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

Percentage Accurate: 90.7% → 97.5%

Time: 8.2s

Alternatives: 9

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: 90.7% 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: 97.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{y}}{1 + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{1}{y \cdot \left(x \cdot z\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 5e+69)
   (/ (/ (/ 1.0 x) y) (+ 1.0 (* z z)))
   (* (/ 1.0 z) (/ 1.0 (* y (* x z))))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e+69) {
		tmp = ((1.0 / x) / y) / (1.0 + (z * z));
	} else {
		tmp = (1.0 / z) * (1.0 / (y * (x * z)));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 5d+69) then
        tmp = ((1.0d0 / x) / y) / (1.0d0 + (z * z))
    else
        tmp = (1.0d0 / z) * (1.0d0 / (y * (x * z)))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e+69) {
		tmp = ((1.0 / x) / y) / (1.0 + (z * z));
	} else {
		tmp = (1.0 / z) * (1.0 / (y * (x * z)));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 5e+69:
		tmp = ((1.0 / x) / y) / (1.0 + (z * z))
	else:
		tmp = (1.0 / z) * (1.0 / (y * (x * z)))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e+69)
		tmp = Float64(Float64(Float64(1.0 / x) / y) / Float64(1.0 + Float64(z * z)));
	else
		tmp = Float64(Float64(1.0 / z) * Float64(1.0 / Float64(y * Float64(x * z))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 5e+69)
		tmp = ((1.0 / x) / y) / (1.0 + (z * z));
	else
		tmp = (1.0 / z) * (1.0 / (y * (x * z)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e+69], N[(N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision] / N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] * N[(1.0 / N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.00000000000000036e69

   1. Initial program 99.0%

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

      1. associate-/r* 99.7%

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

   3. Simplified 99.7%

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

   if 5.00000000000000036e69 < (*.f64 z z)

   1. Initial program 82.7%

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

      1. associate-/r* 82.5%

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

      2. +-commutative 82.5%

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

      3. fma-def 82.5%

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

   3. Simplified 82.5%

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

   4. Taylor expanded in z around inf 83.4%

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

      1. unpow2 83.4%

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

      2. *-commutative 83.4%

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

      3. associate-*l* 79.4%

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

      4. *-commutative 79.4%

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

      5. associate-*l* 85.2%

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

   6. Simplified 85.2%

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

      1. associate-/r* 86.0%

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

      2. div-inv 86.0%

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

      3. *-commutative 86.0%

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

      4. associate-*l* 94.9%

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

   8. Applied egg-rr 94.9%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{y}}{1 + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{1}{y \cdot \left(x \cdot z\right)}\\ \end{array} \]

Alternative 2: 97.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{\frac{\frac{1}{y}}{x \cdot \mathsf{hypot}\left(1, z\right)}}{\mathsf{hypot}\left(1, z\right)} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (/ (/ (/ 1.0 y) (* x (hypot 1.0 z))) (hypot 1.0 z)))

C

assert(x < y);
double code(double x, double y, double z) {
	return ((1.0 / y) / (x * hypot(1.0, z))) / hypot(1.0, z);
}

Java

assert x < y;
public static double code(double x, double y, double z) {
	return ((1.0 / y) / (x * Math.hypot(1.0, z))) / Math.hypot(1.0, z);
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	return ((1.0 / y) / (x * math.hypot(1.0, z))) / math.hypot(1.0, z)

Julia

x, y = sort([x, y])
function code(x, y, z)
	return Float64(Float64(Float64(1.0 / y) / Float64(x * hypot(1.0, z))) / hypot(1.0, z))
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y, z)
	tmp = ((1.0 / y) / (x * hypot(1.0, z))) / hypot(1.0, z);
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(N[(N[(1.0 / y), $MachinePrecision] / N[(x * N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.7%

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

   1. associate-/r* 91.7%

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

   2. +-commutative 91.7%

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

   3. fma-def 91.7%

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

3. Simplified 91.7%

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

   1. fma-udef 91.7%

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

   2. +-commutative 91.7%

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

   3. associate-/r* 91.7%

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

   4. associate-/r* 90.5%

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

   5. add-sqr-sqrt 90.5%

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

   6. *-un-lft-identity 90.5%

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

   7. times-frac 90.4%

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

   8. hypot-1-def 90.4%

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

   9. associate-/l/ 90.4%

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

   10. hypot-1-def 93.3%

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

5. Applied egg-rr 93.3%

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

   1. associate-*l/ 93.3%

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

   2. *-lft-identity 93.3%

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

   3. associate-/r* 93.4%

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

   4. associate-/l/ 97.6%

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

   5. *-commutative 97.6%

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

7. Simplified 97.6%

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

8. Final simplification 97.6%

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

Alternative 3: 96.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{1}{y \cdot \left(x \cdot z\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 5e-5) (/ (/ 1.0 x) y) (* (/ 1.0 z) (/ 1.0 (* y (* x z))))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e-5) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = (1.0 / z) * (1.0 / (y * (x * z)));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 5d-5) then
        tmp = (1.0d0 / x) / y
    else
        tmp = (1.0d0 / z) * (1.0d0 / (y * (x * z)))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e-5) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = (1.0 / z) * (1.0 / (y * (x * z)));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 5e-5:
		tmp = (1.0 / x) / y
	else:
		tmp = (1.0 / z) * (1.0 / (y * (x * z)))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e-5)
		tmp = Float64(Float64(1.0 / x) / y);
	else
		tmp = Float64(Float64(1.0 / z) * Float64(1.0 / Float64(y * Float64(x * z))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 5e-5)
		tmp = (1.0 / x) / y;
	else
		tmp = (1.0 / z) * (1.0 / (y * (x * z)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e-5], N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] * N[(1.0 / N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.00000000000000024e-5

   1. Initial program 99.7%

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

      1. associate-/r* 99.7%

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

   3. Simplified 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. add-sqr-sqrt 44.3%

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

      3. times-frac 44.3%

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

   5. Applied egg-rr 44.3%

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

      1. associate-*l/ 44.3%

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

      2. *-lft-identity 44.3%

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

      3. associate-/r* 44.3%

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

   7. Simplified 44.3%

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{x \cdot \sqrt{y}}}{\sqrt{y}}}}{1 + z \cdot z} \]
   8. Step-by-step derivation

      1. div-inv 44.2%

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

      2. pow1/2 44.2%

         \[\leadsto \frac{\frac{1}{x \cdot \sqrt{y}} \cdot \frac{1}{\color{blue}{{y}^{0.5}}}}{1 + z \cdot z} \]

      3. pow-flip 44.3%

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

      4. metadata-eval 44.3%

         \[\leadsto \frac{\frac{1}{x \cdot \sqrt{y}} \cdot {y}^{\color{blue}{-0.5}}}{1 + z \cdot z} \]

   9. Applied egg-rr 44.3%

      \[\leadsto \frac{\color{blue}{\frac{1}{x \cdot \sqrt{y}} \cdot {y}^{-0.5}}}{1 + z \cdot z} \]
   10. Step-by-step derivation

       1. associate-*l/ 44.3%

          \[\leadsto \frac{\color{blue}{\frac{1 \cdot {y}^{-0.5}}{x \cdot \sqrt{y}}}}{1 + z \cdot z} \]

       2. *-lft-identity 44.3%

          \[\leadsto \frac{\frac{\color{blue}{{y}^{-0.5}}}{x \cdot \sqrt{y}}}{1 + z \cdot z} \]

   11. Simplified 44.3%

       \[\leadsto \frac{\color{blue}{\frac{{y}^{-0.5}}{x \cdot \sqrt{y}}}}{1 + z \cdot z} \]

   12. Taylor expanded in z around 0 99.0%

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

       1. associate-/l/ 99.0%

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

   14. Simplified 99.0%

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

   if 5.00000000000000024e-5 < (*.f64 z z)

   1. Initial program 83.5%

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

      1. associate-/r* 83.4%

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

      2. +-commutative 83.4%

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

      3. fma-def 83.4%

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

   3. Simplified 83.4%

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

   4. Taylor expanded in z around inf 83.8%

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

      1. unpow2 83.8%

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

      2. *-commutative 83.8%

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

      3. associate-*l* 80.9%

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

      4. *-commutative 80.9%

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

      5. associate-*l* 86.2%

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

   6. Simplified 86.2%

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

      1. associate-/r* 86.9%

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

      2. div-inv 86.9%

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

      3. *-commutative 86.9%

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

      4. associate-*l* 95.0%

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

   8. Applied egg-rr 95.0%

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

4. Final simplification 97.0%

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

Alternative 4: 97.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{1}{y \cdot \left(x \cdot z\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 5e+69)
   (/ (/ 1.0 x) (* y (+ 1.0 (* z z))))
   (* (/ 1.0 z) (/ 1.0 (* y (* x z))))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e+69) {
		tmp = (1.0 / x) / (y * (1.0 + (z * z)));
	} else {
		tmp = (1.0 / z) * (1.0 / (y * (x * z)));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 5d+69) then
        tmp = (1.0d0 / x) / (y * (1.0d0 + (z * z)))
    else
        tmp = (1.0d0 / z) * (1.0d0 / (y * (x * z)))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e+69) {
		tmp = (1.0 / x) / (y * (1.0 + (z * z)));
	} else {
		tmp = (1.0 / z) * (1.0 / (y * (x * z)));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 5e+69:
		tmp = (1.0 / x) / (y * (1.0 + (z * z)))
	else:
		tmp = (1.0 / z) * (1.0 / (y * (x * z)))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e+69)
		tmp = Float64(Float64(1.0 / x) / Float64(y * Float64(1.0 + Float64(z * z))));
	else
		tmp = Float64(Float64(1.0 / z) * Float64(1.0 / Float64(y * Float64(x * z))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 5e+69)
		tmp = (1.0 / x) / (y * (1.0 + (z * z)));
	else
		tmp = (1.0 / z) * (1.0 / (y * (x * z)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e+69], N[(N[(1.0 / x), $MachinePrecision] / N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] * N[(1.0 / N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.00000000000000036e69

   1. Initial program 99.0%

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

   if 5.00000000000000036e69 < (*.f64 z z)

   1. Initial program 82.7%

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

      1. associate-/r* 82.5%

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

      2. +-commutative 82.5%

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

      3. fma-def 82.5%

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

   3. Simplified 82.5%

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

   4. Taylor expanded in z around inf 83.4%

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

      1. unpow2 83.4%

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

      2. *-commutative 83.4%

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

      3. associate-*l* 79.4%

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

      4. *-commutative 79.4%

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

      5. associate-*l* 85.2%

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

   6. Simplified 85.2%

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

      1. associate-/r* 86.0%

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

      2. div-inv 86.0%

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

      3. *-commutative 86.0%

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

      4. associate-*l* 94.9%

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

   8. Applied egg-rr 94.9%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{1}{y \cdot \left(x \cdot z\right)}\\ \end{array} \]

Alternative 5: 96.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{--1}{y}}{x \cdot z}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 5e-5) (/ (/ 1.0 x) y) (/ (/ (/ (- -1.0) y) (* x z)) z)))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e-5) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = ((-(-1.0) / y) / (x * z)) / z;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 5d-5) then
        tmp = (1.0d0 / x) / y
    else
        tmp = ((-(-1.0d0) / y) / (x * z)) / z
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e-5) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = ((-(-1.0) / y) / (x * z)) / z;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 5e-5:
		tmp = (1.0 / x) / y
	else:
		tmp = ((-(-1.0) / y) / (x * z)) / z
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e-5)
		tmp = Float64(Float64(1.0 / x) / y);
	else
		tmp = Float64(Float64(Float64(Float64(-(-1.0)) / y) / Float64(x * z)) / z);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 5e-5)
		tmp = (1.0 / x) / y;
	else
		tmp = ((-(-1.0) / y) / (x * z)) / z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e-5], N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision], N[(N[(N[((--1.0) / y), $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.00000000000000024e-5

   1. Initial program 99.7%

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

      1. associate-/r* 99.7%

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

   3. Simplified 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. add-sqr-sqrt 44.3%

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

      3. times-frac 44.3%

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

   5. Applied egg-rr 44.3%

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

      1. associate-*l/ 44.3%

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

      2. *-lft-identity 44.3%

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

      3. associate-/r* 44.3%

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

   7. Simplified 44.3%

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{x \cdot \sqrt{y}}}{\sqrt{y}}}}{1 + z \cdot z} \]
   8. Step-by-step derivation

      1. div-inv 44.2%

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

      2. pow1/2 44.2%

         \[\leadsto \frac{\frac{1}{x \cdot \sqrt{y}} \cdot \frac{1}{\color{blue}{{y}^{0.5}}}}{1 + z \cdot z} \]

      3. pow-flip 44.3%

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

      4. metadata-eval 44.3%

         \[\leadsto \frac{\frac{1}{x \cdot \sqrt{y}} \cdot {y}^{\color{blue}{-0.5}}}{1 + z \cdot z} \]

   9. Applied egg-rr 44.3%

      \[\leadsto \frac{\color{blue}{\frac{1}{x \cdot \sqrt{y}} \cdot {y}^{-0.5}}}{1 + z \cdot z} \]
   10. Step-by-step derivation

       1. associate-*l/ 44.3%

          \[\leadsto \frac{\color{blue}{\frac{1 \cdot {y}^{-0.5}}{x \cdot \sqrt{y}}}}{1 + z \cdot z} \]

       2. *-lft-identity 44.3%

          \[\leadsto \frac{\frac{\color{blue}{{y}^{-0.5}}}{x \cdot \sqrt{y}}}{1 + z \cdot z} \]

   11. Simplified 44.3%

       \[\leadsto \frac{\color{blue}{\frac{{y}^{-0.5}}{x \cdot \sqrt{y}}}}{1 + z \cdot z} \]

   12. Taylor expanded in z around 0 99.0%

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

       1. associate-/l/ 99.0%

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

   14. Simplified 99.0%

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

   if 5.00000000000000024e-5 < (*.f64 z z)

   1. Initial program 83.5%

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

      1. associate-/r* 83.4%

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

      2. +-commutative 83.4%

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

      3. fma-def 83.4%

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

   3. Simplified 83.4%

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

   4. Taylor expanded in z around inf 83.8%

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

      1. unpow2 83.8%

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

      2. *-commutative 83.8%

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

      3. associate-*l* 80.9%

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

      4. *-commutative 80.9%

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

      5. associate-*l* 86.2%

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

   6. Simplified 86.2%

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

   7. Taylor expanded in z around 0 94.3%

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

      1. metadata-eval 94.3%

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

      2. *-commutative 94.3%

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

      3. frac-times 95.0%

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

      4. associate-/r/ 94.2%

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

      5. frac-2neg 94.2%

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

      6. metadata-eval 94.2%

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

      7. div-inv 94.2%

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

      8. distribute-neg-frac 94.2%

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

      9. frac-2neg 94.2%

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

      10. *-commutative 94.2%

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

      11. associate-*r* 94.9%

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

      12. *-commutative 94.9%

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

      13. distribute-rgt-neg-in 94.9%

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

      14. *-commutative 94.9%

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

      15. add-sqr-sqrt 45.7%

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

      16. sqrt-unprod 61.5%

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

      17. sqr-neg 61.5%

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

      18. sqrt-unprod 27.1%

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

      19. add-sqr-sqrt 52.4%

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

      20. associate-*r* 52.4%

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

      21. *-commutative 52.4%

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

   9. Applied egg-rr 95.1%

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

4. Final simplification 97.0%

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

Alternative 6: 93.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(z \cdot \left(y \cdot z\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 5e-5) (/ (/ 1.0 x) y) (/ 1.0 (* x (* z (* y z))))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e-5) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = 1.0 / (x * (z * (y * z)));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 5d-5) then
        tmp = (1.0d0 / x) / y
    else
        tmp = 1.0d0 / (x * (z * (y * z)))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e-5) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = 1.0 / (x * (z * (y * z)));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 5e-5:
		tmp = (1.0 / x) / y
	else:
		tmp = 1.0 / (x * (z * (y * z)))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e-5)
		tmp = Float64(Float64(1.0 / x) / y);
	else
		tmp = Float64(1.0 / Float64(x * Float64(z * Float64(y * z))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 5e-5)
		tmp = (1.0 / x) / y;
	else
		tmp = 1.0 / (x * (z * (y * z)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e-5], N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision], N[(1.0 / N[(x * N[(z * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.00000000000000024e-5

   1. Initial program 99.7%

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

      1. associate-/r* 99.7%

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

   3. Simplified 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. add-sqr-sqrt 44.3%

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

      3. times-frac 44.3%

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

   5. Applied egg-rr 44.3%

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

      1. associate-*l/ 44.3%

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

      2. *-lft-identity 44.3%

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

      3. associate-/r* 44.3%

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

   7. Simplified 44.3%

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{x \cdot \sqrt{y}}}{\sqrt{y}}}}{1 + z \cdot z} \]
   8. Step-by-step derivation

      1. div-inv 44.2%

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

      2. pow1/2 44.2%

         \[\leadsto \frac{\frac{1}{x \cdot \sqrt{y}} \cdot \frac{1}{\color{blue}{{y}^{0.5}}}}{1 + z \cdot z} \]

      3. pow-flip 44.3%

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

      4. metadata-eval 44.3%

         \[\leadsto \frac{\frac{1}{x \cdot \sqrt{y}} \cdot {y}^{\color{blue}{-0.5}}}{1 + z \cdot z} \]

   9. Applied egg-rr 44.3%

      \[\leadsto \frac{\color{blue}{\frac{1}{x \cdot \sqrt{y}} \cdot {y}^{-0.5}}}{1 + z \cdot z} \]
   10. Step-by-step derivation

       1. associate-*l/ 44.3%

          \[\leadsto \frac{\color{blue}{\frac{1 \cdot {y}^{-0.5}}{x \cdot \sqrt{y}}}}{1 + z \cdot z} \]

       2. *-lft-identity 44.3%

          \[\leadsto \frac{\frac{\color{blue}{{y}^{-0.5}}}{x \cdot \sqrt{y}}}{1 + z \cdot z} \]

   11. Simplified 44.3%

       \[\leadsto \frac{\color{blue}{\frac{{y}^{-0.5}}{x \cdot \sqrt{y}}}}{1 + z \cdot z} \]

   12. Taylor expanded in z around 0 99.0%

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

       1. associate-/l/ 99.0%

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

   14. Simplified 99.0%

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

   if 5.00000000000000024e-5 < (*.f64 z z)

   1. Initial program 83.5%

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

      1. associate-/r* 83.4%

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

      2. +-commutative 83.4%

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

      3. fma-def 83.4%

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

   3. Simplified 83.4%

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

   4. Taylor expanded in z around inf 83.0%

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

      1. unpow2 83.0%

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

      2. associate-*r* 89.7%

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

      3. *-commutative 89.7%

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

   6. Simplified 89.7%

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

4. Final simplification 94.4%

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

Alternative 7: 96.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(y \cdot \left(x \cdot z\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 5e-5) (/ (/ 1.0 x) y) (/ 1.0 (* z (* y (* x z))))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e-5) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = 1.0 / (z * (y * (x * z)));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 5d-5) then
        tmp = (1.0d0 / x) / y
    else
        tmp = 1.0d0 / (z * (y * (x * z)))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e-5) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = 1.0 / (z * (y * (x * z)));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 5e-5:
		tmp = (1.0 / x) / y
	else:
		tmp = 1.0 / (z * (y * (x * z)))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e-5)
		tmp = Float64(Float64(1.0 / x) / y);
	else
		tmp = Float64(1.0 / Float64(z * Float64(y * Float64(x * z))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 5e-5)
		tmp = (1.0 / x) / y;
	else
		tmp = 1.0 / (z * (y * (x * z)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e-5], N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision], N[(1.0 / N[(z * N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.00000000000000024e-5

   1. Initial program 99.7%

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

      1. associate-/r* 99.7%

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

   3. Simplified 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. add-sqr-sqrt 44.3%

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

      3. times-frac 44.3%

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

   5. Applied egg-rr 44.3%

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

      1. associate-*l/ 44.3%

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

      2. *-lft-identity 44.3%

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

      3. associate-/r* 44.3%

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

   7. Simplified 44.3%

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{x \cdot \sqrt{y}}}{\sqrt{y}}}}{1 + z \cdot z} \]
   8. Step-by-step derivation

      1. div-inv 44.2%

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

      2. pow1/2 44.2%

         \[\leadsto \frac{\frac{1}{x \cdot \sqrt{y}} \cdot \frac{1}{\color{blue}{{y}^{0.5}}}}{1 + z \cdot z} \]

      3. pow-flip 44.3%

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

      4. metadata-eval 44.3%

         \[\leadsto \frac{\frac{1}{x \cdot \sqrt{y}} \cdot {y}^{\color{blue}{-0.5}}}{1 + z \cdot z} \]

   9. Applied egg-rr 44.3%

      \[\leadsto \frac{\color{blue}{\frac{1}{x \cdot \sqrt{y}} \cdot {y}^{-0.5}}}{1 + z \cdot z} \]
   10. Step-by-step derivation

       1. associate-*l/ 44.3%

          \[\leadsto \frac{\color{blue}{\frac{1 \cdot {y}^{-0.5}}{x \cdot \sqrt{y}}}}{1 + z \cdot z} \]

       2. *-lft-identity 44.3%

          \[\leadsto \frac{\frac{\color{blue}{{y}^{-0.5}}}{x \cdot \sqrt{y}}}{1 + z \cdot z} \]

   11. Simplified 44.3%

       \[\leadsto \frac{\color{blue}{\frac{{y}^{-0.5}}{x \cdot \sqrt{y}}}}{1 + z \cdot z} \]

   12. Taylor expanded in z around 0 99.0%

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

       1. associate-/l/ 99.0%

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

   14. Simplified 99.0%

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

   if 5.00000000000000024e-5 < (*.f64 z z)

   1. Initial program 83.5%

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

      1. associate-/r* 83.4%

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

      2. +-commutative 83.4%

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

      3. fma-def 83.4%

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

   3. Simplified 83.4%

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

   4. Taylor expanded in z around inf 83.8%

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

      1. unpow2 83.8%

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

      2. *-commutative 83.8%

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

      3. associate-*l* 80.9%

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

      4. *-commutative 80.9%

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

      5. associate-*l* 86.2%

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

   6. Simplified 86.2%

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

   7. Taylor expanded in z around 0 94.3%

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

4. Final simplification 96.7%

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

Alternative 8: 58.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{1}{y \cdot x} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (/ 1.0 (* y x)))

C

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

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 / (y * x)
end function

Java

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

Python

[x, y] = sort([x, y])
def code(x, y, z):
	return 1.0 / (y * x)

Julia

x, y = sort([x, y])
function code(x, y, z)
	return Float64(1.0 / Float64(y * x))
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y, z)
	tmp = 1.0 / (y * x);
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(1.0 / N[(y * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.7%

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

   1. associate-/r* 91.7%

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

   2. +-commutative 91.7%

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

   3. fma-def 91.7%

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

3. Simplified 91.7%

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

4. Taylor expanded in z around 0 58.2%

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

5. Final simplification 58.2%

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

Alternative 9: 58.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{\frac{1}{x}}{y} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (/ (/ 1.0 x) y))

C

assert(x < y);
double code(double x, double y, double z) {
	return (1.0 / x) / y;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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
end function

Java

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

Python

[x, y] = sort([x, y])
def code(x, y, z):
	return (1.0 / x) / y

Julia

x, y = sort([x, y])
function code(x, y, z)
	return Float64(Float64(1.0 / x) / y)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y, z)
	tmp = (1.0 / x) / y;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision]

Derivation

1. Initial program 91.7%

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

   1. associate-/r* 90.5%

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

3. Simplified 90.5%

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

   1. *-un-lft-identity 90.5%

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

   2. add-sqr-sqrt 46.9%

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

   3. times-frac 46.9%

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

5. Applied egg-rr 46.9%

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

   1. associate-*l/ 46.9%

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

   2. *-lft-identity 46.9%

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

   3. associate-/r* 46.9%

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

7. Simplified 46.9%

   \[\leadsto \frac{\color{blue}{\frac{\frac{1}{x \cdot \sqrt{y}}}{\sqrt{y}}}}{1 + z \cdot z} \]
8. Step-by-step derivation

   1. div-inv 46.8%

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

   2. pow1/2 46.8%

      \[\leadsto \frac{\frac{1}{x \cdot \sqrt{y}} \cdot \frac{1}{\color{blue}{{y}^{0.5}}}}{1 + z \cdot z} \]

   3. pow-flip 46.9%

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

   4. metadata-eval 46.9%

      \[\leadsto \frac{\frac{1}{x \cdot \sqrt{y}} \cdot {y}^{\color{blue}{-0.5}}}{1 + z \cdot z} \]

9. Applied egg-rr 46.9%

   \[\leadsto \frac{\color{blue}{\frac{1}{x \cdot \sqrt{y}} \cdot {y}^{-0.5}}}{1 + z \cdot z} \]
10. Step-by-step derivation

    1. associate-*l/ 47.0%

       \[\leadsto \frac{\color{blue}{\frac{1 \cdot {y}^{-0.5}}{x \cdot \sqrt{y}}}}{1 + z \cdot z} \]

    2. *-lft-identity 47.0%

       \[\leadsto \frac{\frac{\color{blue}{{y}^{-0.5}}}{x \cdot \sqrt{y}}}{1 + z \cdot z} \]

11. Simplified 47.0%

    \[\leadsto \frac{\color{blue}{\frac{{y}^{-0.5}}{x \cdot \sqrt{y}}}}{1 + z \cdot z} \]

12. Taylor expanded in z around 0 58.2%

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

    1. associate-/l/ 58.2%

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

14. Simplified 58.2%

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

15. Final simplification 58.2%

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

Developer target: 92.8% 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 2023224 
(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)))))