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

Percentage Accurate: 61.0% → 89.9%

Time: 23.0s

Alternatives: 18

Speedup: 18.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.0% 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: 89.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+72}:\\ \;\;\;\;\frac{x \cdot y}{\mathsf{fma}\left(0.5, \frac{a}{\frac{{z}^{2}}{t}}, -1\right)}\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+37}:\\ \;\;\;\;\frac{z \cdot \left(x \cdot y\right)}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -6e+72)
   (/ (* x y) (fma 0.5 (/ a (/ (pow z 2.0) t)) -1.0))
   (if (<= z 1.35e+37) (/ (* z (* x y)) (sqrt (- (* z z) (* a t)))) (* x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -6e+72) {
		tmp = (x * y) / fma(0.5, (a / (pow(z, 2.0) / t)), -1.0);
	} else if (z <= 1.35e+37) {
		tmp = (z * (x * y)) / sqrt(((z * z) - (a * t)));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -6e+72)
		tmp = Float64(Float64(x * y) / fma(0.5, Float64(a / Float64((z ^ 2.0) / t)), -1.0));
	elseif (z <= 1.35e+37)
		tmp = Float64(Float64(z * Float64(x * y)) / sqrt(Float64(Float64(z * z) - Float64(a * t))));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -6e+72], N[(N[(x * y), $MachinePrecision] / N[(0.5 * N[(a / N[(N[Power[z, 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.35e+37], N[(N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.00000000000000006e72

   1. Initial program 27.7%

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

      1. associate-/l* 36.7%

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

   3. Simplified 36.7%

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

   4. Taylor expanded in z around -inf 92.7%

      \[\leadsto \frac{x \cdot y}{\color{blue}{0.5 \cdot \frac{a \cdot t}{{z}^{2}} - 1}} \]
   5. Step-by-step derivation

      1. fma-neg 92.7%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(0.5, \frac{a \cdot t}{{z}^{2}}, -1\right)}} \]

      2. associate-/l* 96.5%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(0.5, \color{blue}{\frac{a}{\frac{{z}^{2}}{t}}}, -1\right)} \]

      3. metadata-eval 96.5%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(0.5, \frac{a}{\frac{{z}^{2}}{t}}, \color{blue}{-1}\right)} \]

   6. Simplified 96.5%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(0.5, \frac{a}{\frac{{z}^{2}}{t}}, -1\right)}} \]

   if -6.00000000000000006e72 < z < 1.34999999999999993e37

   1. Initial program 91.2%

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

   if 1.34999999999999993e37 < z

   1. Initial program 53.4%

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

      1. associate-*l/ 53.5%

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

      2. *-commutative 53.5%

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

      3. associate-*l/ 52.3%

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

      4. associate-*r* 51.4%

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

      5. sub-neg 51.4%

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

      6. distribute-rgt-neg-out 51.4%

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

      7. +-commutative 51.4%

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

      8. fma-def 51.4%

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

   3. Simplified 51.4%

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

   4. Taylor expanded in t around 0 98.8%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+72}:\\ \;\;\;\;\frac{x \cdot y}{\mathsf{fma}\left(0.5, \frac{a}{\frac{{z}^{2}}{t}}, -1\right)}\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+37}:\\ \;\;\;\;\frac{z \cdot \left(x \cdot y\right)}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 2: 90.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+83}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+37}:\\ \;\;\;\;z \cdot \frac{x \cdot y}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2e+83)
   (* y (- x))
   (if (<= z 1.35e+37) (* z (/ (* x y) (sqrt (- (* z z) (* a t))))) (* x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2e+83) {
		tmp = y * -x;
	} else if (z <= 1.35e+37) {
		tmp = z * ((x * y) / sqrt(((z * z) - (a * t))));
	} else {
		tmp = 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 <= (-2d+83)) then
        tmp = y * -x
    else if (z <= 1.35d+37) then
        tmp = z * ((x * y) / sqrt(((z * z) - (a * t))))
    else
        tmp = 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 <= -2e+83) {
		tmp = y * -x;
	} else if (z <= 1.35e+37) {
		tmp = z * ((x * y) / Math.sqrt(((z * z) - (a * t))));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -2e+83:
		tmp = y * -x
	elif z <= 1.35e+37:
		tmp = z * ((x * y) / math.sqrt(((z * z) - (a * t))))
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2e+83)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 1.35e+37)
		tmp = Float64(z * Float64(Float64(x * y) / sqrt(Float64(Float64(z * z) - Float64(a * t)))));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2e+83)
		tmp = y * -x;
	elseif (z <= 1.35e+37)
		tmp = z * ((x * y) / sqrt(((z * z) - (a * t))));
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2e+83], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 1.35e+37], N[(z * N[(N[(x * y), $MachinePrecision] / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.00000000000000006e83

   1. Initial program 26.6%

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

      1. associate-*l/ 30.6%

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

      2. *-commutative 30.6%

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

      3. associate-*l/ 29.0%

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

      4. associate-*r* 24.8%

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

      5. sub-neg 24.8%

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

      6. distribute-rgt-neg-out 24.8%

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

      7. +-commutative 24.8%

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

      8. fma-def 24.8%

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

   3. Simplified 24.8%

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

   4. Taylor expanded in z around -inf 96.4%

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

      1. *-commutative 96.4%

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

      2. mul-1-neg 96.4%

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

      3. *-commutative 96.4%

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

   6. Simplified 96.4%

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

   if -2.00000000000000006e83 < z < 1.34999999999999993e37

   1. Initial program 90.7%

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

      1. associate-*l/ 90.1%

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

   3. Simplified 90.1%

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

   if 1.34999999999999993e37 < z

   1. Initial program 53.4%

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

      1. associate-*l/ 53.5%

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

      2. *-commutative 53.5%

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

      3. associate-*l/ 52.3%

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

      4. associate-*r* 51.4%

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

      5. sub-neg 51.4%

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

      6. distribute-rgt-neg-out 51.4%

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

      7. +-commutative 51.4%

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

      8. fma-def 51.4%

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

   3. Simplified 51.4%

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

   4. Taylor expanded in t around 0 98.8%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+83}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+37}:\\ \;\;\;\;z \cdot \frac{x \cdot y}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 3: 89.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.15 \cdot 10^{+74}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+37}:\\ \;\;\;\;\frac{z \cdot \left(x \cdot y\right)}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.15e+74)
   (* y (- x))
   (if (<= z 1.25e+37) (/ (* z (* x y)) (sqrt (- (* z z) (* a t)))) (* x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.15e+74) {
		tmp = y * -x;
	} else if (z <= 1.25e+37) {
		tmp = (z * (x * y)) / sqrt(((z * z) - (a * t)));
	} else {
		tmp = 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 <= (-2.15d+74)) then
        tmp = y * -x
    else if (z <= 1.25d+37) then
        tmp = (z * (x * y)) / sqrt(((z * z) - (a * t)))
    else
        tmp = 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 <= -2.15e+74) {
		tmp = y * -x;
	} else if (z <= 1.25e+37) {
		tmp = (z * (x * y)) / Math.sqrt(((z * z) - (a * t)));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.15e+74:
		tmp = y * -x
	elif z <= 1.25e+37:
		tmp = (z * (x * y)) / math.sqrt(((z * z) - (a * t)))
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.15e+74)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 1.25e+37)
		tmp = Float64(Float64(z * Float64(x * y)) / sqrt(Float64(Float64(z * z) - Float64(a * t))));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.15e+74)
		tmp = y * -x;
	elseif (z <= 1.25e+37)
		tmp = (z * (x * y)) / sqrt(((z * z) - (a * t)));
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.15e+74], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 1.25e+37], N[(N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.15e74

   1. Initial program 27.7%

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

      1. associate-*l/ 33.2%

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

      2. *-commutative 33.2%

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

      3. associate-*l/ 31.6%

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

      4. associate-*r* 27.5%

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

      5. sub-neg 27.5%

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

      6. distribute-rgt-neg-out 27.5%

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

      7. +-commutative 27.5%

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

      8. fma-def 27.5%

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

   3. Simplified 27.5%

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

   4. Taylor expanded in z around -inf 96.5%

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

      1. *-commutative 96.5%

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

      2. mul-1-neg 96.5%

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

      3. *-commutative 96.5%

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

   6. Simplified 96.5%

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

   if -2.15e74 < z < 1.24999999999999997e37

   1. Initial program 91.2%

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

   if 1.24999999999999997e37 < z

   1. Initial program 53.4%

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

      1. associate-*l/ 53.5%

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

      2. *-commutative 53.5%

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

      3. associate-*l/ 52.3%

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

      4. associate-*r* 51.4%

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

      5. sub-neg 51.4%

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

      6. distribute-rgt-neg-out 51.4%

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

      7. +-commutative 51.4%

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

      8. fma-def 51.4%

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

   3. Simplified 51.4%

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

   4. Taylor expanded in t around 0 98.8%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.15 \cdot 10^{+74}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+37}:\\ \;\;\;\;\frac{z \cdot \left(x \cdot y\right)}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 4: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.3 \cdot 10^{-32}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{+20}:\\ \;\;\;\;{\left(a \cdot \left(-t\right)\right)}^{-0.5} \cdot \left(x \cdot \left(z \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -4.3e-32)
   (* y (- x))
   (if (<= z 8.2e+20) (* (pow (* a (- t)) -0.5) (* x (* z y))) (* x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4.3e-32) {
		tmp = y * -x;
	} else if (z <= 8.2e+20) {
		tmp = pow((a * -t), -0.5) * (x * (z * y));
	} else {
		tmp = 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 <= (-4.3d-32)) then
        tmp = y * -x
    else if (z <= 8.2d+20) then
        tmp = ((a * -t) ** (-0.5d0)) * (x * (z * y))
    else
        tmp = 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 <= -4.3e-32) {
		tmp = y * -x;
	} else if (z <= 8.2e+20) {
		tmp = Math.pow((a * -t), -0.5) * (x * (z * y));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -4.3e-32:
		tmp = y * -x
	elif z <= 8.2e+20:
		tmp = math.pow((a * -t), -0.5) * (x * (z * y))
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -4.3e-32)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 8.2e+20)
		tmp = Float64((Float64(a * Float64(-t)) ^ -0.5) * Float64(x * Float64(z * y)));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -4.3e-32)
		tmp = y * -x;
	elseif (z <= 8.2e+20)
		tmp = ((a * -t) ^ -0.5) * (x * (z * y));
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -4.3e-32], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 8.2e+20], N[(N[Power[N[(a * (-t)), $MachinePrecision], -0.5], $MachinePrecision] * N[(x * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.2999999999999999e-32

   1. Initial program 50.6%

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

      1. associate-*l/ 54.2%

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

      2. *-commutative 54.2%

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

      3. associate-*l/ 53.1%

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

      4. associate-*r* 50.3%

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

      5. sub-neg 50.3%

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

      6. distribute-rgt-neg-out 50.3%

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

      7. +-commutative 50.3%

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

      8. fma-def 50.3%

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

   3. Simplified 50.3%

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

   4. Taylor expanded in z around -inf 93.6%

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

      1. *-commutative 93.6%

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

      2. mul-1-neg 93.6%

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

      3. *-commutative 93.6%

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

   6. Simplified 93.6%

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

   if -4.2999999999999999e-32 < z < 8.2e20

   1. Initial program 89.2%

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

   2. Taylor expanded in x around 0 86.8%

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

      1. div-inv 86.9%

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

      2. *-commutative 86.9%

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

      3. pow1/2 86.9%

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

      4. pow-flip 86.9%

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

      5. pow2 86.9%

         \[\leadsto {\left(\color{blue}{{z}^{2}} - t \cdot a\right)}^{\left(-0.5\right)} \cdot \left(x \cdot \left(y \cdot z\right)\right) \]

      6. metadata-eval 86.9%

         \[\leadsto {\left({z}^{2} - t \cdot a\right)}^{\color{blue}{-0.5}} \cdot \left(x \cdot \left(y \cdot z\right)\right) \]

      7. *-commutative 86.9%

         \[\leadsto {\left({z}^{2} - t \cdot a\right)}^{-0.5} \cdot \left(x \cdot \color{blue}{\left(z \cdot y\right)}\right) \]

   4. Applied egg-rr 86.9%

      \[\leadsto \color{blue}{{\left({z}^{2} - t \cdot a\right)}^{-0.5} \cdot \left(x \cdot \left(z \cdot y\right)\right)} \]

   5. Taylor expanded in z around 0 75.5%

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

      1. associate-*r* 75.5%

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

      2. mul-1-neg 75.5%

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

      3. *-commutative 75.5%

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

   7. Simplified 75.5%

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

   if 8.2e20 < z

   1. Initial program 56.7%

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

      1. associate-*l/ 56.9%

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

      2. *-commutative 56.9%

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

      3. associate-*l/ 55.7%

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

      4. associate-*r* 53.7%

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

      5. sub-neg 53.7%

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

      6. distribute-rgt-neg-out 53.7%

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

      7. +-commutative 53.7%

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

      8. fma-def 53.8%

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

   3. Simplified 53.8%

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

   4. Taylor expanded in t around 0 96.7%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.3 \cdot 10^{-32}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{+20}:\\ \;\;\;\;{\left(a \cdot \left(-t\right)\right)}^{-0.5} \cdot \left(x \cdot \left(z \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 5: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{-31}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 8 \cdot 10^{+20}:\\ \;\;\;\;\frac{y}{\sqrt{a \cdot \left(-t\right)}} \cdot \left(z \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -6.8e-31)
   (* y (- x))
   (if (<= z 8e+20) (* (/ y (sqrt (* a (- t)))) (* z x)) (* x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -6.8e-31) {
		tmp = y * -x;
	} else if (z <= 8e+20) {
		tmp = (y / sqrt((a * -t))) * (z * x);
	} else {
		tmp = 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 <= (-6.8d-31)) then
        tmp = y * -x
    else if (z <= 8d+20) then
        tmp = (y / sqrt((a * -t))) * (z * x)
    else
        tmp = 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 <= -6.8e-31) {
		tmp = y * -x;
	} else if (z <= 8e+20) {
		tmp = (y / Math.sqrt((a * -t))) * (z * x);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -6.8e-31:
		tmp = y * -x
	elif z <= 8e+20:
		tmp = (y / math.sqrt((a * -t))) * (z * x)
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -6.8e-31)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 8e+20)
		tmp = Float64(Float64(y / sqrt(Float64(a * Float64(-t)))) * Float64(z * x));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -6.8e-31)
		tmp = y * -x;
	elseif (z <= 8e+20)
		tmp = (y / sqrt((a * -t))) * (z * x);
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -6.8e-31], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 8e+20], N[(N[(y / N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(z * x), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.8000000000000002e-31

   1. Initial program 50.6%

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

      1. associate-*l/ 54.2%

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

      2. *-commutative 54.2%

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

      3. associate-*l/ 53.1%

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

      4. associate-*r* 50.3%

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

      5. sub-neg 50.3%

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

      6. distribute-rgt-neg-out 50.3%

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

      7. +-commutative 50.3%

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

      8. fma-def 50.3%

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

   3. Simplified 50.3%

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

   4. Taylor expanded in z around -inf 93.6%

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

      1. *-commutative 93.6%

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

      2. mul-1-neg 93.6%

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

      3. *-commutative 93.6%

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

   6. Simplified 93.6%

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

   if -6.8000000000000002e-31 < z < 8e20

   1. Initial program 89.2%

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

      1. associate-*l/ 87.4%

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

      2. *-commutative 87.4%

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

      3. associate-*l/ 87.4%

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

      4. associate-*r* 85.4%

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

      5. sub-neg 85.4%

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

      6. distribute-rgt-neg-out 85.4%

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

      7. +-commutative 85.4%

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

      8. fma-def 84.3%

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

   3. Simplified 84.3%

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

   4. Taylor expanded in t around inf 72.8%

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

      1. *-commutative 72.8%

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

      2. mul-1-neg 72.8%

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

      3. *-commutative 72.8%

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

   6. Simplified 72.8%

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

   if 8e20 < z

   1. Initial program 56.7%

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

      1. associate-*l/ 56.9%

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

      2. *-commutative 56.9%

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

      3. associate-*l/ 55.7%

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

      4. associate-*r* 53.7%

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

      5. sub-neg 53.7%

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

      6. distribute-rgt-neg-out 53.7%

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

      7. +-commutative 53.7%

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

      8. fma-def 53.8%

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

   3. Simplified 53.8%

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

   4. Taylor expanded in t around 0 96.7%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{-31}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 8 \cdot 10^{+20}:\\ \;\;\;\;\frac{y}{\sqrt{a \cdot \left(-t\right)}} \cdot \left(z \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 6: 77.5% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+48}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-172}:\\ \;\;\;\;\frac{y}{z - 0.5 \cdot \frac{t}{\frac{z}{a}}} \cdot \left(z \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.9e+48)
   (* y (- x))
   (if (<= z 4.8e-172)
     (* (/ y (- z (* 0.5 (/ t (/ z a))))) (* z (- x)))
     (* x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.9e+48) {
		tmp = y * -x;
	} else if (z <= 4.8e-172) {
		tmp = (y / (z - (0.5 * (t / (z / a))))) * (z * -x);
	} else {
		tmp = 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 <= (-1.9d+48)) then
        tmp = y * -x
    else if (z <= 4.8d-172) then
        tmp = (y / (z - (0.5d0 * (t / (z / a))))) * (z * -x)
    else
        tmp = 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 <= -1.9e+48) {
		tmp = y * -x;
	} else if (z <= 4.8e-172) {
		tmp = (y / (z - (0.5 * (t / (z / a))))) * (z * -x);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.9e+48:
		tmp = y * -x
	elif z <= 4.8e-172:
		tmp = (y / (z - (0.5 * (t / (z / a))))) * (z * -x)
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.9e+48)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 4.8e-172)
		tmp = Float64(Float64(y / Float64(z - Float64(0.5 * Float64(t / Float64(z / a))))) * Float64(z * Float64(-x)));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.9e+48)
		tmp = y * -x;
	elseif (z <= 4.8e-172)
		tmp = (y / (z - (0.5 * (t / (z / a))))) * (z * -x);
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.9e+48], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 4.8e-172], N[(N[(y / N[(z - N[(0.5 * N[(t / N[(z / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(z * (-x)), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.9e48

   1. Initial program 31.5%

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

      1. associate-*l/ 36.7%

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

      2. *-commutative 36.7%

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

      3. associate-*l/ 35.1%

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

      4. associate-*r* 31.3%

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

      5. sub-neg 31.3%

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

      6. distribute-rgt-neg-out 31.3%

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

      7. +-commutative 31.3%

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

      8. fma-def 31.3%

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

   3. Simplified 31.3%

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

   4. Taylor expanded in z around -inf 96.7%

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

      1. *-commutative 96.7%

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

      2. mul-1-neg 96.7%

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

      3. *-commutative 96.7%

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

   6. Simplified 96.7%

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

   if -1.9e48 < z < 4.8000000000000002e-172

   1. Initial program 89.1%

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

      1. associate-*l/ 89.3%

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

      2. *-commutative 89.3%

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

      3. associate-*l/ 89.4%

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

      4. associate-*r* 84.5%

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

      5. sub-neg 84.5%

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

      6. distribute-rgt-neg-out 84.5%

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

      7. +-commutative 84.5%

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

      8. fma-def 83.4%

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

   3. Simplified 83.4%

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

   4. Taylor expanded in z around -inf 55.4%

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

      1. *-commutative 55.4%

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

      2. div-inv 55.4%

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

      3. associate-*l* 55.4%

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

      4. frac-2neg 55.4%

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

      5. metadata-eval 55.4%

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

      6. mul-1-neg 55.4%

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

      7. add-sqr-sqrt 45.0%

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

      8. sqrt-unprod 53.3%

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

      9. mul-1-neg 53.3%

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

      10. mul-1-neg 53.3%

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

      11. sqr-neg 53.3%

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

      12. sqrt-prod 11.6%

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

      13. add-sqr-sqrt 54.4%

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

   6. Applied egg-rr 54.4%

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

      1. frac-2neg 54.4%

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

      2. neg-sub0 54.4%

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

      3. div-sub 54.4%

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

   8. Applied egg-rr 55.4%

      \[\leadsto \color{blue}{\left(\frac{0}{z - \frac{a}{z} \cdot \left(t \cdot 0.5\right)} - \frac{y}{z - \frac{a}{z} \cdot \left(t \cdot 0.5\right)}\right)} \cdot \left(x \cdot z\right) \]
   9. Step-by-step derivation

      1. div0 55.4%

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

      2. neg-sub0 55.4%

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

      3. distribute-neg-frac 55.4%

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

      4. associate-*r* 55.4%

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

      5. *-commutative 55.4%

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

      6. associate-*r/ 55.4%

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

      7. associate-/l* 55.4%

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

   10. Simplified 55.4%

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

   if 4.8000000000000002e-172 < z

   1. Initial program 65.5%

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

      1. associate-*l/ 63.9%

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

      2. *-commutative 63.9%

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

      3. associate-*l/ 63.0%

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

      4. associate-*r* 63.9%

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

      5. sub-neg 63.9%

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

      6. distribute-rgt-neg-out 63.9%

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

      7. +-commutative 63.9%

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

      8. fma-def 63.9%

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

   3. Simplified 63.9%

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

   4. Taylor expanded in t around 0 86.9%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+48}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-172}:\\ \;\;\;\;\frac{y}{z - 0.5 \cdot \frac{t}{\frac{z}{a}}} \cdot \left(z \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 7: 76.3% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{-91}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-58}:\\ \;\;\;\;\left(z \cdot x\right) \cdot \frac{y}{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3e-91)
   (* y (- x))
   (if (<= z 2.2e-58) (* (* z x) (/ y (+ z (* -0.5 (/ a (/ z t)))))) (* x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3e-91) {
		tmp = y * -x;
	} else if (z <= 2.2e-58) {
		tmp = (z * x) * (y / (z + (-0.5 * (a / (z / t)))));
	} else {
		tmp = 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 <= (-3d-91)) then
        tmp = y * -x
    else if (z <= 2.2d-58) then
        tmp = (z * x) * (y / (z + ((-0.5d0) * (a / (z / t)))))
    else
        tmp = 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 <= -3e-91) {
		tmp = y * -x;
	} else if (z <= 2.2e-58) {
		tmp = (z * x) * (y / (z + (-0.5 * (a / (z / t)))));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -3e-91:
		tmp = y * -x
	elif z <= 2.2e-58:
		tmp = (z * x) * (y / (z + (-0.5 * (a / (z / t)))))
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3e-91)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 2.2e-58)
		tmp = Float64(Float64(z * x) * Float64(y / Float64(z + Float64(-0.5 * Float64(a / Float64(z / t))))));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -3e-91)
		tmp = y * -x;
	elseif (z <= 2.2e-58)
		tmp = (z * x) * (y / (z + (-0.5 * (a / (z / t)))));
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3e-91], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 2.2e-58], N[(N[(z * x), $MachinePrecision] * N[(y / N[(z + N[(-0.5 * N[(a / N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.0000000000000002e-91

   1. Initial program 53.4%

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

      1. associate-*l/ 57.1%

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

      2. *-commutative 57.1%

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

      3. associate-*l/ 56.1%

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

      4. associate-*r* 54.0%

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

      5. sub-neg 54.0%

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

      6. distribute-rgt-neg-out 54.0%

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

      7. +-commutative 54.0%

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

      8. fma-def 54.0%

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

   3. Simplified 54.0%

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

   4. Taylor expanded in z around -inf 85.4%

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

      1. *-commutative 85.4%

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

      2. mul-1-neg 85.4%

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

      3. *-commutative 85.4%

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

   6. Simplified 85.4%

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

   if -3.0000000000000002e-91 < z < 2.20000000000000006e-58

   1. Initial program 90.5%

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

      1. associate-*l/ 87.4%

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

      2. *-commutative 87.4%

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

      3. associate-*l/ 87.4%

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

      4. associate-*r* 84.5%

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

      5. sub-neg 84.5%

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

      6. distribute-rgt-neg-out 84.5%

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

      7. +-commutative 84.5%

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

      8. fma-def 83.0%

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

   3. Simplified 83.0%

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

   4. Taylor expanded in t around 0 48.0%

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

      1. associate-/l* 48.5%

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

   6. Simplified 48.5%

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

   if 2.20000000000000006e-58 < z

   1. Initial program 61.7%

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

      1. associate-*l/ 61.8%

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

      2. *-commutative 61.8%

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

      3. associate-*l/ 60.9%

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

      4. associate-*r* 58.9%

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

      5. sub-neg 58.9%

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

      6. distribute-rgt-neg-out 58.9%

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

      7. +-commutative 58.9%

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

      8. fma-def 59.0%

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

   3. Simplified 59.0%

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

   4. Taylor expanded in t around 0 91.1%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{-91}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-58}:\\ \;\;\;\;\left(z \cdot x\right) \cdot \frac{y}{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 8: 76.5% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{-91}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{+21}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{z + -0.5 \cdot \left(t \cdot \frac{a}{z}\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3e-91)
   (* y (- x))
   (if (<= z 3.7e+21) (/ (* x (* z y)) (+ z (* -0.5 (* t (/ a z))))) (* x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3e-91) {
		tmp = y * -x;
	} else if (z <= 3.7e+21) {
		tmp = (x * (z * y)) / (z + (-0.5 * (t * (a / z))));
	} else {
		tmp = 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 <= (-3d-91)) then
        tmp = y * -x
    else if (z <= 3.7d+21) then
        tmp = (x * (z * y)) / (z + ((-0.5d0) * (t * (a / z))))
    else
        tmp = 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 <= -3e-91) {
		tmp = y * -x;
	} else if (z <= 3.7e+21) {
		tmp = (x * (z * y)) / (z + (-0.5 * (t * (a / z))));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -3e-91:
		tmp = y * -x
	elif z <= 3.7e+21:
		tmp = (x * (z * y)) / (z + (-0.5 * (t * (a / z))))
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3e-91)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 3.7e+21)
		tmp = Float64(Float64(x * Float64(z * y)) / Float64(z + Float64(-0.5 * Float64(t * Float64(a / z)))));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -3e-91)
		tmp = y * -x;
	elseif (z <= 3.7e+21)
		tmp = (x * (z * y)) / (z + (-0.5 * (t * (a / z))));
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3e-91], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 3.7e+21], N[(N[(x * N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(z + N[(-0.5 * N[(t * N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.0000000000000002e-91

   1. Initial program 53.4%

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

      1. associate-*l/ 57.1%

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

      2. *-commutative 57.1%

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

      3. associate-*l/ 56.1%

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

      4. associate-*r* 54.0%

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

      5. sub-neg 54.0%

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

      6. distribute-rgt-neg-out 54.0%

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

      7. +-commutative 54.0%

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

      8. fma-def 54.0%

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

   3. Simplified 54.0%

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

   4. Taylor expanded in z around -inf 85.4%

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

      1. *-commutative 85.4%

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

      2. mul-1-neg 85.4%

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

      3. *-commutative 85.4%

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

   6. Simplified 85.4%

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

   if -3.0000000000000002e-91 < z < 3.7e21

   1. Initial program 91.7%

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

   2. Taylor expanded in x around 0 89.0%

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

   3. Taylor expanded in z around inf 47.0%

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

      1. associate-*l/ 47.4%

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

      2. *-commutative 47.4%

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

   5. Simplified 47.4%

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

   if 3.7e21 < z

   1. Initial program 56.7%

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

      1. associate-*l/ 56.9%

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

      2. *-commutative 56.9%

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

      3. associate-*l/ 55.7%

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

      4. associate-*r* 53.7%

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

      5. sub-neg 53.7%

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

      6. distribute-rgt-neg-out 53.7%

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

      7. +-commutative 53.7%

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

      8. fma-def 53.8%

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

   3. Simplified 53.8%

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

   4. Taylor expanded in t around 0 96.7%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{-91}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{+21}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{z + -0.5 \cdot \left(t \cdot \frac{a}{z}\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 9: 77.6% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.22 \cdot 10^{+48}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{-177}:\\ \;\;\;\;\left(z \cdot x\right) \cdot \frac{y}{0.5 \cdot \frac{a \cdot t}{z} - z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.22e+48)
   (* y (- x))
   (if (<= z 3.7e-177) (* (* z x) (/ y (- (* 0.5 (/ (* a t) z)) z))) (* x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.22e+48) {
		tmp = y * -x;
	} else if (z <= 3.7e-177) {
		tmp = (z * x) * (y / ((0.5 * ((a * t) / z)) - z));
	} else {
		tmp = 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 <= (-1.22d+48)) then
        tmp = y * -x
    else if (z <= 3.7d-177) then
        tmp = (z * x) * (y / ((0.5d0 * ((a * t) / z)) - z))
    else
        tmp = 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 <= -1.22e+48) {
		tmp = y * -x;
	} else if (z <= 3.7e-177) {
		tmp = (z * x) * (y / ((0.5 * ((a * t) / z)) - z));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.22e+48:
		tmp = y * -x
	elif z <= 3.7e-177:
		tmp = (z * x) * (y / ((0.5 * ((a * t) / z)) - z))
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.22e+48)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 3.7e-177)
		tmp = Float64(Float64(z * x) * Float64(y / Float64(Float64(0.5 * Float64(Float64(a * t) / z)) - z)));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.22e+48)
		tmp = y * -x;
	elseif (z <= 3.7e-177)
		tmp = (z * x) * (y / ((0.5 * ((a * t) / z)) - z));
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.22e+48], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 3.7e-177], N[(N[(z * x), $MachinePrecision] * N[(y / N[(N[(0.5 * N[(N[(a * t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.22000000000000004e48

   1. Initial program 31.5%

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

      1. associate-*l/ 36.7%

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

      2. *-commutative 36.7%

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

      3. associate-*l/ 35.1%

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

      4. associate-*r* 31.3%

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

      5. sub-neg 31.3%

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

      6. distribute-rgt-neg-out 31.3%

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

      7. +-commutative 31.3%

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

      8. fma-def 31.3%

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

   3. Simplified 31.3%

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

   4. Taylor expanded in z around -inf 96.7%

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

      1. *-commutative 96.7%

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

      2. mul-1-neg 96.7%

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

      3. *-commutative 96.7%

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

   6. Simplified 96.7%

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

   if -1.22000000000000004e48 < z < 3.69999999999999993e-177

   1. Initial program 89.1%

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

      1. associate-*l/ 89.3%

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

      2. *-commutative 89.3%

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

      3. associate-*l/ 89.4%

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

      4. associate-*r* 84.5%

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

      5. sub-neg 84.5%

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

      6. distribute-rgt-neg-out 84.5%

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

      7. +-commutative 84.5%

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

      8. fma-def 83.4%

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

   3. Simplified 83.4%

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

   4. Taylor expanded in z around -inf 55.4%

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

   if 3.69999999999999993e-177 < z

   1. Initial program 65.5%

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

      1. associate-*l/ 63.9%

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

      2. *-commutative 63.9%

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

      3. associate-*l/ 63.0%

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

      4. associate-*r* 63.9%

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

      5. sub-neg 63.9%

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

      6. distribute-rgt-neg-out 63.9%

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

      7. +-commutative 63.9%

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

      8. fma-def 63.9%

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

   3. Simplified 63.9%

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

   4. Taylor expanded in t around 0 86.9%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.22 \cdot 10^{+48}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{-177}:\\ \;\;\;\;\left(z \cdot x\right) \cdot \frac{y}{0.5 \cdot \frac{a \cdot t}{z} - z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 10: 77.6% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.25 \cdot 10^{+52}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 7.3 \cdot 10^{-177}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{0.5 \cdot \frac{a \cdot t}{z} - z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.25e+52)
   (* y (- x))
   (if (<= z 7.3e-177) (/ (* x (* z y)) (- (* 0.5 (/ (* a t) z)) z)) (* x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.25e+52) {
		tmp = y * -x;
	} else if (z <= 7.3e-177) {
		tmp = (x * (z * y)) / ((0.5 * ((a * t) / z)) - z);
	} else {
		tmp = 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 <= (-2.25d+52)) then
        tmp = y * -x
    else if (z <= 7.3d-177) then
        tmp = (x * (z * y)) / ((0.5d0 * ((a * t) / z)) - z)
    else
        tmp = 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 <= -2.25e+52) {
		tmp = y * -x;
	} else if (z <= 7.3e-177) {
		tmp = (x * (z * y)) / ((0.5 * ((a * t) / z)) - z);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.25e+52:
		tmp = y * -x
	elif z <= 7.3e-177:
		tmp = (x * (z * y)) / ((0.5 * ((a * t) / z)) - z)
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.25e+52)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 7.3e-177)
		tmp = Float64(Float64(x * Float64(z * y)) / Float64(Float64(0.5 * Float64(Float64(a * t) / z)) - z));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.25e+52)
		tmp = y * -x;
	elseif (z <= 7.3e-177)
		tmp = (x * (z * y)) / ((0.5 * ((a * t) / z)) - z);
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.25e+52], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 7.3e-177], N[(N[(x * N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 * N[(N[(a * t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.25e52

   1. Initial program 31.5%

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

      1. associate-*l/ 36.7%

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

      2. *-commutative 36.7%

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

      3. associate-*l/ 35.1%

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

      4. associate-*r* 31.3%

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

      5. sub-neg 31.3%

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

      6. distribute-rgt-neg-out 31.3%

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

      7. +-commutative 31.3%

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

      8. fma-def 31.3%

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

   3. Simplified 31.3%

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

   4. Taylor expanded in z around -inf 96.7%

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

      1. *-commutative 96.7%

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

      2. mul-1-neg 96.7%

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

      3. *-commutative 96.7%

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

   6. Simplified 96.7%

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

   if -2.25e52 < z < 7.2999999999999998e-177

   1. Initial program 89.1%

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

   2. Taylor expanded in x around 0 85.9%

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

   3. Taylor expanded in z around -inf 55.6%

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

   if 7.2999999999999998e-177 < z

   1. Initial program 65.5%

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

      1. associate-*l/ 63.9%

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

      2. *-commutative 63.9%

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

      3. associate-*l/ 63.0%

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

      4. associate-*r* 63.9%

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

      5. sub-neg 63.9%

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

      6. distribute-rgt-neg-out 63.9%

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

      7. +-commutative 63.9%

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

      8. fma-def 63.9%

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

   3. Simplified 63.9%

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

   4. Taylor expanded in t around 0 86.9%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.25 \cdot 10^{+52}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 7.3 \cdot 10^{-177}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{0.5 \cdot \frac{a \cdot t}{z} - z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 11: 75.7% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{-91}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-173}:\\ \;\;\;\;\left(z \cdot x\right) \cdot \left(2 \cdot \frac{z}{\frac{a \cdot t}{-y}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.5e-91)
   (* y (- x))
   (if (<= z 9e-173) (* (* z x) (* 2.0 (/ z (/ (* a t) (- y))))) (* x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.5e-91) {
		tmp = y * -x;
	} else if (z <= 9e-173) {
		tmp = (z * x) * (2.0 * (z / ((a * t) / -y)));
	} else {
		tmp = 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 <= (-3.5d-91)) then
        tmp = y * -x
    else if (z <= 9d-173) then
        tmp = (z * x) * (2.0d0 * (z / ((a * t) / -y)))
    else
        tmp = 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 <= -3.5e-91) {
		tmp = y * -x;
	} else if (z <= 9e-173) {
		tmp = (z * x) * (2.0 * (z / ((a * t) / -y)));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -3.5e-91:
		tmp = y * -x
	elif z <= 9e-173:
		tmp = (z * x) * (2.0 * (z / ((a * t) / -y)))
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.5e-91)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 9e-173)
		tmp = Float64(Float64(z * x) * Float64(2.0 * Float64(z / Float64(Float64(a * t) / Float64(-y)))));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -3.5e-91)
		tmp = y * -x;
	elseif (z <= 9e-173)
		tmp = (z * x) * (2.0 * (z / ((a * t) / -y)));
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.5e-91], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 9e-173], N[(N[(z * x), $MachinePrecision] * N[(2.0 * N[(z / N[(N[(a * t), $MachinePrecision] / (-y)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.4999999999999999e-91

   1. Initial program 53.4%

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

      1. associate-*l/ 57.1%

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

      2. *-commutative 57.1%

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

      3. associate-*l/ 56.1%

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

      4. associate-*r* 54.0%

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

      5. sub-neg 54.0%

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

      6. distribute-rgt-neg-out 54.0%

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

      7. +-commutative 54.0%

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

      8. fma-def 54.0%

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

   3. Simplified 54.0%

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

   4. Taylor expanded in z around -inf 85.4%

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

      1. *-commutative 85.4%

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

      2. mul-1-neg 85.4%

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

      3. *-commutative 85.4%

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

   6. Simplified 85.4%

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

   if -3.4999999999999999e-91 < z < 9.00000000000000037e-173

   1. Initial program 89.9%

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

      1. associate-*l/ 89.4%

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

      2. *-commutative 89.4%

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

      3. associate-*l/ 89.5%

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

      4. associate-*r* 81.0%

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

      5. sub-neg 81.0%

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

      6. distribute-rgt-neg-out 81.0%

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

      7. +-commutative 81.0%

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

      8. fma-def 79.1%

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

   3. Simplified 79.1%

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

   4. Taylor expanded in z around -inf 43.7%

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

   5. Taylor expanded in z around 0 42.0%

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

      1. times-frac 41.5%

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

   7. Simplified 41.5%

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

      1. clear-num 41.5%

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

      2. metadata-eval 41.5%

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

      3. frac-2neg 41.5%

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

      4. mul-1-neg 41.5%

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

      5. frac-times 39.8%

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

      6. metadata-eval 39.8%

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

      7. *-un-lft-identity 39.8%

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

      8. add-sqr-sqrt 24.7%

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

      9. sqrt-unprod 37.8%

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

      10. swap-sqr 37.8%

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

      11. metadata-eval 37.8%

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

      12. unpow2 37.8%

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

      13. *-un-lft-identity 37.8%

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

      14. unpow2 37.8%

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

      15. sqrt-prod 17.1%

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

      16. add-sqr-sqrt 41.4%

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

   9. Applied egg-rr 41.4%

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

       1. add-sqr-sqrt 25.4%

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

       2. frac-2neg 25.4%

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

       3. sqrt-unprod 38.7%

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

       4. sqr-neg 38.7%

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

       5. sqrt-unprod 17.9%

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

       6. add-sqr-sqrt 39.8%

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

       7. associate-*l/ 40.2%

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

       8. distribute-lft-neg-in 40.2%

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

       9. distribute-rgt-neg-in 40.2%

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

       10. add-sqr-sqrt 22.2%

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

       11. sqrt-unprod 33.8%

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

       12. sqr-neg 33.8%

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

       13. sqrt-unprod 16.2%

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

       14. add-sqr-sqrt 41.8%

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

   11. Applied egg-rr 41.8%

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

   if 9.00000000000000037e-173 < z

   1. Initial program 65.5%

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

      1. associate-*l/ 63.9%

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

      2. *-commutative 63.9%

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

      3. associate-*l/ 63.0%

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

      4. associate-*r* 63.9%

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

      5. sub-neg 63.9%

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

      6. distribute-rgt-neg-out 63.9%

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

      7. +-commutative 63.9%

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

      8. fma-def 63.9%

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

   3. Simplified 63.9%

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

   4. Taylor expanded in t around 0 86.9%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{-91}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-173}:\\ \;\;\;\;\left(z \cdot x\right) \cdot \left(2 \cdot \frac{z}{\frac{a \cdot t}{-y}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 12: 75.7% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{-66}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-181}:\\ \;\;\;\;z \cdot \left(x \cdot \left(2 \cdot \left(\frac{z}{a} \cdot \frac{y}{t}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.5e-66)
   (* y (- x))
   (if (<= z 2.3e-181) (* z (* x (* 2.0 (* (/ z a) (/ y t))))) (* x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.5e-66) {
		tmp = y * -x;
	} else if (z <= 2.3e-181) {
		tmp = z * (x * (2.0 * ((z / a) * (y / t))));
	} else {
		tmp = 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 <= (-1.5d-66)) then
        tmp = y * -x
    else if (z <= 2.3d-181) then
        tmp = z * (x * (2.0d0 * ((z / a) * (y / t))))
    else
        tmp = 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 <= -1.5e-66) {
		tmp = y * -x;
	} else if (z <= 2.3e-181) {
		tmp = z * (x * (2.0 * ((z / a) * (y / t))));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.5e-66:
		tmp = y * -x
	elif z <= 2.3e-181:
		tmp = z * (x * (2.0 * ((z / a) * (y / t))))
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.5e-66)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 2.3e-181)
		tmp = Float64(z * Float64(x * Float64(2.0 * Float64(Float64(z / a) * Float64(y / t)))));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.5e-66)
		tmp = y * -x;
	elseif (z <= 2.3e-181)
		tmp = z * (x * (2.0 * ((z / a) * (y / t))));
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.5e-66], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 2.3e-181], N[(z * N[(x * N[(2.0 * N[(N[(z / a), $MachinePrecision] * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.5000000000000001e-66

   1. Initial program 52.3%

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

      1. associate-*l/ 55.6%

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

      2. *-commutative 55.6%

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

      3. associate-*l/ 54.6%

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

      4. associate-*r* 53.1%

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

      5. sub-neg 53.1%

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

      6. distribute-rgt-neg-out 53.1%

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

      7. +-commutative 53.1%

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

      8. fma-def 53.1%

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

   3. Simplified 53.1%

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

   4. Taylor expanded in z around -inf 87.0%

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

      1. *-commutative 87.0%

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

      2. mul-1-neg 87.0%

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

      3. *-commutative 87.0%

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

   6. Simplified 87.0%

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

   if -1.5000000000000001e-66 < z < 2.29999999999999991e-181

   1. Initial program 89.6%

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

      1. associate-*l/ 89.9%

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

      2. *-commutative 89.9%

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

      3. associate-*l/ 90.0%

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

      4. associate-*r* 81.0%

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

      5. sub-neg 81.0%

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

      6. distribute-rgt-neg-out 81.0%

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

      7. +-commutative 81.0%

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

      8. fma-def 79.2%

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

   3. Simplified 79.2%

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

   4. Taylor expanded in z around -inf 45.1%

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

   5. Taylor expanded in z around 0 41.9%

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

      1. times-frac 41.4%

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

   7. Simplified 41.4%

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

      1. pow1 41.4%

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

      2. associate-*r* 41.5%

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

      3. *-commutative 41.5%

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

      4. *-commutative 41.5%

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

      5. frac-times 42.0%

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

      6. *-commutative 42.0%

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

      7. times-frac 40.1%

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

   9. Applied egg-rr 40.1%

      \[\leadsto \color{blue}{{\left(z \cdot \left(x \cdot \left(2 \cdot \left(\frac{z}{a} \cdot \frac{y}{t}\right)\right)\right)\right)}^{1}} \]
   10. Step-by-step derivation

       1. unpow1 40.1%

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

   11. Simplified 40.1%

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

   if 2.29999999999999991e-181 < z

   1. Initial program 65.5%

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

      1. associate-*l/ 63.9%

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

      2. *-commutative 63.9%

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

      3. associate-*l/ 63.0%

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

      4. associate-*r* 63.9%

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

      5. sub-neg 63.9%

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

      6. distribute-rgt-neg-out 63.9%

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

      7. +-commutative 63.9%

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

      8. fma-def 63.9%

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

   3. Simplified 63.9%

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

   4. Taylor expanded in t around 0 86.9%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{-66}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-181}:\\ \;\;\;\;z \cdot \left(x \cdot \left(2 \cdot \left(\frac{z}{a} \cdot \frac{y}{t}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 13: 76.0% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{-91}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{-177}:\\ \;\;\;\;\left(z \cdot x\right) \cdot \left(-2 \cdot \frac{z}{t \cdot \frac{a}{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3e-91)
   (* y (- x))
   (if (<= z 1.65e-177) (* (* z x) (* -2.0 (/ z (* t (/ a y))))) (* x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3e-91) {
		tmp = y * -x;
	} else if (z <= 1.65e-177) {
		tmp = (z * x) * (-2.0 * (z / (t * (a / y))));
	} else {
		tmp = 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 <= (-3d-91)) then
        tmp = y * -x
    else if (z <= 1.65d-177) then
        tmp = (z * x) * ((-2.0d0) * (z / (t * (a / y))))
    else
        tmp = 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 <= -3e-91) {
		tmp = y * -x;
	} else if (z <= 1.65e-177) {
		tmp = (z * x) * (-2.0 * (z / (t * (a / y))));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -3e-91:
		tmp = y * -x
	elif z <= 1.65e-177:
		tmp = (z * x) * (-2.0 * (z / (t * (a / y))))
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3e-91)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 1.65e-177)
		tmp = Float64(Float64(z * x) * Float64(-2.0 * Float64(z / Float64(t * Float64(a / y)))));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -3e-91)
		tmp = y * -x;
	elseif (z <= 1.65e-177)
		tmp = (z * x) * (-2.0 * (z / (t * (a / y))));
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3e-91], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 1.65e-177], N[(N[(z * x), $MachinePrecision] * N[(-2.0 * N[(z / N[(t * N[(a / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.0000000000000002e-91

   1. Initial program 53.4%

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

      1. associate-*l/ 57.1%

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

      2. *-commutative 57.1%

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

      3. associate-*l/ 56.1%

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

      4. associate-*r* 54.0%

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

      5. sub-neg 54.0%

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

      6. distribute-rgt-neg-out 54.0%

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

      7. +-commutative 54.0%

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

      8. fma-def 54.0%

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

   3. Simplified 54.0%

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

   4. Taylor expanded in z around -inf 85.4%

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

      1. *-commutative 85.4%

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

      2. mul-1-neg 85.4%

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

      3. *-commutative 85.4%

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

   6. Simplified 85.4%

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

   if -3.0000000000000002e-91 < z < 1.65e-177

   1. Initial program 89.9%

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

      1. associate-*l/ 89.4%

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

      2. *-commutative 89.4%

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

      3. associate-*l/ 89.5%

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

      4. associate-*r* 81.0%

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

      5. sub-neg 81.0%

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

      6. distribute-rgt-neg-out 81.0%

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

      7. +-commutative 81.0%

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

      8. fma-def 79.1%

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

   3. Simplified 79.1%

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

   4. Taylor expanded in z around -inf 43.7%

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

   5. Taylor expanded in z around 0 42.0%

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

      1. times-frac 41.5%

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

   7. Simplified 41.5%

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

      1. clear-num 41.5%

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

      2. metadata-eval 41.5%

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

      3. frac-2neg 41.5%

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

      4. mul-1-neg 41.5%

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

      5. frac-times 39.8%

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

      6. metadata-eval 39.8%

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

      7. *-un-lft-identity 39.8%

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

      8. add-sqr-sqrt 24.7%

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

      9. sqrt-unprod 37.8%

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

      10. swap-sqr 37.8%

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

      11. metadata-eval 37.8%

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

      12. unpow2 37.8%

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

      13. *-un-lft-identity 37.8%

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

      14. unpow2 37.8%

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

      15. sqrt-prod 17.1%

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

      16. add-sqr-sqrt 41.4%

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

   9. Applied egg-rr 41.4%

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

   10. Taylor expanded in z around 0 43.6%

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

       1. *-commutative 43.6%

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

       2. associate-/l* 41.8%

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

       3. *-commutative 41.8%

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

       4. associate-*r/ 41.4%

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

   12. Simplified 41.4%

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

   if 1.65e-177 < z

   1. Initial program 65.5%

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

      1. associate-*l/ 63.9%

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

      2. *-commutative 63.9%

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

      3. associate-*l/ 63.0%

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

      4. associate-*r* 63.9%

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

      5. sub-neg 63.9%

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

      6. distribute-rgt-neg-out 63.9%

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

      7. +-commutative 63.9%

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

      8. fma-def 63.9%

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

   3. Simplified 63.9%

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

   4. Taylor expanded in t around 0 86.9%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{-91}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{-177}:\\ \;\;\;\;\left(z \cdot x\right) \cdot \left(-2 \cdot \frac{z}{t \cdot \frac{a}{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 14: 76.1% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{-58}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(z \cdot \left(-x\right)\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.45e-58)
   (* y (- x))
   (if (<= z 1.05e-239) (/ (* y (* z (- x))) z) (* x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.45e-58) {
		tmp = y * -x;
	} else if (z <= 1.05e-239) {
		tmp = (y * (z * -x)) / z;
	} else {
		tmp = 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 <= (-1.45d-58)) then
        tmp = y * -x
    else if (z <= 1.05d-239) then
        tmp = (y * (z * -x)) / z
    else
        tmp = 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 <= -1.45e-58) {
		tmp = y * -x;
	} else if (z <= 1.05e-239) {
		tmp = (y * (z * -x)) / z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.45e-58:
		tmp = y * -x
	elif z <= 1.05e-239:
		tmp = (y * (z * -x)) / z
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.45e-58)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 1.05e-239)
		tmp = Float64(Float64(y * Float64(z * Float64(-x))) / z);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.45e-58)
		tmp = y * -x;
	elseif (z <= 1.05e-239)
		tmp = (y * (z * -x)) / z;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.45e-58], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 1.05e-239], N[(N[(y * N[(z * (-x)), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.44999999999999995e-58

   1. Initial program 52.9%

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

      1. associate-*l/ 56.3%

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

      2. *-commutative 56.3%

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

      3. associate-*l/ 55.2%

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

      4. associate-*r* 52.6%

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

      5. sub-neg 52.6%

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

      6. distribute-rgt-neg-out 52.6%

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

      7. +-commutative 52.6%

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

      8. fma-def 52.6%

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

   3. Simplified 52.6%

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

   4. Taylor expanded in z around -inf 88.6%

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

      1. *-commutative 88.6%

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

      2. mul-1-neg 88.6%

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

      3. *-commutative 88.6%

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

   6. Simplified 88.6%

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

   if -1.44999999999999995e-58 < z < 1.0500000000000001e-239

   1. Initial program 87.6%

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

      1. associate-*l/ 88.4%

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

      2. *-commutative 88.4%

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

      3. associate-*l/ 88.5%

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

      4. associate-*r* 79.9%

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

      5. sub-neg 79.9%

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

      6. distribute-rgt-neg-out 79.9%

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

      7. +-commutative 79.9%

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

      8. fma-def 77.9%

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

   3. Simplified 77.9%

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

   4. Taylor expanded in t around 0 19.2%

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

      1. *-commutative 19.2%

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

      2. frac-2neg 19.2%

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

      3. mul-1-neg 19.2%

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

      4. associate-*r/ 29.5%

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

      5. *-commutative 29.5%

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

      6. add-sqr-sqrt 19.1%

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

      7. sqrt-unprod 8.6%

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

      8. mul-1-neg 8.6%

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

      9. mul-1-neg 8.6%

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

      10. sqr-neg 8.6%

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

      11. sqrt-prod 10.2%

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

      12. add-sqr-sqrt 38.2%

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

   6. Applied egg-rr 38.2%

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

   if 1.0500000000000001e-239 < z

   1. Initial program 67.3%

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

      1. associate-*l/ 65.6%

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

      2. *-commutative 65.6%

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

      3. associate-*l/ 64.8%

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

      4. associate-*r* 65.6%

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

      5. sub-neg 65.6%

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

      6. distribute-rgt-neg-out 65.6%

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

      7. +-commutative 65.6%

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

      8. fma-def 65.6%

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

   3. Simplified 65.6%

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

   4. Taylor expanded in t around 0 81.2%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{-58}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(z \cdot \left(-x\right)\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 15: 75.0% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.25 \cdot 10^{-215}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-173}:\\ \;\;\;\;\frac{y}{\frac{z}{z \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.25e-215)
   (* y (- x))
   (if (<= z 5e-173) (/ y (/ z (* z x))) (* x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.25e-215) {
		tmp = y * -x;
	} else if (z <= 5e-173) {
		tmp = y / (z / (z * x));
	} else {
		tmp = 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 <= (-3.25d-215)) then
        tmp = y * -x
    else if (z <= 5d-173) then
        tmp = y / (z / (z * x))
    else
        tmp = 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 <= -3.25e-215) {
		tmp = y * -x;
	} else if (z <= 5e-173) {
		tmp = y / (z / (z * x));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -3.25e-215:
		tmp = y * -x
	elif z <= 5e-173:
		tmp = y / (z / (z * x))
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.25e-215)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 5e-173)
		tmp = Float64(y / Float64(z / Float64(z * x)));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -3.25e-215)
		tmp = y * -x;
	elseif (z <= 5e-173)
		tmp = y / (z / (z * x));
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.25e-215], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 5e-173], N[(y / N[(z / N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.25e-215

   1. Initial program 59.6%

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

      1. associate-*l/ 61.9%

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

      2. *-commutative 61.9%

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

      3. associate-*l/ 61.1%

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

      4. associate-*r* 58.4%

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

      5. sub-neg 58.4%

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

      6. distribute-rgt-neg-out 58.4%

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

      7. +-commutative 58.4%

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

      8. fma-def 57.5%

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

   3. Simplified 57.5%

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

   4. Taylor expanded in z around -inf 75.6%

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

      1. *-commutative 75.6%

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

      2. mul-1-neg 75.6%

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

      3. *-commutative 75.6%

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

   6. Simplified 75.6%

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

   if -3.25e-215 < z < 5.0000000000000002e-173

   1. Initial program 88.3%

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

      1. associate-*l/ 89.8%

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

      2. *-commutative 89.8%

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

      3. associate-*l/ 89.8%

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

      4. associate-*r* 80.3%

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

      5. sub-neg 80.3%

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

      6. distribute-rgt-neg-out 80.3%

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

      7. +-commutative 80.3%

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

      8. fma-def 80.2%

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

   3. Simplified 80.2%

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

   4. Taylor expanded in t around 0 23.6%

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

      1. associate-*l/ 37.9%

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

      2. associate-/l* 29.8%

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

      3. *-commutative 29.8%

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

   6. Applied egg-rr 29.8%

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

   if 5.0000000000000002e-173 < z

   1. Initial program 65.5%

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

      1. associate-*l/ 63.9%

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

      2. *-commutative 63.9%

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

      3. associate-*l/ 63.0%

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

      4. associate-*r* 63.9%

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

      5. sub-neg 63.9%

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

      6. distribute-rgt-neg-out 63.9%

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

      7. +-commutative 63.9%

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

      8. fma-def 63.9%

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

   3. Simplified 63.9%

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

   4. Taylor expanded in t around 0 86.9%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.25 \cdot 10^{-215}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-173}:\\ \;\;\;\;\frac{y}{\frac{z}{z \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 16: 75.8% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{-215}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{-112}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.55e-215)
   (* y (- x))
   (if (<= z 1.45e-112) (/ (* x (* z y)) z) (* x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.55e-215) {
		tmp = y * -x;
	} else if (z <= 1.45e-112) {
		tmp = (x * (z * y)) / z;
	} else {
		tmp = 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 <= (-1.55d-215)) then
        tmp = y * -x
    else if (z <= 1.45d-112) then
        tmp = (x * (z * y)) / z
    else
        tmp = 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 <= -1.55e-215) {
		tmp = y * -x;
	} else if (z <= 1.45e-112) {
		tmp = (x * (z * y)) / z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.55e-215:
		tmp = y * -x
	elif z <= 1.45e-112:
		tmp = (x * (z * y)) / z
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.55e-215)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 1.45e-112)
		tmp = Float64(Float64(x * Float64(z * y)) / z);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.55e-215)
		tmp = y * -x;
	elseif (z <= 1.45e-112)
		tmp = (x * (z * y)) / z;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.55e-215], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 1.45e-112], N[(N[(x * N[(z * y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.54999999999999997e-215

   1. Initial program 60.0%

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

      1. associate-*l/ 62.3%

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

      2. *-commutative 62.3%

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

      3. associate-*l/ 61.4%

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

      4. associate-*r* 58.8%

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

      5. sub-neg 58.8%

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

      6. distribute-rgt-neg-out 58.8%

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

      7. +-commutative 58.8%

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

      8. fma-def 57.9%

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

   3. Simplified 57.9%

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

   4. Taylor expanded in z around -inf 74.9%

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

      1. *-commutative 74.9%

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

      2. mul-1-neg 74.9%

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

      3. *-commutative 74.9%

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

   6. Simplified 74.9%

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

   if -1.54999999999999997e-215 < z < 1.44999999999999996e-112

   1. Initial program 87.2%

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

      1. associate-*l/ 84.3%

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

      2. *-commutative 84.3%

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

      3. associate-*l/ 84.4%

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

      4. associate-*r* 82.1%

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

      5. sub-neg 82.1%

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

      6. distribute-rgt-neg-out 82.1%

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

      7. +-commutative 82.1%

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

      8. fma-def 81.9%

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

   3. Simplified 81.9%

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

   4. Taylor expanded in t around 0 28.6%

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

      1. associate-*l/ 43.7%

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

      2. *-commutative 43.7%

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

      3. associate-*r* 47.4%

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

   6. Applied egg-rr 47.4%

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

   if 1.44999999999999996e-112 < z

   1. Initial program 64.7%

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

      1. associate-*l/ 64.8%

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

      2. *-commutative 64.8%

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

      3. associate-*l/ 63.9%

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

      4. associate-*r* 62.1%

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

      5. sub-neg 62.1%

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

      6. distribute-rgt-neg-out 62.1%

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

      7. +-commutative 62.1%

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

      8. fma-def 62.2%

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

   3. Simplified 62.2%

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

   4. Taylor expanded in t around 0 88.1%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{-215}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{-112}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 17: 73.6% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -5e-310:
		tmp = y * -x
	else:
		tmp = x * y
	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(x * y);
	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 = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < -4.999999999999985e-310

   1. Initial program 64.2%

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

      1. associate-*l/ 66.2%

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

      2. *-commutative 66.2%

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

      3. associate-*l/ 65.5%

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

      4. associate-*r* 61.0%

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

      5. sub-neg 61.0%

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

      6. distribute-rgt-neg-out 61.0%

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

      7. +-commutative 61.0%

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

      8. fma-def 60.2%

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

   3. Simplified 60.2%

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

   4. Taylor expanded in z around -inf 66.5%

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

      1. *-commutative 66.5%

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

      2. mul-1-neg 66.5%

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

      3. *-commutative 66.5%

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

   6. Simplified 66.5%

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

   if -4.999999999999985e-310 < z

   1. Initial program 68.6%

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

      1. associate-*l/ 67.8%

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

      2. *-commutative 67.8%

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

      3. associate-*l/ 67.1%

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

      4. associate-*r* 67.1%

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

      5. sub-neg 67.1%

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

      6. distribute-rgt-neg-out 67.1%

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

      7. +-commutative 67.1%

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

      8. fma-def 67.1%

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

   3. Simplified 67.1%

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

   4. Taylor expanded in t around 0 76.1%

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

4. Final simplification 71.3%

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

Alternative 18: 44.6% accurate, 37.7× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.4%

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

   1. associate-*l/ 67.0%

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

   2. *-commutative 67.0%

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

   3. associate-*l/ 66.3%

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

   4. associate-*r* 64.0%

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

   5. sub-neg 64.0%

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

   6. distribute-rgt-neg-out 64.0%

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

   7. +-commutative 64.0%

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

   8. fma-def 63.6%

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

3. Simplified 63.6%

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

4. Taylor expanded in t around 0 43.7%

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

5. Final simplification 43.7%

   \[\leadsto x \cdot y \]

Developer target: 89.8% 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 2023301 
(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)))))