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

Percentage Accurate: 90.3% → 97.6%

Time: 7.3s

Alternatives: 7

Speedup: 0.9×

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: 90.3% 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: 97.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+284}:\\ \;\;\;\;\frac{\frac{\frac{-1}{x}}{y}}{-\mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(y \cdot z\right) \cdot x\right) \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 4e+284)
   (/ (/ (/ -1.0 x) y) (- (fma z z 1.0)))
   (/ 1.0 (* (* (* y z) x) z))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 4e+284) {
		tmp = ((-1.0 / x) / y) / -fma(z, z, 1.0);
	} else {
		tmp = 1.0 / (((y * z) * x) * z);
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 4e+284)
		tmp = Float64(Float64(Float64(-1.0 / x) / y) / Float64(-fma(z, z, 1.0)));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(y * z) * x) * z));
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 4e+284], N[(N[(N[(-1.0 / x), $MachinePrecision] / y), $MachinePrecision] / (-N[(z * z + 1.0), $MachinePrecision])), $MachinePrecision], N[(1.0 / N[(N[(N[(y * z), $MachinePrecision] * x), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.00000000000000032e284

   1. Initial program 95.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. frac-2neg N/A

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

      5. lower-/.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. lower-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lower-neg.f64 99.1

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 99.1

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

   4. Applied rewrites 99.1%

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

   if 4.00000000000000032e284 < (*.f64 z z)

   1. Initial program 72.1%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 72.1

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

   5. Applied rewrites 72.1%

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

   7. Applied rewrites 71.5%

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

   9. Applied rewrites 97.7%

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

Alternative 2: 97.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+258}:\\ \;\;\;\;\frac{1}{\left(\mathsf{fma}\left(z, z, 1\right) \cdot x\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x \cdot z}}{y \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e+258)
   (/ 1.0 (* (* (fma z z 1.0) x) y))
   (/ (/ 1.0 (* x z)) (* y z))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+258) {
		tmp = 1.0 / ((fma(z, z, 1.0) * x) * y);
	} else {
		tmp = (1.0 / (x * z)) / (y * z);
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+258)
		tmp = Float64(1.0 / Float64(Float64(fma(z, z, 1.0) * x) * y));
	else
		tmp = Float64(Float64(1.0 / Float64(x * z)) / Float64(y * z));
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+258], N[(1.0 / N[(N[(N[(z * z + 1.0), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(x * z), $MachinePrecision]), $MachinePrecision] / N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.00000000000000011e258

   1. Initial program 95.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. frac-2neg N/A

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

      5. lower-/.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. lower-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lower-neg.f64 99.1

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 99.1

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

   4. Applied rewrites 99.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. neg-mul-1 N/A

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

      8. lift-neg.f64 N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. frac-2neg N/A

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

      15. clear-num N/A

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

      16. lower-/.f64 N/A

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

      17. div-inv N/A

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

   6. Applied rewrites 94.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 97.1

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 97.1

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

   8. Applied rewrites 97.1%

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

   if 2.00000000000000011e258 < (*.f64 z z)

   1. Initial program 74.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 74.5

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

   5. Applied rewrites 74.5%

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

   7. Applied rewrites 97.8%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+258}:\\ \;\;\;\;\frac{1}{\left(\mathsf{fma}\left(z, z, 1\right) \cdot x\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x \cdot z}}{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+284}:\\ \;\;\;\;\frac{1}{\left(\mathsf{fma}\left(z, z, 1\right) \cdot x\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(y \cdot z\right) \cdot x\right) \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 4e+284)
   (/ 1.0 (* (* (fma z z 1.0) x) y))
   (/ 1.0 (* (* (* y z) x) z))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 4e+284) {
		tmp = 1.0 / ((fma(z, z, 1.0) * x) * y);
	} else {
		tmp = 1.0 / (((y * z) * x) * z);
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 4e+284)
		tmp = Float64(1.0 / Float64(Float64(fma(z, z, 1.0) * x) * y));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(y * z) * x) * z));
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 4e+284], N[(1.0 / N[(N[(N[(z * z + 1.0), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(y * z), $MachinePrecision] * x), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.00000000000000032e284

   1. Initial program 95.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. frac-2neg N/A

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

      5. lower-/.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. lower-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lower-neg.f64 99.1

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 99.1

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

   4. Applied rewrites 99.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. neg-mul-1 N/A

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

      8. lift-neg.f64 N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. frac-2neg N/A

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

      15. clear-num N/A

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

      16. lower-/.f64 N/A

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

      17. div-inv N/A

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

   6. Applied rewrites 94.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 97.2

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 97.2

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

   8. Applied rewrites 97.2%

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

   if 4.00000000000000032e284 < (*.f64 z z)

   1. Initial program 72.1%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 72.1

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

   5. Applied rewrites 72.1%

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

   7. Applied rewrites 71.5%

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

   9. Applied rewrites 97.7%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+284}:\\ \;\;\;\;\frac{1}{\left(\mathsf{fma}\left(z, z, 1\right) \cdot x\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(y \cdot z\right) \cdot x\right) \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{-1}{y}}{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(y \cdot z\right) \cdot \left(x \cdot z\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 5e-8) (/ (/ -1.0 y) (- x)) (/ 1.0 (* (* y z) (* x z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e-8) {
		tmp = (-1.0 / y) / -x;
	} else {
		tmp = 1.0 / ((y * z) * (x * z));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z 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-8) then
        tmp = ((-1.0d0) / y) / -x
    else
        tmp = 1.0d0 / ((y * z) * (x * z))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e-8) {
		tmp = (-1.0 / y) / -x;
	} else {
		tmp = 1.0 / ((y * z) * (x * z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 5e-8:
		tmp = (-1.0 / y) / -x
	else:
		tmp = 1.0 / ((y * z) * (x * z))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e-8)
		tmp = Float64(Float64(-1.0 / y) / Float64(-x));
	else
		tmp = Float64(1.0 / Float64(Float64(y * z) * Float64(x * z)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 5e-8)
		tmp = (-1.0 / y) / -x;
	else
		tmp = 1.0 / ((y * z) * (x * z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e-8], N[(N[(-1.0 / y), $MachinePrecision] / (-x)), $MachinePrecision], N[(1.0 / N[(N[(y * z), $MachinePrecision] * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.9999999999999998e-8

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 99.1

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

   5. Applied rewrites 99.1%

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

   7. Applied rewrites 99.1%

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

   if 4.9999999999999998e-8 < (*.f64 z z)

   1. Initial program 78.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 75.8

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

   5. Applied rewrites 75.8%

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

   7. Applied rewrites 94.0%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{-1}{y}}{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(y \cdot z\right) \cdot \left(x \cdot z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 58.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x, y, and z 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 && y < z;
public static double code(double x, double y, double z) {
	return (-1.0 / y) / -x;
}

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x, y, and z 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 89.2%

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

3. Taylor expanded in z around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f64 N/A

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

   3. lower-/.f64 59.5

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

5. Applied rewrites 59.5%

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

7. Applied rewrites 59.5%

   \[\leadsto \frac{\frac{-1}{y}}{\color{blue}{-x}} \]
8. Add Preprocessing

Alternative 6: 58.3% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x, y, and z 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 && y < z;
public static double code(double x, double y, double z) {
	return (1.0 / x) / y;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.2%

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

3. Taylor expanded in z around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f64 N/A

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

   3. lower-/.f64 59.5

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

5. Applied rewrites 59.5%

   \[\leadsto \color{blue}{\frac{\frac{1}{x}}{y}} \]
6. Add Preprocessing

Alternative 7: 58.3% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x, y, and z 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 && y < z;
public static double code(double x, double y, double z) {
	return 1.0 / (y * x);
}

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x, y, and z 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 89.2%

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

3. Taylor expanded in z around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f64 N/A

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

   3. lower-/.f64 59.5

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

5. Applied rewrites 59.5%

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

7. Applied rewrites 59.2%

   \[\leadsto \frac{1}{\color{blue}{y \cdot x}} \]
8. Add Preprocessing

Developer Target 1: 92.3% accurate, 0.5× 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 2024296 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (* y (+ 1 (* z z))) -inf.0) (/ (/ 1 y) (* (+ 1 (* z z)) x)) (if (< (* y (+ 1 (* z z))) 868074325056725200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (/ 1 x) (* (+ 1 (* z z)) y)) (/ (/ 1 y) (* (+ 1 (* z z)) x)))))

  (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))