 + z$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z$95$m ^ 2], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 2: 99.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ y_s \cdot \left(x_s \cdot {\left(\frac{{x_m}^{-0.5}}{\sqrt{y_m} \cdot \mathsf{hypot}\left(1, z_m\right)}\right)}^{2}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (* y_s (* x_s (pow (/ (pow x_m -0.5) (* (sqrt y_m) (hypot 1.0 z_m))) 2.0))))

C

z_m = fabs(z);
x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	return y_s * (x_s * pow((pow(x_m, -0.5) / (sqrt(y_m) * hypot(1.0, z_m))), 2.0));
}

Java

z_m = Math.abs(z);
x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	return y_s * (x_s * Math.pow((Math.pow(x_m, -0.5) / (Math.sqrt(y_m) * Math.hypot(1.0, z_m))), 2.0));
}

Python

z_m = math.fabs(z)
x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x_m, y_m, z_m] = sort([x_m, y_m, z_m])
def code(y_s, x_s, x_m, y_m, z_m):
	return y_s * (x_s * math.pow((math.pow(x_m, -0.5) / (math.sqrt(y_m) * math.hypot(1.0, z_m))), 2.0))

Julia

z_m = abs(z)
x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	return Float64(y_s * Float64(x_s * (Float64((x_m ^ -0.5) / Float64(sqrt(y_m) * hypot(1.0, z_m))) ^ 2.0)))
end

MATLAB

z_m = abs(z);
x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
x_m, y_m, z_m = num2cell(sort([x_m, y_m, z_m])){:}
function tmp = code(y_s, x_s, x_m, y_m, z_m)
	tmp = y_s * (x_s * (((x_m ^ -0.5) / (sqrt(y_m) * hypot(1.0, z_m))) ^ 2.0));
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * N[Power[N[(N[Power[x$95$m, -0.5], $MachinePrecision] / N[(N[Sqrt[y$95$m], $MachinePrecision] * N[Sqrt[1.0 ^ 2 + z$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 3: 99.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 2.5 \cdot 10^{+228}:\\ \;\;\;\;\frac{\frac{\frac{1}{x_m \cdot \mathsf{hypot}\left(1, z_m\right)}}{\mathsf{hypot}\left(1, z_m\right)}}{y_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{x_m}}{\mathsf{hypot}\left(1, z_m\right)}}{y_m \cdot \mathsf{hypot}\left(1, z_m\right)}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z_m 2.5e+228)
     (/ (/ (/ 1.0 (* x_m (hypot 1.0 z_m))) (hypot 1.0 z_m)) y_m)
     (/ (/ (/ 1.0 x_m) (hypot 1.0 z_m)) (* y_m (hypot 1.0 z_m)))))))

C

z_m = fabs(z);
x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 2.5e+228) {
		tmp = ((1.0 / (x_m * hypot(1.0, z_m))) / hypot(1.0, z_m)) / y_m;
	} else {
		tmp = ((1.0 / x_m) / hypot(1.0, z_m)) / (y_m * hypot(1.0, z_m));
	}
	return y_s * (x_s * tmp);
}

Java

z_m = Math.abs(z);
x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 2.5e+228) {
		tmp = ((1.0 / (x_m * Math.hypot(1.0, z_m))) / Math.hypot(1.0, z_m)) / y_m;
	} else {
		tmp = ((1.0 / x_m) / Math.hypot(1.0, z_m)) / (y_m * Math.hypot(1.0, z_m));
	}
	return y_s * (x_s * tmp);
}

Python

z_m = math.fabs(z)
x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x_m, y_m, z_m] = sort([x_m, y_m, z_m])
def code(y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if z_m <= 2.5e+228:
		tmp = ((1.0 / (x_m * math.hypot(1.0, z_m))) / math.hypot(1.0, z_m)) / y_m
	else:
		tmp = ((1.0 / x_m) / math.hypot(1.0, z_m)) / (y_m * math.hypot(1.0, z_m))
	return y_s * (x_s * tmp)

Julia

z_m = abs(z)
x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (z_m <= 2.5e+228)
		tmp = Float64(Float64(Float64(1.0 / Float64(x_m * hypot(1.0, z_m))) / hypot(1.0, z_m)) / y_m);
	else
		tmp = Float64(Float64(Float64(1.0 / x_m) / hypot(1.0, z_m)) / Float64(y_m * hypot(1.0, z_m)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

z_m = abs(z);
x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
x_m, y_m, z_m = num2cell(sort([x_m, y_m, z_m])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (z_m <= 2.5e+228)
		tmp = ((1.0 / (x_m * hypot(1.0, z_m))) / hypot(1.0, z_m)) / y_m;
	else
		tmp = ((1.0 / x_m) / hypot(1.0, z_m)) / (y_m * hypot(1.0, z_m));
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 2.5e+228], N[(N[(N[(1.0 / N[(x$95$m * N[Sqrt[1.0 ^ 2 + z$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z$95$m ^ 2], $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(N[(1.0 / x$95$m), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z$95$m ^ 2], $MachinePrecision]), $MachinePrecision] / N[(y$95$m * N[Sqrt[1.0 ^ 2 + z$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 4: 97.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \cdot z_m \leq 2 \cdot 10^{+21}:\\ \;\;\;\;\frac{\frac{1}{x_m}}{y_m \cdot \left(1 + z_m \cdot z_m\right)}\\ \mathbf{elif}\;z_m \cdot z_m \leq 2 \cdot 10^{+269}:\\ \;\;\;\;\frac{\frac{1}{x_m \cdot y_m}}{{z_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y_m}}{z_m} \cdot \frac{1}{x_m \cdot z_m}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z_m z_m) 2e+21)
     (/ (/ 1.0 x_m) (* y_m (+ 1.0 (* z_m z_m))))
     (if (<= (* z_m z_m) 2e+269)
       (/ (/ 1.0 (* x_m y_m)) (pow z_m 2.0))
       (* (/ (/ 1.0 y_m) z_m) (/ 1.0 (* x_m z_m))))))))

C

z_m = fabs(z);
x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 2e+21) {
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z_m * z_m)));
	} else if ((z_m * z_m) <= 2e+269) {
		tmp = (1.0 / (x_m * y_m)) / pow(z_m, 2.0);
	} else {
		tmp = ((1.0 / y_m) / z_m) * (1.0 / (x_m * z_m));
	}
	return y_s * (x_s * tmp);
}

Fortran

z_m = abs(z)
x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if ((z_m * z_m) <= 2d+21) then
        tmp = (1.0d0 / x_m) / (y_m * (1.0d0 + (z_m * z_m)))
    else if ((z_m * z_m) <= 2d+269) then
        tmp = (1.0d0 / (x_m * y_m)) / (z_m ** 2.0d0)
    else
        tmp = ((1.0d0 / y_m) / z_m) * (1.0d0 / (x_m * z_m))
    end if
    code = y_s * (x_s * tmp)
end function

Java

z_m = Math.abs(z);
x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 2e+21) {
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z_m * z_m)));
	} else if ((z_m * z_m) <= 2e+269) {
		tmp = (1.0 / (x_m * y_m)) / Math.pow(z_m, 2.0);
	} else {
		tmp = ((1.0 / y_m) / z_m) * (1.0 / (x_m * z_m));
	}
	return y_s * (x_s * tmp);
}

Python

z_m = math.fabs(z)
x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x_m, y_m, z_m] = sort([x_m, y_m, z_m])
def code(y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if (z_m * z_m) <= 2e+21:
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z_m * z_m)))
	elif (z_m * z_m) <= 2e+269:
		tmp = (1.0 / (x_m * y_m)) / math.pow(z_m, 2.0)
	else:
		tmp = ((1.0 / y_m) / z_m) * (1.0 / (x_m * z_m))
	return y_s * (x_s * tmp)

Julia

z_m = abs(z)
x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(z_m * z_m) <= 2e+21)
		tmp = Float64(Float64(1.0 / x_m) / Float64(y_m * Float64(1.0 + Float64(z_m * z_m))));
	elseif (Float64(z_m * z_m) <= 2e+269)
		tmp = Float64(Float64(1.0 / Float64(x_m * y_m)) / (z_m ^ 2.0));
	else
		tmp = Float64(Float64(Float64(1.0 / y_m) / z_m) * Float64(1.0 / Float64(x_m * z_m)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

z_m = abs(z);
x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
x_m, y_m, z_m = num2cell(sort([x_m, y_m, z_m])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if ((z_m * z_m) <= 2e+21)
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z_m * z_m)));
	elseif ((z_m * z_m) <= 2e+269)
		tmp = (1.0 / (x_m * y_m)) / (z_m ^ 2.0);
	else
		tmp = ((1.0 / y_m) / z_m) * (1.0 / (x_m * z_m));
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 2e+21], N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(y$95$m * N[(1.0 + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 2e+269], N[(N[(1.0 / N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] / N[Power[z$95$m, 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision] * N[(1.0 / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 5: 99.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \cdot z_m \leq 2 \cdot 10^{+21}:\\ \;\;\;\;\frac{\frac{1}{x_m}}{y_m \cdot \left(1 + z_m \cdot z_m\right)}\\ \mathbf{elif}\;z_m \cdot z_m \leq 2 \cdot 10^{+269}:\\ \;\;\;\;\frac{\frac{1}{x_m \cdot {z_m}^{2}}}{y_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y_m}}{z_m} \cdot \frac{1}{x_m \cdot z_m}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z_m z_m) 2e+21)
     (/ (/ 1.0 x_m) (* y_m (+ 1.0 (* z_m z_m))))
     (if (<= (* z_m z_m) 2e+269)
       (/ (/ 1.0 (* x_m (pow z_m 2.0))) y_m)
       (* (/ (/ 1.0 y_m) z_m) (/ 1.0 (* x_m z_m))))))))

C

z_m = fabs(z);
x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 2e+21) {
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z_m * z_m)));
	} else if ((z_m * z_m) <= 2e+269) {
		tmp = (1.0 / (x_m * pow(z_m, 2.0))) / y_m;
	} else {
		tmp = ((1.0 / y_m) / z_m) * (1.0 / (x_m * z_m));
	}
	return y_s * (x_s * tmp);
}

Fortran

z_m = abs(z)
x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if ((z_m * z_m) <= 2d+21) then
        tmp = (1.0d0 / x_m) / (y_m * (1.0d0 + (z_m * z_m)))
    else if ((z_m * z_m) <= 2d+269) then
        tmp = (1.0d0 / (x_m * (z_m ** 2.0d0))) / y_m
    else
        tmp = ((1.0d0 / y_m) / z_m) * (1.0d0 / (x_m * z_m))
    end if
    code = y_s * (x_s * tmp)
end function

Java

z_m = Math.abs(z);
x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 2e+21) {
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z_m * z_m)));
	} else if ((z_m * z_m) <= 2e+269) {
		tmp = (1.0 / (x_m * Math.pow(z_m, 2.0))) / y_m;
	} else {
		tmp = ((1.0 / y_m) / z_m) * (1.0 / (x_m * z_m));
	}
	return y_s * (x_s * tmp);
}

Python

z_m = math.fabs(z)
x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x_m, y_m, z_m] = sort([x_m, y_m, z_m])
def code(y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if (z_m * z_m) <= 2e+21:
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z_m * z_m)))
	elif (z_m * z_m) <= 2e+269:
		tmp = (1.0 / (x_m * math.pow(z_m, 2.0))) / y_m
	else:
		tmp = ((1.0 / y_m) / z_m) * (1.0 / (x_m * z_m))
	return y_s * (x_s * tmp)

Julia

z_m = abs(z)
x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(z_m * z_m) <= 2e+21)
		tmp = Float64(Float64(1.0 / x_m) / Float64(y_m * Float64(1.0 + Float64(z_m * z_m))));
	elseif (Float64(z_m * z_m) <= 2e+269)
		tmp = Float64(Float64(1.0 / Float64(x_m * (z_m ^ 2.0))) / y_m);
	else
		tmp = Float64(Float64(Float64(1.0 / y_m) / z_m) * Float64(1.0 / Float64(x_m * z_m)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

z_m = abs(z);
x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
x_m, y_m, z_m = num2cell(sort([x_m, y_m, z_m])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if ((z_m * z_m) <= 2e+21)
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z_m * z_m)));
	elseif ((z_m * z_m) <= 2e+269)
		tmp = (1.0 / (x_m * (z_m ^ 2.0))) / y_m;
	else
		tmp = ((1.0 / y_m) / z_m) * (1.0 / (x_m * z_m));
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 2e+21], N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(y$95$m * N[(1.0 + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 2e+269], N[(N[(1.0 / N[(x$95$m * N[Power[z$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(N[(1.0 / y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision] * N[(1.0 / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 6: 97.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \cdot z_m \leq 2 \cdot 10^{+21}:\\ \;\;\;\;\frac{\frac{1}{x_m}}{y_m \cdot \left(1 + z_m \cdot z_m\right)}\\ \mathbf{elif}\;z_m \cdot z_m \leq 2 \cdot 10^{+269}:\\ \;\;\;\;\frac{{z_m}^{-2}}{x_m \cdot y_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y_m}}{z_m} \cdot \frac{1}{x_m \cdot z_m}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z_m z_m) 2e+21)
     (/ (/ 1.0 x_m) (* y_m (+ 1.0 (* z_m z_m))))
     (if (<= (* z_m z_m) 2e+269)
       (/ (pow z_m -2.0) (* x_m y_m))
       (* (/ (/ 1.0 y_m) z_m) (/ 1.0 (* x_m z_m))))))))

C

z_m = fabs(z);
x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 2e+21) {
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z_m * z_m)));
	} else if ((z_m * z_m) <= 2e+269) {
		tmp = pow(z_m, -2.0) / (x_m * y_m);
	} else {
		tmp = ((1.0 / y_m) / z_m) * (1.0 / (x_m * z_m));
	}
	return y_s * (x_s * tmp);
}

Fortran

z_m = abs(z)
x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if ((z_m * z_m) <= 2d+21) then
        tmp = (1.0d0 / x_m) / (y_m * (1.0d0 + (z_m * z_m)))
    else if ((z_m * z_m) <= 2d+269) then
        tmp = (z_m ** (-2.0d0)) / (x_m * y_m)
    else
        tmp = ((1.0d0 / y_m) / z_m) * (1.0d0 / (x_m * z_m))
    end if
    code = y_s * (x_s * tmp)
end function

Java

z_m = Math.abs(z);
x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 2e+21) {
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z_m * z_m)));
	} else if ((z_m * z_m) <= 2e+269) {
		tmp = Math.pow(z_m, -2.0) / (x_m * y_m);
	} else {
		tmp = ((1.0 / y_m) / z_m) * (1.0 / (x_m * z_m));
	}
	return y_s * (x_s * tmp);
}

Python

z_m = math.fabs(z)
x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x_m, y_m, z_m] = sort([x_m, y_m, z_m])
def code(y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if (z_m * z_m) <= 2e+21:
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z_m * z_m)))
	elif (z_m * z_m) <= 2e+269:
		tmp = math.pow(z_m, -2.0) / (x_m * y_m)
	else:
		tmp = ((1.0 / y_m) / z_m) * (1.0 / (x_m * z_m))
	return y_s * (x_s * tmp)

Julia

z_m = abs(z)
x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(z_m * z_m) <= 2e+21)
		tmp = Float64(Float64(1.0 / x_m) / Float64(y_m * Float64(1.0 + Float64(z_m * z_m))));
	elseif (Float64(z_m * z_m) <= 2e+269)
		tmp = Float64((z_m ^ -2.0) / Float64(x_m * y_m));
	else
		tmp = Float64(Float64(Float64(1.0 / y_m) / z_m) * Float64(1.0 / Float64(x_m * z_m)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

z_m = abs(z);
x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
x_m, y_m, z_m = num2cell(sort([x_m, y_m, z_m])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if ((z_m * z_m) <= 2e+21)
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z_m * z_m)));
	elseif ((z_m * z_m) <= 2e+269)
		tmp = (z_m ^ -2.0) / (x_m * y_m);
	else
		tmp = ((1.0 / y_m) / z_m) * (1.0 / (x_m * z_m));
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 2e+21], N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(y$95$m * N[(1.0 + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 2e+269], N[(N[Power[z$95$m, -2.0], $MachinePrecision] / N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision] * N[(1.0 / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 7: 99.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \cdot z_m \leq 2 \cdot 10^{+269}:\\ \;\;\;\;\frac{\frac{1}{y_m}}{x_m \cdot \mathsf{fma}\left(z_m, z_m, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y_m}}{z_m} \cdot \frac{1}{x_m \cdot z_m}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z_m z_m) 2e+269)
     (/ (/ 1.0 y_m) (* x_m (fma z_m z_m 1.0)))
     (* (/ (/ 1.0 y_m) z_m) (/ 1.0 (* x_m z_m)))))))

C

z_m = fabs(z);
x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 2e+269) {
		tmp = (1.0 / y_m) / (x_m * fma(z_m, z_m, 1.0));
	} else {
		tmp = ((1.0 / y_m) / z_m) * (1.0 / (x_m * z_m));
	}
	return y_s * (x_s * tmp);
}

Julia

z_m = abs(z)
x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(z_m * z_m) <= 2e+269)
		tmp = Float64(Float64(1.0 / y_m) / Float64(x_m * fma(z_m, z_m, 1.0)));
	else
		tmp = Float64(Float64(Float64(1.0 / y_m) / z_m) * Float64(1.0 / Float64(x_m * z_m)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 2e+269], N[(N[(1.0 / y$95$m), $MachinePrecision] / N[(x$95$m * N[(z$95$m * z$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision] * N[(1.0 / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 8: 99.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \cdot z_m \leq 2 \cdot 10^{+269}:\\ \;\;\;\;\frac{\frac{1}{x_m \cdot \mathsf{fma}\left(z_m, z_m, 1\right)}}{y_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y_m}}{z_m} \cdot \frac{1}{x_m \cdot z_m}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z_m z_m) 2e+269)
     (/ (/ 1.0 (* x_m (fma z_m z_m 1.0))) y_m)
     (* (/ (/ 1.0 y_m) z_m) (/ 1.0 (* x_m z_m)))))))

C

z_m = fabs(z);
x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 2e+269) {
		tmp = (1.0 / (x_m * fma(z_m, z_m, 1.0))) / y_m;
	} else {
		tmp = ((1.0 / y_m) / z_m) * (1.0 / (x_m * z_m));
	}
	return y_s * (x_s * tmp);
}

Julia

z_m = abs(z)
x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(z_m * z_m) <= 2e+269)
		tmp = Float64(Float64(1.0 / Float64(x_m * fma(z_m, z_m, 1.0))) / y_m);
	else
		tmp = Float64(Float64(Float64(1.0 / y_m) / z_m) * Float64(1.0 / Float64(x_m * z_m)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 2e+269], N[(N[(1.0 / N[(x$95$m * N[(z$95$m * z$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(N[(1.0 / y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision] * N[(1.0 / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 9: 99.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 2.6 \cdot 10^{+142}:\\ \;\;\;\;\frac{\frac{1}{x_m + x_m \cdot {z_m}^{2}}}{y_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y_m}}{z_m} \cdot \frac{1}{x_m \cdot z_m}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z_m 2.6e+142)
     (/ (/ 1.0 (+ x_m (* x_m (pow z_m 2.0)))) y_m)
     (* (/ (/ 1.0 y_m) z_m) (/ 1.0 (* x_m z_m)))))))

C

z_m = fabs(z);
x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 2.6e+142) {
		tmp = (1.0 / (x_m + (x_m * pow(z_m, 2.0)))) / y_m;
	} else {
		tmp = ((1.0 / y_m) / z_m) * (1.0 / (x_m * z_m));
	}
	return y_s * (x_s * tmp);
}

Fortran

z_m = abs(z)
x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (z_m <= 2.6d+142) then
        tmp = (1.0d0 / (x_m + (x_m * (z_m ** 2.0d0)))) / y_m
    else
        tmp = ((1.0d0 / y_m) / z_m) * (1.0d0 / (x_m * z_m))
    end if
    code = y_s * (x_s * tmp)
end function

Java

z_m = Math.abs(z);
x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 2.6e+142) {
		tmp = (1.0 / (x_m + (x_m * Math.pow(z_m, 2.0)))) / y_m;
	} else {
		tmp = ((1.0 / y_m) / z_m) * (1.0 / (x_m * z_m));
	}
	return y_s * (x_s * tmp);
}

Python

z_m = math.fabs(z)
x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x_m, y_m, z_m] = sort([x_m, y_m, z_m])
def code(y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if z_m <= 2.6e+142:
		tmp = (1.0 / (x_m + (x_m * math.pow(z_m, 2.0)))) / y_m
	else:
		tmp = ((1.0 / y_m) / z_m) * (1.0 / (x_m * z_m))
	return y_s * (x_s * tmp)

Julia

z_m = abs(z)
x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (z_m <= 2.6e+142)
		tmp = Float64(Float64(1.0 / Float64(x_m + Float64(x_m * (z_m ^ 2.0)))) / y_m);
	else
		tmp = Float64(Float64(Float64(1.0 / y_m) / z_m) * Float64(1.0 / Float64(x_m * z_m)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

z_m = abs(z);
x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
x_m, y_m, z_m = num2cell(sort([x_m, y_m, z_m])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (z_m <= 2.6e+142)
		tmp = (1.0 / (x_m + (x_m * (z_m ^ 2.0)))) / y_m;
	else
		tmp = ((1.0 / y_m) / z_m) * (1.0 / (x_m * z_m));
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 2.6e+142], N[(N[(1.0 / N[(x$95$m + N[(x$95$m * N[Power[z$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(N[(1.0 / y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision] * N[(1.0 / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 10: 97.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \cdot z_m \leq 10^{+38}:\\ \;\;\;\;\frac{\frac{1}{x_m}}{y_m \cdot \left(1 + z_m \cdot z_m\right)}\\ \mathbf{elif}\;z_m \cdot z_m \leq 2 \cdot 10^{+269}:\\ \;\;\;\;\frac{1}{z_m} \cdot \frac{\frac{\frac{1}{x_m}}{y_m}}{z_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y_m}}{z_m} \cdot \frac{1}{x_m \cdot z_m}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z_m z_m) 1e+38)
     (/ (/ 1.0 x_m) (* y_m (+ 1.0 (* z_m z_m))))
     (if (<= (* z_m z_m) 2e+269)
       (* (/ 1.0 z_m) (/ (/ (/ 1.0 x_m) y_m) z_m))
       (* (/ (/ 1.0 y_m) z_m) (/ 1.0 (* x_m z_m))))))))

C

z_m = fabs(z);
x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 1e+38) {
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z_m * z_m)));
	} else if ((z_m * z_m) <= 2e+269) {
		tmp = (1.0 / z_m) * (((1.0 / x_m) / y_m) / z_m);
	} else {
		tmp = ((1.0 / y_m) / z_m) * (1.0 / (x_m * z_m));
	}
	return y_s * (x_s * tmp);
}

Fortran

z_m = abs(z)
x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if ((z_m * z_m) <= 1d+38) then
        tmp = (1.0d0 / x_m) / (y_m * (1.0d0 + (z_m * z_m)))
    else if ((z_m * z_m) <= 2d+269) then
        tmp = (1.0d0 / z_m) * (((1.0d0 / x_m) / y_m) / z_m)
    else
        tmp = ((1.0d0 / y_m) / z_m) * (1.0d0 / (x_m * z_m))
    end if
    code = y_s * (x_s * tmp)
end function

Java

z_m = Math.abs(z);
x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 1e+38) {
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z_m * z_m)));
	} else if ((z_m * z_m) <= 2e+269) {
		tmp = (1.0 / z_m) * (((1.0 / x_m) / y_m) / z_m);
	} else {
		tmp = ((1.0 / y_m) / z_m) * (1.0 / (x_m * z_m));
	}
	return y_s * (x_s * tmp);
}

Python

z_m = math.fabs(z)
x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x_m, y_m, z_m] = sort([x_m, y_m, z_m])
def code(y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if (z_m * z_m) <= 1e+38:
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z_m * z_m)))
	elif (z_m * z_m) <= 2e+269:
		tmp = (1.0 / z_m) * (((1.0 / x_m) / y_m) / z_m)
	else:
		tmp = ((1.0 / y_m) / z_m) * (1.0 / (x_m * z_m))
	return y_s * (x_s * tmp)

Julia

z_m = abs(z)
x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(z_m * z_m) <= 1e+38)
		tmp = Float64(Float64(1.0 / x_m) / Float64(y_m * Float64(1.0 + Float64(z_m * z_m))));
	elseif (Float64(z_m * z_m) <= 2e+269)
		tmp = Float64(Float64(1.0 / z_m) * Float64(Float64(Float64(1.0 / x_m) / y_m) / z_m));
	else
		tmp = Float64(Float64(Float64(1.0 / y_m) / z_m) * Float64(1.0 / Float64(x_m * z_m)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

z_m = abs(z);
x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
x_m, y_m, z_m = num2cell(sort([x_m, y_m, z_m])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if ((z_m * z_m) <= 1e+38)
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z_m * z_m)));
	elseif ((z_m * z_m) <= 2e+269)
		tmp = (1.0 / z_m) * (((1.0 / x_m) / y_m) / z_m);
	else
		tmp = ((1.0 / y_m) / z_m) * (1.0 / (x_m * z_m));
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 1e+38], N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(y$95$m * N[(1.0 + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 2e+269], N[(N[(1.0 / z$95$m), $MachinePrecision] * N[(N[(N[(1.0 / x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision] * N[(1.0 / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 11: 96.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ \begin{array}{l} t_0 := \frac{\frac{1}{x_m}}{y_m}\\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 0.042:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z_m \leq 2.2 \cdot 10^{+135}:\\ \;\;\;\;\frac{1}{z_m} \cdot \frac{t_0}{z_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y_m}}{z_m} \cdot \frac{1}{x_m \cdot z_m}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 x_m) y_m)))
   (*
    y_s
    (*
     x_s
     (if (<= z_m 0.042)
       t_0
       (if (<= z_m 2.2e+135)
         (* (/ 1.0 z_m) (/ t_0 z_m))
         (* (/ (/ 1.0 y_m) z_m) (/ 1.0 (* x_m z_m)))))))))

C

z_m = fabs(z);
x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double t_0 = (1.0 / x_m) / y_m;
	double tmp;
	if (z_m <= 0.042) {
		tmp = t_0;
	} else if (z_m <= 2.2e+135) {
		tmp = (1.0 / z_m) * (t_0 / z_m);
	} else {
		tmp = ((1.0 / y_m) / z_m) * (1.0 / (x_m * z_m));
	}
	return y_s * (x_s * tmp);
}

Fortran

z_m = abs(z)
x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / x_m) / y_m
    if (z_m <= 0.042d0) then
        tmp = t_0
    else if (z_m <= 2.2d+135) then
        tmp = (1.0d0 / z_m) * (t_0 / z_m)
    else
        tmp = ((1.0d0 / y_m) / z_m) * (1.0d0 / (x_m * z_m))
    end if
    code = y_s * (x_s * tmp)
end function

Java

z_m = Math.abs(z);
x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double t_0 = (1.0 / x_m) / y_m;
	double tmp;
	if (z_m <= 0.042) {
		tmp = t_0;
	} else if (z_m <= 2.2e+135) {
		tmp = (1.0 / z_m) * (t_0 / z_m);
	} else {
		tmp = ((1.0 / y_m) / z_m) * (1.0 / (x_m * z_m));
	}
	return y_s * (x_s * tmp);
}

Python

z_m = math.fabs(z)
x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x_m, y_m, z_m] = sort([x_m, y_m, z_m])
def code(y_s, x_s, x_m, y_m, z_m):
	t_0 = (1.0 / x_m) / y_m
	tmp = 0
	if z_m <= 0.042:
		tmp = t_0
	elif z_m <= 2.2e+135:
		tmp = (1.0 / z_m) * (t_0 / z_m)
	else:
		tmp = ((1.0 / y_m) / z_m) * (1.0 / (x_m * z_m))
	return y_s * (x_s * tmp)

Julia

z_m = abs(z)
x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	t_0 = Float64(Float64(1.0 / x_m) / y_m)
	tmp = 0.0
	if (z_m <= 0.042)
		tmp = t_0;
	elseif (z_m <= 2.2e+135)
		tmp = Float64(Float64(1.0 / z_m) * Float64(t_0 / z_m));
	else
		tmp = Float64(Float64(Float64(1.0 / y_m) / z_m) * Float64(1.0 / Float64(x_m * z_m)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

z_m = abs(z);
x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
x_m, y_m, z_m = num2cell(sort([x_m, y_m, z_m])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z_m)
	t_0 = (1.0 / x_m) / y_m;
	tmp = 0.0;
	if (z_m <= 0.042)
		tmp = t_0;
	elseif (z_m <= 2.2e+135)
		tmp = (1.0 / z_m) * (t_0 / z_m);
	else
		tmp = ((1.0 / y_m) / z_m) * (1.0 / (x_m * z_m));
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := Block[{t$95$0 = N[(N[(1.0 / x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 0.042], t$95$0, If[LessEqual[z$95$m, 2.2e+135], N[(N[(1.0 / z$95$m), $MachinePrecision] * N[(t$95$0 / z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision] * N[(1.0 / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 12: 94.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ \begin{array}{l} t_0 := \frac{\frac{1}{x_m}}{y_m}\\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 0.042:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z_m} \cdot \frac{t_0}{z_m}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 x_m) y_m)))
   (* y_s (* x_s (if (<= z_m 0.042) t_0 (* (/ 1.0 z_m) (/ t_0 z_m)))))))

C

z_m = fabs(z);
x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double t_0 = (1.0 / x_m) / y_m;
	double tmp;
	if (z_m <= 0.042) {
		tmp = t_0;
	} else {
		tmp = (1.0 / z_m) * (t_0 / z_m);
	}
	return y_s * (x_s * tmp);
}

Fortran

z_m = abs(z)
x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / x_m) / y_m
    if (z_m <= 0.042d0) then
        tmp = t_0
    else
        tmp = (1.0d0 / z_m) * (t_0 / z_m)
    end if
    code = y_s * (x_s * tmp)
end function

Java

z_m = Math.abs(z);
x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double t_0 = (1.0 / x_m) / y_m;
	double tmp;
	if (z_m <= 0.042) {
		tmp = t_0;
	} else {
		tmp = (1.0 / z_m) * (t_0 / z_m);
	}
	return y_s * (x_s * tmp);
}

Python

z_m = math.fabs(z)
x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x_m, y_m, z_m] = sort([x_m, y_m, z_m])
def code(y_s, x_s, x_m, y_m, z_m):
	t_0 = (1.0 / x_m) / y_m
	tmp = 0
	if z_m <= 0.042:
		tmp = t_0
	else:
		tmp = (1.0 / z_m) * (t_0 / z_m)
	return y_s * (x_s * tmp)

Julia

z_m = abs(z)
x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	t_0 = Float64(Float64(1.0 / x_m) / y_m)
	tmp = 0.0
	if (z_m <= 0.042)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / z_m) * Float64(t_0 / z_m));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

z_m = abs(z);
x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
x_m, y_m, z_m = num2cell(sort([x_m, y_m, z_m])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z_m)
	t_0 = (1.0 / x_m) / y_m;
	tmp = 0.0;
	if (z_m <= 0.042)
		tmp = t_0;
	else
		tmp = (1.0 / z_m) * (t_0 / z_m);
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := Block[{t$95$0 = N[(N[(1.0 / x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 0.042], t$95$0, N[(N[(1.0 / z$95$m), $MachinePrecision] * N[(t$95$0 / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 13: 72.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 0.042:\\ \;\;\;\;\frac{\frac{1}{x_m}}{y_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x_m \cdot \left(y_m \cdot z_m\right)}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (* x_s (if (<= z_m 0.042) (/ (/ 1.0 x_m) y_m) (/ 1.0 (* x_m (* y_m z_m)))))))

C

z_m = fabs(z);
x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 0.042) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = 1.0 / (x_m * (y_m * z_m));
	}
	return y_s * (x_s * tmp);
}

Fortran

z_m = abs(z)
x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (z_m <= 0.042d0) then
        tmp = (1.0d0 / x_m) / y_m
    else
        tmp = 1.0d0 / (x_m * (y_m * z_m))
    end if
    code = y_s * (x_s * tmp)
end function

Java

z_m = Math.abs(z);
x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 0.042) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = 1.0 / (x_m * (y_m * z_m));
	}
	return y_s * (x_s * tmp);
}

Python

z_m = math.fabs(z)
x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x_m, y_m, z_m] = sort([x_m, y_m, z_m])
def code(y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if z_m <= 0.042:
		tmp = (1.0 / x_m) / y_m
	else:
		tmp = 1.0 / (x_m * (y_m * z_m))
	return y_s * (x_s * tmp)

Julia

z_m = abs(z)
x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (z_m <= 0.042)
		tmp = Float64(Float64(1.0 / x_m) / y_m);
	else
		tmp = Float64(1.0 / Float64(x_m * Float64(y_m * z_m)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

z_m = abs(z);
x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
x_m, y_m, z_m = num2cell(sort([x_m, y_m, z_m])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (z_m <= 0.042)
		tmp = (1.0 / x_m) / y_m;
	else
		tmp = 1.0 / (x_m * (y_m * z_m));
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 0.042], N[(N[(1.0 / x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision], N[(1.0 / N[(x$95$m * N[(y$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

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1. Add Preprocessing

Alternative 14: 58.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ y_s \cdot \left(x_s \cdot \frac{1}{x_m \cdot y_m}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (* y_s (* x_s (/ 1.0 (* x_m y_m)))))

C

z_m = fabs(z);
x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	return y_s * (x_s * (1.0 / (x_m * y_m)));
}

Fortran

z_m = abs(z)
x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    code = y_s * (x_s * (1.0d0 / (x_m * y_m)))
end function

Java

z_m = Math.abs(z);
x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	return y_s * (x_s * (1.0 / (x_m * y_m)));
}

Python

z_m = math.fabs(z)
x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x_m, y_m, z_m] = sort([x_m, y_m, z_m])
def code(y_s, x_s, x_m, y_m, z_m):
	return y_s * (x_s * (1.0 / (x_m * y_m)))

Julia

z_m = abs(z)
x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	return Float64(y_s * Float64(x_s * Float64(1.0 / Float64(x_m * y_m))))
end

MATLAB

z_m = abs(z);
x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
x_m, y_m, z_m = num2cell(sort([x_m, y_m, z_m])){:}
function tmp = code(y_s, x_s, x_m, y_m, z_m)
	tmp = y_s * (x_s * (1.0 / (x_m * y_m)));
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * N[(1.0 / N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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1. Add Preprocessing

Alternative 15: 58.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ y_s \cdot \left(x_s \cdot \frac{\frac{1}{x_m}}{y_m}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (* y_s (* x_s (/ (/ 1.0 x_m) y_m))))

C

z_m = fabs(z);
x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	return y_s * (x_s * ((1.0 / x_m) / y_m));
}

Fortran

z_m = abs(z)
x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    code = y_s * (x_s * ((1.0d0 / x_m) / y_m))
end function

Java

z_m = Math.abs(z);
x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	return y_s * (x_s * ((1.0 / x_m) / y_m));
}

Python

z_m = math.fabs(z)
x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x_m, y_m, z_m] = sort([x_m, y_m, z_m])
def code(y_s, x_s, x_m, y_m, z_m):
	return y_s * (x_s * ((1.0 / x_m) / y_m))

Julia

z_m = abs(z)
x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	return Float64(y_s * Float64(x_s * Float64(Float64(1.0 / x_m) / y_m)))
end

MATLAB

z_m = abs(z);
x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
x_m, y_m, z_m = num2cell(sort([x_m, y_m, z_m])){:}
function tmp = code(y_s, x_s, x_m, y_m, z_m)
	tmp = y_s * (x_s * ((1.0 / x_m) / y_m));
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * N[(N[(1.0 / x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Developer target: 93.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + z \cdot z\\ t_1 := y \cdot t_0\\ t_2 := \frac{\frac{1}{y}}{t_0 \cdot x}\\ \mathbf{if}\;t_1 < -\infty:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 < 8.680743250567252 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{t_0 \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* z z))) (t_1 (* y t_0)) (t_2 (/ (/ 1.0 y) (* t_0 x))))
   (if (< t_1 (- INFINITY))
     t_2
     (if (< t_1 8.680743250567252e+305) (/ (/ 1.0 x) (* t_0 y)) t_2))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 + (z * z);
	double t_1 = y * t_0;
	double t_2 = (1.0 / y) / (t_0 * x);
	double tmp;
	if (t_1 < -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 < 8.680743250567252e+305) {
		tmp = (1.0 / x) / (t_0 * y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 + (z * z);
	double t_1 = y * t_0;
	double t_2 = (1.0 / y) / (t_0 * x);
	double tmp;
	if (t_1 < -Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else if (t_1 < 8.680743250567252e+305) {
		tmp = (1.0 / x) / (t_0 * y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 + (z * z)
	t_1 = y * t_0
	t_2 = (1.0 / y) / (t_0 * x)
	tmp = 0
	if t_1 < -math.inf:
		tmp = t_2
	elif t_1 < 8.680743250567252e+305:
		tmp = (1.0 / x) / (t_0 * y)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 + Float64(z * z))
	t_1 = Float64(y * t_0)
	t_2 = Float64(Float64(1.0 / y) / Float64(t_0 * x))
	tmp = 0.0
	if (t_1 < Float64(-Inf))
		tmp = t_2;
	elseif (t_1 < 8.680743250567252e+305)
		tmp = Float64(Float64(1.0 / x) / Float64(t_0 * y));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 + (z * z);
	t_1 = y * t_0;
	t_2 = (1.0 / y) / (t_0 * x);
	tmp = 0.0;
	if (t_1 < -Inf)
		tmp = t_2;
	elseif (t_1 < 8.680743250567252e+305)
		tmp = (1.0 / x) / (t_0 * y);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 / y), $MachinePrecision] / N[(t$95$0 * x), $MachinePrecision]), $MachinePrecision]}, If[Less[t$95$1, (-Infinity)], t$95$2, If[Less[t$95$1, 8.680743250567252e+305], N[(N[(1.0 / x), $MachinePrecision] / N[(t$95$0 * y), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Reproduce

herbie shell --seed 2023350 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< (* y (+ 1.0 (* z z))) (- INFINITY)) (/ (/ 1.0 y) (* (+ 1.0 (* z z)) x)) (if (< (* y (+ 1.0 (* z z))) 8.680743250567252e+305) (/ (/ 1.0 x) (* (+ 1.0 (* z z)) y)) (/ (/ 1.0 y) (* (+ 1.0 (* z z)) x))))

  (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))