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

Percentage Accurate: 90.2% → 97.5%

Time: 9.2s

Alternatives: 11

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: 90.2% 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, 0.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := x \cdot \mathsf{hypot}\left(1, z\right)\\ \mathbf{if}\;x \leq -2.4 \cdot 10^{-114}:\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{t_0}}{\mathsf{hypot}\left(1, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\mathsf{hypot}\left(1, z\right) \cdot t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (hypot 1.0 z))))
   (if (<= x -2.4e-114)
     (/ (/ (/ 1.0 y) t_0) (hypot 1.0 z))
     (/ (/ 1.0 y) (* (hypot 1.0 z) t_0)))))

C

assert(x < y);
double code(double x, double y, double z) {
	double t_0 = x * hypot(1.0, z);
	double tmp;
	if (x <= -2.4e-114) {
		tmp = ((1.0 / y) / t_0) / hypot(1.0, z);
	} else {
		tmp = (1.0 / y) / (hypot(1.0, z) * t_0);
	}
	return tmp;
}

Java

assert x < y;
public static double code(double x, double y, double z) {
	double t_0 = x * Math.hypot(1.0, z);
	double tmp;
	if (x <= -2.4e-114) {
		tmp = ((1.0 / y) / t_0) / Math.hypot(1.0, z);
	} else {
		tmp = (1.0 / y) / (Math.hypot(1.0, z) * t_0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	t_0 = x * math.hypot(1.0, z)
	tmp = 0
	if x <= -2.4e-114:
		tmp = ((1.0 / y) / t_0) / math.hypot(1.0, z)
	else:
		tmp = (1.0 / y) / (math.hypot(1.0, z) * t_0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	t_0 = Float64(x * hypot(1.0, z))
	tmp = 0.0
	if (x <= -2.4e-114)
		tmp = Float64(Float64(Float64(1.0 / y) / t_0) / hypot(1.0, z));
	else
		tmp = Float64(Float64(1.0 / y) / Float64(hypot(1.0, z) * t_0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	t_0 = x * hypot(1.0, z);
	tmp = 0.0;
	if (x <= -2.4e-114)
		tmp = ((1.0 / y) / t_0) / hypot(1.0, z);
	else
		tmp = (1.0 / y) / (hypot(1.0, z) * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.4e-114], N[(N[(N[(1.0 / y), $MachinePrecision] / t$95$0), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] / N[(N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.4000000000000001e-114

   1. Initial program 94.0%

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

      1. associate-/r* 93.5%

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

      2. +-commutative 93.5%

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

      3. fma-def 93.5%

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

   3. Simplified 93.5%

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

      1. fma-udef 93.5%

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

      2. +-commutative 93.5%

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

      3. associate-/r* 94.0%

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

      4. associate-/r* 93.9%

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

      5. add-sqr-sqrt 93.9%

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

      6. *-un-lft-identity 93.9%

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

      7. times-frac 93.8%

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

      8. hypot-1-def 93.8%

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

      9. associate-/l/ 93.3%

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

      10. hypot-1-def 96.8%

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

   5. Applied egg-rr 96.8%

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

      1. associate-*l/ 96.9%

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

      2. *-lft-identity 96.9%

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

      3. associate-/r* 97.4%

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

      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%

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

   if -2.4000000000000001e-114 < x

   1. Initial program 88.0%

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

      1. associate-/r* 87.5%

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

      2. +-commutative 87.5%

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

      3. fma-def 87.5%

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

   3. Simplified 87.5%

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

      1. fma-udef 87.5%

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

      2. +-commutative 87.5%

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

      3. associate-/r* 88.0%

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

      4. associate-/r* 88.2%

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

      5. div-inv 88.2%

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

      6. add-sqr-sqrt 88.1%

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

      7. times-frac 89.6%

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

      8. hypot-1-def 89.6%

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

      9. hypot-1-def 98.2%

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

   5. Applied egg-rr 98.2%

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

      1. clear-num 98.2%

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

      2. frac-times 96.0%

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

      3. *-un-lft-identity 96.0%

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

      4. div-inv 95.9%

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

      5. remove-double-div 96.0%

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

      6. *-commutative 96.0%

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

   7. Applied egg-rr 96.0%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{-114}:\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{x \cdot \mathsf{hypot}\left(1, z\right)}}{\mathsf{hypot}\left(1, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\mathsf{hypot}\left(1, z\right) \cdot \left(x \cdot \mathsf{hypot}\left(1, z\right)\right)}\\ \end{array} \]

Alternative 2: 97.7% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y, z)
	tmp = ((1.0 / x) / hypot(1.0, z)) * ((1.0 / y) / 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 / x), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / y), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.8%

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

   1. associate-/r* 89.3%

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

   2. +-commutative 89.3%

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

   3. fma-def 89.3%

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

3. Simplified 89.3%

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

   1. fma-udef 89.3%

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

   2. +-commutative 89.3%

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

   3. associate-/r* 89.8%

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

   4. associate-/r* 89.9%

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

   5. div-inv 89.9%

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

   6. add-sqr-sqrt 89.9%

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

   7. times-frac 90.9%

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

   8. hypot-1-def 90.9%

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

   9. hypot-1-def 98.0%

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

5. Applied egg-rr 98.0%

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

6. Final simplification 98.0%

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

Alternative 3: 97.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+81}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{z}}{x}}{z \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e+81)
   (/ (/ 1.0 x) (* y (+ 1.0 (* z z))))
   (/ (/ (/ 1.0 z) x) (* z y))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+81) {
		tmp = (1.0 / x) / (y * (1.0 + (z * z)));
	} else {
		tmp = ((1.0 / z) / x) / (z * y);
	}
	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) <= 2d+81) then
        tmp = (1.0d0 / x) / (y * (1.0d0 + (z * z)))
    else
        tmp = ((1.0d0 / z) / x) / (z * y)
    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) <= 2e+81) {
		tmp = (1.0 / x) / (y * (1.0 + (z * z)));
	} else {
		tmp = ((1.0 / z) / x) / (z * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e+81:
		tmp = (1.0 / x) / (y * (1.0 + (z * z)))
	else:
		tmp = ((1.0 / z) / x) / (z * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+81)
		tmp = Float64(Float64(1.0 / x) / Float64(y * Float64(1.0 + Float64(z * z))));
	else
		tmp = Float64(Float64(Float64(1.0 / z) / x) / Float64(z * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 2e+81)
		tmp = (1.0 / x) / (y * (1.0 + (z * z)));
	else
		tmp = ((1.0 / z) / x) / (z * y);
	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], 2e+81], N[(N[(1.0 / x), $MachinePrecision] / N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / z), $MachinePrecision] / x), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.99999999999999984e81

   1. Initial program 99.7%

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

   if 1.99999999999999984e81 < (*.f64 z z)

   1. Initial program 78.7%

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

      1. associate-/r* 78.7%

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

      2. +-commutative 78.7%

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

      3. fma-def 78.7%

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

   3. Simplified 78.7%

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

   4. Taylor expanded in z around inf 77.9%

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

      1. unpow2 77.9%

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

      2. *-commutative 77.9%

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

      3. associate-*l* 78.4%

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

      4. *-commutative 78.4%

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

      5. associate-*l* 86.4%

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

   6. Simplified 86.4%

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

      1. expm1-log1p-u 82.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{z \cdot \left(z \cdot \left(y \cdot x\right)\right)}\right)\right)} \]

      2. expm1-udef 61.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{z \cdot \left(z \cdot \left(y \cdot x\right)\right)}\right)} - 1} \]

      3. associate-/r* 61.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{1}{z}}{z \cdot \left(y \cdot x\right)}}\right)} - 1 \]

      4. associate-*r* 63.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{1}{z}}{\color{blue}{\left(z \cdot y\right) \cdot x}}\right)} - 1 \]

      5. *-commutative 63.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{1}{z}}{\color{blue}{x \cdot \left(z \cdot y\right)}}\right)} - 1 \]

   8. Applied egg-rr 63.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{1}{z}}{x \cdot \left(z \cdot y\right)}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 88.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{z}}{x \cdot \left(z \cdot y\right)}\right)\right)} \]

      2. expm1-log1p 93.5%

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

      3. associate-/r* 96.2%

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

      4. *-commutative 96.2%

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

   10. Simplified 96.2%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+81}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{z}}{x}}{z \cdot y}\\ \end{array} \]

Alternative 4: 93.7% accurate, 0.8× speedup

Math

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

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.0) (not (<= z 1.0)))
   (/ 1.0 (* x (* z (* z y))))
   (/ (/ 1.0 x) y)))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 1.0)) {
		tmp = 1.0 / (x * (z * (z * y)));
	} else {
		tmp = (1.0 / x) / y;
	}
	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 <= (-1.0d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = 1.0d0 / (x * (z * (z * y)))
    else
        tmp = (1.0d0 / x) / y
    end if
    code = tmp
end function

Java

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

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z <= -1.0) or not (z <= 1.0):
		tmp = 1.0 / (x * (z * (z * y)))
	else:
		tmp = (1.0 / x) / y
	return tmp

Julia

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

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.0) || ~((z <= 1.0)))
		tmp = 1.0 / (x * (z * (z * y)));
	else
		tmp = (1.0 / x) / y;
	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[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(1.0 / N[(x * N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 80.6%

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

      1. associate-/r* 80.5%

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

      2. +-commutative 80.5%

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

      3. fma-def 80.5%

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

   3. Simplified 80.5%

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

   4. Taylor expanded in z around inf 80.4%

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

      1. unpow2 80.4%

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

      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%

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

   if -1 < z < 1

   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* 98.6%

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

      2. +-commutative 98.6%

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

      3. fma-def 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in z around 0 98.3%

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

      1. metadata-eval 98.3%

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

      2. frac-times 99.3%

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

   6. Applied egg-rr 99.3%

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

      1. associate-*l/ 99.4%

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

      2. *-un-lft-identity 99.4%

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

   8. Applied egg-rr 99.4%

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

4. Final simplification 94.4%

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

Alternative 5: 93.3% accurate, 0.8× speedup

Math

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

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.0) (not (<= z 1.0)))
   (/ 1.0 (* y (* z (* x z))))
   (/ (/ 1.0 x) y)))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 1.0)) {
		tmp = 1.0 / (y * (z * (x * z)));
	} else {
		tmp = (1.0 / x) / y;
	}
	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 <= (-1.0d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = 1.0d0 / (y * (z * (x * z)))
    else
        tmp = (1.0d0 / x) / y
    end if
    code = tmp
end function

Java

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

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z <= -1.0) or not (z <= 1.0):
		tmp = 1.0 / (y * (z * (x * z)))
	else:
		tmp = (1.0 / x) / y
	return tmp

Julia

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

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.0) || ~((z <= 1.0)))
		tmp = 1.0 / (y * (z * (x * z)));
	else
		tmp = (1.0 / x) / y;
	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[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(1.0 / N[(y * N[(z * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 80.6%

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

      1. associate-/r* 80.5%

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

      2. +-commutative 80.5%

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

      3. fma-def 80.5%

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

   3. Simplified 80.5%

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

      1. fma-udef 80.5%

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

      2. +-commutative 80.5%

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

      3. associate-/r* 80.6%

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

      4. associate-/r* 80.8%

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

      5. div-inv 80.7%

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

      6. add-sqr-sqrt 80.7%

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

      7. times-frac 82.6%

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

      8. hypot-1-def 82.6%

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

      9. hypot-1-def 96.5%

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

   5. Applied egg-rr 96.5%

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

      1. clear-num 96.4%

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

      2. frac-times 91.3%

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

      3. *-un-lft-identity 91.3%

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

      4. div-inv 91.3%

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

      5. remove-double-div 91.3%

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

      6. *-commutative 91.3%

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

   7. Applied egg-rr 91.3%

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

   8. Taylor expanded in z around inf 79.7%

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

      1. unpow2 79.7%

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

      2. associate-*r* 90.5%

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

   10. Simplified 90.5%

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

   if -1 < z < 1

   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* 98.6%

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

      2. +-commutative 98.6%

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

      3. fma-def 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in z around 0 98.3%

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

      1. metadata-eval 98.3%

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

      2. frac-times 99.3%

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

   6. Applied egg-rr 99.3%

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

      1. associate-*l/ 99.4%

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

      2. *-un-lft-identity 99.4%

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

   8. Applied egg-rr 99.4%

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

4. Final simplification 94.8%

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

Alternative 6: 96.5% accurate, 0.8× speedup

Math

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

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.0) (not (<= z 1.0)))
   (/ 1.0 (* z (* y (* x z))))
   (/ (/ 1.0 x) y)))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 1.0)) {
		tmp = 1.0 / (z * (y * (x * z)));
	} else {
		tmp = (1.0 / x) / y;
	}
	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 <= (-1.0d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = 1.0d0 / (z * (y * (x * z)))
    else
        tmp = (1.0d0 / x) / y
    end if
    code = tmp
end function

Java

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

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z <= -1.0) or not (z <= 1.0):
		tmp = 1.0 / (z * (y * (x * z)))
	else:
		tmp = (1.0 / x) / y
	return tmp

Julia

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

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.0) || ~((z <= 1.0)))
		tmp = 1.0 / (z * (y * (x * z)));
	else
		tmp = (1.0 / x) / y;
	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[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(1.0 / N[(z * N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 80.6%

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

      1. associate-/r* 80.5%

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

      2. +-commutative 80.5%

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

      3. fma-def 80.5%

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

   3. Simplified 80.5%

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

   4. Taylor expanded in z around inf 79.7%

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

      1. unpow2 79.7%

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

      2. *-commutative 79.7%

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

      3. associate-*l* 80.1%

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

      4. *-commutative 80.1%

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

      5. associate-*l* 87.4%

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

   6. Simplified 87.4%

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

   7. Taylor expanded in z around 0 91.1%

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

   if -1 < z < 1

   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* 98.6%

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

      2. +-commutative 98.6%

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

      3. fma-def 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in z around 0 98.3%

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

      1. metadata-eval 98.3%

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

      2. frac-times 99.3%

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

   6. Applied egg-rr 99.3%

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

      1. associate-*l/ 99.4%

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

      2. *-un-lft-identity 99.4%

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

   8. Applied egg-rr 99.4%

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

4. Final simplification 95.1%

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

Alternative 7: 96.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.1 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{z}}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.1) (not (<= z 1.0)))
   (/ (/ (/ 1.0 y) z) (* x z))
   (/ (/ 1.0 x) y)))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.1) || !(z <= 1.0)) {
		tmp = ((1.0 / y) / z) / (x * z);
	} else {
		tmp = (1.0 / x) / y;
	}
	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 <= (-1.1d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = ((1.0d0 / y) / z) / (x * z)
    else
        tmp = (1.0d0 / x) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.1) || !(z <= 1.0)) {
		tmp = ((1.0 / y) / z) / (x * z);
	} else {
		tmp = (1.0 / x) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z <= -1.1) or not (z <= 1.0):
		tmp = ((1.0 / y) / z) / (x * z)
	else:
		tmp = (1.0 / x) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.1) || !(z <= 1.0))
		tmp = Float64(Float64(Float64(1.0 / y) / z) / Float64(x * z));
	else
		tmp = Float64(Float64(1.0 / x) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.1) || ~((z <= 1.0)))
		tmp = ((1.0 / y) / z) / (x * z);
	else
		tmp = (1.0 / x) / y;
	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[Or[LessEqual[z, -1.1], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(N[(N[(1.0 / y), $MachinePrecision] / z), $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.1000000000000001 or 1 < z

   1. Initial program 80.6%

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

      1. associate-/r* 80.5%

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

      2. +-commutative 80.5%

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

      3. fma-def 80.5%

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

   3. Simplified 80.5%

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

   4. Taylor expanded in z around inf 79.7%

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

      1. unpow2 79.7%

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

      2. *-commutative 79.7%

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

      3. associate-*l* 80.1%

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

      4. *-commutative 80.1%

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

      5. associate-*l* 87.4%

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

   6. Simplified 87.4%

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

   7. Taylor expanded in z around 0 91.1%

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

      1. associate-/r* 91.8%

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

      2. *-un-lft-identity 91.8%

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

      3. associate-*r* 93.9%

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

      4. *-commutative 93.9%

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

      5. *-commutative 93.9%

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

      6. times-frac 89.6%

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

      7. *-commutative 89.6%

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

   9. Applied egg-rr 89.6%

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

       1. associate-/r* 89.6%

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

       2. frac-times 96.2%

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

       3. *-un-lft-identity 96.2%

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

       4. associate-/l/ 96.1%

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

       5. associate-/r* 96.3%

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

       6. *-commutative 96.3%

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

   11. Applied egg-rr 96.3%

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

   if -1.1000000000000001 < z < 1

   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* 98.6%

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

      2. +-commutative 98.6%

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

      3. fma-def 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in z around 0 98.3%

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

      1. metadata-eval 98.3%

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

      2. frac-times 99.3%

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

   6. Applied egg-rr 99.3%

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

      1. associate-*l/ 99.4%

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

      2. *-un-lft-identity 99.4%

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

   8. Applied egg-rr 99.4%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{z}}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \end{array} \]

Alternative 8: 95.3% accurate, 0.8× speedup

Math

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

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z -1.0)
   (/ (/ 1.0 x) (* z (* z y)))
   (if (<= z 1.0) (/ (/ 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 <= -1.0) {
		tmp = (1.0 / x) / (z * (z * y));
	} else if (z <= 1.0) {
		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 <= (-1.0d0)) then
        tmp = (1.0d0 / x) / (z * (z * y))
    else if (z <= 1.0d0) 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 <= -1.0) {
		tmp = (1.0 / x) / (z * (z * y));
	} else if (z <= 1.0) {
		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 <= -1.0:
		tmp = (1.0 / x) / (z * (z * y))
	elif z <= 1.0:
		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 (z <= -1.0)
		tmp = Float64(Float64(1.0 / x) / Float64(z * Float64(z * y)));
	elseif (z <= 1.0)
		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 <= -1.0)
		tmp = (1.0 / x) / (z * (z * y));
	elseif (z <= 1.0)
		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[z, -1.0], N[(N[(1.0 / x), $MachinePrecision] / N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.0], 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 3 regimes

2. if z < -1

   1. Initial program 77.3%

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

      1. associate-/r* 77.2%

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

      2. +-commutative 77.2%

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

      3. fma-def 77.2%

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

   3. Simplified 77.2%

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

   4. Taylor expanded in z around inf 77.2%

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

      1. unpow2 77.2%

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

      2. *-commutative 77.2%

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

      3. associate-*l* 79.9%

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

      4. *-commutative 79.9%

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

      5. associate-*l* 87.6%

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

   6. Simplified 87.6%

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

   7. Taylor expanded in z around 0 89.3%

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

      1. associate-/r* 90.6%

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

      2. *-un-lft-identity 90.6%

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

      3. associate-*r* 93.5%

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

      4. *-commutative 93.5%

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

      5. *-commutative 93.5%

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

      6. times-frac 92.2%

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

      7. *-commutative 92.2%

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

   9. Applied egg-rr 92.2%

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

       1. associate-/l/ 92.3%

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

       2. un-div-inv 92.4%

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

       3. *-commutative 92.4%

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

       4. *-commutative 92.4%

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

   11. Applied egg-rr 92.4%

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

   if -1 < z < 1

   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* 98.6%

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

      2. +-commutative 98.6%

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

      3. fma-def 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in z around 0 98.3%

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

      1. metadata-eval 98.3%

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

      2. frac-times 99.3%

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

   6. Applied egg-rr 99.3%

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

      1. associate-*l/ 99.4%

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

      2. *-un-lft-identity 99.4%

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

   8. Applied egg-rr 99.4%

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

   if 1 < 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* 83.5%

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

      2. +-commutative 83.5%

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

      3. fma-def 83.5%

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

   3. Simplified 83.5%

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

   4. Taylor expanded in z around inf 81.9%

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

      1. unpow2 81.9%

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

      2. *-commutative 81.9%

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

      3. associate-*l* 80.3%

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

      4. *-commutative 80.3%

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

      5. associate-*l* 87.2%

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

   6. Simplified 87.2%

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

   7. Taylor expanded in z around 0 92.8%

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

4. Final simplification 95.9%

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

Alternative 9: 96.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = ((1.0 / z) / x) / (z * y);
	} else if (z <= 1.0) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = ((1.0 / y) / z) / (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 <= (-1.0d0)) then
        tmp = ((1.0d0 / z) / x) / (z * y)
    else if (z <= 1.0d0) then
        tmp = (1.0d0 / x) / y
    else
        tmp = ((1.0d0 / y) / z) / (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 <= -1.0) {
		tmp = ((1.0 / z) / x) / (z * y);
	} else if (z <= 1.0) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = ((1.0 / y) / z) / (x * z);
	}
	return tmp;
}

Python

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

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(Float64(Float64(1.0 / z) / x) / Float64(z * y));
	elseif (z <= 1.0)
		tmp = Float64(Float64(1.0 / x) / y);
	else
		tmp = Float64(Float64(Float64(1.0 / y) / z) / 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 <= -1.0)
		tmp = ((1.0 / z) / x) / (z * y);
	elseif (z <= 1.0)
		tmp = (1.0 / x) / y;
	else
		tmp = ((1.0 / y) / z) / (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[z, -1.0], N[(N[(N[(1.0 / z), $MachinePrecision] / x), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.0], N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision], N[(N[(N[(1.0 / y), $MachinePrecision] / z), $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 77.3%

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

      1. associate-/r* 77.2%

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

      2. +-commutative 77.2%

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

      3. fma-def 77.2%

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

   3. Simplified 77.2%

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

   4. Taylor expanded in z around inf 77.2%

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

      1. unpow2 77.2%

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

      2. *-commutative 77.2%

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

      3. associate-*l* 79.9%

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

      4. *-commutative 79.9%

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

      5. associate-*l* 87.6%

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

   6. Simplified 87.6%

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

      1. expm1-log1p-u 84.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{z \cdot \left(z \cdot \left(y \cdot x\right)\right)}\right)\right)} \]

      2. expm1-udef 63.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{z \cdot \left(z \cdot \left(y \cdot x\right)\right)}\right)} - 1} \]

      3. associate-/r* 63.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{1}{z}}{z \cdot \left(y \cdot x\right)}}\right)} - 1 \]

      4. associate-*r* 66.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{1}{z}}{\color{blue}{\left(z \cdot y\right) \cdot x}}\right)} - 1 \]

      5. *-commutative 66.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{1}{z}}{\color{blue}{x \cdot \left(z \cdot y\right)}}\right)} - 1 \]

   8. Applied egg-rr 66.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{1}{z}}{x \cdot \left(z \cdot y\right)}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 86.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{z}}{x \cdot \left(z \cdot y\right)}\right)\right)} \]

      2. expm1-log1p 93.5%

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

      3. associate-/r* 94.4%

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

      4. *-commutative 94.4%

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

   10. Simplified 94.4%

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

   if -1 < z < 1

   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* 98.6%

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

      2. +-commutative 98.6%

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

      3. fma-def 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in z around 0 98.3%

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

      1. metadata-eval 98.3%

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

      2. frac-times 99.3%

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

   6. Applied egg-rr 99.3%

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

      1. associate-*l/ 99.4%

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

      2. *-un-lft-identity 99.4%

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

   8. Applied egg-rr 99.4%

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

   if 1 < 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* 83.5%

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

      2. +-commutative 83.5%

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

      3. fma-def 83.5%

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

   3. Simplified 83.5%

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

   4. Taylor expanded in z around inf 81.9%

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

      1. unpow2 81.9%

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

      2. *-commutative 81.9%

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

      3. associate-*l* 80.3%

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

      4. *-commutative 80.3%

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

      5. associate-*l* 87.2%

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

   6. Simplified 87.2%

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

   7. Taylor expanded in z around 0 92.8%

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

      1. associate-/r* 92.9%

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

      2. *-un-lft-identity 92.9%

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

      3. associate-*r* 94.2%

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

      4. *-commutative 94.2%

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

      5. *-commutative 94.2%

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

      6. times-frac 87.3%

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

      7. *-commutative 87.3%

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

   9. Applied egg-rr 87.3%

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

       1. associate-/r* 87.3%

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

       2. frac-times 98.0%

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

       3. *-un-lft-identity 98.0%

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

       4. associate-/l/ 97.9%

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

       5. associate-/r* 98.0%

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

       6. *-commutative 98.0%

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

   11. Applied egg-rr 98.0%

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

4. Final simplification 97.8%

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

Alternative 10: 57.9% accurate, 2.2× speedup

Math

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

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(1.0 / Float64(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[(1.0 / N[(x * y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.8%

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

   1. associate-/r* 89.3%

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

   2. +-commutative 89.3%

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

   3. fma-def 89.3%

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

3. Simplified 89.3%

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

4. Taylor expanded in z around 0 57.8%

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

5. Final simplification 57.8%

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

Alternative 11: 57.9% 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 89.8%

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

   1. associate-/r* 89.3%

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

   2. +-commutative 89.3%

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

   3. fma-def 89.3%

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

3. Simplified 89.3%

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

4. Taylor expanded in z around 0 57.8%

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

   1. metadata-eval 57.8%

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

   2. frac-times 57.9%

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

6. Applied egg-rr 57.9%

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

   1. associate-*l/ 57.9%

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

   2. *-un-lft-identity 57.9%

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

8. Applied egg-rr 57.9%

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

9. Final simplification 57.9%

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

Developer target: 92.1% 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 2023182 
(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)))))