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

Percentage Accurate: 82.8% → 97.3%

Time: 10.6s

Alternatives: 14

Speedup: 1.0×

• 25.6% 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: 82.8% 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: 97.3% 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 82.3%

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

   1. associate-*l* 82.3%

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

   2. times-frac 96.3%

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

   3. distribute-lft-in 96.3%

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

   4. fma-def 96.3%

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

   5. *-rgt-identity 96.3%

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

3. Simplified 96.3%

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

   1. *-commutative 96.3%

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

   2. associate-*l/ 96.1%

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

   3. fma-udef 96.1%

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

   4. distribute-lft1-in 96.1%

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

   5. frac-times 93.9%

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

   6. associate-*r/ 98.6%

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

5. Applied egg-rr 98.6%

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

6. Final simplification 98.6%

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

Alternative 2: 95.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(z + 1\right) \cdot \left(z \cdot z\right)\\ \mathbf{if}\;t_0 \leq -0.5:\\ \;\;\;\;\frac{\frac{y}{z}}{z \cdot \frac{z}{x}}\\ \mathbf{elif}\;t_0 \leq 10^{-71}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z + 1} \cdot \frac{x}{z \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (+ z 1.0) (* z z))))
   (if (<= t_0 -0.5)
     (/ (/ y z) (* z (/ z x)))
     (if (<= t_0 1e-71)
       (/ (* y (/ x z)) z)
       (* (/ y (+ z 1.0)) (/ x (* z z)))))))

C

assert(x < y);
double code(double x, double y, double z) {
	double t_0 = (z + 1.0) * (z * z);
	double tmp;
	if (t_0 <= -0.5) {
		tmp = (y / z) / (z * (z / x));
	} else if (t_0 <= 1e-71) {
		tmp = (y * (x / z)) / z;
	} else {
		tmp = (y / (z + 1.0)) * (x / (z * z));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (z + 1.0d0) * (z * z)
    if (t_0 <= (-0.5d0)) then
        tmp = (y / z) / (z * (z / x))
    else if (t_0 <= 1d-71) then
        tmp = (y * (x / z)) / z
    else
        tmp = (y / (z + 1.0d0)) * (x / (z * z))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double t_0 = (z + 1.0) * (z * z);
	double tmp;
	if (t_0 <= -0.5) {
		tmp = (y / z) / (z * (z / x));
	} else if (t_0 <= 1e-71) {
		tmp = (y * (x / z)) / z;
	} else {
		tmp = (y / (z + 1.0)) * (x / (z * z));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	t_0 = (z + 1.0) * (z * z)
	tmp = 0
	if t_0 <= -0.5:
		tmp = (y / z) / (z * (z / x))
	elif t_0 <= 1e-71:
		tmp = (y * (x / z)) / z
	else:
		tmp = (y / (z + 1.0)) * (x / (z * z))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	t_0 = Float64(Float64(z + 1.0) * Float64(z * z))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = Float64(Float64(y / z) / Float64(z * Float64(z / x)));
	elseif (t_0 <= 1e-71)
		tmp = Float64(Float64(y * Float64(x / z)) / z);
	else
		tmp = Float64(Float64(y / Float64(z + 1.0)) * Float64(x / Float64(z * z)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	t_0 = (z + 1.0) * (z * z);
	tmp = 0.0;
	if (t_0 <= -0.5)
		tmp = (y / z) / (z * (z / x));
	elseif (t_0 <= 1e-71)
		tmp = (y * (x / z)) / z;
	else
		tmp = (y / (z + 1.0)) * (x / (z * z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(N[(y / z), $MachinePrecision] / N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-71], N[(N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(y / N[(z + 1.0), $MachinePrecision]), $MachinePrecision] * N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z 1)) < -0.5

   1. Initial program 81.4%

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

      1. /-rgt-identity 81.4%

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

      2. associate-/l* 81.4%

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

      3. associate-/l/ 86.0%

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

      4. associate-*l* 90.3%

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

      5. associate-*r/ 90.3%

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

      6. *-rgt-identity 90.3%

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

      7. associate-*l* 90.3%

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

      8. associate-*r/ 90.4%

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

      9. distribute-lft-in 90.3%

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

      10. fma-def 90.4%

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

      11. *-rgt-identity 90.4%

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

   3. Simplified 90.4%

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

   4. Taylor expanded in z around inf 89.1%

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

      1. unpow2 89.1%

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

   6. Simplified 89.1%

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

      1. *-un-lft-identity 89.1%

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

      2. *-commutative 89.1%

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

      3. times-frac 93.0%

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

      4. clear-num 93.0%

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

   8. Applied egg-rr 93.0%

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

      1. associate-/r* 97.2%

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

      2. clear-num 97.2%

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

      3. frac-times 98.5%

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

      4. *-commutative 98.5%

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

      5. *-un-lft-identity 98.5%

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

   10. Applied egg-rr 98.5%

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

   if -0.5 < (*.f64 (*.f64 z z) (+.f64 z 1)) < 9.9999999999999992e-72

   1. Initial program 78.8%

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

      1. associate-*l* 78.8%

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

      2. times-frac 98.1%

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

      3. distribute-lft-in 98.1%

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

      4. fma-def 98.1%

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

      5. *-rgt-identity 98.1%

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

   3. Simplified 98.1%

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

      1. *-commutative 98.1%

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

      2. associate-*l/ 98.1%

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

      3. fma-udef 98.1%

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

      4. distribute-lft1-in 98.1%

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

      5. frac-times 88.6%

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

      6. associate-*r/ 98.1%

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

   5. Applied egg-rr 98.1%

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

   6. Taylor expanded in z around 0 90.4%

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

      1. associate-*r/ 98.1%

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

   8. Simplified 98.1%

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

   if 9.9999999999999992e-72 < (*.f64 (*.f64 z z) (+.f64 z 1))

   1. Initial program 87.9%

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

      1. times-frac 96.1%

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

   3. Simplified 96.1%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq -0.5:\\ \;\;\;\;\frac{\frac{y}{z}}{z \cdot \frac{z}{x}}\\ \mathbf{elif}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 10^{-71}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z + 1} \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 3: 92.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 5.8 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.0) (not (<= z 5.8e-12)))
   (* (/ x z) (/ y (* z z)))
   (/ (* y (/ x z)) z)))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 5.8e-12)) {
		tmp = (x / z) * (y / (z * z));
	} else {
		tmp = (y * (x / z)) / z;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.0d0)) .or. (.not. (z <= 5.8d-12))) then
        tmp = (x / z) * (y / (z * z))
    else
        tmp = (y * (x / z)) / z
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 5.8e-12)) {
		tmp = (x / z) * (y / (z * z));
	} else {
		tmp = (y * (x / z)) / z;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z <= -1.0) or not (z <= 5.8e-12):
		tmp = (x / z) * (y / (z * z))
	else:
		tmp = (y * (x / z)) / z
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.0) || !(z <= 5.8e-12))
		tmp = Float64(Float64(x / z) * Float64(y / Float64(z * z)));
	else
		tmp = Float64(Float64(y * Float64(x / z)) / z);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.0) || ~((z <= 5.8e-12)))
		tmp = (x / z) * (y / (z * z));
	else
		tmp = (y * (x / z)) / z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 5.8e-12]], $MachinePrecision]], N[(N[(x / z), $MachinePrecision] * N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 5.8000000000000003e-12 < z

   1. Initial program 83.2%

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

      1. associate-*l* 83.2%

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

      2. times-frac 95.2%

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

      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. Taylor expanded in z around inf 93.6%

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

      1. unpow2 93.6%

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

   6. Simplified 93.6%

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

   if -1 < z < 5.8000000000000003e-12

   1. Initial program 81.3%

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

      1. associate-*l* 81.3%

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

      2. times-frac 97.5%

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

      3. distribute-lft-in 97.5%

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

      4. fma-def 97.5%

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

      5. *-rgt-identity 97.5%

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

   3. Simplified 97.5%

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

      1. *-commutative 97.5%

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

      2. associate-*l/ 97.5%

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

      3. fma-udef 97.5%

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

      4. distribute-lft1-in 97.5%

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

      5. frac-times 89.2%

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

      6. associate-*r/ 97.5%

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

   5. Applied egg-rr 97.5%

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

   6. Taylor expanded in z around 0 90.1%

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

      1. associate-*r/ 96.1%

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

   8. Simplified 96.1%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 5.8 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z}\\ \end{array} \]

Alternative 4: 92.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 5.8 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.0) (not (<= z 5.8e-12)))
   (* (/ y z) (/ x (* z z)))
   (/ (* y (/ x z)) z)))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 5.8e-12)) {
		tmp = (y / z) * (x / (z * z));
	} else {
		tmp = (y * (x / z)) / z;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.0d0)) .or. (.not. (z <= 5.8d-12))) then
        tmp = (y / z) * (x / (z * z))
    else
        tmp = (y * (x / z)) / z
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 5.8e-12)) {
		tmp = (y / z) * (x / (z * z));
	} else {
		tmp = (y * (x / z)) / z;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z <= -1.0) or not (z <= 5.8e-12):
		tmp = (y / z) * (x / (z * z))
	else:
		tmp = (y * (x / z)) / z
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.0) || !(z <= 5.8e-12))
		tmp = Float64(Float64(y / z) * Float64(x / Float64(z * z)));
	else
		tmp = Float64(Float64(y * Float64(x / z)) / z);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.0) || ~((z <= 5.8e-12)))
		tmp = (y / z) * (x / (z * z));
	else
		tmp = (y * (x / z)) / z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 5.8e-12]], $MachinePrecision]], N[(N[(y / z), $MachinePrecision] * N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 5.8000000000000003e-12 < z

   1. Initial program 83.2%

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

      1. /-rgt-identity 83.2%

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

      2. associate-/l* 83.2%

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

      3. associate-/l/ 87.2%

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

      4. associate-*l* 91.5%

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

      5. associate-*r/ 91.5%

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

      6. *-rgt-identity 91.5%

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

      7. associate-*l* 91.5%

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

      8. associate-*r/ 91.5%

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

      9. distribute-lft-in 91.5%

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

      10. fma-def 91.5%

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

      11. *-rgt-identity 91.5%

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

   3. Simplified 91.5%

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

   4. Taylor expanded in z around inf 89.8%

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

      1. unpow2 89.8%

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

   6. Simplified 89.8%

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

      1. associate-/r* 93.1%

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

      2. associate-/l* 95.2%

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

      3. associate-/r/ 96.7%

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

      4. associate-/r* 93.9%

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

   8. Applied egg-rr 93.9%

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

   if -1 < z < 5.8000000000000003e-12

   1. Initial program 81.3%

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

      1. associate-*l* 81.3%

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

      2. times-frac 97.5%

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

      3. distribute-lft-in 97.5%

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

      4. fma-def 97.5%

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

      5. *-rgt-identity 97.5%

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

   3. Simplified 97.5%

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

      1. *-commutative 97.5%

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

      2. associate-*l/ 97.5%

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

      3. fma-udef 97.5%

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

      4. distribute-lft1-in 97.5%

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

      5. frac-times 89.2%

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

      6. associate-*r/ 97.5%

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

   5. Applied egg-rr 97.5%

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

   6. Taylor expanded in z around 0 90.1%

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

      1. associate-*r/ 96.1%

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

   8. Simplified 96.1%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 5.8 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z}\\ \end{array} \]

Alternative 5: 94.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 5.8 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{\frac{y}{z}}{z \cdot \frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.0) (not (<= z 5.8e-12)))
   (/ (/ y z) (* z (/ z x)))
   (/ (* y (/ x z)) z)))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 5.8e-12)) {
		tmp = (y / z) / (z * (z / x));
	} else {
		tmp = (y * (x / z)) / z;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.0d0)) .or. (.not. (z <= 5.8d-12))) then
        tmp = (y / z) / (z * (z / x))
    else
        tmp = (y * (x / z)) / z
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 5.8e-12)) {
		tmp = (y / z) / (z * (z / x));
	} else {
		tmp = (y * (x / z)) / z;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z <= -1.0) or not (z <= 5.8e-12):
		tmp = (y / z) / (z * (z / x))
	else:
		tmp = (y * (x / z)) / z
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.0) || !(z <= 5.8e-12))
		tmp = Float64(Float64(y / z) / Float64(z * Float64(z / x)));
	else
		tmp = Float64(Float64(y * Float64(x / z)) / z);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.0) || ~((z <= 5.8e-12)))
		tmp = (y / z) / (z * (z / x));
	else
		tmp = (y * (x / z)) / z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 5.8e-12]], $MachinePrecision]], N[(N[(y / z), $MachinePrecision] / N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 5.8000000000000003e-12 < z

   1. Initial program 83.2%

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

      1. /-rgt-identity 83.2%

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

      2. associate-/l* 83.2%

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

      3. associate-/l/ 87.2%

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

      4. associate-*l* 91.5%

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

      5. associate-*r/ 91.5%

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

      6. *-rgt-identity 91.5%

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

      7. associate-*l* 91.5%

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

      8. associate-*r/ 91.5%

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

      9. distribute-lft-in 91.5%

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

      10. fma-def 91.5%

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

      11. *-rgt-identity 91.5%

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

   3. Simplified 91.5%

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

   4. Taylor expanded in z around inf 89.8%

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

      1. unpow2 89.8%

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

   6. Simplified 89.8%

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

      1. *-un-lft-identity 89.8%

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

      2. *-commutative 89.8%

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

      3. times-frac 93.0%

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

      4. clear-num 93.6%

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

   8. Applied egg-rr 93.6%

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

      1. associate-/r* 95.7%

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

      2. clear-num 95.1%

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

      3. frac-times 96.7%

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

      4. *-commutative 96.7%

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

      5. *-un-lft-identity 96.7%

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

   10. Applied egg-rr 96.7%

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

   if -1 < z < 5.8000000000000003e-12

   1. Initial program 81.3%

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

      1. associate-*l* 81.3%

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

      2. times-frac 97.5%

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

      3. distribute-lft-in 97.5%

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

      4. fma-def 97.5%

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

      5. *-rgt-identity 97.5%

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

   3. Simplified 97.5%

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

      1. *-commutative 97.5%

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

      2. associate-*l/ 97.5%

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

      3. fma-udef 97.5%

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

      4. distribute-lft1-in 97.5%

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

      5. frac-times 89.2%

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

      6. associate-*r/ 97.5%

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

   5. Applied egg-rr 97.5%

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

   6. Taylor expanded in z around 0 90.1%

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

      1. associate-*r/ 96.1%

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

   8. Simplified 96.1%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 5.8 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{\frac{y}{z}}{z \cdot \frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z}\\ \end{array} \]

Alternative 6: 92.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{x}{z \cdot \frac{z \cdot z}{y}}\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-12}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z -1.0)
   (/ x (* z (/ (* z z) y)))
   (if (<= z 5.8e-12) (/ (* y (/ x z)) z) (* (/ y z) (/ x (* z z))))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = x / (z * ((z * z) / y));
	} else if (z <= 5.8e-12) {
		tmp = (y * (x / z)) / z;
	} else {
		tmp = (y / z) * (x / (z * z));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.0d0)) then
        tmp = x / (z * ((z * z) / y))
    else if (z <= 5.8d-12) then
        tmp = (y * (x / z)) / z
    else
        tmp = (y / z) * (x / (z * z))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = x / (z * ((z * z) / y));
	} else if (z <= 5.8e-12) {
		tmp = (y * (x / z)) / z;
	} else {
		tmp = (y / z) * (x / (z * z));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if z <= -1.0:
		tmp = x / (z * ((z * z) / y))
	elif z <= 5.8e-12:
		tmp = (y * (x / z)) / z
	else:
		tmp = (y / z) * (x / (z * z))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(x / Float64(z * Float64(Float64(z * z) / y)));
	elseif (z <= 5.8e-12)
		tmp = Float64(Float64(y * Float64(x / z)) / z);
	else
		tmp = Float64(Float64(y / z) * 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 (z <= -1.0)
		tmp = x / (z * ((z * z) / y));
	elseif (z <= 5.8e-12)
		tmp = (y * (x / z)) / z;
	else
		tmp = (y / z) * (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[z, -1.0], N[(x / N[(z * N[(N[(z * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.8e-12], N[(N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 81.4%

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

      1. /-rgt-identity 81.4%

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

      2. associate-/l* 81.4%

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

      3. associate-/l/ 86.0%

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

      4. associate-*l* 90.3%

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

      5. associate-*r/ 90.3%

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

      6. *-rgt-identity 90.3%

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

      7. associate-*l* 90.3%

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

      8. associate-*r/ 90.4%

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

      9. distribute-lft-in 90.3%

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

      10. fma-def 90.4%

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

      11. *-rgt-identity 90.4%

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

   3. Simplified 90.4%

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

   4. Taylor expanded in z around inf 89.1%

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

      1. unpow2 89.1%

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

   6. Simplified 89.1%

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

   if -1 < z < 5.8000000000000003e-12

   1. Initial program 81.3%

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

      1. associate-*l* 81.3%

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

      2. times-frac 97.5%

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

      3. distribute-lft-in 97.5%

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

      4. fma-def 97.5%

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

      5. *-rgt-identity 97.5%

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

   3. Simplified 97.5%

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

      1. *-commutative 97.5%

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

      2. associate-*l/ 97.5%

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

      3. fma-udef 97.5%

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

      4. distribute-lft1-in 97.5%

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

      5. frac-times 89.2%

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

      6. associate-*r/ 97.5%

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

   5. Applied egg-rr 97.5%

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

   6. Taylor expanded in z around 0 90.1%

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

      1. associate-*r/ 96.1%

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

   8. Simplified 96.1%

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

   if 5.8000000000000003e-12 < z

   1. Initial program 85.1%

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

      1. /-rgt-identity 85.1%

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

      2. associate-/l* 85.0%

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

      3. associate-/l/ 88.3%

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

      4. associate-*l* 92.7%

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

      5. associate-*r/ 92.8%

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

      6. *-rgt-identity 92.8%

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

      7. associate-*l* 92.7%

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

      8. associate-*r/ 92.7%

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

      9. distribute-lft-in 92.7%

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

      10. fma-def 92.7%

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

      11. *-rgt-identity 92.7%

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

   3. Simplified 92.7%

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

   4. Taylor expanded in z around inf 90.7%

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

      1. unpow2 90.7%

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

   6. Simplified 90.7%

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

      1. associate-/r* 93.1%

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

      2. associate-/l* 93.1%

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

      3. associate-/r/ 94.8%

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

      4. associate-/r* 94.8%

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

   8. Applied egg-rr 94.8%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{x}{z \cdot \frac{z \cdot z}{y}}\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-12}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 7: 93.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{x \cdot \frac{\frac{y}{z}}{z}}{z}\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-12}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z -1.0)
   (/ (* x (/ (/ y z) z)) z)
   (if (<= z 5.8e-12) (/ (* y (/ x z)) z) (* (/ y z) (/ x (* z z))))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = (x * ((y / z) / z)) / z;
	} else if (z <= 5.8e-12) {
		tmp = (y * (x / z)) / z;
	} else {
		tmp = (y / z) * (x / (z * z));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.0d0)) then
        tmp = (x * ((y / z) / z)) / z
    else if (z <= 5.8d-12) then
        tmp = (y * (x / z)) / z
    else
        tmp = (y / z) * (x / (z * z))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = (x * ((y / z) / z)) / z;
	} else if (z <= 5.8e-12) {
		tmp = (y * (x / z)) / z;
	} else {
		tmp = (y / z) * (x / (z * z));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if z <= -1.0:
		tmp = (x * ((y / z) / z)) / z
	elif z <= 5.8e-12:
		tmp = (y * (x / z)) / z
	else:
		tmp = (y / z) * (x / (z * z))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(Float64(x * Float64(Float64(y / z) / z)) / z);
	elseif (z <= 5.8e-12)
		tmp = Float64(Float64(y * Float64(x / z)) / z);
	else
		tmp = Float64(Float64(y / z) * 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 (z <= -1.0)
		tmp = (x * ((y / z) / z)) / z;
	elseif (z <= 5.8e-12)
		tmp = (y * (x / z)) / z;
	else
		tmp = (y / z) * (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[z, -1.0], N[(N[(x * N[(N[(y / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 5.8e-12], N[(N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 81.4%

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

      1. associate-*l* 81.4%

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

      2. times-frac 94.3%

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

      3. distribute-lft-in 94.3%

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

      4. fma-def 94.3%

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

      5. *-rgt-identity 94.3%

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

   3. Simplified 94.3%

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

      1. *-commutative 94.3%

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

      2. associate-*l/ 92.9%

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

      3. fma-udef 92.9%

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

      4. distribute-lft1-in 92.9%

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

      5. frac-times 99.7%

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

      6. associate-*r/ 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in z around inf 85.6%

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

      1. unpow2 85.6%

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

      2. *-commutative 85.6%

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

      3. associate-*r/ 93.1%

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

      4. associate-/r* 97.2%

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

   8. Simplified 97.2%

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

   if -1 < z < 5.8000000000000003e-12

   1. Initial program 81.3%

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

      1. associate-*l* 81.3%

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

      2. times-frac 97.5%

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

      3. distribute-lft-in 97.5%

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

      4. fma-def 97.5%

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

      5. *-rgt-identity 97.5%

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

   3. Simplified 97.5%

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

      1. *-commutative 97.5%

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

      2. associate-*l/ 97.5%

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

      3. fma-udef 97.5%

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

      4. distribute-lft1-in 97.5%

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

      5. frac-times 89.2%

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

      6. associate-*r/ 97.5%

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

   5. Applied egg-rr 97.5%

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

   6. Taylor expanded in z around 0 90.1%

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

      1. associate-*r/ 96.1%

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

   8. Simplified 96.1%

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

   if 5.8000000000000003e-12 < z

   1. Initial program 85.1%

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

      1. /-rgt-identity 85.1%

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

      2. associate-/l* 85.0%

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

      3. associate-/l/ 88.3%

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

      4. associate-*l* 92.7%

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

      5. associate-*r/ 92.8%

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

      6. *-rgt-identity 92.8%

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

      7. associate-*l* 92.7%

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

      8. associate-*r/ 92.7%

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

      9. distribute-lft-in 92.7%

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

      10. fma-def 92.7%

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

      11. *-rgt-identity 92.7%

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

   3. Simplified 92.7%

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

   4. Taylor expanded in z around inf 90.7%

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

      1. unpow2 90.7%

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

   6. Simplified 90.7%

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

      1. associate-/r* 93.1%

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

      2. associate-/l* 93.1%

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

      3. associate-/r/ 94.8%

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

      4. associate-/r* 94.8%

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

   8. Applied egg-rr 94.8%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{x \cdot \frac{\frac{y}{z}}{z}}{z}\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-12}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 8: 93.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{\frac{x}{z}}{z \cdot \frac{z}{y}}\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-12}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z -1.0)
   (/ (/ x z) (* z (/ z y)))
   (if (<= z 5.8e-12) (/ (* y (/ x z)) z) (* (/ y z) (/ x (* z z))))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = (x / z) / (z * (z / y));
	} else if (z <= 5.8e-12) {
		tmp = (y * (x / z)) / z;
	} else {
		tmp = (y / z) * (x / (z * z));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.0d0)) then
        tmp = (x / z) / (z * (z / y))
    else if (z <= 5.8d-12) then
        tmp = (y * (x / z)) / z
    else
        tmp = (y / z) * (x / (z * z))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = (x / z) / (z * (z / y));
	} else if (z <= 5.8e-12) {
		tmp = (y * (x / z)) / z;
	} else {
		tmp = (y / z) * (x / (z * z));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if z <= -1.0:
		tmp = (x / z) / (z * (z / y))
	elif z <= 5.8e-12:
		tmp = (y * (x / z)) / z
	else:
		tmp = (y / z) * (x / (z * z))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(Float64(x / z) / Float64(z * Float64(z / y)));
	elseif (z <= 5.8e-12)
		tmp = Float64(Float64(y * Float64(x / z)) / z);
	else
		tmp = Float64(Float64(y / z) * 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 (z <= -1.0)
		tmp = (x / z) / (z * (z / y));
	elseif (z <= 5.8e-12)
		tmp = (y * (x / z)) / z;
	else
		tmp = (y / z) * (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[z, -1.0], N[(N[(x / z), $MachinePrecision] / N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.8e-12], N[(N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 81.4%

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

      1. /-rgt-identity 81.4%

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

      2. associate-/l* 81.4%

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

      3. associate-/l/ 86.0%

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

      4. associate-*l* 90.3%

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

      5. associate-*r/ 90.3%

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

      6. *-rgt-identity 90.3%

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

      7. associate-*l* 90.3%

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

      8. associate-*r/ 90.4%

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

      9. distribute-lft-in 90.3%

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

      10. fma-def 90.4%

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

      11. *-rgt-identity 90.4%

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

   3. Simplified 90.4%

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

   4. Taylor expanded in z around inf 89.1%

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

      1. unpow2 89.1%

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

   6. Simplified 89.1%

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

      1. associate-/r* 93.1%

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

      2. add-sqr-sqrt 49.7%

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

      3. *-un-lft-identity 49.7%

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

      4. times-frac 49.7%

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

      5. div-inv 49.7%

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

      6. associate-*l* 52.4%

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

      7. div-inv 52.4%

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

   8. Applied egg-rr 52.4%

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

      1. times-frac 52.4%

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

      2. *-lft-identity 52.4%

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

      3. rem-square-sqrt 97.2%

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

   10. Simplified 97.2%

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

   if -1 < z < 5.8000000000000003e-12

   1. Initial program 81.3%

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

      1. associate-*l* 81.3%

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

      2. times-frac 97.5%

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

      3. distribute-lft-in 97.5%

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

      4. fma-def 97.5%

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

      5. *-rgt-identity 97.5%

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

   3. Simplified 97.5%

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

      1. *-commutative 97.5%

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

      2. associate-*l/ 97.5%

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

      3. fma-udef 97.5%

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

      4. distribute-lft1-in 97.5%

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

      5. frac-times 89.2%

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

      6. associate-*r/ 97.5%

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

   5. Applied egg-rr 97.5%

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

   6. Taylor expanded in z around 0 90.1%

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

      1. associate-*r/ 96.1%

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

   8. Simplified 96.1%

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

   if 5.8000000000000003e-12 < z

   1. Initial program 85.1%

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

      1. /-rgt-identity 85.1%

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

      2. associate-/l* 85.0%

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

      3. associate-/l/ 88.3%

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

      4. associate-*l* 92.7%

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

      5. associate-*r/ 92.8%

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

      6. *-rgt-identity 92.8%

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

      7. associate-*l* 92.7%

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

      8. associate-*r/ 92.7%

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

      9. distribute-lft-in 92.7%

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

      10. fma-def 92.7%

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

      11. *-rgt-identity 92.7%

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

   3. Simplified 92.7%

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

   4. Taylor expanded in z around inf 90.7%

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

      1. unpow2 90.7%

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

   6. Simplified 90.7%

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

      1. associate-/r* 93.1%

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

      2. associate-/l* 93.1%

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

      3. associate-/r/ 94.8%

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

      4. associate-/r* 94.8%

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

   8. Applied egg-rr 94.8%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{\frac{x}{z}}{z \cdot \frac{z}{y}}\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-12}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 9: 40.7% accurate, 1.1× speedup

Math

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

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.0) (not (<= z -1e-310))) (/ y (/ z x)) (* (/ y z) (- x))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= -1e-310)) {
		tmp = y / (z / x);
	} else {
		tmp = (y / z) * -x;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.0d0)) .or. (.not. (z <= (-1d-310)))) then
        tmp = y / (z / x)
    else
        tmp = (y / z) * -x
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= -1e-310)) {
		tmp = y / (z / x);
	} else {
		tmp = (y / z) * -x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z <= -1.0) or not (z <= -1e-310):
		tmp = y / (z / x)
	else:
		tmp = (y / z) * -x
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.0) || !(z <= -1e-310))
		tmp = Float64(y / Float64(z / x));
	else
		tmp = Float64(Float64(y / z) * Float64(-x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.0) || ~((z <= -1e-310)))
		tmp = y / (z / x);
	else
		tmp = (y / 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[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, -1e-310]], $MachinePrecision]], N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * (-x)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or -9.999999999999969e-311 < z

   1. Initial program 83.2%

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

      1. associate-*l* 83.2%

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

      2. times-frac 95.6%

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

      3. distribute-lft-in 95.6%

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

      4. fma-def 95.6%

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

      5. *-rgt-identity 95.6%

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

   3. Simplified 95.6%

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

   4. Taylor expanded in z around 0 52.8%

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

      1. neg-mul-1 52.8%

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

      2. +-commutative 52.8%

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

      3. unsub-neg 52.8%

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

   6. Simplified 52.8%

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

   7. Taylor expanded in z around inf 19.9%

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

      1. associate-*r/ 23.8%

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

      2. neg-mul-1 23.8%

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

      3. distribute-rgt-neg-in 23.8%

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

      4. distribute-neg-frac 23.8%

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

   9. Simplified 23.8%

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

       1. add-sqr-sqrt 19.3%

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

       2. sqrt-unprod 36.8%

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

       3. distribute-frac-neg 36.8%

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

       4. distribute-frac-neg 36.8%

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

       5. sqr-neg 36.8%

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

       6. sqrt-unprod 22.4%

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

       7. add-sqr-sqrt 35.3%

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

       8. clear-num 35.8%

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

       9. div-inv 35.8%

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

   11. Applied egg-rr 35.8%

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

   if -1 < z < -9.999999999999969e-311

   1. Initial program 79.9%

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

      1. associate-*l* 79.9%

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

      2. times-frac 98.3%

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

      3. distribute-lft-in 98.3%

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

      4. fma-def 98.3%

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

      5. *-rgt-identity 98.3%

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

   3. Simplified 98.3%

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

   4. Taylor expanded in z around 0 97.9%

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

      1. neg-mul-1 97.9%

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

      2. +-commutative 97.9%

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

      3. unsub-neg 97.9%

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

   6. Simplified 97.9%

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

   7. Taylor expanded in z around inf 36.4%

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

      1. associate-*r/ 36.4%

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

      2. neg-mul-1 36.4%

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

      3. distribute-rgt-neg-in 36.4%

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

      4. associate-*l/ 39.2%

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

      5. distribute-rgt-neg-out 39.2%

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

   9. Simplified 39.2%

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

4. Final simplification 36.7%

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

Alternative 10: 77.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -6000000000:\\ \;\;\;\;\frac{x}{z \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= x -6000000000.0) (/ x (* z (/ z y))) (* (/ x z) (/ y z))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (x <= -6000000000.0) {
		tmp = x / (z * (z / y));
	} 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 <= (-6000000000.0d0)) then
        tmp = x / (z * (z / y))
    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 <= -6000000000.0) {
		tmp = x / (z * (z / y));
	} else {
		tmp = (x / z) * (y / z);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if x <= -6000000000.0:
		tmp = x / (z * (z / y))
	else:
		tmp = (x / z) * (y / z)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (x <= -6000000000.0)
		tmp = Float64(x / Float64(z * Float64(z / y)));
	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 <= -6000000000.0)
		tmp = x / (z * (z / y));
	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, -6000000000.0], N[(x / N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6e9

   1. Initial program 82.7%

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

      1. /-rgt-identity 82.7%

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

      2. associate-/l* 82.6%

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

      3. associate-/l/ 87.7%

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

      4. associate-*l* 89.3%

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

      5. associate-*r/ 89.3%

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

      6. *-rgt-identity 89.3%

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

      7. associate-*l* 89.4%

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

      8. associate-*r/ 89.3%

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

      9. distribute-lft-in 89.3%

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

      10. fma-def 89.3%

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

      11. *-rgt-identity 89.3%

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

   3. Simplified 89.3%

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

   4. Taylor expanded in z around 0 76.1%

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

   if -6e9 < x

   1. Initial program 82.2%

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

      1. associate-*l* 82.2%

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

      2. times-frac 97.8%

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

      3. distribute-lft-in 97.8%

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

      4. fma-def 97.8%

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

      5. *-rgt-identity 97.8%

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

   3. Simplified 97.8%

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

   4. Taylor expanded in z around 0 68.8%

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

      1. unpow2 68.8%

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

      2. associate-/l/ 72.7%

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

      3. associate-*r/ 76.9%

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

      4. associate-*l/ 77.8%

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

   6. Simplified 77.8%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6000000000:\\ \;\;\;\;\frac{x}{z \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \end{array} \]

Alternative 11: 79.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-56}:\\ \;\;\;\;\frac{x}{z \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 2e-56) (/ x (* z (/ z y))) (/ y (* z (/ z x)))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e-56) {
		tmp = x / (z * (z / y));
	} else {
		tmp = y / (z * (z / x));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 2d-56) then
        tmp = x / (z * (z / y))
    else
        tmp = y / (z * (z / x))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e-56) {
		tmp = x / (z * (z / y));
	} else {
		tmp = y / (z * (z / x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if y <= 2e-56:
		tmp = x / (z * (z / y))
	else:
		tmp = y / (z * (z / x))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (y <= 2e-56)
		tmp = Float64(x / Float64(z * Float64(z / y)));
	else
		tmp = Float64(y / Float64(z * Float64(z / x)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2e-56)
		tmp = x / (z * (z / y));
	else
		tmp = y / (z * (z / x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 2e-56], N[(x / N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.0000000000000001e-56

   1. Initial program 83.5%

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

      1. /-rgt-identity 83.5%

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

      2. associate-/l* 83.5%

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

      3. associate-/l/ 85.2%

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

      4. associate-*l* 85.7%

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

      5. associate-*r/ 85.7%

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

      6. *-rgt-identity 85.7%

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

      7. associate-*l* 91.2%

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

      8. associate-*r/ 91.2%

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

      9. distribute-lft-in 91.2%

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

      10. fma-def 91.2%

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

      11. *-rgt-identity 91.2%

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

   3. Simplified 91.2%

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

   4. Taylor expanded in z around 0 80.5%

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

   if 2.0000000000000001e-56 < y

   1. Initial program 79.1%

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

      1. associate-*l* 79.1%

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

      2. times-frac 96.9%

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

      3. distribute-lft-in 96.8%

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

      4. fma-def 96.9%

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

      5. *-rgt-identity 96.9%

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

   3. Simplified 96.9%

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

   4. Taylor expanded in z around 0 59.6%

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

      1. unpow2 59.6%

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

      2. associate-/l* 68.1%

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

   6. Simplified 68.1%

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

      1. associate-/l* 68.1%

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

      2. associate-/r/ 68.1%

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

   8. Applied egg-rr 68.1%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-56}:\\ \;\;\;\;\frac{x}{z \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 12: 39.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{\frac{z}{x}}\\ \mathbf{if}\;z \leq -1 \cdot 10^{-310}:\\ \;\;\;\;-t_0\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ y (/ z x)))) (if (<= z -1e-310) (- t_0) t_0)))

C

assert(x < y);
double code(double x, double y, double z) {
	double t_0 = y / (z / x);
	double tmp;
	if (z <= -1e-310) {
		tmp = -t_0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y / (z / x)
    if (z <= (-1d-310)) then
        tmp = -t_0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double t_0 = y / (z / x);
	double tmp;
	if (z <= -1e-310) {
		tmp = -t_0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	t_0 = y / (z / x)
	tmp = 0
	if z <= -1e-310:
		tmp = -t_0
	else:
		tmp = t_0
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	t_0 = Float64(y / Float64(z / x))
	tmp = 0.0
	if (z <= -1e-310)
		tmp = Float64(-t_0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	t_0 = y / (z / x);
	tmp = 0.0;
	if (z <= -1e-310)
		tmp = -t_0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := Block[{t$95$0 = N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1e-310], (-t$95$0), t$95$0]]

Derivation

1. Split input into 2 regimes

2. if z < -9.999999999999969e-311

   1. Initial program 80.6%

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

      1. associate-*l* 80.6%

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

      2. times-frac 96.3%

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

      3. distribute-lft-in 96.3%

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

      4. fma-def 96.3%

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

      5. *-rgt-identity 96.3%

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

   3. Simplified 96.3%

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

   4. Taylor expanded in z around 0 64.8%

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

      1. neg-mul-1 64.8%

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

      2. +-commutative 64.8%

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

      3. unsub-neg 64.8%

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

   6. Simplified 64.8%

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

   7. Taylor expanded in z around inf 32.0%

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

      1. associate-*r/ 32.0%

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

      2. neg-mul-1 32.0%

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

      3. distribute-rgt-neg-in 32.0%

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

      4. associate-*l/ 36.2%

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

      5. distribute-rgt-neg-out 36.2%

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

   9. Simplified 36.2%

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

       1. associate-*l/ 32.0%

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

       2. associate-/l* 35.4%

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

   11. Applied egg-rr 35.4%

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

   if -9.999999999999969e-311 < z

   1. Initial program 84.2%

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

      1. associate-*l* 84.2%

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

      2. times-frac 96.3%

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

      3. distribute-lft-in 96.3%

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

      4. fma-def 96.3%

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

      5. *-rgt-identity 96.3%

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

   3. Simplified 96.3%

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

   4. Taylor expanded in z around 0 64.7%

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

      1. neg-mul-1 64.7%

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

      2. +-commutative 64.7%

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

      3. unsub-neg 64.7%

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

   6. Simplified 64.7%

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

   7. Taylor expanded in z around inf 15.5%

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

      1. associate-*r/ 19.3%

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

      2. neg-mul-1 19.3%

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

      3. distribute-rgt-neg-in 19.3%

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

      4. distribute-neg-frac 19.3%

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

   9. Simplified 19.3%

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

       1. add-sqr-sqrt 15.3%

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

       2. sqrt-unprod 30.0%

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

       3. distribute-frac-neg 30.0%

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

       4. distribute-frac-neg 30.0%

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

       5. sqr-neg 30.0%

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

       6. sqrt-unprod 21.6%

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

       7. add-sqr-sqrt 35.1%

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

       8. clear-num 35.8%

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

       9. div-inv 35.8%

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

   11. Applied egg-rr 35.8%

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

4. Final simplification 35.6%

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

Alternative 13: 73.9% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

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-*l* 82.3%

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

   2. times-frac 96.3%

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

   3. distribute-lft-in 96.3%

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

   4. fma-def 96.3%

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

   5. *-rgt-identity 96.3%

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

3. Simplified 96.3%

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

4. Taylor expanded in z around 0 69.8%

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

   1. unpow2 69.8%

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

   2. associate-/l/ 71.3%

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

   3. associate-*r/ 74.5%

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

   4. associate-*l/ 75.6%

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

6. Simplified 75.6%

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

7. Final simplification 75.6%

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

Alternative 14: 31.4% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = y / (z / x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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-*l* 82.3%

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

   2. times-frac 96.3%

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

   3. distribute-lft-in 96.3%

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

   4. fma-def 96.3%

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

   5. *-rgt-identity 96.3%

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

3. Simplified 96.3%

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

4. Taylor expanded in z around 0 64.8%

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

   1. neg-mul-1 64.8%

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

   2. +-commutative 64.8%

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

   3. unsub-neg 64.8%

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

6. Simplified 64.8%

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

7. Taylor expanded in z around inf 24.3%

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

   1. associate-*r/ 27.9%

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

   2. neg-mul-1 27.9%

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

   3. distribute-rgt-neg-in 27.9%

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

   4. distribute-neg-frac 27.9%

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

9. Simplified 27.9%

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

    1. add-sqr-sqrt 20.5%

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

    2. sqrt-unprod 35.1%

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

    3. distribute-frac-neg 35.1%

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

    4. distribute-frac-neg 35.1%

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

    5. sqr-neg 35.1%

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

    6. sqrt-unprod 16.9%

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

    7. add-sqr-sqrt 26.6%

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

    8. clear-num 26.9%

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

    9. div-inv 26.9%

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

11. Applied egg-rr 26.9%

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

12. Final simplification 26.9%

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

Developer target: 95.9% 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 2023185 
(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))))