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

Percentage Accurate: 82.9% → 97.0%

Time: 9.0s

Alternatives: 12

Speedup: 1.0×

• 25.2% 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: 82.9% 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 80.4%

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

   1. associate-*r/ 81.8%

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

   2. sqr-neg 81.8%

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

   3. *-commutative 81.8%

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

   4. distribute-rgt1-in 65.3%

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

   5. sqr-neg 65.3%

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

   6. fma-def 81.8%

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

   7. sqr-neg 81.8%

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

   8. cube-unmult 81.8%

      \[\leadsto x \cdot \frac{y}{\mathsf{fma}\left(z, z, \color{blue}{{z}^{3}}\right)} \]

3. Simplified 81.8%

   \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]
4. Step-by-step derivation

   1. associate-*r/ 80.4%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]

   2. fma-udef 64.8%

      \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot z + {z}^{3}}} \]

   3. cube-mult 64.8%

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

   4. distribute-rgt1-in 80.4%

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

   5. *-commutative 80.4%

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

   6. frac-times 86.4%

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

   7. *-commutative 86.4%

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

   8. associate-/r* 91.6%

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

   9. associate-*r/ 96.7%

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

5. Applied egg-rr 96.7%

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

6. Final simplification 96.7%

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

Alternative 2: 94.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -130000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\frac{x}{z} \cdot \frac{\frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= z -130000.0) (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 <= -130000.0) || !(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 <= (-130000.0d0)) .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 <= -130000.0) || !(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 <= -130000.0) 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 <= -130000.0) || !(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 <= -130000.0) || ~((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, -130000.0], 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 < -1.3e5 or 1 < z

   1. Initial program 78.7%

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

      1. frac-times 93.0%

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

      2. associate-*l/ 91.0%

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

      3. times-frac 98.3%

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

   3. Applied egg-rr 98.3%

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

   4. Taylor expanded in z around inf 97.1%

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

   if -1.3e5 < z < 1

   1. Initial program 82.3%

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

      1. associate-*r/ 79.4%

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

      2. sqr-neg 79.4%

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

      3. *-commutative 79.4%

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

      4. distribute-rgt1-in 79.4%

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

      5. sqr-neg 79.4%

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

      6. fma-def 79.4%

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

      7. sqr-neg 79.4%

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

      8. cube-unmult 79.4%

         \[\leadsto x \cdot \frac{y}{\mathsf{fma}\left(z, z, \color{blue}{{z}^{3}}\right)} \]

   3. Simplified 79.4%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 82.3%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]

      2. fma-udef 82.3%

         \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot z + {z}^{3}}} \]

      3. cube-mult 82.3%

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

      4. distribute-rgt1-in 82.3%

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

      5. *-commutative 82.3%

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

      6. frac-times 79.2%

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

      7. *-commutative 79.2%

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

      8. associate-/r* 84.7%

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

      9. associate-*r/ 94.5%

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

   5. Applied egg-rr 94.5%

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

   6. Taylor expanded in z around 0 88.6%

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

      1. *-commutative 88.6%

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

      2. associate-*r/ 93.6%

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

   8. Simplified 93.6%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -130000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\frac{x}{z} \cdot \frac{\frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z}\\ \end{array} \]

Alternative 3: 95.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -130000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\frac{\frac{x}{z} \cdot \frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= z -130000.0) (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 <= -130000.0) || !(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 <= (-130000.0d0)) .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 <= -130000.0) || !(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 <= -130000.0) 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 <= -130000.0) || !(z <= 1.0))
		tmp = Float64(Float64(Float64(x / z) * 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 <= -130000.0) || ~((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, -130000.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(N[(N[(x / z), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.3e5 or 1 < z

   1. Initial program 78.7%

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

      1. sqr-neg 78.7%

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

      2. times-frac 93.0%

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

      3. sqr-neg 93.0%

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

   3. Simplified 93.0%

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

   4. Taylor expanded in z around inf 91.7%

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

      1. *-commutative 91.7%

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

      2. associate-/r* 96.7%

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

      3. associate-*r/ 97.3%

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

   6. Applied egg-rr 97.3%

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

   if -1.3e5 < z < 1

   1. Initial program 82.3%

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

      1. associate-*r/ 79.4%

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

      2. sqr-neg 79.4%

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

      3. *-commutative 79.4%

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

      4. distribute-rgt1-in 79.4%

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

      5. sqr-neg 79.4%

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

      6. fma-def 79.4%

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

      7. sqr-neg 79.4%

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

      8. cube-unmult 79.4%

         \[\leadsto x \cdot \frac{y}{\mathsf{fma}\left(z, z, \color{blue}{{z}^{3}}\right)} \]

   3. Simplified 79.4%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 82.3%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]

      2. fma-udef 82.3%

         \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot z + {z}^{3}}} \]

      3. cube-mult 82.3%

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

      4. distribute-rgt1-in 82.3%

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

      5. *-commutative 82.3%

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

      6. frac-times 79.2%

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

      7. *-commutative 79.2%

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

      8. associate-/r* 84.7%

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

      9. associate-*r/ 94.5%

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

   5. Applied egg-rr 94.5%

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

   6. Taylor expanded in z around 0 88.6%

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

      1. *-commutative 88.6%

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

      2. associate-*r/ 93.6%

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

   8. Simplified 93.6%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -130000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\frac{\frac{x}{z} \cdot \frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z}\\ \end{array} \]

Alternative 4: 94.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -130000:\\ \;\;\;\;\frac{\frac{x}{z} \cdot \frac{y}{z}}{z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{z \cdot \frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z -130000.0)
   (/ (* (/ 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 <= -130000.0) {
		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 <= (-130000.0d0)) 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 <= -130000.0) {
		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 <= -130000.0:
		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 <= -130000.0)
		tmp = Float64(Float64(Float64(x / z) * 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 <= -130000.0)
		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, -130000.0], N[(N[(N[(x / z), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision] / z), $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 < -1.3e5

   1. Initial program 79.4%

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

      1. sqr-neg 79.4%

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

      2. times-frac 93.0%

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

      3. sqr-neg 93.0%

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

   3. Simplified 93.0%

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

   4. Taylor expanded in z around inf 91.8%

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

      1. *-commutative 91.8%

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

      2. associate-/r* 96.9%

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

      3. associate-*r/ 97.3%

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

   6. Applied egg-rr 97.3%

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

   if -1.3e5 < z < 1

   1. Initial program 82.3%

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

      1. associate-*r/ 79.4%

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

      2. sqr-neg 79.4%

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

      3. *-commutative 79.4%

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

      4. distribute-rgt1-in 79.4%

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

      5. sqr-neg 79.4%

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

      6. fma-def 79.4%

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

      7. sqr-neg 79.4%

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

      8. cube-unmult 79.4%

         \[\leadsto x \cdot \frac{y}{\mathsf{fma}\left(z, z, \color{blue}{{z}^{3}}\right)} \]

   3. Simplified 79.4%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 82.3%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]

      2. fma-udef 82.3%

         \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot z + {z}^{3}}} \]

      3. cube-mult 82.3%

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

      4. distribute-rgt1-in 82.3%

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

      5. *-commutative 82.3%

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

      6. frac-times 79.2%

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

      7. *-commutative 79.2%

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

      8. associate-/r* 84.7%

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

      9. associate-*r/ 94.5%

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

   5. Applied egg-rr 94.5%

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

   6. Taylor expanded in z around 0 88.6%

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

      1. *-commutative 88.6%

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

      2. associate-*r/ 93.6%

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

   8. Simplified 93.6%

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

   if 1 < z

   1. Initial program 78.0%

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

      1. frac-times 92.9%

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

      2. associate-*l/ 90.2%

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

      3. times-frac 98.1%

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

   3. Applied egg-rr 98.1%

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

   4. Taylor expanded in z around inf 96.9%

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

      1. *-commutative 96.9%

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

      2. associate-*l/ 97.4%

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

      3. associate-/l* 96.6%

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

      4. div-inv 96.5%

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

      5. clear-num 96.5%

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

   6. Applied egg-rr 96.5%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -130000:\\ \;\;\;\;\frac{\frac{x}{z} \cdot \frac{y}{z}}{z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{z \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 5: 95.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -130000:\\ \;\;\;\;\frac{\frac{x}{z} \cdot \frac{y}{z}}{z}\\ \mathbf{elif}\;z \leq 0.76:\\ \;\;\;\;\frac{y \cdot \left(\frac{x}{z} - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{z \cdot \frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z -130000.0)
   (/ (* (/ x z) (/ y z)) z)
   (if (<= z 0.76) (/ (* y (- (/ x z) x)) z) (/ (/ y z) (* z (/ z x))))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= -130000.0) {
		tmp = ((x / z) * (y / z)) / z;
	} else if (z <= 0.76) {
		tmp = (y * ((x / z) - x)) / 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 <= (-130000.0d0)) then
        tmp = ((x / z) * (y / z)) / z
    else if (z <= 0.76d0) then
        tmp = (y * ((x / z) - x)) / 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 <= -130000.0) {
		tmp = ((x / z) * (y / z)) / z;
	} else if (z <= 0.76) {
		tmp = (y * ((x / z) - x)) / 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 <= -130000.0:
		tmp = ((x / z) * (y / z)) / z
	elif z <= 0.76:
		tmp = (y * ((x / z) - x)) / 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 <= -130000.0)
		tmp = Float64(Float64(Float64(x / z) * Float64(y / z)) / z);
	elseif (z <= 0.76)
		tmp = Float64(Float64(y * Float64(Float64(x / z) - x)) / 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 <= -130000.0)
		tmp = ((x / z) * (y / z)) / z;
	elseif (z <= 0.76)
		tmp = (y * ((x / z) - x)) / 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, -130000.0], N[(N[(N[(x / z), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 0.76], N[(N[(y * N[(N[(x / z), $MachinePrecision] - x), $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 < -1.3e5

   1. Initial program 79.4%

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

      1. sqr-neg 79.4%

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

      2. times-frac 93.0%

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

      3. sqr-neg 93.0%

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

   3. Simplified 93.0%

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

   4. Taylor expanded in z around inf 91.8%

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

      1. *-commutative 91.8%

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

      2. associate-/r* 96.9%

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

      3. associate-*r/ 97.3%

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

   6. Applied egg-rr 97.3%

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

   if -1.3e5 < z < 0.76000000000000001

   1. Initial program 82.3%

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

      1. associate-*r/ 79.4%

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

      2. sqr-neg 79.4%

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

      3. *-commutative 79.4%

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

      4. distribute-rgt1-in 79.4%

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

      5. sqr-neg 79.4%

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

      6. fma-def 79.4%

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

      7. sqr-neg 79.4%

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

      8. cube-unmult 79.4%

         \[\leadsto x \cdot \frac{y}{\mathsf{fma}\left(z, z, \color{blue}{{z}^{3}}\right)} \]

   3. Simplified 79.4%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 82.3%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]

      2. fma-udef 82.3%

         \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot z + {z}^{3}}} \]

      3. cube-mult 82.3%

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

      4. distribute-rgt1-in 82.3%

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

      5. *-commutative 82.3%

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

      6. frac-times 79.2%

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

      7. *-commutative 79.2%

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

      8. associate-/r* 84.7%

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

      9. associate-*r/ 94.5%

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

   5. Applied egg-rr 94.5%

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

   6. Taylor expanded in z around 0 86.6%

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

      1. mul-1-neg 86.6%

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

      2. distribute-lft-neg-out 86.6%

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

      3. +-commutative 86.6%

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

      4. *-commutative 86.6%

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

      5. associate-*r/ 91.6%

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

      6. *-commutative 91.6%

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

      7. distribute-lft-out 94.0%

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

   8. Simplified 94.0%

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

   if 0.76000000000000001 < z

   1. Initial program 78.0%

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

      1. frac-times 92.9%

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

      2. associate-*l/ 90.2%

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

      3. times-frac 98.1%

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

   3. Applied egg-rr 98.1%

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

   4. Taylor expanded in z around inf 96.9%

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

      1. *-commutative 96.9%

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

      2. associate-*l/ 97.4%

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

      3. associate-/l* 96.6%

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

      4. div-inv 96.5%

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

      5. clear-num 96.5%

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

   6. Applied egg-rr 96.5%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -130000:\\ \;\;\;\;\frac{\frac{x}{z} \cdot \frac{y}{z}}{z}\\ \mathbf{elif}\;z \leq 0.76:\\ \;\;\;\;\frac{y \cdot \left(\frac{x}{z} - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{z \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 6: 78.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{-23} \lor \neg \left(y \leq 2.4 \cdot 10^{-52}\right):\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.45e-23) (not (<= y 2.4e-52)))
   (* y (/ x (* z z)))
   (* (/ x z) (/ y z))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.45e-23) || !(y <= 2.4e-52)) {
		tmp = y * (x / (z * z));
	} else {
		tmp = (x / z) * (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 ((y <= (-1.45d-23)) .or. (.not. (y <= 2.4d-52))) then
        tmp = y * (x / (z * z))
    else
        tmp = (x / z) * (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 ((y <= -1.45e-23) || !(y <= 2.4e-52)) {
		tmp = y * (x / (z * z));
	} else {
		tmp = (x / z) * (y / z);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (y <= -1.45e-23) or not (y <= 2.4e-52):
		tmp = y * (x / (z * z))
	else:
		tmp = (x / z) * (y / z)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.45e-23) || !(y <= 2.4e-52))
		tmp = Float64(y * Float64(x / Float64(z * z)));
	else
		tmp = Float64(Float64(x / z) * 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 ((y <= -1.45e-23) || ~((y <= 2.4e-52)))
		tmp = y * (x / (z * z));
	else
		tmp = (x / z) * (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[Or[LessEqual[y, -1.45e-23], N[Not[LessEqual[y, 2.4e-52]], $MachinePrecision]], N[(y * N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.4500000000000001e-23 or 2.4000000000000002e-52 < y

   1. Initial program 80.3%

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

      1. sqr-neg 80.3%

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

      2. times-frac 92.2%

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

      3. sqr-neg 92.2%

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

   3. Simplified 92.2%

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

   4. Taylor expanded in z around 0 75.2%

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

   if -1.4500000000000001e-23 < y < 2.4000000000000002e-52

   1. Initial program 80.5%

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

      1. frac-times 78.8%

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

      2. associate-*l/ 82.2%

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

      3. times-frac 94.1%

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

   3. Applied egg-rr 94.1%

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

   4. Taylor expanded in z around 0 86.6%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{-23} \lor \neg \left(y \leq 2.4 \cdot 10^{-52}\right):\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \end{array} \]

Alternative 7: 96.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

assert(x < y);
double code(double x, double y, double z) {
	return (x / z) * ((y / (z + 1.0)) / 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 / z) * ((y / (z + 1.0d0)) / z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.4%

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

   1. frac-times 86.4%

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

   2. associate-*l/ 86.8%

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

   3. times-frac 95.4%

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

3. Applied egg-rr 95.4%

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

4. Final simplification 95.4%

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

Alternative 8: 77.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{-53}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 3e-53) (* (/ x z) (/ y z)) (* y (/ (/ x z) z))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 3e-53) {
		tmp = (x / z) * (y / 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 <= 3d-53) then
        tmp = (x / z) * (y / 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 <= 3e-53) {
		tmp = (x / z) * (y / z);
	} else {
		tmp = y * ((x / z) / z);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if y <= 3e-53:
		tmp = (x / z) * (y / 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 <= 3e-53)
		tmp = Float64(Float64(x / z) * Float64(y / 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 <= 3e-53)
		tmp = (x / z) * (y / 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, 3e-53], N[(N[(x / z), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(x / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.0000000000000002e-53

   1. Initial program 82.6%

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

      1. frac-times 85.7%

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

      2. associate-*l/ 87.7%

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

      3. times-frac 95.7%

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

   3. Applied egg-rr 95.7%

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

   4. Taylor expanded in z around 0 78.7%

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

   if 3.0000000000000002e-53 < y

   1. Initial program 75.1%

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

      1. associate-*r/ 75.4%

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

      2. sqr-neg 75.4%

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

      3. *-commutative 75.4%

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

      4. distribute-rgt1-in 69.4%

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

      5. sqr-neg 69.4%

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

      6. fma-def 75.4%

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

      7. sqr-neg 75.4%

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

      8. cube-unmult 75.5%

         \[\leadsto x \cdot \frac{y}{\mathsf{fma}\left(z, z, \color{blue}{{z}^{3}}\right)} \]

   3. Simplified 75.5%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 75.1%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]

      2. fma-udef 69.4%

         \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot z + {z}^{3}}} \]

      3. cube-mult 69.4%

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

      4. distribute-rgt1-in 75.1%

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

      5. *-commutative 75.1%

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

      6. frac-times 88.3%

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

      7. *-commutative 88.3%

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

      8. associate-/r* 98.4%

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

      9. associate-*r/ 97.4%

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

   5. Applied egg-rr 97.4%

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

   6. Taylor expanded in z around 0 63.2%

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

      1. *-commutative 63.2%

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

      2. associate-*r/ 63.9%

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

   8. Simplified 63.9%

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

      1. *-un-lft-identity 63.9%

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

      2. times-frac 69.1%

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

      3. /-rgt-identity 69.1%

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

   10. Applied egg-rr 69.1%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{-53}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \]

Alternative 9: 79.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -9.6 \cdot 10^{-60}:\\ \;\;\;\;\frac{x}{z \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= x -9.6e-60) (/ x (* z (/ z y))) (* y (/ (/ x z) z))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (x <= -9.6e-60) {
		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 (x <= (-9.6d-60)) 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 (x <= -9.6e-60) {
		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 x <= -9.6e-60:
		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 (x <= -9.6e-60)
		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 (x <= -9.6e-60)
		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[x, -9.6e-60], 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 x < -9.60000000000000038e-60

   1. Initial program 77.4%

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

      1. frac-times 83.8%

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

      2. associate-*l/ 87.3%

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

      3. times-frac 93.9%

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

   3. Applied egg-rr 93.9%

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

   4. Taylor expanded in z around 0 63.0%

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

      1. associate-*l/ 66.3%

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

      2. associate-/l* 72.0%

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

      3. div-inv 72.0%

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

      4. clear-num 72.0%

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

   6. Applied egg-rr 72.0%

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

   if -9.60000000000000038e-60 < x

   1. Initial program 81.8%

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

      1. associate-*r/ 81.8%

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

      2. sqr-neg 81.8%

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

      3. *-commutative 81.8%

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

      4. distribute-rgt1-in 63.2%

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

      5. sqr-neg 63.2%

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

      6. fma-def 81.8%

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

      7. sqr-neg 81.8%

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

      8. cube-unmult 81.9%

         \[\leadsto x \cdot \frac{y}{\mathsf{fma}\left(z, z, \color{blue}{{z}^{3}}\right)} \]

   3. Simplified 81.9%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 81.8%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]

      2. fma-udef 63.1%

         \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot z + {z}^{3}}} \]

      3. cube-mult 63.1%

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

      4. distribute-rgt1-in 81.8%

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

      5. *-commutative 81.8%

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

      6. frac-times 87.5%

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

      7. *-commutative 87.5%

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

      8. associate-/r* 92.4%

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

      9. associate-*r/ 96.8%

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

   5. Applied egg-rr 96.8%

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

   6. Taylor expanded in z around 0 72.9%

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

      1. *-commutative 72.9%

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

      2. associate-*r/ 78.6%

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

   8. Simplified 78.6%

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

      1. *-un-lft-identity 78.6%

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

      2. times-frac 79.1%

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

      3. /-rgt-identity 79.1%

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

   10. Applied egg-rr 79.1%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.6 \cdot 10^{-60}:\\ \;\;\;\;\frac{x}{z \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \]

Alternative 10: 79.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-60}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{y}{z}}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= x -6.5e-60) (/ x (/ z (/ y z))) (* y (/ (/ x z) z))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (x <= -6.5e-60) {
		tmp = x / (z / (y / 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 (x <= (-6.5d-60)) then
        tmp = x / (z / (y / 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 (x <= -6.5e-60) {
		tmp = x / (z / (y / z));
	} else {
		tmp = y * ((x / z) / z);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if x <= -6.5e-60:
		tmp = x / (z / (y / z))
	else:
		tmp = y * ((x / z) / z)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (x <= -6.5e-60)
		tmp = Float64(x / Float64(z / Float64(y / 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 (x <= -6.5e-60)
		tmp = x / (z / (y / 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[x, -6.5e-60], N[(x / N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(x / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.49999999999999995e-60

   1. Initial program 77.4%

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

      1. frac-times 83.8%

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

      2. associate-*l/ 87.3%

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

      3. times-frac 93.9%

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

   3. Applied egg-rr 93.9%

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

   4. Taylor expanded in z around 0 63.0%

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

      1. associate-*l/ 66.3%

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

      2. associate-/l* 72.0%

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

      3. div-inv 72.0%

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

      4. clear-num 72.0%

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

   6. Applied egg-rr 72.0%

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

      1. clear-num 72.0%

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

      2. un-div-inv 72.0%

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

   8. Applied egg-rr 72.0%

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

   if -6.49999999999999995e-60 < x

   1. Initial program 81.8%

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

      1. associate-*r/ 81.8%

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

      2. sqr-neg 81.8%

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

      3. *-commutative 81.8%

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

      4. distribute-rgt1-in 63.2%

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

      5. sqr-neg 63.2%

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

      6. fma-def 81.8%

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

      7. sqr-neg 81.8%

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

      8. cube-unmult 81.9%

         \[\leadsto x \cdot \frac{y}{\mathsf{fma}\left(z, z, \color{blue}{{z}^{3}}\right)} \]

   3. Simplified 81.9%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 81.8%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]

      2. fma-udef 63.1%

         \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot z + {z}^{3}}} \]

      3. cube-mult 63.1%

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

      4. distribute-rgt1-in 81.8%

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

      5. *-commutative 81.8%

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

      6. frac-times 87.5%

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

      7. *-commutative 87.5%

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

      8. associate-/r* 92.4%

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

      9. associate-*r/ 96.8%

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

   5. Applied egg-rr 96.8%

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

   6. Taylor expanded in z around 0 72.9%

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

      1. *-commutative 72.9%

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

      2. associate-*r/ 78.6%

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

   8. Simplified 78.6%

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

      1. *-un-lft-identity 78.6%

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

      2. times-frac 79.1%

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

      3. /-rgt-identity 79.1%

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

   10. Applied egg-rr 79.1%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-60}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{y}{z}}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \]

Alternative 11: 79.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-60}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{y}{z}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= x -1.7e-60) (/ x (/ z (/ y z))) (/ y (* z (/ z x)))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.7e-60) {
		tmp = x / (z / (y / z));
	} 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 (x <= (-1.7d-60)) then
        tmp = x / (z / (y / z))
    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 (x <= -1.7e-60) {
		tmp = x / (z / (y / z));
	} else {
		tmp = y / (z * (z / x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if x <= -1.7e-60:
		tmp = x / (z / (y / z))
	else:
		tmp = y / (z * (z / x))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (x <= -1.7e-60)
		tmp = Float64(x / Float64(z / Float64(y / z)));
	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 (x <= -1.7e-60)
		tmp = x / (z / (y / z));
	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[x, -1.7e-60], N[(x / N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.70000000000000003e-60

   1. Initial program 77.4%

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

      1. frac-times 83.8%

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

      2. associate-*l/ 87.3%

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

      3. times-frac 93.9%

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

   3. Applied egg-rr 93.9%

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

   4. Taylor expanded in z around 0 63.0%

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

      1. associate-*l/ 66.3%

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

      2. associate-/l* 72.0%

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

      3. div-inv 72.0%

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

      4. clear-num 72.0%

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

   6. Applied egg-rr 72.0%

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

      1. clear-num 72.0%

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

      2. un-div-inv 72.0%

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

   8. Applied egg-rr 72.0%

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

   if -1.70000000000000003e-60 < x

   1. Initial program 81.8%

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

      1. frac-times 87.5%

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

      2. associate-*l/ 86.7%

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

      3. times-frac 96.2%

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

   3. Applied egg-rr 96.2%

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

   4. Taylor expanded in z around 0 77.5%

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

      1. *-commutative 77.5%

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

      2. associate-/r/ 80.6%

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

      3. div-inv 80.6%

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

      4. clear-num 80.6%

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

   6. Applied egg-rr 80.6%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-60}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{y}{z}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 12: 73.9% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.4%

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

   1. associate-*r/ 81.8%

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

   2. sqr-neg 81.8%

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

   3. *-commutative 81.8%

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

   4. distribute-rgt1-in 65.3%

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

   5. sqr-neg 65.3%

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

   6. fma-def 81.8%

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

   7. sqr-neg 81.8%

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

   8. cube-unmult 81.8%

      \[\leadsto x \cdot \frac{y}{\mathsf{fma}\left(z, z, \color{blue}{{z}^{3}}\right)} \]

3. Simplified 81.8%

   \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]
4. Step-by-step derivation

   1. associate-*r/ 80.4%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]

   2. fma-udef 64.8%

      \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot z + {z}^{3}}} \]

   3. cube-mult 64.8%

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

   4. distribute-rgt1-in 80.4%

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

   5. *-commutative 80.4%

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

   6. frac-times 86.4%

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

   7. *-commutative 86.4%

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

   8. associate-/r* 91.6%

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

   9. associate-*r/ 96.7%

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

5. Applied egg-rr 96.7%

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

6. Taylor expanded in z around 0 70.3%

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

   1. *-commutative 70.3%

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

   2. associate-*r/ 73.3%

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

8. Simplified 73.3%

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

   1. *-un-lft-identity 73.3%

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

   2. times-frac 73.5%

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

   3. /-rgt-identity 73.5%

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

10. Applied egg-rr 73.5%

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

11. Final simplification 73.5%

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

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 2023310 
(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))))