Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2

Percentage Accurate: 83.4% → 97.0%

Time: 11.5s

Alternatives: 18

Speedup: 1.0×

• 25.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (x y z) :precision binary64 (/ (* x y) (* (* z z) (+ z 1.0))))

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * y) / ((z * z) * (z + 1.0d0))
end function

Java

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

Python

def code(x, y, z):
	return (x * y) / ((z * z) * (z + 1.0))

Julia

function code(x, y, z)
	return Float64(Float64(x * y) / Float64(Float64(z * z) * Float64(z + 1.0)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * y) / ((z * z) * (z + 1.0));
end

Wolfram

code[x_, y_, z_] := N[(N[(x * y), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 83.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (/ (* x y) (* (* z z) (+ z 1.0))))

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * y) / ((z * z) * (z + 1.0d0))
end function

Java

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

Python

def code(x, y, z):
	return (x * y) / ((z * z) * (z + 1.0))

Julia

function code(x, y, z)
	return Float64(Float64(x * y) / Float64(Float64(z * z) * Float64(z + 1.0)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * y) / ((z * z) * (z + 1.0));
end

Wolfram

code[x_, y_, z_] := N[(N[(x * y), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 97.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 = ((y / (z + 1.0d0)) * (x / z)) / z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.3%

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

   1. associate-*l* 84.3%

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

   2. times-frac 94.2%

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

   3. distribute-lft-in 94.2%

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

   4. fma-def 94.2%

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

   5. *-rgt-identity 94.2%

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

3. Simplified 94.2%

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

   1. *-commutative 94.2%

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

   2. associate-*l/ 95.7%

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

   3. fma-udef 95.7%

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

   4. distribute-lft1-in 95.7%

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

   5. frac-times 92.2%

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

   6. associate-*r/ 97.6%

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

5. Applied egg-rr 97.6%

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

6. Final simplification 97.6%

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

Alternative 2: 94.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(z + 1\right) \cdot \left(z \cdot z\right)\\ \mathbf{if}\;t_0 \leq -400000000000:\\ \;\;\;\;\frac{\frac{x \cdot \frac{y}{z}}{z}}{z}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{-51}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z + 1} \cdot \frac{x}{z \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (+ z 1.0) (* z z))))
   (if (<= t_0 -400000000000.0)
     (/ (/ (* x (/ y z)) z) z)
     (if (<= t_0 2e-51)
       (/ (* y (/ x z)) z)
       (* (/ y (+ z 1.0)) (/ x (* z z)))))))

C

assert(x < y);
double code(double x, double y, double z) {
	double t_0 = (z + 1.0) * (z * z);
	double tmp;
	if (t_0 <= -400000000000.0) {
		tmp = ((x * (y / z)) / z) / z;
	} else if (t_0 <= 2e-51) {
		tmp = (y * (x / z)) / z;
	} else {
		tmp = (y / (z + 1.0)) * (x / (z * 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 = (z + 1.0d0) * (z * z)
    if (t_0 <= (-400000000000.0d0)) then
        tmp = ((x * (y / z)) / z) / z
    else if (t_0 <= 2d-51) then
        tmp = (y * (x / z)) / z
    else
        tmp = (y / (z + 1.0d0)) * (x / (z * z))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double t_0 = (z + 1.0) * (z * z);
	double tmp;
	if (t_0 <= -400000000000.0) {
		tmp = ((x * (y / z)) / z) / z;
	} else if (t_0 <= 2e-51) {
		tmp = (y * (x / z)) / z;
	} else {
		tmp = (y / (z + 1.0)) * (x / (z * z));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	t_0 = (z + 1.0) * (z * z)
	tmp = 0
	if t_0 <= -400000000000.0:
		tmp = ((x * (y / z)) / z) / z
	elif t_0 <= 2e-51:
		tmp = (y * (x / z)) / z
	else:
		tmp = (y / (z + 1.0)) * (x / (z * z))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	t_0 = Float64(Float64(z + 1.0) * Float64(z * z))
	tmp = 0.0
	if (t_0 <= -400000000000.0)
		tmp = Float64(Float64(Float64(x * Float64(y / z)) / z) / z);
	elseif (t_0 <= 2e-51)
		tmp = Float64(Float64(y * Float64(x / z)) / z);
	else
		tmp = Float64(Float64(y / Float64(z + 1.0)) * Float64(x / Float64(z * z)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	t_0 = (z + 1.0) * (z * z);
	tmp = 0.0;
	if (t_0 <= -400000000000.0)
		tmp = ((x * (y / z)) / z) / z;
	elseif (t_0 <= 2e-51)
		tmp = (y * (x / z)) / z;
	else
		tmp = (y / (z + 1.0)) * (x / (z * 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[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -400000000000.0], N[(N[(N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$0, 2e-51], N[(N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(y / N[(z + 1.0), $MachinePrecision]), $MachinePrecision] * N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z 1)) < -4e11

   1. Initial program 82.1%

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

      1. associate-*l* 82.1%

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

      2. times-frac 93.1%

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

      3. distribute-lft-in 93.1%

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

      4. fma-def 93.1%

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

      5. *-rgt-identity 93.1%

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

   3. Simplified 93.1%

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

      1. associate-*r/ 93.1%

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

      2. fma-udef 93.1%

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

      3. *-rgt-identity 93.1%

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

      4. distribute-lft-in 93.1%

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

      5. frac-times 94.8%

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

      6. associate-/r* 90.9%

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

      7. associate-*l/ 91.9%

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

      8. associate-/r* 95.9%

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

      9. associate-*r/ 90.2%

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

      10. *-commutative 90.2%

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

   5. Applied egg-rr 90.2%

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

   6. Taylor expanded in z around inf 87.5%

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

      1. associate-*l/ 93.3%

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

      2. *-commutative 93.3%

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

   8. Simplified 93.3%

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

   if -4e11 < (*.f64 (*.f64 z z) (+.f64 z 1)) < 2e-51

   1. Initial program 84.5%

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

      1. times-frac 82.0%

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

   3. Simplified 82.0%

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

   4. Taylor expanded in z around 0 82.0%

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

      1. *-commutative 82.0%

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

      2. associate-/r* 88.2%

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

      3. associate-*r/ 97.4%

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

   6. Applied egg-rr 97.4%

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

   if 2e-51 < (*.f64 (*.f64 z z) (+.f64 z 1))

   1. Initial program 86.2%

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

      1. times-frac 93.7%

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

   3. Simplified 93.7%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq -400000000000:\\ \;\;\;\;\frac{\frac{x \cdot \frac{y}{z}}{z}}{z}\\ \mathbf{elif}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 2 \cdot 10^{-51}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z + 1} \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 3: 94.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(z + 1\right) \cdot \left(z \cdot z\right)\\ \mathbf{if}\;t_0 \leq -400000000000:\\ \;\;\;\;\frac{\frac{x \cdot \frac{y}{z}}{z}}{z}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{-197}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z + 1} \cdot x}{z \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (+ z 1.0) (* z z))))
   (if (<= t_0 -400000000000.0)
     (/ (/ (* x (/ y z)) z) z)
     (if (<= t_0 5e-197)
       (/ (* y (/ x z)) z)
       (/ (* (/ y (+ z 1.0)) x) (* z z))))))

C

assert(x < y);
double code(double x, double y, double z) {
	double t_0 = (z + 1.0) * (z * z);
	double tmp;
	if (t_0 <= -400000000000.0) {
		tmp = ((x * (y / z)) / z) / z;
	} else if (t_0 <= 5e-197) {
		tmp = (y * (x / z)) / z;
	} else {
		tmp = ((y / (z + 1.0)) * x) / (z * 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 = (z + 1.0d0) * (z * z)
    if (t_0 <= (-400000000000.0d0)) then
        tmp = ((x * (y / z)) / z) / z
    else if (t_0 <= 5d-197) then
        tmp = (y * (x / z)) / z
    else
        tmp = ((y / (z + 1.0d0)) * x) / (z * z)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double t_0 = (z + 1.0) * (z * z);
	double tmp;
	if (t_0 <= -400000000000.0) {
		tmp = ((x * (y / z)) / z) / z;
	} else if (t_0 <= 5e-197) {
		tmp = (y * (x / z)) / z;
	} else {
		tmp = ((y / (z + 1.0)) * x) / (z * z);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	t_0 = (z + 1.0) * (z * z)
	tmp = 0
	if t_0 <= -400000000000.0:
		tmp = ((x * (y / z)) / z) / z
	elif t_0 <= 5e-197:
		tmp = (y * (x / z)) / z
	else:
		tmp = ((y / (z + 1.0)) * x) / (z * z)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	t_0 = Float64(Float64(z + 1.0) * Float64(z * z))
	tmp = 0.0
	if (t_0 <= -400000000000.0)
		tmp = Float64(Float64(Float64(x * Float64(y / z)) / z) / z);
	elseif (t_0 <= 5e-197)
		tmp = Float64(Float64(y * Float64(x / z)) / z);
	else
		tmp = Float64(Float64(Float64(y / Float64(z + 1.0)) * x) / Float64(z * z));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	t_0 = (z + 1.0) * (z * z);
	tmp = 0.0;
	if (t_0 <= -400000000000.0)
		tmp = ((x * (y / z)) / z) / z;
	elseif (t_0 <= 5e-197)
		tmp = (y * (x / z)) / z;
	else
		tmp = ((y / (z + 1.0)) * x) / (z * 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[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -400000000000.0], N[(N[(N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$0, 5e-197], N[(N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(y / N[(z + 1.0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z 1)) < -4e11

   1. Initial program 82.1%

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

      1. associate-*l* 82.1%

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

      2. times-frac 93.1%

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

      3. distribute-lft-in 93.1%

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

      4. fma-def 93.1%

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

      5. *-rgt-identity 93.1%

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

   3. Simplified 93.1%

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

      1. associate-*r/ 93.1%

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

      2. fma-udef 93.1%

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

      3. *-rgt-identity 93.1%

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

      4. distribute-lft-in 93.1%

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

      5. frac-times 94.8%

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

      6. associate-/r* 90.9%

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

      7. associate-*l/ 91.9%

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

      8. associate-/r* 95.9%

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

      9. associate-*r/ 90.2%

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

      10. *-commutative 90.2%

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

   5. Applied egg-rr 90.2%

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

   6. Taylor expanded in z around inf 87.5%

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

      1. associate-*l/ 93.3%

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

      2. *-commutative 93.3%

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

   8. Simplified 93.3%

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

   if -4e11 < (*.f64 (*.f64 z z) (+.f64 z 1)) < 5.0000000000000002e-197

   1. Initial program 77.6%

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

      1. times-frac 77.0%

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

   3. Simplified 77.0%

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

   4. Taylor expanded in z around 0 77.0%

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

      1. *-commutative 77.0%

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

      2. associate-/r* 86.1%

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

      3. associate-*r/ 97.5%

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

   6. Applied egg-rr 97.5%

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

   if 5.0000000000000002e-197 < (*.f64 (*.f64 z z) (+.f64 z 1))

   1. Initial program 91.0%

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

      1. associate-*l* 91.1%

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

      2. times-frac 93.2%

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

      3. distribute-lft-in 93.2%

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

      4. fma-def 93.2%

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

      5. *-rgt-identity 93.2%

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

   3. Simplified 93.2%

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

      1. *-commutative 93.2%

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

      2. associate-*l/ 96.1%

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

      3. fma-udef 96.1%

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

      4. distribute-lft1-in 96.0%

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

      5. frac-times 95.1%

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

      6. associate-/r* 93.3%

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

      7. associate-*r/ 96.8%

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

   5. Applied egg-rr 96.8%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq -400000000000:\\ \;\;\;\;\frac{\frac{x \cdot \frac{y}{z}}{z}}{z}\\ \mathbf{elif}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 5 \cdot 10^{-197}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z + 1} \cdot x}{z \cdot z}\\ \end{array} \]

Alternative 4: 92.7% accurate, 0.8× speedup

Math

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

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.0) (not (<= z 1.0)))
   (* (/ x z) (/ y (* z z)))
   (/ (* y (/ x z)) z)))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 1.0)) {
		tmp = (x / z) * (y / (z * z));
	} else {
		tmp = (y * (x / z)) / 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)) .or. (.not. (z <= 1.0d0))) then
        tmp = (x / z) * (y / (z * z))
    else
        tmp = (y * (x / z)) / 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) || !(z <= 1.0)) {
		tmp = (x / z) * (y / (z * z));
	} else {
		tmp = (y * (x / z)) / z;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z <= -1.0) or not (z <= 1.0):
		tmp = (x / z) * (y / (z * z))
	else:
		tmp = (y * (x / z)) / z
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.0) || !(z <= 1.0))
		tmp = Float64(Float64(x / z) * Float64(y / Float64(z * z)));
	else
		tmp = Float64(Float64(y * Float64(x / z)) / 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) || ~((z <= 1.0)))
		tmp = (x / z) * (y / (z * z));
	else
		tmp = (y * (x / z)) / 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[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(N[(x / z), $MachinePrecision] * N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 83.1%

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

      1. associate-*l* 83.1%

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

      2. times-frac 92.9%

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

      3. distribute-lft-in 92.9%

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

      4. fma-def 92.9%

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

      5. *-rgt-identity 92.9%

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

   3. Simplified 92.9%

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

   4. Taylor expanded in z around inf 90.6%

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

      1. unpow2 90.6%

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

   6. Simplified 90.6%

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

   if -1 < z < 1

   1. Initial program 85.4%

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

      1. times-frac 83.1%

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

   3. Simplified 83.1%

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

   4. Taylor expanded in z around 0 80.9%

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

      1. *-commutative 80.9%

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

      2. associate-/r* 86.7%

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

      3. associate-*r/ 95.4%

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

   6. Applied egg-rr 95.4%

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

4. Final simplification 93.0%

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

Alternative 5: 94.7% accurate, 0.8× speedup

Math

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

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.0) (not (<= z 1.0)))
   (* (/ x z) (/ (/ y z) z))
   (/ (* y (/ x z)) z)))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 1.0)) {
		tmp = (x / z) * ((y / z) / z);
	} else {
		tmp = (y * (x / z)) / 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)) .or. (.not. (z <= 1.0d0))) then
        tmp = (x / z) * ((y / z) / z)
    else
        tmp = (y * (x / z)) / 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) || !(z <= 1.0)) {
		tmp = (x / z) * ((y / z) / z);
	} else {
		tmp = (y * (x / z)) / z;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z <= -1.0) or not (z <= 1.0):
		tmp = (x / z) * ((y / z) / z)
	else:
		tmp = (y * (x / z)) / z
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.0) || !(z <= 1.0))
		tmp = Float64(Float64(x / z) * Float64(Float64(y / z) / z));
	else
		tmp = Float64(Float64(y * Float64(x / z)) / 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) || ~((z <= 1.0)))
		tmp = (x / z) * ((y / z) / z);
	else
		tmp = (y * (x / z)) / 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[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(N[(x / z), $MachinePrecision] * N[(N[(y / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 83.1%

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

      1. associate-*l* 83.1%

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

      2. times-frac 92.9%

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

      3. distribute-lft-in 92.9%

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

      4. fma-def 92.9%

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

      5. *-rgt-identity 92.9%

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

   3. Simplified 92.9%

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

   4. Taylor expanded in z around inf 90.6%

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

      1. unpow2 90.6%

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

      2. associate-/r* 93.6%

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

   6. Simplified 93.6%

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

   if -1 < z < 1

   1. Initial program 85.4%

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

      1. times-frac 83.1%

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

   3. Simplified 83.1%

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

   4. Taylor expanded in z around 0 80.9%

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

      1. *-commutative 80.9%

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

      2. associate-/r* 86.7%

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

      3. associate-*r/ 95.4%

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

   6. Applied egg-rr 95.4%

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

4. Final simplification 94.5%

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

Alternative 6: 94.9% accurate, 0.8× speedup

Math

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

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.0) (not (<= z 1.0)))
   (/ (/ (* x (/ y z)) z) z)
   (/ (* y (/ x z)) z)))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 1.0)) {
		tmp = ((x * (y / z)) / z) / z;
	} else {
		tmp = (y * (x / z)) / 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)) .or. (.not. (z <= 1.0d0))) then
        tmp = ((x * (y / z)) / z) / z
    else
        tmp = (y * (x / z)) / 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) || !(z <= 1.0)) {
		tmp = ((x * (y / z)) / z) / z;
	} else {
		tmp = (y * (x / z)) / z;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z <= -1.0) or not (z <= 1.0):
		tmp = ((x * (y / z)) / z) / z
	else:
		tmp = (y * (x / z)) / z
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.0) || !(z <= 1.0))
		tmp = Float64(Float64(Float64(x * Float64(y / z)) / z) / z);
	else
		tmp = Float64(Float64(y * Float64(x / z)) / 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) || ~((z <= 1.0)))
		tmp = ((x * (y / z)) / z) / z;
	else
		tmp = (y * (x / z)) / 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[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(N[(N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / z), $MachinePrecision], N[(N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 83.1%

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

      1. associate-*l* 83.1%

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

      2. times-frac 92.9%

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

      3. distribute-lft-in 92.9%

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

      4. fma-def 92.9%

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

      5. *-rgt-identity 92.9%

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

   3. Simplified 92.9%

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

      1. associate-*r/ 93.9%

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

      2. fma-udef 93.9%

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

      3. *-rgt-identity 93.9%

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

      4. distribute-lft-in 93.9%

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

      5. frac-times 95.4%

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

      6. associate-/r* 91.8%

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

      7. associate-*l/ 93.2%

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

      8. associate-/r* 96.9%

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

      9. associate-*r/ 89.9%

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

      10. *-commutative 89.9%

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

   5. Applied egg-rr 89.9%

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

   6. Taylor expanded in z around inf 87.7%

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

      1. associate-*l/ 94.6%

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

      2. *-commutative 94.6%

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

   8. Simplified 94.6%

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

   if -1 < z < 1

   1. Initial program 85.4%

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

      1. times-frac 83.1%

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

   3. Simplified 83.1%

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

   4. Taylor expanded in z around 0 80.9%

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

      1. *-commutative 80.9%

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

      2. associate-/r* 86.7%

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

      3. associate-*r/ 95.4%

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

   6. Applied egg-rr 95.4%

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

4. Final simplification 95.0%

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

Alternative 7: 93.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{x}{z} \cdot \frac{\frac{y}{z}}{z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{t_0}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{z \cdot z}\\ \end{array} \end{array} \]

FPCore

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

C

assert(x < y);
double code(double x, double y, double z) {
	double t_0 = y * (x / z);
	double tmp;
	if (z <= -1.0) {
		tmp = (x / z) * ((y / z) / z);
	} else if (z <= 1.0) {
		tmp = t_0 / z;
	} else {
		tmp = t_0 / (z * 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 * (x / z)
    if (z <= (-1.0d0)) then
        tmp = (x / z) * ((y / z) / z)
    else if (z <= 1.0d0) then
        tmp = t_0 / z
    else
        tmp = t_0 / (z * 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 * (x / z);
	double tmp;
	if (z <= -1.0) {
		tmp = (x / z) * ((y / z) / z);
	} else if (z <= 1.0) {
		tmp = t_0 / z;
	} else {
		tmp = t_0 / (z * z);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	t_0 = y * (x / z)
	tmp = 0
	if z <= -1.0:
		tmp = (x / z) * ((y / z) / z)
	elif z <= 1.0:
		tmp = t_0 / z
	else:
		tmp = t_0 / (z * z)
	return tmp

Julia

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

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	t_0 = y * (x / z);
	tmp = 0.0;
	if (z <= -1.0)
		tmp = (x / z) * ((y / z) / z);
	elseif (z <= 1.0)
		tmp = t_0 / z;
	else
		tmp = t_0 / (z * 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[(x / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.0], N[(N[(x / z), $MachinePrecision] * N[(N[(y / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.0], N[(t$95$0 / z), $MachinePrecision], N[(t$95$0 / N[(z * z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 82.1%

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

      1. associate-*l* 82.1%

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

      2. times-frac 93.1%

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

      3. distribute-lft-in 93.1%

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

      4. fma-def 93.1%

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

      5. *-rgt-identity 93.1%

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

   3. Simplified 93.1%

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

   4. Taylor expanded in z around inf 90.4%

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

      1. unpow2 90.4%

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

      2. associate-/r* 94.4%

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

   6. Simplified 94.4%

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

   if -1 < z < 1

   1. Initial program 85.4%

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

      1. times-frac 83.1%

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

   3. Simplified 83.1%

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

   4. Taylor expanded in z around 0 80.9%

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

      1. *-commutative 80.9%

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

      2. associate-/r* 86.7%

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

      3. associate-*r/ 95.4%

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

   6. Applied egg-rr 95.4%

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

   if 1 < z

   1. Initial program 84.3%

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

      1. associate-*l* 84.4%

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

      2. times-frac 92.7%

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

      3. distribute-lft-in 92.7%

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

      4. fma-def 92.7%

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

      5. *-rgt-identity 92.7%

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

   3. Simplified 92.7%

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

   4. Taylor expanded in z around inf 90.9%

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

      1. unpow2 90.9%

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

   6. Simplified 90.9%

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

      1. associate-*r/ 93.0%

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

   8. Applied egg-rr 93.0%

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

4. Final simplification 94.6%

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

Alternative 8: 93.9% accurate, 0.8× speedup

Math

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

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)
   (* (/ x z) (/ (/ y z) z))
   (if (<= z 1.0) (/ (* y (/ x z)) z) (/ (* x (/ y z)) (* z z)))))

C

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

Python

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

Julia

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

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 82.1%

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

      1. associate-*l* 82.1%

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

      2. times-frac 93.1%

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

      3. distribute-lft-in 93.1%

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

      4. fma-def 93.1%

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

      5. *-rgt-identity 93.1%

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

   3. Simplified 93.1%

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

   4. Taylor expanded in z around inf 90.4%

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

      1. unpow2 90.4%

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

      2. associate-/r* 94.4%

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

   6. Simplified 94.4%

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

   if -1 < z < 1

   1. Initial program 85.4%

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

      1. times-frac 83.1%

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

   3. Simplified 83.1%

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

   4. Taylor expanded in z around 0 80.9%

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

      1. *-commutative 80.9%

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

      2. associate-/r* 86.7%

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

      3. associate-*r/ 95.4%

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

   6. Applied egg-rr 95.4%

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

   if 1 < z

   1. Initial program 84.3%

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

      1. associate-*l* 84.4%

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

      2. times-frac 92.7%

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

      3. distribute-lft-in 92.7%

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

      4. fma-def 92.7%

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

      5. *-rgt-identity 92.7%

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

   3. Simplified 92.7%

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

   4. Taylor expanded in z around inf 90.9%

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

      1. unpow2 90.9%

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

   6. Simplified 90.9%

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

      1. associate-/r* 92.7%

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

      2. *-commutative 92.7%

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

      3. frac-times 93.0%

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

   8. Applied egg-rr 93.0%

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

4. Final simplification 94.6%

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

Alternative 9: 94.6% accurate, 0.8× speedup

Math

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

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)
   (* (/ x z) (/ (/ y z) z))
   (if (<= z 1.0) (/ (* y (/ x z)) z) (/ (/ y z) (* z (/ z x))))))

C

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

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if z <= -1.0:
		tmp = (x / z) * ((y / z) / z)
	elif z <= 1.0:
		tmp = (y * (x / z)) / z
	else:
		tmp = (y / z) / (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(x / z) * Float64(Float64(y / z) / z));
	elseif (z <= 1.0)
		tmp = Float64(Float64(y * Float64(x / z)) / z);
	else
		tmp = Float64(Float64(y / z) / 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 = (x / z) * ((y / z) / z);
	elseif (z <= 1.0)
		tmp = (y * (x / z)) / z;
	else
		tmp = (y / z) / (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[(x / z), $MachinePrecision] * N[(N[(y / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.0], N[(N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(y / z), $MachinePrecision] / N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 82.1%

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

      1. associate-*l* 82.1%

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

      2. times-frac 93.1%

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

      3. distribute-lft-in 93.1%

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

      4. fma-def 93.1%

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

      5. *-rgt-identity 93.1%

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

   3. Simplified 93.1%

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

   4. Taylor expanded in z around inf 90.4%

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

      1. unpow2 90.4%

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

      2. associate-/r* 94.4%

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

   6. Simplified 94.4%

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

   if -1 < z < 1

   1. Initial program 85.4%

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

      1. times-frac 83.1%

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

   3. Simplified 83.1%

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

   4. Taylor expanded in z around 0 80.9%

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

      1. *-commutative 80.9%

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

      2. associate-/r* 86.7%

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

      3. associate-*r/ 95.4%

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

   6. Applied egg-rr 95.4%

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

   if 1 < z

   1. Initial program 84.3%

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

      1. associate-*l* 84.4%

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

      2. times-frac 92.7%

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

      3. distribute-lft-in 92.7%

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

      4. fma-def 92.7%

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

      5. *-rgt-identity 92.7%

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

   3. Simplified 92.7%

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

   4. Taylor expanded in z around inf 90.9%

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

      1. unpow2 90.9%

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

   6. Simplified 90.9%

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

      1. clear-num 90.9%

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

      2. associate-/r* 92.6%

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

      3. frac-times 93.0%

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

      4. *-un-lft-identity 93.0%

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

      5. associate-/r/ 93.0%

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

      6. div-inv 93.0%

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

      7. clear-num 93.0%

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

   8. Applied egg-rr 93.0%

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

4. Final simplification 94.6%

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

Alternative 10: 79.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 8.6 \cdot 10^{+24}:\\ \;\;\;\;x \cdot \frac{\frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 8.6e+24) (* x (/ (/ y z) z)) (* y (/ x (* z z)))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 8.6e+24) {
		tmp = x * ((y / z) / z);
	} else {
		tmp = y * (x / (z * 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 (y <= 8.6d+24) then
        tmp = x * ((y / z) / z)
    else
        tmp = y * (x / (z * z))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 8.6e+24) {
		tmp = x * ((y / z) / z);
	} else {
		tmp = y * (x / (z * z));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if y <= 8.6e+24:
		tmp = x * ((y / z) / z)
	else:
		tmp = y * (x / (z * z))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (y <= 8.6e+24)
		tmp = Float64(x * Float64(Float64(y / z) / z));
	else
		tmp = Float64(y * Float64(x / Float64(z * z)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 8.6e+24)
		tmp = x * ((y / z) / z);
	else
		tmp = y * (x / (z * 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[y, 8.6e+24], N[(x * N[(N[(y / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.59999999999999975e24

   1. Initial program 84.7%

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

      1. associate-*l* 84.7%

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

      2. times-frac 94.2%

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

      3. distribute-lft-in 94.2%

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

      4. fma-def 94.2%

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

      5. *-rgt-identity 94.2%

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

   3. Simplified 94.2%

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

      1. *-commutative 94.2%

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

      2. associate-*l/ 96.8%

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

      3. fma-udef 96.8%

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

      4. distribute-lft1-in 96.8%

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

      5. frac-times 90.4%

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

      6. associate-*r/ 97.8%

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

   5. Applied egg-rr 97.8%

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

   6. Taylor expanded in z around 0 53.8%

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

      1. unpow2 53.8%

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

      2. times-frac 60.5%

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

      3. associate-*r/ 61.9%

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

      4. associate-*l/ 58.5%

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

      5. associate-*r/ 58.5%

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

      6. neg-mul-1 58.5%

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

      7. distribute-lft-neg-in 58.5%

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

      8. associate-*l/ 60.4%

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

      9. distribute-rgt-out 62.0%

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

      10. distribute-frac-neg 62.0%

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

      11. sub-neg 62.0%

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

      12. div-sub 65.7%

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

   8. Simplified 65.7%

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

   9. Taylor expanded in z around 0 76.9%

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

   if 8.59999999999999975e24 < y

   1. Initial program 83.1%

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

      1. times-frac 91.5%

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

   3. Simplified 91.5%

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

   4. Taylor expanded in z around 0 73.0%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.6 \cdot 10^{+24}:\\ \;\;\;\;x \cdot \frac{\frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 11: 79.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-39}:\\ \;\;\;\;x \cdot \frac{\frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 5.2e-39) (* x (/ (/ y z) z)) (* y (/ (/ x z) z))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 5.2e-39) {
		tmp = x * ((y / z) / z);
	} else {
		tmp = y * ((x / z) / 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 (y <= 5.2d-39) then
        tmp = x * ((y / z) / z)
    else
        tmp = y * ((x / z) / z)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 5.2e-39) {
		tmp = x * ((y / z) / z);
	} else {
		tmp = y * ((x / z) / z);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if y <= 5.2e-39:
		tmp = x * ((y / z) / z)
	else:
		tmp = y * ((x / z) / z)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (y <= 5.2e-39)
		tmp = Float64(x * Float64(Float64(y / z) / z));
	else
		tmp = Float64(y * Float64(Float64(x / z) / z));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 5.2e-39)
		tmp = x * ((y / z) / z);
	else
		tmp = y * ((x / z) / 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[y, 5.2e-39], N[(x * N[(N[(y / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(x / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.2e-39

   1. Initial program 84.8%

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

      1. associate-*l* 84.8%

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

      2. times-frac 93.9%

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

      3. distribute-lft-in 93.9%

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

      4. fma-def 93.9%

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

      5. *-rgt-identity 93.9%

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

   3. Simplified 93.9%

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

      1. *-commutative 93.9%

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

      2. associate-*l/ 96.6%

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

      3. fma-udef 96.6%

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

      4. distribute-lft1-in 96.6%

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

      5. frac-times 89.9%

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

      6. associate-*r/ 97.7%

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

   5. Applied egg-rr 97.7%

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

   6. Taylor expanded in z around 0 53.1%

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

      1. unpow2 53.1%

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

      2. times-frac 60.3%

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

      3. associate-*r/ 61.8%

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

      4. associate-*l/ 58.2%

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

      5. associate-*r/ 58.2%

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

      6. neg-mul-1 58.2%

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

      7. distribute-lft-neg-in 58.2%

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

      8. associate-*l/ 60.2%

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

      9. distribute-rgt-out 61.9%

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

      10. distribute-frac-neg 61.9%

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

      11. sub-neg 61.9%

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

      12. div-sub 65.7%

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

   8. Simplified 65.7%

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

   9. Taylor expanded in z around 0 77.1%

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

   if 5.2e-39 < y

   1. Initial program 83.1%

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

      1. times-frac 92.7%

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

   3. Simplified 92.7%

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

   4. Taylor expanded in z around 0 73.1%

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

   5. Taylor expanded in x around 0 73.1%

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

      1. unpow2 73.1%

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

      2. associate-/l/ 71.7%

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

   7. Simplified 71.7%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-39}:\\ \;\;\;\;x \cdot \frac{\frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \]

Alternative 12: 80.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{-38}:\\ \;\;\;\;\frac{x}{z \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 1.6e-38) (/ x (* z (/ z y))) (* y (/ (/ x z) z))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.6e-38) {
		tmp = x / (z * (z / y));
	} else {
		tmp = y * ((x / z) / 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 (y <= 1.6d-38) then
        tmp = x / (z * (z / y))
    else
        tmp = y * ((x / z) / z)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.6e-38) {
		tmp = x / (z * (z / y));
	} else {
		tmp = y * ((x / z) / z);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if y <= 1.6e-38:
		tmp = x / (z * (z / y))
	else:
		tmp = y * ((x / z) / z)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (y <= 1.6e-38)
		tmp = Float64(x / Float64(z * Float64(z / y)));
	else
		tmp = Float64(y * Float64(Float64(x / z) / z));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.6e-38)
		tmp = x / (z * (z / y));
	else
		tmp = y * ((x / z) / 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[y, 1.6e-38], N[(x / N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(x / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.59999999999999989e-38

   1. Initial program 84.8%

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

      1. /-rgt-identity 84.8%

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

      2. associate-/l* 84.7%

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

      3. associate-/l/ 81.3%

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

      4. associate-*l* 83.5%

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

      5. associate-*r/ 83.4%

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

      6. *-rgt-identity 83.4%

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

      7. associate-*l* 89.8%

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

      8. associate-*r/ 89.9%

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

      9. distribute-lft-in 89.9%

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

      10. fma-def 89.9%

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

      11. *-rgt-identity 89.9%

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

   3. Simplified 89.9%

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

   4. Taylor expanded in z around 0 77.1%

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

   if 1.59999999999999989e-38 < y

   1. Initial program 83.1%

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

      1. times-frac 92.7%

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

   3. Simplified 92.7%

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

   4. Taylor expanded in z around 0 73.1%

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

   5. Taylor expanded in x around 0 73.1%

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

      1. unpow2 73.1%

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

      2. associate-/l/ 71.7%

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

   7. Simplified 71.7%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{-38}:\\ \;\;\;\;\frac{x}{z \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \]

Alternative 13: 80.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.9 \cdot 10^{-31}:\\ \;\;\;\;\frac{x}{z \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 4.9e-31) (/ x (* z (/ z y))) (/ y (* z (/ z x)))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.9e-31) {
		tmp = x / (z * (z / y));
	} else {
		tmp = 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 (y <= 4.9d-31) then
        tmp = x / (z * (z / y))
    else
        tmp = 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 (y <= 4.9e-31) {
		tmp = x / (z * (z / y));
	} else {
		tmp = y / (z * (z / x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if y <= 4.9e-31:
		tmp = x / (z * (z / y))
	else:
		tmp = y / (z * (z / x))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (y <= 4.9e-31)
		tmp = Float64(x / Float64(z * Float64(z / y)));
	else
		tmp = Float64(y / Float64(z * Float64(z / x)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 4.9e-31)
		tmp = x / (z * (z / y));
	else
		tmp = 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[y, 4.9e-31], N[(x / N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.90000000000000023e-31

   1. Initial program 85.0%

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

      1. /-rgt-identity 85.0%

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

      2. associate-/l* 85.0%

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

      3. associate-/l/ 81.6%

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

      4. associate-*l* 83.7%

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

      5. associate-*r/ 83.7%

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

      6. *-rgt-identity 83.7%

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

      7. associate-*l* 90.0%

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

      8. associate-*r/ 90.0%

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

      9. distribute-lft-in 90.0%

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

      10. fma-def 90.0%

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

      11. *-rgt-identity 90.0%

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

   3. Simplified 90.0%

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

   4. Taylor expanded in z around 0 77.5%

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

   if 4.90000000000000023e-31 < y

   1. Initial program 82.4%

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

      1. associate-*l* 82.4%

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

      2. times-frac 94.6%

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

      3. distribute-lft-in 94.6%

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

      4. fma-def 94.6%

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

      5. *-rgt-identity 94.6%

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

   3. Simplified 94.6%

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

   4. Taylor expanded in z around 0 67.1%

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

      1. unpow2 67.1%

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

      2. associate-/l* 72.0%

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

      3. associate-/l* 71.8%

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

   6. Simplified 71.8%

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

      1. associate-/r/ 71.8%

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

   8. Applied egg-rr 71.8%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.9 \cdot 10^{-31}:\\ \;\;\;\;\frac{x}{z \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 14: 40.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \frac{-y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z -1e-310) (* x (/ (- y) z)) (* x (/ y z))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= -1e-310) {
		tmp = x * (-y / z);
	} else {
		tmp = 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 <= (-1d-310)) then
        tmp = x * (-y / z)
    else
        tmp = 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 <= -1e-310) {
		tmp = x * (-y / z);
	} else {
		tmp = x * (y / z);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if z <= -1e-310:
		tmp = x * (-y / z)
	else:
		tmp = x * (y / z)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (z <= -1e-310)
		tmp = Float64(x * Float64(Float64(-y) / z));
	else
		tmp = Float64(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 <= -1e-310)
		tmp = x * (-y / z);
	else
		tmp = 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, -1e-310], N[(x * N[((-y) / z), $MachinePrecision]), $MachinePrecision], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -9.999999999999969e-311

   1. Initial program 84.6%

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

      1. associate-*r/ 82.5%

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

      2. associate-*l* 82.5%

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

      3. distribute-lft-in 82.6%

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

      4. fma-def 82.5%

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

      5. *-rgt-identity 82.5%

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

   3. Simplified 82.5%

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

   4. Taylor expanded in z around 0 53.7%

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

      1. +-commutative 53.7%

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

      2. mul-1-neg 53.7%

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

      3. unsub-neg 53.7%

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

      4. unpow2 53.7%

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

      5. associate-/r* 56.8%

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

   6. Simplified 56.8%

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

   7. Taylor expanded in z around inf 27.8%

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

      1. associate-*r/ 27.8%

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

      2. neg-mul-1 27.8%

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

      3. distribute-lft-neg-in 27.8%

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

      4. associate-*l/ 32.2%

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

      5. *-commutative 32.2%

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

   9. Simplified 32.2%

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

   if -9.999999999999969e-311 < z

   1. Initial program 83.9%

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

      1. associate-*r/ 83.3%

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

      2. associate-*l* 83.3%

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

      3. distribute-lft-in 83.3%

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

      4. fma-def 83.3%

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

      5. *-rgt-identity 83.3%

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

   3. Simplified 83.3%

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

   4. Taylor expanded in z around 0 45.3%

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

      1. +-commutative 45.3%

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

      2. mul-1-neg 45.3%

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

      3. unsub-neg 45.3%

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

      4. unpow2 45.3%

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

      5. associate-/r* 51.3%

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

   6. Simplified 51.3%

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

   7. Taylor expanded in z around inf 16.8%

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

      1. associate-*r/ 16.8%

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

      2. neg-mul-1 16.8%

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

      3. distribute-lft-neg-in 16.8%

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

      4. associate-*l/ 19.0%

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

      5. *-commutative 19.0%

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

   9. Simplified 19.0%

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

       1. clear-num 19.8%

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

       2. un-div-inv 19.8%

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

       3. add-sqr-sqrt 8.2%

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

       4. sqrt-unprod 34.2%

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

       5. sqr-neg 34.2%

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

       6. sqrt-unprod 25.4%

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

       7. add-sqr-sqrt 41.8%

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

   11. Applied egg-rr 41.8%

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

       1. clear-num 42.2%

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

       2. associate-/r/ 42.6%

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

       3. clear-num 41.9%

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

   13. Applied egg-rr 41.9%

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

4. Final simplification 37.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \frac{-y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]

Alternative 15: 40.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{x}{\frac{-z}{y}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z -1e-310) (/ x (/ (- z) y)) (* x (/ y z))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= -1e-310) {
		tmp = x / (-z / y);
	} else {
		tmp = 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 <= (-1d-310)) then
        tmp = x / (-z / y)
    else
        tmp = 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 <= -1e-310) {
		tmp = x / (-z / y);
	} else {
		tmp = x * (y / z);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if z <= -1e-310:
		tmp = x / (-z / y)
	else:
		tmp = x * (y / z)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (z <= -1e-310)
		tmp = Float64(x / Float64(Float64(-z) / y));
	else
		tmp = Float64(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 <= -1e-310)
		tmp = x / (-z / y);
	else
		tmp = 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, -1e-310], N[(x / N[((-z) / y), $MachinePrecision]), $MachinePrecision], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -9.999999999999969e-311

   1. Initial program 84.6%

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

      1. associate-*r/ 82.5%

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

      2. associate-*l* 82.5%

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

      3. distribute-lft-in 82.6%

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

      4. fma-def 82.5%

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

      5. *-rgt-identity 82.5%

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

   3. Simplified 82.5%

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

   4. Taylor expanded in z around 0 53.7%

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

      1. +-commutative 53.7%

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

      2. mul-1-neg 53.7%

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

      3. unsub-neg 53.7%

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

      4. unpow2 53.7%

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

      5. associate-/r* 56.8%

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

   6. Simplified 56.8%

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

   7. Taylor expanded in z around inf 27.8%

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

      1. associate-*r/ 27.8%

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

      2. neg-mul-1 27.8%

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

      3. distribute-lft-neg-in 27.8%

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

      4. associate-*l/ 32.2%

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

      5. *-commutative 32.2%

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

   9. Simplified 32.2%

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

       1. associate-*r/ 27.8%

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

       2. frac-2neg 27.8%

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

       3. add-sqr-sqrt 13.5%

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

       4. sqrt-unprod 22.6%

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

       5. sqr-neg 22.6%

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

       6. sqrt-unprod 8.1%

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

       7. add-sqr-sqrt 17.3%

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

       8. *-commutative 17.3%

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

       9. distribute-lft-neg-in 17.3%

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

       10. add-sqr-sqrt 9.2%

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

       11. sqrt-unprod 27.4%

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

       12. sqr-neg 27.4%

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

       13. sqrt-unprod 14.2%

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

       14. add-sqr-sqrt 27.8%

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

   11. Applied egg-rr 27.8%

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

       1. *-commutative 27.8%

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

       2. associate-/l* 33.7%

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

   13. Simplified 33.7%

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

   if -9.999999999999969e-311 < z

   1. Initial program 83.9%

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

      1. associate-*r/ 83.3%

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

      2. associate-*l* 83.3%

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

      3. distribute-lft-in 83.3%

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

      4. fma-def 83.3%

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

      5. *-rgt-identity 83.3%

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

   3. Simplified 83.3%

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

   4. Taylor expanded in z around 0 45.3%

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

      1. +-commutative 45.3%

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

      2. mul-1-neg 45.3%

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

      3. unsub-neg 45.3%

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

      4. unpow2 45.3%

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

      5. associate-/r* 51.3%

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

   6. Simplified 51.3%

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

   7. Taylor expanded in z around inf 16.8%

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

      1. associate-*r/ 16.8%

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

      2. neg-mul-1 16.8%

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

      3. distribute-lft-neg-in 16.8%

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

      4. associate-*l/ 19.0%

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

      5. *-commutative 19.0%

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

   9. Simplified 19.0%

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

       1. clear-num 19.8%

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

       2. un-div-inv 19.8%

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

       3. add-sqr-sqrt 8.2%

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

       4. sqrt-unprod 34.2%

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

       5. sqr-neg 34.2%

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

       6. sqrt-unprod 25.4%

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

       7. add-sqr-sqrt 41.8%

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

   11. Applied egg-rr 41.8%

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

       1. clear-num 42.2%

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

       2. associate-/r/ 42.6%

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

       3. clear-num 41.9%

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

   13. Applied egg-rr 41.9%

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

4. Final simplification 37.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{x}{\frac{-z}{y}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]

Alternative 16: 74.3% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

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 = x * ((y / z) / z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.3%

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

   1. associate-*l* 84.3%

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

   2. times-frac 94.2%

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

   3. distribute-lft-in 94.2%

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

   4. fma-def 94.2%

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

   5. *-rgt-identity 94.2%

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

3. Simplified 94.2%

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

   1. *-commutative 94.2%

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

   2. associate-*l/ 95.7%

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

   3. fma-udef 95.7%

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

   4. distribute-lft1-in 95.7%

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

   5. frac-times 92.2%

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

   6. associate-*r/ 97.6%

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

5. Applied egg-rr 97.6%

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

6. Taylor expanded in z around 0 49.8%

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

   1. unpow2 49.8%

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

   2. times-frac 54.9%

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

   3. associate-*r/ 56.0%

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

   4. associate-*l/ 52.7%

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

   5. associate-*r/ 52.7%

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

   6. neg-mul-1 52.7%

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

   7. distribute-lft-neg-in 52.7%

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

   8. associate-*l/ 51.7%

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

   9. distribute-rgt-out 54.1%

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

   10. distribute-frac-neg 54.1%

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

   11. sub-neg 54.1%

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

   12. div-sub 61.9%

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

8. Simplified 61.9%

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

9. Taylor expanded in z around 0 74.7%

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

10. Final simplification 74.7%

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

Alternative 17: 31.1% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

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 = y * (x / z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.3%

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

   1. associate-*r/ 82.9%

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

   2. associate-*l* 82.9%

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

   3. distribute-lft-in 82.9%

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

   4. fma-def 82.9%

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

   5. *-rgt-identity 82.9%

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

3. Simplified 82.9%

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

4. Taylor expanded in z around 0 49.5%

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

   1. +-commutative 49.5%

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

   2. mul-1-neg 49.5%

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

   3. unsub-neg 49.5%

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

   4. unpow2 49.5%

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

   5. associate-/r* 54.1%

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

6. Simplified 54.1%

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

7. Taylor expanded in z around inf 22.3%

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

   1. associate-*r/ 22.3%

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

   2. neg-mul-1 22.3%

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

   3. distribute-lft-neg-in 22.3%

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

   4. associate-*l/ 25.7%

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

   5. *-commutative 25.7%

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

9. Simplified 25.7%

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

    1. clear-num 26.8%

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

    2. un-div-inv 26.8%

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

    3. add-sqr-sqrt 12.0%

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

    4. sqrt-unprod 29.1%

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

    5. sqr-neg 29.1%

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

    6. sqrt-unprod 17.5%

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

    7. add-sqr-sqrt 31.3%

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

11. Applied egg-rr 31.3%

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

    1. associate-/r/ 32.3%

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

13. Applied egg-rr 32.3%

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

14. Final simplification 32.3%

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

Alternative 18: 31.1% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

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 = x * (y / z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.3%

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

   1. associate-*r/ 82.9%

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

   2. associate-*l* 82.9%

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

   3. distribute-lft-in 82.9%

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

   4. fma-def 82.9%

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

   5. *-rgt-identity 82.9%

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

3. Simplified 82.9%

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

4. Taylor expanded in z around 0 49.5%

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

   1. +-commutative 49.5%

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

   2. mul-1-neg 49.5%

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

   3. unsub-neg 49.5%

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

   4. unpow2 49.5%

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

   5. associate-/r* 54.1%

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

6. Simplified 54.1%

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

7. Taylor expanded in z around inf 22.3%

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

   1. associate-*r/ 22.3%

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

   2. neg-mul-1 22.3%

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

   3. distribute-lft-neg-in 22.3%

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

   4. associate-*l/ 25.7%

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

   5. *-commutative 25.7%

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

9. Simplified 25.7%

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

    1. clear-num 26.8%

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

    2. un-div-inv 26.8%

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

    3. add-sqr-sqrt 12.0%

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

    4. sqrt-unprod 29.1%

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

    5. sqr-neg 29.1%

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

    6. sqrt-unprod 17.5%

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

    7. add-sqr-sqrt 31.3%

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

11. Applied egg-rr 31.3%

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

    1. clear-num 31.5%

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

    2. associate-/r/ 31.7%

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

    3. clear-num 30.6%

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

13. Applied egg-rr 30.6%

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

14. Final simplification 30.6%

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

Developer target: 95.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z < 249.6182814532307:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (< z 249.6182814532307)
   (/ (* y (/ x z)) (+ z (* z z)))
   (/ (* (/ (/ y z) (+ 1.0 z)) x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z < 249.6182814532307) {
		tmp = (y * (x / z)) / (z + (z * z));
	} else {
		tmp = (((y / z) / (1.0 + z)) * x) / z;
	}
	return tmp;
}

Fortran

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 < 249.6182814532307d0) then
        tmp = (y * (x / z)) / (z + (z * z))
    else
        tmp = (((y / z) / (1.0d0 + z)) * x) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z < 249.6182814532307) {
		tmp = (y * (x / z)) / (z + (z * z));
	} else {
		tmp = (((y / z) / (1.0 + z)) * x) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z < 249.6182814532307:
		tmp = (y * (x / z)) / (z + (z * z))
	else:
		tmp = (((y / z) / (1.0 + z)) * x) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z < 249.6182814532307)
		tmp = Float64(Float64(y * Float64(x / z)) / Float64(z + Float64(z * z)));
	else
		tmp = Float64(Float64(Float64(Float64(y / z) / Float64(1.0 + z)) * x) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z < 249.6182814532307)
		tmp = (y * (x / z)) / (z + (z * z));
	else
		tmp = (((y / z) / (1.0 + z)) * x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Less[z, 249.6182814532307], N[(N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision] / N[(z + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y / z), $MachinePrecision] / N[(1.0 + z), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision]]

Reproduce

herbie shell --seed 2023192 
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< z 249.6182814532307) (/ (* y (/ x z)) (+ z (* z z))) (/ (* (/ (/ y z) (+ 1.0 z)) x) z))

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