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

Percentage Accurate: 82.7% → 96.3%

Time: 10.3s

Alternatives: 13

Speedup: 1.0×

• 25.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 + 1.0d0))) / z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. frac-times 87.1%

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

   2. associate-*l/ 86.0%

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

   3. times-frac 96.0%

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

3. Applied egg-rr 96.0%

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

   1. associate-*l/ 96.7%

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

   2. associate-/l/ 94.9%

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

   3. associate-/r* 96.7%

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

5. Applied egg-rr 96.7%

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

6. Final simplification 96.7%

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

Alternative 2: 94.9% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.0) (not (<= z 1.0)))
   (* (/ x z) (/ (/ y z) z))
   (/ (/ x (/ z y)) z)))

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.0d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = (x / z) * ((y / z) / z)
    else
        tmp = (x / (z / y)) / z
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(N[(x / z), $MachinePrecision] * N[(N[(y / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 83.1%

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

      1. frac-times 92.4%

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

      2. associate-*l/ 90.3%

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

      3. times-frac 97.0%

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

   3. Applied egg-rr 97.0%

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

   4. Taylor expanded in z around inf 95.8%

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

   if -1 < z < 1

   1. Initial program 82.8%

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

      1. frac-times 83.2%

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

      2. associate-*l/ 82.8%

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

      3. times-frac 95.3%

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

   3. Applied egg-rr 95.3%

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

      1. associate-*l/ 96.5%

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

      2. associate-/l/ 96.5%

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

      3. associate-/r* 96.5%

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

   5. Applied egg-rr 96.5%

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

   6. Taylor expanded in x around 0 93.2%

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

      1. associate-/r* 93.2%

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

      2. associate-*r/ 96.5%

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

      3. +-commutative 96.5%

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

      4. associate-/l* 96.5%

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

   8. Simplified 96.5%

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

   9. Taylor expanded in z around 0 94.0%

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

4. Final simplification 94.8%

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

Alternative 3: 96.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.9%

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

   1. frac-times 87.1%

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

   2. associate-*l/ 86.0%

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

   3. times-frac 96.0%

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

3. Applied egg-rr 96.0%

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

4. Final simplification 96.0%

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

Alternative 4: 75.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{+79}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.35e+79) (* (/ y z) (/ x z)) (* y (/ x (* z z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.35e+79) {
		tmp = (y / z) * (x / z);
	} else {
		tmp = y * (x / (z * 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 (y <= 1.35d+79) then
        tmp = (y / z) * (x / z)
    else
        tmp = y * (x / (z * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.35e+79) {
		tmp = (y / z) * (x / z);
	} else {
		tmp = y * (x / (z * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.35e+79:
		tmp = (y / z) * (x / z)
	else:
		tmp = y * (x / (z * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.35e+79)
		tmp = Float64(Float64(y / z) * Float64(x / z));
	else
		tmp = Float64(y * Float64(x / Float64(z * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.35e+79)
		tmp = (y / z) * (x / z);
	else
		tmp = y * (x / (z * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.35e+79], N[(N[(y / z), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.35e79

   1. Initial program 83.4%

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

      1. frac-times 87.1%

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

      2. associate-*l/ 86.6%

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

      3. times-frac 97.0%

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

   3. Applied egg-rr 97.0%

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

   4. Taylor expanded in z around 0 77.4%

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

   if 1.35e79 < y

   1. Initial program 80.4%

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

      1. sqr-neg 80.4%

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

      2. times-frac 87.1%

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

      3. sqr-neg 87.1%

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

   3. Simplified 87.1%

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

   4. Taylor expanded in z around 0 77.5%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{+79}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 5: 75.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-84}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2e-84) (* (/ y z) (/ x z)) (/ y (* z (/ z x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e-84) {
		tmp = (y / z) * (x / z);
	} else {
		tmp = y / (z * (z / x));
	}
	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 (y <= 2d-84) then
        tmp = (y / z) * (x / z)
    else
        tmp = y / (z * (z / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e-84) {
		tmp = (y / z) * (x / z);
	} else {
		tmp = y / (z * (z / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2e-84:
		tmp = (y / z) * (x / z)
	else:
		tmp = y / (z * (z / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2e-84)
		tmp = Float64(Float64(y / z) * Float64(x / z));
	else
		tmp = Float64(y / Float64(z * Float64(z / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2e-84)
		tmp = (y / z) * (x / z);
	else
		tmp = y / (z * (z / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2e-84], N[(N[(y / z), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(y / N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.0000000000000001e-84

   1. Initial program 82.2%

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

      1. frac-times 86.2%

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

      2. associate-*l/ 86.1%

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

      3. times-frac 96.4%

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

   3. Applied egg-rr 96.4%

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

   4. Taylor expanded in z around 0 76.3%

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

   if 2.0000000000000001e-84 < y

   1. Initial program 84.4%

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

      1. frac-times 89.2%

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

      2. associate-*l/ 85.7%

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

      3. times-frac 95.1%

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

   3. Applied egg-rr 95.1%

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

   4. Taylor expanded in z around 0 77.8%

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

      1. clear-num 77.7%

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

      2. frac-times 82.6%

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

      3. *-un-lft-identity 82.6%

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

   6. Applied egg-rr 82.6%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-84}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 6: 75.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.02 \cdot 10^{-84}:\\ \;\;\;\;\frac{x \cdot \frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.02e-84) (/ (* x (/ y z)) z) (/ y (* z (/ z x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.02e-84) {
		tmp = (x * (y / z)) / z;
	} else {
		tmp = y / (z * (z / x));
	}
	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 (y <= 1.02d-84) then
        tmp = (x * (y / z)) / z
    else
        tmp = y / (z * (z / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.02e-84) {
		tmp = (x * (y / z)) / z;
	} else {
		tmp = y / (z * (z / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.02e-84:
		tmp = (x * (y / z)) / z
	else:
		tmp = y / (z * (z / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.02e-84)
		tmp = Float64(Float64(x * Float64(y / z)) / z);
	else
		tmp = Float64(y / Float64(z * Float64(z / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.02e-84)
		tmp = (x * (y / z)) / z;
	else
		tmp = y / (z * (z / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.02e-84], N[(N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(y / N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.02000000000000004e-84

   1. Initial program 82.7%

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

      1. frac-times 86.4%

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

      2. associate-*l/ 86.6%

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

      3. times-frac 96.4%

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

   3. Applied egg-rr 96.4%

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

      1. associate-*l/ 97.3%

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

      2. associate-/l/ 95.8%

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

      3. associate-/r* 97.3%

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

   5. Applied egg-rr 97.3%

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

   6. Taylor expanded in z around 0 72.7%

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

      1. associate-*r/ 77.1%

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

   8. Simplified 77.1%

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

   if 1.02000000000000004e-84 < y

   1. Initial program 83.4%

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

      1. frac-times 88.6%

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

      2. associate-*l/ 84.6%

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

      3. times-frac 95.2%

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

   3. Applied egg-rr 95.2%

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

   4. Taylor expanded in z around 0 78.0%

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

      1. clear-num 78.0%

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

      2. frac-times 82.8%

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

      3. *-un-lft-identity 82.8%

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

   6. Applied egg-rr 82.8%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.02 \cdot 10^{-84}:\\ \;\;\;\;\frac{x \cdot \frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 7: 75.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2e-83) (/ (/ x z) (/ z y)) (/ y (* z (/ z x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e-83) {
		tmp = (x / z) / (z / y);
	} else {
		tmp = y / (z * (z / x));
	}
	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 (y <= 2d-83) then
        tmp = (x / z) / (z / y)
    else
        tmp = y / (z * (z / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e-83) {
		tmp = (x / z) / (z / y);
	} else {
		tmp = y / (z * (z / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2e-83:
		tmp = (x / z) / (z / y)
	else:
		tmp = y / (z * (z / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2e-83)
		tmp = Float64(Float64(x / z) / Float64(z / y));
	else
		tmp = Float64(y / Float64(z * Float64(z / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2e-83)
		tmp = (x / z) / (z / y);
	else
		tmp = y / (z * (z / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2e-83], N[(N[(x / z), $MachinePrecision] / N[(z / y), $MachinePrecision]), $MachinePrecision], N[(y / N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.0000000000000001e-83

   1. Initial program 82.3%

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

      1. associate-*r/ 82.8%

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

      2. sqr-neg 82.8%

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

      3. *-commutative 82.8%

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

      4. distribute-rgt1-in 70.8%

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

      5. sqr-neg 70.8%

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

      6. fma-def 82.8%

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

      7. sqr-neg 82.8%

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

      8. cube-unmult 82.8%

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

   3. Simplified 82.8%

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

      1. associate-*r/ 82.3%

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

      2. fma-udef 70.4%

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

      3. cube-mult 70.4%

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

      4. distribute-rgt1-in 82.3%

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

      5. *-commutative 82.3%

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

      6. frac-times 86.2%

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

      7. *-commutative 86.2%

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

      8. associate-/r* 92.1%

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

      9. clear-num 92.0%

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

      10. frac-times 96.1%

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

      11. *-un-lft-identity 96.1%

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

   5. Applied egg-rr 96.1%

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

   6. Taylor expanded in z around 0 75.9%

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

   if 2.0000000000000001e-83 < y

   1. Initial program 84.2%

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

      1. frac-times 89.1%

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

      2. associate-*l/ 85.5%

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

      3. times-frac 95.1%

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

   3. Applied egg-rr 95.1%

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

   4. Taylor expanded in z around 0 78.7%

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

      1. clear-num 78.7%

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

      2. frac-times 83.6%

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

      3. *-un-lft-identity 83.6%

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

   6. Applied egg-rr 83.6%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 8: 75.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2e-52) (/ (/ x (/ z y)) z) (/ y (* z (/ z x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e-52) {
		tmp = (x / (z / y)) / z;
	} else {
		tmp = y / (z * (z / x));
	}
	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 (y <= 2d-52) then
        tmp = (x / (z / y)) / z
    else
        tmp = y / (z * (z / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e-52) {
		tmp = (x / (z / y)) / z;
	} else {
		tmp = y / (z * (z / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2e-52:
		tmp = (x / (z / y)) / z
	else:
		tmp = y / (z * (z / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2e-52)
		tmp = Float64(Float64(x / Float64(z / y)) / z);
	else
		tmp = Float64(y / Float64(z * Float64(z / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2e-52)
		tmp = (x / (z / y)) / z;
	else
		tmp = y / (z * (z / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2e-52], N[(N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(y / N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2e-52

   1. Initial program 82.2%

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

      1. frac-times 86.5%

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

      2. associate-*l/ 85.9%

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

      3. times-frac 96.5%

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

   3. Applied egg-rr 96.5%

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

      1. associate-*l/ 97.4%

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

      2. associate-/l/ 95.9%

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

      3. associate-/r* 97.4%

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

   5. Applied egg-rr 97.4%

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

   6. Taylor expanded in x around 0 90.1%

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

      1. associate-/r* 90.7%

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

      2. associate-*r/ 97.6%

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

      3. +-commutative 97.6%

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

      4. associate-/l* 97.1%

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

   8. Simplified 97.1%

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

   9. Taylor expanded in z around 0 77.0%

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

   if 2e-52 < y

   1. Initial program 84.7%

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

      1. frac-times 88.5%

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

      2. associate-*l/ 86.0%

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

      3. times-frac 94.9%

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

   3. Applied egg-rr 94.9%

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

   4. Taylor expanded in z around 0 78.5%

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

      1. clear-num 78.4%

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

      2. frac-times 83.6%

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

      3. *-un-lft-identity 83.6%

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

   6. Applied egg-rr 83.6%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 9: 39.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \frac{-y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5e-310) (* x (/ (- y) z)) (/ y (/ z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5e-310) {
		tmp = x * (-y / z);
	} else {
		tmp = y / (z / x);
	}
	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 <= (-5d-310)) then
        tmp = x * (-y / z)
    else
        tmp = y / (z / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -5e-310) {
		tmp = x * (-y / z);
	} else {
		tmp = y / (z / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -5e-310:
		tmp = x * (-y / z)
	else:
		tmp = y / (z / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5e-310)
		tmp = Float64(x * Float64(Float64(-y) / z));
	else
		tmp = Float64(y / Float64(z / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -5e-310)
		tmp = x * (-y / z);
	else
		tmp = y / (z / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5e-310], N[(x * N[((-y) / z), $MachinePrecision]), $MachinePrecision], N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.999999999999985e-310

   1. Initial program 82.2%

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

      1. frac-times 86.6%

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

      2. associate-*l/ 85.3%

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

      3. times-frac 96.5%

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

   3. Applied egg-rr 96.5%

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

   4. Taylor expanded in z around 0 68.6%

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

      1. neg-mul-1 68.6%

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

      2. +-commutative 68.6%

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

      3. unsub-neg 68.6%

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

   6. Simplified 68.6%

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

   7. Taylor expanded in z around inf 28.2%

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

      1. associate-*r/ 35.8%

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

      2. neg-mul-1 35.8%

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

      3. distribute-rgt-neg-in 35.8%

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

      4. distribute-frac-neg 35.8%

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

   9. Simplified 35.8%

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

   if -4.999999999999985e-310 < z

   1. Initial program 83.6%

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

      1. frac-times 87.6%

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

      2. associate-*l/ 86.6%

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

      3. times-frac 95.6%

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

   3. Applied egg-rr 95.6%

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

   4. Taylor expanded in z around 0 70.4%

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

      1. neg-mul-1 70.4%

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

      2. +-commutative 70.4%

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

      3. unsub-neg 70.4%

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

   6. Simplified 70.4%

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

   7. Taylor expanded in z around inf 11.1%

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

      1. associate-*r/ 14.7%

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

      2. neg-mul-1 14.7%

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

      3. distribute-rgt-neg-in 14.7%

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

      4. distribute-frac-neg 14.7%

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

   9. Simplified 14.7%

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

       1. associate-*r/ 11.1%

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

       2. add-cube-cbrt 11.1%

          \[\leadsto \frac{x \cdot \left(-y\right)}{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}} \]

       3. times-frac 14.0%

          \[\leadsto \color{blue}{\frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{-y}{\sqrt[3]{z}}} \]

       4. add-sqr-sqrt 6.0%

          \[\leadsto \frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{\color{blue}{\sqrt{-y} \cdot \sqrt{-y}}}{\sqrt[3]{z}} \]

       5. sqrt-unprod 23.3%

          \[\leadsto \frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{\color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}}}{\sqrt[3]{z}} \]

       6. sqr-neg 23.3%

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

       7. sqrt-unprod 19.2%

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

       8. add-sqr-sqrt 36.1%

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

       9. times-frac 28.9%

          \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}} \]

       10. *-commutative 28.9%

           \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}} \]

       11. add-cube-cbrt 28.9%

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

       12. associate-/l* 38.9%

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

   11. Applied egg-rr 38.9%

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

4. Final simplification 37.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \frac{-y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternative 10: 39.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5e-310) (/ (- x) (/ z y)) (/ y (/ z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5e-310) {
		tmp = -x / (z / y);
	} else {
		tmp = y / (z / x);
	}
	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 <= (-5d-310)) then
        tmp = -x / (z / y)
    else
        tmp = y / (z / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -5e-310) {
		tmp = -x / (z / y);
	} else {
		tmp = y / (z / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -5e-310:
		tmp = -x / (z / y)
	else:
		tmp = y / (z / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5e-310)
		tmp = Float64(Float64(-x) / Float64(z / y));
	else
		tmp = Float64(y / Float64(z / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -5e-310)
		tmp = -x / (z / y);
	else
		tmp = y / (z / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5e-310], N[((-x) / N[(z / y), $MachinePrecision]), $MachinePrecision], N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.999999999999985e-310

   1. Initial program 82.2%

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

      1. frac-times 86.6%

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

      2. associate-*l/ 85.3%

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

      3. times-frac 96.5%

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

   3. Applied egg-rr 96.5%

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

   4. Taylor expanded in z around 0 68.6%

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

      1. neg-mul-1 68.6%

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

      2. +-commutative 68.6%

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

      3. unsub-neg 68.6%

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

   6. Simplified 68.6%

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

   7. Taylor expanded in z around inf 28.2%

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

      1. associate-*r/ 35.8%

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

      2. neg-mul-1 35.8%

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

      3. distribute-rgt-neg-in 35.8%

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

      4. distribute-frac-neg 35.8%

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

   9. Simplified 35.8%

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

       1. associate-*r/ 28.2%

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

       2. distribute-rgt-neg-in 28.2%

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

       3. distribute-neg-frac 28.2%

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

       4. associate-/l* 35.8%

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

       5. distribute-neg-frac 35.8%

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

   11. Applied egg-rr 35.8%

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

   if -4.999999999999985e-310 < z

   1. Initial program 83.6%

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

      1. frac-times 87.6%

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

      2. associate-*l/ 86.6%

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

      3. times-frac 95.6%

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

   3. Applied egg-rr 95.6%

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

   4. Taylor expanded in z around 0 70.4%

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

      1. neg-mul-1 70.4%

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

      2. +-commutative 70.4%

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

      3. unsub-neg 70.4%

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

   6. Simplified 70.4%

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

   7. Taylor expanded in z around inf 11.1%

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

      1. associate-*r/ 14.7%

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

      2. neg-mul-1 14.7%

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

      3. distribute-rgt-neg-in 14.7%

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

      4. distribute-frac-neg 14.7%

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

   9. Simplified 14.7%

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

       1. associate-*r/ 11.1%

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

       2. add-cube-cbrt 11.1%

          \[\leadsto \frac{x \cdot \left(-y\right)}{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}} \]

       3. times-frac 14.0%

          \[\leadsto \color{blue}{\frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{-y}{\sqrt[3]{z}}} \]

       4. add-sqr-sqrt 6.0%

          \[\leadsto \frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{\color{blue}{\sqrt{-y} \cdot \sqrt{-y}}}{\sqrt[3]{z}} \]

       5. sqrt-unprod 23.3%

          \[\leadsto \frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{\color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}}}{\sqrt[3]{z}} \]

       6. sqr-neg 23.3%

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

       7. sqrt-unprod 19.2%

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

       8. add-sqr-sqrt 36.1%

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

       9. times-frac 28.9%

          \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}} \]

       10. *-commutative 28.9%

           \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}} \]

       11. add-cube-cbrt 28.9%

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

       12. associate-/l* 38.9%

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

   11. Applied egg-rr 38.9%

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

4. Final simplification 37.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternative 11: 32.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{+30}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1e+30) (/ x (/ z y)) (/ y (/ z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1e+30) {
		tmp = x / (z / y);
	} else {
		tmp = y / (z / x);
	}
	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 (y <= 1d+30) then
        tmp = x / (z / y)
    else
        tmp = y / (z / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1e+30) {
		tmp = x / (z / y);
	} else {
		tmp = y / (z / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1e+30:
		tmp = x / (z / y)
	else:
		tmp = y / (z / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1e+30)
		tmp = Float64(x / Float64(z / y));
	else
		tmp = Float64(y / Float64(z / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1e+30)
		tmp = x / (z / y);
	else
		tmp = y / (z / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1e+30], N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision], N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1e30

   1. Initial program 83.4%

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

      1. frac-times 87.3%

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

      2. associate-*l/ 86.8%

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

      3. times-frac 96.8%

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

   3. Applied egg-rr 96.8%

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

   4. Taylor expanded in z around 0 68.9%

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

      1. neg-mul-1 68.9%

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

      2. +-commutative 68.9%

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

      3. unsub-neg 68.9%

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

   6. Simplified 68.9%

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

   7. Taylor expanded in z around inf 19.2%

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

      1. associate-*r/ 24.8%

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

      2. neg-mul-1 24.8%

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

      3. distribute-rgt-neg-in 24.8%

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

      4. distribute-frac-neg 24.8%

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

   9. Simplified 24.8%

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

       1. add-sqr-sqrt 13.0%

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

       2. sqrt-unprod 26.7%

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

       3. sqr-neg 26.7%

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

       4. sqrt-unprod 13.1%

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

       5. add-sqr-sqrt 28.7%

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

       6. clear-num 28.7%

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

       7. div-inv 28.2%

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

   11. Applied egg-rr 28.2%

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

   if 1e30 < y

   1. Initial program 81.2%

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

      1. frac-times 86.4%

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

      2. associate-*l/ 83.0%

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

      3. times-frac 93.0%

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

   3. Applied egg-rr 93.0%

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

   4. Taylor expanded in z around 0 71.7%

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

      1. neg-mul-1 71.7%

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

      2. +-commutative 71.7%

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

      3. unsub-neg 71.7%

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

   6. Simplified 71.7%

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

   7. Taylor expanded in z around inf 20.9%

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

      1. associate-*r/ 26.0%

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

      2. neg-mul-1 26.0%

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

      3. distribute-rgt-neg-in 26.0%

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

      4. distribute-frac-neg 26.0%

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

   9. Simplified 26.0%

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

       1. associate-*r/ 20.9%

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

       2. add-cube-cbrt 20.9%

          \[\leadsto \frac{x \cdot \left(-y\right)}{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}} \]

       3. times-frac 25.9%

          \[\leadsto \color{blue}{\frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{-y}{\sqrt[3]{z}}} \]

       4. add-sqr-sqrt 0.0%

          \[\leadsto \frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{\color{blue}{\sqrt{-y} \cdot \sqrt{-y}}}{\sqrt[3]{z}} \]

       5. sqrt-unprod 19.2%

          \[\leadsto \frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{\color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}}}{\sqrt[3]{z}} \]

       6. sqr-neg 19.2%

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

       7. sqrt-unprod 24.8%

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

       8. add-sqr-sqrt 24.8%

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

       9. times-frac 18.3%

          \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}} \]

       10. *-commutative 18.3%

           \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}} \]

       11. add-cube-cbrt 18.3%

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

       12. associate-/l* 29.8%

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

   11. Applied egg-rr 29.8%

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

4. Final simplification 28.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{+30}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternative 12: 73.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \frac{y}{z} \cdot \frac{x}{z} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. frac-times 87.1%

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

   2. associate-*l/ 86.0%

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

   3. times-frac 96.0%

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

3. Applied egg-rr 96.0%

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

4. Taylor expanded in z around 0 76.8%

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

5. Final simplification 76.8%

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

Alternative 13: 30.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \frac{x}{\frac{z}{y}} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ x (/ z y)))

C

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

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 / (z / y)
end function

Java

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

Python

def code(x, y, z):
	return x / (z / y)

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. frac-times 87.1%

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

   2. associate-*l/ 86.0%

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

   3. times-frac 96.0%

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

3. Applied egg-rr 96.0%

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

4. Taylor expanded in z around 0 69.5%

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

   1. neg-mul-1 69.5%

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

   2. +-commutative 69.5%

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

   3. unsub-neg 69.5%

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

6. Simplified 69.5%

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

7. Taylor expanded in z around inf 19.5%

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

   1. associate-*r/ 25.1%

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

   2. neg-mul-1 25.1%

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

   3. distribute-rgt-neg-in 25.1%

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

   4. distribute-frac-neg 25.1%

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

9. Simplified 25.1%

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

    1. add-sqr-sqrt 10.2%

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

    2. sqrt-unprod 25.1%

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

    3. sqr-neg 25.1%

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

    4. sqrt-unprod 14.6%

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

    5. add-sqr-sqrt 26.8%

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

    6. clear-num 26.8%

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

    7. div-inv 26.5%

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

11. Applied egg-rr 26.5%

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

12. Final simplification 26.5%

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

Developer target: 95.1% 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 2023308 
(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))))