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

Percentage Accurate: 90.5% → 97.4%

Time: 8.4s

Alternatives: 8

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: 90.5% 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: 97.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y \cdot \left(1 + z \cdot z\right)\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{+303}:\\ \;\;\;\;\frac{\frac{1}{x}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{y \cdot \left(x \cdot z\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (+ 1.0 (* z z)))))
   (if (<= t_0 5e+303) (/ (/ 1.0 x) t_0) (/ (/ 1.0 z) (* y (* x z))))))

C

assert(x < y);
double code(double x, double y, double z) {
	double t_0 = y * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 5e+303) {
		tmp = (1.0 / x) / t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = y * (1.0d0 + (z * z))
    if (t_0 <= 5d+303) then
        tmp = (1.0d0 / x) / t_0
    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 t_0 = y * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 5e+303) {
		tmp = (1.0 / x) / t_0;
	} else {
		tmp = (1.0 / z) / (y * (x * z));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	t_0 = y * (1.0 + (z * z))
	tmp = 0
	if t_0 <= 5e+303:
		tmp = (1.0 / x) / t_0
	else:
		tmp = (1.0 / z) / (y * (x * z))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	t_0 = Float64(y * Float64(1.0 + Float64(z * z)))
	tmp = 0.0
	if (t_0 <= 5e+303)
		tmp = Float64(Float64(1.0 / x) / t_0);
	else
		tmp = Float64(Float64(1.0 / z) / Float64(y * Float64(x * z)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	t_0 = y * (1.0 + (z * z));
	tmp = 0.0;
	if (t_0 <= 5e+303)
		tmp = (1.0 / x) / t_0;
	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_] := Block[{t$95$0 = N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e+303], N[(N[(1.0 / x), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] / N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 1 (*.f64 z z))) < 4.9999999999999997e303

   1. Initial program 95.8%

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

   if 4.9999999999999997e303 < (*.f64 y (+.f64 1 (*.f64 z z)))

   1. Initial program 78.2%

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

      1. associate-/r* 78.1%

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

      2. +-commutative 78.1%

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

      3. fma-def 78.1%

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

   3. Simplified 78.1%

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

      1. fma-udef 78.1%

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

      2. +-commutative 78.1%

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

      3. associate-/r* 78.2%

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

      4. associate-/r* 78.1%

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

      5. div-inv 78.1%

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

      6. add-sqr-sqrt 78.1%

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

      7. times-frac 78.1%

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

      8. hypot-1-def 78.1%

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

      9. hypot-1-def 98.8%

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

   5. Applied egg-rr 98.8%

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

   6. Taylor expanded in z around inf 78.1%

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

      1. unpow2 78.1%

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

   8. Simplified 78.1%

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

      1. associate-/r* 78.1%

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

      2. *-un-lft-identity 78.1%

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

      3. associate-*l* 86.3%

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

      4. times-frac 99.7%

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

   10. Applied egg-rr 99.7%

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

       1. associate-/l/ 99.9%

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

       2. un-div-inv 99.9%

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

       3. *-commutative 99.9%

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

   12. Applied egg-rr 99.9%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 5 \cdot 10^{+303}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{y \cdot \left(x \cdot z\right)}\\ \end{array} \]

Alternative 2: 97.8% 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 93.3%

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

   1. associate-/r* 92.5%

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

   2. +-commutative 92.5%

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

   3. fma-def 92.5%

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

3. Simplified 92.5%

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

   1. fma-udef 92.5%

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

   2. +-commutative 92.5%

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

   3. associate-/r* 93.3%

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

   4. associate-/r* 91.5%

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

   5. div-inv 91.4%

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

   6. add-sqr-sqrt 91.4%

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

   7. times-frac 92.5%

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

   8. hypot-1-def 92.5%

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

   9. hypot-1-def 97.6%

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

5. Applied egg-rr 97.6%

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

6. Final simplification 97.6%

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

Alternative 3: 96.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+301}:\\ \;\;\;\;\frac{1}{x \cdot \left(y + y \cdot \left(z \cdot z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{y \cdot \left(x \cdot z\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e+301)
   (/ 1.0 (* x (+ y (* y (* z z)))))
   (/ (/ 1.0 z) (* y (* x z)))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+301) {
		tmp = 1.0 / (x * (y + (y * (z * z))));
	} 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) <= 2d+301) then
        tmp = 1.0d0 / (x * (y + (y * (z * z))))
    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) <= 2e+301) {
		tmp = 1.0 / (x * (y + (y * (z * z))));
	} 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) <= 2e+301:
		tmp = 1.0 / (x * (y + (y * (z * z))))
	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) <= 2e+301)
		tmp = Float64(1.0 / Float64(x * Float64(y + Float64(y * Float64(z * z)))));
	else
		tmp = Float64(Float64(1.0 / 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) <= 2e+301)
		tmp = 1.0 / (x * (y + (y * (z * z))));
	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], 2e+301], N[(1.0 / N[(x * N[(y + N[(y * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] / N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.00000000000000011e301

   1. Initial program 98.2%

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

      1. associate-/r* 97.2%

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

      2. +-commutative 97.2%

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

      3. fma-def 97.2%

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

   3. Simplified 97.2%

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

      1. fma-udef 97.2%

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

      2. distribute-lft-in 97.2%

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

      3. *-rgt-identity 97.2%

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

   5. Applied egg-rr 97.2%

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

   if 2.00000000000000011e301 < (*.f64 z z)

   1. Initial program 76.6%

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

      1. associate-/r* 76.6%

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

      2. +-commutative 76.6%

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

      3. fma-def 76.6%

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

   3. Simplified 76.6%

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

      1. fma-udef 76.6%

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

      2. +-commutative 76.6%

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

      3. associate-/r* 76.6%

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

      4. associate-/r* 76.3%

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

      5. div-inv 76.3%

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

      6. add-sqr-sqrt 76.3%

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

      7. times-frac 76.6%

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

      8. hypot-1-def 76.6%

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

      9. hypot-1-def 99.2%

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

   5. Applied egg-rr 99.2%

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

   6. Taylor expanded in z around inf 76.6%

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

      1. unpow2 76.6%

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

   8. Simplified 76.6%

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

      1. associate-/r* 76.6%

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

      2. *-un-lft-identity 76.6%

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

      3. associate-*l* 89.9%

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

      4. times-frac 98.1%

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

   10. Applied egg-rr 98.1%

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

       1. associate-/l/ 98.2%

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

       2. un-div-inv 99.8%

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

       3. *-commutative 99.8%

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

   12. Applied egg-rr 99.8%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+301}:\\ \;\;\;\;\frac{1}{x \cdot \left(y + y \cdot \left(z \cdot z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{y \cdot \left(x \cdot z\right)}\\ \end{array} \]

Alternative 4: 93.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.2:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(z \cdot \left(z \cdot y\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 0.2) (/ (/ 1.0 x) y) (/ 1.0 (* x (* z (* z y))))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 0.2) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = 1.0 / (x * (z * (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) <= 0.2d0) then
        tmp = (1.0d0 / x) / y
    else
        tmp = 1.0d0 / (x * (z * (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) <= 0.2) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = 1.0 / (x * (z * (z * y)));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 0.2:
		tmp = (1.0 / x) / y
	else:
		tmp = 1.0 / (x * (z * (z * y)))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 0.2)
		tmp = Float64(Float64(1.0 / x) / y);
	else
		tmp = Float64(1.0 / Float64(x * Float64(z * 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) <= 0.2)
		tmp = (1.0 / x) / y;
	else
		tmp = 1.0 / (x * (z * (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], 0.2], N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision], N[(1.0 / N[(x * N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 0.20000000000000001

   1. Initial program 99.6%

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

      1. associate-/r* 98.4%

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

      2. +-commutative 98.4%

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

      3. fma-def 98.4%

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

   3. Simplified 98.4%

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

      1. fma-udef 98.4%

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

      2. +-commutative 98.4%

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

      3. associate-/r* 99.6%

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

      4. associate-/r* 99.0%

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

      5. div-inv 98.9%

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

      6. add-sqr-sqrt 98.9%

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

      7. times-frac 99.5%

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

      8. hypot-1-def 99.5%

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

      9. hypot-1-def 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in x around 0 98.3%

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

      1. associate-/r* 99.7%

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

      2. *-commutative 99.7%

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

      3. unpow2 99.7%

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

      4. fma-udef 99.7%

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

   8. Simplified 99.7%

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

      1. fma-udef 99.7%

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

      2. distribute-rgt-in 99.7%

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

      3. associate-*l* 99.7%

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

      4. *-un-lft-identity 99.7%

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

   10. Applied egg-rr 99.7%

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

   11. Taylor expanded in z around 0 96.7%

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

       1. associate-/l/ 97.8%

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

   13. Simplified 97.8%

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

   if 0.20000000000000001 < (*.f64 z z)

   1. Initial program 87.0%

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

      1. associate-/r* 86.7%

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

      2. +-commutative 86.7%

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

      3. fma-def 86.7%

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

   3. Simplified 86.7%

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

   4. Taylor expanded in z around inf 85.5%

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

      1. unpow2 85.5%

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

      2. *-commutative 85.5%

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

      3. associate-*r* 92.2%

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

   6. Simplified 92.2%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.2:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(z \cdot \left(z \cdot y\right)\right)}\\ \end{array} \]

Alternative 5: 96.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.2:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot z\right) \cdot \left(z \cdot y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 0.2) (/ (/ 1.0 x) y) (/ 1.0 (* (* x z) (* z y)))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 0.2) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = 1.0 / ((x * z) * (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) <= 0.2d0) then
        tmp = (1.0d0 / x) / y
    else
        tmp = 1.0d0 / ((x * z) * (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) <= 0.2) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = 1.0 / ((x * z) * (z * y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 0.2:
		tmp = (1.0 / x) / y
	else:
		tmp = 1.0 / ((x * z) * (z * y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 0.2)
		tmp = Float64(Float64(1.0 / x) / y);
	else
		tmp = Float64(1.0 / Float64(Float64(x * z) * 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) <= 0.2)
		tmp = (1.0 / x) / y;
	else
		tmp = 1.0 / ((x * z) * (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], 0.2], N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision], N[(1.0 / N[(N[(x * z), $MachinePrecision] * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 0.20000000000000001

   1. Initial program 99.6%

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

      1. associate-/r* 98.4%

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

      2. +-commutative 98.4%

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

      3. fma-def 98.4%

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

   3. Simplified 98.4%

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

      1. fma-udef 98.4%

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

      2. +-commutative 98.4%

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

      3. associate-/r* 99.6%

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

      4. associate-/r* 99.0%

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

      5. div-inv 98.9%

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

      6. add-sqr-sqrt 98.9%

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

      7. times-frac 99.5%

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

      8. hypot-1-def 99.5%

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

      9. hypot-1-def 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in x around 0 98.3%

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

      1. associate-/r* 99.7%

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

      2. *-commutative 99.7%

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

      3. unpow2 99.7%

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

      4. fma-udef 99.7%

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

   8. Simplified 99.7%

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

      1. fma-udef 99.7%

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

      2. distribute-rgt-in 99.7%

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

      3. associate-*l* 99.7%

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

      4. *-un-lft-identity 99.7%

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

   10. Applied egg-rr 99.7%

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

   11. Taylor expanded in z around 0 96.7%

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

       1. associate-/l/ 97.8%

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

   13. Simplified 97.8%

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

   if 0.20000000000000001 < (*.f64 z z)

   1. Initial program 87.0%

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

      1. associate-/r* 86.7%

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

      2. +-commutative 86.7%

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

      3. fma-def 86.7%

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

   3. Simplified 86.7%

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

      1. fma-udef 86.7%

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

      2. +-commutative 86.7%

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

      3. associate-/r* 87.0%

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

      4. associate-/r* 83.9%

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

      5. div-inv 83.9%

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

      6. add-sqr-sqrt 83.9%

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

      7. times-frac 85.4%

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

      8. hypot-1-def 85.4%

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

      9. hypot-1-def 95.7%

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

   5. Applied egg-rr 95.7%

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

   6. Taylor expanded in z around inf 83.3%

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

      1. unpow2 83.3%

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

   8. Simplified 83.3%

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

      1. associate-/r* 83.5%

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

      2. *-un-lft-identity 83.5%

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

      3. associate-*l* 89.6%

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

      4. times-frac 92.6%

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

   10. Applied egg-rr 92.6%

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

       1. associate-*r/ 94.5%

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

       2. frac-times 94.0%

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

       3. metadata-eval 94.0%

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

       4. *-commutative 94.0%

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

       5. associate-/r* 93.7%

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

       6. *-commutative 93.7%

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

   12. Applied egg-rr 93.7%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.2:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot z\right) \cdot \left(z \cdot y\right)}\\ \end{array} \]

Alternative 6: 97.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.2:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{y \cdot \left(x \cdot z\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 0.2) (/ (/ 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) <= 0.2) {
		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) <= 0.2d0) 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) <= 0.2) {
		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) <= 0.2:
		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) <= 0.2)
		tmp = Float64(Float64(1.0 / x) / y);
	else
		tmp = Float64(Float64(1.0 / 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) <= 0.2)
		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], 0.2], N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] / N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 0.20000000000000001

   1. Initial program 99.6%

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

      1. associate-/r* 98.4%

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

      2. +-commutative 98.4%

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

      3. fma-def 98.4%

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

   3. Simplified 98.4%

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

      1. fma-udef 98.4%

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

      2. +-commutative 98.4%

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

      3. associate-/r* 99.6%

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

      4. associate-/r* 99.0%

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

      5. div-inv 98.9%

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

      6. add-sqr-sqrt 98.9%

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

      7. times-frac 99.5%

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

      8. hypot-1-def 99.5%

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

      9. hypot-1-def 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in x around 0 98.3%

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

      1. associate-/r* 99.7%

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

      2. *-commutative 99.7%

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

      3. unpow2 99.7%

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

      4. fma-udef 99.7%

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

   8. Simplified 99.7%

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

      1. fma-udef 99.7%

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

      2. distribute-rgt-in 99.7%

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

      3. associate-*l* 99.7%

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

      4. *-un-lft-identity 99.7%

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

   10. Applied egg-rr 99.7%

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

   11. Taylor expanded in z around 0 96.7%

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

       1. associate-/l/ 97.8%

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

   13. Simplified 97.8%

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

   if 0.20000000000000001 < (*.f64 z z)

   1. Initial program 87.0%

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

      1. associate-/r* 86.7%

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

      2. +-commutative 86.7%

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

      3. fma-def 86.7%

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

   3. Simplified 86.7%

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

      1. fma-udef 86.7%

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

      2. +-commutative 86.7%

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

      3. associate-/r* 87.0%

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

      4. associate-/r* 83.9%

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

      5. div-inv 83.9%

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

      6. add-sqr-sqrt 83.9%

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

      7. times-frac 85.4%

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

      8. hypot-1-def 85.4%

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

      9. hypot-1-def 95.7%

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

   5. Applied egg-rr 95.7%

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

   6. Taylor expanded in z around inf 83.3%

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

      1. unpow2 83.3%

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

   8. Simplified 83.3%

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

      1. associate-/r* 83.5%

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

      2. *-un-lft-identity 83.5%

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

      3. associate-*l* 89.6%

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

      4. times-frac 92.6%

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

   10. Applied egg-rr 92.6%

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

       1. associate-/l/ 92.6%

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

       2. un-div-inv 94.9%

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

       3. *-commutative 94.9%

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

   12. Applied egg-rr 94.9%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.2:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{y \cdot \left(x \cdot z\right)}\\ \end{array} \]

Alternative 7: 58.8% 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 93.3%

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

   1. associate-/r* 92.5%

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

   2. +-commutative 92.5%

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

   3. fma-def 92.5%

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

3. Simplified 92.5%

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

4. Taylor expanded in z around 0 58.0%

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

5. Final simplification 58.0%

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

Alternative 8: 58.8% 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 93.3%

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

   1. associate-/r* 92.5%

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

   2. +-commutative 92.5%

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

   3. fma-def 92.5%

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

3. Simplified 92.5%

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

   1. fma-udef 92.5%

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

   2. +-commutative 92.5%

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

   3. associate-/r* 93.3%

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

   4. associate-/r* 91.5%

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

   5. div-inv 91.4%

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

   6. add-sqr-sqrt 91.4%

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

   7. times-frac 92.5%

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

   8. hypot-1-def 92.5%

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

   9. hypot-1-def 97.6%

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

5. Applied egg-rr 97.6%

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

6. Taylor expanded in x around 0 91.4%

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

   1. associate-/r* 92.2%

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

   2. *-commutative 92.2%

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

   3. unpow2 92.2%

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

   4. fma-udef 92.2%

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

8. Simplified 92.2%

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

   1. fma-udef 92.2%

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

   2. distribute-rgt-in 92.2%

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

   3. associate-*l* 95.3%

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

   4. *-un-lft-identity 95.3%

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

10. Applied egg-rr 95.3%

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

11. Taylor expanded in z around 0 58.0%

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

    1. associate-/l/ 58.4%

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

13. Simplified 58.4%

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

14. Final simplification 58.4%

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

Developer target: 92.6% 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 2023214 
(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)))))