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

Percentage Accurate: 90.8% → 98.1%

Time: 11.7s

Alternatives: 9

Speedup: 0.6×

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: 90.8% 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: 98.1% accurate, 0.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y \cdot \left(1 + z \cdot z\right)\\ \mathbf{if}\;t_0 \leq 10^{-210}:\\ \;\;\;\;\frac{\frac{1}{x}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{y}^{-0.5}}{\mathsf{hypot}\left(1, z\right)}}{x \cdot \left(\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (+ 1.0 (* z z)))))
   (if (<= t_0 1e-210)
     (/ (/ 1.0 x) t_0)
     (/ (/ (pow y -0.5) (hypot 1.0 z)) (* x (* (hypot 1.0 z) (sqrt y)))))))

C

assert(x < y);
double code(double x, double y, double z) {
	double t_0 = y * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 1e-210) {
		tmp = (1.0 / x) / t_0;
	} else {
		tmp = (pow(y, -0.5) / hypot(1.0, z)) / (x * (hypot(1.0, z) * sqrt(y)));
	}
	return tmp;
}

Java

assert x < y;
public static double code(double x, double y, double z) {
	double t_0 = y * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 1e-210) {
		tmp = (1.0 / x) / t_0;
	} else {
		tmp = (Math.pow(y, -0.5) / Math.hypot(1.0, z)) / (x * (Math.hypot(1.0, z) * Math.sqrt(y)));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	t_0 = y * (1.0 + (z * z))
	tmp = 0
	if t_0 <= 1e-210:
		tmp = (1.0 / x) / t_0
	else:
		tmp = (math.pow(y, -0.5) / math.hypot(1.0, z)) / (x * (math.hypot(1.0, z) * math.sqrt(y)))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	t_0 = Float64(y * Float64(1.0 + Float64(z * z)))
	tmp = 0.0
	if (t_0 <= 1e-210)
		tmp = Float64(Float64(1.0 / x) / t_0);
	else
		tmp = Float64(Float64((y ^ -0.5) / hypot(1.0, z)) / Float64(x * Float64(hypot(1.0, z) * sqrt(y))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	t_0 = y * (1.0 + (z * z));
	tmp = 0.0;
	if (t_0 <= 1e-210)
		tmp = (1.0 / x) / t_0;
	else
		tmp = ((y ^ -0.5) / hypot(1.0, z)) / (x * (hypot(1.0, z) * sqrt(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_] := Block[{t$95$0 = N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-210], N[(N[(1.0 / x), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[Power[y, -0.5], $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision] / N[(x * N[(N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 1 (*.f64 z z))) < 1e-210

   1. Initial program 91.4%

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

   if 1e-210 < (*.f64 y (+.f64 1 (*.f64 z z)))

   1. Initial program 86.5%

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

      1. *-un-lft-identity 86.5%

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

      2. times-frac 88.0%

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

      3. associate-/r* 87.9%

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

      4. +-commutative 87.9%

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

      5. fma-udef 87.9%

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

      6. *-commutative 87.9%

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

      7. fma-udef 87.9%

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

      8. +-commutative 87.9%

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

      9. *-commutative 87.9%

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

      10. +-commutative 87.9%

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

      11. fma-udef 87.9%

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

   3. Applied egg-rr 87.9%

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

      1. associate-*l/ 88.0%

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

      2. *-commutative 88.0%

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

      3. times-frac 86.4%

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

      4. associate-/l/ 86.5%

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

      5. metadata-eval 86.5%

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

      6. fma-udef 86.5%

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

      7. unpow2 86.5%

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

      8. +-commutative 86.5%

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

      9. unpow2 86.5%

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

      10. metadata-eval 86.5%

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

      11. rem-square-sqrt 86.4%

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

      12. hypot-udef 86.5%

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

      13. hypot-udef 86.5%

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

      14. add-sqr-sqrt 86.2%

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

      15. swap-sqr 91.4%

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

      16. *-commutative 91.4%

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

      17. *-commutative 91.4%

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

      18. frac-times 91.5%

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

   5. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\frac{{y}^{-0.5}}{\mathsf{hypot}\left(1, z\right)}}{x \cdot \left(\sqrt{y} \cdot \mathsf{hypot}\left(1, z\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 10^{-210}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{y}^{-0.5}}{\mathsf{hypot}\left(1, z\right)}}{x \cdot \left(\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}\right)}\\ \end{array} \]

Alternative 2: 93.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y \cdot \left(1 + z \cdot z\right)\\ \mathbf{if}\;t_0 \leq 10^{+301}:\\ \;\;\;\;\frac{\frac{1}{x}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{1}{y}} \cdot \frac{1}{z}\right) \cdot \frac{\frac{1}{x}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (+ 1.0 (* z z)))))
   (if (<= t_0 1e+301)
     (/ (/ 1.0 x) t_0)
     (*
      (* (sqrt (/ 1.0 y)) (/ 1.0 z))
      (/ (/ 1.0 x) (* (hypot 1.0 z) (sqrt y)))))))

C

assert(x < y);
double code(double x, double y, double z) {
	double t_0 = y * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 1e+301) {
		tmp = (1.0 / x) / t_0;
	} else {
		tmp = (sqrt((1.0 / y)) * (1.0 / z)) * ((1.0 / x) / (hypot(1.0, z) * sqrt(y)));
	}
	return tmp;
}

Java

assert x < y;
public static double code(double x, double y, double z) {
	double t_0 = y * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 1e+301) {
		tmp = (1.0 / x) / t_0;
	} else {
		tmp = (Math.sqrt((1.0 / y)) * (1.0 / z)) * ((1.0 / x) / (Math.hypot(1.0, z) * Math.sqrt(y)));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	t_0 = y * (1.0 + (z * z))
	tmp = 0
	if t_0 <= 1e+301:
		tmp = (1.0 / x) / t_0
	else:
		tmp = (math.sqrt((1.0 / y)) * (1.0 / z)) * ((1.0 / x) / (math.hypot(1.0, z) * math.sqrt(y)))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	t_0 = Float64(y * Float64(1.0 + Float64(z * z)))
	tmp = 0.0
	if (t_0 <= 1e+301)
		tmp = Float64(Float64(1.0 / x) / t_0);
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 / y)) * Float64(1.0 / z)) * Float64(Float64(1.0 / x) / Float64(hypot(1.0, z) * sqrt(y))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	t_0 = y * (1.0 + (z * z));
	tmp = 0.0;
	if (t_0 <= 1e+301)
		tmp = (1.0 / x) / t_0;
	else
		tmp = (sqrt((1.0 / y)) * (1.0 / z)) * ((1.0 / x) / (hypot(1.0, z) * sqrt(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_] := Block[{t$95$0 = N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e+301], N[(N[(1.0 / x), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] / N[(N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 1 (*.f64 z z))) < 1.00000000000000005e301

   1. Initial program 94.6%

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

   if 1.00000000000000005e301 < (*.f64 y (+.f64 1 (*.f64 z z)))

   1. Initial program 63.0%

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

      1. *-un-lft-identity 63.0%

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

      2. add-sqr-sqrt 63.0%

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

      3. times-frac 63.0%

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

      4. sqrt-prod 63.0%

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

      5. hypot-1-def 63.0%

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

      6. sqrt-prod 73.2%

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

      7. hypot-1-def 99.5%

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

   3. Applied egg-rr 99.5%

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

   4. Taylor expanded in z around inf 89.4%

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

4. Final simplification 93.7%

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

Alternative 3: 93.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y \cdot \left(1 + z \cdot z\right)\\ \mathbf{if}\;t_0 \leq 10^{+301}:\\ \;\;\;\;\frac{\frac{1}{x}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}} \cdot \frac{\sqrt{\frac{1}{y}}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (+ 1.0 (* z z)))))
   (if (<= t_0 1e+301)
     (/ (/ 1.0 x) t_0)
     (* (/ (/ 1.0 x) (* (hypot 1.0 z) (sqrt y))) (/ (sqrt (/ 1.0 y)) z)))))

C

assert(x < y);
double code(double x, double y, double z) {
	double t_0 = y * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 1e+301) {
		tmp = (1.0 / x) / t_0;
	} else {
		tmp = ((1.0 / x) / (hypot(1.0, z) * sqrt(y))) * (sqrt((1.0 / y)) / z);
	}
	return tmp;
}

Java

assert x < y;
public static double code(double x, double y, double z) {
	double t_0 = y * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 1e+301) {
		tmp = (1.0 / x) / t_0;
	} else {
		tmp = ((1.0 / x) / (Math.hypot(1.0, z) * Math.sqrt(y))) * (Math.sqrt((1.0 / y)) / z);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	t_0 = y * (1.0 + (z * z))
	tmp = 0
	if t_0 <= 1e+301:
		tmp = (1.0 / x) / t_0
	else:
		tmp = ((1.0 / x) / (math.hypot(1.0, z) * math.sqrt(y))) * (math.sqrt((1.0 / y)) / z)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	t_0 = Float64(y * Float64(1.0 + Float64(z * z)))
	tmp = 0.0
	if (t_0 <= 1e+301)
		tmp = Float64(Float64(1.0 / x) / t_0);
	else
		tmp = Float64(Float64(Float64(1.0 / x) / Float64(hypot(1.0, z) * sqrt(y))) * Float64(sqrt(Float64(1.0 / y)) / z));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	t_0 = y * (1.0 + (z * z));
	tmp = 0.0;
	if (t_0 <= 1e+301)
		tmp = (1.0 / x) / t_0;
	else
		tmp = ((1.0 / x) / (hypot(1.0, z) * sqrt(y))) * (sqrt((1.0 / y)) / z);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e+301], N[(N[(1.0 / x), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(1.0 / x), $MachinePrecision] / N[(N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 1 (*.f64 z z))) < 1.00000000000000005e301

   1. Initial program 94.6%

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

   if 1.00000000000000005e301 < (*.f64 y (+.f64 1 (*.f64 z z)))

   1. Initial program 63.0%

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

      1. *-un-lft-identity 63.0%

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

      2. add-sqr-sqrt 63.0%

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

      3. times-frac 63.0%

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

      4. sqrt-prod 63.0%

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

      5. hypot-1-def 63.0%

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

      6. sqrt-prod 73.2%

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

      7. hypot-1-def 99.5%

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

   3. Applied egg-rr 99.5%

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

   4. Taylor expanded in z around inf 89.4%

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

      1. associate-*r/ 89.4%

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

      2. *-rgt-identity 89.4%

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

   6. Simplified 89.4%

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

4. Final simplification 93.7%

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

Alternative 4: 97.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y \cdot \left(1 + z \cdot z\right)\\ \mathbf{if}\;t_0 \leq 10^{+301}:\\ \;\;\;\;\frac{\frac{1}{x}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z \cdot x}}{\sqrt{y}} \cdot \frac{\frac{1}{z}}{\sqrt{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (+ 1.0 (* z z)))))
   (if (<= t_0 1e+301)
     (/ (/ 1.0 x) t_0)
     (* (/ (/ 1.0 (* z x)) (sqrt y)) (/ (/ 1.0 z) (sqrt y))))))

C

assert(x < y);
double code(double x, double y, double z) {
	double t_0 = y * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 1e+301) {
		tmp = (1.0 / x) / t_0;
	} else {
		tmp = ((1.0 / (z * x)) / sqrt(y)) * ((1.0 / z) / sqrt(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) :: t_0
    real(8) :: tmp
    t_0 = y * (1.0d0 + (z * z))
    if (t_0 <= 1d+301) then
        tmp = (1.0d0 / x) / t_0
    else
        tmp = ((1.0d0 / (z * x)) / sqrt(y)) * ((1.0d0 / z) / sqrt(y))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double t_0 = y * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 1e+301) {
		tmp = (1.0 / x) / t_0;
	} else {
		tmp = ((1.0 / (z * x)) / Math.sqrt(y)) * ((1.0 / z) / Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	t_0 = y * (1.0 + (z * z))
	tmp = 0
	if t_0 <= 1e+301:
		tmp = (1.0 / x) / t_0
	else:
		tmp = ((1.0 / (z * x)) / math.sqrt(y)) * ((1.0 / z) / math.sqrt(y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	t_0 = Float64(y * Float64(1.0 + Float64(z * z)))
	tmp = 0.0
	if (t_0 <= 1e+301)
		tmp = Float64(Float64(1.0 / x) / t_0);
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(z * x)) / sqrt(y)) * Float64(Float64(1.0 / z) / sqrt(y)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	t_0 = y * (1.0 + (z * z));
	tmp = 0.0;
	if (t_0 <= 1e+301)
		tmp = (1.0 / x) / t_0;
	else
		tmp = ((1.0 / (z * x)) / sqrt(y)) * ((1.0 / z) / sqrt(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_] := Block[{t$95$0 = N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e+301], N[(N[(1.0 / x), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(1.0 / N[(z * x), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / z), $MachinePrecision] / N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 1 (*.f64 z z))) < 1.00000000000000005e301

   1. Initial program 94.6%

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

   if 1.00000000000000005e301 < (*.f64 y (+.f64 1 (*.f64 z z)))

   1. Initial program 63.0%

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

      1. associate-/r* 63.0%

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

      2. associate-/l/ 63.0%

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

      3. associate-/r* 63.0%

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

      4. associate-/l/ 73.2%

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

      5. sqr-neg 73.2%

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

      6. +-commutative 73.2%

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

      7. sqr-neg 73.2%

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

      8. fma-def 73.2%

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

   3. Simplified 73.2%

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

   4. Taylor expanded in z around inf 73.2%

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

      1. *-commutative 73.2%

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

   6. Simplified 73.2%

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

      1. add-sqr-sqrt 39.2%

         \[\leadsto \frac{\frac{1}{y}}{\color{blue}{\sqrt{{z}^{2} \cdot x} \cdot \sqrt{{z}^{2} \cdot x}}} \]

      2. pow2 39.2%

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

      3. sqrt-prod 39.2%

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

      4. unpow2 39.2%

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

      5. sqrt-prod 23.8%

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

      6. add-sqr-sqrt 43.4%

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

   8. Applied egg-rr 43.4%

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

      1. associate-/l/ 43.4%

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

      2. associate-/r* 43.3%

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

      3. metadata-eval 43.3%

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

      4. unpow2 43.3%

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

      5. *-commutative 43.3%

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

      6. associate-*r* 43.3%

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

      7. associate-*r* 43.4%

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

      8. add-sqr-sqrt 87.7%

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

      9. frac-times 88.2%

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

      10. associate-/l/ 88.2%

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

      11. add-sqr-sqrt 88.2%

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

      12. times-frac 99.5%

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

      13. associate-/l/ 99.5%

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

   10. Applied egg-rr 99.5%

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

4. Final simplification 95.5%

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

Alternative 5: 97.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y \cdot \left(1 + z \cdot z\right)\\ \mathbf{if}\;t_0 \leq 10^{+301}:\\ \;\;\;\;\frac{\frac{1}{x}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y \cdot \left(z \cdot x\right)}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (+ 1.0 (* z z)))))
   (if (<= t_0 1e+301) (/ (/ 1.0 x) t_0) (/ (/ 1.0 (* y (* z x))) z))))

C

assert(x < y);
double code(double x, double y, double z) {
	double t_0 = y * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 1e+301) {
		tmp = (1.0 / x) / t_0;
	} else {
		tmp = (1.0 / (y * (z * x))) / z;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (1.0d0 + (z * z))
    if (t_0 <= 1d+301) then
        tmp = (1.0d0 / x) / t_0
    else
        tmp = (1.0d0 / (y * (z * x))) / z
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double t_0 = y * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 1e+301) {
		tmp = (1.0 / x) / t_0;
	} else {
		tmp = (1.0 / (y * (z * x))) / z;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	t_0 = y * (1.0 + (z * z))
	tmp = 0
	if t_0 <= 1e+301:
		tmp = (1.0 / x) / t_0
	else:
		tmp = (1.0 / (y * (z * x))) / z
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	t_0 = Float64(y * Float64(1.0 + Float64(z * z)))
	tmp = 0.0
	if (t_0 <= 1e+301)
		tmp = Float64(Float64(1.0 / x) / t_0);
	else
		tmp = Float64(Float64(1.0 / Float64(y * Float64(z * x))) / z);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	t_0 = y * (1.0 + (z * z));
	tmp = 0.0;
	if (t_0 <= 1e+301)
		tmp = (1.0 / x) / t_0;
	else
		tmp = (1.0 / (y * (z * x))) / z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e+301], N[(N[(1.0 / x), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(1.0 / N[(y * N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 1 (*.f64 z z))) < 1.00000000000000005e301

   1. Initial program 94.6%

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

   if 1.00000000000000005e301 < (*.f64 y (+.f64 1 (*.f64 z z)))

   1. Initial program 63.0%

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

      1. *-un-lft-identity 63.0%

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

      2. times-frac 73.2%

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

      3. associate-/r* 73.2%

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

      4. +-commutative 73.2%

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

      5. fma-udef 73.2%

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

      6. *-commutative 73.2%

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

      7. fma-udef 73.2%

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

      8. +-commutative 73.2%

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

      9. *-commutative 73.2%

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

      10. +-commutative 73.2%

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

      11. fma-udef 73.2%

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

   3. Applied egg-rr 73.2%

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

   4. Taylor expanded in z around inf 73.2%

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

      1. associate-/r* 73.2%

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

   6. Simplified 73.2%

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

      1. *-un-lft-identity 73.2%

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

      2. unpow2 73.2%

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

      3. times-frac 88.2%

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

   8. Applied egg-rr 88.2%

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

      1. associate-*l* 97.6%

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

      2. associate-*l/ 97.7%

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

      3. *-un-lft-identity 97.7%

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

      4. associate-/l/ 97.8%

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

      5. frac-times 97.8%

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

      6. metadata-eval 97.8%

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

   10. Applied egg-rr 97.8%

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

4. Final simplification 95.2%

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

Alternative 6: 78.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 1:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{+162}:\\ \;\;\;\;\frac{\frac{\frac{1}{y \cdot x}}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z \cdot x}}{y \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z 1.0)
   (/ (/ 1.0 y) x)
   (if (<= z 2.6e+162)
     (/ (/ (/ 1.0 (* y x)) z) z)
     (/ (/ 1.0 (* z x)) (* y z)))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.0) {
		tmp = (1.0 / y) / x;
	} else if (z <= 2.6e+162) {
		tmp = ((1.0 / (y * x)) / z) / z;
	} else {
		tmp = (1.0 / (z * x)) / (y * z);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 1.0d0) then
        tmp = (1.0d0 / y) / x
    else if (z <= 2.6d+162) then
        tmp = ((1.0d0 / (y * x)) / z) / z
    else
        tmp = (1.0d0 / (z * x)) / (y * z)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.0) {
		tmp = (1.0 / y) / x;
	} else if (z <= 2.6e+162) {
		tmp = ((1.0 / (y * x)) / z) / z;
	} else {
		tmp = (1.0 / (z * x)) / (y * z);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if z <= 1.0:
		tmp = (1.0 / y) / x
	elif z <= 2.6e+162:
		tmp = ((1.0 / (y * x)) / z) / z
	else:
		tmp = (1.0 / (z * x)) / (y * z)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (z <= 1.0)
		tmp = Float64(Float64(1.0 / y) / x);
	elseif (z <= 2.6e+162)
		tmp = Float64(Float64(Float64(1.0 / Float64(y * x)) / z) / z);
	else
		tmp = Float64(Float64(1.0 / Float64(z * x)) / Float64(y * z));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 1.0)
		tmp = (1.0 / y) / x;
	elseif (z <= 2.6e+162)
		tmp = ((1.0 / (y * x)) / z) / z;
	else
		tmp = (1.0 / (z * x)) / (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[z, 1.0], N[(N[(1.0 / y), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[z, 2.6e+162], N[(N[(N[(1.0 / N[(y * x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / z), $MachinePrecision], N[(N[(1.0 / N[(z * x), $MachinePrecision]), $MachinePrecision] / N[(y * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 1

   1. Initial program 94.2%

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

      1. associate-/r* 93.9%

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

      2. associate-/l/ 94.2%

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

      3. associate-/r* 94.3%

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

      4. associate-/l/ 92.8%

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

      5. sqr-neg 92.8%

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

      6. +-commutative 92.8%

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

      7. sqr-neg 92.8%

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

      8. fma-def 92.8%

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

   3. Simplified 92.8%

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

   4. Taylor expanded in z around 0 65.9%

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

   if 1 < z < 2.6e162

   1. Initial program 79.2%

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

      1. *-un-lft-identity 79.2%

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

      2. times-frac 93.6%

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

      3. associate-/r* 93.6%

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

      4. +-commutative 93.6%

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

      5. fma-udef 93.6%

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

      6. *-commutative 93.6%

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

      7. fma-udef 93.6%

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

      8. +-commutative 93.6%

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

      9. *-commutative 93.6%

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

      10. +-commutative 93.6%

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

      11. fma-udef 93.6%

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

   3. Applied egg-rr 93.6%

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

   4. Taylor expanded in z around inf 91.3%

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

      1. associate-/r* 91.3%

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

   6. Simplified 91.3%

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

      1. associate-*l/ 91.4%

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

      2. div-inv 91.6%

         \[\leadsto \frac{\color{blue}{\frac{\frac{1}{x}}{y}}}{{z}^{2}} \]

      3. unpow2 91.6%

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

      4. associate-/r* 94.4%

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

      5. associate-/l/ 94.3%

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

   8. Applied egg-rr 94.3%

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

   if 2.6e162 < z

   1. Initial program 67.0%

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

      1. *-un-lft-identity 67.0%

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

      2. times-frac 67.0%

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

      3. associate-/r* 67.0%

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

      4. +-commutative 67.0%

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

      5. fma-udef 67.0%

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

      6. *-commutative 67.0%

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

      7. fma-udef 67.0%

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

      8. +-commutative 67.0%

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

      9. *-commutative 67.0%

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

      10. +-commutative 67.0%

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

      11. fma-udef 67.0%

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

   3. Applied egg-rr 67.0%

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

   4. Taylor expanded in z around inf 67.0%

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

      1. associate-/r* 67.0%

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

   6. Simplified 67.0%

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

      1. *-un-lft-identity 67.0%

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

      2. unpow2 67.0%

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

      3. times-frac 85.1%

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

   8. Applied egg-rr 85.1%

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

      1. *-commutative 85.1%

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

      2. associate-*l/ 85.2%

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

      3. *-un-lft-identity 85.2%

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

      4. frac-times 99.8%

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

      5. *-un-lft-identity 99.8%

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

      6. associate-/l/ 100.0%

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

   10. Applied egg-rr 100.0%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{+162}:\\ \;\;\;\;\frac{\frac{\frac{1}{y \cdot x}}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z \cdot x}}{y \cdot z}\\ \end{array} \]

Alternative 7: 77.0% accurate, 1.0× speedup

Math

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

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 y) (* z (* z x)))))

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

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 94.2%

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

      1. associate-/r* 93.9%

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

      2. associate-/l/ 94.2%

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

      3. associate-/r* 94.3%

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

      4. associate-/l/ 92.8%

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

      5. sqr-neg 92.8%

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

      6. +-commutative 92.8%

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

      7. sqr-neg 92.8%

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

      8. fma-def 92.8%

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

   3. Simplified 92.8%

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

   4. Taylor expanded in z around 0 65.9%

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

   if 1 < z

   1. Initial program 73.1%

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

      1. associate-/r* 72.8%

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

      2. associate-/l/ 73.1%

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

      3. associate-/r* 73.2%

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

      4. associate-/l/ 80.3%

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

      5. sqr-neg 80.3%

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

      6. +-commutative 80.3%

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

      7. sqr-neg 80.3%

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

      8. fma-def 80.3%

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

   3. Simplified 80.3%

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

   4. Taylor expanded in z around inf 79.2%

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

      1. *-commutative 79.2%

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

   6. Simplified 79.2%

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

      1. add-sqr-sqrt 38.1%

         \[\leadsto \frac{\frac{1}{y}}{\color{blue}{\sqrt{{z}^{2} \cdot x} \cdot \sqrt{{z}^{2} \cdot x}}} \]

      2. pow2 38.1%

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

      3. sqrt-prod 38.1%

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

      4. unpow2 38.1%

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

      5. sqrt-prod 45.3%

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

      6. add-sqr-sqrt 45.5%

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

   8. Applied egg-rr 45.5%

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

      1. unpow2 45.5%

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

      2. *-commutative 45.5%

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

      3. associate-*r* 45.4%

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

      4. associate-*r* 45.4%

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

      5. add-sqr-sqrt 88.3%

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

   10. Applied egg-rr 88.3%

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

4. Final simplification 71.5%

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

Alternative 8: 78.5% accurate, 1.0× speedup

Math

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

FPCore

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

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 / (z * x)) / (y * z);
	}
	return tmp;
}

Fortran

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

Java

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 94.2%

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

      1. associate-/r* 93.9%

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

      2. associate-/l/ 94.2%

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

      3. associate-/r* 94.3%

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

      4. associate-/l/ 92.8%

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

      5. sqr-neg 92.8%

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

      6. +-commutative 92.8%

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

      7. sqr-neg 92.8%

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

      8. fma-def 92.8%

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

   3. Simplified 92.8%

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

   4. Taylor expanded in z around 0 65.9%

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

   if 1 < z

   1. Initial program 73.1%

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

      1. *-un-lft-identity 73.1%

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

      2. times-frac 80.3%

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

      3. associate-/r* 80.3%

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

      4. +-commutative 80.3%

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

      5. fma-udef 80.3%

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

      6. *-commutative 80.3%

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

      7. fma-udef 80.3%

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

      8. +-commutative 80.3%

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

      9. *-commutative 80.3%

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

      10. +-commutative 80.3%

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

      11. fma-udef 80.3%

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

   3. Applied egg-rr 80.3%

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

   4. Taylor expanded in z around inf 79.2%

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

      1. associate-/r* 79.2%

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

   6. Simplified 79.2%

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

      1. *-un-lft-identity 79.2%

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

      2. unpow2 79.2%

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

      3. times-frac 88.2%

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

   8. Applied egg-rr 88.2%

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

      1. *-commutative 88.2%

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

      2. associate-*l/ 88.3%

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

      3. *-un-lft-identity 88.3%

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

      4. frac-times 94.1%

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

      5. *-un-lft-identity 94.1%

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

      6. associate-/l/ 94.2%

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

   10. Applied egg-rr 94.2%

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

4. Final simplification 73.0%

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

Alternative 9: 59.1% 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 89.0%

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

2. Taylor expanded in z around 0 53.6%

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

   1. *-commutative 53.6%

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

4. Simplified 53.6%

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

5. Final simplification 53.6%

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

Developer target: 92.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + z \cdot z\\ t_1 := y \cdot t_0\\ t_2 := \frac{\frac{1}{y}}{t_0 \cdot x}\\ \mathbf{if}\;t_1 < -\infty:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 < 8.680743250567252 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{t_0 \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* z z))) (t_1 (* y t_0)) (t_2 (/ (/ 1.0 y) (* t_0 x))))
   (if (< t_1 (- INFINITY))
     t_2
     (if (< t_1 8.680743250567252e+305) (/ (/ 1.0 x) (* t_0 y)) t_2))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 + (z * z);
	double t_1 = y * t_0;
	double t_2 = (1.0 / y) / (t_0 * x);
	double tmp;
	if (t_1 < -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 < 8.680743250567252e+305) {
		tmp = (1.0 / x) / (t_0 * y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 + (z * z);
	double t_1 = y * t_0;
	double t_2 = (1.0 / y) / (t_0 * x);
	double tmp;
	if (t_1 < -Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else if (t_1 < 8.680743250567252e+305) {
		tmp = (1.0 / x) / (t_0 * y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 + (z * z)
	t_1 = y * t_0
	t_2 = (1.0 / y) / (t_0 * x)
	tmp = 0
	if t_1 < -math.inf:
		tmp = t_2
	elif t_1 < 8.680743250567252e+305:
		tmp = (1.0 / x) / (t_0 * y)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 + Float64(z * z))
	t_1 = Float64(y * t_0)
	t_2 = Float64(Float64(1.0 / y) / Float64(t_0 * x))
	tmp = 0.0
	if (t_1 < Float64(-Inf))
		tmp = t_2;
	elseif (t_1 < 8.680743250567252e+305)
		tmp = Float64(Float64(1.0 / x) / Float64(t_0 * y));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 + (z * z);
	t_1 = y * t_0;
	t_2 = (1.0 / y) / (t_0 * x);
	tmp = 0.0;
	if (t_1 < -Inf)
		tmp = t_2;
	elseif (t_1 < 8.680743250567252e+305)
		tmp = (1.0 / x) / (t_0 * y);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 / y), $MachinePrecision] / N[(t$95$0 * x), $MachinePrecision]), $MachinePrecision]}, If[Less[t$95$1, (-Infinity)], t$95$2, If[Less[t$95$1, 8.680743250567252e+305], N[(N[(1.0 / x), $MachinePrecision] / N[(t$95$0 * y), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Reproduce

herbie shell --seed 2023311 
(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)))))