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

Percentage Accurate: 83.3% → 96.8%

Time: 9.1s

Alternatives: 10

Speedup: 1.0×

• 26.1% 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: 83.3% 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: 96.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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) / (z / y)) / (z + 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

   1. associate-*l* 81.9%

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

   2. times-frac 94.7%

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

   3. distribute-lft-in 94.7%

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

   4. fma-def 94.8%

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

   5. *-rgt-identity 94.8%

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

3. Simplified 94.8%

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

   1. fma-udef 94.7%

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

   2. *-rgt-identity 94.7%

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

   3. distribute-lft-in 94.7%

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

   4. times-frac 81.9%

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

   5. associate-*l* 81.9%

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

   6. associate-/r* 84.5%

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

   7. times-frac 97.2%

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

5. Applied egg-rr 97.2%

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

   1. clear-num 97.2%

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

   2. un-div-inv 97.3%

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

7. Applied egg-rr 97.3%

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

8. Final simplification 97.3%

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

Alternative 2: 94.7% accurate, 0.8× speedup

Math

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

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.0) (not (<= z 1.0)))
   (* (/ x z) (/ (/ y z) z))
   (/ (/ x (/ z y)) 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 = (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 ((z <= (-1.0d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = (x / z) * ((y / 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 ((z <= -1.0) || !(z <= 1.0)) {
		tmp = (x / z) * ((y / z) / z);
	} else {
		tmp = (x / (z / y)) / 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 = (x / (z / y)) / 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(x / Float64(z / y)) / z);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.0) || ~((z <= 1.0)))
		tmp = (x / z) * ((y / 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[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[(x / N[(z / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 80.0%

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

      1. associate-*l* 80.0%

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

      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. Taylor expanded in z around inf 91.8%

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

      1. unpow2 91.8%

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

      2. associate-/r* 96.9%

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

   6. Simplified 96.9%

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

   if -1 < z < 1

   1. Initial program 83.7%

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

      1. associate-*l* 83.7%

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

      2. times-frac 96.2%

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

      3. distribute-lft-in 96.2%

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

      4. fma-def 96.2%

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

      5. *-rgt-identity 96.2%

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

   3. Simplified 96.2%

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

   4. Taylor expanded in z around 0 82.4%

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

      1. unpow2 82.4%

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

      2. associate-/l/ 93.9%

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

      3. associate-*r/ 95.4%

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

      4. associate-*l/ 94.9%

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

   6. Simplified 94.9%

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

      1. *-commutative 94.9%

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

      2. associate-*l/ 97.5%

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

   8. Applied egg-rr 97.5%

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

      1. clear-num 97.4%

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

      2. un-div-inv 97.5%

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

   10. Applied egg-rr 97.5%

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

4. Final simplification 97.2%

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

Alternative 3: 96.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

   1. associate-*l* 81.9%

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

   2. times-frac 94.7%

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

   3. distribute-lft-in 94.7%

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

   4. fma-def 94.8%

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

   5. *-rgt-identity 94.8%

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

3. Simplified 94.8%

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

   1. fma-udef 94.7%

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

   2. *-rgt-identity 94.7%

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

   3. distribute-lft-in 94.7%

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

   4. times-frac 81.9%

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

   5. associate-*l* 81.9%

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

   6. associate-/r* 84.5%

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

   7. times-frac 97.2%

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

5. Applied egg-rr 97.2%

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

   1. associate-/l* 97.2%

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

   2. associate-/r/ 96.6%

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

7. Applied egg-rr 96.6%

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

8. Final simplification 96.6%

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

Alternative 4: 96.8% accurate, 1.0× speedup

Math

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

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)) (+ z 1.0)))

C

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

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)) / (z + 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

   1. associate-*l* 81.9%

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

   2. times-frac 94.7%

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

   3. distribute-lft-in 94.7%

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

   4. fma-def 94.8%

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

   5. *-rgt-identity 94.8%

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

3. Simplified 94.8%

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

   1. fma-udef 94.7%

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

   2. *-rgt-identity 94.7%

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

   3. distribute-lft-in 94.7%

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

   4. times-frac 81.9%

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

   5. associate-*l* 81.9%

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

   6. associate-/r* 84.5%

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

   7. times-frac 97.2%

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

5. Applied egg-rr 97.2%

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

6. Final simplification 97.2%

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

Alternative 5: 78.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{+67}:\\ \;\;\;\;x \cdot \frac{\frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 7.5e+67) (* x (/ (/ y z) z)) (* y (/ x (* z z)))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 7.5e+67) {
		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 <= 7.5d+67) 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 <= 7.5e+67) {
		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 <= 7.5e+67:
		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 <= 7.5e+67)
		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 <= 7.5e+67)
		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, 7.5e+67], 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 < 7.5000000000000005e67

   1. Initial program 84.4%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 95.1%

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

      3. distribute-lft-in 95.2%

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

      4. fma-def 95.2%

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

      5. *-rgt-identity 95.2%

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

   3. Simplified 95.2%

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

      1. fma-udef 95.2%

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

      2. *-rgt-identity 95.2%

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

      3. distribute-lft-in 95.1%

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

      4. times-frac 84.4%

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

      5. associate-*l* 84.4%

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

      6. associate-/r* 85.9%

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

      7. times-frac 97.0%

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

   5. Applied egg-rr 97.0%

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

      1. associate-/l* 97.0%

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

      2. associate-/r/ 96.4%

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

   7. Applied egg-rr 96.4%

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

   8. Taylor expanded in z around 0 74.7%

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

      1. unpow2 74.7%

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

      2. *-commutative 74.7%

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

      3. associate-*r/ 76.1%

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

      4. associate-/r* 79.9%

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

   10. Simplified 79.9%

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

   if 7.5000000000000005e67 < y

   1. Initial program 73.0%

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

      1. times-frac 91.8%

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

   3. Simplified 91.8%

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

   4. Taylor expanded in z around 0 65.7%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{+67}:\\ \;\;\;\;x \cdot \frac{\frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 6: 78.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -0.02:\\ \;\;\;\;x \cdot \frac{\frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= x -0.02) (* x (/ (/ y z) z)) (* (/ x z) (/ y z))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.02) {
		tmp = x * ((y / 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 (x <= (-0.02d0)) then
        tmp = x * ((y / 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 (x <= -0.02) {
		tmp = x * ((y / z) / z);
	} else {
		tmp = (x / z) * (y / z);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if x <= -0.02:
		tmp = x * ((y / 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 (x <= -0.02)
		tmp = Float64(x * Float64(Float64(y / 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 (x <= -0.02)
		tmp = x * ((y / 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[LessEqual[x, -0.02], N[(x * N[(N[(y / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0200000000000000004

   1. Initial program 76.4%

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

      1. associate-*l* 76.4%

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

      2. times-frac 87.1%

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

      3. distribute-lft-in 87.1%

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

      4. fma-def 87.1%

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

      5. *-rgt-identity 87.1%

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

   3. Simplified 87.1%

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

      1. fma-udef 87.1%

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

      2. *-rgt-identity 87.1%

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

      3. distribute-lft-in 87.1%

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

      4. times-frac 76.4%

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

      5. associate-*l* 76.4%

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

      6. associate-/r* 77.9%

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

      7. times-frac 94.1%

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

   5. Applied egg-rr 94.1%

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

      1. associate-/l* 94.3%

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

      2. associate-/r/ 93.5%

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

   7. Applied egg-rr 93.5%

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

   8. Taylor expanded in z around 0 68.3%

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

      1. unpow2 68.3%

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

      2. *-commutative 68.3%

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

      3. associate-*r/ 71.8%

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

      4. associate-/r* 77.4%

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

   10. Simplified 77.4%

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

   if -0.0200000000000000004 < x

   1. Initial program 83.7%

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

      1. associate-*l* 83.7%

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

      2. times-frac 97.4%

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

      3. distribute-lft-in 97.4%

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

      4. fma-def 97.4%

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

      5. *-rgt-identity 97.4%

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

   3. Simplified 97.4%

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

   4. Taylor expanded in z around 0 73.0%

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

      1. unpow2 73.0%

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

      2. associate-/l/ 76.3%

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

      3. associate-*r/ 79.2%

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

      4. associate-*l/ 79.2%

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

   6. Simplified 79.2%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.02:\\ \;\;\;\;x \cdot \frac{\frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \end{array} \]

Alternative 7: 79.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{-59}:\\ \;\;\;\;x \cdot \frac{\frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= x -7.6e-59) (* x (/ (/ y z) z)) (/ y (* z (/ z x)))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (x <= -7.6e-59) {
		tmp = x * ((y / z) / 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 <= (-7.6d-59)) then
        tmp = x * ((y / z) / 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 <= -7.6e-59) {
		tmp = x * ((y / z) / z);
	} else {
		tmp = y / (z * (z / x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if x <= -7.6e-59:
		tmp = x * ((y / z) / 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 <= -7.6e-59)
		tmp = Float64(x * Float64(Float64(y / z) / 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 <= -7.6e-59)
		tmp = x * ((y / z) / 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, -7.6e-59], N[(x * N[(N[(y / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(y / N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.59999999999999966e-59

   1. Initial program 77.3%

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

      1. associate-*l* 77.3%

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

      2. times-frac 88.9%

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

      3. distribute-lft-in 89.0%

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

      4. fma-def 89.0%

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

      5. *-rgt-identity 89.0%

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

   3. Simplified 89.0%

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

      1. fma-udef 89.0%

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

      2. *-rgt-identity 89.0%

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

      3. distribute-lft-in 88.9%

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

      4. times-frac 77.3%

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

      5. associate-*l* 77.3%

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

      6. associate-/r* 81.0%

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

      7. times-frac 94.9%

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

   5. Applied egg-rr 94.9%

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

      1. associate-/l* 95.0%

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

      2. associate-/r/ 94.4%

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

   7. Applied egg-rr 94.4%

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

   8. Taylor expanded in z around 0 68.0%

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

      1. unpow2 68.0%

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

      2. *-commutative 68.0%

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

      3. associate-*r/ 71.0%

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

      4. associate-/r* 75.7%

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

   10. Simplified 75.7%

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

   if -7.59999999999999966e-59 < x

   1. Initial program 83.8%

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

      1. associate-*l* 83.8%

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

      2. times-frac 97.2%

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

      3. distribute-lft-in 97.2%

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

      4. fma-def 97.2%

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

      5. *-rgt-identity 97.2%

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

   3. Simplified 97.2%

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

   4. Taylor expanded in z around 0 73.4%

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

      1. unpow2 73.4%

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

      2. associate-/l* 74.1%

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

      3. associate-/l* 80.7%

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

   6. Simplified 80.7%

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

      1. associate-/r/ 80.7%

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

   8. Applied egg-rr 80.7%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{-59}:\\ \;\;\;\;x \cdot \frac{\frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 8: 79.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-58}:\\ \;\;\;\;x \cdot \frac{\frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{\frac{x}{z}}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= x -1.4e-58) (* x (/ (/ y z) z)) (/ y (/ z (/ x z)))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.4e-58) {
		tmp = x * ((y / z) / z);
	} else {
		tmp = y / (z / (x / z));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.4d-58)) then
        tmp = x * ((y / z) / z)
    else
        tmp = y / (z / (x / z))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.4e-58) {
		tmp = x * ((y / z) / z);
	} else {
		tmp = y / (z / (x / z));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if x <= -1.4e-58:
		tmp = x * ((y / z) / z)
	else:
		tmp = y / (z / (x / z))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (x <= -1.4e-58)
		tmp = Float64(x * Float64(Float64(y / z) / z));
	else
		tmp = Float64(y / Float64(z / Float64(x / z)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.4e-58)
		tmp = x * ((y / z) / z);
	else
		tmp = y / (z / (x / z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[x, -1.4e-58], N[(x * N[(N[(y / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(y / N[(z / N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.4e-58

   1. Initial program 77.3%

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

      1. associate-*l* 77.3%

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

      2. times-frac 88.9%

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

      3. distribute-lft-in 89.0%

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

      4. fma-def 89.0%

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

      5. *-rgt-identity 89.0%

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

   3. Simplified 89.0%

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

      1. fma-udef 89.0%

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

      2. *-rgt-identity 89.0%

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

      3. distribute-lft-in 88.9%

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

      4. times-frac 77.3%

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

      5. associate-*l* 77.3%

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

      6. associate-/r* 81.0%

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

      7. times-frac 94.9%

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

   5. Applied egg-rr 94.9%

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

      1. associate-/l* 95.0%

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

      2. associate-/r/ 94.4%

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

   7. Applied egg-rr 94.4%

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

   8. Taylor expanded in z around 0 68.0%

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

      1. unpow2 68.0%

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

      2. *-commutative 68.0%

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

      3. associate-*r/ 71.0%

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

      4. associate-/r* 75.7%

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

   10. Simplified 75.7%

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

   if -1.4e-58 < x

   1. Initial program 83.8%

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

      1. associate-*l* 83.8%

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

      2. times-frac 97.2%

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

      3. distribute-lft-in 97.2%

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

      4. fma-def 97.2%

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

      5. *-rgt-identity 97.2%

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

   3. Simplified 97.2%

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

   4. Taylor expanded in z around 0 73.4%

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

      1. unpow2 73.4%

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

      2. associate-/l* 74.1%

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

      3. associate-/l* 80.7%

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

   6. Simplified 80.7%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-58}:\\ \;\;\;\;x \cdot \frac{\frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{\frac{x}{z}}}\\ \end{array} \]

Alternative 9: 74.0% 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 81.9%

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

   1. associate-*l* 81.9%

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

   2. times-frac 94.7%

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

   3. distribute-lft-in 94.7%

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

   4. fma-def 94.8%

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

   5. *-rgt-identity 94.8%

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

3. Simplified 94.8%

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

   1. fma-udef 94.7%

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

   2. *-rgt-identity 94.7%

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

   3. distribute-lft-in 94.7%

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

   4. times-frac 81.9%

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

   5. associate-*l* 81.9%

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

   6. associate-/r* 84.5%

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

   7. times-frac 97.2%

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

5. Applied egg-rr 97.2%

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

   1. associate-/l* 97.2%

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

   2. associate-/r/ 96.6%

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

7. Applied egg-rr 96.6%

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

8. Taylor expanded in z around 0 71.8%

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

   1. unpow2 71.8%

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

   2. *-commutative 71.8%

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

   3. associate-*r/ 72.3%

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

   4. associate-/r* 74.5%

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

10. Simplified 74.5%

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

11. Final simplification 74.5%

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

Alternative 10: 29.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 81.9%

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

   1. associate-*l* 81.9%

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

   2. times-frac 94.7%

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

   3. distribute-lft-in 94.7%

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

   4. fma-def 94.8%

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

   5. *-rgt-identity 94.8%

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

3. Simplified 94.8%

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

4. Taylor expanded in z around 0 68.2%

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

   1. neg-mul-1 68.2%

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

   2. +-commutative 68.2%

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

   3. unsub-neg 68.2%

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

6. Simplified 68.2%

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

7. Taylor expanded in z around inf 26.5%

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

   1. associate-*r/ 26.5%

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

   2. *-commutative 26.5%

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

   3. neg-mul-1 26.5%

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

   4. distribute-rgt-neg-in 26.5%

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

   5. associate-*l/ 30.5%

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

9. Simplified 30.5%

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

10. Final simplification 30.5%

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

Developer target: 95.2% 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 2023199 
(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))))