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

Percentage Accurate: 84.0% → 94.4%

Time: 10.7s

Alternatives: 16

Speedup: 1.0×

• 26.3% 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: 84.0% 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: 94.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(z + 1\right) \cdot \left(z \cdot z\right)\\ \mathbf{if}\;t_0 \leq -20000000:\\ \;\;\;\;\frac{\frac{y}{z}}{\frac{z}{\frac{x}{z}}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{-287}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot z} \cdot \frac{x}{z + 1}\\ \end{array} \end{array} \]

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 -20000000.0)
     (/ (/ y z) (/ z (/ x z)))
     (if (<= t_0 5e-287)
       (/ (/ x z) (/ z y))
       (* (/ y (* z z)) (/ x (+ z 1.0)))))))

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 <= -20000000.0) {
		tmp = (y / z) / (z / (x / z));
	} else if (t_0 <= 5e-287) {
		tmp = (x / z) / (z / y);
	} else {
		tmp = (y / (z * z)) * (x / (z + 1.0));
	}
	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 = (z + 1.0d0) * (z * z)
    if (t_0 <= (-20000000.0d0)) then
        tmp = (y / z) / (z / (x / z))
    else if (t_0 <= 5d-287) then
        tmp = (x / z) / (z / y)
    else
        tmp = (y / (z * z)) * (x / (z + 1.0d0))
    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 <= -20000000.0) {
		tmp = (y / z) / (z / (x / z));
	} else if (t_0 <= 5e-287) {
		tmp = (x / z) / (z / y);
	} else {
		tmp = (y / (z * z)) * (x / (z + 1.0));
	}
	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 <= -20000000.0:
		tmp = (y / z) / (z / (x / z))
	elif t_0 <= 5e-287:
		tmp = (x / z) / (z / y)
	else:
		tmp = (y / (z * z)) * (x / (z + 1.0))
	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 <= -20000000.0)
		tmp = Float64(Float64(y / z) / Float64(z / Float64(x / z)));
	elseif (t_0 <= 5e-287)
		tmp = Float64(Float64(x / z) / Float64(z / y));
	else
		tmp = Float64(Float64(y / Float64(z * z)) * Float64(x / Float64(z + 1.0)));
	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 <= -20000000.0)
		tmp = (y / z) / (z / (x / z));
	elseif (t_0 <= 5e-287)
		tmp = (x / z) / (z / y);
	else
		tmp = (y / (z * z)) * (x / (z + 1.0));
	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[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -20000000.0], N[(N[(y / z), $MachinePrecision] / N[(z / N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-287], N[(N[(x / z), $MachinePrecision] / N[(z / y), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision] * N[(x / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z 1)) < -2e7

   1. Initial program 84.0%

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

      1. *-commutative 84.0%

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

      2. sqr-neg 84.0%

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

      3. times-frac 94.3%

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

      4. sqr-neg 94.3%

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

   3. Simplified 94.3%

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

      1. *-commutative 94.3%

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

      2. clear-num 94.3%

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

      3. associate-/r* 96.5%

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

      4. frac-times 97.3%

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

      5. *-un-lft-identity 97.3%

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

   5. Applied egg-rr 97.3%

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

   6. Taylor expanded in z around inf 93.6%

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

      1. unpow2 93.6%

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

      2. associate-*r/ 95.9%

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

   8. Simplified 95.9%

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

      1. clear-num 95.9%

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

      2. un-div-inv 96.0%

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

   10. Applied egg-rr 96.0%

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

   if -2e7 < (*.f64 (*.f64 z z) (+.f64 z 1)) < 5.00000000000000025e-287

   1. Initial program 78.4%

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

      1. *-commutative 78.4%

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

      2. sqr-neg 78.4%

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

      3. times-frac 78.7%

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

      4. sqr-neg 78.7%

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

   3. Simplified 78.7%

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

      1. frac-times 78.4%

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

      2. *-commutative 78.4%

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

      3. associate-/r* 78.4%

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

      4. frac-times 99.9%

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

      5. associate-/l* 99.8%

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

      6. div-inv 99.8%

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

      7. clear-num 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in z around 0 99.8%

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

   if 5.00000000000000025e-287 < (*.f64 (*.f64 z z) (+.f64 z 1))

   1. Initial program 91.8%

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

      1. *-commutative 91.8%

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

      2. sqr-neg 91.8%

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

      3. times-frac 95.7%

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

      4. sqr-neg 95.7%

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

   3. Simplified 95.7%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq -20000000:\\ \;\;\;\;\frac{\frac{y}{z}}{\frac{z}{\frac{x}{z}}}\\ \mathbf{elif}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 5 \cdot 10^{-287}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot z} \cdot \frac{x}{z + 1}\\ \end{array} \]

Alternative 2: 91.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 88.4%

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

      1. *-commutative 88.4%

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

      2. sqr-neg 88.4%

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

      3. times-frac 94.5%

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

      4. sqr-neg 94.5%

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

   3. Simplified 94.5%

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

      1. *-commutative 94.5%

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

      2. clear-num 94.3%

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

      3. associate-/r* 95.4%

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

      4. frac-times 96.8%

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

      5. *-un-lft-identity 96.8%

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

   5. Applied egg-rr 96.8%

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

   6. Taylor expanded in z around inf 92.3%

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

      1. unpow2 92.3%

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

      2. associate-*r/ 94.0%

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

   8. Simplified 94.0%

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

      1. associate-/r* 92.7%

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

      2. associate-/r/ 90.7%

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

      3. associate-/l/ 90.7%

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

   10. Applied egg-rr 90.7%

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

   if -1 < z < 1

   1. Initial program 82.9%

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

      1. *-commutative 82.9%

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

      2. sqr-neg 82.9%

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

      3. times-frac 85.6%

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

      4. sqr-neg 85.6%

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

   3. Simplified 85.6%

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

      1. frac-times 82.9%

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

      2. *-commutative 82.9%

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

      3. associate-/r* 82.9%

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

      4. frac-times 98.3%

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

      5. associate-/l* 98.2%

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

      6. div-inv 98.2%

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

      7. clear-num 98.3%

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

   5. Applied egg-rr 98.3%

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

   6. Taylor expanded in z around 0 97.7%

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

4. Final simplification 94.1%

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

Alternative 3: 94.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 88.4%

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

      1. *-commutative 88.4%

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

      2. sqr-neg 88.4%

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

      3. times-frac 94.5%

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

      4. sqr-neg 94.5%

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

   3. Simplified 94.5%

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

      1. frac-times 88.4%

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

      2. *-commutative 88.4%

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

      3. associate-/r* 90.5%

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

      4. frac-times 96.9%

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

      5. associate-/l* 95.4%

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

      6. div-inv 95.3%

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

      7. clear-num 95.4%

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

   5. Applied egg-rr 95.4%

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

   6. Taylor expanded in z around inf 91.5%

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

      1. unpow2 91.5%

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

      2. associate-/l* 92.7%

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

   8. Simplified 92.7%

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

      1. associate-/r/ 92.7%

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

   10. Applied egg-rr 92.7%

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

   if -1 < z < 1

   1. Initial program 82.9%

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

      1. *-commutative 82.9%

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

      2. sqr-neg 82.9%

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

      3. times-frac 85.6%

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

      4. sqr-neg 85.6%

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

   3. Simplified 85.6%

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

      1. frac-times 82.9%

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

      2. *-commutative 82.9%

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

      3. associate-/r* 82.9%

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

      4. frac-times 98.3%

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

      5. associate-/l* 98.2%

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

      6. div-inv 98.2%

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

      7. clear-num 98.3%

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

   5. Applied egg-rr 98.3%

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

   6. Taylor expanded in z around 0 97.7%

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

4. Final simplification 95.1%

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

Alternative 4: 94.5% accurate, 0.8× speedup

Math

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

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

C

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

Python

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

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 84.0%

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

      1. *-commutative 84.0%

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

      2. sqr-neg 84.0%

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

      3. times-frac 94.3%

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

      4. sqr-neg 94.3%

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

   3. Simplified 94.3%

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

      1. frac-times 84.0%

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

      2. *-commutative 84.0%

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

      3. associate-/r* 85.9%

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

      4. frac-times 95.9%

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

      5. associate-/l* 96.6%

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

      6. div-inv 96.5%

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

      7. clear-num 96.6%

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

   5. Applied egg-rr 96.6%

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

   6. Taylor expanded in z around inf 92.9%

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

      1. unpow2 92.9%

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

      2. associate-/l* 95.2%

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

   8. Simplified 95.2%

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

   if -1 < z < 1

   1. Initial program 82.9%

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

      1. *-commutative 82.9%

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

      2. sqr-neg 82.9%

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

      3. times-frac 85.6%

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

      4. sqr-neg 85.6%

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

   3. Simplified 85.6%

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

      1. frac-times 82.9%

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

      2. *-commutative 82.9%

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

      3. associate-/r* 82.9%

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

      4. frac-times 98.3%

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

      5. associate-/l* 98.2%

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

      6. div-inv 98.2%

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

      7. clear-num 98.3%

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

   5. Applied egg-rr 98.3%

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

   6. Taylor expanded in z around 0 97.7%

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

   if 1 < z

   1. Initial program 92.4%

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

      1. *-commutative 92.4%

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

      2. sqr-neg 92.4%

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

      3. times-frac 94.6%

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

      4. sqr-neg 94.6%

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

   3. Simplified 94.6%

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

      1. frac-times 92.4%

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

      2. *-commutative 92.4%

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

      3. associate-/r* 94.5%

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

      4. frac-times 97.8%

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

      5. associate-/l* 94.3%

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

      6. div-inv 94.3%

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

      7. clear-num 94.3%

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

   5. Applied egg-rr 94.3%

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

   6. Taylor expanded in z around inf 90.4%

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

      1. unpow2 90.4%

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

      2. associate-/l* 90.4%

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

   8. Simplified 90.4%

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

      1. associate-/r/ 90.4%

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

   10. Applied egg-rr 90.4%

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

4. Final simplification 95.1%

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

Alternative 5: 94.5% accurate, 0.8× speedup

Math

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

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

C

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

Python

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

Julia

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

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 84.0%

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

      1. *-commutative 84.0%

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

      2. sqr-neg 84.0%

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

      3. times-frac 94.3%

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

      4. sqr-neg 94.3%

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

   3. Simplified 94.3%

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

      1. *-commutative 94.3%

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

      2. clear-num 94.3%

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

      3. associate-/r* 96.5%

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

      4. frac-times 97.3%

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

      5. *-un-lft-identity 97.3%

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

   5. Applied egg-rr 97.3%

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

   6. Taylor expanded in z around inf 93.6%

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

      1. unpow2 93.6%

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

      2. associate-*r/ 95.9%

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

   8. Simplified 95.9%

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

   if -1 < z < 1

   1. Initial program 82.9%

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

      1. *-commutative 82.9%

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

      2. sqr-neg 82.9%

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

      3. times-frac 85.6%

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

      4. sqr-neg 85.6%

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

   3. Simplified 85.6%

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

      1. frac-times 82.9%

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

      2. *-commutative 82.9%

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

      3. associate-/r* 82.9%

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

      4. frac-times 98.3%

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

      5. associate-/l* 98.2%

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

      6. div-inv 98.2%

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

      7. clear-num 98.3%

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

   5. Applied egg-rr 98.3%

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

   6. Taylor expanded in z around 0 97.7%

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

   if 1 < z

   1. Initial program 92.4%

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

      1. *-commutative 92.4%

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

      2. sqr-neg 92.4%

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

      3. times-frac 94.6%

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

      4. sqr-neg 94.6%

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

   3. Simplified 94.6%

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

      1. frac-times 92.4%

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

      2. *-commutative 92.4%

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

      3. associate-/r* 94.5%

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

      4. frac-times 97.8%

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

      5. associate-/l* 94.3%

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

      6. div-inv 94.3%

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

      7. clear-num 94.3%

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

   5. Applied egg-rr 94.3%

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

   6. Taylor expanded in z around inf 90.4%

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

      1. unpow2 90.4%

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

      2. associate-/l* 90.4%

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

   8. Simplified 90.4%

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

      1. associate-/r/ 90.4%

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

   10. Applied egg-rr 90.4%

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

4. Final simplification 95.3%

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

Alternative 6: 94.5% accurate, 0.8× speedup

Math

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

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

C

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

Python

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

Julia

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

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 84.0%

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

      1. *-commutative 84.0%

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

      2. sqr-neg 84.0%

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

      3. times-frac 94.3%

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

      4. sqr-neg 94.3%

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

   3. Simplified 94.3%

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

      1. *-commutative 94.3%

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

      2. clear-num 94.3%

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

      3. associate-/r* 96.5%

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

      4. frac-times 97.3%

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

      5. *-un-lft-identity 97.3%

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

   5. Applied egg-rr 97.3%

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

   6. Taylor expanded in z around inf 93.6%

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

      1. unpow2 93.6%

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

      2. associate-*r/ 95.9%

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

   8. Simplified 95.9%

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

      1. clear-num 95.9%

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

      2. un-div-inv 96.0%

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

   10. Applied egg-rr 96.0%

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

   if -1 < z < 1

   1. Initial program 82.9%

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

      1. *-commutative 82.9%

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

      2. sqr-neg 82.9%

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

      3. times-frac 85.6%

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

      4. sqr-neg 85.6%

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

   3. Simplified 85.6%

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

      1. frac-times 82.9%

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

      2. *-commutative 82.9%

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

      3. associate-/r* 82.9%

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

      4. frac-times 98.3%

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

      5. associate-/l* 98.2%

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

      6. div-inv 98.2%

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

      7. clear-num 98.3%

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

   5. Applied egg-rr 98.3%

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

   6. Taylor expanded in z around 0 97.7%

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

   if 1 < z

   1. Initial program 92.4%

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

      1. *-commutative 92.4%

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

      2. sqr-neg 92.4%

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

      3. times-frac 94.6%

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

      4. sqr-neg 94.6%

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

   3. Simplified 94.6%

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

      1. frac-times 92.4%

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

      2. *-commutative 92.4%

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

      3. associate-/r* 94.5%

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

      4. frac-times 97.8%

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

      5. associate-/l* 94.3%

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

      6. div-inv 94.3%

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

      7. clear-num 94.3%

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

   5. Applied egg-rr 94.3%

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

   6. Taylor expanded in z around inf 90.4%

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

      1. unpow2 90.4%

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

      2. associate-/l* 90.4%

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

   8. Simplified 90.4%

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

      1. associate-/r/ 90.4%

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

   10. Applied egg-rr 90.4%

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

4. Final simplification 95.3%

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

Alternative 7: 94.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{\frac{y}{z}}{\frac{z}{\frac{x}{z}}}\\ \mathbf{elif}\;z \leq 0.75:\\ \;\;\;\;\frac{x}{z} \cdot \left(\frac{y}{z} - y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z \cdot \frac{z}{y}}\\ \end{array} \end{array} \]

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)
   (/ (/ y z) (/ z (/ x z)))
   (if (<= z 0.75) (* (/ x z) (- (/ y z) y)) (/ (/ x z) (* z (/ z y))))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = (y / z) / (z / (x / z));
	} else if (z <= 0.75) {
		tmp = (x / z) * ((y / z) - y);
	} else {
		tmp = (x / z) / (z * (z / y));
	}
	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 = (y / z) / (z / (x / z))
    else if (z <= 0.75d0) then
        tmp = (x / z) * ((y / z) - y)
    else
        tmp = (x / z) / (z * (z / y))
    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 = (y / z) / (z / (x / z));
	} else if (z <= 0.75) {
		tmp = (x / z) * ((y / z) - y);
	} else {
		tmp = (x / z) / (z * (z / y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if z <= -1.0:
		tmp = (y / z) / (z / (x / z))
	elif z <= 0.75:
		tmp = (x / z) * ((y / z) - y)
	else:
		tmp = (x / z) / (z * (z / y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(Float64(y / z) / Float64(z / Float64(x / z)));
	elseif (z <= 0.75)
		tmp = Float64(Float64(x / z) * Float64(Float64(y / z) - y));
	else
		tmp = Float64(Float64(x / z) / Float64(z * Float64(z / y)));
	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 = (y / z) / (z / (x / z));
	elseif (z <= 0.75)
		tmp = (x / z) * ((y / z) - y);
	else
		tmp = (x / z) / (z * (z / y));
	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[(y / z), $MachinePrecision] / N[(z / N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.75], N[(N[(x / z), $MachinePrecision] * N[(N[(y / z), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] / N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 84.0%

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

      1. *-commutative 84.0%

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

      2. sqr-neg 84.0%

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

      3. times-frac 94.3%

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

      4. sqr-neg 94.3%

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

   3. Simplified 94.3%

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

      1. *-commutative 94.3%

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

      2. clear-num 94.3%

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

      3. associate-/r* 96.5%

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

      4. frac-times 97.3%

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

      5. *-un-lft-identity 97.3%

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

   5. Applied egg-rr 97.3%

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

   6. Taylor expanded in z around inf 93.6%

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

      1. unpow2 93.6%

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

      2. associate-*r/ 95.9%

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

   8. Simplified 95.9%

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

      1. clear-num 95.9%

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

      2. un-div-inv 96.0%

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

   10. Applied egg-rr 96.0%

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

   if -1 < z < 0.75

   1. Initial program 82.9%

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

      1. associate-*l* 82.9%

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

      2. times-frac 98.3%

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

      3. associate-/r* 98.3%

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

      4. associate-*r/ 98.3%

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

   3. Simplified 98.3%

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

   4. Taylor expanded in z around 0 61.3%

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

      1. +-commutative 61.3%

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

      2. unpow2 61.3%

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

      3. times-frac 76.6%

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

      4. mul-1-neg 76.6%

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

      5. associate-*l/ 68.4%

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

      6. distribute-rgt-neg-in 68.4%

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

      7. distribute-lft-out 97.8%

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

   6. Simplified 97.8%

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

   if 0.75 < z

   1. Initial program 92.4%

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

      1. *-commutative 92.4%

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

      2. sqr-neg 92.4%

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

      3. times-frac 94.6%

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

      4. sqr-neg 94.6%

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

   3. Simplified 94.6%

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

      1. frac-times 92.4%

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

      2. *-commutative 92.4%

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

      3. associate-/r* 94.5%

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

      4. frac-times 97.8%

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

      5. associate-/l* 94.3%

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

      6. div-inv 94.3%

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

      7. clear-num 94.3%

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

   5. Applied egg-rr 94.3%

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

   6. Taylor expanded in z around inf 90.4%

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

      1. unpow2 90.4%

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

      2. associate-/l* 90.4%

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

   8. Simplified 90.4%

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

      1. associate-/r/ 90.4%

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

   10. Applied egg-rr 90.4%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{\frac{y}{z}}{\frac{z}{\frac{x}{z}}}\\ \mathbf{elif}\;z \leq 0.75:\\ \;\;\;\;\frac{x}{z} \cdot \left(\frac{y}{z} - y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z \cdot \frac{z}{y}}\\ \end{array} \]

Alternative 8: 79.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{z \cdot z}\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t_0 \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{-144}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot t_0\\ \end{array} \end{array} \]

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 z))))
   (if (<= z -1.0)
     (* t_0 (- x))
     (if (<= z 6.4e-144) (/ (/ x z) (/ z y)) (* x t_0)))))

C

assert(x < y);
double code(double x, double y, double z) {
	double t_0 = y / (z * z);
	double tmp;
	if (z <= -1.0) {
		tmp = t_0 * -x;
	} else if (z <= 6.4e-144) {
		tmp = (x / z) / (z / y);
	} else {
		tmp = x * 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 * z)
    if (z <= (-1.0d0)) then
        tmp = t_0 * -x
    else if (z <= 6.4d-144) then
        tmp = (x / z) / (z / y)
    else
        tmp = x * 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 * z);
	double tmp;
	if (z <= -1.0) {
		tmp = t_0 * -x;
	} else if (z <= 6.4e-144) {
		tmp = (x / z) / (z / y);
	} else {
		tmp = x * t_0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	t_0 = y / (z * z)
	tmp = 0
	if z <= -1.0:
		tmp = t_0 * -x
	elif z <= 6.4e-144:
		tmp = (x / z) / (z / y)
	else:
		tmp = x * t_0
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	t_0 = Float64(y / Float64(z * z))
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(t_0 * Float64(-x));
	elseif (z <= 6.4e-144)
		tmp = Float64(Float64(x / z) / Float64(z / y));
	else
		tmp = Float64(x * t_0);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	t_0 = y / (z * z);
	tmp = 0.0;
	if (z <= -1.0)
		tmp = t_0 * -x;
	elseif (z <= 6.4e-144)
		tmp = (x / z) / (z / y);
	else
		tmp = x * 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 * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.0], N[(t$95$0 * (-x)), $MachinePrecision], If[LessEqual[z, 6.4e-144], N[(N[(x / z), $MachinePrecision] / N[(z / y), $MachinePrecision]), $MachinePrecision], N[(x * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 84.0%

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

      1. *-commutative 84.0%

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

      2. sqr-neg 84.0%

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

      3. times-frac 94.3%

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

      4. sqr-neg 94.3%

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

   3. Simplified 94.3%

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

      1. frac-times 84.0%

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

      2. *-commutative 84.0%

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

      3. associate-/r* 85.9%

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

      4. frac-times 95.9%

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

      5. associate-/l* 96.6%

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

      6. div-inv 96.5%

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

      7. clear-num 96.6%

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

   5. Applied egg-rr 96.6%

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

   6. Taylor expanded in z around 0 48.2%

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

      1. associate-/r* 59.7%

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

      2. frac-2neg 59.7%

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

      3. div-inv 59.7%

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

      4. associate-*r/ 63.4%

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

      5. distribute-neg-frac 63.4%

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

      6. add-sqr-sqrt 34.1%

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

      7. sqrt-unprod 51.1%

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

      8. sqr-neg 51.1%

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

      9. sqrt-unprod 30.1%

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

      10. add-sqr-sqrt 68.1%

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

      11. frac-2neg 68.1%

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

      12. associate-*r/ 64.3%

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

      13. associate-/l/ 64.3%

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

      14. clear-num 64.3%

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

      15. associate-/l/ 68.1%

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

   8. Applied egg-rr 68.1%

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

      1. *-commutative 68.1%

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

   10. Simplified 68.1%

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

   if -1 < z < 6.39999999999999946e-144

   1. Initial program 81.8%

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

      1. *-commutative 81.8%

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

      2. sqr-neg 81.8%

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

      3. times-frac 82.0%

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

      4. sqr-neg 82.0%

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

   3. Simplified 82.0%

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

      1. frac-times 81.8%

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

      2. *-commutative 81.8%

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

      3. associate-/r* 81.8%

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

      4. frac-times 99.8%

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

      5. associate-/l* 99.8%

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

      6. div-inv 99.8%

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

      7. clear-num 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in z around 0 99.1%

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

   if 6.39999999999999946e-144 < z

   1. Initial program 91.1%

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

      1. *-commutative 91.1%

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

      2. sqr-neg 91.1%

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

      3. times-frac 95.9%

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

      4. sqr-neg 95.9%

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

   3. Simplified 95.9%

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

      1. frac-times 91.1%

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

      2. *-commutative 91.1%

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

      3. associate-/r* 92.7%

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

      4. frac-times 96.3%

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

      5. associate-/l* 93.8%

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

      6. div-inv 93.8%

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

      7. clear-num 93.8%

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

   5. Applied egg-rr 93.8%

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

   6. Taylor expanded in z around 0 70.6%

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

      1. unpow2 70.6%

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

      2. associate-*r/ 74.5%

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

   8. Simplified 74.5%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{y}{z \cdot z} \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{-144}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \end{array} \]

Alternative 9: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{-9}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{y}{z}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* x y) -1e-9) (* x (/ y (* z z))) (/ (* x (/ y z)) z)))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((x * y) <= -1e-9) {
		tmp = x * (y / (z * z));
	} else {
		tmp = (x * (y / z)) / z;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x * y) <= (-1d-9)) then
        tmp = x * (y / (z * z))
    else
        tmp = (x * (y / z)) / z
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((x * y) <= -1e-9) {
		tmp = x * (y / (z * z));
	} else {
		tmp = (x * (y / z)) / z;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (x * y) <= -1e-9:
		tmp = x * (y / (z * z))
	else:
		tmp = (x * (y / z)) / z
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(x * y) <= -1e-9)
		tmp = Float64(x * Float64(y / Float64(z * z)));
	else
		tmp = Float64(Float64(x * Float64(y / z)) / z);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * y) <= -1e-9)
		tmp = x * (y / (z * z));
	else
		tmp = (x * (y / 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[N[(x * y), $MachinePrecision], -1e-9], N[(x * N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1.00000000000000006e-9

   1. Initial program 87.1%

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

      1. *-commutative 87.1%

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

      2. sqr-neg 87.1%

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

      3. times-frac 94.7%

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

      4. sqr-neg 94.7%

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

   3. Simplified 94.7%

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

      1. frac-times 87.1%

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

      2. *-commutative 87.1%

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

      3. associate-/r* 88.9%

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

      4. frac-times 94.2%

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

      5. associate-/l* 91.9%

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

      6. div-inv 91.8%

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

      7. clear-num 91.8%

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

   5. Applied egg-rr 91.8%

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

   6. Taylor expanded in z around 0 61.8%

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

      1. unpow2 61.8%

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

      2. associate-*r/ 68.5%

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

   8. Simplified 68.5%

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

   if -1.00000000000000006e-9 < (*.f64 x y)

   1. Initial program 85.3%

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

      1. *-commutative 85.3%

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

      2. associate-*r/ 85.5%

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

      3. sqr-neg 85.5%

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

      4. associate-*l* 85.5%

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

      5. associate-*l* 85.5%

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

      6. sqr-neg 85.5%

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

      7. associate-*l* 85.5%

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

      8. distribute-lft-in 85.5%

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

      9. fma-def 85.5%

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

      10. *-rgt-identity 85.5%

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

   3. Simplified 85.5%

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

   4. Taylor expanded in z around 0 74.8%

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

      1. unpow2 74.8%

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

   6. Simplified 74.8%

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

      1. associate-*r/ 73.7%

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

      2. *-commutative 73.7%

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

      3. associate-/r* 78.6%

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

      4. associate-*r/ 83.7%

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

   8. Applied egg-rr 83.7%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{-9}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{y}{z}}{z}\\ \end{array} \]

Alternative 10: 76.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{-9}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{z}{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* x y) -1e-9) (* x (/ y (* z z))) (/ (/ x z) (/ z y))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((x * y) <= -1e-9) {
		tmp = x * (y / (z * z));
	} else {
		tmp = (x / z) / (z / y);
	}
	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 * y) <= (-1d-9)) then
        tmp = x * (y / (z * z))
    else
        tmp = (x / z) / (z / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((x * y) <= -1e-9) {
		tmp = x * (y / (z * z));
	} else {
		tmp = (x / z) / (z / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (x * y) <= -1e-9:
		tmp = x * (y / (z * z))
	else:
		tmp = (x / z) / (z / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(x * y) <= -1e-9)
		tmp = Float64(x * Float64(y / Float64(z * z)));
	else
		tmp = Float64(Float64(x / z) / Float64(z / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * y) <= -1e-9)
		tmp = x * (y / (z * z));
	else
		tmp = (x / z) / (z / y);
	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[N[(x * y), $MachinePrecision], -1e-9], N[(x * N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] / N[(z / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1.00000000000000006e-9

   1. Initial program 87.1%

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

      1. *-commutative 87.1%

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

      2. sqr-neg 87.1%

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

      3. times-frac 94.7%

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

      4. sqr-neg 94.7%

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

   3. Simplified 94.7%

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

      1. frac-times 87.1%

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

      2. *-commutative 87.1%

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

      3. associate-/r* 88.9%

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

      4. frac-times 94.2%

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

      5. associate-/l* 91.9%

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

      6. div-inv 91.8%

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

      7. clear-num 91.8%

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

   5. Applied egg-rr 91.8%

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

   6. Taylor expanded in z around 0 61.8%

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

      1. unpow2 61.8%

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

      2. associate-*r/ 68.5%

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

   8. Simplified 68.5%

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

   if -1.00000000000000006e-9 < (*.f64 x y)

   1. Initial program 85.3%

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

      1. *-commutative 85.3%

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

      2. sqr-neg 85.3%

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

      3. times-frac 88.5%

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

      4. sqr-neg 88.5%

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

   3. Simplified 88.5%

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

      1. frac-times 85.3%

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

      2. *-commutative 85.3%

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

      3. associate-/r* 86.1%

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

      4. frac-times 98.8%

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

      5. associate-/l* 98.6%

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

      6. div-inv 98.6%

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

      7. clear-num 98.6%

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

   5. Applied egg-rr 98.6%

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

   6. Taylor expanded in z around 0 85.3%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{-9}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{z}{y}}\\ \end{array} \]

Alternative 11: 96.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x / z) / ((z + 1.0d0) * (z / y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.8%

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

   1. *-commutative 85.8%

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

   2. sqr-neg 85.8%

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

   3. times-frac 90.2%

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

   4. sqr-neg 90.2%

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

3. Simplified 90.2%

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

   1. frac-times 85.8%

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

   2. *-commutative 85.8%

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

   3. associate-/r* 86.9%

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

   4. frac-times 97.6%

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

   5. associate-/l* 96.8%

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

   6. div-inv 96.7%

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

   7. clear-num 96.8%

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

5. Applied egg-rr 96.8%

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

6. Final simplification 96.8%

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

Alternative 12: 96.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((x / z) * (y / z)) / (z + 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.8%

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

   1. associate-*l* 85.8%

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

   2. times-frac 96.3%

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

   3. associate-/r* 96.8%

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

   4. associate-*r/ 97.6%

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

3. Simplified 97.6%

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

4. Final simplification 97.6%

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

Alternative 13: 78.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -20000000000:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= x -20000000000.0) (* x (/ y (* z z))) (* y (/ (/ x z) z))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (x <= -20000000000.0) {
		tmp = x * (y / (z * z));
	} else {
		tmp = y * ((x / z) / z);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-20000000000.0d0)) then
        tmp = x * (y / (z * z))
    else
        tmp = y * ((x / z) / z)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -20000000000.0) {
		tmp = x * (y / (z * z));
	} else {
		tmp = y * ((x / z) / z);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if x <= -20000000000.0:
		tmp = x * (y / (z * z))
	else:
		tmp = y * ((x / z) / z)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (x <= -20000000000.0)
		tmp = Float64(x * Float64(y / Float64(z * z)));
	else
		tmp = Float64(y * Float64(Float64(x / z) / z));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -20000000000.0)
		tmp = x * (y / (z * z));
	else
		tmp = y * ((x / z) / z);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[x, -20000000000.0], N[(x * N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(x / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2e10

   1. Initial program 93.2%

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

      1. *-commutative 93.2%

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

      2. sqr-neg 93.2%

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

      3. times-frac 95.5%

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

      4. sqr-neg 95.5%

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

   3. Simplified 95.5%

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

      1. frac-times 93.2%

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

      2. *-commutative 93.2%

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

      3. associate-/r* 95.5%

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

      4. frac-times 95.6%

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

      5. associate-/l* 94.9%

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

      6. div-inv 94.8%

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

      7. clear-num 94.9%

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

   5. Applied egg-rr 94.9%

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

   6. Taylor expanded in z around 0 77.6%

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

      1. unpow2 77.6%

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

      2. associate-*r/ 84.4%

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

   8. Simplified 84.4%

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

   if -2e10 < x

   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. *-commutative 83.2%

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

      2. associate-*r/ 84.7%

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

      3. sqr-neg 84.7%

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

      4. associate-*l* 84.7%

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

      5. associate-*l* 84.7%

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

      6. sqr-neg 84.7%

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

      7. associate-*l* 84.7%

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

      8. distribute-lft-in 84.7%

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

      9. fma-def 84.7%

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

      10. *-rgt-identity 84.7%

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

   3. Simplified 84.7%

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

   4. Taylor expanded in z around 0 70.3%

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

      1. unpow2 70.3%

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

      2. associate-/r* 73.0%

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

   6. Simplified 73.0%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -20000000000:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \]

Alternative 14: 79.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1000000000000:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= x -1000000000000.0) (* x (/ y (* z z))) (/ y (* z (/ z x)))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (x <= -1000000000000.0) {
		tmp = x * (y / (z * z));
	} else {
		tmp = y / (z * (z / x));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1000000000000.0d0)) then
        tmp = x * (y / (z * z))
    else
        tmp = y / (z * (z / x))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1000000000000.0) {
		tmp = x * (y / (z * z));
	} else {
		tmp = y / (z * (z / x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if x <= -1000000000000.0:
		tmp = x * (y / (z * z))
	else:
		tmp = y / (z * (z / x))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (x <= -1000000000000.0)
		tmp = Float64(x * Float64(y / Float64(z * z)));
	else
		tmp = Float64(y / Float64(z * Float64(z / x)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1000000000000.0)
		tmp = x * (y / (z * z));
	else
		tmp = y / (z * (z / x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[x, -1000000000000.0], N[(x * N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1e12

   1. Initial program 93.2%

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

      1. *-commutative 93.2%

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

      2. sqr-neg 93.2%

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

      3. times-frac 95.5%

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

      4. sqr-neg 95.5%

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

   3. Simplified 95.5%

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

      1. frac-times 93.2%

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

      2. *-commutative 93.2%

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

      3. associate-/r* 95.5%

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

      4. frac-times 95.6%

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

      5. associate-/l* 94.9%

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

      6. div-inv 94.8%

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

      7. clear-num 94.9%

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

   5. Applied egg-rr 94.9%

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

   6. Taylor expanded in z around 0 77.6%

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

      1. unpow2 77.6%

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

      2. associate-*r/ 84.4%

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

   8. Simplified 84.4%

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

   if -1e12 < x

   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. *-commutative 83.2%

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

      2. associate-*r/ 84.7%

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

      3. sqr-neg 84.7%

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

      4. associate-*l* 84.7%

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

      5. associate-*l* 84.7%

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

      6. sqr-neg 84.7%

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

      7. associate-*l* 84.7%

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

      8. distribute-lft-in 84.7%

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

      9. fma-def 84.7%

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

      10. *-rgt-identity 84.7%

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

   3. Simplified 84.7%

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

   4. Taylor expanded in z around 0 70.3%

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

      1. unpow2 70.3%

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

   6. Simplified 70.3%

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

      1. associate-*r/ 67.9%

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

      2. *-commutative 67.9%

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

      3. times-frac 76.9%

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

      4. clear-num 76.9%

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

      5. frac-times 73.6%

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

      6. *-un-lft-identity 73.6%

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

   8. Applied egg-rr 73.6%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1000000000000:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 15: 72.9% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.8%

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

   1. *-commutative 85.8%

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

   2. sqr-neg 85.8%

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

   3. times-frac 90.2%

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

   4. sqr-neg 90.2%

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

3. Simplified 90.2%

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

   1. frac-times 85.8%

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

   2. *-commutative 85.8%

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

   3. associate-/r* 86.9%

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

   4. frac-times 97.6%

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

   5. associate-/l* 96.8%

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

   6. div-inv 96.7%

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

   7. clear-num 96.8%

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

5. Applied egg-rr 96.8%

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

6. Taylor expanded in z around 0 70.5%

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

   1. unpow2 70.5%

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

   2. associate-*r/ 74.3%

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

8. Simplified 74.3%

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

9. Final simplification 74.3%

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

Alternative 16: 30.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.8%

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

   1. associate-*l* 85.8%

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

   2. times-frac 96.3%

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

   3. associate-/r* 96.8%

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

   4. associate-*r/ 97.6%

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

3. Simplified 97.6%

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

4. Taylor expanded in z around 0 47.7%

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

   1. +-commutative 47.7%

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

   2. unpow2 47.7%

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

   3. times-frac 55.1%

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

   4. mul-1-neg 55.1%

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

   5. associate-*l/ 52.7%

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

   6. distribute-rgt-neg-in 52.7%

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

   7. distribute-lft-out 66.8%

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

6. Simplified 66.8%

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

7. Taylor expanded in z around inf 27.5%

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

   1. associate-*l/ 30.0%

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

   2. associate-*r* 30.0%

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

   3. neg-mul-1 30.0%

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

   4. distribute-frac-neg 30.0%

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

   5. *-commutative 30.0%

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

9. Simplified 30.0%

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

10. Final simplification 30.0%

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

Developer target: 95.4% 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 2023297 
(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))))