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

Percentage Accurate: 91.3% → 97.9%

Time: 12.3s

Alternatives: 10

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: 91.3% 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: 97.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+41}:\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{x}}{\mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{x \cdot \left(z \cdot y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 4e+41)
   (/ (/ (/ 1.0 y) x) (fma z z 1.0))
   (/ (/ 1.0 z) (* x (* z y)))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 4e+41) {
		tmp = ((1.0 / y) / x) / fma(z, z, 1.0);
	} 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) <= 4e+41)
		tmp = Float64(Float64(Float64(1.0 / y) / x) / fma(z, z, 1.0));
	else
		tmp = Float64(Float64(1.0 / z) / Float64(x * Float64(z * y)));
	end
	return 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], 4e+41], N[(N[(N[(1.0 / y), $MachinePrecision] / x), $MachinePrecision] / N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] / N[(x * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.00000000000000002e41

   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* 99.6%

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

      2. associate-/r* 98.6%

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

      3. sqr-neg 98.6%

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

      4. +-commutative 98.6%

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

      5. sqr-neg 98.6%

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

      6. fma-def 98.6%

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

   3. Simplified 98.6%

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

      1. add-log-exp 26.8%

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

      2. *-un-lft-identity 26.8%

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

      3. log-prod 26.8%

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

      4. metadata-eval 26.8%

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

      5. add-log-exp 98.6%

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

      6. *-commutative 98.6%

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

      7. associate-/r* 99.7%

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

   5. Applied egg-rr 99.7%

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

   if 4.00000000000000002e41 < (*.f64 z z)

   1. Initial program 80.5%

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

      1. associate-/r* 80.4%

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

      2. *-commutative 80.4%

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

      3. sqr-neg 80.4%

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

      4. +-commutative 80.4%

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

      5. distribute-lft1-in 80.4%

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

      6. *-commutative 80.4%

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

      7. fma-def 80.4%

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

      8. sqr-neg 80.4%

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

   3. Simplified 80.4%

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

   4. Taylor expanded in z around inf 80.4%

      \[\leadsto \frac{1}{\color{blue}{x \cdot \left(y \cdot {z}^{2}\right)}} \]
   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* 81.0%

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

      3. *-commutative 81.0%

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

   6. Simplified 81.0%

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

      1. associate-/r* 80.1%

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

      2. associate-/r* 80.2%

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

      3. div-inv 80.1%

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

      4. times-frac 95.2%

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

   8. Applied egg-rr 95.2%

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

      1. frac-times 80.1%

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

      2. div-inv 80.2%

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

      3. associate-/l/ 80.1%

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

      4. associate-/l/ 81.0%

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

      5. associate-*r* 91.7%

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

      6. associate-*l* 95.1%

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

      7. associate-/r* 95.1%

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

      8. *-commutative 95.1%

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

      9. associate-*l* 97.0%

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

      10. *-commutative 97.0%

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

   10. Applied egg-rr 97.0%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+41}:\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{x}}{\mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{x \cdot \left(z \cdot y\right)}\\ \end{array} \]

Alternative 2: 94.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.004:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+293}:\\ \;\;\;\;\frac{1}{x \cdot \left(\left(z \cdot z\right) \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(z \cdot \left(z \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 0.004)
   (/ (/ 1.0 y) x)
   (if (<= (* z z) 5e+293)
     (/ 1.0 (* x (* (* z z) y)))
     (/ 1.0 (* y (* z (* z x)))))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 0.004) {
		tmp = (1.0 / y) / x;
	} else if ((z * z) <= 5e+293) {
		tmp = 1.0 / (x * ((z * z) * y));
	} else {
		tmp = 1.0 / (y * (z * (z * x)));
	}
	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) <= 0.004d0) then
        tmp = (1.0d0 / y) / x
    else if ((z * z) <= 5d+293) then
        tmp = 1.0d0 / (x * ((z * z) * y))
    else
        tmp = 1.0d0 / (y * (z * (z * x)))
    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) <= 0.004) {
		tmp = (1.0 / y) / x;
	} else if ((z * z) <= 5e+293) {
		tmp = 1.0 / (x * ((z * z) * y));
	} else {
		tmp = 1.0 / (y * (z * (z * x)));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 0.004:
		tmp = (1.0 / y) / x
	elif (z * z) <= 5e+293:
		tmp = 1.0 / (x * ((z * z) * y))
	else:
		tmp = 1.0 / (y * (z * (z * x)))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 0.004)
		tmp = Float64(Float64(1.0 / y) / x);
	elseif (Float64(z * z) <= 5e+293)
		tmp = Float64(1.0 / Float64(x * Float64(Float64(z * z) * y)));
	else
		tmp = Float64(1.0 / Float64(y * Float64(z * Float64(z * x))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 0.004)
		tmp = (1.0 / y) / x;
	elseif ((z * z) <= 5e+293)
		tmp = 1.0 / (x * ((z * z) * y));
	else
		tmp = 1.0 / (y * (z * (z * x)));
	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], 0.004], N[(N[(1.0 / y), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 5e+293], N[(1.0 / N[(x * N[(N[(z * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(y * N[(z * N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 0.0040000000000000001

   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* 98.5%

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

      2. *-commutative 98.5%

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

      3. sqr-neg 98.5%

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

      4. +-commutative 98.5%

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

      5. distribute-lft1-in 98.5%

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

      6. *-commutative 98.5%

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

      7. fma-def 98.5%

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

      8. sqr-neg 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in z around 0 97.3%

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

      1. *-commutative 97.3%

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

   6. Simplified 97.3%

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

      1. associate-/r* 98.5%

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

      2. div-inv 98.4%

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

   8. Applied egg-rr 98.4%

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

      1. div-inv 98.5%

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

   10. Applied egg-rr 98.5%

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

   if 0.0040000000000000001 < (*.f64 z z) < 5.00000000000000033e293

   1. Initial program 96.5%

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

      1. associate-/r* 95.9%

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

      2. *-commutative 95.9%

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

      3. sqr-neg 95.9%

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

      4. +-commutative 95.9%

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

      5. distribute-lft1-in 95.9%

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

      6. *-commutative 95.9%

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

      7. fma-def 95.9%

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

      8. sqr-neg 95.9%

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

   3. Simplified 95.9%

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

   4. Taylor expanded in z around inf 92.9%

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

      1. unpow2 92.9%

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

   6. Simplified 92.9%

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

   if 5.00000000000000033e293 < (*.f64 z z)

   1. Initial program 68.6%

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

      1. associate-/r* 68.6%

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

      2. *-commutative 68.6%

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

      3. sqr-neg 68.6%

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

      4. +-commutative 68.6%

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

      5. distribute-lft1-in 68.6%

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

      6. *-commutative 68.6%

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

      7. fma-def 68.6%

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

      8. sqr-neg 68.6%

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

   3. Simplified 68.6%

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

   4. Taylor expanded in z around inf 68.6%

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

      1. *-commutative 68.6%

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

      2. unpow2 68.6%

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

      3. associate-*l* 68.6%

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

      4. associate-*l* 88.3%

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

   6. Simplified 88.3%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.004:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+293}:\\ \;\;\;\;\frac{1}{x \cdot \left(\left(z \cdot z\right) \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(z \cdot \left(z \cdot x\right)\right)}\\ \end{array} \]

Alternative 3: 97.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 1000000:\\ \;\;\;\;\frac{1}{x \cdot \left(y + \left(z \cdot z\right) \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{x \cdot \left(z \cdot y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 1000000.0)
   (/ 1.0 (* x (+ y (* (* z z) y))))
   (/ (/ 1.0 z) (* x (* z y)))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1000000.0) {
		tmp = 1.0 / (x * (y + ((z * z) * y)));
	} 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) <= 1000000.0d0) then
        tmp = 1.0d0 / (x * (y + ((z * z) * y)))
    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) <= 1000000.0) {
		tmp = 1.0 / (x * (y + ((z * z) * y)));
	} 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) <= 1000000.0:
		tmp = 1.0 / (x * (y + ((z * z) * y)))
	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) <= 1000000.0)
		tmp = Float64(1.0 / Float64(x * Float64(y + Float64(Float64(z * z) * y))));
	else
		tmp = Float64(Float64(1.0 / z) / Float64(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) <= 1000000.0)
		tmp = 1.0 / (x * (y + ((z * z) * y)));
	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], 1000000.0], N[(1.0 / N[(x * N[(y + N[(N[(z * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] / N[(x * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1e6

   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* 98.6%

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

      2. *-commutative 98.6%

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

      3. sqr-neg 98.6%

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

      4. +-commutative 98.6%

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

      5. distribute-lft1-in 98.6%

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

      6. *-commutative 98.6%

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

      7. fma-def 98.6%

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

      8. sqr-neg 98.6%

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

   3. Simplified 98.6%

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

      1. fma-udef 98.6%

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

   5. Applied egg-rr 98.6%

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

   if 1e6 < (*.f64 z z)

   1. Initial program 82.3%

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

      1. associate-/r* 82.0%

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

      2. *-commutative 82.0%

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

      3. sqr-neg 82.0%

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

      4. +-commutative 82.0%

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

      5. distribute-lft1-in 82.0%

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

      6. *-commutative 82.0%

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

      7. fma-def 82.0%

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

      8. sqr-neg 82.0%

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

   3. Simplified 82.0%

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

   4. Taylor expanded in z around inf 82.0%

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

      1. unpow2 82.0%

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

      2. associate-*r* 82.6%

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

      3. *-commutative 82.6%

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

   6. Simplified 82.6%

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

      1. associate-/r* 82.0%

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

      2. associate-/r* 82.1%

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

      3. div-inv 82.0%

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

      4. times-frac 95.6%

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

   8. Applied egg-rr 95.6%

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

      1. frac-times 82.0%

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

      2. div-inv 82.1%

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

      3. associate-/l/ 82.0%

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

      4. associate-/l/ 82.6%

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

      5. associate-*r* 92.2%

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

      6. associate-*l* 95.3%

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

      7. associate-/r* 95.6%

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

      8. *-commutative 95.6%

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

      9. associate-*l* 97.3%

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

      10. *-commutative 97.3%

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

   10. Applied egg-rr 97.3%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 1000000:\\ \;\;\;\;\frac{1}{x \cdot \left(y + \left(z \cdot z\right) \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{x \cdot \left(z \cdot y\right)}\\ \end{array} \]

Alternative 4: 97.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+35}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(z \cdot z + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{x \cdot \left(z \cdot y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 5e+35)
   (/ (/ 1.0 x) (* y (+ (* z z) 1.0)))
   (/ (/ 1.0 z) (* x (* z y)))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e+35) {
		tmp = (1.0 / x) / (y * ((z * z) + 1.0));
	} 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) <= 5d+35) then
        tmp = (1.0d0 / x) / (y * ((z * z) + 1.0d0))
    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) <= 5e+35) {
		tmp = (1.0 / x) / (y * ((z * z) + 1.0));
	} 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) <= 5e+35:
		tmp = (1.0 / x) / (y * ((z * z) + 1.0))
	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) <= 5e+35)
		tmp = Float64(Float64(1.0 / x) / Float64(y * Float64(Float64(z * z) + 1.0)));
	else
		tmp = Float64(Float64(1.0 / z) / Float64(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) <= 5e+35)
		tmp = (1.0 / x) / (y * ((z * z) + 1.0));
	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], 5e+35], N[(N[(1.0 / x), $MachinePrecision] / N[(y * N[(N[(z * z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] / N[(x * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.00000000000000021e35

   1. Initial program 99.6%

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

   if 5.00000000000000021e35 < (*.f64 z z)

   1. Initial program 81.2%

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

      1. associate-/r* 81.1%

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

      2. *-commutative 81.1%

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

      3. sqr-neg 81.1%

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

      4. +-commutative 81.1%

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

      5. distribute-lft1-in 81.1%

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

      6. *-commutative 81.1%

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

      7. fma-def 81.1%

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

      8. sqr-neg 81.1%

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

   3. Simplified 81.1%

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

   4. Taylor expanded in z around inf 81.1%

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

      1. unpow2 81.1%

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

      2. associate-*r* 81.7%

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

      3. *-commutative 81.7%

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

   6. Simplified 81.7%

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

      1. associate-/r* 80.9%

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

      2. associate-/r* 80.9%

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

      3. div-inv 80.9%

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

      4. times-frac 95.3%

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

   8. Applied egg-rr 95.3%

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

      1. frac-times 80.9%

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

      2. div-inv 80.9%

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

      3. associate-/l/ 80.9%

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

      4. associate-/l/ 81.7%

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

      5. associate-*r* 92.0%

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

      6. associate-*l* 95.3%

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

      7. associate-/r* 95.3%

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

      8. *-commutative 95.3%

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

      9. associate-*l* 97.1%

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

      10. *-commutative 97.1%

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

   10. Applied egg-rr 97.1%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+35}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(z \cdot z + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{x \cdot \left(z \cdot y\right)}\\ \end{array} \]

Alternative 5: 97.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.004:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(x \cdot \left(z \cdot y\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 0.004) (/ (/ 1.0 y) x) (/ 1.0 (* z (* x (* z y))))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 0.004) {
		tmp = (1.0 / y) / x;
	} 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) <= 0.004d0) then
        tmp = (1.0d0 / y) / x
    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) <= 0.004) {
		tmp = (1.0 / y) / x;
	} 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) <= 0.004:
		tmp = (1.0 / y) / x
	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) <= 0.004)
		tmp = Float64(Float64(1.0 / y) / x);
	else
		tmp = Float64(1.0 / Float64(z * Float64(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) <= 0.004)
		tmp = (1.0 / y) / x;
	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], 0.004], N[(N[(1.0 / y), $MachinePrecision] / x), $MachinePrecision], N[(1.0 / N[(z * N[(x * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 0.0040000000000000001

   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* 98.5%

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

      2. *-commutative 98.5%

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

      3. sqr-neg 98.5%

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

      4. +-commutative 98.5%

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

      5. distribute-lft1-in 98.5%

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

      6. *-commutative 98.5%

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

      7. fma-def 98.5%

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

      8. sqr-neg 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in z around 0 97.3%

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

      1. *-commutative 97.3%

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

   6. Simplified 97.3%

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

      1. associate-/r* 98.5%

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

      2. div-inv 98.4%

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

   8. Applied egg-rr 98.4%

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

      1. div-inv 98.5%

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

   10. Applied egg-rr 98.5%

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

   if 0.0040000000000000001 < (*.f64 z z)

   1. Initial program 82.8%

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

      1. associate-/r* 82.4%

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

      2. *-commutative 82.4%

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

      3. sqr-neg 82.4%

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

      4. +-commutative 82.4%

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

      5. distribute-lft1-in 82.4%

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

      6. *-commutative 82.4%

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

      7. fma-def 82.4%

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

      8. sqr-neg 82.4%

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

   3. Simplified 82.4%

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

   4. Taylor expanded in z around inf 80.9%

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

      1. unpow2 80.9%

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

      2. associate-*r* 81.5%

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

      3. *-commutative 81.5%

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

   6. Simplified 81.5%

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

      1. associate-/r* 80.9%

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

      2. associate-/r* 81.0%

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

      3. *-un-lft-identity 81.0%

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

      4. times-frac 90.0%

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

      5. associate-/l/ 90.0%

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

   8. Applied egg-rr 90.0%

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

      1. clear-num 90.0%

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

      2. frac-times 89.5%

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

      3. metadata-eval 89.5%

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

      4. associate-/r/ 90.9%

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

      5. /-rgt-identity 90.9%

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

      6. associate-*r* 93.9%

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

   10. Applied egg-rr 93.9%

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

   11. Taylor expanded in z around 0 95.4%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.004:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(x \cdot \left(z \cdot y\right)\right)}\\ \end{array} \]

Alternative 6: 97.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.004:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{x \cdot \left(z \cdot y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 0.004) (/ (/ 1.0 y) x) (/ (/ 1.0 z) (* x (* z y)))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 0.004) {
		tmp = (1.0 / y) / x;
	} 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) <= 0.004d0) then
        tmp = (1.0d0 / y) / x
    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) <= 0.004) {
		tmp = (1.0 / y) / x;
	} 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) <= 0.004:
		tmp = (1.0 / y) / x
	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) <= 0.004)
		tmp = Float64(Float64(1.0 / y) / x);
	else
		tmp = Float64(Float64(1.0 / z) / Float64(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) <= 0.004)
		tmp = (1.0 / y) / x;
	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], 0.004], N[(N[(1.0 / y), $MachinePrecision] / x), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] / N[(x * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 0.0040000000000000001

   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* 98.5%

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

      2. *-commutative 98.5%

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

      3. sqr-neg 98.5%

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

      4. +-commutative 98.5%

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

      5. distribute-lft1-in 98.5%

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

      6. *-commutative 98.5%

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

      7. fma-def 98.5%

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

      8. sqr-neg 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in z around 0 97.3%

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

      1. *-commutative 97.3%

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

   6. Simplified 97.3%

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

      1. associate-/r* 98.5%

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

      2. div-inv 98.4%

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

   8. Applied egg-rr 98.4%

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

      1. div-inv 98.5%

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

   10. Applied egg-rr 98.5%

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

   if 0.0040000000000000001 < (*.f64 z z)

   1. Initial program 82.8%

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

      1. associate-/r* 82.4%

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

      2. *-commutative 82.4%

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

      3. sqr-neg 82.4%

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

      4. +-commutative 82.4%

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

      5. distribute-lft1-in 82.4%

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

      6. *-commutative 82.4%

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

      7. fma-def 82.4%

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

      8. sqr-neg 82.4%

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

   3. Simplified 82.4%

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

   4. Taylor expanded in z around inf 80.9%

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

      1. unpow2 80.9%

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

      2. associate-*r* 81.5%

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

      3. *-commutative 81.5%

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

   6. Simplified 81.5%

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

      1. associate-/r* 80.9%

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

      2. associate-/r* 81.0%

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

      3. div-inv 81.0%

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

      4. times-frac 94.2%

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

   8. Applied egg-rr 94.2%

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

      1. frac-times 81.0%

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

      2. div-inv 81.0%

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

      3. associate-/l/ 80.9%

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

      4. associate-/l/ 81.5%

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

      5. associate-*r* 90.9%

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

      6. associate-*l* 93.9%

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

      7. associate-/r* 94.1%

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

      8. *-commutative 94.1%

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

      9. associate-*l* 95.8%

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

      10. *-commutative 95.8%

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

   10. Applied egg-rr 95.8%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.004:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{x \cdot \left(z \cdot y\right)}\\ \end{array} \]

Alternative 7: 74.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 94.9%

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

      1. associate-/r* 94.0%

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

      2. *-commutative 94.0%

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

      3. sqr-neg 94.0%

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

      4. +-commutative 94.0%

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

      5. distribute-lft1-in 94.0%

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

      6. *-commutative 94.0%

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

      7. fma-def 94.0%

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

      8. sqr-neg 94.0%

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

   3. Simplified 94.0%

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

   4. Taylor expanded in z around 0 73.1%

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

      1. *-commutative 73.1%

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

   6. Simplified 73.1%

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

      1. associate-/r* 73.8%

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

      2. div-inv 73.7%

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

   8. Applied egg-rr 73.7%

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

      1. div-inv 73.8%

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

   10. Applied egg-rr 73.8%

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

   if 1 < z

   1. Initial program 81.2%

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

      1. associate-/r* 81.2%

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

      2. *-commutative 81.2%

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

      3. sqr-neg 81.2%

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

      4. +-commutative 81.2%

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

      5. distribute-lft1-in 81.2%

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

      6. *-commutative 81.2%

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

      7. fma-def 81.2%

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

      8. sqr-neg 81.2%

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

   3. Simplified 81.2%

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

   4. Taylor expanded in z around inf 78.1%

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

      1. unpow2 78.1%

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

   6. Simplified 78.1%

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

4. Final simplification 74.7%

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

Alternative 8: 58.9% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.8%

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

   1. associate-/r* 91.1%

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

   2. *-commutative 91.1%

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

   3. sqr-neg 91.1%

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

   4. +-commutative 91.1%

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

   5. distribute-lft1-in 91.1%

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

   6. *-commutative 91.1%

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

   7. fma-def 91.1%

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

   8. sqr-neg 91.1%

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

3. Simplified 91.1%

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

4. Taylor expanded in z around 0 61.0%

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

   1. *-commutative 61.0%

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

6. Simplified 61.0%

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

7. Final simplification 61.0%

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

Alternative 9: 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 91.8%

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

   1. distribute-lft-in 91.9%

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

   2. *-rgt-identity 91.9%

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

   3. +-commutative 91.9%

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

   4. associate-*r* 97.0%

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

   5. fma-def 97.0%

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

3. Applied egg-rr 97.0%

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

4. Taylor expanded in z around 0 61.0%

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

   1. associate-/r* 61.2%

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

6. Simplified 61.2%

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

7. Final simplification 61.2%

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

Alternative 10: 58.8% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (1.0d0 / y) / x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.8%

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

   1. associate-/r* 91.1%

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

   2. *-commutative 91.1%

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

   3. sqr-neg 91.1%

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

   4. +-commutative 91.1%

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

   5. distribute-lft1-in 91.1%

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

   6. *-commutative 91.1%

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

   7. fma-def 91.1%

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

   8. sqr-neg 91.1%

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

3. Simplified 91.1%

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

4. Taylor expanded in z around 0 61.0%

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

   1. *-commutative 61.0%

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

6. Simplified 61.0%

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

   1. associate-/r* 61.3%

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

   2. div-inv 61.2%

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

8. Applied egg-rr 61.2%

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

   1. div-inv 61.3%

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

10. Applied egg-rr 61.3%

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

11. Final simplification 61.3%

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

Developer target: 93.4% 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 2023275 
(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)))))