Statistics.Math.RootFinding:ridders from math-functions-0.1.5.2

Percentage Accurate: 61.5% → 89.7%

Time: 32.5s

Alternatives: 20

Speedup: 18.6×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (/ (* (* x y) z) (sqrt (- (* z z) (* t a)))))

C

double code(double x, double y, double z, double t, double a) {
	return ((x * y) * z) / sqrt(((z * z) - (t * a)));
}

Fortran

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

Java

public static double code(double x, double y, double z, double t, double a) {
	return ((x * y) * z) / Math.sqrt(((z * z) - (t * a)));
}

Python

def code(x, y, z, t, a):
	return ((x * y) * z) / math.sqrt(((z * z) - (t * a)))

Julia

function code(x, y, z, t, a)
	return Float64(Float64(Float64(x * y) * z) / sqrt(Float64(Float64(z * z) - Float64(t * a))))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = ((x * y) * z) / sqrt(((z * z) - (t * a)));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(x * y), $MachinePrecision] * z), $MachinePrecision] / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 61.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (/ (* (* x y) z) (sqrt (- (* z z) (* t a)))))

C

double code(double x, double y, double z, double t, double a) {
	return ((x * y) * z) / sqrt(((z * z) - (t * a)));
}

Fortran

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

Java

public static double code(double x, double y, double z, double t, double a) {
	return ((x * y) * z) / Math.sqrt(((z * z) - (t * a)));
}

Python

def code(x, y, z, t, a):
	return ((x * y) * z) / math.sqrt(((z * z) - (t * a)))

Julia

function code(x, y, z, t, a)
	return Float64(Float64(Float64(x * y) * z) / sqrt(Float64(Float64(z * z) - Float64(t * a))))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = ((x * y) * z) / sqrt(((z * z) - (t * a)));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(x * y), $MachinePrecision] * z), $MachinePrecision] / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 89.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.4 \cdot 10^{+53}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+21}:\\ \;\;\;\;\frac{z \cdot \left(y \cdot x\right)}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -7.4e+53)
   (* y (- x))
   (if (<= z 8.5e+21) (/ (* z (* y x)) (sqrt (- (* z z) (* t a)))) (* y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -7.4e+53) {
		tmp = y * -x;
	} else if (z <= 8.5e+21) {
		tmp = (z * (y * x)) / sqrt(((z * z) - (t * a)));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-7.4d+53)) then
        tmp = y * -x
    else if (z <= 8.5d+21) then
        tmp = (z * (y * x)) / sqrt(((z * z) - (t * a)))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -7.4e+53) {
		tmp = y * -x;
	} else if (z <= 8.5e+21) {
		tmp = (z * (y * x)) / Math.sqrt(((z * z) - (t * a)));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -7.4e+53:
		tmp = y * -x
	elif z <= 8.5e+21:
		tmp = (z * (y * x)) / math.sqrt(((z * z) - (t * a)))
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -7.4e+53)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 8.5e+21)
		tmp = Float64(Float64(z * Float64(y * x)) / sqrt(Float64(Float64(z * z) - Float64(t * a))));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -7.4e+53)
		tmp = y * -x;
	elseif (z <= 8.5e+21)
		tmp = (z * (y * x)) / sqrt(((z * z) - (t * a)));
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -7.4e+53], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 8.5e+21], N[(N[(z * N[(y * x), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.4e53

   1. Initial program 39.1%

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

      1. associate-*l* 37.0%

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

      2. associate-*r/ 46.2%

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

      3. *-commutative 46.2%

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

      4. associate-/l* 48.2%

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

   3. Simplified 48.2%

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

   4. Taylor expanded in z around -inf 94.0%

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

      1. associate-*r* 94.0%

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

      2. neg-mul-1 94.0%

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

      3. *-commutative 94.0%

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

   6. Simplified 94.0%

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

   if -7.4e53 < z < 8.5e21

   1. Initial program 87.4%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]

   if 8.5e21 < z

   1. Initial program 36.0%

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

      1. associate-*l* 34.3%

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

      2. associate-*r/ 39.3%

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

      3. *-commutative 39.3%

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

      4. associate-/l* 41.1%

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

   3. Simplified 41.1%

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

   4. Taylor expanded in z around inf 94.0%

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

      1. *-commutative 94.0%

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

   6. Simplified 94.0%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.4 \cdot 10^{+53}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+21}:\\ \;\;\;\;\frac{z \cdot \left(y \cdot x\right)}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 2: 89.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{+146}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+21}:\\ \;\;\;\;x \cdot \frac{z}{\frac{\sqrt{z \cdot z - t \cdot a}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.75e+146)
   (* y (- x))
   (if (<= z 8.5e+21) (* x (/ z (/ (sqrt (- (* z z) (* t a))) y))) (* y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.75e+146) {
		tmp = y * -x;
	} else if (z <= 8.5e+21) {
		tmp = x * (z / (sqrt(((z * z) - (t * a))) / y));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.75d+146)) then
        tmp = y * -x
    else if (z <= 8.5d+21) then
        tmp = x * (z / (sqrt(((z * z) - (t * a))) / y))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.75e+146) {
		tmp = y * -x;
	} else if (z <= 8.5e+21) {
		tmp = x * (z / (Math.sqrt(((z * z) - (t * a))) / y));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.75e+146:
		tmp = y * -x
	elif z <= 8.5e+21:
		tmp = x * (z / (math.sqrt(((z * z) - (t * a))) / y))
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.75e+146)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 8.5e+21)
		tmp = Float64(x * Float64(z / Float64(sqrt(Float64(Float64(z * z) - Float64(t * a))) / y)));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.75e+146)
		tmp = y * -x;
	elseif (z <= 8.5e+21)
		tmp = x * (z / (sqrt(((z * z) - (t * a))) / y));
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.75e+146], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 8.5e+21], N[(x * N[(z / N[(N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.7500000000000001e146

   1. Initial program 24.2%

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

      1. associate-*l* 23.4%

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

      2. associate-*r/ 23.5%

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

      3. *-commutative 23.5%

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

      4. associate-/l* 22.0%

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

   3. Simplified 22.0%

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

   4. Taylor expanded in z around -inf 97.6%

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

      1. associate-*r* 97.6%

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

      2. neg-mul-1 97.6%

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

      3. *-commutative 97.6%

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

   6. Simplified 97.6%

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

   if -1.7500000000000001e146 < z < 8.5e21

   1. Initial program 83.5%

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

      1. associate-*l* 82.4%

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

      2. associate-*r/ 83.3%

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

      3. *-commutative 83.3%

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

      4. associate-/l* 83.5%

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

   3. Simplified 83.5%

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

   if 8.5e21 < z

   1. Initial program 36.0%

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

      1. associate-*l* 34.3%

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

      2. associate-*r/ 39.3%

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

      3. *-commutative 39.3%

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

      4. associate-/l* 41.1%

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

   3. Simplified 41.1%

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

   4. Taylor expanded in z around inf 94.0%

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

      1. *-commutative 94.0%

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

   6. Simplified 94.0%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{+146}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+21}:\\ \;\;\;\;x \cdot \frac{z}{\frac{\sqrt{z \cdot z - t \cdot a}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 3: 90.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+54}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{+21}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -6e+54)
   (* y (- x))
   (if (<= z 1.02e+21) (* x (/ (* z y) (sqrt (- (* z z) (* t a))))) (* y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -6e+54) {
		tmp = y * -x;
	} else if (z <= 1.02e+21) {
		tmp = x * ((z * y) / sqrt(((z * z) - (t * a))));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-6d+54)) then
        tmp = y * -x
    else if (z <= 1.02d+21) then
        tmp = x * ((z * y) / sqrt(((z * z) - (t * a))))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -6e+54) {
		tmp = y * -x;
	} else if (z <= 1.02e+21) {
		tmp = x * ((z * y) / Math.sqrt(((z * z) - (t * a))));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -6e+54:
		tmp = y * -x
	elif z <= 1.02e+21:
		tmp = x * ((z * y) / math.sqrt(((z * z) - (t * a))))
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -6e+54)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 1.02e+21)
		tmp = Float64(x * Float64(Float64(z * y) / sqrt(Float64(Float64(z * z) - Float64(t * a)))));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -6e+54)
		tmp = y * -x;
	elseif (z <= 1.02e+21)
		tmp = x * ((z * y) / sqrt(((z * z) - (t * a))));
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -6e+54], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 1.02e+21], N[(x * N[(N[(z * y), $MachinePrecision] / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.9999999999999998e54

   1. Initial program 39.1%

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

      1. associate-*l* 37.0%

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

      2. associate-*r/ 46.2%

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

      3. *-commutative 46.2%

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

      4. associate-/l* 48.2%

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

   3. Simplified 48.2%

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

   4. Taylor expanded in z around -inf 94.0%

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

      1. associate-*r* 94.0%

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

      2. neg-mul-1 94.0%

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

      3. *-commutative 94.0%

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

   6. Simplified 94.0%

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

   if -5.9999999999999998e54 < z < 1.02e21

   1. Initial program 87.4%

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

      1. associate-*l* 86.9%

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

      2. associate-*r/ 83.2%

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

   3. Simplified 83.2%

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

   if 1.02e21 < z

   1. Initial program 36.0%

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

      1. associate-*l* 34.3%

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

      2. associate-*r/ 39.3%

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

      3. *-commutative 39.3%

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

      4. associate-/l* 41.1%

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

   3. Simplified 41.1%

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

   4. Taylor expanded in z around inf 94.0%

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

      1. *-commutative 94.0%

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

   6. Simplified 94.0%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+54}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{+21}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 4: 89.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.02 \cdot 10^{+153}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+21}:\\ \;\;\;\;z \cdot \frac{y \cdot x}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.02e+153)
   (* y (- x))
   (if (<= z 8.5e+21) (* z (/ (* y x) (sqrt (- (* z z) (* t a))))) (* y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.02e+153) {
		tmp = y * -x;
	} else if (z <= 8.5e+21) {
		tmp = z * ((y * x) / sqrt(((z * z) - (t * a))));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.02d+153)) then
        tmp = y * -x
    else if (z <= 8.5d+21) then
        tmp = z * ((y * x) / sqrt(((z * z) - (t * a))))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.02e+153) {
		tmp = y * -x;
	} else if (z <= 8.5e+21) {
		tmp = z * ((y * x) / Math.sqrt(((z * z) - (t * a))));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.02e+153:
		tmp = y * -x
	elif z <= 8.5e+21:
		tmp = z * ((y * x) / math.sqrt(((z * z) - (t * a))))
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.02e+153)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 8.5e+21)
		tmp = Float64(z * Float64(Float64(y * x) / sqrt(Float64(Float64(z * z) - Float64(t * a)))));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.02e+153)
		tmp = y * -x;
	elseif (z <= 8.5e+21)
		tmp = z * ((y * x) / sqrt(((z * z) - (t * a))));
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.02e+153], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 8.5e+21], N[(z * N[(N[(y * x), $MachinePrecision] / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.0199999999999999e153

   1. Initial program 15.2%

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

      1. associate-*l* 14.4%

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

      2. associate-*r/ 14.5%

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

      3. *-commutative 14.5%

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

      4. associate-/l* 15.6%

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

   3. Simplified 15.6%

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

   4. Taylor expanded in z around -inf 97.4%

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

      1. associate-*r* 97.4%

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

      2. neg-mul-1 97.4%

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

      3. *-commutative 97.4%

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

   6. Simplified 97.4%

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

   if -1.0199999999999999e153 < z < 8.5e21

   1. Initial program 83.9%

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

      1. associate-*l/ 86.6%

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

   3. Simplified 86.6%

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

   if 8.5e21 < z

   1. Initial program 36.0%

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

      1. associate-*l* 34.3%

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

      2. associate-*r/ 39.3%

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

      3. *-commutative 39.3%

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

      4. associate-/l* 41.1%

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

   3. Simplified 41.1%

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

   4. Taylor expanded in z around inf 94.0%

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

      1. *-commutative 94.0%

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

   6. Simplified 94.0%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.02 \cdot 10^{+153}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+21}:\\ \;\;\;\;z \cdot \frac{y \cdot x}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 5: 83.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-51}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 3.05 \cdot 10^{-120}:\\ \;\;\;\;x \cdot \frac{z}{\frac{\sqrt{t \cdot \left(-a\right)}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.1e-51)
   (* y (- x))
   (if (<= z 3.05e-120) (* x (/ z (/ (sqrt (* t (- a))) y))) (* y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.1e-51) {
		tmp = y * -x;
	} else if (z <= 3.05e-120) {
		tmp = x * (z / (sqrt((t * -a)) / y));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.1d-51)) then
        tmp = y * -x
    else if (z <= 3.05d-120) then
        tmp = x * (z / (sqrt((t * -a)) / y))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.1e-51) {
		tmp = y * -x;
	} else if (z <= 3.05e-120) {
		tmp = x * (z / (Math.sqrt((t * -a)) / y));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.1e-51:
		tmp = y * -x
	elif z <= 3.05e-120:
		tmp = x * (z / (math.sqrt((t * -a)) / y))
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.1e-51)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 3.05e-120)
		tmp = Float64(x * Float64(z / Float64(sqrt(Float64(t * Float64(-a))) / y)));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.1e-51)
		tmp = y * -x;
	elseif (z <= 3.05e-120)
		tmp = x * (z / (sqrt((t * -a)) / y));
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.1e-51], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 3.05e-120], N[(x * N[(z / N[(N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.1e-51

   1. Initial program 56.0%

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

      1. associate-*l* 54.3%

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

      2. associate-*r/ 59.5%

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

      3. *-commutative 59.5%

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

      4. associate-/l* 61.1%

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

   3. Simplified 61.1%

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

   4. Taylor expanded in z around -inf 85.7%

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

      1. associate-*r* 85.7%

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

      2. neg-mul-1 85.7%

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

      3. *-commutative 85.7%

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

   6. Simplified 85.7%

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

   if -1.1e-51 < z < 3.05e-120

   1. Initial program 82.7%

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

      1. associate-*l* 83.9%

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

      2. associate-*r/ 79.6%

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

      3. *-commutative 79.6%

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

      4. associate-/l* 78.6%

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

   3. Simplified 78.6%

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

   4. Taylor expanded in z around 0 77.1%

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

      1. mul-1-neg 77.1%

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

      2. distribute-rgt-neg-out 77.1%

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

   6. Simplified 77.1%

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

   if 3.05e-120 < z

   1. Initial program 50.0%

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

      1. associate-*l* 47.7%

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

      2. associate-*r/ 50.6%

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

      3. *-commutative 50.6%

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

      4. associate-/l* 51.1%

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

   3. Simplified 51.1%

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

   4. Taylor expanded in z around inf 87.2%

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

      1. *-commutative 87.2%

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

   6. Simplified 87.2%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-51}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 3.05 \cdot 10^{-120}:\\ \;\;\;\;x \cdot \frac{z}{\frac{\sqrt{t \cdot \left(-a\right)}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 6: 83.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-55}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-122}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1e-55)
   (* y (- x))
   (if (<= z 9e-122) (* x (/ (* z y) (sqrt (* t (- a))))) (* y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1e-55) {
		tmp = y * -x;
	} else if (z <= 9e-122) {
		tmp = x * ((z * y) / sqrt((t * -a)));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1d-55)) then
        tmp = y * -x
    else if (z <= 9d-122) then
        tmp = x * ((z * y) / sqrt((t * -a)))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1e-55) {
		tmp = y * -x;
	} else if (z <= 9e-122) {
		tmp = x * ((z * y) / Math.sqrt((t * -a)));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1e-55:
		tmp = y * -x
	elif z <= 9e-122:
		tmp = x * ((z * y) / math.sqrt((t * -a)))
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1e-55)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 9e-122)
		tmp = Float64(x * Float64(Float64(z * y) / sqrt(Float64(t * Float64(-a)))));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1e-55)
		tmp = y * -x;
	elseif (z <= 9e-122)
		tmp = x * ((z * y) / sqrt((t * -a)));
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1e-55], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 9e-122], N[(x * N[(N[(z * y), $MachinePrecision] / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -9.99999999999999995e-56

   1. Initial program 56.0%

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

      1. associate-*l* 54.3%

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

      2. associate-*r/ 59.5%

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

      3. *-commutative 59.5%

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

      4. associate-/l* 61.1%

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

   3. Simplified 61.1%

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

   4. Taylor expanded in z around -inf 85.7%

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

      1. associate-*r* 85.7%

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

      2. neg-mul-1 85.7%

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

      3. *-commutative 85.7%

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

   6. Simplified 85.7%

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

   if -9.99999999999999995e-56 < z < 8.99999999999999959e-122

   1. Initial program 82.7%

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

      1. associate-*l* 83.9%

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

      2. associate-*r/ 79.6%

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

   3. Simplified 79.6%

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

   4. Taylor expanded in z around 0 78.1%

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

      1. mul-1-neg 77.1%

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

      2. distribute-rgt-neg-out 77.1%

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

   6. Simplified 78.1%

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

   if 8.99999999999999959e-122 < z

   1. Initial program 50.0%

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

      1. associate-*l* 47.7%

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

      2. associate-*r/ 50.6%

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

      3. *-commutative 50.6%

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

      4. associate-/l* 51.1%

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

   3. Simplified 51.1%

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

   4. Taylor expanded in z around inf 87.2%

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

      1. *-commutative 87.2%

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

   6. Simplified 87.2%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-55}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-122}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 7: 83.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{-58}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-84}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{z}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.9e-58)
   (* y (- x))
   (if (<= z 1.05e-84) (* (* y x) (/ z (sqrt (* t (- a))))) (* y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.9e-58) {
		tmp = y * -x;
	} else if (z <= 1.05e-84) {
		tmp = (y * x) * (z / sqrt((t * -a)));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-2.9d-58)) then
        tmp = y * -x
    else if (z <= 1.05d-84) then
        tmp = (y * x) * (z / sqrt((t * -a)))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.9e-58) {
		tmp = y * -x;
	} else if (z <= 1.05e-84) {
		tmp = (y * x) * (z / Math.sqrt((t * -a)));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.9e-58:
		tmp = y * -x
	elif z <= 1.05e-84:
		tmp = (y * x) * (z / math.sqrt((t * -a)))
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.9e-58)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 1.05e-84)
		tmp = Float64(Float64(y * x) * Float64(z / sqrt(Float64(t * Float64(-a)))));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.9e-58)
		tmp = y * -x;
	elseif (z <= 1.05e-84)
		tmp = (y * x) * (z / sqrt((t * -a)));
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.9e-58], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 1.05e-84], N[(N[(y * x), $MachinePrecision] * N[(z / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.8999999999999999e-58

   1. Initial program 56.0%

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

      1. associate-*l* 54.3%

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

      2. associate-*r/ 59.5%

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

      3. *-commutative 59.5%

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

      4. associate-/l* 61.1%

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

   3. Simplified 61.1%

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

   4. Taylor expanded in z around -inf 85.7%

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

      1. associate-*r* 85.7%

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

      2. neg-mul-1 85.7%

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

      3. *-commutative 85.7%

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

   6. Simplified 85.7%

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

   if -2.8999999999999999e-58 < z < 1.04999999999999999e-84

   1. Initial program 84.4%

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

      1. *-commutative 84.4%

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

      2. associate-/l* 81.8%

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

   3. Simplified 81.8%

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

   4. Taylor expanded in z around 0 76.1%

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

      1. mul-1-neg 72.4%

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

      2. distribute-rgt-neg-out 72.4%

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

   6. Simplified 76.1%

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

      1. associate-/r/ 77.5%

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

      2. *-commutative 77.5%

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

   8. Applied egg-rr 77.5%

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

   if 1.04999999999999999e-84 < z

   1. Initial program 46.3%

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

      1. associate-*l* 44.9%

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

      2. associate-*r/ 49.0%

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

      3. *-commutative 49.0%

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

      4. associate-/l* 50.5%

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

   3. Simplified 50.5%

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

   4. Taylor expanded in z around inf 90.2%

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

      1. *-commutative 90.2%

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

   6. Simplified 90.2%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{-58}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-84}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{z}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 8: 83.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.4 \cdot 10^{-59}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-81}:\\ \;\;\;\;\frac{z}{\frac{\frac{\sqrt{t \cdot \left(-a\right)}}{x}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -4.4e-59)
   (* y (- x))
   (if (<= z 3.2e-81) (/ z (/ (/ (sqrt (* t (- a))) x) y)) (* y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4.4e-59) {
		tmp = y * -x;
	} else if (z <= 3.2e-81) {
		tmp = z / ((sqrt((t * -a)) / x) / y);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-4.4d-59)) then
        tmp = y * -x
    else if (z <= 3.2d-81) then
        tmp = z / ((sqrt((t * -a)) / x) / y)
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4.4e-59) {
		tmp = y * -x;
	} else if (z <= 3.2e-81) {
		tmp = z / ((Math.sqrt((t * -a)) / x) / y);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -4.4e-59:
		tmp = y * -x
	elif z <= 3.2e-81:
		tmp = z / ((math.sqrt((t * -a)) / x) / y)
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -4.4e-59)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 3.2e-81)
		tmp = Float64(z / Float64(Float64(sqrt(Float64(t * Float64(-a))) / x) / y));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -4.4e-59)
		tmp = y * -x;
	elseif (z <= 3.2e-81)
		tmp = z / ((sqrt((t * -a)) / x) / y);
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -4.4e-59], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 3.2e-81], N[(z / N[(N[(N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.3999999999999998e-59

   1. Initial program 56.0%

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

      1. associate-*l* 54.3%

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

      2. associate-*r/ 59.5%

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

      3. *-commutative 59.5%

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

      4. associate-/l* 61.1%

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

   3. Simplified 61.1%

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

   4. Taylor expanded in z around -inf 85.7%

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

      1. associate-*r* 85.7%

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

      2. neg-mul-1 85.7%

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

      3. *-commutative 85.7%

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

   6. Simplified 85.7%

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

   if -4.3999999999999998e-59 < z < 3.2e-81

   1. Initial program 84.4%

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

      1. *-commutative 84.4%

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

      2. associate-/l* 81.8%

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

   3. Simplified 81.8%

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

      1. *-un-lft-identity 81.8%

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

      2. *-commutative 81.8%

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

      3. times-frac 84.5%

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

   5. Applied egg-rr 84.5%

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

      1. associate-*l/ 84.6%

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

      2. *-lft-identity 84.6%

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

      3. *-commutative 84.6%

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

   7. Simplified 84.6%

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

   8. Taylor expanded in z around 0 79.0%

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

      1. neg-mul-1 79.0%

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

      2. distribute-rgt-neg-in 79.0%

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

   10. Simplified 79.0%

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

   if 3.2e-81 < z

   1. Initial program 46.3%

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

      1. associate-*l* 44.9%

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

      2. associate-*r/ 49.0%

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

      3. *-commutative 49.0%

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

      4. associate-/l* 50.5%

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

   3. Simplified 50.5%

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

   4. Taylor expanded in z around inf 90.2%

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

      1. *-commutative 90.2%

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

   6. Simplified 90.2%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.4 \cdot 10^{-59}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-81}:\\ \;\;\;\;\frac{z}{\frac{\frac{\sqrt{t \cdot \left(-a\right)}}{x}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 9: 83.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{-60}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-85}:\\ \;\;\;\;\frac{z \cdot \left(y \cdot x\right)}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.5e-60)
   (* y (- x))
   (if (<= z 4.6e-85) (/ (* z (* y x)) (sqrt (* t (- a)))) (* y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.5e-60) {
		tmp = y * -x;
	} else if (z <= 4.6e-85) {
		tmp = (z * (y * x)) / sqrt((t * -a));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-2.5d-60)) then
        tmp = y * -x
    else if (z <= 4.6d-85) then
        tmp = (z * (y * x)) / sqrt((t * -a))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.5e-60) {
		tmp = y * -x;
	} else if (z <= 4.6e-85) {
		tmp = (z * (y * x)) / Math.sqrt((t * -a));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.5e-60:
		tmp = y * -x
	elif z <= 4.6e-85:
		tmp = (z * (y * x)) / math.sqrt((t * -a))
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.5e-60)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 4.6e-85)
		tmp = Float64(Float64(z * Float64(y * x)) / sqrt(Float64(t * Float64(-a))));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.5e-60)
		tmp = y * -x;
	elseif (z <= 4.6e-85)
		tmp = (z * (y * x)) / sqrt((t * -a));
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.5e-60], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 4.6e-85], N[(N[(z * N[(y * x), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.5000000000000001e-60

   1. Initial program 56.0%

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

      1. associate-*l* 54.3%

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

      2. associate-*r/ 59.5%

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

      3. *-commutative 59.5%

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

      4. associate-/l* 61.1%

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

   3. Simplified 61.1%

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

   4. Taylor expanded in z around -inf 85.7%

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

      1. associate-*r* 85.7%

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

      2. neg-mul-1 85.7%

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

      3. *-commutative 85.7%

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

   6. Simplified 85.7%

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

   if -2.5000000000000001e-60 < z < 4.6000000000000001e-85

   1. Initial program 84.4%

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

      1. *-commutative 84.4%

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

      2. associate-/l* 81.8%

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

   3. Simplified 81.8%

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

   4. Taylor expanded in z around 0 76.1%

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

      1. mul-1-neg 72.4%

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

      2. distribute-rgt-neg-out 72.4%

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

   6. Simplified 76.1%

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

      1. expm1-log1p-u 68.4%

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

      2. expm1-udef 43.9%

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

      3. associate-/r/ 46.3%

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

      4. *-commutative 46.3%

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

   8. Applied egg-rr 46.3%

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

      1. expm1-def 69.5%

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

      2. expm1-log1p 77.5%

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

      3. associate-*l/ 78.7%

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

   10. Simplified 78.7%

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

   if 4.6000000000000001e-85 < z

   1. Initial program 46.3%

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

      1. associate-*l* 44.9%

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

      2. associate-*r/ 49.0%

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

      3. *-commutative 49.0%

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

      4. associate-/l* 50.5%

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

   3. Simplified 50.5%

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

   4. Taylor expanded in z around inf 90.2%

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

      1. *-commutative 90.2%

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

   6. Simplified 90.2%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{-60}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-85}:\\ \;\;\;\;\frac{z \cdot \left(y \cdot x\right)}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 10: 77.2% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{-109}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-87}:\\ \;\;\;\;\frac{-y}{\frac{\frac{t}{\frac{x}{a}} \cdot \frac{0.5}{z} - \frac{z}{x}}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -8.2e-109)
   (* y (- x))
   (if (<= z 4.4e-87)
     (/ (- y) (/ (- (* (/ t (/ x a)) (/ 0.5 z)) (/ z x)) z))
     (* y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -8.2e-109) {
		tmp = y * -x;
	} else if (z <= 4.4e-87) {
		tmp = -y / ((((t / (x / a)) * (0.5 / z)) - (z / x)) / z);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-8.2d-109)) then
        tmp = y * -x
    else if (z <= 4.4d-87) then
        tmp = -y / ((((t / (x / a)) * (0.5d0 / z)) - (z / x)) / z)
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -8.2e-109) {
		tmp = y * -x;
	} else if (z <= 4.4e-87) {
		tmp = -y / ((((t / (x / a)) * (0.5 / z)) - (z / x)) / z);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -8.2e-109:
		tmp = y * -x
	elif z <= 4.4e-87:
		tmp = -y / ((((t / (x / a)) * (0.5 / z)) - (z / x)) / z)
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -8.2e-109)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 4.4e-87)
		tmp = Float64(Float64(-y) / Float64(Float64(Float64(Float64(t / Float64(x / a)) * Float64(0.5 / z)) - Float64(z / x)) / z));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -8.2e-109)
		tmp = y * -x;
	elseif (z <= 4.4e-87)
		tmp = -y / ((((t / (x / a)) * (0.5 / z)) - (z / x)) / z);
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -8.2e-109], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 4.4e-87], N[((-y) / N[(N[(N[(N[(t / N[(x / a), $MachinePrecision]), $MachinePrecision] * N[(0.5 / z), $MachinePrecision]), $MachinePrecision] - N[(z / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -8.2000000000000004e-109

   1. Initial program 58.2%

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

      1. associate-*l* 56.6%

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

      2. associate-*r/ 61.4%

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

      3. *-commutative 61.4%

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

      4. associate-/l* 62.8%

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

   3. Simplified 62.8%

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

   4. Taylor expanded in z around -inf 83.8%

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

      1. associate-*r* 83.8%

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

      2. neg-mul-1 83.8%

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

      3. *-commutative 83.8%

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

   6. Simplified 83.8%

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

   if -8.2000000000000004e-109 < z < 4.39999999999999976e-87

   1. Initial program 84.2%

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

      1. *-commutative 84.2%

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

      2. associate-/l* 81.3%

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

   3. Simplified 81.3%

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

   4. Taylor expanded in z around inf 46.3%

      \[\leadsto \frac{z}{\color{blue}{-0.5 \cdot \frac{a \cdot t}{x \cdot \left(y \cdot z\right)} + \frac{z}{x \cdot y}}} \]

   5. Taylor expanded in y around -inf 46.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot z}{-1 \cdot \frac{z}{x} + 0.5 \cdot \frac{a \cdot t}{x \cdot z}}} \]
   6. Step-by-step derivation

      1. mul-1-neg 46.4%

         \[\leadsto \color{blue}{-\frac{y \cdot z}{-1 \cdot \frac{z}{x} + 0.5 \cdot \frac{a \cdot t}{x \cdot z}}} \]

      2. associate-/l* 46.4%

         \[\leadsto -\color{blue}{\frac{y}{\frac{-1 \cdot \frac{z}{x} + 0.5 \cdot \frac{a \cdot t}{x \cdot z}}{z}}} \]

      3. distribute-neg-frac 46.4%

         \[\leadsto \color{blue}{\frac{-y}{\frac{-1 \cdot \frac{z}{x} + 0.5 \cdot \frac{a \cdot t}{x \cdot z}}{z}}} \]

      4. mul-1-neg 46.4%

         \[\leadsto \frac{-y}{\frac{\color{blue}{\left(-\frac{z}{x}\right)} + 0.5 \cdot \frac{a \cdot t}{x \cdot z}}{z}} \]

      5. distribute-frac-neg 46.4%

         \[\leadsto \frac{-y}{\frac{\color{blue}{\frac{-z}{x}} + 0.5 \cdot \frac{a \cdot t}{x \cdot z}}{z}} \]

      6. +-commutative 46.4%

         \[\leadsto \frac{-y}{\frac{\color{blue}{0.5 \cdot \frac{a \cdot t}{x \cdot z} + \frac{-z}{x}}}{z}} \]

      7. distribute-frac-neg 46.4%

         \[\leadsto \frac{-y}{\frac{0.5 \cdot \frac{a \cdot t}{x \cdot z} + \color{blue}{\left(-\frac{z}{x}\right)}}{z}} \]

      8. unsub-neg 46.4%

         \[\leadsto \frac{-y}{\frac{\color{blue}{0.5 \cdot \frac{a \cdot t}{x \cdot z} - \frac{z}{x}}}{z}} \]

      9. associate-*r/ 46.4%

         \[\leadsto \frac{-y}{\frac{\color{blue}{\frac{0.5 \cdot \left(a \cdot t\right)}{x \cdot z}} - \frac{z}{x}}{z}} \]

      10. *-commutative 46.4%

          \[\leadsto \frac{-y}{\frac{\frac{\color{blue}{\left(a \cdot t\right) \cdot 0.5}}{x \cdot z} - \frac{z}{x}}{z}} \]

      11. times-frac 48.0%

          \[\leadsto \frac{-y}{\frac{\color{blue}{\frac{a \cdot t}{x} \cdot \frac{0.5}{z}} - \frac{z}{x}}{z}} \]

      12. *-commutative 48.0%

          \[\leadsto \frac{-y}{\frac{\frac{\color{blue}{t \cdot a}}{x} \cdot \frac{0.5}{z} - \frac{z}{x}}{z}} \]

      13. associate-/l* 47.9%

          \[\leadsto \frac{-y}{\frac{\color{blue}{\frac{t}{\frac{x}{a}}} \cdot \frac{0.5}{z} - \frac{z}{x}}{z}} \]

   7. Simplified 47.9%

      \[\leadsto \color{blue}{\frac{-y}{\frac{\frac{t}{\frac{x}{a}} \cdot \frac{0.5}{z} - \frac{z}{x}}{z}}} \]

   if 4.39999999999999976e-87 < z

   1. Initial program 46.3%

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

      1. associate-*l* 44.9%

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

      2. associate-*r/ 49.0%

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

      3. *-commutative 49.0%

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

      4. associate-/l* 50.5%

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

   3. Simplified 50.5%

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

   4. Taylor expanded in z around inf 90.2%

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

      1. *-commutative 90.2%

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

   6. Simplified 90.2%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{-109}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-87}:\\ \;\;\;\;\frac{-y}{\frac{\frac{t}{\frac{x}{a}} \cdot \frac{0.5}{z} - \frac{z}{x}}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 11: 77.3% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{-108}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+20}:\\ \;\;\;\;x \cdot \frac{z}{\frac{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.9e-108)
   (* y (- x))
   (if (<= z 2.5e+20) (* x (/ z (/ (+ z (* -0.5 (/ a (/ z t)))) y))) (* y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.9e-108) {
		tmp = y * -x;
	} else if (z <= 2.5e+20) {
		tmp = x * (z / ((z + (-0.5 * (a / (z / t)))) / y));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-3.9d-108)) then
        tmp = y * -x
    else if (z <= 2.5d+20) then
        tmp = x * (z / ((z + ((-0.5d0) * (a / (z / t)))) / y))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.9e-108) {
		tmp = y * -x;
	} else if (z <= 2.5e+20) {
		tmp = x * (z / ((z + (-0.5 * (a / (z / t)))) / y));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -3.9e-108:
		tmp = y * -x
	elif z <= 2.5e+20:
		tmp = x * (z / ((z + (-0.5 * (a / (z / t)))) / y))
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.9e-108)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 2.5e+20)
		tmp = Float64(x * Float64(z / Float64(Float64(z + Float64(-0.5 * Float64(a / Float64(z / t)))) / y)));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -3.9e-108)
		tmp = y * -x;
	elseif (z <= 2.5e+20)
		tmp = x * (z / ((z + (-0.5 * (a / (z / t)))) / y));
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.9e-108], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 2.5e+20], N[(x * N[(z / N[(N[(z + N[(-0.5 * N[(a / N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.89999999999999995e-108

   1. Initial program 58.2%

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

      1. associate-*l* 56.6%

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

      2. associate-*r/ 61.4%

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

      3. *-commutative 61.4%

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

      4. associate-/l* 62.8%

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

   3. Simplified 62.8%

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

   4. Taylor expanded in z around -inf 83.8%

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

      1. associate-*r* 83.8%

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

      2. neg-mul-1 83.8%

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

      3. *-commutative 83.8%

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

   6. Simplified 83.8%

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

   if -3.89999999999999995e-108 < z < 2.5e20

   1. Initial program 86.4%

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

      1. associate-*l* 86.0%

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

      2. associate-*r/ 81.7%

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

      3. *-commutative 81.7%

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

      4. associate-/l* 79.6%

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

   3. Simplified 79.6%

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

   4. Taylor expanded in z around inf 52.0%

      \[\leadsto x \cdot \frac{z}{\frac{\color{blue}{z + -0.5 \cdot \frac{a \cdot t}{z}}}{y}} \]
   5. Step-by-step derivation

      1. associate-/l* 52.7%

         \[\leadsto x \cdot \frac{z}{\frac{z + -0.5 \cdot \color{blue}{\frac{a}{\frac{z}{t}}}}{y}} \]

   6. Simplified 52.7%

      \[\leadsto x \cdot \frac{z}{\frac{\color{blue}{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}}{y}} \]

   if 2.5e20 < z

   1. Initial program 36.0%

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

      1. associate-*l* 34.3%

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

      2. associate-*r/ 39.3%

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

      3. *-commutative 39.3%

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

      4. associate-/l* 41.1%

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

   3. Simplified 41.1%

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

   4. Taylor expanded in z around inf 94.0%

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

      1. *-commutative 94.0%

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

   6. Simplified 94.0%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{-108}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+20}:\\ \;\;\;\;x \cdot \frac{z}{\frac{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 12: 77.5% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{-109}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.06 \cdot 10^{+38}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -8.2e-109)
   (* y (- x))
   (if (<= z 1.06e+38)
     (* x (/ (* z y) (+ z (* -0.5 (/ a (/ z t))))))
     (* y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -8.2e-109) {
		tmp = y * -x;
	} else if (z <= 1.06e+38) {
		tmp = x * ((z * y) / (z + (-0.5 * (a / (z / t)))));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-8.2d-109)) then
        tmp = y * -x
    else if (z <= 1.06d+38) then
        tmp = x * ((z * y) / (z + ((-0.5d0) * (a / (z / t)))))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -8.2e-109) {
		tmp = y * -x;
	} else if (z <= 1.06e+38) {
		tmp = x * ((z * y) / (z + (-0.5 * (a / (z / t)))));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -8.2e-109:
		tmp = y * -x
	elif z <= 1.06e+38:
		tmp = x * ((z * y) / (z + (-0.5 * (a / (z / t)))))
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -8.2e-109)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 1.06e+38)
		tmp = Float64(x * Float64(Float64(z * y) / Float64(z + Float64(-0.5 * Float64(a / Float64(z / t))))));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -8.2e-109)
		tmp = y * -x;
	elseif (z <= 1.06e+38)
		tmp = x * ((z * y) / (z + (-0.5 * (a / (z / t)))));
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -8.2e-109], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 1.06e+38], N[(x * N[(N[(z * y), $MachinePrecision] / N[(z + N[(-0.5 * N[(a / N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -8.2000000000000004e-109

   1. Initial program 58.2%

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

      1. associate-*l* 56.6%

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

      2. associate-*r/ 61.4%

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

      3. *-commutative 61.4%

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

      4. associate-/l* 62.8%

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

   3. Simplified 62.8%

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

   4. Taylor expanded in z around -inf 83.8%

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

      1. associate-*r* 83.8%

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

      2. neg-mul-1 83.8%

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

      3. *-commutative 83.8%

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

   6. Simplified 83.8%

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

   if -8.2000000000000004e-109 < z < 1.06e38

   1. Initial program 86.0%

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

      1. associate-*l* 85.7%

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

      2. associate-*r/ 81.6%

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

   3. Simplified 81.6%

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

   4. Taylor expanded in z around inf 55.5%

      \[\leadsto x \cdot \frac{y \cdot z}{\color{blue}{z + -0.5 \cdot \frac{a \cdot t}{z}}} \]
   5. Step-by-step derivation

      1. associate-/l* 53.2%

         \[\leadsto x \cdot \frac{z}{\frac{z + -0.5 \cdot \color{blue}{\frac{a}{\frac{z}{t}}}}{y}} \]

   6. Simplified 55.6%

      \[\leadsto x \cdot \frac{y \cdot z}{\color{blue}{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}} \]

   if 1.06e38 < z

   1. Initial program 33.0%

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

      1. associate-*l* 31.2%

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

      2. associate-*r/ 36.5%

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

      3. *-commutative 36.5%

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

      4. associate-/l* 39.7%

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

   3. Simplified 39.7%

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

   4. Taylor expanded in z around inf 94.9%

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

      1. *-commutative 94.9%

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

   6. Simplified 94.9%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{-109}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.06 \cdot 10^{+38}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 13: 77.1% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{-109}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-60}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{z + -0.5 \cdot \frac{t \cdot a}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -8.2e-109)
   (* y (- x))
   (if (<= z 5.8e-60) (/ (* x (* z y)) (+ z (* -0.5 (/ (* t a) z)))) (* y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -8.2e-109) {
		tmp = y * -x;
	} else if (z <= 5.8e-60) {
		tmp = (x * (z * y)) / (z + (-0.5 * ((t * a) / z)));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-8.2d-109)) then
        tmp = y * -x
    else if (z <= 5.8d-60) then
        tmp = (x * (z * y)) / (z + ((-0.5d0) * ((t * a) / z)))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -8.2e-109) {
		tmp = y * -x;
	} else if (z <= 5.8e-60) {
		tmp = (x * (z * y)) / (z + (-0.5 * ((t * a) / z)));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -8.2e-109:
		tmp = y * -x
	elif z <= 5.8e-60:
		tmp = (x * (z * y)) / (z + (-0.5 * ((t * a) / z)))
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -8.2e-109)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 5.8e-60)
		tmp = Float64(Float64(x * Float64(z * y)) / Float64(z + Float64(-0.5 * Float64(Float64(t * a) / z))));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -8.2e-109)
		tmp = y * -x;
	elseif (z <= 5.8e-60)
		tmp = (x * (z * y)) / (z + (-0.5 * ((t * a) / z)));
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -8.2e-109], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 5.8e-60], N[(N[(x * N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(z + N[(-0.5 * N[(N[(t * a), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -8.2000000000000004e-109

   1. Initial program 58.2%

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

      1. associate-*l* 56.6%

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

      2. associate-*r/ 61.4%

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

      3. *-commutative 61.4%

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

      4. associate-/l* 62.8%

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

   3. Simplified 62.8%

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

   4. Taylor expanded in z around -inf 83.8%

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

      1. associate-*r* 83.8%

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

      2. neg-mul-1 83.8%

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

      3. *-commutative 83.8%

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

   6. Simplified 83.8%

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

   if -8.2000000000000004e-109 < z < 5.7999999999999999e-60

   1. Initial program 85.0%

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

      1. associate-*l* 84.6%

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

      2. associate-*r/ 79.2%

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

   3. Simplified 79.2%

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

   4. Taylor expanded in z around inf 49.8%

      \[\leadsto x \cdot \frac{y \cdot z}{\color{blue}{z + -0.5 \cdot \frac{a \cdot t}{z}}} \]
   5. Step-by-step derivation

      1. associate-/l* 48.3%

         \[\leadsto x \cdot \frac{z}{\frac{z + -0.5 \cdot \color{blue}{\frac{a}{\frac{z}{t}}}}{y}} \]

   6. Simplified 49.8%

      \[\leadsto x \cdot \frac{y \cdot z}{\color{blue}{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}} \]

   7. Taylor expanded in x around 0 49.2%

      \[\leadsto \color{blue}{\frac{x \cdot \left(y \cdot z\right)}{z + -0.5 \cdot \frac{a \cdot t}{z}}} \]

   if 5.7999999999999999e-60 < z

   1. Initial program 44.6%

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

      1. associate-*l* 43.1%

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

      2. associate-*r/ 47.3%

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

      3. *-commutative 47.3%

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

      4. associate-/l* 48.9%

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

   3. Simplified 48.9%

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

   4. Taylor expanded in z around inf 90.9%

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

      1. *-commutative 90.9%

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

   6. Simplified 90.9%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{-109}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-60}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{z + -0.5 \cdot \frac{t \cdot a}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 14: 76.3% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.4 \cdot 10^{-109}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-153}:\\ \;\;\;\;-2 \cdot \frac{z \cdot \left(z \cdot y\right)}{\frac{t}{\frac{x}{a}}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -8.4e-109)
   (* y (- x))
   (if (<= z 1.6e-153) (* -2.0 (/ (* z (* z y)) (/ t (/ x a)))) (* y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -8.4e-109) {
		tmp = y * -x;
	} else if (z <= 1.6e-153) {
		tmp = -2.0 * ((z * (z * y)) / (t / (x / a)));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-8.4d-109)) then
        tmp = y * -x
    else if (z <= 1.6d-153) then
        tmp = (-2.0d0) * ((z * (z * y)) / (t / (x / a)))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -8.4e-109) {
		tmp = y * -x;
	} else if (z <= 1.6e-153) {
		tmp = -2.0 * ((z * (z * y)) / (t / (x / a)));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -8.4e-109:
		tmp = y * -x
	elif z <= 1.6e-153:
		tmp = -2.0 * ((z * (z * y)) / (t / (x / a)))
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -8.4e-109)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 1.6e-153)
		tmp = Float64(-2.0 * Float64(Float64(z * Float64(z * y)) / Float64(t / Float64(x / a))));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -8.4e-109)
		tmp = y * -x;
	elseif (z <= 1.6e-153)
		tmp = -2.0 * ((z * (z * y)) / (t / (x / a)));
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -8.4e-109], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 1.6e-153], N[(-2.0 * N[(N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(t / N[(x / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -8.39999999999999984e-109

   1. Initial program 58.2%

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

      1. associate-*l* 56.6%

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

      2. associate-*r/ 61.4%

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

      3. *-commutative 61.4%

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

      4. associate-/l* 62.8%

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

   3. Simplified 62.8%

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

   4. Taylor expanded in z around -inf 83.8%

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

      1. associate-*r* 83.8%

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

      2. neg-mul-1 83.8%

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

      3. *-commutative 83.8%

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

   6. Simplified 83.8%

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

   if -8.39999999999999984e-109 < z < 1.6e-153

   1. Initial program 80.2%

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

      1. *-commutative 80.2%

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

      2. associate-/l* 78.3%

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

   3. Simplified 78.3%

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

   4. Taylor expanded in z around inf 45.6%

      \[\leadsto \frac{z}{\color{blue}{-0.5 \cdot \frac{a \cdot t}{x \cdot \left(y \cdot z\right)} + \frac{z}{x \cdot y}}} \]

   5. Taylor expanded in z around 0 45.2%

      \[\leadsto \color{blue}{-2 \cdot \frac{x \cdot \left(y \cdot {z}^{2}\right)}{a \cdot t}} \]
   6. Step-by-step derivation

      1. *-commutative 45.2%

         \[\leadsto -2 \cdot \frac{\color{blue}{\left(y \cdot {z}^{2}\right) \cdot x}}{a \cdot t} \]

      2. associate-/l* 45.0%

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

      3. *-commutative 45.0%

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

      4. unpow2 45.0%

         \[\leadsto -2 \cdot \frac{\color{blue}{\left(z \cdot z\right)} \cdot y}{\frac{a \cdot t}{x}} \]

      5. associate-*l* 45.2%

         \[\leadsto -2 \cdot \frac{\color{blue}{z \cdot \left(z \cdot y\right)}}{\frac{a \cdot t}{x}} \]

      6. *-commutative 45.2%

         \[\leadsto -2 \cdot \frac{z \cdot \color{blue}{\left(y \cdot z\right)}}{\frac{a \cdot t}{x}} \]

      7. *-commutative 45.2%

         \[\leadsto -2 \cdot \frac{z \cdot \left(y \cdot z\right)}{\frac{\color{blue}{t \cdot a}}{x}} \]

      8. associate-/l* 46.4%

         \[\leadsto -2 \cdot \frac{z \cdot \left(y \cdot z\right)}{\color{blue}{\frac{t}{\frac{x}{a}}}} \]

   7. Simplified 46.4%

      \[\leadsto \color{blue}{-2 \cdot \frac{z \cdot \left(y \cdot z\right)}{\frac{t}{\frac{x}{a}}}} \]

   if 1.6e-153 < z

   1. Initial program 52.7%

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

      1. associate-*l* 49.4%

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

      2. associate-*r/ 52.2%

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

      3. *-commutative 52.2%

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

      4. associate-/l* 52.6%

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

   3. Simplified 52.6%

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

   4. Taylor expanded in z around inf 83.6%

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

      1. *-commutative 83.6%

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

   6. Simplified 83.6%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.4 \cdot 10^{-109}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-153}:\\ \;\;\;\;-2 \cdot \frac{z \cdot \left(z \cdot y\right)}{\frac{t}{\frac{x}{a}}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 15: 76.2% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{-109}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-161}:\\ \;\;\;\;2 \cdot \left(\frac{x}{a} \cdot \frac{y \cdot \left(z \cdot z\right)}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -8.2e-109)
   (* y (- x))
   (if (<= z 4.2e-161) (* 2.0 (* (/ x a) (/ (* y (* z z)) t))) (* y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -8.2e-109) {
		tmp = y * -x;
	} else if (z <= 4.2e-161) {
		tmp = 2.0 * ((x / a) * ((y * (z * z)) / t));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-8.2d-109)) then
        tmp = y * -x
    else if (z <= 4.2d-161) then
        tmp = 2.0d0 * ((x / a) * ((y * (z * z)) / t))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -8.2e-109) {
		tmp = y * -x;
	} else if (z <= 4.2e-161) {
		tmp = 2.0 * ((x / a) * ((y * (z * z)) / t));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -8.2e-109:
		tmp = y * -x
	elif z <= 4.2e-161:
		tmp = 2.0 * ((x / a) * ((y * (z * z)) / t))
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -8.2e-109)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 4.2e-161)
		tmp = Float64(2.0 * Float64(Float64(x / a) * Float64(Float64(y * Float64(z * z)) / t)));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -8.2e-109)
		tmp = y * -x;
	elseif (z <= 4.2e-161)
		tmp = 2.0 * ((x / a) * ((y * (z * z)) / t));
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -8.2e-109], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 4.2e-161], N[(2.0 * N[(N[(x / a), $MachinePrecision] * N[(N[(y * N[(z * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -8.2000000000000004e-109

   1. Initial program 58.2%

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

      1. associate-*l* 56.6%

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

      2. associate-*r/ 61.4%

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

      3. *-commutative 61.4%

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

      4. associate-/l* 62.8%

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

   3. Simplified 62.8%

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

   4. Taylor expanded in z around -inf 83.8%

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

      1. associate-*r* 83.8%

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

      2. neg-mul-1 83.8%

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

      3. *-commutative 83.8%

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

   6. Simplified 83.8%

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

   if -8.2000000000000004e-109 < z < 4.2000000000000001e-161

   1. Initial program 79.3%

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

      1. associate-*l/ 77.4%

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

   3. Simplified 77.4%

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

   4. Taylor expanded in z around -inf 50.4%

      \[\leadsto \frac{x \cdot y}{\color{blue}{-1 \cdot z + 0.5 \cdot \frac{a \cdot t}{z}}} \cdot z \]

   5. Taylor expanded in z around 0 49.2%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{x \cdot \left(y \cdot z\right)}{a \cdot t}\right)} \cdot z \]
   6. Step-by-step derivation

      1. times-frac 50.3%

         \[\leadsto \left(2 \cdot \color{blue}{\left(\frac{x}{a} \cdot \frac{y \cdot z}{t}\right)}\right) \cdot z \]

      2. *-commutative 50.3%

         \[\leadsto \left(2 \cdot \left(\frac{x}{a} \cdot \frac{\color{blue}{z \cdot y}}{t}\right)\right) \cdot z \]

   7. Simplified 50.3%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\frac{x}{a} \cdot \frac{z \cdot y}{t}\right)\right)} \cdot z \]

   8. Taylor expanded in x around 0 47.1%

      \[\leadsto \color{blue}{2 \cdot \frac{x \cdot \left(y \cdot {z}^{2}\right)}{a \cdot t}} \]
   9. Step-by-step derivation

      1. times-frac 51.4%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{x}{a} \cdot \frac{y \cdot {z}^{2}}{t}\right)} \]

      2. unpow2 51.4%

         \[\leadsto 2 \cdot \left(\frac{x}{a} \cdot \frac{y \cdot \color{blue}{\left(z \cdot z\right)}}{t}\right) \]

   10. Simplified 51.4%

       \[\leadsto \color{blue}{2 \cdot \left(\frac{x}{a} \cdot \frac{y \cdot \left(z \cdot z\right)}{t}\right)} \]

   if 4.2000000000000001e-161 < z

   1. Initial program 53.5%

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

      1. associate-*l* 50.3%

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

      2. associate-*r/ 53.1%

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

      3. *-commutative 53.1%

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

      4. associate-/l* 53.5%

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

   3. Simplified 53.5%

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

   4. Taylor expanded in z around inf 82.1%

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

      1. *-commutative 82.1%

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

   6. Simplified 82.1%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{-109}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-161}:\\ \;\;\;\;2 \cdot \left(\frac{x}{a} \cdot \frac{y \cdot \left(z \cdot z\right)}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 16: 76.2% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{-108}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-221}:\\ \;\;\;\;z \cdot \left(2 \cdot \left(\frac{x}{a} \cdot \left(z \cdot \frac{y}{t}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -4e-108)
   (* y (- x))
   (if (<= z 5e-221) (* z (* 2.0 (* (/ x a) (* z (/ y t))))) (* y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4e-108) {
		tmp = y * -x;
	} else if (z <= 5e-221) {
		tmp = z * (2.0 * ((x / a) * (z * (y / t))));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-4d-108)) then
        tmp = y * -x
    else if (z <= 5d-221) then
        tmp = z * (2.0d0 * ((x / a) * (z * (y / t))))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4e-108) {
		tmp = y * -x;
	} else if (z <= 5e-221) {
		tmp = z * (2.0 * ((x / a) * (z * (y / t))));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -4e-108:
		tmp = y * -x
	elif z <= 5e-221:
		tmp = z * (2.0 * ((x / a) * (z * (y / t))))
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -4e-108)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 5e-221)
		tmp = Float64(z * Float64(2.0 * Float64(Float64(x / a) * Float64(z * Float64(y / t)))));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -4e-108)
		tmp = y * -x;
	elseif (z <= 5e-221)
		tmp = z * (2.0 * ((x / a) * (z * (y / t))));
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -4e-108], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 5e-221], N[(z * N[(2.0 * N[(N[(x / a), $MachinePrecision] * N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.00000000000000016e-108

   1. Initial program 58.2%

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

      1. associate-*l* 56.6%

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

      2. associate-*r/ 61.4%

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

      3. *-commutative 61.4%

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

      4. associate-/l* 62.8%

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

   3. Simplified 62.8%

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

   4. Taylor expanded in z around -inf 83.8%

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

      1. associate-*r* 83.8%

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

      2. neg-mul-1 83.8%

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

      3. *-commutative 83.8%

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

   6. Simplified 83.8%

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

   if -4.00000000000000016e-108 < z < 4.99999999999999996e-221

   1. Initial program 83.9%

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

      1. associate-*l/ 81.2%

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

   3. Simplified 81.2%

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

   4. Taylor expanded in z around -inf 54.9%

      \[\leadsto \frac{x \cdot y}{\color{blue}{-1 \cdot z + 0.5 \cdot \frac{a \cdot t}{z}}} \cdot z \]

   5. Taylor expanded in z around 0 54.9%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{x \cdot \left(y \cdot z\right)}{a \cdot t}\right)} \cdot z \]
   6. Step-by-step derivation

      1. times-frac 56.3%

         \[\leadsto \left(2 \cdot \color{blue}{\left(\frac{x}{a} \cdot \frac{y \cdot z}{t}\right)}\right) \cdot z \]

      2. *-commutative 56.3%

         \[\leadsto \left(2 \cdot \left(\frac{x}{a} \cdot \frac{\color{blue}{z \cdot y}}{t}\right)\right) \cdot z \]

   7. Simplified 56.3%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\frac{x}{a} \cdot \frac{z \cdot y}{t}\right)\right)} \cdot z \]

   8. Taylor expanded in x around 0 54.9%

      \[\leadsto \left(2 \cdot \color{blue}{\frac{x \cdot \left(y \cdot z\right)}{a \cdot t}}\right) \cdot z \]
   9. Step-by-step derivation

      1. times-frac 56.3%

         \[\leadsto \left(2 \cdot \color{blue}{\left(\frac{x}{a} \cdot \frac{y \cdot z}{t}\right)}\right) \cdot z \]

      2. *-rgt-identity 56.3%

         \[\leadsto \left(2 \cdot \left(\frac{x}{a} \cdot \frac{\color{blue}{\left(y \cdot z\right) \cdot 1}}{t}\right)\right) \cdot z \]

      3. associate-*r/ 56.3%

         \[\leadsto \left(2 \cdot \left(\frac{x}{a} \cdot \color{blue}{\left(\left(y \cdot z\right) \cdot \frac{1}{t}\right)}\right)\right) \cdot z \]

      4. *-commutative 56.3%

         \[\leadsto \left(2 \cdot \left(\frac{x}{a} \cdot \left(\color{blue}{\left(z \cdot y\right)} \cdot \frac{1}{t}\right)\right)\right) \cdot z \]

      5. associate-*l* 56.2%

         \[\leadsto \left(2 \cdot \left(\frac{x}{a} \cdot \color{blue}{\left(z \cdot \left(y \cdot \frac{1}{t}\right)\right)}\right)\right) \cdot z \]

      6. associate-*r/ 56.2%

         \[\leadsto \left(2 \cdot \left(\frac{x}{a} \cdot \left(z \cdot \color{blue}{\frac{y \cdot 1}{t}}\right)\right)\right) \cdot z \]

      7. *-rgt-identity 56.2%

         \[\leadsto \left(2 \cdot \left(\frac{x}{a} \cdot \left(z \cdot \frac{\color{blue}{y}}{t}\right)\right)\right) \cdot z \]

   10. Simplified 56.2%

       \[\leadsto \left(2 \cdot \color{blue}{\left(\frac{x}{a} \cdot \left(z \cdot \frac{y}{t}\right)\right)}\right) \cdot z \]

   if 4.99999999999999996e-221 < z

   1. Initial program 54.5%

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

      1. associate-*l* 53.1%

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

      2. associate-*r/ 55.0%

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

      3. *-commutative 55.0%

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

      4. associate-/l* 54.6%

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

   3. Simplified 54.6%

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

   4. Taylor expanded in z around inf 77.5%

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

      1. *-commutative 77.5%

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

   6. Simplified 77.5%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{-108}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-221}:\\ \;\;\;\;z \cdot \left(2 \cdot \left(\frac{x}{a} \cdot \left(z \cdot \frac{y}{t}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 17: 76.6% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{-108}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-161}:\\ \;\;\;\;z \cdot \left(2 \cdot \left(\frac{x}{a} \cdot \frac{z \cdot y}{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.85e-108)
   (* y (- x))
   (if (<= z 6e-161) (* z (* 2.0 (* (/ x a) (/ (* z y) t)))) (* y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.85e-108) {
		tmp = y * -x;
	} else if (z <= 6e-161) {
		tmp = z * (2.0 * ((x / a) * ((z * y) / t)));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.85d-108)) then
        tmp = y * -x
    else if (z <= 6d-161) then
        tmp = z * (2.0d0 * ((x / a) * ((z * y) / t)))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.85e-108) {
		tmp = y * -x;
	} else if (z <= 6e-161) {
		tmp = z * (2.0 * ((x / a) * ((z * y) / t)));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.85e-108:
		tmp = y * -x
	elif z <= 6e-161:
		tmp = z * (2.0 * ((x / a) * ((z * y) / t)))
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.85e-108)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 6e-161)
		tmp = Float64(z * Float64(2.0 * Float64(Float64(x / a) * Float64(Float64(z * y) / t))));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.85e-108)
		tmp = y * -x;
	elseif (z <= 6e-161)
		tmp = z * (2.0 * ((x / a) * ((z * y) / t)));
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.85e-108], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 6e-161], N[(z * N[(2.0 * N[(N[(x / a), $MachinePrecision] * N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.85e-108

   1. Initial program 58.2%

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

      1. associate-*l* 56.6%

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

      2. associate-*r/ 61.4%

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

      3. *-commutative 61.4%

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

      4. associate-/l* 62.8%

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

   3. Simplified 62.8%

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

   4. Taylor expanded in z around -inf 83.8%

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

      1. associate-*r* 83.8%

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

      2. neg-mul-1 83.8%

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

      3. *-commutative 83.8%

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

   6. Simplified 83.8%

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

   if -1.85e-108 < z < 5.99999999999999977e-161

   1. Initial program 79.3%

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

      1. associate-*l/ 77.4%

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

   3. Simplified 77.4%

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

   4. Taylor expanded in z around -inf 50.4%

      \[\leadsto \frac{x \cdot y}{\color{blue}{-1 \cdot z + 0.5 \cdot \frac{a \cdot t}{z}}} \cdot z \]

   5. Taylor expanded in z around 0 49.2%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{x \cdot \left(y \cdot z\right)}{a \cdot t}\right)} \cdot z \]
   6. Step-by-step derivation

      1. times-frac 50.3%

         \[\leadsto \left(2 \cdot \color{blue}{\left(\frac{x}{a} \cdot \frac{y \cdot z}{t}\right)}\right) \cdot z \]

      2. *-commutative 50.3%

         \[\leadsto \left(2 \cdot \left(\frac{x}{a} \cdot \frac{\color{blue}{z \cdot y}}{t}\right)\right) \cdot z \]

   7. Simplified 50.3%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\frac{x}{a} \cdot \frac{z \cdot y}{t}\right)\right)} \cdot z \]

   if 5.99999999999999977e-161 < z

   1. Initial program 53.5%

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

      1. associate-*l* 50.3%

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

      2. associate-*r/ 53.1%

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

      3. *-commutative 53.1%

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

      4. associate-/l* 53.5%

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

   3. Simplified 53.5%

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

   4. Taylor expanded in z around inf 82.1%

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

      1. *-commutative 82.1%

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

   6. Simplified 82.1%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{-108}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-161}:\\ \;\;\;\;z \cdot \left(2 \cdot \left(\frac{x}{a} \cdot \frac{z \cdot y}{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 18: 75.0% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{-188}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-92}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -4e-188) (* y (- x)) (if (<= z 6e-92) (* x (/ (* z y) z)) (* y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4e-188) {
		tmp = y * -x;
	} else if (z <= 6e-92) {
		tmp = x * ((z * y) / z);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-4d-188)) then
        tmp = y * -x
    else if (z <= 6d-92) then
        tmp = x * ((z * y) / z)
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4e-188) {
		tmp = y * -x;
	} else if (z <= 6e-92) {
		tmp = x * ((z * y) / z);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -4e-188:
		tmp = y * -x
	elif z <= 6e-92:
		tmp = x * ((z * y) / z)
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -4e-188)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 6e-92)
		tmp = Float64(x * Float64(Float64(z * y) / z));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -4e-188)
		tmp = y * -x;
	elseif (z <= 6e-92)
		tmp = x * ((z * y) / z);
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -4e-188], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 6e-92], N[(x * N[(N[(z * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.9999999999999998e-188

   1. Initial program 61.3%

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

      1. associate-*l* 60.0%

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

      2. associate-*r/ 63.4%

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

      3. *-commutative 63.4%

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

      4. associate-/l* 64.7%

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

   3. Simplified 64.7%

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

   4. Taylor expanded in z around -inf 76.2%

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

      1. associate-*r* 76.2%

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

      2. neg-mul-1 76.2%

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

      3. *-commutative 76.2%

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

   6. Simplified 76.2%

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

   if -3.9999999999999998e-188 < z < 6.00000000000000027e-92

   1. Initial program 83.7%

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

      1. associate-*l* 83.1%

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

      2. associate-*r/ 79.6%

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

   3. Simplified 79.6%

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

   4. Taylor expanded in z around inf 40.1%

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

   if 6.00000000000000027e-92 < z

   1. Initial program 46.9%

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

      1. associate-*l* 45.4%

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

      2. associate-*r/ 48.6%

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

      3. *-commutative 48.6%

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

      4. associate-/l* 50.0%

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

   3. Simplified 50.0%

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

   4. Taylor expanded in z around inf 89.3%

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

      1. *-commutative 89.3%

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

   6. Simplified 89.3%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{-188}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-92}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 19: 73.4% accurate, 18.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2.6 \cdot 10^{-300}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z 2.6e-300) (* y (- x)) (* y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= 2.6e-300) {
		tmp = y * -x;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= 2.6d-300) then
        tmp = y * -x
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= 2.6e-300) {
		tmp = y * -x;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= 2.6e-300:
		tmp = y * -x
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= 2.6e-300)
		tmp = Float64(y * Float64(-x));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= 2.6e-300)
		tmp = y * -x;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, 2.6e-300], N[(y * (-x)), $MachinePrecision], N[(y * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.59999999999999997e-300

   1. Initial program 64.8%

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

      1. associate-*l* 62.8%

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

      2. associate-*r/ 65.2%

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

      3. *-commutative 65.2%

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

      4. associate-/l* 67.0%

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

   3. Simplified 67.0%

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

   4. Taylor expanded in z around -inf 71.4%

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

      1. associate-*r* 71.4%

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

      2. neg-mul-1 71.4%

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

      3. *-commutative 71.4%

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

   6. Simplified 71.4%

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

   if 2.59999999999999997e-300 < z

   1. Initial program 55.7%

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

      1. associate-*l* 55.2%

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

      2. associate-*r/ 56.9%

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

      3. *-commutative 56.9%

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

      4. associate-/l* 56.1%

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

   3. Simplified 56.1%

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

   4. Taylor expanded in z around inf 71.9%

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

      1. *-commutative 71.9%

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

   6. Simplified 71.9%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.6 \cdot 10^{-300}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 20: 43.6% accurate, 37.7× speedup

Math

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

FPCore

(FPCore (x y z t a) :precision binary64 (* y x))

C

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

Fortran

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

Java

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

Python

def code(x, y, z, t, a):
	return y * x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.0%

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

   1. associate-*l* 58.8%

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

   2. associate-*r/ 60.8%

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

   3. *-commutative 60.8%

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

   4. associate-/l* 61.3%

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

3. Simplified 61.3%

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

4. Taylor expanded in z around inf 43.2%

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

   1. *-commutative 43.2%

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

6. Simplified 43.2%

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

7. Final simplification 43.2%

   \[\leadsto y \cdot x \]

Developer target: 89.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z < -3.1921305903852764 \cdot 10^{+46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z < 5.976268120920894 \cdot 10^{+90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (< z -3.1921305903852764e+46)
   (- (* y x))
   (if (< z 5.976268120920894e+90)
     (/ (* x z) (/ (sqrt (- (* z z) (* a t))) y))
     (* y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z < -3.1921305903852764e+46) {
		tmp = -(y * x);
	} else if (z < 5.976268120920894e+90) {
		tmp = (x * z) / (sqrt(((z * z) - (a * t))) / y);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z < (-3.1921305903852764d+46)) then
        tmp = -(y * x)
    else if (z < 5.976268120920894d+90) then
        tmp = (x * z) / (sqrt(((z * z) - (a * t))) / y)
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z < -3.1921305903852764e+46) {
		tmp = -(y * x);
	} else if (z < 5.976268120920894e+90) {
		tmp = (x * z) / (Math.sqrt(((z * z) - (a * t))) / y);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z < -3.1921305903852764e+46:
		tmp = -(y * x)
	elif z < 5.976268120920894e+90:
		tmp = (x * z) / (math.sqrt(((z * z) - (a * t))) / y)
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z < -3.1921305903852764e+46)
		tmp = Float64(-Float64(y * x));
	elseif (z < 5.976268120920894e+90)
		tmp = Float64(Float64(x * z) / Float64(sqrt(Float64(Float64(z * z) - Float64(a * t))) / y));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z < -3.1921305903852764e+46)
		tmp = -(y * x);
	elseif (z < 5.976268120920894e+90)
		tmp = (x * z) / (sqrt(((z * z) - (a * t))) / y);
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Less[z, -3.1921305903852764e+46], (-N[(y * x), $MachinePrecision]), If[Less[z, 5.976268120920894e+90], N[(N[(x * z), $MachinePrecision] / N[(N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Reproduce

herbie shell --seed 2023275 
(FPCore (x y z t a)
  :name "Statistics.Math.RootFinding:ridders from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< z -3.1921305903852764e+46) (- (* y x)) (if (< z 5.976268120920894e+90) (/ (* x z) (/ (sqrt (- (* z z) (* a t))) y)) (* y x)))

  (/ (* (* x y) z) (sqrt (- (* z z) (* t a)))))