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

Percentage Accurate: 88.7% → 98.6%

Time: 13.2s

Alternatives: 10

Speedup: 0.6×

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: 88.7% 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: 98.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+27}:\\ \;\;\;\;\frac{\frac{1}{y\_m}}{x\_m \cdot \mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z} \cdot \frac{1}{z \cdot x\_m}}{y\_m}\\ \end{array}\right) \end{array} \]

FPCore

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 should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z z) 1e+27)
     (/ (/ 1.0 y_m) (* x_m (fma z z 1.0)))
     (/ (* (/ 1.0 z) (/ 1.0 (* z x_m))) y_m)))))

C

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);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 1e+27) {
		tmp = (1.0 / y_m) / (x_m * fma(z, z, 1.0));
	} else {
		tmp = ((1.0 / z) * (1.0 / (z * x_m))) / y_m;
	}
	return y_s * (x_s * tmp);
}

Julia

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 = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e+27)
		tmp = Float64(Float64(1.0 / y_m) / Float64(x_m * fma(z, z, 1.0)));
	else
		tmp = Float64(Float64(Float64(1.0 / z) * Float64(1.0 / Float64(z * x_m))) / y_m);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

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 should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 1e+27], N[(N[(1.0 / y$95$m), $MachinePrecision] / N[(x$95$m * N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / z), $MachinePrecision] * N[(1.0 / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1e27

   1. Initial program 99.7%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/l/ 99.7%

         \[\leadsto \color{blue}{\frac{\frac{\frac{1}{x}}{1 + z \cdot z}}{y}} \]

      2. +-commutative 99.7%

         \[\leadsto \frac{\frac{\frac{1}{x}}{\color{blue}{z \cdot z + 1}}}{y} \]

      3. fma-undefine 99.7%

         \[\leadsto \frac{\frac{\frac{1}{x}}{\color{blue}{\mathsf{fma}\left(z, z, 1\right)}}}{y} \]

      4. associate-/r* 99.7%

         \[\leadsto \frac{\color{blue}{\frac{1}{x \cdot \mathsf{fma}\left(z, z, 1\right)}}}{y} \]

      5. div-inv 99.5%

         \[\leadsto \color{blue}{\frac{1}{x \cdot \mathsf{fma}\left(z, z, 1\right)} \cdot \frac{1}{y}} \]

      6. add0 99.5%

         \[\leadsto \color{blue}{\frac{1}{x \cdot \mathsf{fma}\left(z, z, 1\right)} \cdot \frac{1}{y} + 0} \]

      7. associate-*l/ 99.5%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{1}{y}}{x \cdot \mathsf{fma}\left(z, z, 1\right)}} + 0 \]

      8. *-un-lft-identity 99.5%

         \[\leadsto \frac{\color{blue}{\frac{1}{y}}}{x \cdot \mathsf{fma}\left(z, z, 1\right)} + 0 \]

   4. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\frac{\frac{1}{y}}{x \cdot \mathsf{fma}\left(z, z, 1\right)} + 0} \]

   if 1e27 < (*.f64 z z)

   1. Initial program 77.4%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 79.0%

         \[\leadsto \color{blue}{\frac{\frac{\frac{1}{x}}{1 + z \cdot z}}{y}} \]

      2. metadata-eval 79.0%

         \[\leadsto \frac{\frac{\frac{\color{blue}{\frac{-1}{-1}}}{x}}{1 + z \cdot z}}{y} \]

      3. associate-/r* 79.0%

         \[\leadsto \frac{\frac{\color{blue}{\frac{-1}{-1 \cdot x}}}{1 + z \cdot z}}{y} \]

      4. neg-mul-1 79.0%

         \[\leadsto \frac{\frac{\frac{-1}{\color{blue}{-x}}}{1 + z \cdot z}}{y} \]

      5. metadata-eval 79.0%

         \[\leadsto \frac{\frac{\frac{\color{blue}{-1}}{-x}}{1 + z \cdot z}}{y} \]

      6. distribute-neg-frac 79.0%

         \[\leadsto \frac{\frac{\color{blue}{-\frac{1}{-x}}}{1 + z \cdot z}}{y} \]

      7. distribute-frac-neg 79.0%

         \[\leadsto \frac{\color{blue}{-\frac{\frac{1}{-x}}{1 + z \cdot z}}}{y} \]

   3. Simplified 78.9%

      \[\leadsto \color{blue}{\frac{\frac{1}{x \cdot \mathsf{fma}\left(z, z, 1\right)}}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 78.9%

      \[\leadsto \frac{\frac{1}{\color{blue}{x \cdot {z}^{2}}}}{y} \]
   6. Step-by-step derivation

      1. associate-/r* 79.0%

         \[\leadsto \frac{\color{blue}{\frac{\frac{1}{x}}{{z}^{2}}}}{y} \]

      2. *-un-lft-identity 79.0%

         \[\leadsto \frac{\frac{\color{blue}{1 \cdot \frac{1}{x}}}{{z}^{2}}}{y} \]

      3. unpow2 79.0%

         \[\leadsto \frac{\frac{1 \cdot \frac{1}{x}}{\color{blue}{z \cdot z}}}{y} \]

      4. times-frac 90.3%

         \[\leadsto \frac{\color{blue}{\frac{1}{z} \cdot \frac{\frac{1}{x}}{z}}}{y} \]

      5. associate-/r* 90.4%

         \[\leadsto \frac{\frac{1}{z} \cdot \color{blue}{\frac{1}{x \cdot z}}}{y} \]

      6. *-commutative 90.4%

         \[\leadsto \frac{\frac{1}{z} \cdot \frac{1}{\color{blue}{z \cdot x}}}{y} \]

   7. Applied egg-rr 90.4%

      \[\leadsto \frac{\color{blue}{\frac{1}{z} \cdot \frac{1}{z \cdot x}}}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+27}:\\ \;\;\;\;\frac{\frac{1}{y}}{x \cdot \mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z} \cdot \frac{1}{z \cdot x}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.6% accurate, 0.1× speedup

Math

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

FPCore

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 should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (* y_s (* x_s (/ (/ (/ 1.0 (hypot 1.0 z)) (* (hypot 1.0 z) x_m)) y_m))))

C

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);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	return y_s * (x_s * (((1.0 / hypot(1.0, z)) / (hypot(1.0, z) * x_m)) / y_m));
}

Java

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;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	return y_s * (x_s * (((1.0 / Math.hypot(1.0, z)) / (Math.hypot(1.0, z) * x_m)) / y_m));
}

Python

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] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	return y_s * (x_s * (((1.0 / math.hypot(1.0, z)) / (math.hypot(1.0, z) * x_m)) / y_m))

Julia

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 = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	return Float64(y_s * Float64(x_s * Float64(Float64(Float64(1.0 / hypot(1.0, z)) / Float64(hypot(1.0, z) * x_m)) / y_m)))
end

MATLAB

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

Wolfram

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 should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * N[(N[(N[(1.0 / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.7%

   \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
2. Step-by-step derivation

   1. associate-/l/ 89.5%

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{x}}{1 + z \cdot z}}{y}} \]

   2. metadata-eval 89.5%

      \[\leadsto \frac{\frac{\frac{\color{blue}{\frac{-1}{-1}}}{x}}{1 + z \cdot z}}{y} \]

   3. associate-/r* 89.5%

      \[\leadsto \frac{\frac{\color{blue}{\frac{-1}{-1 \cdot x}}}{1 + z \cdot z}}{y} \]

   4. neg-mul-1 89.5%

      \[\leadsto \frac{\frac{\frac{-1}{\color{blue}{-x}}}{1 + z \cdot z}}{y} \]

   5. metadata-eval 89.5%

      \[\leadsto \frac{\frac{\frac{\color{blue}{-1}}{-x}}{1 + z \cdot z}}{y} \]

   6. distribute-neg-frac 89.5%

      \[\leadsto \frac{\frac{\color{blue}{-\frac{1}{-x}}}{1 + z \cdot z}}{y} \]

   7. distribute-frac-neg 89.5%

      \[\leadsto \frac{\color{blue}{-\frac{\frac{1}{-x}}{1 + z \cdot z}}}{y} \]

3. Simplified 89.5%

   \[\leadsto \color{blue}{\frac{\frac{1}{x \cdot \mathsf{fma}\left(z, z, 1\right)}}{y}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-/r* 89.5%

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{x}}{\mathsf{fma}\left(z, z, 1\right)}}}{y} \]

   2. *-un-lft-identity 89.5%

      \[\leadsto \frac{\frac{\color{blue}{1 \cdot \frac{1}{x}}}{\mathsf{fma}\left(z, z, 1\right)}}{y} \]

   3. fma-undefine 89.5%

      \[\leadsto \frac{\frac{1 \cdot \frac{1}{x}}{\color{blue}{z \cdot z + 1}}}{y} \]

   4. +-commutative 89.5%

      \[\leadsto \frac{\frac{1 \cdot \frac{1}{x}}{\color{blue}{1 + z \cdot z}}}{y} \]

   5. add-sqr-sqrt 89.5%

      \[\leadsto \frac{\frac{1 \cdot \frac{1}{x}}{\color{blue}{\sqrt{1 + z \cdot z} \cdot \sqrt{1 + z \cdot z}}}}{y} \]

   6. times-frac 89.4%

      \[\leadsto \frac{\color{blue}{\frac{1}{\sqrt{1 + z \cdot z}} \cdot \frac{\frac{1}{x}}{\sqrt{1 + z \cdot z}}}}{y} \]

   7. hypot-1-def 89.4%

      \[\leadsto \frac{\frac{1}{\color{blue}{\mathsf{hypot}\left(1, z\right)}} \cdot \frac{\frac{1}{x}}{\sqrt{1 + z \cdot z}}}{y} \]

   8. hypot-1-def 95.1%

      \[\leadsto \frac{\frac{1}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{\frac{1}{x}}{\color{blue}{\mathsf{hypot}\left(1, z\right)}}}{y} \]

6. Applied egg-rr 95.1%

   \[\leadsto \frac{\color{blue}{\frac{1}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{\frac{1}{x}}{\mathsf{hypot}\left(1, z\right)}}}{y} \]
7. Step-by-step derivation

   1. associate-/l/ 95.1%

      \[\leadsto \frac{\frac{1}{\mathsf{hypot}\left(1, z\right)} \cdot \color{blue}{\frac{1}{\mathsf{hypot}\left(1, z\right) \cdot x}}}{y} \]

   2. un-div-inv 95.2%

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{\mathsf{hypot}\left(1, z\right)}}{\mathsf{hypot}\left(1, z\right) \cdot x}}}{y} \]

8. Applied egg-rr 95.2%

   \[\leadsto \frac{\color{blue}{\frac{\frac{1}{\mathsf{hypot}\left(1, z\right)}}{\mathsf{hypot}\left(1, z\right) \cdot x}}}{y} \]

9. Final simplification 95.2%

   \[\leadsto \frac{\frac{\frac{1}{\mathsf{hypot}\left(1, z\right)}}{\mathsf{hypot}\left(1, z\right) \cdot x}}{y} \]
10. Add Preprocessing

Alternative 3: 98.8% accurate, 0.1× speedup

Math

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

FPCore

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 should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (* y_s (* x_s (/ (/ 1.0 (hypot 1.0 z)) (* (* (hypot 1.0 z) x_m) y_m)))))

C

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);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	return y_s * (x_s * ((1.0 / hypot(1.0, z)) / ((hypot(1.0, z) * x_m) * y_m)));
}

Java

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;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	return y_s * (x_s * ((1.0 / Math.hypot(1.0, z)) / ((Math.hypot(1.0, z) * x_m) * y_m)));
}

Python

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] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	return y_s * (x_s * ((1.0 / math.hypot(1.0, z)) / ((math.hypot(1.0, z) * x_m) * y_m)))

Julia

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 = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	return Float64(y_s * Float64(x_s * Float64(Float64(1.0 / hypot(1.0, z)) / Float64(Float64(hypot(1.0, z) * x_m) * y_m))))
end

MATLAB

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

Wolfram

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 should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * N[(N[(1.0 / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision] * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.7%

   \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-/l/ 89.5%

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{x}}{1 + z \cdot z}}{y}} \]

   2. +-commutative 89.5%

      \[\leadsto \frac{\frac{\frac{1}{x}}{\color{blue}{z \cdot z + 1}}}{y} \]

   3. fma-undefine 89.5%

      \[\leadsto \frac{\frac{\frac{1}{x}}{\color{blue}{\mathsf{fma}\left(z, z, 1\right)}}}{y} \]

   4. associate-/r* 89.5%

      \[\leadsto \frac{\color{blue}{\frac{1}{x \cdot \mathsf{fma}\left(z, z, 1\right)}}}{y} \]

   5. div-inv 89.4%

      \[\leadsto \color{blue}{\frac{1}{x \cdot \mathsf{fma}\left(z, z, 1\right)} \cdot \frac{1}{y}} \]

   6. add-sqr-sqrt 62.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{x \cdot \mathsf{fma}\left(z, z, 1\right)}} \cdot \sqrt{\frac{1}{x \cdot \mathsf{fma}\left(z, z, 1\right)}}\right)} \cdot \frac{1}{y} \]

   7. associate-*l* 62.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{x \cdot \mathsf{fma}\left(z, z, 1\right)}} \cdot \left(\sqrt{\frac{1}{x \cdot \mathsf{fma}\left(z, z, 1\right)}} \cdot \frac{1}{y}\right)} \]

   8. associate-/r* 61.6%

      \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{x}}{\mathsf{fma}\left(z, z, 1\right)}}} \cdot \left(\sqrt{\frac{1}{x \cdot \mathsf{fma}\left(z, z, 1\right)}} \cdot \frac{1}{y}\right) \]

   9. fma-undefine 61.6%

      \[\leadsto \sqrt{\frac{\frac{1}{x}}{\color{blue}{z \cdot z + 1}}} \cdot \left(\sqrt{\frac{1}{x \cdot \mathsf{fma}\left(z, z, 1\right)}} \cdot \frac{1}{y}\right) \]

   10. +-commutative 61.6%

       \[\leadsto \sqrt{\frac{\frac{1}{x}}{\color{blue}{1 + z \cdot z}}} \cdot \left(\sqrt{\frac{1}{x \cdot \mathsf{fma}\left(z, z, 1\right)}} \cdot \frac{1}{y}\right) \]

   11. sqrt-div 51.4%

       \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{x}}}{\sqrt{1 + z \cdot z}}} \cdot \left(\sqrt{\frac{1}{x \cdot \mathsf{fma}\left(z, z, 1\right)}} \cdot \frac{1}{y}\right) \]

   12. inv-pow 51.4%

       \[\leadsto \frac{\sqrt{\color{blue}{{x}^{-1}}}}{\sqrt{1 + z \cdot z}} \cdot \left(\sqrt{\frac{1}{x \cdot \mathsf{fma}\left(z, z, 1\right)}} \cdot \frac{1}{y}\right) \]

   13. sqrt-pow1 51.5%

       \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{-1}{2}\right)}}}{\sqrt{1 + z \cdot z}} \cdot \left(\sqrt{\frac{1}{x \cdot \mathsf{fma}\left(z, z, 1\right)}} \cdot \frac{1}{y}\right) \]

   14. metadata-eval 51.5%

       \[\leadsto \frac{{x}^{\color{blue}{-0.5}}}{\sqrt{1 + z \cdot z}} \cdot \left(\sqrt{\frac{1}{x \cdot \mathsf{fma}\left(z, z, 1\right)}} \cdot \frac{1}{y}\right) \]

   15. hypot-1-def 51.4%

       \[\leadsto \frac{{x}^{-0.5}}{\color{blue}{\mathsf{hypot}\left(1, z\right)}} \cdot \left(\sqrt{\frac{1}{x \cdot \mathsf{fma}\left(z, z, 1\right)}} \cdot \frac{1}{y}\right) \]

4. Applied egg-rr 55.5%

   \[\leadsto \color{blue}{\frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right)} \cdot \left(\frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{1}{y}\right)} \]
5. Step-by-step derivation

   1. associate-*r* 54.0%

      \[\leadsto \color{blue}{\left(\frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right)}\right) \cdot \frac{1}{y}} \]

   2. unpow2 54.0%

      \[\leadsto \color{blue}{{\left(\frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right)}\right)}^{2}} \cdot \frac{1}{y} \]

6. Simplified 54.0%

   \[\leadsto \color{blue}{{\left(\frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right)}\right)}^{2} \cdot \frac{1}{y}} \]
7. Step-by-step derivation

   1. unpow2 54.0%

      \[\leadsto \color{blue}{\left(\frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right)}\right)} \cdot \frac{1}{y} \]

   2. frac-times 51.5%

      \[\leadsto \color{blue}{\frac{{x}^{-0.5} \cdot {x}^{-0.5}}{\mathsf{hypot}\left(1, z\right) \cdot \mathsf{hypot}\left(1, z\right)}} \cdot \frac{1}{y} \]

   3. pow-prod-up 89.4%

      \[\leadsto \frac{\color{blue}{{x}^{\left(-0.5 + -0.5\right)}}}{\mathsf{hypot}\left(1, z\right) \cdot \mathsf{hypot}\left(1, z\right)} \cdot \frac{1}{y} \]

   4. metadata-eval 89.4%

      \[\leadsto \frac{{x}^{\color{blue}{-1}}}{\mathsf{hypot}\left(1, z\right) \cdot \mathsf{hypot}\left(1, z\right)} \cdot \frac{1}{y} \]

   5. inv-pow 89.4%

      \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{\mathsf{hypot}\left(1, z\right) \cdot \mathsf{hypot}\left(1, z\right)} \cdot \frac{1}{y} \]

   6. *-un-lft-identity 89.4%

      \[\leadsto \frac{\color{blue}{1 \cdot \frac{1}{x}}}{\mathsf{hypot}\left(1, z\right) \cdot \mathsf{hypot}\left(1, z\right)} \cdot \frac{1}{y} \]

   7. frac-times 94.9%

      \[\leadsto \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{\frac{1}{x}}{\mathsf{hypot}\left(1, z\right)}\right)} \cdot \frac{1}{y} \]

   8. *-commutative 94.9%

      \[\leadsto \color{blue}{\left(\frac{\frac{1}{x}}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{1}{\mathsf{hypot}\left(1, z\right)}\right)} \cdot \frac{1}{y} \]

   9. *-commutative 94.9%

      \[\leadsto \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{\frac{1}{x}}{\mathsf{hypot}\left(1, z\right)}\right)} \cdot \frac{1}{y} \]

   10. div-inv 95.1%

       \[\leadsto \color{blue}{\frac{\frac{1}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{\frac{1}{x}}{\mathsf{hypot}\left(1, z\right)}}{y}} \]

   11. associate-/l* 96.7%

       \[\leadsto \color{blue}{\frac{\frac{1}{\mathsf{hypot}\left(1, z\right)}}{\frac{y}{\frac{\frac{1}{x}}{\mathsf{hypot}\left(1, z\right)}}}} \]

   12. div-inv 96.4%

       \[\leadsto \frac{\frac{1}{\mathsf{hypot}\left(1, z\right)}}{\color{blue}{y \cdot \frac{1}{\frac{\frac{1}{x}}{\mathsf{hypot}\left(1, z\right)}}}} \]

   13. clear-num 96.4%

       \[\leadsto \frac{\frac{1}{\mathsf{hypot}\left(1, z\right)}}{y \cdot \color{blue}{\frac{\mathsf{hypot}\left(1, z\right)}{\frac{1}{x}}}} \]

   14. div-inv 96.4%

       \[\leadsto \frac{\frac{1}{\mathsf{hypot}\left(1, z\right)}}{y \cdot \color{blue}{\left(\mathsf{hypot}\left(1, z\right) \cdot \frac{1}{\frac{1}{x}}\right)}} \]

   15. clear-num 96.5%

       \[\leadsto \frac{\frac{1}{\mathsf{hypot}\left(1, z\right)}}{y \cdot \left(\mathsf{hypot}\left(1, z\right) \cdot \color{blue}{\frac{x}{1}}\right)} \]

   16. /-rgt-identity 96.5%

       \[\leadsto \frac{\frac{1}{\mathsf{hypot}\left(1, z\right)}}{y \cdot \left(\mathsf{hypot}\left(1, z\right) \cdot \color{blue}{x}\right)} \]

8. Applied egg-rr 96.5%

   \[\leadsto \color{blue}{\frac{\frac{1}{\mathsf{hypot}\left(1, z\right)}}{y \cdot \left(\mathsf{hypot}\left(1, z\right) \cdot x\right)}} \]

9. Final simplification 96.5%

   \[\leadsto \frac{\frac{1}{\mathsf{hypot}\left(1, z\right)}}{\left(\mathsf{hypot}\left(1, z\right) \cdot x\right) \cdot y} \]
10. Add Preprocessing

Alternative 4: 98.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+27}:\\ \;\;\;\;\frac{\frac{\frac{-1}{x\_m}}{y\_m}}{-1 - {z}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z} \cdot \frac{1}{z \cdot x\_m}}{y\_m}\\ \end{array}\right) \end{array} \]

FPCore

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 should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z z) 1e+27)
     (/ (/ (/ -1.0 x_m) y_m) (- -1.0 (pow z 2.0)))
     (/ (* (/ 1.0 z) (/ 1.0 (* z x_m))) y_m)))))

C

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);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 1e+27) {
		tmp = ((-1.0 / x_m) / y_m) / (-1.0 - pow(z, 2.0));
	} else {
		tmp = ((1.0 / z) * (1.0 / (z * x_m))) / y_m;
	}
	return y_s * (x_s * tmp);
}

Fortran

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 should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    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
    real(8) :: tmp
    if ((z * z) <= 1d+27) then
        tmp = (((-1.0d0) / x_m) / y_m) / ((-1.0d0) - (z ** 2.0d0))
    else
        tmp = ((1.0d0 / z) * (1.0d0 / (z * x_m))) / y_m
    end if
    code = y_s * (x_s * tmp)
end function

Java

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;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 1e+27) {
		tmp = ((-1.0 / x_m) / y_m) / (-1.0 - Math.pow(z, 2.0));
	} else {
		tmp = ((1.0 / z) * (1.0 / (z * x_m))) / y_m;
	}
	return y_s * (x_s * tmp);
}

Python

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] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if (z * z) <= 1e+27:
		tmp = ((-1.0 / x_m) / y_m) / (-1.0 - math.pow(z, 2.0))
	else:
		tmp = ((1.0 / z) * (1.0 / (z * x_m))) / y_m
	return y_s * (x_s * tmp)

Julia

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 = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e+27)
		tmp = Float64(Float64(Float64(-1.0 / x_m) / y_m) / Float64(-1.0 - (z ^ 2.0)));
	else
		tmp = Float64(Float64(Float64(1.0 / z) * Float64(1.0 / Float64(z * x_m))) / y_m);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

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 = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if ((z * z) <= 1e+27)
		tmp = ((-1.0 / x_m) / y_m) / (-1.0 - (z ^ 2.0));
	else
		tmp = ((1.0 / z) * (1.0 / (z * x_m))) / y_m;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

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 should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 1e+27], N[(N[(N[(-1.0 / x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision] / N[(-1.0 - N[Power[z, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / z), $MachinePrecision] * N[(1.0 / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1e27

   1. Initial program 99.7%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. frac-2neg 99.7%

         \[\leadsto \color{blue}{\frac{-\frac{1}{x}}{-y \cdot \left(1 + z \cdot z\right)}} \]

      2. div-inv 99.5%

         \[\leadsto \color{blue}{\left(-\frac{1}{x}\right) \cdot \frac{1}{-y \cdot \left(1 + z \cdot z\right)}} \]

      3. distribute-neg-frac 99.5%

         \[\leadsto \color{blue}{\frac{-1}{x}} \cdot \frac{1}{-y \cdot \left(1 + z \cdot z\right)} \]

      4. metadata-eval 99.5%

         \[\leadsto \frac{\color{blue}{-1}}{x} \cdot \frac{1}{-y \cdot \left(1 + z \cdot z\right)} \]

      5. distribute-rgt-neg-in 99.5%

         \[\leadsto \frac{-1}{x} \cdot \frac{1}{\color{blue}{y \cdot \left(-\left(1 + z \cdot z\right)\right)}} \]

      6. +-commutative 99.5%

         \[\leadsto \frac{-1}{x} \cdot \frac{1}{y \cdot \left(-\color{blue}{\left(z \cdot z + 1\right)}\right)} \]

      7. fma-undefine 99.5%

         \[\leadsto \frac{-1}{x} \cdot \frac{1}{y \cdot \left(-\color{blue}{\mathsf{fma}\left(z, z, 1\right)}\right)} \]

   4. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\frac{-1}{x} \cdot \frac{1}{y \cdot \left(-\mathsf{fma}\left(z, z, 1\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-*r/ 99.7%

         \[\leadsto \color{blue}{\frac{\frac{-1}{x} \cdot 1}{y \cdot \left(-\mathsf{fma}\left(z, z, 1\right)\right)}} \]

      2. *-rgt-identity 99.7%

         \[\leadsto \frac{\color{blue}{\frac{-1}{x}}}{y \cdot \left(-\mathsf{fma}\left(z, z, 1\right)\right)} \]

      3. associate-/r* 99.7%

         \[\leadsto \color{blue}{\frac{\frac{\frac{-1}{x}}{y}}{-\mathsf{fma}\left(z, z, 1\right)}} \]

      4. neg-sub0 99.7%

         \[\leadsto \frac{\frac{\frac{-1}{x}}{y}}{\color{blue}{0 - \mathsf{fma}\left(z, z, 1\right)}} \]

      5. fma-undefine 99.7%

         \[\leadsto \frac{\frac{\frac{-1}{x}}{y}}{0 - \color{blue}{\left(z \cdot z + 1\right)}} \]

      6. unpow2 99.7%

         \[\leadsto \frac{\frac{\frac{-1}{x}}{y}}{0 - \left(\color{blue}{{z}^{2}} + 1\right)} \]

      7. +-commutative 99.7%

         \[\leadsto \frac{\frac{\frac{-1}{x}}{y}}{0 - \color{blue}{\left(1 + {z}^{2}\right)}} \]

      8. associate--r+ 99.7%

         \[\leadsto \frac{\frac{\frac{-1}{x}}{y}}{\color{blue}{\left(0 - 1\right) - {z}^{2}}} \]

      9. metadata-eval 99.7%

         \[\leadsto \frac{\frac{\frac{-1}{x}}{y}}{\color{blue}{-1} - {z}^{2}} \]

   6. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\frac{\frac{-1}{x}}{y}}{-1 - {z}^{2}}} \]

   if 1e27 < (*.f64 z z)

   1. Initial program 77.4%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 79.0%

         \[\leadsto \color{blue}{\frac{\frac{\frac{1}{x}}{1 + z \cdot z}}{y}} \]

      2. metadata-eval 79.0%

         \[\leadsto \frac{\frac{\frac{\color{blue}{\frac{-1}{-1}}}{x}}{1 + z \cdot z}}{y} \]

      3. associate-/r* 79.0%

         \[\leadsto \frac{\frac{\color{blue}{\frac{-1}{-1 \cdot x}}}{1 + z \cdot z}}{y} \]

      4. neg-mul-1 79.0%

         \[\leadsto \frac{\frac{\frac{-1}{\color{blue}{-x}}}{1 + z \cdot z}}{y} \]

      5. metadata-eval 79.0%

         \[\leadsto \frac{\frac{\frac{\color{blue}{-1}}{-x}}{1 + z \cdot z}}{y} \]

      6. distribute-neg-frac 79.0%

         \[\leadsto \frac{\frac{\color{blue}{-\frac{1}{-x}}}{1 + z \cdot z}}{y} \]

      7. distribute-frac-neg 79.0%

         \[\leadsto \frac{\color{blue}{-\frac{\frac{1}{-x}}{1 + z \cdot z}}}{y} \]

   3. Simplified 78.9%

      \[\leadsto \color{blue}{\frac{\frac{1}{x \cdot \mathsf{fma}\left(z, z, 1\right)}}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 78.9%

      \[\leadsto \frac{\frac{1}{\color{blue}{x \cdot {z}^{2}}}}{y} \]
   6. Step-by-step derivation

      1. associate-/r* 79.0%

         \[\leadsto \frac{\color{blue}{\frac{\frac{1}{x}}{{z}^{2}}}}{y} \]

      2. *-un-lft-identity 79.0%

         \[\leadsto \frac{\frac{\color{blue}{1 \cdot \frac{1}{x}}}{{z}^{2}}}{y} \]

      3. unpow2 79.0%

         \[\leadsto \frac{\frac{1 \cdot \frac{1}{x}}{\color{blue}{z \cdot z}}}{y} \]

      4. times-frac 90.3%

         \[\leadsto \frac{\color{blue}{\frac{1}{z} \cdot \frac{\frac{1}{x}}{z}}}{y} \]

      5. associate-/r* 90.4%

         \[\leadsto \frac{\frac{1}{z} \cdot \color{blue}{\frac{1}{x \cdot z}}}{y} \]

      6. *-commutative 90.4%

         \[\leadsto \frac{\frac{1}{z} \cdot \frac{1}{\color{blue}{z \cdot x}}}{y} \]

   7. Applied egg-rr 90.4%

      \[\leadsto \frac{\color{blue}{\frac{1}{z} \cdot \frac{1}{z \cdot x}}}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+27}:\\ \;\;\;\;\frac{\frac{\frac{-1}{x}}{y}}{-1 - {z}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z} \cdot \frac{1}{z \cdot x}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+24}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{y\_m \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z} \cdot \frac{1}{z \cdot x\_m}}{y\_m}\\ \end{array}\right) \end{array} \]

FPCore

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 should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z z) 5e+24)
     (/ (/ 1.0 x_m) (* y_m (+ 1.0 (* z z))))
     (/ (* (/ 1.0 z) (/ 1.0 (* z x_m))) y_m)))))

C

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);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 5e+24) {
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z * z)));
	} else {
		tmp = ((1.0 / z) * (1.0 / (z * x_m))) / y_m;
	}
	return y_s * (x_s * tmp);
}

Fortran

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 should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    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
    real(8) :: tmp
    if ((z * z) <= 5d+24) then
        tmp = (1.0d0 / x_m) / (y_m * (1.0d0 + (z * z)))
    else
        tmp = ((1.0d0 / z) * (1.0d0 / (z * x_m))) / y_m
    end if
    code = y_s * (x_s * tmp)
end function

Java

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;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 5e+24) {
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z * z)));
	} else {
		tmp = ((1.0 / z) * (1.0 / (z * x_m))) / y_m;
	}
	return y_s * (x_s * tmp);
}

Python

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] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if (z * z) <= 5e+24:
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z * z)))
	else:
		tmp = ((1.0 / z) * (1.0 / (z * x_m))) / y_m
	return y_s * (x_s * tmp)

Julia

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 = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e+24)
		tmp = Float64(Float64(1.0 / x_m) / Float64(y_m * Float64(1.0 + Float64(z * z))));
	else
		tmp = Float64(Float64(Float64(1.0 / z) * Float64(1.0 / Float64(z * x_m))) / y_m);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

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 = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if ((z * z) <= 5e+24)
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z * z)));
	else
		tmp = ((1.0 / z) * (1.0 / (z * x_m))) / y_m;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

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 should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 5e+24], N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(y$95$m * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / z), $MachinePrecision] * N[(1.0 / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.00000000000000045e24

   1. Initial program 99.7%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Add Preprocessing

   if 5.00000000000000045e24 < (*.f64 z z)

   1. Initial program 77.6%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 79.1%

         \[\leadsto \color{blue}{\frac{\frac{\frac{1}{x}}{1 + z \cdot z}}{y}} \]

      2. metadata-eval 79.1%

         \[\leadsto \frac{\frac{\frac{\color{blue}{\frac{-1}{-1}}}{x}}{1 + z \cdot z}}{y} \]

      3. associate-/r* 79.1%

         \[\leadsto \frac{\frac{\color{blue}{\frac{-1}{-1 \cdot x}}}{1 + z \cdot z}}{y} \]

      4. neg-mul-1 79.1%

         \[\leadsto \frac{\frac{\frac{-1}{\color{blue}{-x}}}{1 + z \cdot z}}{y} \]

      5. metadata-eval 79.1%

         \[\leadsto \frac{\frac{\frac{\color{blue}{-1}}{-x}}{1 + z \cdot z}}{y} \]

      6. distribute-neg-frac 79.1%

         \[\leadsto \frac{\frac{\color{blue}{-\frac{1}{-x}}}{1 + z \cdot z}}{y} \]

      7. distribute-frac-neg 79.1%

         \[\leadsto \frac{\color{blue}{-\frac{\frac{1}{-x}}{1 + z \cdot z}}}{y} \]

   3. Simplified 79.1%

      \[\leadsto \color{blue}{\frac{\frac{1}{x \cdot \mathsf{fma}\left(z, z, 1\right)}}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 79.1%

      \[\leadsto \frac{\frac{1}{\color{blue}{x \cdot {z}^{2}}}}{y} \]
   6. Step-by-step derivation

      1. associate-/r* 79.1%

         \[\leadsto \frac{\color{blue}{\frac{\frac{1}{x}}{{z}^{2}}}}{y} \]

      2. *-un-lft-identity 79.1%

         \[\leadsto \frac{\frac{\color{blue}{1 \cdot \frac{1}{x}}}{{z}^{2}}}{y} \]

      3. unpow2 79.1%

         \[\leadsto \frac{\frac{1 \cdot \frac{1}{x}}{\color{blue}{z \cdot z}}}{y} \]

      4. times-frac 90.4%

         \[\leadsto \frac{\color{blue}{\frac{1}{z} \cdot \frac{\frac{1}{x}}{z}}}{y} \]

      5. associate-/r* 90.5%

         \[\leadsto \frac{\frac{1}{z} \cdot \color{blue}{\frac{1}{x \cdot z}}}{y} \]

      6. *-commutative 90.5%

         \[\leadsto \frac{\frac{1}{z} \cdot \frac{1}{\color{blue}{z \cdot x}}}{y} \]

   7. Applied egg-rr 90.5%

      \[\leadsto \frac{\color{blue}{\frac{1}{z} \cdot \frac{1}{z \cdot x}}}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+24}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z} \cdot \frac{1}{z \cdot x}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.0% accurate, 0.7× speedup

Math

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

FPCore

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 should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z 1.0)
     (/ (/ 1.0 y_m) x_m)
     (/ (* (/ 1.0 z) (/ 1.0 (* z x_m))) y_m)))))

C

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);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 1.0) {
		tmp = (1.0 / y_m) / x_m;
	} else {
		tmp = ((1.0 / z) * (1.0 / (z * x_m))) / y_m;
	}
	return y_s * (x_s * tmp);
}

Fortran

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 should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    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
    real(8) :: tmp
    if (z <= 1.0d0) then
        tmp = (1.0d0 / y_m) / x_m
    else
        tmp = ((1.0d0 / z) * (1.0d0 / (z * x_m))) / y_m
    end if
    code = y_s * (x_s * tmp)
end function

Java

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;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 1.0) {
		tmp = (1.0 / y_m) / x_m;
	} else {
		tmp = ((1.0 / z) * (1.0 / (z * x_m))) / y_m;
	}
	return y_s * (x_s * tmp);
}

Python

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] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if z <= 1.0:
		tmp = (1.0 / y_m) / x_m
	else:
		tmp = ((1.0 / z) * (1.0 / (z * x_m))) / y_m
	return y_s * (x_s * tmp)

Julia

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

MATLAB

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

Wolfram

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 should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z, 1.0], N[(N[(1.0 / y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision], N[(N[(N[(1.0 / z), $MachinePrecision] * N[(1.0 / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 93.1%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 68.3%

      \[\leadsto \color{blue}{\frac{1}{x \cdot y}} \]
   4. Step-by-step derivation

      1. associate-/l/ 68.3%

         \[\leadsto \color{blue}{\frac{\frac{1}{y}}{x}} \]

   5. Simplified 68.3%

      \[\leadsto \color{blue}{\frac{\frac{1}{y}}{x}} \]

   if 1 < z

   1. Initial program 73.9%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 75.6%

         \[\leadsto \color{blue}{\frac{\frac{\frac{1}{x}}{1 + z \cdot z}}{y}} \]

      2. metadata-eval 75.6%

         \[\leadsto \frac{\frac{\frac{\color{blue}{\frac{-1}{-1}}}{x}}{1 + z \cdot z}}{y} \]

      3. associate-/r* 75.6%

         \[\leadsto \frac{\frac{\color{blue}{\frac{-1}{-1 \cdot x}}}{1 + z \cdot z}}{y} \]

      4. neg-mul-1 75.6%

         \[\leadsto \frac{\frac{\frac{-1}{\color{blue}{-x}}}{1 + z \cdot z}}{y} \]

      5. metadata-eval 75.6%

         \[\leadsto \frac{\frac{\frac{\color{blue}{-1}}{-x}}{1 + z \cdot z}}{y} \]

      6. distribute-neg-frac 75.6%

         \[\leadsto \frac{\frac{\color{blue}{-\frac{1}{-x}}}{1 + z \cdot z}}{y} \]

      7. distribute-frac-neg 75.6%

         \[\leadsto \frac{\color{blue}{-\frac{\frac{1}{-x}}{1 + z \cdot z}}}{y} \]

   3. Simplified 75.5%

      \[\leadsto \color{blue}{\frac{\frac{1}{x \cdot \mathsf{fma}\left(z, z, 1\right)}}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 74.0%

      \[\leadsto \frac{\frac{1}{\color{blue}{x \cdot {z}^{2}}}}{y} \]
   6. Step-by-step derivation

      1. associate-/r* 74.1%

         \[\leadsto \frac{\color{blue}{\frac{\frac{1}{x}}{{z}^{2}}}}{y} \]

      2. *-un-lft-identity 74.1%

         \[\leadsto \frac{\frac{\color{blue}{1 \cdot \frac{1}{x}}}{{z}^{2}}}{y} \]

      3. unpow2 74.1%

         \[\leadsto \frac{\frac{1 \cdot \frac{1}{x}}{\color{blue}{z \cdot z}}}{y} \]

      4. times-frac 90.2%

         \[\leadsto \frac{\color{blue}{\frac{1}{z} \cdot \frac{\frac{1}{x}}{z}}}{y} \]

      5. associate-/r* 90.4%

         \[\leadsto \frac{\frac{1}{z} \cdot \color{blue}{\frac{1}{x \cdot z}}}{y} \]

      6. *-commutative 90.4%

         \[\leadsto \frac{\frac{1}{z} \cdot \frac{1}{\color{blue}{z \cdot x}}}{y} \]

   7. Applied egg-rr 90.4%

      \[\leadsto \frac{\color{blue}{\frac{1}{z} \cdot \frac{1}{z \cdot x}}}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z} \cdot \frac{1}{z \cdot x}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.9% accurate, 0.8× speedup

Math

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

FPCore

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 should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (* x_s (if (<= z 1.0) (/ (/ 1.0 y_m) x_m) (/ 1.0 (* z (* z (* x_m y_m))))))))

C

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);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 1.0) {
		tmp = (1.0 / y_m) / x_m;
	} else {
		tmp = 1.0 / (z * (z * (x_m * y_m)));
	}
	return y_s * (x_s * tmp);
}

Fortran

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 should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    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
    real(8) :: tmp
    if (z <= 1.0d0) then
        tmp = (1.0d0 / y_m) / x_m
    else
        tmp = 1.0d0 / (z * (z * (x_m * y_m)))
    end if
    code = y_s * (x_s * tmp)
end function

Java

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;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 1.0) {
		tmp = (1.0 / y_m) / x_m;
	} else {
		tmp = 1.0 / (z * (z * (x_m * y_m)));
	}
	return y_s * (x_s * tmp);
}

Python

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] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if z <= 1.0:
		tmp = (1.0 / y_m) / x_m
	else:
		tmp = 1.0 / (z * (z * (x_m * y_m)))
	return y_s * (x_s * tmp)

Julia

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

MATLAB

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

Wolfram

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 should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z, 1.0], N[(N[(1.0 / y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision], N[(1.0 / N[(z * N[(z * N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 93.1%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 68.3%

      \[\leadsto \color{blue}{\frac{1}{x \cdot y}} \]
   4. Step-by-step derivation

      1. associate-/l/ 68.3%

         \[\leadsto \color{blue}{\frac{\frac{1}{y}}{x}} \]

   5. Simplified 68.3%

      \[\leadsto \color{blue}{\frac{\frac{1}{y}}{x}} \]

   if 1 < z

   1. Initial program 73.9%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 71.5%

      \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot {z}^{2}\right)}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 30.2%

         \[\leadsto \frac{1}{\color{blue}{\sqrt{x \cdot \left(y \cdot {z}^{2}\right)} \cdot \sqrt{x \cdot \left(y \cdot {z}^{2}\right)}}} \]

      2. pow2 30.2%

         \[\leadsto \frac{1}{\color{blue}{{\left(\sqrt{x \cdot \left(y \cdot {z}^{2}\right)}\right)}^{2}}} \]

      3. associate-*r* 33.4%

         \[\leadsto \frac{1}{{\left(\sqrt{\color{blue}{\left(x \cdot y\right) \cdot {z}^{2}}}\right)}^{2}} \]

      4. sqrt-prod 33.3%

         \[\leadsto \frac{1}{{\color{blue}{\left(\sqrt{x \cdot y} \cdot \sqrt{{z}^{2}}\right)}}^{2}} \]

      5. unpow2 33.3%

         \[\leadsto \frac{1}{{\left(\sqrt{x \cdot y} \cdot \sqrt{\color{blue}{z \cdot z}}\right)}^{2}} \]

      6. sqrt-prod 46.1%

         \[\leadsto \frac{1}{{\left(\sqrt{x \cdot y} \cdot \color{blue}{\left(\sqrt{z} \cdot \sqrt{z}\right)}\right)}^{2}} \]

      7. add-sqr-sqrt 46.2%

         \[\leadsto \frac{1}{{\left(\sqrt{x \cdot y} \cdot \color{blue}{z}\right)}^{2}} \]

   5. Applied egg-rr 46.2%

      \[\leadsto \frac{1}{\color{blue}{{\left(\sqrt{x \cdot y} \cdot z\right)}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 46.2%

         \[\leadsto \frac{1}{\color{blue}{\left(\sqrt{x \cdot y} \cdot z\right) \cdot \left(\sqrt{x \cdot y} \cdot z\right)}} \]

      2. swap-sqr 33.3%

         \[\leadsto \frac{1}{\color{blue}{\left(\sqrt{x \cdot y} \cdot \sqrt{x \cdot y}\right) \cdot \left(z \cdot z\right)}} \]

      3. add-sqr-sqrt 71.4%

         \[\leadsto \frac{1}{\color{blue}{\left(x \cdot y\right)} \cdot \left(z \cdot z\right)} \]

      4. associate-*r* 87.9%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(x \cdot y\right) \cdot z\right) \cdot z}} \]

   7. Applied egg-rr 87.9%

      \[\leadsto \frac{1}{\color{blue}{\left(\left(x \cdot y\right) \cdot z\right) \cdot z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(z \cdot \left(x \cdot y\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.0% accurate, 0.9× speedup

Math

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

FPCore

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 should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (* y_s (* x_s (if (<= z 1.0) (/ (/ 1.0 y_m) x_m) (/ 1.0 (* x_m (* z y_m)))))))

C

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);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 1.0) {
		tmp = (1.0 / y_m) / x_m;
	} else {
		tmp = 1.0 / (x_m * (z * y_m));
	}
	return y_s * (x_s * tmp);
}

Fortran

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 should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    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
    real(8) :: tmp
    if (z <= 1.0d0) then
        tmp = (1.0d0 / y_m) / x_m
    else
        tmp = 1.0d0 / (x_m * (z * y_m))
    end if
    code = y_s * (x_s * tmp)
end function

Java

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;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 1.0) {
		tmp = (1.0 / y_m) / x_m;
	} else {
		tmp = 1.0 / (x_m * (z * y_m));
	}
	return y_s * (x_s * tmp);
}

Python

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] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if z <= 1.0:
		tmp = (1.0 / y_m) / x_m
	else:
		tmp = 1.0 / (x_m * (z * y_m))
	return y_s * (x_s * tmp)

Julia

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

MATLAB

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

Wolfram

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 should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z, 1.0], N[(N[(1.0 / y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision], N[(1.0 / N[(x$95$m * N[(z * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 93.1%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 68.3%

      \[\leadsto \color{blue}{\frac{1}{x \cdot y}} \]
   4. Step-by-step derivation

      1. associate-/l/ 68.3%

         \[\leadsto \color{blue}{\frac{\frac{1}{y}}{x}} \]

   5. Simplified 68.3%

      \[\leadsto \color{blue}{\frac{\frac{1}{y}}{x}} \]

   if 1 < z

   1. Initial program 73.9%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 75.6%

         \[\leadsto \color{blue}{\frac{\frac{\frac{1}{x}}{1 + z \cdot z}}{y}} \]

      2. metadata-eval 75.6%

         \[\leadsto \frac{\frac{\frac{\color{blue}{\frac{-1}{-1}}}{x}}{1 + z \cdot z}}{y} \]

      3. associate-/r* 75.6%

         \[\leadsto \frac{\frac{\color{blue}{\frac{-1}{-1 \cdot x}}}{1 + z \cdot z}}{y} \]

      4. neg-mul-1 75.6%

         \[\leadsto \frac{\frac{\frac{-1}{\color{blue}{-x}}}{1 + z \cdot z}}{y} \]

      5. metadata-eval 75.6%

         \[\leadsto \frac{\frac{\frac{\color{blue}{-1}}{-x}}{1 + z \cdot z}}{y} \]

      6. distribute-neg-frac 75.6%

         \[\leadsto \frac{\frac{\color{blue}{-\frac{1}{-x}}}{1 + z \cdot z}}{y} \]

      7. distribute-frac-neg 75.6%

         \[\leadsto \frac{\color{blue}{-\frac{\frac{1}{-x}}{1 + z \cdot z}}}{y} \]

   3. Simplified 75.5%

      \[\leadsto \color{blue}{\frac{\frac{1}{x \cdot \mathsf{fma}\left(z, z, 1\right)}}{y}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 75.6%

         \[\leadsto \frac{\color{blue}{\frac{\frac{1}{x}}{\mathsf{fma}\left(z, z, 1\right)}}}{y} \]

      2. *-un-lft-identity 75.6%

         \[\leadsto \frac{\frac{\color{blue}{1 \cdot \frac{1}{x}}}{\mathsf{fma}\left(z, z, 1\right)}}{y} \]

      3. fma-undefine 75.6%

         \[\leadsto \frac{\frac{1 \cdot \frac{1}{x}}{\color{blue}{z \cdot z + 1}}}{y} \]

      4. +-commutative 75.6%

         \[\leadsto \frac{\frac{1 \cdot \frac{1}{x}}{\color{blue}{1 + z \cdot z}}}{y} \]

      5. add-sqr-sqrt 75.6%

         \[\leadsto \frac{\frac{1 \cdot \frac{1}{x}}{\color{blue}{\sqrt{1 + z \cdot z} \cdot \sqrt{1 + z \cdot z}}}}{y} \]

      6. times-frac 75.5%

         \[\leadsto \frac{\color{blue}{\frac{1}{\sqrt{1 + z \cdot z}} \cdot \frac{\frac{1}{x}}{\sqrt{1 + z \cdot z}}}}{y} \]

      7. hypot-1-def 75.5%

         \[\leadsto \frac{\frac{1}{\color{blue}{\mathsf{hypot}\left(1, z\right)}} \cdot \frac{\frac{1}{x}}{\sqrt{1 + z \cdot z}}}{y} \]

      8. hypot-1-def 91.7%

         \[\leadsto \frac{\frac{1}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{\frac{1}{x}}{\color{blue}{\mathsf{hypot}\left(1, z\right)}}}{y} \]

   6. Applied egg-rr 91.7%

      \[\leadsto \frac{\color{blue}{\frac{1}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{\frac{1}{x}}{\mathsf{hypot}\left(1, z\right)}}}{y} \]

   7. Taylor expanded in z around inf 90.4%

      \[\leadsto \frac{\frac{1}{\mathsf{hypot}\left(1, z\right)} \cdot \color{blue}{\frac{1}{x \cdot z}}}{y} \]

   8. Taylor expanded in z around 0 33.7%

      \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot z\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(z \cdot y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.5% accurate, 2.2× speedup

Math

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

FPCore

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 should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (* y_s (* x_s (/ 1.0 (* x_m y_m)))))

C

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);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	return y_s * (x_s * (1.0 / (x_m * y_m)));
}

Fortran

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 should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    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
    code = y_s * (x_s * (1.0d0 / (x_m * y_m)))
end function

Java

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;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	return y_s * (x_s * (1.0 / (x_m * y_m)));
}

Python

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] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	return y_s * (x_s * (1.0 / (x_m * y_m)))

Julia

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

MATLAB

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

Wolfram

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 should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * N[(1.0 / N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.7%

   \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
2. Add Preprocessing

3. Taylor expanded in z around 0 57.3%

   \[\leadsto \color{blue}{\frac{1}{x \cdot y}} \]

4. Final simplification 57.3%

   \[\leadsto \frac{1}{x \cdot y} \]
5. Add Preprocessing

Alternative 10: 58.5% accurate, 2.2× speedup

Math

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

FPCore

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 should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (* y_s (* x_s (/ (/ 1.0 y_m) x_m))))

C

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);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	return y_s * (x_s * ((1.0 / y_m) / x_m));
}

Fortran

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 should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    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
    code = y_s * (x_s * ((1.0d0 / y_m) / x_m))
end function

Java

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;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	return y_s * (x_s * ((1.0 / y_m) / x_m));
}

Python

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] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	return y_s * (x_s * ((1.0 / y_m) / x_m))

Julia

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

MATLAB

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

Wolfram

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 should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * N[(N[(1.0 / y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.7%

   \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
2. Add Preprocessing

3. Taylor expanded in z around 0 57.3%

   \[\leadsto \color{blue}{\frac{1}{x \cdot y}} \]
4. Step-by-step derivation

   1. associate-/l/ 57.2%

      \[\leadsto \color{blue}{\frac{\frac{1}{y}}{x}} \]

5. Simplified 57.2%

   \[\leadsto \color{blue}{\frac{\frac{1}{y}}{x}} \]

6. Final simplification 57.2%

   \[\leadsto \frac{\frac{1}{y}}{x} \]
7. Add Preprocessing

Developer target: 92.6% accurate, 0.3× 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 2024034 
(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)))))