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

Percentage Accurate: 82.7% → 96.3%

Time: 11.7s

Alternatives: 18

Speedup: 1.0×

• 25.8% 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, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{-117} \lor \neg \left(z \leq 2 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{\frac{x}{z}}{z} \cdot \frac{y}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\frac{y}{z} - y\right)}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= z -8e-117) (not (<= z 2e-11)))
   (* (/ (/ x z) z) (/ y (+ z 1.0)))
   (/ (* x (- (/ y z) y)) z)))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z <= -8e-117) || !(z <= 2e-11)) {
		tmp = ((x / z) / z) * (y / (z + 1.0));
	} else {
		tmp = (x * ((y / z) - y)) / z;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-8d-117)) .or. (.not. (z <= 2d-11))) then
        tmp = ((x / z) / z) * (y / (z + 1.0d0))
    else
        tmp = (x * ((y / z) - y)) / z
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -8e-117) || !(z <= 2e-11)) {
		tmp = ((x / z) / z) * (y / (z + 1.0));
	} else {
		tmp = (x * ((y / z) - y)) / z;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z <= -8e-117) or not (z <= 2e-11):
		tmp = ((x / z) / z) * (y / (z + 1.0))
	else:
		tmp = (x * ((y / z) - y)) / z
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if ((z <= -8e-117) || !(z <= 2e-11))
		tmp = Float64(Float64(Float64(x / z) / z) * Float64(y / Float64(z + 1.0)));
	else
		tmp = Float64(Float64(x * Float64(Float64(y / z) - y)) / z);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -8e-117) || ~((z <= 2e-11)))
		tmp = ((x / z) / z) * (y / (z + 1.0));
	else
		tmp = (x * ((y / z) - y)) / z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[Or[LessEqual[z, -8e-117], N[Not[LessEqual[z, 2e-11]], $MachinePrecision]], N[(N[(N[(x / z), $MachinePrecision] / z), $MachinePrecision] * N[(y / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(N[(y / z), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.00000000000000024e-117 or 1.99999999999999988e-11 < z

   1. Initial program 86.7%

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

      1. associate-*l* 86.7%

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

      2. times-frac 94.7%

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

      3. distribute-lft-in 94.7%

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

      4. fma-def 94.7%

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

      5. *-rgt-identity 94.7%

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

   3. Simplified 94.7%

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

      1. frac-times 86.7%

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

      2. fma-udef 86.7%

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

      3. distribute-lft-in 68.4%

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

      4. *-un-lft-identity 68.4%

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

      5. distribute-rgt-in 86.7%

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

      6. associate-/r* 88.5%

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

      7. times-frac 97.0%

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

   5. Applied egg-rr 97.0%

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

      1. frac-2neg 97.0%

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

      2. associate-*r/ 97.5%

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

      3. distribute-frac-neg 97.5%

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

      4. distribute-rgt-neg-out 97.5%

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

      5. associate-/r* 95.6%

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

      6. times-frac 95.8%

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

      7. frac-2neg 95.8%

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

   7. Applied egg-rr 95.8%

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

   if -8.00000000000000024e-117 < z < 1.99999999999999988e-11

   1. Initial program 81.3%

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

      1. associate-*l* 81.3%

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

      2. times-frac 94.7%

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

      3. distribute-lft-in 94.7%

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

      4. fma-def 94.7%

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

      5. *-rgt-identity 94.7%

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

   3. Simplified 94.7%

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

   4. Taylor expanded in z around 0 94.7%

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

      1. neg-mul-1 94.7%

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

      2. +-commutative 94.7%

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

      3. unsub-neg 94.7%

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

   6. Simplified 94.7%

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

   7. Taylor expanded in x around 0 95.1%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{-117} \lor \neg \left(z \leq 2 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{\frac{x}{z}}{z} \cdot \frac{y}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\frac{y}{z} - y\right)}{z}\\ \end{array} \]

Alternative 2: 94.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{-119} \lor \neg \left(z \leq 9.5 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{y}{z + 1} \cdot \frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\frac{y}{z} - y\right)}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.65e-119) (not (<= z 9.5e-12)))
   (* (/ y (+ z 1.0)) (/ x (* z z)))
   (/ (* x (- (/ y z) y)) z)))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.65e-119) || !(z <= 9.5e-12)) {
		tmp = (y / (z + 1.0)) * (x / (z * z));
	} else {
		tmp = (x * ((y / z) - y)) / z;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.65d-119)) .or. (.not. (z <= 9.5d-12))) then
        tmp = (y / (z + 1.0d0)) * (x / (z * z))
    else
        tmp = (x * ((y / z) - y)) / z
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.65e-119) || !(z <= 9.5e-12)) {
		tmp = (y / (z + 1.0)) * (x / (z * z));
	} else {
		tmp = (x * ((y / z) - y)) / z;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z <= -1.65e-119) or not (z <= 9.5e-12):
		tmp = (y / (z + 1.0)) * (x / (z * z))
	else:
		tmp = (x * ((y / z) - y)) / z
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.65e-119) || !(z <= 9.5e-12))
		tmp = Float64(Float64(y / Float64(z + 1.0)) * Float64(x / Float64(z * z)));
	else
		tmp = Float64(Float64(x * Float64(Float64(y / z) - y)) / z);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.65e-119) || ~((z <= 9.5e-12)))
		tmp = (y / (z + 1.0)) * (x / (z * z));
	else
		tmp = (x * ((y / z) - y)) / z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[Or[LessEqual[z, -1.65e-119], N[Not[LessEqual[z, 9.5e-12]], $MachinePrecision]], N[(N[(y / N[(z + 1.0), $MachinePrecision]), $MachinePrecision] * N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(N[(y / z), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.65000000000000004e-119 or 9.4999999999999995e-12 < z

   1. Initial program 86.8%

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

      1. times-frac 94.1%

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

   3. Simplified 94.1%

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

   if -1.65000000000000004e-119 < z < 9.4999999999999995e-12

   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. associate-*l* 81.2%

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

      2. times-frac 94.6%

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

      3. distribute-lft-in 94.6%

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

      4. fma-def 94.6%

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

      5. *-rgt-identity 94.6%

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

   3. Simplified 94.6%

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

   4. Taylor expanded in z around 0 94.6%

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

      1. neg-mul-1 94.6%

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

      2. +-commutative 94.6%

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

      3. unsub-neg 94.6%

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

   6. Simplified 94.6%

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

   7. Taylor expanded in x around 0 95.1%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{-119} \lor \neg \left(z \leq 9.5 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{y}{z + 1} \cdot \frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\frac{y}{z} - y\right)}{z}\\ \end{array} \]

Alternative 3: 93.0% accurate, 0.8× speedup

Math

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

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.0) (not (<= z 0.75)))
   (* (/ x z) (/ y (* z z)))
   (* (/ x z) (- (/ y z) y))))

C

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

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z <= -1.0) or not (z <= 0.75):
		tmp = (x / z) * (y / (z * z))
	else:
		tmp = (x / z) * ((y / z) - y)
	return tmp

Julia

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

Derivation

1. Split input into 2 regimes

2. if z < -1 or 0.75 < z

   1. Initial program 87.2%

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

      1. associate-*l* 87.2%

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

      2. times-frac 94.0%

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

      3. distribute-lft-in 94.0%

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

      4. fma-def 94.0%

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

      5. *-rgt-identity 94.0%

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

   3. Simplified 94.0%

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

   4. Taylor expanded in z around inf 92.0%

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

      1. unpow2 92.0%

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

   6. Simplified 92.0%

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

   if -1 < z < 0.75

   1. Initial program 81.6%

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

      1. associate-*l* 81.6%

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

      2. times-frac 95.3%

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

      3. distribute-lft-in 95.3%

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

      4. fma-def 95.3%

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

      5. *-rgt-identity 95.3%

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

   3. Simplified 95.3%

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

   4. Taylor expanded in z around 0 94.5%

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

      1. neg-mul-1 94.5%

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

      2. +-commutative 94.5%

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

      3. unsub-neg 94.5%

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

   6. Simplified 94.5%

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

4. Final simplification 93.2%

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

Alternative 4: 93.1% accurate, 0.8× speedup

Math

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

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z -1.0)
   (* (/ y z) (/ x (* z z)))
   (if (<= z 0.75) (* (/ x z) (- (/ y z) y)) (* (/ x z) (/ y (* z z))))))

C

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

Python

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

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(Float64(y / z) * Float64(x / Float64(z * z)));
	elseif (z <= 0.75)
		tmp = Float64(Float64(x / z) * Float64(Float64(y / z) - y));
	else
		tmp = Float64(Float64(x / z) * Float64(y / Float64(z * z)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.0)
		tmp = (y / z) * (x / (z * z));
	elseif (z <= 0.75)
		tmp = (x / z) * ((y / z) - y);
	else
		tmp = (x / z) * (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[z, -1.0], N[(N[(y / z), $MachinePrecision] * N[(x / N[(z * 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[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 85.6%

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

      1. associate-*l* 85.6%

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

      2. times-frac 97.3%

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

      3. distribute-lft-in 97.3%

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

      4. fma-def 97.3%

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

      5. *-rgt-identity 97.3%

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

   3. Simplified 97.3%

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

      1. *-commutative 97.3%

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

      2. frac-times 85.6%

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

      3. fma-udef 85.6%

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

      4. distribute-lft1-in 85.6%

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

      5. associate-*r* 85.6%

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

      6. frac-times 95.7%

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

      7. clear-num 95.7%

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

      8. frac-times 97.4%

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

      9. *-un-lft-identity 97.4%

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

   5. Applied egg-rr 97.4%

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

   6. Taylor expanded in z around inf 96.4%

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

      1. *-un-lft-identity 96.4%

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

      2. times-frac 94.6%

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

      3. clear-num 94.7%

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

   8. Applied egg-rr 94.7%

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

   if -1 < z < 0.75

   1. Initial program 81.6%

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

      1. associate-*l* 81.6%

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

      2. times-frac 95.3%

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

      3. distribute-lft-in 95.3%

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

      4. fma-def 95.3%

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

      5. *-rgt-identity 95.3%

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

   3. Simplified 95.3%

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

   4. Taylor expanded in z around 0 94.5%

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

      1. neg-mul-1 94.5%

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

      2. +-commutative 94.5%

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

      3. unsub-neg 94.5%

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

   6. Simplified 94.5%

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

   if 0.75 < z

   1. Initial program 88.5%

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

      1. associate-*l* 88.5%

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

      2. times-frac 91.0%

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

      3. distribute-lft-in 91.0%

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

      4. fma-def 91.0%

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

      5. *-rgt-identity 91.0%

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

   3. Simplified 91.0%

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

   4. Taylor expanded in z around inf 88.1%

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

      1. unpow2 88.1%

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

   6. Simplified 88.1%

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

4. Final simplification 92.8%

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

Alternative 5: 93.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z \cdot z}\\ \mathbf{elif}\;z \leq 0.78:\\ \;\;\;\;\frac{x \cdot \left(\frac{y}{z} - y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z -1.0)
   (* (/ y z) (/ x (* z z)))
   (if (<= z 0.78) (/ (* x (- (/ y z) y)) z) (* (/ x z) (/ y (* z z))))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = (y / z) * (x / (z * z));
	} else if (z <= 0.78) {
		tmp = (x * ((y / z) - y)) / z;
	} else {
		tmp = (x / z) * (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 (z <= (-1.0d0)) then
        tmp = (y / z) * (x / (z * z))
    else if (z <= 0.78d0) then
        tmp = (x * ((y / z) - y)) / z
    else
        tmp = (x / z) * (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 (z <= -1.0) {
		tmp = (y / z) * (x / (z * z));
	} else if (z <= 0.78) {
		tmp = (x * ((y / z) - y)) / z;
	} else {
		tmp = (x / z) * (y / (z * z));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if z <= -1.0:
		tmp = (y / z) * (x / (z * z))
	elif z <= 0.78:
		tmp = (x * ((y / z) - y)) / z
	else:
		tmp = (x / z) * (y / (z * z))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(Float64(y / z) * Float64(x / Float64(z * z)));
	elseif (z <= 0.78)
		tmp = Float64(Float64(x * Float64(Float64(y / z) - y)) / z);
	else
		tmp = Float64(Float64(x / z) * Float64(y / Float64(z * z)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.0)
		tmp = (y / z) * (x / (z * z));
	elseif (z <= 0.78)
		tmp = (x * ((y / z) - y)) / z;
	else
		tmp = (x / z) * (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[z, -1.0], N[(N[(y / z), $MachinePrecision] * N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.78], N[(N[(x * N[(N[(y / z), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 85.6%

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

      1. associate-*l* 85.6%

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

      2. times-frac 97.3%

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

      3. distribute-lft-in 97.3%

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

      4. fma-def 97.3%

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

      5. *-rgt-identity 97.3%

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

   3. Simplified 97.3%

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

      1. *-commutative 97.3%

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

      2. frac-times 85.6%

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

      3. fma-udef 85.6%

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

      4. distribute-lft1-in 85.6%

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

      5. associate-*r* 85.6%

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

      6. frac-times 95.7%

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

      7. clear-num 95.7%

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

      8. frac-times 97.4%

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

      9. *-un-lft-identity 97.4%

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

   5. Applied egg-rr 97.4%

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

   6. Taylor expanded in z around inf 96.4%

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

      1. *-un-lft-identity 96.4%

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

      2. times-frac 94.6%

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

      3. clear-num 94.7%

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

   8. Applied egg-rr 94.7%

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

   if -1 < z < 0.78000000000000003

   1. Initial program 81.6%

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

      1. associate-*l* 81.6%

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

      2. times-frac 95.3%

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

      3. distribute-lft-in 95.3%

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

      4. fma-def 95.3%

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

      5. *-rgt-identity 95.3%

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

   3. Simplified 95.3%

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

   4. Taylor expanded in z around 0 94.5%

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

      1. neg-mul-1 94.5%

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

      2. +-commutative 94.5%

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

      3. unsub-neg 94.5%

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

   6. Simplified 94.5%

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

   7. Taylor expanded in x around 0 94.1%

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

   if 0.78000000000000003 < z

   1. Initial program 88.5%

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

      1. associate-*l* 88.5%

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

      2. times-frac 91.0%

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

      3. distribute-lft-in 91.0%

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

      4. fma-def 91.0%

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

      5. *-rgt-identity 91.0%

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

   3. Simplified 91.0%

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

   4. Taylor expanded in z around inf 88.1%

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

      1. unpow2 88.1%

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

   6. Simplified 88.1%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z \cdot z}\\ \mathbf{elif}\;z \leq 0.78:\\ \;\;\;\;\frac{x \cdot \left(\frac{y}{z} - y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z \cdot z}\\ \end{array} \]

Alternative 6: 96.4% accurate, 1.0× speedup

Math

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

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(Float64(x / z) * 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[(N[(x / z), $MachinePrecision] * y), $MachinePrecision] / z), $MachinePrecision] / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.4%

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

   1. associate-*l* 84.4%

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

   2. times-frac 94.7%

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

   3. distribute-lft-in 94.7%

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

   4. fma-def 94.7%

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

   5. *-rgt-identity 94.7%

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

3. Simplified 94.7%

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

   1. frac-times 84.4%

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

   2. fma-udef 84.4%

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

   3. distribute-lft-in 73.9%

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

   4. *-un-lft-identity 73.9%

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

   5. distribute-rgt-in 84.4%

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

   6. associate-/r* 85.5%

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

   7. times-frac 96.0%

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

5. Applied egg-rr 96.0%

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

   1. associate-*r/ 97.2%

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

7. Applied egg-rr 97.2%

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

8. Final simplification 97.2%

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

Alternative 7: 94.4% accurate, 1.0× speedup

Math

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

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 (* z z))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.4%

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

   1. associate-*l* 84.4%

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

   2. times-frac 94.7%

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

   3. distribute-lft-in 94.7%

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

   4. fma-def 94.7%

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

   5. *-rgt-identity 94.7%

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

3. Simplified 94.7%

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

   1. frac-times 84.4%

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

   2. fma-udef 84.4%

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

   3. distribute-lft-in 73.9%

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

   4. *-un-lft-identity 73.9%

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

   5. distribute-rgt-in 84.4%

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

   6. associate-/r* 85.5%

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

   7. times-frac 96.0%

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

5. Applied egg-rr 96.0%

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

   1. frac-2neg 96.0%

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

   2. associate-*r/ 97.2%

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

   3. distribute-frac-neg 97.2%

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

   4. distribute-rgt-neg-out 97.2%

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

   5. associate-/r* 96.1%

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

   6. times-frac 93.5%

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

   7. frac-2neg 93.5%

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

7. Applied egg-rr 93.5%

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

   1. frac-times 96.1%

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

   2. associate-*l/ 88.3%

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

   3. associate-*r/ 94.8%

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

   4. +-commutative 94.8%

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

   5. distribute-lft-in 94.8%

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

   6. *-rgt-identity 94.8%

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

9. Applied egg-rr 94.8%

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

10. Final simplification 94.8%

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

Alternative 8: 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 84.4%

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

   1. associate-*l* 84.4%

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

   2. times-frac 94.7%

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

   3. distribute-lft-in 94.7%

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

   4. fma-def 94.7%

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

   5. *-rgt-identity 94.7%

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

3. Simplified 94.7%

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

   1. frac-times 84.4%

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

   2. fma-udef 84.4%

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

   3. distribute-lft-in 73.9%

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

   4. *-un-lft-identity 73.9%

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

   5. distribute-rgt-in 84.4%

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

   6. associate-/r* 85.5%

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

   7. times-frac 96.0%

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

5. Applied egg-rr 96.0%

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

6. Final simplification 96.0%

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

Alternative 9: 77.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{+76}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= x -3.1e+76) (* x (/ y (* z z))) (* (/ x z) (/ y z))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.1e+76) {
		tmp = x * (y / (z * z));
	} else {
		tmp = (x / z) * (y / z);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-3.1d+76)) then
        tmp = x * (y / (z * z))
    else
        tmp = (x / z) * (y / z)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.1e+76) {
		tmp = x * (y / (z * z));
	} else {
		tmp = (x / z) * (y / z);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if x <= -3.1e+76:
		tmp = x * (y / (z * z))
	else:
		tmp = (x / z) * (y / z)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (x <= -3.1e+76)
		tmp = Float64(x * Float64(y / Float64(z * z)));
	else
		tmp = Float64(Float64(x / z) * Float64(y / z));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -3.1e+76)
		tmp = x * (y / (z * z));
	else
		tmp = (x / z) * (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[x, -3.1e+76], N[(x * N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.10000000000000011e76

   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. associate-*l* 83.6%

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

      2. times-frac 93.1%

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

      3. distribute-lft-in 93.1%

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

      4. fma-def 93.1%

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

      5. *-rgt-identity 93.1%

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

   3. Simplified 93.1%

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

   4. Taylor expanded in z around 0 71.3%

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

      1. *-commutative 71.3%

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

      2. unpow2 71.3%

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

      3. associate-*r/ 78.8%

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

   6. Simplified 78.8%

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

   if -3.10000000000000011e76 < x

   1. Initial program 84.6%

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

      1. associate-*l* 84.6%

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

      2. times-frac 95.0%

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

      3. distribute-lft-in 95.0%

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

      4. fma-def 95.0%

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

      5. *-rgt-identity 95.0%

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

   3. Simplified 95.0%

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

   4. Taylor expanded in z around 0 77.5%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{+76}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \end{array} \]

Alternative 10: 77.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{-16}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 1.6e-16) (* (/ x z) (/ y z)) (* y (/ x (* z z)))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.6e-16) {
		tmp = (x / z) * (y / z);
	} else {
		tmp = y * (x / (z * z));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1.6d-16) then
        tmp = (x / z) * (y / z)
    else
        tmp = y * (x / (z * z))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.6e-16) {
		tmp = (x / z) * (y / z);
	} else {
		tmp = y * (x / (z * z));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if y <= 1.6e-16:
		tmp = (x / z) * (y / z)
	else:
		tmp = y * (x / (z * z))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (y <= 1.6e-16)
		tmp = Float64(Float64(x / z) * Float64(y / z));
	else
		tmp = Float64(y * Float64(x / Float64(z * z)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.6e-16)
		tmp = (x / z) * (y / z);
	else
		tmp = y * (x / (z * z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 1.6e-16], N[(N[(x / z), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.60000000000000011e-16

   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. associate-*l* 84.0%

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

      2. times-frac 95.1%

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

      3. distribute-lft-in 95.2%

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

      4. fma-def 95.2%

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

      5. *-rgt-identity 95.2%

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

   3. Simplified 95.2%

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

   4. Taylor expanded in z around 0 77.5%

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

   if 1.60000000000000011e-16 < y

   1. Initial program 85.9%

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

      1. times-frac 93.3%

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

   3. Simplified 93.3%

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

   4. Taylor expanded in z around 0 80.6%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{-16}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 11: 78.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+76}:\\ \;\;\;\;\frac{x}{z \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 2e+76) (/ x (* z (/ z y))) (* y (/ x (* z z)))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e+76) {
		tmp = x / (z * (z / y));
	} else {
		tmp = y * (x / (z * z));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 2d+76) then
        tmp = x / (z * (z / y))
    else
        tmp = y * (x / (z * z))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e+76) {
		tmp = x / (z * (z / y));
	} else {
		tmp = y * (x / (z * z));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if y <= 2e+76:
		tmp = x / (z * (z / y))
	else:
		tmp = y * (x / (z * z))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (y <= 2e+76)
		tmp = Float64(x / Float64(z * Float64(z / y)));
	else
		tmp = Float64(y * Float64(x / Float64(z * z)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2e+76)
		tmp = x / (z * (z / y));
	else
		tmp = y * (x / (z * z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 2e+76], N[(x / N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.0000000000000001e76

   1. Initial program 85.1%

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

      1. /-rgt-identity 85.1%

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

      2. associate-/l* 85.0%

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

      3. associate-/l/ 85.6%

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

      4. associate-*l* 86.5%

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

      5. associate-*r/ 86.5%

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

      6. *-rgt-identity 86.5%

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

      7. associate-*l* 91.2%

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

      8. associate-*r/ 91.2%

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

      9. distribute-lft-in 91.2%

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

      10. fma-def 91.2%

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

      11. *-rgt-identity 91.2%

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

   3. Simplified 91.2%

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

   4. Taylor expanded in z around 0 75.1%

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

   if 2.0000000000000001e76 < y

   1. Initial program 81.1%

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

      1. times-frac 90.7%

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

   3. Simplified 90.7%

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

   4. Taylor expanded in z around 0 80.1%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+76}:\\ \;\;\;\;\frac{x}{z \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 12: 79.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-79}:\\ \;\;\;\;\frac{x}{z \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= x -2.3e-79) (/ x (* z (/ z y))) (/ y (* z (/ z x)))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.3e-79) {
		tmp = x / (z * (z / y));
	} else {
		tmp = y / (z * (z / x));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-2.3d-79)) then
        tmp = x / (z * (z / y))
    else
        tmp = y / (z * (z / x))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.3e-79) {
		tmp = x / (z * (z / y));
	} else {
		tmp = y / (z * (z / x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if x <= -2.3e-79:
		tmp = x / (z * (z / y))
	else:
		tmp = y / (z * (z / x))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (x <= -2.3e-79)
		tmp = Float64(x / Float64(z * Float64(z / y)));
	else
		tmp = Float64(y / Float64(z * Float64(z / x)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2.3e-79)
		tmp = x / (z * (z / y));
	else
		tmp = y / (z * (z / x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[x, -2.3e-79], N[(x / N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.30000000000000012e-79

   1. Initial program 82.1%

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

      1. /-rgt-identity 82.1%

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

      2. associate-/l* 82.1%

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

      3. associate-/l/ 86.7%

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

      4. associate-*l* 88.0%

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

      5. associate-*r/ 88.0%

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

      6. *-rgt-identity 88.0%

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

      7. associate-*l* 94.2%

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

      8. associate-*r/ 92.9%

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

      9. distribute-lft-in 92.9%

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

      10. fma-def 92.9%

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

      11. *-rgt-identity 92.9%

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

   3. Simplified 92.9%

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

   4. Taylor expanded in z around 0 76.9%

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

   if -2.30000000000000012e-79 < x

   1. Initial program 85.4%

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

      1. associate-*l* 85.4%

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

      2. times-frac 94.9%

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

      3. distribute-lft-in 94.9%

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

      4. fma-def 94.9%

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

      5. *-rgt-identity 94.9%

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

   3. Simplified 94.9%

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

   4. Taylor expanded in z around 0 69.9%

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

      1. *-commutative 69.9%

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

      2. unpow2 69.9%

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

      3. associate-*r/ 68.9%

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

   6. Simplified 68.9%

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

      1. frac-2neg 68.9%

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

      2. div-inv 68.6%

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

      3. add-sqr-sqrt 37.1%

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

      4. sqrt-unprod 44.3%

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

      5. sqr-neg 44.3%

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

      6. sqrt-unprod 16.2%

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

      7. add-sqr-sqrt 35.1%

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

      8. distribute-rgt-neg-in 35.1%

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

   8. Applied egg-rr 35.1%

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

      1. associate-*r/ 35.1%

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

      2. *-rgt-identity 35.1%

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

      3. associate-/r* 32.1%

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

   10. Simplified 32.1%

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

   11. Taylor expanded in x around 0 33.4%

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

       1. mul-1-neg 33.4%

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

       2. unpow2 33.4%

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

       3. times-frac 32.5%

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

       4. distribute-rgt-neg-in 32.5%

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

       5. distribute-frac-neg 32.5%

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

   13. Simplified 32.5%

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

       1. frac-times 33.4%

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

       2. *-commutative 33.4%

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

       3. add-sqr-sqrt 14.1%

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

       4. sqrt-unprod 54.2%

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

       5. sqr-neg 54.2%

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

       6. sqrt-unprod 48.1%

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

       7. add-sqr-sqrt 69.9%

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

       8. frac-times 76.5%

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

       9. clear-num 76.9%

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

       10. frac-times 78.2%

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

       11. *-un-lft-identity 78.2%

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

   15. Applied egg-rr 78.2%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-79}:\\ \;\;\;\;\frac{x}{z \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 13: 40.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (z <= -1e-310)
		tmp = Float64(Float64(x / z) * Float64(-y));
	else
		tmp = Float64(x / 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 <= -1e-310)
		tmp = (x / z) * -y;
	else
		tmp = x / (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, -1e-310], N[(N[(x / z), $MachinePrecision] * (-y)), $MachinePrecision], N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -9.999999999999969e-311

   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. associate-*l* 85.3%

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

      2. times-frac 97.8%

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

      3. distribute-lft-in 97.8%

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

      4. fma-def 97.8%

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

      5. *-rgt-identity 97.8%

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

   3. Simplified 97.8%

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

   4. Taylor expanded in z around 0 67.6%

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

      1. neg-mul-1 67.6%

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

      2. +-commutative 67.6%

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

      3. unsub-neg 67.6%

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

   6. Simplified 67.6%

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

   7. Taylor expanded in z around inf 44.4%

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

      1. neg-mul-1 44.4%

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

   9. Simplified 44.4%

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

   if -9.999999999999969e-311 < 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. associate-*l* 83.6%

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

      2. times-frac 91.8%

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

      3. distribute-lft-in 91.8%

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

      4. fma-def 91.8%

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

      5. *-rgt-identity 91.8%

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

   3. Simplified 91.8%

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

   4. Taylor expanded in z around 0 63.7%

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

      1. neg-mul-1 63.7%

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

      2. +-commutative 63.7%

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

      3. unsub-neg 63.7%

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

   6. Simplified 63.7%

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

   7. Taylor expanded in z around inf 19.7%

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

      1. neg-mul-1 19.7%

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

   9. Simplified 19.7%

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

       1. associate-*l/ 18.3%

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

       2. div-inv 18.3%

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

       3. add-sqr-sqrt 10.6%

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

       4. sqrt-unprod 24.2%

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

       5. sqr-neg 24.2%

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

       6. sqrt-unprod 13.1%

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

       7. add-sqr-sqrt 33.2%

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

       8. associate-*r* 37.5%

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

       9. div-inv 37.5%

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

       10. clear-num 38.9%

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

       11. un-div-inv 38.9%

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

   11. Applied egg-rr 38.9%

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

4. Final simplification 41.5%

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

Alternative 14: 40.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (z <= -1e-310)
		tmp = Float64(Float64(y / z) * Float64(-x));
	else
		tmp = Float64(x / 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 <= -1e-310)
		tmp = (y / z) * -x;
	else
		tmp = x / (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, -1e-310], N[(N[(y / z), $MachinePrecision] * (-x)), $MachinePrecision], N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -9.999999999999969e-311

   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. associate-*l* 85.3%

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

      2. times-frac 97.8%

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

      3. distribute-lft-in 97.8%

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

      4. fma-def 97.8%

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

      5. *-rgt-identity 97.8%

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

   3. Simplified 97.8%

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

   4. Taylor expanded in z around 0 67.6%

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

      1. neg-mul-1 67.6%

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

      2. +-commutative 67.6%

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

      3. unsub-neg 67.6%

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

   6. Simplified 67.6%

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

   7. Taylor expanded in z around inf 39.7%

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

      1. mul-1-neg 39.7%

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

      2. associate-*l/ 45.2%

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

      3. distribute-rgt-neg-in 45.2%

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

   9. Simplified 45.2%

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

   if -9.999999999999969e-311 < 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. associate-*l* 83.6%

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

      2. times-frac 91.8%

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

      3. distribute-lft-in 91.8%

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

      4. fma-def 91.8%

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

      5. *-rgt-identity 91.8%

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

   3. Simplified 91.8%

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

   4. Taylor expanded in z around 0 63.7%

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

      1. neg-mul-1 63.7%

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

      2. +-commutative 63.7%

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

      3. unsub-neg 63.7%

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

   6. Simplified 63.7%

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

   7. Taylor expanded in z around inf 19.7%

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

      1. neg-mul-1 19.7%

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

   9. Simplified 19.7%

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

       1. associate-*l/ 18.3%

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

       2. div-inv 18.3%

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

       3. add-sqr-sqrt 10.6%

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

       4. sqrt-unprod 24.2%

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

       5. sqr-neg 24.2%

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

       6. sqrt-unprod 13.1%

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

       7. add-sqr-sqrt 33.2%

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

       8. associate-*r* 37.5%

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

       9. div-inv 37.5%

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

       10. clear-num 38.9%

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

       11. un-div-inv 38.9%

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

   11. Applied egg-rr 38.9%

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

4. Final simplification 41.9%

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

Alternative 15: 34.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+51}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= x -1e+51) (/ x (/ z y)) (/ y (/ z x))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (x <= -1e+51) {
		tmp = x / (z / y);
	} else {
		tmp = y / (z / x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1d+51)) then
        tmp = x / (z / y)
    else
        tmp = y / (z / x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1e+51) {
		tmp = x / (z / y);
	} else {
		tmp = y / (z / x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if x <= -1e+51:
		tmp = x / (z / y)
	else:
		tmp = y / (z / x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (x <= -1e+51)
		tmp = Float64(x / Float64(z / y));
	else
		tmp = Float64(y / 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 <= -1e+51)
		tmp = x / (z / y);
	else
		tmp = y / (z / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[x, -1e+51], N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision], N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1e51

   1. Initial program 84.8%

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

      1. associate-*l* 84.8%

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

      2. times-frac 93.6%

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

      3. distribute-lft-in 93.6%

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

      4. fma-def 93.6%

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

      5. *-rgt-identity 93.6%

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

   3. Simplified 93.6%

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

   4. Taylor expanded in z around 0 48.0%

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

      1. neg-mul-1 48.0%

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

      2. +-commutative 48.0%

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

      3. unsub-neg 48.0%

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

   6. Simplified 48.0%

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

   7. Taylor expanded in z around inf 32.5%

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

      1. neg-mul-1 32.5%

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

   9. Simplified 32.5%

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

       1. associate-*l/ 30.6%

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

       2. div-inv 30.6%

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

       3. add-sqr-sqrt 18.2%

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

       4. sqrt-unprod 27.7%

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

       5. sqr-neg 27.7%

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

       6. sqrt-unprod 5.2%

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

       7. add-sqr-sqrt 12.6%

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

       8. associate-*r* 21.8%

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

       9. div-inv 21.8%

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

       10. clear-num 25.4%

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

       11. un-div-inv 25.4%

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

   11. Applied egg-rr 25.4%

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

   if -1e51 < x

   1. Initial program 84.3%

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

      1. associate-*l* 84.3%

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

      2. times-frac 94.9%

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

      3. distribute-lft-in 94.9%

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

      4. fma-def 94.9%

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

      5. *-rgt-identity 94.9%

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

   3. Simplified 94.9%

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

   4. Taylor expanded in z around 0 70.0%

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

      1. neg-mul-1 70.0%

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

      2. +-commutative 70.0%

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

      3. unsub-neg 70.0%

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

   6. Simplified 70.0%

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

   7. Taylor expanded in z around inf 31.2%

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

      1. neg-mul-1 31.2%

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

   9. Simplified 31.2%

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

       1. *-commutative 31.2%

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

       2. clear-num 32.2%

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

       3. un-div-inv 32.2%

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

       4. add-sqr-sqrt 17.9%

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

       5. sqrt-unprod 28.9%

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

       6. sqr-neg 28.9%

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

       7. sqrt-unprod 14.2%

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

       8. add-sqr-sqrt 33.0%

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

   11. Applied egg-rr 33.0%

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

4. Final simplification 31.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+51}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternative 16: 72.4% 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 84.4%

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

   1. associate-*l* 84.4%

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

   2. times-frac 94.7%

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

   3. distribute-lft-in 94.7%

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

   4. fma-def 94.7%

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

   5. *-rgt-identity 94.7%

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

3. Simplified 94.7%

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

4. Taylor expanded in z around 0 70.0%

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

   1. *-commutative 70.0%

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

   2. unpow2 70.0%

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

   3. associate-*r/ 70.6%

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

6. Simplified 70.6%

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

7. Final simplification 70.6%

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

Alternative 17: 31.7% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 84.4%

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

   1. associate-*l* 84.4%

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

   2. times-frac 94.7%

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

   3. distribute-lft-in 94.7%

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

   4. fma-def 94.7%

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

   5. *-rgt-identity 94.7%

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

3. Simplified 94.7%

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

4. Taylor expanded in z around 0 65.6%

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

   1. neg-mul-1 65.6%

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

   2. +-commutative 65.6%

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

   3. unsub-neg 65.6%

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

6. Simplified 65.6%

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

7. Taylor expanded in z around inf 31.5%

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

   1. neg-mul-1 31.5%

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

9. Simplified 31.5%

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

    1. expm1-log1p-u 25.0%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{z} \cdot \left(-y\right)\right)\right)} \]

    2. expm1-udef 36.5%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{z} \cdot \left(-y\right)\right)} - 1} \]

    3. add-sqr-sqrt 20.7%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{z} \cdot \color{blue}{\left(\sqrt{-y} \cdot \sqrt{-y}\right)}\right)} - 1 \]

    4. sqrt-unprod 32.9%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{z} \cdot \color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}}\right)} - 1 \]

    5. sqr-neg 32.9%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{z} \cdot \sqrt{\color{blue}{y \cdot y}}\right)} - 1 \]

    6. sqrt-unprod 15.8%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{z} \cdot \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)}\right)} - 1 \]

    7. add-sqr-sqrt 34.7%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{z} \cdot \color{blue}{y}\right)} - 1 \]

11. Applied egg-rr 34.7%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{z} \cdot y\right)} - 1} \]
12. Step-by-step derivation

    1. expm1-def 23.5%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{z} \cdot y\right)\right)} \]

    2. expm1-log1p 28.5%

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

    3. associate-*l/ 25.9%

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

    4. associate-*r/ 30.3%

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

13. Simplified 30.3%

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

14. Final simplification 30.3%

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

Alternative 18: 31.9% accurate, 2.2× speedup

Math

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

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (/ x (/ z 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(x / Float64(z / 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[(x / N[(z / y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.4%

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

   1. associate-*l* 84.4%

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

   2. times-frac 94.7%

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

   3. distribute-lft-in 94.7%

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

   4. fma-def 94.7%

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

   5. *-rgt-identity 94.7%

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

3. Simplified 94.7%

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

4. Taylor expanded in z around 0 65.6%

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

   1. neg-mul-1 65.6%

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

   2. +-commutative 65.6%

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

   3. unsub-neg 65.6%

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

6. Simplified 65.6%

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

7. Taylor expanded in z around inf 31.5%

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

   1. neg-mul-1 31.5%

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

9. Simplified 31.5%

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

    1. associate-*l/ 28.5%

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

    2. div-inv 28.5%

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

    3. add-sqr-sqrt 15.7%

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

    4. sqrt-unprod 28.0%

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

    5. sqr-neg 28.0%

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

    6. sqrt-unprod 11.3%

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

    7. add-sqr-sqrt 25.9%

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

    8. associate-*r* 30.3%

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

    9. div-inv 30.3%

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

    10. clear-num 31.0%

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

    11. un-div-inv 31.0%

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

11. Applied egg-rr 31.0%

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

12. Final simplification 31.0%

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

Developer target: 95.5% 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 2023230 
(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))))