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

Percentage Accurate: 89.4% → 99.3%

Time: 10.9s

Alternatives: 12

Speedup: 0.7×

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: 89.4% 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: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \cdot \left(1 + z \cdot z\right) \leq 4 \cdot 10^{+306}:\\ \;\;\;\;\frac{\frac{1}{y\_m + y\_m \cdot \left(z \cdot z\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y\_m \cdot \left(x \cdot z\right)}}{z}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (* y_m (+ 1.0 (* z z))) 4e+306)
    (/ (/ 1.0 (+ y_m (* y_m (* z z)))) x)
    (/ (/ 1.0 (* y_m (* x z))) z))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((y_m * (1.0 + (z * z))) <= 4e+306) {
		tmp = (1.0 / (y_m + (y_m * (z * z)))) / x;
	} else {
		tmp = (1.0 / (y_m * (x * z))) / z;
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y_m * (1.0d0 + (z * z))) <= 4d+306) then
        tmp = (1.0d0 / (y_m + (y_m * (z * z)))) / x
    else
        tmp = (1.0d0 / (y_m * (x * z))) / z
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z;
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((y_m * (1.0 + (z * z))) <= 4e+306) {
		tmp = (1.0 / (y_m + (y_m * (z * z)))) / x;
	} else {
		tmp = (1.0 / (y_m * (x * z))) / z;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	tmp = 0
	if (y_m * (1.0 + (z * z))) <= 4e+306:
		tmp = (1.0 / (y_m + (y_m * (z * z)))) / x
	else:
		tmp = (1.0 / (y_m * (x * z))) / z
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(y_m * Float64(1.0 + Float64(z * z))) <= 4e+306)
		tmp = Float64(Float64(1.0 / Float64(y_m + Float64(y_m * Float64(z * z)))) / x);
	else
		tmp = Float64(Float64(1.0 / Float64(y_m * Float64(x * z))) / z);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if ((y_m * (1.0 + (z * z))) <= 4e+306)
		tmp = (1.0 / (y_m + (y_m * (z * z)))) / x;
	else
		tmp = (1.0 / (y_m * (x * z))) / z;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(y$95$m * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e+306], N[(N[(1.0 / N[(y$95$m + N[(y$95$m * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(1.0 / N[(y$95$m * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 4.00000000000000007e306

   1. Initial program 93.5%

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

      1. associate-/l/ 92.5%

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

      2. associate-/r* 93.6%

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

      3. distribute-lft-in 93.6%

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

      4. *-rgt-identity 93.6%

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

   4. Applied egg-rr 93.6%

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

   if 4.00000000000000007e306 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 72.5%

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

   3. Taylor expanded in z around inf 72.5%

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

      1. unpow2 72.5%

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

   5. Simplified 72.5%

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

      1. associate-/r* 77.9%

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

      2. div-inv 77.9%

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

      3. times-frac 98.6%

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

   7. Applied egg-rr 98.6%

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

      1. associate-*r/ 99.8%

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

      2. associate-/l/ 99.8%

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

      3. frac-times 99.8%

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

      4. metadata-eval 99.8%

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

   9. Applied egg-rr 99.8%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 4 \cdot 10^{+306}:\\ \;\;\;\;\frac{\frac{1}{y + y \cdot \left(z \cdot z\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y \cdot \left(x \cdot z\right)}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(1, z\right) \cdot \sqrt{y\_m}\\ y\_s \cdot \frac{\frac{\frac{1}{x}}{t\_0}}{t\_0} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (* (hypot 1.0 z) (sqrt y_m)))) (* y_s (/ (/ (/ 1.0 x) t_0) t_0))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = hypot(1.0, z) * sqrt(y_m);
	return y_s * (((1.0 / x) / t_0) / t_0);
}

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z;
public static double code(double y_s, double x, double y_m, double z) {
	double t_0 = Math.hypot(1.0, z) * Math.sqrt(y_m);
	return y_s * (((1.0 / x) / t_0) / t_0);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	t_0 = math.hypot(1.0, z) * math.sqrt(y_m)
	return y_s * (((1.0 / x) / t_0) / t_0)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	t_0 = Float64(hypot(1.0, z) * sqrt(y_m))
	return Float64(y_s * Float64(Float64(Float64(1.0 / x) / t_0) / t_0))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp = code(y_s, x, y_m, z)
	t_0 = hypot(1.0, z) * sqrt(y_m);
	tmp = y_s * (((1.0 / x) / t_0) / t_0);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision] * N[Sqrt[y$95$m], $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(N[(N[(1.0 / x), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 90.3%

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

   1. add-sqr-sqrt 46.7%

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

   2. associate-/r* 46.7%

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

   3. *-commutative 46.7%

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

   4. sqrt-prod 46.7%

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

   5. hypot-1-def 46.7%

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

   6. *-commutative 46.7%

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

   7. sqrt-prod 47.6%

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

   8. hypot-1-def 50.9%

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

4. Applied egg-rr 50.9%

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

Alternative 3: 77.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 1:\\ \;\;\;\;\frac{\frac{1}{y\_m}}{x}\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+201}:\\ \;\;\;\;\frac{\frac{1}{y\_m}}{z \cdot \left(x \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot z\right) \cdot \left(z \cdot y\_m\right)}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= z 1.0)
    (/ (/ 1.0 y_m) x)
    (if (<= z 5e+201)
      (/ (/ 1.0 y_m) (* z (* x z)))
      (/ 1.0 (* (* x z) (* z y_m)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (z <= 1.0) {
		tmp = (1.0 / y_m) / x;
	} else if (z <= 5e+201) {
		tmp = (1.0 / y_m) / (z * (x * z));
	} else {
		tmp = 1.0 / ((x * z) * (z * y_m));
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 1.0d0) then
        tmp = (1.0d0 / y_m) / x
    else if (z <= 5d+201) then
        tmp = (1.0d0 / y_m) / (z * (x * z))
    else
        tmp = 1.0d0 / ((x * z) * (z * y_m))
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z;
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (z <= 1.0) {
		tmp = (1.0 / y_m) / x;
	} else if (z <= 5e+201) {
		tmp = (1.0 / y_m) / (z * (x * z));
	} else {
		tmp = 1.0 / ((x * z) * (z * y_m));
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	tmp = 0
	if z <= 1.0:
		tmp = (1.0 / y_m) / x
	elif z <= 5e+201:
		tmp = (1.0 / y_m) / (z * (x * z))
	else:
		tmp = 1.0 / ((x * z) * (z * y_m))
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (z <= 1.0)
		tmp = Float64(Float64(1.0 / y_m) / x);
	elseif (z <= 5e+201)
		tmp = Float64(Float64(1.0 / y_m) / Float64(z * Float64(x * z)));
	else
		tmp = Float64(1.0 / Float64(Float64(x * z) * Float64(z * y_m)));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (z <= 1.0)
		tmp = (1.0 / y_m) / x;
	elseif (z <= 5e+201)
		tmp = (1.0 / y_m) / (z * (x * z));
	else
		tmp = 1.0 / ((x * z) * (z * y_m));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[z, 1.0], N[(N[(1.0 / y$95$m), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[z, 5e+201], N[(N[(1.0 / y$95$m), $MachinePrecision] / N[(z * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(x * z), $MachinePrecision] * N[(z * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 1

   1. Initial program 93.2%

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

      1. associate-/l/ 92.1%

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

      2. associate-/r* 93.3%

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

      3. distribute-lft-in 93.3%

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

      4. *-rgt-identity 93.3%

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

   4. Applied egg-rr 93.3%

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

   5. Taylor expanded in z around 0 66.7%

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

   if 1 < z < 4.9999999999999995e201

   1. Initial program 82.4%

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

   3. Taylor expanded in z around inf 82.4%

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

      1. unpow2 82.4%

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

   5. Simplified 82.4%

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

      1. associate-/r* 90.0%

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

      2. div-inv 89.9%

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

      3. times-frac 94.1%

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

   7. Applied egg-rr 94.1%

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

      1. associate-/l/ 91.7%

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

      2. frac-times 87.2%

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

      3. *-un-lft-identity 87.2%

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

   9. Applied egg-rr 87.2%

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

   if 4.9999999999999995e201 < z

   1. Initial program 75.0%

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

   3. Taylor expanded in z around inf 75.0%

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

      1. unpow2 75.0%

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

   5. Simplified 75.0%

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

      1. associate-/r* 73.9%

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

      2. div-inv 73.9%

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

      3. times-frac 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. associate-/l/ 99.7%

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

      2. associate-/l/ 99.8%

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

      3. *-commutative 99.8%

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

      4. frac-times 100.0%

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

      5. metadata-eval 100.0%

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+201}:\\ \;\;\;\;\frac{\frac{1}{y}}{z \cdot \left(x \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot z\right) \cdot \left(z \cdot y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+65}:\\ \;\;\;\;\frac{\frac{1}{x}}{y\_m + z \cdot \left(z \cdot y\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y\_m \cdot \left(x \cdot z\right)}}{z}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (* z z) 5e+65)
    (/ (/ 1.0 x) (+ y_m (* z (* z y_m))))
    (/ (/ 1.0 (* y_m (* x z))) z))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((z * z) <= 5e+65) {
		tmp = (1.0 / x) / (y_m + (z * (z * y_m)));
	} else {
		tmp = (1.0 / (y_m * (x * z))) / z;
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 5d+65) then
        tmp = (1.0d0 / x) / (y_m + (z * (z * y_m)))
    else
        tmp = (1.0d0 / (y_m * (x * z))) / z
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z;
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((z * z) <= 5e+65) {
		tmp = (1.0 / x) / (y_m + (z * (z * y_m)));
	} else {
		tmp = (1.0 / (y_m * (x * z))) / z;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	tmp = 0
	if (z * z) <= 5e+65:
		tmp = (1.0 / x) / (y_m + (z * (z * y_m)))
	else:
		tmp = (1.0 / (y_m * (x * z))) / z
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e+65)
		tmp = Float64(Float64(1.0 / x) / Float64(y_m + Float64(z * Float64(z * y_m))));
	else
		tmp = Float64(Float64(1.0 / Float64(y_m * Float64(x * z))) / z);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if ((z * z) <= 5e+65)
		tmp = (1.0 / x) / (y_m + (z * (z * y_m)));
	else
		tmp = (1.0 / (y_m * (x * z))) / z;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 5e+65], N[(N[(1.0 / x), $MachinePrecision] / N[(y$95$m + N[(z * N[(z * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(y$95$m * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.99999999999999973e65

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. distribute-lft-in 99.6%

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

      3. *-rgt-identity 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. associate-*r* 99.6%

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

   6. Applied egg-rr 99.6%

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

   if 4.99999999999999973e65 < (*.f64 z z)

   1. Initial program 80.1%

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

   3. Taylor expanded in z around inf 80.1%

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

      1. unpow2 80.1%

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

   5. Simplified 80.1%

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

      1. associate-/r* 81.7%

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

      2. div-inv 81.7%

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

      3. times-frac 97.1%

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

   7. Applied egg-rr 97.1%

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

      1. associate-*r/ 98.6%

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

      2. associate-/l/ 98.0%

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

      3. frac-times 98.1%

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

      4. metadata-eval 98.1%

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

   9. Applied egg-rr 98.1%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+65}:\\ \;\;\;\;\frac{\frac{1}{x}}{y + z \cdot \left(z \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y \cdot \left(x \cdot z\right)}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+60}:\\ \;\;\;\;\frac{\frac{1}{x}}{y\_m + y\_m \cdot \left(z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y\_m \cdot \left(x \cdot z\right)}}{z}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (* z z) 2e+60)
    (/ (/ 1.0 x) (+ y_m (* y_m (* z z))))
    (/ (/ 1.0 (* y_m (* x z))) z))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((z * z) <= 2e+60) {
		tmp = (1.0 / x) / (y_m + (y_m * (z * z)));
	} else {
		tmp = (1.0 / (y_m * (x * z))) / z;
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 2d+60) then
        tmp = (1.0d0 / x) / (y_m + (y_m * (z * z)))
    else
        tmp = (1.0d0 / (y_m * (x * z))) / z
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z;
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((z * z) <= 2e+60) {
		tmp = (1.0 / x) / (y_m + (y_m * (z * z)));
	} else {
		tmp = (1.0 / (y_m * (x * z))) / z;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	tmp = 0
	if (z * z) <= 2e+60:
		tmp = (1.0 / x) / (y_m + (y_m * (z * z)))
	else:
		tmp = (1.0 / (y_m * (x * z))) / z
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+60)
		tmp = Float64(Float64(1.0 / x) / Float64(y_m + Float64(y_m * Float64(z * z))));
	else
		tmp = Float64(Float64(1.0 / Float64(y_m * Float64(x * z))) / z);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if ((z * z) <= 2e+60)
		tmp = (1.0 / x) / (y_m + (y_m * (z * z)));
	else
		tmp = (1.0 / (y_m * (x * z))) / z;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 2e+60], N[(N[(1.0 / x), $MachinePrecision] / N[(y$95$m + N[(y$95$m * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(y$95$m * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.9999999999999999e60

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. distribute-lft-in 99.6%

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

      3. *-rgt-identity 99.6%

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

   4. Applied egg-rr 99.6%

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

   if 1.9999999999999999e60 < (*.f64 z z)

   1. Initial program 80.3%

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

   3. Taylor expanded in z around inf 80.3%

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

      1. unpow2 80.3%

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

   5. Simplified 80.3%

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

      1. associate-/r* 81.8%

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

      2. div-inv 81.8%

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

      3. times-frac 97.1%

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

   7. Applied egg-rr 97.1%

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

      1. associate-*r/ 98.6%

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

      2. associate-/l/ 98.0%

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

      3. frac-times 98.1%

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

      4. metadata-eval 98.1%

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

   9. Applied egg-rr 98.1%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+60}:\\ \;\;\;\;\frac{\frac{1}{x}}{y + y \cdot \left(z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y \cdot \left(x \cdot z\right)}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+60}:\\ \;\;\;\;\frac{\frac{1}{x}}{y\_m \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y\_m \cdot \left(x \cdot z\right)}}{z}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (* z z) 2e+60)
    (/ (/ 1.0 x) (* y_m (+ 1.0 (* z z))))
    (/ (/ 1.0 (* y_m (* x z))) z))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((z * z) <= 2e+60) {
		tmp = (1.0 / x) / (y_m * (1.0 + (z * z)));
	} else {
		tmp = (1.0 / (y_m * (x * z))) / z;
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 2d+60) then
        tmp = (1.0d0 / x) / (y_m * (1.0d0 + (z * z)))
    else
        tmp = (1.0d0 / (y_m * (x * z))) / z
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z;
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((z * z) <= 2e+60) {
		tmp = (1.0 / x) / (y_m * (1.0 + (z * z)));
	} else {
		tmp = (1.0 / (y_m * (x * z))) / z;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	tmp = 0
	if (z * z) <= 2e+60:
		tmp = (1.0 / x) / (y_m * (1.0 + (z * z)))
	else:
		tmp = (1.0 / (y_m * (x * z))) / z
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+60)
		tmp = Float64(Float64(1.0 / x) / Float64(y_m * Float64(1.0 + Float64(z * z))));
	else
		tmp = Float64(Float64(1.0 / Float64(y_m * Float64(x * z))) / z);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if ((z * z) <= 2e+60)
		tmp = (1.0 / x) / (y_m * (1.0 + (z * z)));
	else
		tmp = (1.0 / (y_m * (x * z))) / z;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 2e+60], N[(N[(1.0 / x), $MachinePrecision] / N[(y$95$m * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(y$95$m * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.9999999999999999e60

   1. Initial program 99.6%

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

   if 1.9999999999999999e60 < (*.f64 z z)

   1. Initial program 80.3%

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

   3. Taylor expanded in z around inf 80.3%

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

      1. unpow2 80.3%

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

   5. Simplified 80.3%

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

      1. associate-/r* 81.8%

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

      2. div-inv 81.8%

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

      3. times-frac 97.1%

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

   7. Applied egg-rr 97.1%

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

      1. associate-*r/ 98.6%

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

      2. associate-/l/ 98.0%

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

      3. frac-times 98.1%

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

      4. metadata-eval 98.1%

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

   9. Applied egg-rr 98.1%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+60}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y \cdot \left(x \cdot z\right)}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.0001:\\ \;\;\;\;\frac{\frac{1}{y\_m}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y\_m \cdot \left(x \cdot z\right)}}{z}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (* z z) 0.0001) (/ (/ 1.0 y_m) x) (/ (/ 1.0 (* y_m (* x z))) z))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((z * z) <= 0.0001) {
		tmp = (1.0 / y_m) / x;
	} else {
		tmp = (1.0 / (y_m * (x * z))) / z;
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 0.0001d0) then
        tmp = (1.0d0 / y_m) / x
    else
        tmp = (1.0d0 / (y_m * (x * z))) / z
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z;
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((z * z) <= 0.0001) {
		tmp = (1.0 / y_m) / x;
	} else {
		tmp = (1.0 / (y_m * (x * z))) / z;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	tmp = 0
	if (z * z) <= 0.0001:
		tmp = (1.0 / y_m) / x
	else:
		tmp = (1.0 / (y_m * (x * z))) / z
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 0.0001)
		tmp = Float64(Float64(1.0 / y_m) / x);
	else
		tmp = Float64(Float64(1.0 / Float64(y_m * Float64(x * z))) / z);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if ((z * z) <= 0.0001)
		tmp = (1.0 / y_m) / x;
	else
		tmp = (1.0 / (y_m * (x * z))) / z;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 0.0001], N[(N[(1.0 / y$95$m), $MachinePrecision] / x), $MachinePrecision], N[(N[(1.0 / N[(y$95$m * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.00000000000000005e-4

   1. Initial program 99.6%

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

      1. associate-/l/ 97.8%

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

      2. associate-/r* 99.6%

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

      3. distribute-lft-in 99.6%

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

      4. *-rgt-identity 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in z around 0 98.6%

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

   if 1.00000000000000005e-4 < (*.f64 z z)

   1. Initial program 81.6%

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

   3. Taylor expanded in z around inf 81.6%

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

      1. unpow2 81.6%

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

   5. Simplified 81.6%

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

      1. associate-/r* 83.1%

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

      2. div-inv 83.1%

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

      3. times-frac 97.2%

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

   7. Applied egg-rr 97.2%

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

      1. associate-*r/ 98.7%

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

      2. associate-/l/ 98.1%

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

      3. frac-times 98.2%

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

      4. metadata-eval 98.2%

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

   9. Applied egg-rr 98.2%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.0001:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y \cdot \left(x \cdot z\right)}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.6% accurate, 0.7× speedup

Math

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

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= (* z z) 1.0) (/ (/ 1.0 y_m) x) (/ 1.0 (* x (* y_m (* z z)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((z * z) <= 1.0) {
		tmp = (1.0 / y_m) / x;
	} else {
		tmp = 1.0 / (x * (y_m * (z * z)));
	}
	return y_s * tmp;
}

Fortran

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

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z;
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((z * z) <= 1.0) {
		tmp = (1.0 / y_m) / x;
	} else {
		tmp = 1.0 / (x * (y_m * (z * z)));
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	tmp = 0
	if (z * z) <= 1.0:
		tmp = (1.0 / y_m) / x
	else:
		tmp = 1.0 / (x * (y_m * (z * z)))
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 1.0)
		tmp = Float64(Float64(1.0 / y_m) / x);
	else
		tmp = Float64(1.0 / Float64(x * Float64(y_m * Float64(z * z))));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if ((z * z) <= 1.0)
		tmp = (1.0 / y_m) / x;
	else
		tmp = 1.0 / (x * (y_m * (z * z)));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 1.0], N[(N[(1.0 / y$95$m), $MachinePrecision] / x), $MachinePrecision], N[(1.0 / N[(x * N[(y$95$m * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.6%

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

      1. associate-/l/ 97.8%

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

      2. associate-/r* 99.6%

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

      3. distribute-lft-in 99.6%

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

      4. *-rgt-identity 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in z around 0 98.6%

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

   if 1 < (*.f64 z z)

   1. Initial program 81.6%

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

   3. Taylor expanded in z around inf 81.6%

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

      1. unpow2 81.6%

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

   5. Simplified 81.6%

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

Alternative 9: 77.6% accurate, 0.8× speedup

Math

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

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= z 1.0) (/ (/ 1.0 y_m) x) (/ 1.0 (* (* x z) (* z y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (z <= 1.0) {
		tmp = (1.0 / y_m) / x;
	} else {
		tmp = 1.0 / ((x * z) * (z * y_m));
	}
	return y_s * tmp;
}

Fortran

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

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z;
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (z <= 1.0) {
		tmp = (1.0 / y_m) / x;
	} else {
		tmp = 1.0 / ((x * z) * (z * y_m));
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	tmp = 0
	if z <= 1.0:
		tmp = (1.0 / y_m) / x
	else:
		tmp = 1.0 / ((x * z) * (z * y_m))
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (z <= 1.0)
		tmp = Float64(Float64(1.0 / y_m) / x);
	else
		tmp = Float64(1.0 / Float64(Float64(x * z) * Float64(z * y_m)));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (z <= 1.0)
		tmp = (1.0 / y_m) / x;
	else
		tmp = 1.0 / ((x * z) * (z * y_m));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[z, 1.0], N[(N[(1.0 / y$95$m), $MachinePrecision] / x), $MachinePrecision], N[(1.0 / N[(N[(x * z), $MachinePrecision] * N[(z * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 93.2%

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

      1. associate-/l/ 92.1%

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

      2. associate-/r* 93.3%

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

      3. distribute-lft-in 93.3%

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

      4. *-rgt-identity 93.3%

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

   4. Applied egg-rr 93.3%

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

   5. Taylor expanded in z around 0 66.7%

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

   if 1 < z

   1. Initial program 79.2%

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

   3. Taylor expanded in z around inf 79.2%

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

      1. unpow2 79.2%

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

   5. Simplified 79.2%

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

      1. associate-/r* 83.0%

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

      2. div-inv 83.0%

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

      3. times-frac 96.6%

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

   7. Applied egg-rr 96.6%

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

      1. associate-/l/ 95.1%

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

      2. associate-/l/ 95.2%

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

      3. *-commutative 95.2%

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

      4. frac-times 95.2%

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

      5. metadata-eval 95.2%

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

   9. Applied egg-rr 95.2%

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

4. Final simplification 72.6%

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

Alternative 10: 58.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \frac{\frac{1}{y\_m}}{x} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z) :precision binary64 (* y_s (/ (/ 1.0 y_m) x)))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	return y_s * ((1.0 / y_m) / x);
}

Fortran

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

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z;
public static double code(double y_s, double x, double y_m, double z) {
	return y_s * ((1.0 / y_m) / x);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	return y_s * ((1.0 / y_m) / x)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	return Float64(y_s * Float64(Float64(1.0 / y_m) / x))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp = code(y_s, x, y_m, z)
	tmp = y_s * ((1.0 / y_m) / x);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * N[(N[(1.0 / y$95$m), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.3%

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

   1. associate-/l/ 89.4%

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

   2. associate-/r* 90.4%

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

   3. distribute-lft-in 90.4%

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

   4. *-rgt-identity 90.4%

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

4. Applied egg-rr 90.4%

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

5. Taylor expanded in z around 0 54.9%

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

Alternative 11: 58.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \frac{\frac{1}{x}}{y\_m} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z) :precision binary64 (* y_s (/ (/ 1.0 x) y_m)))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	return y_s * ((1.0 / x) / y_m);
}

Fortran

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

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z;
public static double code(double y_s, double x, double y_m, double z) {
	return y_s * ((1.0 / x) / y_m);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	return y_s * ((1.0 / x) / y_m)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	return Float64(y_s * Float64(Float64(1.0 / x) / y_m))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp = code(y_s, x, y_m, z)
	tmp = y_s * ((1.0 / x) / y_m);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * N[(N[(1.0 / x), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.3%

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

3. Taylor expanded in z around 0 54.9%

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

Alternative 12: 58.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \frac{1}{x \cdot y\_m} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z) :precision binary64 (* y_s (/ 1.0 (* x y_m))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	return y_s * (1.0 / (x * y_m));
}

Fortran

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

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z;
public static double code(double y_s, double x, double y_m, double z) {
	return y_s * (1.0 / (x * y_m));
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	return y_s * (1.0 / (x * y_m))

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	return Float64(y_s * Float64(1.0 / Float64(x * y_m)))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp = code(y_s, x, y_m, z)
	tmp = y_s * (1.0 / (x * y_m));
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * N[(1.0 / N[(x * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.3%

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

3. Taylor expanded in z around 0 54.0%

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

Developer target: 92.3% accurate, 0.3× 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 2024097 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (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)))))