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

Percentage Accurate: 60.9% → 89.5%

Time: 22.3s

Alternatives: 16

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: 60.9% 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: 89.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+139}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2.05 \cdot 10^{+108}:\\ \;\;\;\;\frac{z \cdot \left(y \cdot x\right)}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1e+139)
   (* y (- x))
   (if (<= z 2.05e+108) (/ (* 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 <= -1e+139) {
		tmp = y * -x;
	} else if (z <= 2.05e+108) {
		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 <= (-1d+139)) then
        tmp = y * -x
    else if (z <= 2.05d+108) 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 <= -1e+139) {
		tmp = y * -x;
	} else if (z <= 2.05e+108) {
		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 <= -1e+139:
		tmp = y * -x
	elif z <= 2.05e+108:
		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 <= -1e+139)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 2.05e+108)
		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 <= -1e+139)
		tmp = y * -x;
	elseif (z <= 2.05e+108)
		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, -1e+139], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 2.05e+108], 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 < -1.00000000000000003e139

   1. Initial program 17.6%

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

      1. associate-*l* 15.2%

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

      2. associate-*r/ 15.3%

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

      3. *-commutative 15.3%

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

      4. associate-/l* 12.0%

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

   3. Simplified 12.0%

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

   4. Taylor expanded in z around -inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

   6. Simplified 100.0%

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

   if -1.00000000000000003e139 < z < 2.05e108

   1. Initial program 85.5%

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

   if 2.05e108 < z

   1. Initial program 28.6%

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

      1. associate-*l* 28.0%

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

      2. associate-*r/ 30.1%

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

      3. *-commutative 30.1%

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

      4. associate-/l* 25.6%

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

   3. Simplified 25.6%

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

   4. Taylor expanded in z around inf 100.0%

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

      1. *-commutative 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+139}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2.05 \cdot 10^{+108}:\\ \;\;\;\;\frac{z \cdot \left(y \cdot x\right)}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 2: 89.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+148}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{+50}:\\ \;\;\;\;x \cdot \frac{z}{\frac{\sqrt{z \cdot z - t \cdot a}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.3e+148)
   (* y (- x))
   (if (<= z 1.55e+50) (* 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 <= -1.3e+148) {
		tmp = y * -x;
	} else if (z <= 1.55e+50) {
		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 <= (-1.3d+148)) then
        tmp = y * -x
    else if (z <= 1.55d+50) 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 <= -1.3e+148) {
		tmp = y * -x;
	} else if (z <= 1.55e+50) {
		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 <= -1.3e+148:
		tmp = y * -x
	elif z <= 1.55e+50:
		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 <= -1.3e+148)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 1.55e+50)
		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 <= -1.3e+148)
		tmp = y * -x;
	elseif (z <= 1.55e+50)
		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, -1.3e+148], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 1.55e+50], 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 < -1.3e148

   1. Initial program 14.0%

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

      1. associate-*l* 11.5%

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

      2. associate-*r/ 11.6%

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

      3. *-commutative 11.6%

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

      4. associate-/l* 12.4%

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

   3. Simplified 12.4%

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

   4. Taylor expanded in z around -inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

   6. Simplified 100.0%

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

   if -1.3e148 < z < 1.55000000000000001e50

   1. Initial program 84.8%

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

      1. associate-*l* 87.8%

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

      2. associate-*r/ 88.6%

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

      3. *-commutative 88.6%

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

      4. associate-/l* 82.8%

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

   3. Simplified 82.8%

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

   if 1.55000000000000001e50 < z

   1. Initial program 38.5%

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

      1. associate-*l* 38.0%

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

      2. associate-*r/ 39.8%

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

      3. *-commutative 39.8%

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

      4. associate-/l* 35.9%

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

   3. Simplified 35.9%

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

   4. Taylor expanded in z around inf 98.6%

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

      1. *-commutative 98.6%

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

   6. Simplified 98.6%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+148}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{+50}:\\ \;\;\;\;x \cdot \frac{z}{\frac{\sqrt{z \cdot z - t \cdot a}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 3: 88.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{+124}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 9.8 \cdot 10^{+107}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.7e+124)
   (* y (- x))
   (if (<= z 9.8e+107) (/ (* 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 <= -3.7e+124) {
		tmp = y * -x;
	} else if (z <= 9.8e+107) {
		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 <= (-3.7d+124)) then
        tmp = y * -x
    else if (z <= 9.8d+107) 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 <= -3.7e+124) {
		tmp = y * -x;
	} else if (z <= 9.8e+107) {
		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 <= -3.7e+124:
		tmp = y * -x
	elif z <= 9.8e+107:
		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 <= -3.7e+124)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 9.8e+107)
		tmp = Float64(Float64(x * 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 <= -3.7e+124)
		tmp = y * -x;
	elseif (z <= 9.8e+107)
		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, -3.7e+124], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 9.8e+107], N[(N[(x * N[(z * y), $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 < -3.70000000000000008e124

   1. Initial program 20.9%

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

      1. associate-*l* 18.6%

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

      2. associate-*r/ 18.8%

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

      3. *-commutative 18.8%

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

      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 100.0%

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

      1. mul-1-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

   6. Simplified 100.0%

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

   if -3.70000000000000008e124 < z < 9.8000000000000003e107

   1. Initial program 85.3%

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

      1. associate-/l* 85.3%

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

      2. *-commutative 85.3%

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

      3. associate-/l* 85.3%

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

      4. associate-*l* 82.2%

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

   3. Simplified 82.2%

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

   4. Taylor expanded in y around 0 88.3%

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

   if 9.8000000000000003e107 < z

   1. Initial program 28.6%

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

      1. associate-*l* 28.0%

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

      2. associate-*r/ 30.1%

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

      3. *-commutative 30.1%

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

      4. associate-/l* 25.6%

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

   3. Simplified 25.6%

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

   4. Taylor expanded in z around inf 100.0%

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

      1. *-commutative 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{+124}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 9.8 \cdot 10^{+107}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 4: 88.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.6 \cdot 10^{+77}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{+107}:\\ \;\;\;\;\frac{y \cdot \left(z \cdot x\right)}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -7.6e+77)
   (* y (- x))
   (if (<= z 9.2e+107) (/ (* y (* z 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 <= -7.6e+77) {
		tmp = y * -x;
	} else if (z <= 9.2e+107) {
		tmp = (y * (z * 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 <= (-7.6d+77)) then
        tmp = y * -x
    else if (z <= 9.2d+107) then
        tmp = (y * (z * 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 <= -7.6e+77) {
		tmp = y * -x;
	} else if (z <= 9.2e+107) {
		tmp = (y * (z * 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 <= -7.6e+77:
		tmp = y * -x
	elif z <= 9.2e+107:
		tmp = (y * (z * 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 <= -7.6e+77)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 9.2e+107)
		tmp = Float64(Float64(y * Float64(z * 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 <= -7.6e+77)
		tmp = y * -x;
	elseif (z <= 9.2e+107)
		tmp = (y * (z * 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, -7.6e+77], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 9.2e+107], N[(N[(y * N[(z * 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 < -7.6000000000000002e77

   1. Initial program 32.0%

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

      1. associate-*l* 30.0%

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

      2. associate-*r/ 30.2%

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

      3. *-commutative 30.2%

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

      4. associate-/l* 24.2%

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

   3. Simplified 24.2%

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

   4. Taylor expanded in z around -inf 98.4%

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

      1. mul-1-neg 98.4%

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

      2. *-commutative 98.4%

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

      3. distribute-rgt-neg-in 98.4%

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

   6. Simplified 98.4%

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

   if -7.6000000000000002e77 < z < 9.2000000000000001e107

   1. Initial program 84.5%

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

      1. associate-/l* 84.5%

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

      2. *-commutative 84.5%

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

      3. associate-/l* 84.5%

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

      4. associate-*l* 81.9%

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

   3. Simplified 81.9%

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

   if 9.2000000000000001e107 < z

   1. Initial program 28.6%

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

      1. associate-*l* 28.0%

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

      2. associate-*r/ 30.1%

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

      3. *-commutative 30.1%

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

      4. associate-/l* 25.6%

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

   3. Simplified 25.6%

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

   4. Taylor expanded in z around inf 100.0%

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

      1. *-commutative 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.6 \cdot 10^{+77}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{+107}:\\ \;\;\;\;\frac{y \cdot \left(z \cdot x\right)}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 5: 83.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.24 \cdot 10^{-60}:\\ \;\;\;\;\frac{1}{\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right)}{z} + -1}{y \cdot x}}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-113}:\\ \;\;\;\;x \cdot \frac{z}{\frac{\sqrt{t \cdot \left(-a\right)}}{y}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{z + 0.5 \cdot \left(t \cdot \frac{a}{z}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.24e-60)
   (/ 1.0 (/ (+ (/ (* 0.5 (* a (/ t z))) z) -1.0) (* y x)))
   (if (<= z 1.1e-113)
     (* x (/ z (/ (sqrt (* t (- a))) y)))
     (* x (* y (/ z (+ z (* 0.5 (* t (/ a z))))))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.24e-60) {
		tmp = 1.0 / ((((0.5 * (a * (t / z))) / z) + -1.0) / (y * x));
	} else if (z <= 1.1e-113) {
		tmp = x * (z / (sqrt((t * -a)) / y));
	} else {
		tmp = x * (y * (z / (z + (0.5 * (t * (a / z))))));
	}
	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 <= (-1.24d-60)) then
        tmp = 1.0d0 / ((((0.5d0 * (a * (t / z))) / z) + (-1.0d0)) / (y * x))
    else if (z <= 1.1d-113) then
        tmp = x * (z / (sqrt((t * -a)) / y))
    else
        tmp = x * (y * (z / (z + (0.5d0 * (t * (a / z))))))
    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 <= -1.24e-60) {
		tmp = 1.0 / ((((0.5 * (a * (t / z))) / z) + -1.0) / (y * x));
	} else if (z <= 1.1e-113) {
		tmp = x * (z / (Math.sqrt((t * -a)) / y));
	} else {
		tmp = x * (y * (z / (z + (0.5 * (t * (a / z))))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.24e-60:
		tmp = 1.0 / ((((0.5 * (a * (t / z))) / z) + -1.0) / (y * x))
	elif z <= 1.1e-113:
		tmp = x * (z / (math.sqrt((t * -a)) / y))
	else:
		tmp = x * (y * (z / (z + (0.5 * (t * (a / z))))))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.24e-60)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(0.5 * Float64(a * Float64(t / z))) / z) + -1.0) / Float64(y * x)));
	elseif (z <= 1.1e-113)
		tmp = Float64(x * Float64(z / Float64(sqrt(Float64(t * Float64(-a))) / y)));
	else
		tmp = Float64(x * Float64(y * Float64(z / Float64(z + Float64(0.5 * Float64(t * Float64(a / z)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.24e-60)
		tmp = 1.0 / ((((0.5 * (a * (t / z))) / z) + -1.0) / (y * x));
	elseif (z <= 1.1e-113)
		tmp = x * (z / (sqrt((t * -a)) / y));
	else
		tmp = x * (y * (z / (z + (0.5 * (t * (a / z))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.24e-60], N[(1.0 / N[(N[(N[(N[(0.5 * N[(a * N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] + -1.0), $MachinePrecision] / N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.1e-113], N[(x * N[(z / N[(N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * N[(z / N[(z + N[(0.5 * N[(t * N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.23999999999999998e-60

   1. Initial program 48.1%

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

   2. Taylor expanded in z around -inf 81.4%

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

      1. neg-mul-1 81.4%

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

      2. +-commutative 81.4%

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

      3. unsub-neg 81.4%

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

   4. Simplified 81.4%

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

   5. Taylor expanded in x around 0 76.3%

      \[\leadsto \color{blue}{\frac{x \cdot \left(y \cdot z\right)}{0.5 \cdot \frac{a \cdot t}{z} - z}} \]
   6. Step-by-step derivation

      1. associate-/l* 78.5%

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

      2. associate-*r/ 79.8%

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

      3. *-commutative 79.8%

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

   7. Simplified 79.8%

      \[\leadsto \color{blue}{\frac{x}{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right) - z}{z \cdot y}}} \]
   8. Step-by-step derivation

      1. clear-num 78.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right) - z}{z \cdot y}}{x}}} \]

      2. inv-pow 78.9%

         \[\leadsto \color{blue}{{\left(\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right) - z}{z \cdot y}}{x}\right)}^{-1}} \]

      3. associate-/r* 90.7%

         \[\leadsto {\left(\frac{\color{blue}{\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right) - z}{z}}{y}}}{x}\right)}^{-1} \]

      4. div-sub 90.7%

         \[\leadsto {\left(\frac{\frac{\color{blue}{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right)}{z} - \frac{z}{z}}}{y}}{x}\right)}^{-1} \]

      5. pow1 90.7%

         \[\leadsto {\left(\frac{\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right)}{z} - \frac{\color{blue}{{z}^{1}}}{z}}{y}}{x}\right)}^{-1} \]

      6. pow1 90.7%

         \[\leadsto {\left(\frac{\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right)}{z} - \frac{{z}^{1}}{\color{blue}{{z}^{1}}}}{y}}{x}\right)}^{-1} \]

      7. pow-div 90.7%

         \[\leadsto {\left(\frac{\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right)}{z} - \color{blue}{{z}^{\left(1 - 1\right)}}}{y}}{x}\right)}^{-1} \]

      8. metadata-eval 90.7%

         \[\leadsto {\left(\frac{\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right)}{z} - {z}^{\color{blue}{0}}}{y}}{x}\right)}^{-1} \]

      9. metadata-eval 90.7%

         \[\leadsto {\left(\frac{\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right)}{z} - \color{blue}{1}}{y}}{x}\right)}^{-1} \]

   9. Applied egg-rr 90.7%

      \[\leadsto \color{blue}{{\left(\frac{\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right)}{z} - 1}{y}}{x}\right)}^{-1}} \]
   10. Step-by-step derivation

       1. unpow-1 90.7%

          \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right)}{z} - 1}{y}}{x}}} \]

       2. associate-/l/ 92.1%

          \[\leadsto \frac{1}{\color{blue}{\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right)}{z} - 1}{x \cdot y}}} \]

       3. sub-neg 92.1%

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

       4. metadata-eval 92.1%

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

   11. Simplified 92.1%

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

   if -1.23999999999999998e-60 < z < 1.10000000000000002e-113

   1. Initial program 77.1%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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/ 83.1%

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

      3. *-commutative 83.1%

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

      4. associate-/l* 77.1%

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

   3. Simplified 77.1%

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

   4. Taylor expanded in z around 0 72.0%

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

      1. mul-1-neg 72.0%

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

      2. *-commutative 72.0%

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

      3. distribute-rgt-neg-in 72.0%

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

   6. Simplified 72.0%

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

   if 1.10000000000000002e-113 < z

   1. Initial program 57.2%

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

      1. associate-*l* 55.9%

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

      2. associate-*r/ 58.1%

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

      3. *-commutative 58.1%

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

      4. associate-/l* 55.5%

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

   3. Simplified 55.5%

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

   4. Taylor expanded in z around -inf 19.4%

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

      1. fma-def 19.4%

         \[\leadsto x \cdot \frac{z}{\frac{\color{blue}{\mathsf{fma}\left(-1, z, 0.5 \cdot \frac{a \cdot t}{z}\right)}}{y}} \]

      2. associate-/l* 19.4%

         \[\leadsto x \cdot \frac{z}{\frac{\mathsf{fma}\left(-1, z, 0.5 \cdot \color{blue}{\frac{a}{\frac{z}{t}}}\right)}{y}} \]

   6. Simplified 19.4%

      \[\leadsto x \cdot \frac{z}{\frac{\color{blue}{\mathsf{fma}\left(-1, z, 0.5 \cdot \frac{a}{\frac{z}{t}}\right)}}{y}} \]
   7. Step-by-step derivation

      1. associate-/r/ 19.1%

         \[\leadsto x \cdot \color{blue}{\left(\frac{z}{\mathsf{fma}\left(-1, z, 0.5 \cdot \frac{a}{\frac{z}{t}}\right)} \cdot y\right)} \]

      2. fma-udef 19.1%

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

      3. neg-mul-1 19.1%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 49.7%

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

      6. sqr-neg 49.7%

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

      7. sqrt-prod 89.6%

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

      8. add-sqr-sqrt 90.0%

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

      9. associate-/r/ 90.0%

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

   8. Applied egg-rr 90.0%

      \[\leadsto x \cdot \color{blue}{\left(\frac{z}{z + 0.5 \cdot \left(\frac{a}{z} \cdot t\right)} \cdot y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.24 \cdot 10^{-60}:\\ \;\;\;\;\frac{1}{\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right)}{z} + -1}{y \cdot x}}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-113}:\\ \;\;\;\;x \cdot \frac{z}{\frac{\sqrt{t \cdot \left(-a\right)}}{y}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{z + 0.5 \cdot \left(t \cdot \frac{a}{z}\right)}\right)\\ \end{array} \]

Alternative 6: 83.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.4 \cdot 10^{-60}:\\ \;\;\;\;\frac{1}{\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right)}{z} + -1}{y \cdot x}}\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{-114}:\\ \;\;\;\;\left(z \cdot x\right) \cdot \frac{y}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{z + 0.5 \cdot \left(t \cdot \frac{a}{z}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -6.4e-60)
   (/ 1.0 (/ (+ (/ (* 0.5 (* a (/ t z))) z) -1.0) (* y x)))
   (if (<= z 6.4e-114)
     (* (* z x) (/ y (sqrt (* t (- a)))))
     (* x (* y (/ z (+ z (* 0.5 (* t (/ a z))))))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -6.4e-60) {
		tmp = 1.0 / ((((0.5 * (a * (t / z))) / z) + -1.0) / (y * x));
	} else if (z <= 6.4e-114) {
		tmp = (z * x) * (y / sqrt((t * -a)));
	} else {
		tmp = x * (y * (z / (z + (0.5 * (t * (a / z))))));
	}
	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 <= (-6.4d-60)) then
        tmp = 1.0d0 / ((((0.5d0 * (a * (t / z))) / z) + (-1.0d0)) / (y * x))
    else if (z <= 6.4d-114) then
        tmp = (z * x) * (y / sqrt((t * -a)))
    else
        tmp = x * (y * (z / (z + (0.5d0 * (t * (a / z))))))
    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 <= -6.4e-60) {
		tmp = 1.0 / ((((0.5 * (a * (t / z))) / z) + -1.0) / (y * x));
	} else if (z <= 6.4e-114) {
		tmp = (z * x) * (y / Math.sqrt((t * -a)));
	} else {
		tmp = x * (y * (z / (z + (0.5 * (t * (a / z))))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -6.4e-60:
		tmp = 1.0 / ((((0.5 * (a * (t / z))) / z) + -1.0) / (y * x))
	elif z <= 6.4e-114:
		tmp = (z * x) * (y / math.sqrt((t * -a)))
	else:
		tmp = x * (y * (z / (z + (0.5 * (t * (a / z))))))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -6.4e-60)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(0.5 * Float64(a * Float64(t / z))) / z) + -1.0) / Float64(y * x)));
	elseif (z <= 6.4e-114)
		tmp = Float64(Float64(z * x) * Float64(y / sqrt(Float64(t * Float64(-a)))));
	else
		tmp = Float64(x * Float64(y * Float64(z / Float64(z + Float64(0.5 * Float64(t * Float64(a / z)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -6.4e-60)
		tmp = 1.0 / ((((0.5 * (a * (t / z))) / z) + -1.0) / (y * x));
	elseif (z <= 6.4e-114)
		tmp = (z * x) * (y / sqrt((t * -a)));
	else
		tmp = x * (y * (z / (z + (0.5 * (t * (a / z))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -6.4e-60], N[(1.0 / N[(N[(N[(N[(0.5 * N[(a * N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] + -1.0), $MachinePrecision] / N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6.4e-114], N[(N[(z * x), $MachinePrecision] * N[(y / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * N[(z / N[(z + N[(0.5 * N[(t * N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.4000000000000003e-60

   1. Initial program 48.1%

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

   2. Taylor expanded in z around -inf 81.4%

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

      1. neg-mul-1 81.4%

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

      2. +-commutative 81.4%

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

      3. unsub-neg 81.4%

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

   4. Simplified 81.4%

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

   5. Taylor expanded in x around 0 76.3%

      \[\leadsto \color{blue}{\frac{x \cdot \left(y \cdot z\right)}{0.5 \cdot \frac{a \cdot t}{z} - z}} \]
   6. Step-by-step derivation

      1. associate-/l* 78.5%

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

      2. associate-*r/ 79.8%

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

      3. *-commutative 79.8%

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

   7. Simplified 79.8%

      \[\leadsto \color{blue}{\frac{x}{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right) - z}{z \cdot y}}} \]
   8. Step-by-step derivation

      1. clear-num 78.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right) - z}{z \cdot y}}{x}}} \]

      2. inv-pow 78.9%

         \[\leadsto \color{blue}{{\left(\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right) - z}{z \cdot y}}{x}\right)}^{-1}} \]

      3. associate-/r* 90.7%

         \[\leadsto {\left(\frac{\color{blue}{\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right) - z}{z}}{y}}}{x}\right)}^{-1} \]

      4. div-sub 90.7%

         \[\leadsto {\left(\frac{\frac{\color{blue}{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right)}{z} - \frac{z}{z}}}{y}}{x}\right)}^{-1} \]

      5. pow1 90.7%

         \[\leadsto {\left(\frac{\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right)}{z} - \frac{\color{blue}{{z}^{1}}}{z}}{y}}{x}\right)}^{-1} \]

      6. pow1 90.7%

         \[\leadsto {\left(\frac{\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right)}{z} - \frac{{z}^{1}}{\color{blue}{{z}^{1}}}}{y}}{x}\right)}^{-1} \]

      7. pow-div 90.7%

         \[\leadsto {\left(\frac{\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right)}{z} - \color{blue}{{z}^{\left(1 - 1\right)}}}{y}}{x}\right)}^{-1} \]

      8. metadata-eval 90.7%

         \[\leadsto {\left(\frac{\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right)}{z} - {z}^{\color{blue}{0}}}{y}}{x}\right)}^{-1} \]

      9. metadata-eval 90.7%

         \[\leadsto {\left(\frac{\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right)}{z} - \color{blue}{1}}{y}}{x}\right)}^{-1} \]

   9. Applied egg-rr 90.7%

      \[\leadsto \color{blue}{{\left(\frac{\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right)}{z} - 1}{y}}{x}\right)}^{-1}} \]
   10. Step-by-step derivation

       1. unpow-1 90.7%

          \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right)}{z} - 1}{y}}{x}}} \]

       2. associate-/l/ 92.1%

          \[\leadsto \frac{1}{\color{blue}{\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right)}{z} - 1}{x \cdot y}}} \]

       3. sub-neg 92.1%

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

       4. metadata-eval 92.1%

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

   11. Simplified 92.1%

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

   if -6.4000000000000003e-60 < z < 6.4000000000000003e-114

   1. Initial program 77.1%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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. *-commutative 83.9%

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

      3. associate-*l* 77.2%

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

      4. associate-*r/ 73.5%

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

      5. *-commutative 73.5%

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

   3. Simplified 73.5%

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

   4. Taylor expanded in z around 0 68.4%

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

      1. mul-1-neg 72.0%

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

      2. *-commutative 72.0%

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

      3. distribute-rgt-neg-in 72.0%

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

   6. Simplified 68.4%

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

   if 6.4000000000000003e-114 < z

   1. Initial program 57.2%

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

      1. associate-*l* 55.9%

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

      2. associate-*r/ 58.1%

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

      3. *-commutative 58.1%

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

      4. associate-/l* 55.5%

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

   3. Simplified 55.5%

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

   4. Taylor expanded in z around -inf 19.4%

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

      1. fma-def 19.4%

         \[\leadsto x \cdot \frac{z}{\frac{\color{blue}{\mathsf{fma}\left(-1, z, 0.5 \cdot \frac{a \cdot t}{z}\right)}}{y}} \]

      2. associate-/l* 19.4%

         \[\leadsto x \cdot \frac{z}{\frac{\mathsf{fma}\left(-1, z, 0.5 \cdot \color{blue}{\frac{a}{\frac{z}{t}}}\right)}{y}} \]

   6. Simplified 19.4%

      \[\leadsto x \cdot \frac{z}{\frac{\color{blue}{\mathsf{fma}\left(-1, z, 0.5 \cdot \frac{a}{\frac{z}{t}}\right)}}{y}} \]
   7. Step-by-step derivation

      1. associate-/r/ 19.1%

         \[\leadsto x \cdot \color{blue}{\left(\frac{z}{\mathsf{fma}\left(-1, z, 0.5 \cdot \frac{a}{\frac{z}{t}}\right)} \cdot y\right)} \]

      2. fma-udef 19.1%

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

      3. neg-mul-1 19.1%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 49.7%

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

      6. sqr-neg 49.7%

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

      7. sqrt-prod 89.6%

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

      8. add-sqr-sqrt 90.0%

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

      9. associate-/r/ 90.0%

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

   8. Applied egg-rr 90.0%

      \[\leadsto x \cdot \color{blue}{\left(\frac{z}{z + 0.5 \cdot \left(\frac{a}{z} \cdot t\right)} \cdot y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.4 \cdot 10^{-60}:\\ \;\;\;\;\frac{1}{\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right)}{z} + -1}{y \cdot x}}\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{-114}:\\ \;\;\;\;\left(z \cdot x\right) \cdot \frac{y}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{z + 0.5 \cdot \left(t \cdot \frac{a}{z}\right)}\right)\\ \end{array} \]

Alternative 7: 82.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-54}:\\ \;\;\;\;\frac{1}{\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right)}{z} + -1}{y \cdot x}}\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-120}:\\ \;\;\;\;\frac{y \cdot \left(z \cdot x\right)}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{z + 0.5 \cdot \left(t \cdot \frac{a}{z}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5e-54)
   (/ 1.0 (/ (+ (/ (* 0.5 (* a (/ t z))) z) -1.0) (* y x)))
   (if (<= z 6e-120)
     (/ (* y (* z x)) (sqrt (* t (- a))))
     (* x (* y (/ z (+ z (* 0.5 (* t (/ a z))))))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5e-54) {
		tmp = 1.0 / ((((0.5 * (a * (t / z))) / z) + -1.0) / (y * x));
	} else if (z <= 6e-120) {
		tmp = (y * (z * x)) / sqrt((t * -a));
	} else {
		tmp = x * (y * (z / (z + (0.5 * (t * (a / z))))));
	}
	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 <= (-5d-54)) then
        tmp = 1.0d0 / ((((0.5d0 * (a * (t / z))) / z) + (-1.0d0)) / (y * x))
    else if (z <= 6d-120) then
        tmp = (y * (z * x)) / sqrt((t * -a))
    else
        tmp = x * (y * (z / (z + (0.5d0 * (t * (a / z))))))
    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 <= -5e-54) {
		tmp = 1.0 / ((((0.5 * (a * (t / z))) / z) + -1.0) / (y * x));
	} else if (z <= 6e-120) {
		tmp = (y * (z * x)) / Math.sqrt((t * -a));
	} else {
		tmp = x * (y * (z / (z + (0.5 * (t * (a / z))))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -5e-54:
		tmp = 1.0 / ((((0.5 * (a * (t / z))) / z) + -1.0) / (y * x))
	elif z <= 6e-120:
		tmp = (y * (z * x)) / math.sqrt((t * -a))
	else:
		tmp = x * (y * (z / (z + (0.5 * (t * (a / z))))))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5e-54)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(0.5 * Float64(a * Float64(t / z))) / z) + -1.0) / Float64(y * x)));
	elseif (z <= 6e-120)
		tmp = Float64(Float64(y * Float64(z * x)) / sqrt(Float64(t * Float64(-a))));
	else
		tmp = Float64(x * Float64(y * Float64(z / Float64(z + Float64(0.5 * Float64(t * Float64(a / z)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -5e-54)
		tmp = 1.0 / ((((0.5 * (a * (t / z))) / z) + -1.0) / (y * x));
	elseif (z <= 6e-120)
		tmp = (y * (z * x)) / sqrt((t * -a));
	else
		tmp = x * (y * (z / (z + (0.5 * (t * (a / z))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5e-54], N[(1.0 / N[(N[(N[(N[(0.5 * N[(a * N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] + -1.0), $MachinePrecision] / N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6e-120], N[(N[(y * N[(z * x), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[(y * N[(z / N[(z + N[(0.5 * N[(t * N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.00000000000000015e-54

   1. Initial program 48.1%

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

   2. Taylor expanded in z around -inf 81.4%

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

      1. neg-mul-1 81.4%

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

      2. +-commutative 81.4%

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

      3. unsub-neg 81.4%

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

   4. Simplified 81.4%

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

   5. Taylor expanded in x around 0 76.3%

      \[\leadsto \color{blue}{\frac{x \cdot \left(y \cdot z\right)}{0.5 \cdot \frac{a \cdot t}{z} - z}} \]
   6. Step-by-step derivation

      1. associate-/l* 78.5%

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

      2. associate-*r/ 79.8%

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

      3. *-commutative 79.8%

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

   7. Simplified 79.8%

      \[\leadsto \color{blue}{\frac{x}{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right) - z}{z \cdot y}}} \]
   8. Step-by-step derivation

      1. clear-num 78.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right) - z}{z \cdot y}}{x}}} \]

      2. inv-pow 78.9%

         \[\leadsto \color{blue}{{\left(\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right) - z}{z \cdot y}}{x}\right)}^{-1}} \]

      3. associate-/r* 90.7%

         \[\leadsto {\left(\frac{\color{blue}{\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right) - z}{z}}{y}}}{x}\right)}^{-1} \]

      4. div-sub 90.7%

         \[\leadsto {\left(\frac{\frac{\color{blue}{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right)}{z} - \frac{z}{z}}}{y}}{x}\right)}^{-1} \]

      5. pow1 90.7%

         \[\leadsto {\left(\frac{\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right)}{z} - \frac{\color{blue}{{z}^{1}}}{z}}{y}}{x}\right)}^{-1} \]

      6. pow1 90.7%

         \[\leadsto {\left(\frac{\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right)}{z} - \frac{{z}^{1}}{\color{blue}{{z}^{1}}}}{y}}{x}\right)}^{-1} \]

      7. pow-div 90.7%

         \[\leadsto {\left(\frac{\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right)}{z} - \color{blue}{{z}^{\left(1 - 1\right)}}}{y}}{x}\right)}^{-1} \]

      8. metadata-eval 90.7%

         \[\leadsto {\left(\frac{\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right)}{z} - {z}^{\color{blue}{0}}}{y}}{x}\right)}^{-1} \]

      9. metadata-eval 90.7%

         \[\leadsto {\left(\frac{\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right)}{z} - \color{blue}{1}}{y}}{x}\right)}^{-1} \]

   9. Applied egg-rr 90.7%

      \[\leadsto \color{blue}{{\left(\frac{\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right)}{z} - 1}{y}}{x}\right)}^{-1}} \]
   10. Step-by-step derivation

       1. unpow-1 90.7%

          \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right)}{z} - 1}{y}}{x}}} \]

       2. associate-/l/ 92.1%

          \[\leadsto \frac{1}{\color{blue}{\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right)}{z} - 1}{x \cdot y}}} \]

       3. sub-neg 92.1%

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

       4. metadata-eval 92.1%

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

   11. Simplified 92.1%

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

   if -5.00000000000000015e-54 < z < 6.00000000000000022e-120

   1. Initial program 77.1%

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

      1. associate-/l* 75.9%

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

      2. *-commutative 75.9%

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

      3. associate-/l* 77.1%

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

      4. associate-*l* 77.2%

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

   3. Simplified 77.2%

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

   4. Taylor expanded in z around 0 70.9%

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

      1. mul-1-neg 72.0%

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

      2. *-commutative 72.0%

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

      3. distribute-rgt-neg-in 72.0%

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

   6. Simplified 70.9%

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

   if 6.00000000000000022e-120 < z

   1. Initial program 57.2%

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

      1. associate-*l* 55.9%

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

      2. associate-*r/ 58.1%

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

      3. *-commutative 58.1%

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

      4. associate-/l* 55.5%

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

   3. Simplified 55.5%

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

   4. Taylor expanded in z around -inf 19.4%

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

      1. fma-def 19.4%

         \[\leadsto x \cdot \frac{z}{\frac{\color{blue}{\mathsf{fma}\left(-1, z, 0.5 \cdot \frac{a \cdot t}{z}\right)}}{y}} \]

      2. associate-/l* 19.4%

         \[\leadsto x \cdot \frac{z}{\frac{\mathsf{fma}\left(-1, z, 0.5 \cdot \color{blue}{\frac{a}{\frac{z}{t}}}\right)}{y}} \]

   6. Simplified 19.4%

      \[\leadsto x \cdot \frac{z}{\frac{\color{blue}{\mathsf{fma}\left(-1, z, 0.5 \cdot \frac{a}{\frac{z}{t}}\right)}}{y}} \]
   7. Step-by-step derivation

      1. associate-/r/ 19.1%

         \[\leadsto x \cdot \color{blue}{\left(\frac{z}{\mathsf{fma}\left(-1, z, 0.5 \cdot \frac{a}{\frac{z}{t}}\right)} \cdot y\right)} \]

      2. fma-udef 19.1%

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

      3. neg-mul-1 19.1%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 49.7%

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

      6. sqr-neg 49.7%

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

      7. sqrt-prod 89.6%

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

      8. add-sqr-sqrt 90.0%

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

      9. associate-/r/ 90.0%

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

   8. Applied egg-rr 90.0%

      \[\leadsto x \cdot \color{blue}{\left(\frac{z}{z + 0.5 \cdot \left(\frac{a}{z} \cdot t\right)} \cdot y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-54}:\\ \;\;\;\;\frac{1}{\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right)}{z} + -1}{y \cdot x}}\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-120}:\\ \;\;\;\;\frac{y \cdot \left(z \cdot x\right)}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{z + 0.5 \cdot \left(t \cdot \frac{a}{z}\right)}\right)\\ \end{array} \]

Alternative 8: 78.6% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-292}:\\ \;\;\;\;\frac{1}{\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right)}{z} + -1}{y \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{z + 0.5 \cdot \left(t \cdot \frac{a}{z}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1e-292)
   (/ 1.0 (/ (+ (/ (* 0.5 (* a (/ t z))) z) -1.0) (* y x)))
   (* x (* y (/ z (+ z (* 0.5 (* t (/ a z)))))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1e-292) {
		tmp = 1.0 / ((((0.5 * (a * (t / z))) / z) + -1.0) / (y * x));
	} else {
		tmp = x * (y * (z / (z + (0.5 * (t * (a / z))))));
	}
	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-292)) then
        tmp = 1.0d0 / ((((0.5d0 * (a * (t / z))) / z) + (-1.0d0)) / (y * x))
    else
        tmp = x * (y * (z / (z + (0.5d0 * (t * (a / z))))))
    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-292) {
		tmp = 1.0 / ((((0.5 * (a * (t / z))) / z) + -1.0) / (y * x));
	} else {
		tmp = x * (y * (z / (z + (0.5 * (t * (a / z))))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1e-292:
		tmp = 1.0 / ((((0.5 * (a * (t / z))) / z) + -1.0) / (y * x))
	else:
		tmp = x * (y * (z / (z + (0.5 * (t * (a / z))))))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1e-292)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(0.5 * Float64(a * Float64(t / z))) / z) + -1.0) / Float64(y * x)));
	else
		tmp = Float64(x * Float64(y * Float64(z / Float64(z + Float64(0.5 * Float64(t * Float64(a / z)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1e-292)
		tmp = 1.0 / ((((0.5 * (a * (t / z))) / z) + -1.0) / (y * x));
	else
		tmp = x * (y * (z / (z + (0.5 * (t * (a / z))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1e-292], N[(1.0 / N[(N[(N[(N[(0.5 * N[(a * N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] + -1.0), $MachinePrecision] / N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * N[(z / N[(z + N[(0.5 * N[(t * N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.0000000000000001e-292

   1. Initial program 58.5%

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

   2. Taylor expanded in z around -inf 67.7%

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

      1. neg-mul-1 67.7%

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

      2. +-commutative 67.7%

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

      3. unsub-neg 67.7%

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

   4. Simplified 67.7%

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

   5. Taylor expanded in x around 0 64.2%

      \[\leadsto \color{blue}{\frac{x \cdot \left(y \cdot z\right)}{0.5 \cdot \frac{a \cdot t}{z} - z}} \]
   6. Step-by-step derivation

      1. associate-/l* 65.7%

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

      2. associate-*r/ 66.5%

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

      3. *-commutative 66.5%

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

   7. Simplified 66.5%

      \[\leadsto \color{blue}{\frac{x}{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right) - z}{z \cdot y}}} \]
   8. Step-by-step derivation

      1. clear-num 65.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right) - z}{z \cdot y}}{x}}} \]

      2. inv-pow 65.9%

         \[\leadsto \color{blue}{{\left(\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right) - z}{z \cdot y}}{x}\right)}^{-1}} \]

      3. associate-/r* 73.8%

         \[\leadsto {\left(\frac{\color{blue}{\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right) - z}{z}}{y}}}{x}\right)}^{-1} \]

      4. div-sub 73.8%

         \[\leadsto {\left(\frac{\frac{\color{blue}{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right)}{z} - \frac{z}{z}}}{y}}{x}\right)}^{-1} \]

      5. pow1 73.8%

         \[\leadsto {\left(\frac{\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right)}{z} - \frac{\color{blue}{{z}^{1}}}{z}}{y}}{x}\right)}^{-1} \]

      6. pow1 73.8%

         \[\leadsto {\left(\frac{\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right)}{z} - \frac{{z}^{1}}{\color{blue}{{z}^{1}}}}{y}}{x}\right)}^{-1} \]

      7. pow-div 73.8%

         \[\leadsto {\left(\frac{\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right)}{z} - \color{blue}{{z}^{\left(1 - 1\right)}}}{y}}{x}\right)}^{-1} \]

      8. metadata-eval 73.8%

         \[\leadsto {\left(\frac{\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right)}{z} - {z}^{\color{blue}{0}}}{y}}{x}\right)}^{-1} \]

      9. metadata-eval 73.8%

         \[\leadsto {\left(\frac{\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right)}{z} - \color{blue}{1}}{y}}{x}\right)}^{-1} \]

   9. Applied egg-rr 73.8%

      \[\leadsto \color{blue}{{\left(\frac{\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right)}{z} - 1}{y}}{x}\right)}^{-1}} \]
   10. Step-by-step derivation

       1. unpow-1 73.7%

          \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right)}{z} - 1}{y}}{x}}} \]

       2. associate-/l/ 74.7%

          \[\leadsto \frac{1}{\color{blue}{\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right)}{z} - 1}{x \cdot y}}} \]

       3. sub-neg 74.7%

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

       4. metadata-eval 74.7%

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

   11. Simplified 74.7%

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

   if -1.0000000000000001e-292 < z

   1. Initial program 62.5%

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

      1. associate-*l* 65.0%

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

      2. associate-*r/ 65.4%

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

      3. *-commutative 65.4%

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

      4. associate-/l* 62.2%

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

   3. Simplified 62.2%

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

   4. Taylor expanded in z around -inf 27.5%

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

      1. fma-def 27.5%

         \[\leadsto x \cdot \frac{z}{\frac{\color{blue}{\mathsf{fma}\left(-1, z, 0.5 \cdot \frac{a \cdot t}{z}\right)}}{y}} \]

      2. associate-/l* 27.5%

         \[\leadsto x \cdot \frac{z}{\frac{\mathsf{fma}\left(-1, z, 0.5 \cdot \color{blue}{\frac{a}{\frac{z}{t}}}\right)}{y}} \]

   6. Simplified 27.5%

      \[\leadsto x \cdot \frac{z}{\frac{\color{blue}{\mathsf{fma}\left(-1, z, 0.5 \cdot \frac{a}{\frac{z}{t}}\right)}}{y}} \]
   7. Step-by-step derivation

      1. associate-/r/ 27.3%

         \[\leadsto x \cdot \color{blue}{\left(\frac{z}{\mathsf{fma}\left(-1, z, 0.5 \cdot \frac{a}{\frac{z}{t}}\right)} \cdot y\right)} \]

      2. fma-udef 27.3%

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

      3. neg-mul-1 27.3%

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

      4. add-sqr-sqrt 0.8%

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

      5. sqrt-unprod 49.1%

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

      6. sqr-neg 49.1%

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

      7. sqrt-prod 78.8%

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

      8. add-sqr-sqrt 79.9%

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

      9. associate-/r/ 79.9%

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

   8. Applied egg-rr 79.9%

      \[\leadsto x \cdot \color{blue}{\left(\frac{z}{z + 0.5 \cdot \left(\frac{a}{z} \cdot t\right)} \cdot y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-292}:\\ \;\;\;\;\frac{1}{\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right)}{z} + -1}{y \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{z + 0.5 \cdot \left(t \cdot \frac{a}{z}\right)}\right)\\ \end{array} \]

Alternative 9: 75.9% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{-155}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-209}:\\ \;\;\;\;x \cdot \frac{z}{0.5 \cdot \left(\frac{t}{z} \cdot \frac{a}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -6.5e-155)
   (* y (- x))
   (if (<= z 7.5e-209) (* x (/ z (* 0.5 (* (/ t z) (/ a y))))) (* y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -6.5e-155) {
		tmp = y * -x;
	} else if (z <= 7.5e-209) {
		tmp = x * (z / (0.5 * ((t / z) * (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 <= (-6.5d-155)) then
        tmp = y * -x
    else if (z <= 7.5d-209) then
        tmp = x * (z / (0.5d0 * ((t / z) * (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 <= -6.5e-155) {
		tmp = y * -x;
	} else if (z <= 7.5e-209) {
		tmp = x * (z / (0.5 * ((t / z) * (a / y))));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -6.5e-155:
		tmp = y * -x
	elif z <= 7.5e-209:
		tmp = x * (z / (0.5 * ((t / z) * (a / y))))
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -6.5e-155)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 7.5e-209)
		tmp = Float64(x * Float64(z / Float64(0.5 * Float64(Float64(t / z) * 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 <= -6.5e-155)
		tmp = y * -x;
	elseif (z <= 7.5e-209)
		tmp = x * (z / (0.5 * ((t / z) * (a / y))));
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -6.5e-155], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 7.5e-209], N[(x * N[(z / N[(0.5 * N[(N[(t / z), $MachinePrecision] * N[(a / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.5e-155

   1. Initial program 55.5%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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/ 56.8%

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

      3. *-commutative 56.8%

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

      4. associate-/l* 52.0%

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

   3. Simplified 52.0%

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

   4. Taylor expanded in z around -inf 81.1%

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

      1. mul-1-neg 81.1%

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

      2. *-commutative 81.1%

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

      3. distribute-rgt-neg-in 81.1%

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

   6. Simplified 81.1%

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

   if -6.5e-155 < z < 7.49999999999999965e-209

   1. Initial program 77.7%

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

      1. associate-*l* 83.2%

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

      2. associate-*r/ 80.2%

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

      3. *-commutative 80.2%

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

      4. associate-/l* 73.1%

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

   3. Simplified 73.1%

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

   4. Taylor expanded in z around -inf 45.9%

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

      1. fma-def 45.9%

         \[\leadsto x \cdot \frac{z}{\frac{\color{blue}{\mathsf{fma}\left(-1, z, 0.5 \cdot \frac{a \cdot t}{z}\right)}}{y}} \]

      2. associate-/l* 46.0%

         \[\leadsto x \cdot \frac{z}{\frac{\mathsf{fma}\left(-1, z, 0.5 \cdot \color{blue}{\frac{a}{\frac{z}{t}}}\right)}{y}} \]

   6. Simplified 46.0%

      \[\leadsto x \cdot \frac{z}{\frac{\color{blue}{\mathsf{fma}\left(-1, z, 0.5 \cdot \frac{a}{\frac{z}{t}}\right)}}{y}} \]

   7. Taylor expanded in z around 0 43.9%

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

      1. times-frac 45.7%

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

   9. Simplified 45.7%

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

   if 7.49999999999999965e-209 < z

   1. Initial program 57.5%

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

      1. associate-*l* 58.1%

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

      2. associate-*r/ 60.2%

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

      3. *-commutative 60.2%

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

      4. associate-/l* 57.9%

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

   3. Simplified 57.9%

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

   4. Taylor expanded in z around inf 85.4%

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

      1. *-commutative 85.4%

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

   6. Simplified 85.4%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{-155}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-209}:\\ \;\;\;\;x \cdot \frac{z}{0.5 \cdot \left(\frac{t}{z} \cdot \frac{a}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 10: 77.0% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{-155}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{z + 0.5 \cdot \left(t \cdot \frac{a}{z}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -7.5e-155)
   (* y (- x))
   (* x (* y (/ z (+ z (* 0.5 (* t (/ a z)))))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -7.5e-155) {
		tmp = y * -x;
	} else {
		tmp = x * (y * (z / (z + (0.5 * (t * (a / z))))));
	}
	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 <= (-7.5d-155)) then
        tmp = y * -x
    else
        tmp = x * (y * (z / (z + (0.5d0 * (t * (a / z))))))
    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 <= -7.5e-155) {
		tmp = y * -x;
	} else {
		tmp = x * (y * (z / (z + (0.5 * (t * (a / z))))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -7.5e-155:
		tmp = y * -x
	else:
		tmp = x * (y * (z / (z + (0.5 * (t * (a / z))))))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -7.5e-155)
		tmp = Float64(y * Float64(-x));
	else
		tmp = Float64(x * Float64(y * Float64(z / Float64(z + Float64(0.5 * Float64(t * Float64(a / z)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -7.5e-155)
		tmp = y * -x;
	else
		tmp = x * (y * (z / (z + (0.5 * (t * (a / z))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -7.5e-155], N[(y * (-x)), $MachinePrecision], N[(x * N[(y * N[(z / N[(z + N[(0.5 * N[(t * N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -7.5000000000000006e-155

   1. Initial program 55.5%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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/ 56.8%

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

      3. *-commutative 56.8%

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

      4. associate-/l* 52.0%

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

   3. Simplified 52.0%

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

   4. Taylor expanded in z around -inf 81.1%

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

      1. mul-1-neg 81.1%

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

      2. *-commutative 81.1%

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

      3. distribute-rgt-neg-in 81.1%

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

   6. Simplified 81.1%

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

   if -7.5000000000000006e-155 < z

   1. Initial program 63.8%

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

      1. associate-*l* 65.9%

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

      2. associate-*r/ 66.4%

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

      3. *-commutative 66.4%

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

      4. associate-/l* 62.6%

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

   3. Simplified 62.6%

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

   4. Taylor expanded in z around -inf 28.8%

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

      1. fma-def 28.8%

         \[\leadsto x \cdot \frac{z}{\frac{\color{blue}{\mathsf{fma}\left(-1, z, 0.5 \cdot \frac{a \cdot t}{z}\right)}}{y}} \]

      2. associate-/l* 28.8%

         \[\leadsto x \cdot \frac{z}{\frac{\mathsf{fma}\left(-1, z, 0.5 \cdot \color{blue}{\frac{a}{\frac{z}{t}}}\right)}{y}} \]

   6. Simplified 28.8%

      \[\leadsto x \cdot \frac{z}{\frac{\color{blue}{\mathsf{fma}\left(-1, z, 0.5 \cdot \frac{a}{\frac{z}{t}}\right)}}{y}} \]
   7. Step-by-step derivation

      1. associate-/r/ 28.6%

         \[\leadsto x \cdot \color{blue}{\left(\frac{z}{\mathsf{fma}\left(-1, z, 0.5 \cdot \frac{a}{\frac{z}{t}}\right)} \cdot y\right)} \]

      2. fma-udef 28.6%

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

      3. neg-mul-1 28.6%

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

      4. add-sqr-sqrt 5.8%

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

      5. sqrt-unprod 47.4%

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

      6. sqr-neg 47.4%

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

      7. sqrt-prod 67.8%

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

      8. add-sqr-sqrt 73.9%

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

      9. associate-/r/ 73.9%

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

   8. Applied egg-rr 73.9%

      \[\leadsto x \cdot \color{blue}{\left(\frac{z}{z + 0.5 \cdot \left(\frac{a}{z} \cdot t\right)} \cdot y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{-155}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{z + 0.5 \cdot \left(t \cdot \frac{a}{z}\right)}\right)\\ \end{array} \]

Alternative 11: 78.4% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 5 \cdot 10^{-210}:\\ \;\;\;\;\frac{x}{\frac{-1 + \frac{0.5}{\frac{z}{a \cdot \frac{t}{z}}}}{y}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{z + 0.5 \cdot \left(t \cdot \frac{a}{z}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z 5e-210)
   (/ x (/ (+ -1.0 (/ 0.5 (/ z (* a (/ t z))))) y))
   (* x (* y (/ z (+ z (* 0.5 (* t (/ a z)))))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= 5e-210) {
		tmp = x / ((-1.0 + (0.5 / (z / (a * (t / z))))) / y);
	} else {
		tmp = x * (y * (z / (z + (0.5 * (t * (a / z))))));
	}
	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 <= 5d-210) then
        tmp = x / (((-1.0d0) + (0.5d0 / (z / (a * (t / z))))) / y)
    else
        tmp = x * (y * (z / (z + (0.5d0 * (t * (a / z))))))
    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 <= 5e-210) {
		tmp = x / ((-1.0 + (0.5 / (z / (a * (t / z))))) / y);
	} else {
		tmp = x * (y * (z / (z + (0.5 * (t * (a / z))))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= 5e-210:
		tmp = x / ((-1.0 + (0.5 / (z / (a * (t / z))))) / y)
	else:
		tmp = x * (y * (z / (z + (0.5 * (t * (a / z))))))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= 5e-210)
		tmp = Float64(x / Float64(Float64(-1.0 + Float64(0.5 / Float64(z / Float64(a * Float64(t / z))))) / y));
	else
		tmp = Float64(x * Float64(y * Float64(z / Float64(z + Float64(0.5 * Float64(t * Float64(a / z)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= 5e-210)
		tmp = x / ((-1.0 + (0.5 / (z / (a * (t / z))))) / y);
	else
		tmp = x * (y * (z / (z + (0.5 * (t * (a / z))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, 5e-210], N[(x / N[(N[(-1.0 + N[(0.5 / N[(z / N[(a * N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * N[(z / N[(z + N[(0.5 * N[(t * N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 5.0000000000000002e-210

   1. Initial program 62.6%

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

   2. Taylor expanded in z around -inf 64.9%

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

      1. neg-mul-1 64.9%

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

      2. +-commutative 64.9%

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

      3. unsub-neg 64.9%

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

   4. Simplified 64.9%

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

   5. Taylor expanded in x around 0 62.0%

      \[\leadsto \color{blue}{\frac{x \cdot \left(y \cdot z\right)}{0.5 \cdot \frac{a \cdot t}{z} - z}} \]
   6. Step-by-step derivation

      1. associate-/l* 63.2%

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

      2. associate-*r/ 63.9%

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

      3. *-commutative 63.9%

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

   7. Simplified 63.9%

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

   8. Taylor expanded in y around 0 63.2%

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

      1. associate-*r/ 63.9%

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

      2. rem-square-sqrt 55.4%

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

      3. times-frac 57.3%

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

      4. associate-*l/ 61.8%

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

      5. associate-*r/ 61.7%

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

      6. rem-square-sqrt 70.4%

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

      7. div-sub 70.4%

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

      8. *-lft-identity 70.4%

         \[\leadsto \frac{x}{\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right)}{z} - \frac{\color{blue}{1 \cdot z}}{z}}{y}} \]

      9. associate-*l/ 70.4%

         \[\leadsto \frac{x}{\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right)}{z} - \color{blue}{\frac{1}{z} \cdot z}}{y}} \]

      10. lft-mult-inverse 70.4%

          \[\leadsto \frac{x}{\frac{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right)}{z} - \color{blue}{1}}{y}} \]

      11. sub-neg 70.4%

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

      12. associate-/l* 70.4%

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

      13. metadata-eval 70.4%

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

   10. Simplified 70.4%

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

   if 5.0000000000000002e-210 < z

   1. Initial program 57.9%

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

      1. associate-*l* 58.4%

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

      2. associate-*r/ 60.5%

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

      3. *-commutative 60.5%

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

      4. associate-/l* 58.2%

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

   3. Simplified 58.2%

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

   4. Taylor expanded in z around -inf 21.7%

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

      1. fma-def 21.7%

         \[\leadsto x \cdot \frac{z}{\frac{\color{blue}{\mathsf{fma}\left(-1, z, 0.5 \cdot \frac{a \cdot t}{z}\right)}}{y}} \]

      2. associate-/l* 21.7%

         \[\leadsto x \cdot \frac{z}{\frac{\mathsf{fma}\left(-1, z, 0.5 \cdot \color{blue}{\frac{a}{\frac{z}{t}}}\right)}{y}} \]

   6. Simplified 21.7%

      \[\leadsto x \cdot \frac{z}{\frac{\color{blue}{\mathsf{fma}\left(-1, z, 0.5 \cdot \frac{a}{\frac{z}{t}}\right)}}{y}} \]
   7. Step-by-step derivation

      1. associate-/r/ 21.5%

         \[\leadsto x \cdot \color{blue}{\left(\frac{z}{\mathsf{fma}\left(-1, z, 0.5 \cdot \frac{a}{\frac{z}{t}}\right)} \cdot y\right)} \]

      2. fma-udef 21.5%

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

      3. neg-mul-1 21.5%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 48.5%

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

      6. sqr-neg 48.5%

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

      7. sqrt-prod 86.3%

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

      8. add-sqr-sqrt 86.6%

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

      9. associate-/r/ 86.6%

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

   8. Applied egg-rr 86.6%

      \[\leadsto x \cdot \color{blue}{\left(\frac{z}{z + 0.5 \cdot \left(\frac{a}{z} \cdot t\right)} \cdot y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5 \cdot 10^{-210}:\\ \;\;\;\;\frac{x}{\frac{-1 + \frac{0.5}{\frac{z}{a \cdot \frac{t}{z}}}}{y}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{z + 0.5 \cdot \left(t \cdot \frac{a}{z}\right)}\right)\\ \end{array} \]

Alternative 12: 73.8% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{-292}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-209}:\\ \;\;\;\;\left(z \cdot x\right) \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -6.5e-292)
   (* y (- x))
   (if (<= z 3.6e-209) (* (* z x) (/ y z)) (* y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -6.5e-292) {
		tmp = y * -x;
	} else if (z <= 3.6e-209) {
		tmp = (z * x) * (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 <= (-6.5d-292)) then
        tmp = y * -x
    else if (z <= 3.6d-209) then
        tmp = (z * x) * (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 <= -6.5e-292) {
		tmp = y * -x;
	} else if (z <= 3.6e-209) {
		tmp = (z * x) * (y / z);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -6.5e-292:
		tmp = y * -x
	elif z <= 3.6e-209:
		tmp = (z * x) * (y / z)
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -6.5e-292)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 3.6e-209)
		tmp = Float64(Float64(z * x) * Float64(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 <= -6.5e-292)
		tmp = y * -x;
	elseif (z <= 3.6e-209)
		tmp = (z * x) * (y / z);
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -6.5e-292], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 3.6e-209], N[(N[(z * x), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.4999999999999997e-292

   1. Initial program 58.5%

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

      1. associate-*l* 58.2%

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

      2. associate-*r/ 59.6%

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

      3. *-commutative 59.6%

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

      4. associate-/l* 54.4%

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

   3. Simplified 54.4%

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

   4. Taylor expanded in z around -inf 71.1%

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

      1. mul-1-neg 71.1%

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

      2. *-commutative 71.1%

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

      3. distribute-rgt-neg-in 71.1%

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

   6. Simplified 71.1%

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

   if -6.4999999999999997e-292 < z < 3.60000000000000016e-209

   1. Initial program 82.4%

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

      1. associate-*l* 92.5%

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

      2. *-commutative 92.5%

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

      3. associate-*l* 89.2%

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

      4. associate-*r/ 82.5%

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

      5. *-commutative 82.5%

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

   3. Simplified 82.5%

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

   4. Taylor expanded in z around inf 38.8%

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

   if 3.60000000000000016e-209 < z

   1. Initial program 57.5%

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

      1. associate-*l* 58.1%

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

      2. associate-*r/ 60.2%

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

      3. *-commutative 60.2%

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

      4. associate-/l* 57.9%

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

   3. Simplified 57.9%

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

   4. Taylor expanded in z around inf 85.4%

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

      1. *-commutative 85.4%

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

   6. Simplified 85.4%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{-292}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-209}:\\ \;\;\;\;\left(z \cdot x\right) \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 13: 75.5% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{-263}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-140}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.15e-263)
   (* y (- x))
   (if (<= z 6.5e-140) (/ (* x (* z y)) z) (* y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.15e-263) {
		tmp = y * -x;
	} else if (z <= 6.5e-140) {
		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 <= (-1.15d-263)) then
        tmp = y * -x
    else if (z <= 6.5d-140) 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 <= -1.15e-263) {
		tmp = y * -x;
	} else if (z <= 6.5e-140) {
		tmp = (x * (z * y)) / z;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.15e-263:
		tmp = y * -x
	elif z <= 6.5e-140:
		tmp = (x * (z * y)) / z
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.15e-263)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 6.5e-140)
		tmp = Float64(Float64(x * 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 <= -1.15e-263)
		tmp = y * -x;
	elseif (z <= 6.5e-140)
		tmp = (x * (z * y)) / z;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.15e-263], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 6.5e-140], N[(N[(x * N[(z * y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.15000000000000001e-263

   1. Initial program 57.6%

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

      1. associate-*l* 58.2%

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

      2. associate-*r/ 59.6%

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

      3. *-commutative 59.6%

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

      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 73.3%

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

      1. mul-1-neg 73.3%

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

      2. *-commutative 73.3%

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

      3. distribute-rgt-neg-in 73.3%

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

   6. Simplified 73.3%

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

   if -1.15000000000000001e-263 < z < 6.4999999999999995e-140

   1. Initial program 76.0%

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

      1. associate-/l* 72.2%

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

      2. *-commutative 72.2%

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

      3. associate-/l* 76.0%

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

      4. associate-*l* 78.7%

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

   3. Simplified 78.7%

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

   4. Taylor expanded in y around 0 80.3%

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

   5. Taylor expanded in z around inf 51.1%

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

   if 6.4999999999999995e-140 < z

   1. Initial program 57.5%

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

      1. associate-*l* 58.0%

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

      2. associate-*r/ 60.2%

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

      3. *-commutative 60.2%

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

      4. associate-/l* 57.7%

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

   3. Simplified 57.7%

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

   4. Taylor expanded in z around inf 86.3%

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

      1. *-commutative 86.3%

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

   6. Simplified 86.3%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{-263}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-140}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 14: 73.2% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{-307}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{x}}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.7e-307) (* y (- x)) (/ 1.0 (/ (/ 1.0 x) y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.7e-307) {
		tmp = y * -x;
	} else {
		tmp = 1.0 / ((1.0 / x) / y);
	}
	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 <= (-2.7d-307)) then
        tmp = y * -x
    else
        tmp = 1.0d0 / ((1.0d0 / x) / y)
    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 <= -2.7e-307) {
		tmp = y * -x;
	} else {
		tmp = 1.0 / ((1.0 / x) / y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.7e-307:
		tmp = y * -x
	else:
		tmp = 1.0 / ((1.0 / x) / y)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.7e-307)
		tmp = Float64(y * Float64(-x));
	else
		tmp = Float64(1.0 / Float64(Float64(1.0 / x) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.7e-307)
		tmp = y * -x;
	else
		tmp = 1.0 / ((1.0 / x) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.7e-307], N[(y * (-x)), $MachinePrecision], N[(1.0 / N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.69999999999999985e-307

   1. Initial program 58.7%

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

      1. associate-*l* 59.2%

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

      2. associate-*r/ 60.0%

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

      3. *-commutative 60.0%

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

      4. associate-/l* 54.9%

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

   3. Simplified 54.9%

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

   4. Taylor expanded in z around -inf 69.5%

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

      1. mul-1-neg 69.5%

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

      2. *-commutative 69.5%

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

      3. distribute-rgt-neg-in 69.5%

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

   6. Simplified 69.5%

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

   if -2.69999999999999985e-307 < z

   1. Initial program 62.4%

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

      1. associate-*l* 64.2%

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

      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* 62.0%

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

   3. Simplified 62.0%

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

      1. associate-*r/ 58.0%

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

      2. clear-num 57.9%

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

      3. associate-/r* 60.0%

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

      4. associate-*r* 62.3%

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

      5. *-commutative 62.3%

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

      6. pow2 62.3%

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

      7. *-commutative 62.3%

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

      8. *-commutative 62.3%

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

   5. Applied egg-rr 62.3%

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

   6. Taylor expanded in z around inf 74.2%

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

      1. associate-/r* 74.0%

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

   8. Simplified 74.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{1}{x}}{y}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{-307}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{x}}{y}}\\ \end{array} \]

Alternative 15: 73.4% accurate, 18.6× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5e-310) {
		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 <= (-5d-310)) 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 <= -5e-310) {
		tmp = y * -x;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < -4.999999999999985e-310

   1. Initial program 58.7%

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

      1. associate-*l* 59.2%

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

      2. associate-*r/ 60.0%

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

      3. *-commutative 60.0%

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

      4. associate-/l* 54.9%

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

   3. Simplified 54.9%

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

   4. Taylor expanded in z around -inf 69.5%

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

      1. mul-1-neg 69.5%

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

      2. *-commutative 69.5%

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

      3. distribute-rgt-neg-in 69.5%

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

   6. Simplified 69.5%

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

   if -4.999999999999985e-310 < z

   1. Initial program 62.4%

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

      1. associate-*l* 64.2%

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

      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* 62.0%

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

   3. Simplified 62.0%

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

   4. Taylor expanded in z around inf 73.9%

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

      1. *-commutative 73.9%

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

   6. Simplified 73.9%

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

4. Final simplification 71.8%

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

Alternative 16: 43.0% 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 60.6%

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

   1. associate-*l* 61.8%

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

   2. associate-*r/ 62.7%

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

   3. *-commutative 62.7%

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

   4. associate-/l* 58.5%

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

3. Simplified 58.5%

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

4. Taylor expanded in z around inf 43.9%

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

   1. *-commutative 43.9%

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

6. Simplified 43.9%

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

7. Final simplification 43.9%

   \[\leadsto y \cdot x \]

Developer target: 89.3% 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 2023318 
(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)))))