Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.5% → 96.7%

Time: 10.3s

Alternatives: 14

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \end{array} \]

FPCore

(FPCore (c0 A V l) :precision binary64 (* c0 (sqrt (/ A (* V l)))))

C

double code(double c0, double A, double V, double l) {
	return c0 * sqrt((A / (V * l)));
}

Fortran

real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    code = c0 * sqrt((a / (v * l)))
end function

Java

public static double code(double c0, double A, double V, double l) {
	return c0 * Math.sqrt((A / (V * l)));
}

Python

def code(c0, A, V, l):
	return c0 * math.sqrt((A / (V * l)))

Julia

function code(c0, A, V, l)
	return Float64(c0 * sqrt(Float64(A / Float64(V * l))))
end

MATLAB

function tmp = code(c0, A, V, l)
	tmp = c0 * sqrt((A / (V * l)));
end

Wolfram

code[c0_, A_, V_, l_] := N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 73.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \end{array} \]

FPCore

(FPCore (c0 A V l) :precision binary64 (* c0 (sqrt (/ A (* V l)))))

C

double code(double c0, double A, double V, double l) {
	return c0 * sqrt((A / (V * l)));
}

Fortran

real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    code = c0 * sqrt((a / (v * l)))
end function

Java

public static double code(double c0, double A, double V, double l) {
	return c0 * Math.sqrt((A / (V * l)));
}

Python

def code(c0, A, V, l):
	return c0 * math.sqrt((A / (V * l)))

Julia

function code(c0, A, V, l)
	return Float64(c0 * sqrt(Float64(A / Float64(V * l))))
end

MATLAB

function tmp = code(c0, A, V, l)
	tmp = c0 * sqrt((A / (V * l)));
end

Wolfram

code[c0_, A_, V_, l_] := N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 96.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ c0 \cdot {\left(\frac{\sqrt[3]{A}}{\sqrt[3]{V} \cdot \sqrt[3]{\ell}}\right)}^{1.5} \end{array} \]

FPCore

(FPCore (c0 A V l)
 :precision binary64
 (* c0 (pow (/ (cbrt A) (* (cbrt V) (cbrt l))) 1.5)))

C

double code(double c0, double A, double V, double l) {
	return c0 * pow((cbrt(A) / (cbrt(V) * cbrt(l))), 1.5);
}

Java

public static double code(double c0, double A, double V, double l) {
	return c0 * Math.pow((Math.cbrt(A) / (Math.cbrt(V) * Math.cbrt(l))), 1.5);
}

Julia

function code(c0, A, V, l)
	return Float64(c0 * (Float64(cbrt(A) / Float64(cbrt(V) * cbrt(l))) ^ 1.5))
end

Wolfram

code[c0_, A_, V_, l_] := N[(c0 * N[Power[N[(N[Power[A, 1/3], $MachinePrecision] / N[(N[Power[V, 1/3], $MachinePrecision] * N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.5%

   \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
2. Step-by-step derivation

   1. associate-/r* 70.6%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

   2. div-inv 70.6%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V} \cdot \frac{1}{\ell}}} \]

3. Applied egg-rr 70.6%

   \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V} \cdot \frac{1}{\ell}}} \]
4. Step-by-step derivation

   1. pow1/2 70.6%

      \[\leadsto c0 \cdot \color{blue}{{\left(\frac{A}{V} \cdot \frac{1}{\ell}\right)}^{0.5}} \]

   2. metadata-eval 70.6%

      \[\leadsto c0 \cdot {\left(\frac{A}{V} \cdot \frac{1}{\ell}\right)}^{\color{blue}{\left(0.3333333333333333 \cdot 1.5\right)}} \]

   3. pow-pow 66.4%

      \[\leadsto c0 \cdot \color{blue}{{\left({\left(\frac{A}{V} \cdot \frac{1}{\ell}\right)}^{0.3333333333333333}\right)}^{1.5}} \]

   4. un-div-inv 66.4%

      \[\leadsto c0 \cdot {\left({\color{blue}{\left(\frac{\frac{A}{V}}{\ell}\right)}}^{0.3333333333333333}\right)}^{1.5} \]

   5. associate-/r* 64.5%

      \[\leadsto c0 \cdot {\left({\color{blue}{\left(\frac{A}{V \cdot \ell}\right)}}^{0.3333333333333333}\right)}^{1.5} \]

   6. pow1/3 68.0%

      \[\leadsto c0 \cdot {\color{blue}{\left(\sqrt[3]{\frac{A}{V \cdot \ell}}\right)}}^{1.5} \]

5. Applied egg-rr 68.0%

   \[\leadsto c0 \cdot \color{blue}{{\left(\sqrt[3]{\frac{A}{V \cdot \ell}}\right)}^{1.5}} \]
6. Step-by-step derivation

   1. associate-/r* 70.1%

      \[\leadsto c0 \cdot {\left(\sqrt[3]{\color{blue}{\frac{\frac{A}{V}}{\ell}}}\right)}^{1.5} \]

7. Simplified 70.1%

   \[\leadsto c0 \cdot \color{blue}{{\left(\sqrt[3]{\frac{\frac{A}{V}}{\ell}}\right)}^{1.5}} \]
8. Step-by-step derivation

   1. cbrt-div 83.7%

      \[\leadsto c0 \cdot {\color{blue}{\left(\frac{\sqrt[3]{\frac{A}{V}}}{\sqrt[3]{\ell}}\right)}}^{1.5} \]

   2. div-inv 83.6%

      \[\leadsto c0 \cdot {\color{blue}{\left(\sqrt[3]{\frac{A}{V}} \cdot \frac{1}{\sqrt[3]{\ell}}\right)}}^{1.5} \]

   3. cbrt-div 97.4%

      \[\leadsto c0 \cdot {\left(\color{blue}{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}} \cdot \frac{1}{\sqrt[3]{\ell}}\right)}^{1.5} \]

   4. frac-times 97.5%

      \[\leadsto c0 \cdot {\color{blue}{\left(\frac{\sqrt[3]{A} \cdot 1}{\sqrt[3]{V} \cdot \sqrt[3]{\ell}}\right)}}^{1.5} \]

9. Applied egg-rr 97.5%

   \[\leadsto c0 \cdot {\color{blue}{\left(\frac{\sqrt[3]{A} \cdot 1}{\sqrt[3]{V} \cdot \sqrt[3]{\ell}}\right)}}^{1.5} \]
10. Step-by-step derivation

    1. *-rgt-identity 97.5%

       \[\leadsto c0 \cdot {\left(\frac{\color{blue}{\sqrt[3]{A}}}{\sqrt[3]{V} \cdot \sqrt[3]{\ell}}\right)}^{1.5} \]

11. Simplified 97.5%

    \[\leadsto c0 \cdot {\color{blue}{\left(\frac{\sqrt[3]{A}}{\sqrt[3]{V} \cdot \sqrt[3]{\ell}}\right)}}^{1.5} \]

12. Final simplification 97.5%

    \[\leadsto c0 \cdot {\left(\frac{\sqrt[3]{A}}{\sqrt[3]{V} \cdot \sqrt[3]{\ell}}\right)}^{1.5} \]

Alternative 2: 89.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-290}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-316} \lor \neg \left(V \cdot \ell \leq \infty\right):\\ \;\;\;\;c0 \cdot {\left(\frac{\sqrt[3]{\frac{A}{V}}}{\sqrt[3]{\ell}}\right)}^{1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) (- INFINITY))
   (/ c0 (* (sqrt (/ V A)) (sqrt l)))
   (if (<= (* V l) -1e-290)
     (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
     (if (or (<= (* V l) 5e-316) (not (<= (* V l) INFINITY)))
       (* c0 (pow (/ (cbrt (/ A V)) (cbrt l)) 1.5))
       (/ c0 (/ (sqrt (* V l)) (sqrt A)))))))

C

double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = c0 / (sqrt((V / A)) * sqrt(l));
	} else if ((V * l) <= -1e-290) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if (((V * l) <= 5e-316) || !((V * l) <= ((double) INFINITY))) {
		tmp = c0 * pow((cbrt((A / V)) / cbrt(l)), 1.5);
	} else {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	}
	return tmp;
}

Java

public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = c0 / (Math.sqrt((V / A)) * Math.sqrt(l));
	} else if ((V * l) <= -1e-290) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if (((V * l) <= 5e-316) || !((V * l) <= Double.POSITIVE_INFINITY)) {
		tmp = c0 * Math.pow((Math.cbrt((A / V)) / Math.cbrt(l)), 1.5);
	} else {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	}
	return tmp;
}

Julia

function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(c0 / Float64(sqrt(Float64(V / A)) * sqrt(l)));
	elseif (Float64(V * l) <= -1e-290)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif ((Float64(V * l) <= 5e-316) || !(Float64(V * l) <= Inf))
		tmp = Float64(c0 * (Float64(cbrt(Float64(A / V)) / cbrt(l)) ^ 1.5));
	else
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	end
	return tmp
end

Wolfram

code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(c0 / N[(N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-290], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(V * l), $MachinePrecision], 5e-316], N[Not[LessEqual[N[(V * l), $MachinePrecision], Infinity]], $MachinePrecision]], N[(c0 * N[Power[N[(N[Power[N[(A / V), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision]), $MachinePrecision], N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -inf.0

   1. Initial program 23.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. sqrt-div 0.0%

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}} \]

      2. associate-*r/ 0.0%

         \[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}} \]

   3. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}} \]
   4. Step-by-step derivation

      1. associate-/l* 0.0%

         \[\leadsto \color{blue}{\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}} \]

   5. Simplified 0.0%

      \[\leadsto \color{blue}{\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}} \]
   6. Step-by-step derivation

      1. sqrt-undiv 23.7%

         \[\leadsto \frac{c0}{\color{blue}{\sqrt{\frac{V \cdot \ell}{A}}}} \]

      2. associate-*l/ 55.8%

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{V}{A} \cdot \ell}}} \]

      3. associate-/r/ 55.7%

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{V}{\frac{A}{\ell}}}}} \]

      4. associate-/r/ 55.8%

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{V}{A} \cdot \ell}}} \]

      5. sqrt-prod 52.5%

         \[\leadsto \frac{c0}{\color{blue}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}} \]

   7. Applied egg-rr 52.5%

      \[\leadsto \frac{c0}{\color{blue}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}} \]

   if -inf.0 < (*.f64 V l) < -1.0000000000000001e-290

   1. Initial program 83.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. frac-2neg 83.7%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{-A}{-V \cdot \ell}}} \]

      2. sqrt-div 99.4%

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{-A}}{\sqrt{-V \cdot \ell}}} \]

      3. *-commutative 99.4%

         \[\leadsto c0 \cdot \frac{\sqrt{-A}}{\sqrt{-\color{blue}{\ell \cdot V}}} \]

      4. distribute-rgt-neg-in 99.4%

         \[\leadsto c0 \cdot \frac{\sqrt{-A}}{\sqrt{\color{blue}{\ell \cdot \left(-V\right)}}} \]

   3. Applied egg-rr 99.4%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}} \]

   if -1.0000000000000001e-290 < (*.f64 V l) < 5.000000017e-316 or +inf.0 < (*.f64 V l)

   1. Initial program 24.4%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. pow1/2 24.4%

         \[\leadsto c0 \cdot \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{0.5}} \]

      2. add-cube-cbrt 24.3%

         \[\leadsto c0 \cdot {\color{blue}{\left(\left(\sqrt[3]{\frac{A}{V \cdot \ell}} \cdot \sqrt[3]{\frac{A}{V \cdot \ell}}\right) \cdot \sqrt[3]{\frac{A}{V \cdot \ell}}\right)}}^{0.5} \]

      3. pow3 24.3%

         \[\leadsto c0 \cdot {\color{blue}{\left({\left(\sqrt[3]{\frac{A}{V \cdot \ell}}\right)}^{3}\right)}}^{0.5} \]

      4. pow-pow 24.3%

         \[\leadsto c0 \cdot \color{blue}{{\left(\sqrt[3]{\frac{A}{V \cdot \ell}}\right)}^{\left(3 \cdot 0.5\right)}} \]

      5. metadata-eval 24.3%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\frac{A}{V \cdot \ell}}\right)}^{\color{blue}{1.5}} \]

   3. Applied egg-rr 24.3%

      \[\leadsto c0 \cdot \color{blue}{{\left(\sqrt[3]{\frac{A}{V \cdot \ell}}\right)}^{1.5}} \]
   4. Step-by-step derivation

      1. associate-/r* 54.9%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\color{blue}{\frac{\frac{A}{V}}{\ell}}}\right)}^{1.5} \]

      2. un-div-inv 54.8%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\color{blue}{\frac{A}{V} \cdot \frac{1}{\ell}}}\right)}^{1.5} \]

      3. cbrt-prod 84.1%

         \[\leadsto c0 \cdot {\color{blue}{\left(\sqrt[3]{\frac{A}{V}} \cdot \sqrt[3]{\frac{1}{\ell}}\right)}}^{1.5} \]

      4. cbrt-div 83.8%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\frac{A}{V}} \cdot \color{blue}{\frac{\sqrt[3]{1}}{\sqrt[3]{\ell}}}\right)}^{1.5} \]

      5. metadata-eval 83.8%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\frac{A}{V}} \cdot \frac{\color{blue}{1}}{\sqrt[3]{\ell}}\right)}^{1.5} \]

   5. Applied egg-rr 83.8%

      \[\leadsto c0 \cdot {\color{blue}{\left(\sqrt[3]{\frac{A}{V}} \cdot \frac{1}{\sqrt[3]{\ell}}\right)}}^{1.5} \]
   6. Step-by-step derivation

      1. associate-*r/ 83.8%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{\sqrt[3]{\frac{A}{V}} \cdot 1}{\sqrt[3]{\ell}}\right)}}^{1.5} \]

      2. *-rgt-identity 83.8%

         \[\leadsto c0 \cdot {\left(\frac{\color{blue}{\sqrt[3]{\frac{A}{V}}}}{\sqrt[3]{\ell}}\right)}^{1.5} \]

   7. Simplified 83.8%

      \[\leadsto c0 \cdot {\color{blue}{\left(\frac{\sqrt[3]{\frac{A}{V}}}{\sqrt[3]{\ell}}\right)}}^{1.5} \]

   if 5.000000017e-316 < (*.f64 V l) < +inf.0

   1. Initial program 76.8%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. sqrt-div 88.3%

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}} \]

      2. associate-*r/ 86.6%

         \[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}} \]

   3. Applied egg-rr 86.6%

      \[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}} \]
   4. Step-by-step derivation

      1. associate-/l* 88.4%

         \[\leadsto \color{blue}{\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}} \]

   5. Simplified 88.4%

      \[\leadsto \color{blue}{\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-290}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-316} \lor \neg \left(V \cdot \ell \leq \infty\right):\\ \;\;\;\;c0 \cdot {\left(\frac{\sqrt[3]{\frac{A}{V}}}{\sqrt[3]{\ell}}\right)}^{1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \end{array} \]

Alternative 3: 89.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-290}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-316}:\\ \;\;\;\;c0 \cdot {\left(\frac{\sqrt[3]{\frac{A}{V}}}{\sqrt[3]{\ell}}\right)}^{1.5}\\ \mathbf{elif}\;V \cdot \ell \leq \infty:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(\frac{\sqrt[3]{\frac{A}{\ell}}}{\sqrt[3]{V}}\right)}^{1.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) (- INFINITY))
   (/ c0 (* (sqrt (/ V A)) (sqrt l)))
   (if (<= (* V l) -1e-290)
     (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
     (if (<= (* V l) 5e-316)
       (* c0 (pow (/ (cbrt (/ A V)) (cbrt l)) 1.5))
       (if (<= (* V l) INFINITY)
         (/ c0 (/ (sqrt (* V l)) (sqrt A)))
         (* c0 (pow (/ (cbrt (/ A l)) (cbrt V)) 1.5)))))))

C

double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = c0 / (sqrt((V / A)) * sqrt(l));
	} else if ((V * l) <= -1e-290) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 5e-316) {
		tmp = c0 * pow((cbrt((A / V)) / cbrt(l)), 1.5);
	} else if ((V * l) <= ((double) INFINITY)) {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	} else {
		tmp = c0 * pow((cbrt((A / l)) / cbrt(V)), 1.5);
	}
	return tmp;
}

Java

public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = c0 / (Math.sqrt((V / A)) * Math.sqrt(l));
	} else if ((V * l) <= -1e-290) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 5e-316) {
		tmp = c0 * Math.pow((Math.cbrt((A / V)) / Math.cbrt(l)), 1.5);
	} else if ((V * l) <= Double.POSITIVE_INFINITY) {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	} else {
		tmp = c0 * Math.pow((Math.cbrt((A / l)) / Math.cbrt(V)), 1.5);
	}
	return tmp;
}

Julia

function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(c0 / Float64(sqrt(Float64(V / A)) * sqrt(l)));
	elseif (Float64(V * l) <= -1e-290)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 5e-316)
		tmp = Float64(c0 * (Float64(cbrt(Float64(A / V)) / cbrt(l)) ^ 1.5));
	elseif (Float64(V * l) <= Inf)
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	else
		tmp = Float64(c0 * (Float64(cbrt(Float64(A / l)) / cbrt(V)) ^ 1.5));
	end
	return tmp
end

Wolfram

code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(c0 / N[(N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-290], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e-316], N[(c0 * N[Power[N[(N[Power[N[(A / V), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], Infinity], N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Power[N[(N[Power[N[(A / l), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[V, 1/3], $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -inf.0

   1. Initial program 23.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. sqrt-div 0.0%

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}} \]

      2. associate-*r/ 0.0%

         \[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}} \]

   3. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}} \]
   4. Step-by-step derivation

      1. associate-/l* 0.0%

         \[\leadsto \color{blue}{\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}} \]

   5. Simplified 0.0%

      \[\leadsto \color{blue}{\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}} \]
   6. Step-by-step derivation

      1. sqrt-undiv 23.7%

         \[\leadsto \frac{c0}{\color{blue}{\sqrt{\frac{V \cdot \ell}{A}}}} \]

      2. associate-*l/ 55.8%

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{V}{A} \cdot \ell}}} \]

      3. associate-/r/ 55.7%

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{V}{\frac{A}{\ell}}}}} \]

      4. associate-/r/ 55.8%

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{V}{A} \cdot \ell}}} \]

      5. sqrt-prod 52.5%

         \[\leadsto \frac{c0}{\color{blue}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}} \]

   7. Applied egg-rr 52.5%

      \[\leadsto \frac{c0}{\color{blue}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}} \]

   if -inf.0 < (*.f64 V l) < -1.0000000000000001e-290

   1. Initial program 83.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. frac-2neg 83.7%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{-A}{-V \cdot \ell}}} \]

      2. sqrt-div 99.4%

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{-A}}{\sqrt{-V \cdot \ell}}} \]

      3. *-commutative 99.4%

         \[\leadsto c0 \cdot \frac{\sqrt{-A}}{\sqrt{-\color{blue}{\ell \cdot V}}} \]

      4. distribute-rgt-neg-in 99.4%

         \[\leadsto c0 \cdot \frac{\sqrt{-A}}{\sqrt{\color{blue}{\ell \cdot \left(-V\right)}}} \]

   3. Applied egg-rr 99.4%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}} \]

   if -1.0000000000000001e-290 < (*.f64 V l) < 5.000000017e-316

   1. Initial program 24.4%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. pow1/2 24.4%

         \[\leadsto c0 \cdot \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{0.5}} \]

      2. add-cube-cbrt 24.3%

         \[\leadsto c0 \cdot {\color{blue}{\left(\left(\sqrt[3]{\frac{A}{V \cdot \ell}} \cdot \sqrt[3]{\frac{A}{V \cdot \ell}}\right) \cdot \sqrt[3]{\frac{A}{V \cdot \ell}}\right)}}^{0.5} \]

      3. pow3 24.3%

         \[\leadsto c0 \cdot {\color{blue}{\left({\left(\sqrt[3]{\frac{A}{V \cdot \ell}}\right)}^{3}\right)}}^{0.5} \]

      4. pow-pow 24.3%

         \[\leadsto c0 \cdot \color{blue}{{\left(\sqrt[3]{\frac{A}{V \cdot \ell}}\right)}^{\left(3 \cdot 0.5\right)}} \]

      5. metadata-eval 24.3%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\frac{A}{V \cdot \ell}}\right)}^{\color{blue}{1.5}} \]

   3. Applied egg-rr 24.3%

      \[\leadsto c0 \cdot \color{blue}{{\left(\sqrt[3]{\frac{A}{V \cdot \ell}}\right)}^{1.5}} \]
   4. Step-by-step derivation

      1. associate-/r* 54.9%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\color{blue}{\frac{\frac{A}{V}}{\ell}}}\right)}^{1.5} \]

      2. un-div-inv 54.8%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\color{blue}{\frac{A}{V} \cdot \frac{1}{\ell}}}\right)}^{1.5} \]

      3. cbrt-prod 84.1%

         \[\leadsto c0 \cdot {\color{blue}{\left(\sqrt[3]{\frac{A}{V}} \cdot \sqrt[3]{\frac{1}{\ell}}\right)}}^{1.5} \]

      4. cbrt-div 83.8%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\frac{A}{V}} \cdot \color{blue}{\frac{\sqrt[3]{1}}{\sqrt[3]{\ell}}}\right)}^{1.5} \]

      5. metadata-eval 83.8%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\frac{A}{V}} \cdot \frac{\color{blue}{1}}{\sqrt[3]{\ell}}\right)}^{1.5} \]

   5. Applied egg-rr 83.8%

      \[\leadsto c0 \cdot {\color{blue}{\left(\sqrt[3]{\frac{A}{V}} \cdot \frac{1}{\sqrt[3]{\ell}}\right)}}^{1.5} \]
   6. Step-by-step derivation

      1. associate-*r/ 83.8%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{\sqrt[3]{\frac{A}{V}} \cdot 1}{\sqrt[3]{\ell}}\right)}}^{1.5} \]

      2. *-rgt-identity 83.8%

         \[\leadsto c0 \cdot {\left(\frac{\color{blue}{\sqrt[3]{\frac{A}{V}}}}{\sqrt[3]{\ell}}\right)}^{1.5} \]

   7. Simplified 83.8%

      \[\leadsto c0 \cdot {\color{blue}{\left(\frac{\sqrt[3]{\frac{A}{V}}}{\sqrt[3]{\ell}}\right)}}^{1.5} \]

   if 5.000000017e-316 < (*.f64 V l) < +inf.0

   1. Initial program 76.8%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. sqrt-div 88.3%

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}} \]

      2. associate-*r/ 86.6%

         \[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}} \]

   3. Applied egg-rr 86.6%

      \[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}} \]
   4. Step-by-step derivation

      1. associate-/l* 88.4%

         \[\leadsto \color{blue}{\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}} \]

   5. Simplified 88.4%

      \[\leadsto \color{blue}{\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}} \]

   if +inf.0 < (*.f64 V l)

   1. Initial program 68.5%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. pow1/2 68.5%

         \[\leadsto c0 \cdot \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{0.5}} \]

      2. add-cube-cbrt 68.0%

         \[\leadsto c0 \cdot {\color{blue}{\left(\left(\sqrt[3]{\frac{A}{V \cdot \ell}} \cdot \sqrt[3]{\frac{A}{V \cdot \ell}}\right) \cdot \sqrt[3]{\frac{A}{V \cdot \ell}}\right)}}^{0.5} \]

      3. pow3 68.0%

         \[\leadsto c0 \cdot {\color{blue}{\left({\left(\sqrt[3]{\frac{A}{V \cdot \ell}}\right)}^{3}\right)}}^{0.5} \]

      4. pow-pow 68.0%

         \[\leadsto c0 \cdot \color{blue}{{\left(\sqrt[3]{\frac{A}{V \cdot \ell}}\right)}^{\left(3 \cdot 0.5\right)}} \]

      5. metadata-eval 68.0%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\frac{A}{V \cdot \ell}}\right)}^{\color{blue}{1.5}} \]

   3. Applied egg-rr 68.0%

      \[\leadsto c0 \cdot \color{blue}{{\left(\sqrt[3]{\frac{A}{V \cdot \ell}}\right)}^{1.5}} \]
   4. Step-by-step derivation

      1. associate-/r* 70.1%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\color{blue}{\frac{\frac{A}{V}}{\ell}}}\right)}^{1.5} \]

      2. un-div-inv 70.1%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\color{blue}{\frac{A}{V} \cdot \frac{1}{\ell}}}\right)}^{1.5} \]

      3. associate-*l/ 66.1%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\color{blue}{\frac{A \cdot \frac{1}{\ell}}{V}}}\right)}^{1.5} \]

      4. div-inv 66.1%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\frac{\color{blue}{\frac{A}{\ell}}}{V}}\right)}^{1.5} \]

      5. cbrt-div 79.9%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{\sqrt[3]{\frac{A}{\ell}}}{\sqrt[3]{V}}\right)}}^{1.5} \]

   5. Applied egg-rr 79.9%

      \[\leadsto c0 \cdot {\color{blue}{\left(\frac{\sqrt[3]{\frac{A}{\ell}}}{\sqrt[3]{V}}\right)}}^{1.5} \]
3. Recombined 5 regimes into one program.

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-290}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-316}:\\ \;\;\;\;c0 \cdot {\left(\frac{\sqrt[3]{\frac{A}{V}}}{\sqrt[3]{\ell}}\right)}^{1.5}\\ \mathbf{elif}\;V \cdot \ell \leq \infty:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(\frac{\sqrt[3]{\frac{A}{\ell}}}{\sqrt[3]{V}}\right)}^{1.5}\\ \end{array} \]

Alternative 4: 83.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-295}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \left(c0 \cdot {\ell}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) (- INFINITY))
   (/ c0 (* (sqrt (/ V A)) (sqrt l)))
   (if (<= (* V l) -1e-295)
     (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
     (if (<= (* V l) 0.0)
       (* (sqrt (/ A V)) (* c0 (pow l -0.5)))
       (* c0 (* (sqrt A) (sqrt (/ (/ 1.0 V) l))))))))

C

double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = c0 / (sqrt((V / A)) * sqrt(l));
	} else if ((V * l) <= -1e-295) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = sqrt((A / V)) * (c0 * pow(l, -0.5));
	} else {
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	}
	return tmp;
}

Java

public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = c0 / (Math.sqrt((V / A)) * Math.sqrt(l));
	} else if ((V * l) <= -1e-295) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = Math.sqrt((A / V)) * (c0 * Math.pow(l, -0.5));
	} else {
		tmp = c0 * (Math.sqrt(A) * Math.sqrt(((1.0 / V) / l)));
	}
	return tmp;
}

Python

def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = c0 / (math.sqrt((V / A)) * math.sqrt(l))
	elif (V * l) <= -1e-295:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 0.0:
		tmp = math.sqrt((A / V)) * (c0 * math.pow(l, -0.5))
	else:
		tmp = c0 * (math.sqrt(A) * math.sqrt(((1.0 / V) / l)))
	return tmp

Julia

function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(c0 / Float64(sqrt(Float64(V / A)) * sqrt(l)));
	elseif (Float64(V * l) <= -1e-295)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(sqrt(Float64(A / V)) * Float64(c0 * (l ^ -0.5)));
	else
		tmp = Float64(c0 * Float64(sqrt(A) * sqrt(Float64(Float64(1.0 / V) / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = c0 / (sqrt((V / A)) * sqrt(l));
	elseif ((V * l) <= -1e-295)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 0.0)
		tmp = sqrt((A / V)) * (c0 * (l ^ -0.5));
	else
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(c0 / N[(N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-295], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] * N[(c0 * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Sqrt[N[(N[(1.0 / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -inf.0

   1. Initial program 23.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. sqrt-div 0.0%

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}} \]

      2. associate-*r/ 0.0%

         \[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}} \]

   3. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}} \]
   4. Step-by-step derivation

      1. associate-/l* 0.0%

         \[\leadsto \color{blue}{\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}} \]

   5. Simplified 0.0%

      \[\leadsto \color{blue}{\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}} \]
   6. Step-by-step derivation

      1. sqrt-undiv 23.7%

         \[\leadsto \frac{c0}{\color{blue}{\sqrt{\frac{V \cdot \ell}{A}}}} \]

      2. associate-*l/ 55.8%

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{V}{A} \cdot \ell}}} \]

      3. associate-/r/ 55.7%

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{V}{\frac{A}{\ell}}}}} \]

      4. associate-/r/ 55.8%

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{V}{A} \cdot \ell}}} \]

      5. sqrt-prod 52.5%

         \[\leadsto \frac{c0}{\color{blue}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}} \]

   7. Applied egg-rr 52.5%

      \[\leadsto \frac{c0}{\color{blue}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}} \]

   if -inf.0 < (*.f64 V l) < -1.00000000000000006e-295

   1. Initial program 83.9%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. frac-2neg 83.9%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{-A}{-V \cdot \ell}}} \]

      2. sqrt-div 99.4%

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{-A}}{\sqrt{-V \cdot \ell}}} \]

      3. *-commutative 99.4%

         \[\leadsto c0 \cdot \frac{\sqrt{-A}}{\sqrt{-\color{blue}{\ell \cdot V}}} \]

      4. distribute-rgt-neg-in 99.4%

         \[\leadsto c0 \cdot \frac{\sqrt{-A}}{\sqrt{\color{blue}{\ell \cdot \left(-V\right)}}} \]

   3. Applied egg-rr 99.4%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}} \]

   if -1.00000000000000006e-295 < (*.f64 V l) < 0.0

   1. Initial program 20.6%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. sqrt-div 15.5%

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}} \]

      2. associate-*r/ 15.5%

         \[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}} \]

   3. Applied egg-rr 15.5%

      \[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}} \]
   4. Step-by-step derivation

      1. associate-/l* 15.5%

         \[\leadsto \color{blue}{\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}} \]

   5. Simplified 15.5%

      \[\leadsto \color{blue}{\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}} \]
   6. Step-by-step derivation

      1. div-inv 15.5%

         \[\leadsto \color{blue}{c0 \cdot \frac{1}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}} \]

      2. clear-num 15.5%

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}} \]

      3. sqrt-div 20.6%

         \[\leadsto c0 \cdot \color{blue}{\sqrt{\frac{A}{V \cdot \ell}}} \]

      4. associate-/r* 52.3%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      5. un-div-inv 52.3%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V} \cdot \frac{1}{\ell}}} \]

      6. *-commutative 52.3%

         \[\leadsto \color{blue}{\sqrt{\frac{A}{V} \cdot \frac{1}{\ell}} \cdot c0} \]

      7. sqrt-prod 56.9%

         \[\leadsto \color{blue}{\left(\sqrt{\frac{A}{V}} \cdot \sqrt{\frac{1}{\ell}}\right)} \cdot c0 \]

      8. associate-*l* 53.8%

         \[\leadsto \color{blue}{\sqrt{\frac{A}{V}} \cdot \left(\sqrt{\frac{1}{\ell}} \cdot c0\right)} \]

      9. inv-pow 53.8%

         \[\leadsto \sqrt{\frac{A}{V}} \cdot \left(\sqrt{\color{blue}{{\ell}^{-1}}} \cdot c0\right) \]

      10. sqrt-pow1 53.8%

          \[\leadsto \sqrt{\frac{A}{V}} \cdot \left(\color{blue}{{\ell}^{\left(\frac{-1}{2}\right)}} \cdot c0\right) \]

      11. metadata-eval 53.8%

          \[\leadsto \sqrt{\frac{A}{V}} \cdot \left({\ell}^{\color{blue}{-0.5}} \cdot c0\right) \]

   7. Applied egg-rr 53.8%

      \[\leadsto \color{blue}{\sqrt{\frac{A}{V}} \cdot \left({\ell}^{-0.5} \cdot c0\right)} \]

   if 0.0 < (*.f64 V l)

   1. Initial program 76.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. pow1/2 76.7%

         \[\leadsto c0 \cdot \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{0.5}} \]

      2. clear-num 76.1%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)}}^{0.5} \]

      3. inv-pow 76.1%

         \[\leadsto c0 \cdot {\color{blue}{\left({\left(\frac{V \cdot \ell}{A}\right)}^{-1}\right)}}^{0.5} \]

      4. pow-pow 76.6%

         \[\leadsto c0 \cdot \color{blue}{{\left(\frac{V \cdot \ell}{A}\right)}^{\left(-1 \cdot 0.5\right)}} \]

      5. associate-/l* 67.5%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{V}{\frac{A}{\ell}}\right)}}^{\left(-1 \cdot 0.5\right)} \]

      6. metadata-eval 67.5%

         \[\leadsto c0 \cdot {\left(\frac{V}{\frac{A}{\ell}}\right)}^{\color{blue}{-0.5}} \]

   3. Applied egg-rr 67.5%

      \[\leadsto c0 \cdot \color{blue}{{\left(\frac{V}{\frac{A}{\ell}}\right)}^{-0.5}} \]
   4. Step-by-step derivation

      1. associate-/l* 76.6%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{V \cdot \ell}{A}\right)}}^{-0.5} \]

      2. *-lft-identity 76.6%

         \[\leadsto c0 \cdot {\left(\frac{V \cdot \ell}{\color{blue}{1 \cdot A}}\right)}^{-0.5} \]

      3. times-frac 67.8%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{V}{1} \cdot \frac{\ell}{A}\right)}}^{-0.5} \]

      4. /-rgt-identity 67.8%

         \[\leadsto c0 \cdot {\left(\color{blue}{V} \cdot \frac{\ell}{A}\right)}^{-0.5} \]

   5. Simplified 67.8%

      \[\leadsto c0 \cdot \color{blue}{{\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 67.5%

         \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{{\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}} \cdot \sqrt{{\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}}\right)} \]

      2. sqrt-unprod 67.8%

         \[\leadsto c0 \cdot \color{blue}{\sqrt{{\left(V \cdot \frac{\ell}{A}\right)}^{-0.5} \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}}} \]

      3. pow-prod-up 67.8%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{{\left(V \cdot \frac{\ell}{A}\right)}^{\left(-0.5 + -0.5\right)}}} \]

      4. metadata-eval 67.8%

         \[\leadsto c0 \cdot \sqrt{{\left(V \cdot \frac{\ell}{A}\right)}^{\color{blue}{-1}}} \]

      5. inv-pow 67.8%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{V \cdot \frac{\ell}{A}}}} \]

      6. associate-*r/ 76.1%

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{\frac{V \cdot \ell}{A}}}} \]

      7. clear-num 76.7%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      8. div-inv 76.2%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{A \cdot \frac{1}{V \cdot \ell}}} \]

      9. sqrt-prod 87.5%

         \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{A} \cdot \sqrt{\frac{1}{V \cdot \ell}}\right)} \]

   7. Applied egg-rr 87.5%

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{A} \cdot \sqrt{\frac{1}{V \cdot \ell}}\right)} \]
   8. Step-by-step derivation

      1. associate-/r* 88.2%

         \[\leadsto c0 \cdot \left(\sqrt{A} \cdot \sqrt{\color{blue}{\frac{\frac{1}{V}}{\ell}}}\right) \]

   9. Simplified 88.2%

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-295}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \left(c0 \cdot {\ell}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \]

Alternative 5: 74.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0 \lor \neg \left(t_0 \leq 5 \cdot 10^{+307}\right):\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (or (<= t_0 0.0) (not (<= t_0 5e+307)))
     (* c0 (/ (sqrt (/ A V)) (sqrt l)))
     (* c0 (sqrt t_0)))))

C

double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 5e+307)) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else {
		tmp = c0 * sqrt(t_0);
	}
	return tmp;
}

Fortran

real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if ((t_0 <= 0.0d0) .or. (.not. (t_0 <= 5d+307))) then
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    else
        tmp = c0 * sqrt(t_0)
    end if
    code = tmp
end function

Java

public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 5e+307)) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else {
		tmp = c0 * Math.sqrt(t_0);
	}
	return tmp;
}

Python

def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 5e+307):
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	else:
		tmp = c0 * math.sqrt(t_0)
	return tmp

Julia

function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 5e+307))
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	else
		tmp = Float64(c0 * sqrt(t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 5e+307)))
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	else
		tmp = c0 * sqrt(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 5e+307]], $MachinePrecision]], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 0.0 or 5e307 < (/.f64 A (*.f64 V l))

   1. Initial program 27.5%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-/r* 45.7%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      2. sqrt-div 41.7%

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}} \]

   3. Applied egg-rr 41.7%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}} \]

   if 0.0 < (/.f64 A (*.f64 V l)) < 5e307

   1. Initial program 98.8%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0 \lor \neg \left(\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{+307}\right):\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]

Alternative 6: 74.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0 \lor \neg \left(t_0 \leq 5 \cdot 10^{+307}\right):\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (or (<= t_0 0.0) (not (<= t_0 5e+307)))
     (* (sqrt (/ A V)) (/ c0 (sqrt l)))
     (* c0 (sqrt t_0)))))

C

double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 5e+307)) {
		tmp = sqrt((A / V)) * (c0 / sqrt(l));
	} else {
		tmp = c0 * sqrt(t_0);
	}
	return tmp;
}

Fortran

real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if ((t_0 <= 0.0d0) .or. (.not. (t_0 <= 5d+307))) then
        tmp = sqrt((a / v)) * (c0 / sqrt(l))
    else
        tmp = c0 * sqrt(t_0)
    end if
    code = tmp
end function

Java

public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 5e+307)) {
		tmp = Math.sqrt((A / V)) * (c0 / Math.sqrt(l));
	} else {
		tmp = c0 * Math.sqrt(t_0);
	}
	return tmp;
}

Python

def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 5e+307):
		tmp = math.sqrt((A / V)) * (c0 / math.sqrt(l))
	else:
		tmp = c0 * math.sqrt(t_0)
	return tmp

Julia

function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 5e+307))
		tmp = Float64(sqrt(Float64(A / V)) * Float64(c0 / sqrt(l)));
	else
		tmp = Float64(c0 * sqrt(t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 5e+307)))
		tmp = sqrt((A / V)) * (c0 / sqrt(l));
	else
		tmp = c0 * sqrt(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 5e+307]], $MachinePrecision]], N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] * N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 0.0 or 5e307 < (/.f64 A (*.f64 V l))

   1. Initial program 27.5%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-/r* 45.7%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      2. div-inv 45.8%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V} \cdot \frac{1}{\ell}}} \]

   3. Applied egg-rr 45.8%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V} \cdot \frac{1}{\ell}}} \]
   4. Step-by-step derivation

      1. pow1/2 45.8%

         \[\leadsto c0 \cdot \color{blue}{{\left(\frac{A}{V} \cdot \frac{1}{\ell}\right)}^{0.5}} \]

      2. metadata-eval 45.8%

         \[\leadsto c0 \cdot {\left(\frac{A}{V} \cdot \frac{1}{\ell}\right)}^{\color{blue}{\left(0.3333333333333333 \cdot 1.5\right)}} \]

      3. pow-pow 44.1%

         \[\leadsto c0 \cdot \color{blue}{{\left({\left(\frac{A}{V} \cdot \frac{1}{\ell}\right)}^{0.3333333333333333}\right)}^{1.5}} \]

      4. un-div-inv 44.1%

         \[\leadsto c0 \cdot {\left({\color{blue}{\left(\frac{\frac{A}{V}}{\ell}\right)}}^{0.3333333333333333}\right)}^{1.5} \]

      5. associate-/r* 27.5%

         \[\leadsto c0 \cdot {\left({\color{blue}{\left(\frac{A}{V \cdot \ell}\right)}}^{0.3333333333333333}\right)}^{1.5} \]

      6. pow1/3 27.5%

         \[\leadsto c0 \cdot {\color{blue}{\left(\sqrt[3]{\frac{A}{V \cdot \ell}}\right)}}^{1.5} \]

   5. Applied egg-rr 27.5%

      \[\leadsto c0 \cdot \color{blue}{{\left(\sqrt[3]{\frac{A}{V \cdot \ell}}\right)}^{1.5}} \]
   6. Step-by-step derivation

      1. associate-/r* 45.6%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\color{blue}{\frac{\frac{A}{V}}{\ell}}}\right)}^{1.5} \]

   7. Simplified 45.6%

      \[\leadsto c0 \cdot \color{blue}{{\left(\sqrt[3]{\frac{\frac{A}{V}}{\ell}}\right)}^{1.5}} \]
   8. Step-by-step derivation

      1. cbrt-div 77.4%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{\sqrt[3]{\frac{A}{V}}}{\sqrt[3]{\ell}}\right)}}^{1.5} \]

      2. div-inv 77.3%

         \[\leadsto c0 \cdot {\color{blue}{\left(\sqrt[3]{\frac{A}{V}} \cdot \frac{1}{\sqrt[3]{\ell}}\right)}}^{1.5} \]

      3. cbrt-div 96.3%

         \[\leadsto c0 \cdot {\left(\color{blue}{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}} \cdot \frac{1}{\sqrt[3]{\ell}}\right)}^{1.5} \]

      4. frac-times 96.5%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{\sqrt[3]{A} \cdot 1}{\sqrt[3]{V} \cdot \sqrt[3]{\ell}}\right)}}^{1.5} \]

   9. Applied egg-rr 96.5%

      \[\leadsto c0 \cdot {\color{blue}{\left(\frac{\sqrt[3]{A} \cdot 1}{\sqrt[3]{V} \cdot \sqrt[3]{\ell}}\right)}}^{1.5} \]
   10. Step-by-step derivation

       1. *-rgt-identity 96.5%

          \[\leadsto c0 \cdot {\left(\frac{\color{blue}{\sqrt[3]{A}}}{\sqrt[3]{V} \cdot \sqrt[3]{\ell}}\right)}^{1.5} \]

   11. Simplified 96.5%

       \[\leadsto c0 \cdot {\color{blue}{\left(\frac{\sqrt[3]{A}}{\sqrt[3]{V} \cdot \sqrt[3]{\ell}}\right)}}^{1.5} \]
   12. Step-by-step derivation

       1. add-sqr-sqrt 96.4%

          \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{{\left(\frac{\sqrt[3]{A}}{\sqrt[3]{V} \cdot \sqrt[3]{\ell}}\right)}^{1.5}} \cdot \sqrt{{\left(\frac{\sqrt[3]{A}}{\sqrt[3]{V} \cdot \sqrt[3]{\ell}}\right)}^{1.5}}\right)} \]

       2. sqrt-unprod 45.5%

          \[\leadsto c0 \cdot \color{blue}{\sqrt{{\left(\frac{\sqrt[3]{A}}{\sqrt[3]{V} \cdot \sqrt[3]{\ell}}\right)}^{1.5} \cdot {\left(\frac{\sqrt[3]{A}}{\sqrt[3]{V} \cdot \sqrt[3]{\ell}}\right)}^{1.5}}} \]

       3. pow-prod-up 45.5%

          \[\leadsto c0 \cdot \sqrt{\color{blue}{{\left(\frac{\sqrt[3]{A}}{\sqrt[3]{V} \cdot \sqrt[3]{\ell}}\right)}^{\left(1.5 + 1.5\right)}}} \]

       4. cbrt-unprod 27.5%

          \[\leadsto c0 \cdot \sqrt{{\left(\frac{\sqrt[3]{A}}{\color{blue}{\sqrt[3]{V \cdot \ell}}}\right)}^{\left(1.5 + 1.5\right)}} \]

       5. cbrt-undiv 27.5%

          \[\leadsto c0 \cdot \sqrt{{\color{blue}{\left(\sqrt[3]{\frac{A}{V \cdot \ell}}\right)}}^{\left(1.5 + 1.5\right)}} \]

       6. *-un-lft-identity 27.5%

          \[\leadsto c0 \cdot \sqrt{{\left(\sqrt[3]{\frac{\color{blue}{1 \cdot A}}{V \cdot \ell}}\right)}^{\left(1.5 + 1.5\right)}} \]

       7. *-commutative 27.5%

          \[\leadsto c0 \cdot \sqrt{{\left(\sqrt[3]{\frac{1 \cdot A}{\color{blue}{\ell \cdot V}}}\right)}^{\left(1.5 + 1.5\right)}} \]

       8. frac-times 45.6%

          \[\leadsto c0 \cdot \sqrt{{\left(\sqrt[3]{\color{blue}{\frac{1}{\ell} \cdot \frac{A}{V}}}\right)}^{\left(1.5 + 1.5\right)}} \]

       9. associate-/r/ 44.9%

          \[\leadsto c0 \cdot \sqrt{{\left(\sqrt[3]{\color{blue}{\frac{1}{\frac{\ell}{\frac{A}{V}}}}}\right)}^{\left(1.5 + 1.5\right)}} \]

       10. metadata-eval 44.9%

           \[\leadsto c0 \cdot \sqrt{{\left(\sqrt[3]{\frac{1}{\frac{\ell}{\frac{A}{V}}}}\right)}^{\color{blue}{3}}} \]

       11. pow3 44.9%

           \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\sqrt[3]{\frac{1}{\frac{\ell}{\frac{A}{V}}}} \cdot \sqrt[3]{\frac{1}{\frac{\ell}{\frac{A}{V}}}}\right) \cdot \sqrt[3]{\frac{1}{\frac{\ell}{\frac{A}{V}}}}}} \]

       12. add-cube-cbrt 45.1%

           \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{\frac{A}{V}}}}} \]

       13. sqrt-div 46.2%

           \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{\frac{A}{V}}}}} \]

       14. metadata-eval 46.2%

           \[\leadsto c0 \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{\frac{A}{V}}}} \]

       15. frac-2neg 46.2%

           \[\leadsto c0 \cdot \frac{1}{\sqrt{\color{blue}{\frac{-\ell}{-\frac{A}{V}}}}} \]

   13. Applied egg-rr 40.9%

       \[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{\frac{A}{V}}}{\sqrt{\ell}}} \]
   14. Step-by-step derivation

       1. *-commutative 40.9%

          \[\leadsto \frac{\color{blue}{\sqrt{\frac{A}{V}} \cdot c0}}{\sqrt{\ell}} \]

       2. associate-*r/ 40.9%

          \[\leadsto \color{blue}{\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}} \]

   15. Simplified 40.9%

       \[\leadsto \color{blue}{\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}} \]

   if 0.0 < (/.f64 A (*.f64 V l)) < 5e307

   1. Initial program 98.8%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0 \lor \neg \left(\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{+307}\right):\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]

Alternative 7: 74.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (<= t_0 0.0)
     (* (sqrt (/ A V)) (/ c0 (sqrt l)))
     (if (<= t_0 5e+307)
       (* c0 (sqrt t_0))
       (/ c0 (* (sqrt (/ V A)) (sqrt l)))))))

C

double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = sqrt((A / V)) * (c0 / sqrt(l));
	} else if (t_0 <= 5e+307) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = c0 / (sqrt((V / A)) * sqrt(l));
	}
	return tmp;
}

Fortran

real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if (t_0 <= 0.0d0) then
        tmp = sqrt((a / v)) * (c0 / sqrt(l))
    else if (t_0 <= 5d+307) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = c0 / (sqrt((v / a)) * sqrt(l))
    end if
    code = tmp
end function

Java

public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = Math.sqrt((A / V)) * (c0 / Math.sqrt(l));
	} else if (t_0 <= 5e+307) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = c0 / (Math.sqrt((V / A)) * Math.sqrt(l));
	}
	return tmp;
}

Python

def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 0.0:
		tmp = math.sqrt((A / V)) * (c0 / math.sqrt(l))
	elif t_0 <= 5e+307:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = c0 / (math.sqrt((V / A)) * math.sqrt(l))
	return tmp

Julia

function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(sqrt(Float64(A / V)) * Float64(c0 / sqrt(l)));
	elseif (t_0 <= 5e+307)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 / Float64(sqrt(Float64(V / A)) * sqrt(l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = sqrt((A / V)) * (c0 / sqrt(l));
	elseif (t_0 <= 5e+307)
		tmp = c0 * sqrt(t_0);
	else
		tmp = c0 / (sqrt((V / A)) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] * N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+307], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 / N[(N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 A (*.f64 V l)) < 0.0

   1. Initial program 31.0%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-/r* 47.7%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      2. div-inv 47.7%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V} \cdot \frac{1}{\ell}}} \]

   3. Applied egg-rr 47.7%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V} \cdot \frac{1}{\ell}}} \]
   4. Step-by-step derivation

      1. pow1/2 47.7%

         \[\leadsto c0 \cdot \color{blue}{{\left(\frac{A}{V} \cdot \frac{1}{\ell}\right)}^{0.5}} \]

      2. metadata-eval 47.7%

         \[\leadsto c0 \cdot {\left(\frac{A}{V} \cdot \frac{1}{\ell}\right)}^{\color{blue}{\left(0.3333333333333333 \cdot 1.5\right)}} \]

      3. pow-pow 46.3%

         \[\leadsto c0 \cdot \color{blue}{{\left({\left(\frac{A}{V} \cdot \frac{1}{\ell}\right)}^{0.3333333333333333}\right)}^{1.5}} \]

      4. un-div-inv 46.3%

         \[\leadsto c0 \cdot {\left({\color{blue}{\left(\frac{\frac{A}{V}}{\ell}\right)}}^{0.3333333333333333}\right)}^{1.5} \]

      5. associate-/r* 31.0%

         \[\leadsto c0 \cdot {\left({\color{blue}{\left(\frac{A}{V \cdot \ell}\right)}}^{0.3333333333333333}\right)}^{1.5} \]

      6. pow1/3 31.0%

         \[\leadsto c0 \cdot {\color{blue}{\left(\sqrt[3]{\frac{A}{V \cdot \ell}}\right)}}^{1.5} \]

   5. Applied egg-rr 31.0%

      \[\leadsto c0 \cdot \color{blue}{{\left(\sqrt[3]{\frac{A}{V \cdot \ell}}\right)}^{1.5}} \]
   6. Step-by-step derivation

      1. associate-/r* 47.6%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\color{blue}{\frac{\frac{A}{V}}{\ell}}}\right)}^{1.5} \]

   7. Simplified 47.6%

      \[\leadsto c0 \cdot \color{blue}{{\left(\sqrt[3]{\frac{\frac{A}{V}}{\ell}}\right)}^{1.5}} \]
   8. Step-by-step derivation

      1. cbrt-div 81.0%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{\sqrt[3]{\frac{A}{V}}}{\sqrt[3]{\ell}}\right)}}^{1.5} \]

      2. div-inv 80.9%

         \[\leadsto c0 \cdot {\color{blue}{\left(\sqrt[3]{\frac{A}{V}} \cdot \frac{1}{\sqrt[3]{\ell}}\right)}}^{1.5} \]

      3. cbrt-div 94.7%

         \[\leadsto c0 \cdot {\left(\color{blue}{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}} \cdot \frac{1}{\sqrt[3]{\ell}}\right)}^{1.5} \]

      4. frac-times 94.9%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{\sqrt[3]{A} \cdot 1}{\sqrt[3]{V} \cdot \sqrt[3]{\ell}}\right)}}^{1.5} \]

   9. Applied egg-rr 94.9%

      \[\leadsto c0 \cdot {\color{blue}{\left(\frac{\sqrt[3]{A} \cdot 1}{\sqrt[3]{V} \cdot \sqrt[3]{\ell}}\right)}}^{1.5} \]
   10. Step-by-step derivation

       1. *-rgt-identity 94.9%

          \[\leadsto c0 \cdot {\left(\frac{\color{blue}{\sqrt[3]{A}}}{\sqrt[3]{V} \cdot \sqrt[3]{\ell}}\right)}^{1.5} \]

   11. Simplified 94.9%

       \[\leadsto c0 \cdot {\color{blue}{\left(\frac{\sqrt[3]{A}}{\sqrt[3]{V} \cdot \sqrt[3]{\ell}}\right)}}^{1.5} \]
   12. Step-by-step derivation

       1. add-sqr-sqrt 94.9%

          \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{{\left(\frac{\sqrt[3]{A}}{\sqrt[3]{V} \cdot \sqrt[3]{\ell}}\right)}^{1.5}} \cdot \sqrt{{\left(\frac{\sqrt[3]{A}}{\sqrt[3]{V} \cdot \sqrt[3]{\ell}}\right)}^{1.5}}\right)} \]

       2. sqrt-unprod 47.5%

          \[\leadsto c0 \cdot \color{blue}{\sqrt{{\left(\frac{\sqrt[3]{A}}{\sqrt[3]{V} \cdot \sqrt[3]{\ell}}\right)}^{1.5} \cdot {\left(\frac{\sqrt[3]{A}}{\sqrt[3]{V} \cdot \sqrt[3]{\ell}}\right)}^{1.5}}} \]

       3. pow-prod-up 47.5%

          \[\leadsto c0 \cdot \sqrt{\color{blue}{{\left(\frac{\sqrt[3]{A}}{\sqrt[3]{V} \cdot \sqrt[3]{\ell}}\right)}^{\left(1.5 + 1.5\right)}}} \]

       4. cbrt-unprod 31.0%

          \[\leadsto c0 \cdot \sqrt{{\left(\frac{\sqrt[3]{A}}{\color{blue}{\sqrt[3]{V \cdot \ell}}}\right)}^{\left(1.5 + 1.5\right)}} \]

       5. cbrt-undiv 31.0%

          \[\leadsto c0 \cdot \sqrt{{\color{blue}{\left(\sqrt[3]{\frac{A}{V \cdot \ell}}\right)}}^{\left(1.5 + 1.5\right)}} \]

       6. *-un-lft-identity 31.0%

          \[\leadsto c0 \cdot \sqrt{{\left(\sqrt[3]{\frac{\color{blue}{1 \cdot A}}{V \cdot \ell}}\right)}^{\left(1.5 + 1.5\right)}} \]

       7. *-commutative 31.0%

          \[\leadsto c0 \cdot \sqrt{{\left(\sqrt[3]{\frac{1 \cdot A}{\color{blue}{\ell \cdot V}}}\right)}^{\left(1.5 + 1.5\right)}} \]

       8. frac-times 47.6%

          \[\leadsto c0 \cdot \sqrt{{\left(\sqrt[3]{\color{blue}{\frac{1}{\ell} \cdot \frac{A}{V}}}\right)}^{\left(1.5 + 1.5\right)}} \]

       9. associate-/r/ 46.3%

          \[\leadsto c0 \cdot \sqrt{{\left(\sqrt[3]{\color{blue}{\frac{1}{\frac{\ell}{\frac{A}{V}}}}}\right)}^{\left(1.5 + 1.5\right)}} \]

       10. metadata-eval 46.3%

           \[\leadsto c0 \cdot \sqrt{{\left(\sqrt[3]{\frac{1}{\frac{\ell}{\frac{A}{V}}}}\right)}^{\color{blue}{3}}} \]

       11. pow3 46.3%

           \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\sqrt[3]{\frac{1}{\frac{\ell}{\frac{A}{V}}}} \cdot \sqrt[3]{\frac{1}{\frac{\ell}{\frac{A}{V}}}}\right) \cdot \sqrt[3]{\frac{1}{\frac{\ell}{\frac{A}{V}}}}}} \]

       12. add-cube-cbrt 46.5%

           \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{\frac{A}{V}}}}} \]

       13. sqrt-div 46.5%

           \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{\frac{A}{V}}}}} \]

       14. metadata-eval 46.5%

           \[\leadsto c0 \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{\frac{A}{V}}}} \]

       15. frac-2neg 46.5%

           \[\leadsto c0 \cdot \frac{1}{\sqrt{\color{blue}{\frac{-\ell}{-\frac{A}{V}}}}} \]

   13. Applied egg-rr 37.6%

       \[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{\frac{A}{V}}}{\sqrt{\ell}}} \]
   14. Step-by-step derivation

       1. *-commutative 37.6%

          \[\leadsto \frac{\color{blue}{\sqrt{\frac{A}{V}} \cdot c0}}{\sqrt{\ell}} \]

       2. associate-*r/ 39.2%

          \[\leadsto \color{blue}{\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}} \]

   15. Simplified 39.2%

       \[\leadsto \color{blue}{\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}} \]

   if 0.0 < (/.f64 A (*.f64 V l)) < 5e307

   1. Initial program 98.8%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]

   if 5e307 < (/.f64 A (*.f64 V l))

   1. Initial program 22.9%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. sqrt-div 22.6%

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}} \]

      2. associate-*r/ 22.6%

         \[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}} \]

   3. Applied egg-rr 22.6%

      \[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}} \]
   4. Step-by-step derivation

      1. associate-/l* 22.6%

         \[\leadsto \color{blue}{\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}} \]

   5. Simplified 22.6%

      \[\leadsto \color{blue}{\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}} \]
   6. Step-by-step derivation

      1. sqrt-undiv 25.6%

         \[\leadsto \frac{c0}{\color{blue}{\sqrt{\frac{V \cdot \ell}{A}}}} \]

      2. associate-*l/ 45.9%

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{V}{A} \cdot \ell}}} \]

      3. associate-/r/ 43.1%

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{V}{\frac{A}{\ell}}}}} \]

      4. associate-/r/ 45.9%

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{V}{A} \cdot \ell}}} \]

      5. sqrt-prod 46.8%

         \[\leadsto \frac{c0}{\color{blue}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}} \]

   7. Applied egg-rr 46.8%

      \[\leadsto \frac{c0}{\color{blue}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{+307}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 8: 62.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 A V l)
 :precision binary64
 (if (<= l -2e-310)
   (* c0 (* (sqrt A) (sqrt (/ (/ 1.0 V) l))))
   (/ c0 (* (sqrt (/ V A)) (sqrt l)))))

C

double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= -2e-310) {
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	} else {
		tmp = c0 / (sqrt((V / A)) * sqrt(l));
	}
	return tmp;
}

Fortran

real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= (-2d-310)) then
        tmp = c0 * (sqrt(a) * sqrt(((1.0d0 / v) / l)))
    else
        tmp = c0 / (sqrt((v / a)) * sqrt(l))
    end if
    code = tmp
end function

Java

public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= -2e-310) {
		tmp = c0 * (Math.sqrt(A) * Math.sqrt(((1.0 / V) / l)));
	} else {
		tmp = c0 / (Math.sqrt((V / A)) * Math.sqrt(l));
	}
	return tmp;
}

Python

def code(c0, A, V, l):
	tmp = 0
	if l <= -2e-310:
		tmp = c0 * (math.sqrt(A) * math.sqrt(((1.0 / V) / l)))
	else:
		tmp = c0 / (math.sqrt((V / A)) * math.sqrt(l))
	return tmp

Julia

function code(c0, A, V, l)
	tmp = 0.0
	if (l <= -2e-310)
		tmp = Float64(c0 * Float64(sqrt(A) * sqrt(Float64(Float64(1.0 / V) / l))));
	else
		tmp = Float64(c0 / Float64(sqrt(Float64(V / A)) * sqrt(l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (l <= -2e-310)
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	else
		tmp = c0 / (sqrt((V / A)) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, A_, V_, l_] := If[LessEqual[l, -2e-310], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Sqrt[N[(N[(1.0 / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[(N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.999999999999994e-310

   1. Initial program 70.3%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. pow1/2 70.3%

         \[\leadsto c0 \cdot \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{0.5}} \]

      2. clear-num 69.8%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)}}^{0.5} \]

      3. inv-pow 69.8%

         \[\leadsto c0 \cdot {\color{blue}{\left({\left(\frac{V \cdot \ell}{A}\right)}^{-1}\right)}}^{0.5} \]

      4. pow-pow 70.4%

         \[\leadsto c0 \cdot \color{blue}{{\left(\frac{V \cdot \ell}{A}\right)}^{\left(-1 \cdot 0.5\right)}} \]

      5. associate-/l* 64.9%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{V}{\frac{A}{\ell}}\right)}}^{\left(-1 \cdot 0.5\right)} \]

      6. metadata-eval 64.9%

         \[\leadsto c0 \cdot {\left(\frac{V}{\frac{A}{\ell}}\right)}^{\color{blue}{-0.5}} \]

   3. Applied egg-rr 64.9%

      \[\leadsto c0 \cdot \color{blue}{{\left(\frac{V}{\frac{A}{\ell}}\right)}^{-0.5}} \]
   4. Step-by-step derivation

      1. associate-/l* 70.4%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{V \cdot \ell}{A}\right)}}^{-0.5} \]

      2. *-lft-identity 70.4%

         \[\leadsto c0 \cdot {\left(\frac{V \cdot \ell}{\color{blue}{1 \cdot A}}\right)}^{-0.5} \]

      3. times-frac 66.8%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{V}{1} \cdot \frac{\ell}{A}\right)}}^{-0.5} \]

      4. /-rgt-identity 66.8%

         \[\leadsto c0 \cdot {\left(\color{blue}{V} \cdot \frac{\ell}{A}\right)}^{-0.5} \]

   5. Simplified 66.8%

      \[\leadsto c0 \cdot \color{blue}{{\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 66.5%

         \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{{\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}} \cdot \sqrt{{\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}}\right)} \]

      2. sqrt-unprod 66.1%

         \[\leadsto c0 \cdot \color{blue}{\sqrt{{\left(V \cdot \frac{\ell}{A}\right)}^{-0.5} \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}}} \]

      3. pow-prod-up 66.1%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{{\left(V \cdot \frac{\ell}{A}\right)}^{\left(-0.5 + -0.5\right)}}} \]

      4. metadata-eval 66.1%

         \[\leadsto c0 \cdot \sqrt{{\left(V \cdot \frac{\ell}{A}\right)}^{\color{blue}{-1}}} \]

      5. inv-pow 66.1%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{V \cdot \frac{\ell}{A}}}} \]

      6. associate-*r/ 69.8%

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{\frac{V \cdot \ell}{A}}}} \]

      7. clear-num 70.3%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      8. div-inv 69.8%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{A \cdot \frac{1}{V \cdot \ell}}} \]

      9. sqrt-prod 39.6%

         \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{A} \cdot \sqrt{\frac{1}{V \cdot \ell}}\right)} \]

   7. Applied egg-rr 39.6%

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{A} \cdot \sqrt{\frac{1}{V \cdot \ell}}\right)} \]
   8. Step-by-step derivation

      1. associate-/r* 40.3%

         \[\leadsto c0 \cdot \left(\sqrt{A} \cdot \sqrt{\color{blue}{\frac{\frac{1}{V}}{\ell}}}\right) \]

   9. Simplified 40.3%

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)} \]

   if -1.999999999999994e-310 < l

   1. Initial program 66.6%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. sqrt-div 39.2%

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}} \]

      2. associate-*r/ 38.5%

         \[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}} \]

   3. Applied egg-rr 38.5%

      \[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}} \]
   4. Step-by-step derivation

      1. associate-/l* 39.3%

         \[\leadsto \color{blue}{\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}} \]

   5. Simplified 39.3%

      \[\leadsto \color{blue}{\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}} \]
   6. Step-by-step derivation

      1. sqrt-undiv 66.6%

         \[\leadsto \frac{c0}{\color{blue}{\sqrt{\frac{V \cdot \ell}{A}}}} \]

      2. associate-*l/ 70.3%

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{V}{A} \cdot \ell}}} \]

      3. associate-/r/ 66.8%

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{V}{\frac{A}{\ell}}}}} \]

      4. associate-/r/ 70.3%

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{V}{A} \cdot \ell}}} \]

      5. sqrt-prod 83.0%

         \[\leadsto \frac{c0}{\color{blue}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}} \]

   7. Applied egg-rr 83.0%

      \[\leadsto \frac{c0}{\color{blue}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 9: 62.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 A V l)
 :precision binary64
 (if (<= l -2e-310)
   (/ c0 (/ (sqrt (* V l)) (sqrt A)))
   (/ c0 (* (sqrt (/ V A)) (sqrt l)))))

C

double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= -2e-310) {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	} else {
		tmp = c0 / (sqrt((V / A)) * sqrt(l));
	}
	return tmp;
}

Fortran

real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= (-2d-310)) then
        tmp = c0 / (sqrt((v * l)) / sqrt(a))
    else
        tmp = c0 / (sqrt((v / a)) * sqrt(l))
    end if
    code = tmp
end function

Java

public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= -2e-310) {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	} else {
		tmp = c0 / (Math.sqrt((V / A)) * Math.sqrt(l));
	}
	return tmp;
}

Python

def code(c0, A, V, l):
	tmp = 0
	if l <= -2e-310:
		tmp = c0 / (math.sqrt((V * l)) / math.sqrt(A))
	else:
		tmp = c0 / (math.sqrt((V / A)) * math.sqrt(l))
	return tmp

Julia

function code(c0, A, V, l)
	tmp = 0.0
	if (l <= -2e-310)
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	else
		tmp = Float64(c0 / Float64(sqrt(Float64(V / A)) * sqrt(l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (l <= -2e-310)
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	else
		tmp = c0 / (sqrt((V / A)) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, A_, V_, l_] := If[LessEqual[l, -2e-310], N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[(N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.999999999999994e-310

   1. Initial program 70.3%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. sqrt-div 40.1%

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}} \]

      2. associate-*r/ 39.4%

         \[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}} \]

   3. Applied egg-rr 39.4%

      \[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}} \]
   4. Step-by-step derivation

      1. associate-/l* 40.1%

         \[\leadsto \color{blue}{\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}} \]

   5. Simplified 40.1%

      \[\leadsto \color{blue}{\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}} \]

   if -1.999999999999994e-310 < l

   1. Initial program 66.6%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. sqrt-div 39.2%

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}} \]

      2. associate-*r/ 38.5%

         \[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}} \]

   3. Applied egg-rr 38.5%

      \[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}} \]
   4. Step-by-step derivation

      1. associate-/l* 39.3%

         \[\leadsto \color{blue}{\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}} \]

   5. Simplified 39.3%

      \[\leadsto \color{blue}{\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}} \]
   6. Step-by-step derivation

      1. sqrt-undiv 66.6%

         \[\leadsto \frac{c0}{\color{blue}{\sqrt{\frac{V \cdot \ell}{A}}}} \]

      2. associate-*l/ 70.3%

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{V}{A} \cdot \ell}}} \]

      3. associate-/r/ 66.8%

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{V}{\frac{A}{\ell}}}}} \]

      4. associate-/r/ 70.3%

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{V}{A} \cdot \ell}}} \]

      5. sqrt-prod 83.0%

         \[\leadsto \frac{c0}{\color{blue}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}} \]

   7. Applied egg-rr 83.0%

      \[\leadsto \frac{c0}{\color{blue}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 10: 80.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+297}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (<= t_0 0.0)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 5e+297) (* c0 (sqrt t_0)) (* c0 (pow (* V (/ l A)) -0.5))))))

C

double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 5e+297) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = c0 * pow((V * (l / A)), -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if (t_0 <= 0.0d0) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 5d+297) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = c0 * ((v * (l / a)) ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 5e+297) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = c0 * Math.pow((V * (l / A)), -0.5);
	}
	return tmp;
}

Python

def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 5e+297:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = c0 * math.pow((V * (l / A)), -0.5)
	return tmp

Julia

function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 5e+297)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 * (Float64(V * Float64(l / A)) ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 5e+297)
		tmp = c0 * sqrt(t_0);
	else
		tmp = c0 * ((V * (l / A)) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+297], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 * N[Power[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 A (*.f64 V l)) < 0.0

   1. Initial program 31.0%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. pow1/2 31.0%

         \[\leadsto c0 \cdot \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{0.5}} \]

      2. clear-num 31.0%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)}}^{0.5} \]

      3. inv-pow 31.0%

         \[\leadsto c0 \cdot {\color{blue}{\left({\left(\frac{V \cdot \ell}{A}\right)}^{-1}\right)}}^{0.5} \]

      4. pow-pow 31.0%

         \[\leadsto c0 \cdot \color{blue}{{\left(\frac{V \cdot \ell}{A}\right)}^{\left(-1 \cdot 0.5\right)}} \]

      5. associate-/l* 46.5%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{V}{\frac{A}{\ell}}\right)}}^{\left(-1 \cdot 0.5\right)} \]

      6. metadata-eval 46.5%

         \[\leadsto c0 \cdot {\left(\frac{V}{\frac{A}{\ell}}\right)}^{\color{blue}{-0.5}} \]

   3. Applied egg-rr 46.5%

      \[\leadsto c0 \cdot \color{blue}{{\left(\frac{V}{\frac{A}{\ell}}\right)}^{-0.5}} \]
   4. Step-by-step derivation

      1. associate-/l* 31.0%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{V \cdot \ell}{A}\right)}}^{-0.5} \]

      2. *-lft-identity 31.0%

         \[\leadsto c0 \cdot {\left(\frac{V \cdot \ell}{\color{blue}{1 \cdot A}}\right)}^{-0.5} \]

      3. times-frac 46.5%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{V}{1} \cdot \frac{\ell}{A}\right)}}^{-0.5} \]

      4. /-rgt-identity 46.5%

         \[\leadsto c0 \cdot {\left(\color{blue}{V} \cdot \frac{\ell}{A}\right)}^{-0.5} \]

   5. Simplified 46.5%

      \[\leadsto c0 \cdot \color{blue}{{\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}} \]

   6. Taylor expanded in c0 around 0 31.0%

      \[\leadsto \color{blue}{\sqrt{\frac{A}{V \cdot \ell}} \cdot c0} \]
   7. Step-by-step derivation

      1. associate-/r* 47.7%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \cdot c0 \]

   8. Simplified 47.7%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0} \]

   if 0.0 < (/.f64 A (*.f64 V l)) < 4.9999999999999998e297

   1. Initial program 98.8%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]

   if 4.9999999999999998e297 < (/.f64 A (*.f64 V l))

   1. Initial program 24.5%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. pow1/2 24.5%

         \[\leadsto c0 \cdot \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{0.5}} \]

      2. clear-num 24.5%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)}}^{0.5} \]

      3. inv-pow 24.5%

         \[\leadsto c0 \cdot {\color{blue}{\left({\left(\frac{V \cdot \ell}{A}\right)}^{-1}\right)}}^{0.5} \]

      4. pow-pow 27.2%

         \[\leadsto c0 \cdot \color{blue}{{\left(\frac{V \cdot \ell}{A}\right)}^{\left(-1 \cdot 0.5\right)}} \]

      5. associate-/l* 44.3%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{V}{\frac{A}{\ell}}\right)}}^{\left(-1 \cdot 0.5\right)} \]

      6. metadata-eval 44.3%

         \[\leadsto c0 \cdot {\left(\frac{V}{\frac{A}{\ell}}\right)}^{\color{blue}{-0.5}} \]

   3. Applied egg-rr 44.3%

      \[\leadsto c0 \cdot \color{blue}{{\left(\frac{V}{\frac{A}{\ell}}\right)}^{-0.5}} \]
   4. Step-by-step derivation

      1. associate-/l* 27.2%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{V \cdot \ell}{A}\right)}}^{-0.5} \]

      2. *-lft-identity 27.2%

         \[\leadsto c0 \cdot {\left(\frac{V \cdot \ell}{\color{blue}{1 \cdot A}}\right)}^{-0.5} \]

      3. times-frac 46.0%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{V}{1} \cdot \frac{\ell}{A}\right)}}^{-0.5} \]

      4. /-rgt-identity 46.0%

         \[\leadsto c0 \cdot {\left(\color{blue}{V} \cdot \frac{\ell}{A}\right)}^{-0.5} \]

   5. Simplified 46.0%

      \[\leadsto c0 \cdot \color{blue}{{\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{+297}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\ \end{array} \]

Alternative 11: 80.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;\sqrt{\frac{c0}{V} \cdot \left(A \cdot \frac{c0}{\ell}\right)}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+297}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (<= t_0 0.0)
     (sqrt (* (/ c0 V) (* A (/ c0 l))))
     (if (<= t_0 5e+297) (* c0 (sqrt t_0)) (* c0 (pow (* V (/ l A)) -0.5))))))

C

double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = sqrt(((c0 / V) * (A * (c0 / l))));
	} else if (t_0 <= 5e+297) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = c0 * pow((V * (l / A)), -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if (t_0 <= 0.0d0) then
        tmp = sqrt(((c0 / v) * (a * (c0 / l))))
    else if (t_0 <= 5d+297) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = c0 * ((v * (l / a)) ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = Math.sqrt(((c0 / V) * (A * (c0 / l))));
	} else if (t_0 <= 5e+297) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = c0 * Math.pow((V * (l / A)), -0.5);
	}
	return tmp;
}

Python

def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 0.0:
		tmp = math.sqrt(((c0 / V) * (A * (c0 / l))))
	elif t_0 <= 5e+297:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = c0 * math.pow((V * (l / A)), -0.5)
	return tmp

Julia

function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = sqrt(Float64(Float64(c0 / V) * Float64(A * Float64(c0 / l))));
	elseif (t_0 <= 5e+297)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 * (Float64(V * Float64(l / A)) ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = sqrt(((c0 / V) * (A * (c0 / l))));
	elseif (t_0 <= 5e+297)
		tmp = c0 * sqrt(t_0);
	else
		tmp = c0 * ((V * (l / A)) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[Sqrt[N[(N[(c0 / V), $MachinePrecision] * N[(A * N[(c0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 5e+297], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 * N[Power[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 A (*.f64 V l)) < 0.0

   1. Initial program 31.0%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-/r* 47.7%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      2. div-inv 47.7%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V} \cdot \frac{1}{\ell}}} \]

   3. Applied egg-rr 47.7%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V} \cdot \frac{1}{\ell}}} \]
   4. Step-by-step derivation

      1. pow1/2 47.7%

         \[\leadsto c0 \cdot \color{blue}{{\left(\frac{A}{V} \cdot \frac{1}{\ell}\right)}^{0.5}} \]

      2. metadata-eval 47.7%

         \[\leadsto c0 \cdot {\left(\frac{A}{V} \cdot \frac{1}{\ell}\right)}^{\color{blue}{\left(0.3333333333333333 \cdot 1.5\right)}} \]

      3. pow-pow 46.3%

         \[\leadsto c0 \cdot \color{blue}{{\left({\left(\frac{A}{V} \cdot \frac{1}{\ell}\right)}^{0.3333333333333333}\right)}^{1.5}} \]

      4. un-div-inv 46.3%

         \[\leadsto c0 \cdot {\left({\color{blue}{\left(\frac{\frac{A}{V}}{\ell}\right)}}^{0.3333333333333333}\right)}^{1.5} \]

      5. associate-/r* 31.0%

         \[\leadsto c0 \cdot {\left({\color{blue}{\left(\frac{A}{V \cdot \ell}\right)}}^{0.3333333333333333}\right)}^{1.5} \]

      6. pow1/3 31.0%

         \[\leadsto c0 \cdot {\color{blue}{\left(\sqrt[3]{\frac{A}{V \cdot \ell}}\right)}}^{1.5} \]

   5. Applied egg-rr 31.0%

      \[\leadsto c0 \cdot \color{blue}{{\left(\sqrt[3]{\frac{A}{V \cdot \ell}}\right)}^{1.5}} \]
   6. Step-by-step derivation

      1. associate-/r* 47.6%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\color{blue}{\frac{\frac{A}{V}}{\ell}}}\right)}^{1.5} \]

   7. Simplified 47.6%

      \[\leadsto c0 \cdot \color{blue}{{\left(\sqrt[3]{\frac{\frac{A}{V}}{\ell}}\right)}^{1.5}} \]
   8. Step-by-step derivation

      1. cbrt-div 81.0%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{\sqrt[3]{\frac{A}{V}}}{\sqrt[3]{\ell}}\right)}}^{1.5} \]

      2. div-inv 80.9%

         \[\leadsto c0 \cdot {\color{blue}{\left(\sqrt[3]{\frac{A}{V}} \cdot \frac{1}{\sqrt[3]{\ell}}\right)}}^{1.5} \]

      3. cbrt-div 94.7%

         \[\leadsto c0 \cdot {\left(\color{blue}{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}} \cdot \frac{1}{\sqrt[3]{\ell}}\right)}^{1.5} \]

      4. frac-times 94.9%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{\sqrt[3]{A} \cdot 1}{\sqrt[3]{V} \cdot \sqrt[3]{\ell}}\right)}}^{1.5} \]

   9. Applied egg-rr 94.9%

      \[\leadsto c0 \cdot {\color{blue}{\left(\frac{\sqrt[3]{A} \cdot 1}{\sqrt[3]{V} \cdot \sqrt[3]{\ell}}\right)}}^{1.5} \]
   10. Step-by-step derivation

       1. *-rgt-identity 94.9%

          \[\leadsto c0 \cdot {\left(\frac{\color{blue}{\sqrt[3]{A}}}{\sqrt[3]{V} \cdot \sqrt[3]{\ell}}\right)}^{1.5} \]

   11. Simplified 94.9%

       \[\leadsto c0 \cdot {\color{blue}{\left(\frac{\sqrt[3]{A}}{\sqrt[3]{V} \cdot \sqrt[3]{\ell}}\right)}}^{1.5} \]
   12. Step-by-step derivation

       1. add-sqr-sqrt 94.9%

          \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{{\left(\frac{\sqrt[3]{A}}{\sqrt[3]{V} \cdot \sqrt[3]{\ell}}\right)}^{1.5}} \cdot \sqrt{{\left(\frac{\sqrt[3]{A}}{\sqrt[3]{V} \cdot \sqrt[3]{\ell}}\right)}^{1.5}}\right)} \]

       2. sqrt-unprod 47.5%

          \[\leadsto c0 \cdot \color{blue}{\sqrt{{\left(\frac{\sqrt[3]{A}}{\sqrt[3]{V} \cdot \sqrt[3]{\ell}}\right)}^{1.5} \cdot {\left(\frac{\sqrt[3]{A}}{\sqrt[3]{V} \cdot \sqrt[3]{\ell}}\right)}^{1.5}}} \]

       3. pow-prod-up 47.5%

          \[\leadsto c0 \cdot \sqrt{\color{blue}{{\left(\frac{\sqrt[3]{A}}{\sqrt[3]{V} \cdot \sqrt[3]{\ell}}\right)}^{\left(1.5 + 1.5\right)}}} \]

       4. cbrt-unprod 31.0%

          \[\leadsto c0 \cdot \sqrt{{\left(\frac{\sqrt[3]{A}}{\color{blue}{\sqrt[3]{V \cdot \ell}}}\right)}^{\left(1.5 + 1.5\right)}} \]

       5. cbrt-undiv 31.0%

          \[\leadsto c0 \cdot \sqrt{{\color{blue}{\left(\sqrt[3]{\frac{A}{V \cdot \ell}}\right)}}^{\left(1.5 + 1.5\right)}} \]

       6. *-un-lft-identity 31.0%

          \[\leadsto c0 \cdot \sqrt{{\left(\sqrt[3]{\frac{\color{blue}{1 \cdot A}}{V \cdot \ell}}\right)}^{\left(1.5 + 1.5\right)}} \]

       7. *-commutative 31.0%

          \[\leadsto c0 \cdot \sqrt{{\left(\sqrt[3]{\frac{1 \cdot A}{\color{blue}{\ell \cdot V}}}\right)}^{\left(1.5 + 1.5\right)}} \]

       8. frac-times 47.6%

          \[\leadsto c0 \cdot \sqrt{{\left(\sqrt[3]{\color{blue}{\frac{1}{\ell} \cdot \frac{A}{V}}}\right)}^{\left(1.5 + 1.5\right)}} \]

       9. associate-/r/ 46.3%

          \[\leadsto c0 \cdot \sqrt{{\left(\sqrt[3]{\color{blue}{\frac{1}{\frac{\ell}{\frac{A}{V}}}}}\right)}^{\left(1.5 + 1.5\right)}} \]

       10. metadata-eval 46.3%

           \[\leadsto c0 \cdot \sqrt{{\left(\sqrt[3]{\frac{1}{\frac{\ell}{\frac{A}{V}}}}\right)}^{\color{blue}{3}}} \]

       11. pow3 46.3%

           \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\sqrt[3]{\frac{1}{\frac{\ell}{\frac{A}{V}}}} \cdot \sqrt[3]{\frac{1}{\frac{\ell}{\frac{A}{V}}}}\right) \cdot \sqrt[3]{\frac{1}{\frac{\ell}{\frac{A}{V}}}}}} \]

       12. add-cube-cbrt 46.5%

           \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{\frac{A}{V}}}}} \]

   13. Applied egg-rr 31.8%

       \[\leadsto \color{blue}{\sqrt{\frac{c0 \cdot c0}{V \cdot \frac{\ell}{A}}}} \]
   14. Step-by-step derivation

       1. times-frac 44.8%

          \[\leadsto \sqrt{\color{blue}{\frac{c0}{V} \cdot \frac{c0}{\frac{\ell}{A}}}} \]

       2. associate-/r/ 49.3%

          \[\leadsto \sqrt{\frac{c0}{V} \cdot \color{blue}{\left(\frac{c0}{\ell} \cdot A\right)}} \]

   15. Simplified 49.3%

       \[\leadsto \color{blue}{\sqrt{\frac{c0}{V} \cdot \left(\frac{c0}{\ell} \cdot A\right)}} \]

   if 0.0 < (/.f64 A (*.f64 V l)) < 4.9999999999999998e297

   1. Initial program 98.8%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]

   if 4.9999999999999998e297 < (/.f64 A (*.f64 V l))

   1. Initial program 24.5%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. pow1/2 24.5%

         \[\leadsto c0 \cdot \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{0.5}} \]

      2. clear-num 24.5%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)}}^{0.5} \]

      3. inv-pow 24.5%

         \[\leadsto c0 \cdot {\color{blue}{\left({\left(\frac{V \cdot \ell}{A}\right)}^{-1}\right)}}^{0.5} \]

      4. pow-pow 27.2%

         \[\leadsto c0 \cdot \color{blue}{{\left(\frac{V \cdot \ell}{A}\right)}^{\left(-1 \cdot 0.5\right)}} \]

      5. associate-/l* 44.3%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{V}{\frac{A}{\ell}}\right)}}^{\left(-1 \cdot 0.5\right)} \]

      6. metadata-eval 44.3%

         \[\leadsto c0 \cdot {\left(\frac{V}{\frac{A}{\ell}}\right)}^{\color{blue}{-0.5}} \]

   3. Applied egg-rr 44.3%

      \[\leadsto c0 \cdot \color{blue}{{\left(\frac{V}{\frac{A}{\ell}}\right)}^{-0.5}} \]
   4. Step-by-step derivation

      1. associate-/l* 27.2%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{V \cdot \ell}{A}\right)}}^{-0.5} \]

      2. *-lft-identity 27.2%

         \[\leadsto c0 \cdot {\left(\frac{V \cdot \ell}{\color{blue}{1 \cdot A}}\right)}^{-0.5} \]

      3. times-frac 46.0%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{V}{1} \cdot \frac{\ell}{A}\right)}}^{-0.5} \]

      4. /-rgt-identity 46.0%

         \[\leadsto c0 \cdot {\left(\color{blue}{V} \cdot \frac{\ell}{A}\right)}^{-0.5} \]

   5. Simplified 46.0%

      \[\leadsto c0 \cdot \color{blue}{{\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0:\\ \;\;\;\;\sqrt{\frac{c0}{V} \cdot \left(A \cdot \frac{c0}{\ell}\right)}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{+297}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\ \end{array} \]

Alternative 12: 79.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0 \lor \neg \left(t_0 \leq 5 \cdot 10^{+307}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (or (<= t_0 0.0) (not (<= t_0 5e+307)))
     (* c0 (sqrt (/ (/ A V) l)))
     (* c0 (sqrt t_0)))))

C

double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 5e+307)) {
		tmp = c0 * sqrt(((A / V) / l));
	} else {
		tmp = c0 * sqrt(t_0);
	}
	return tmp;
}

Fortran

real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if ((t_0 <= 0.0d0) .or. (.not. (t_0 <= 5d+307))) then
        tmp = c0 * sqrt(((a / v) / l))
    else
        tmp = c0 * sqrt(t_0)
    end if
    code = tmp
end function

Java

public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 5e+307)) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else {
		tmp = c0 * Math.sqrt(t_0);
	}
	return tmp;
}

Python

def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 5e+307):
		tmp = c0 * math.sqrt(((A / V) / l))
	else:
		tmp = c0 * math.sqrt(t_0)
	return tmp

Julia

function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 5e+307))
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	else
		tmp = Float64(c0 * sqrt(t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 5e+307)))
		tmp = c0 * sqrt(((A / V) / l));
	else
		tmp = c0 * sqrt(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 5e+307]], $MachinePrecision]], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 0.0 or 5e307 < (/.f64 A (*.f64 V l))

   1. Initial program 27.5%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. pow1/2 27.5%

         \[\leadsto c0 \cdot \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{0.5}} \]

      2. clear-num 27.5%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)}}^{0.5} \]

      3. inv-pow 27.5%

         \[\leadsto c0 \cdot {\color{blue}{\left({\left(\frac{V \cdot \ell}{A}\right)}^{-1}\right)}}^{0.5} \]

      4. pow-pow 28.7%

         \[\leadsto c0 \cdot \color{blue}{{\left(\frac{V \cdot \ell}{A}\right)}^{\left(-1 \cdot 0.5\right)}} \]

      5. associate-/l* 45.1%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{V}{\frac{A}{\ell}}\right)}}^{\left(-1 \cdot 0.5\right)} \]

      6. metadata-eval 45.1%

         \[\leadsto c0 \cdot {\left(\frac{V}{\frac{A}{\ell}}\right)}^{\color{blue}{-0.5}} \]

   3. Applied egg-rr 45.1%

      \[\leadsto c0 \cdot \color{blue}{{\left(\frac{V}{\frac{A}{\ell}}\right)}^{-0.5}} \]
   4. Step-by-step derivation

      1. associate-/l* 28.7%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{V \cdot \ell}{A}\right)}}^{-0.5} \]

      2. *-lft-identity 28.7%

         \[\leadsto c0 \cdot {\left(\frac{V \cdot \ell}{\color{blue}{1 \cdot A}}\right)}^{-0.5} \]

      3. times-frac 45.8%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{V}{1} \cdot \frac{\ell}{A}\right)}}^{-0.5} \]

      4. /-rgt-identity 45.8%

         \[\leadsto c0 \cdot {\left(\color{blue}{V} \cdot \frac{\ell}{A}\right)}^{-0.5} \]

   5. Simplified 45.8%

      \[\leadsto c0 \cdot \color{blue}{{\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}} \]

   6. Taylor expanded in c0 around 0 27.5%

      \[\leadsto \color{blue}{\sqrt{\frac{A}{V \cdot \ell}} \cdot c0} \]
   7. Step-by-step derivation

      1. associate-/r* 45.7%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \cdot c0 \]

   8. Simplified 45.7%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0} \]

   if 0.0 < (/.f64 A (*.f64 V l)) < 5e307

   1. Initial program 98.8%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0 \lor \neg \left(\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{+307}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]

Alternative 13: 80.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+297}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (<= t_0 0.0)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 5e+297) (* c0 (sqrt t_0)) (/ c0 (sqrt (* V (/ l A))))))))

C

double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 5e+297) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = c0 / sqrt((V * (l / A)));
	}
	return tmp;
}

Fortran

real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if (t_0 <= 0.0d0) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 5d+297) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = c0 / sqrt((v * (l / a)))
    end if
    code = tmp
end function

Java

public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 5e+297) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = c0 / Math.sqrt((V * (l / A)));
	}
	return tmp;
}

Python

def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 5e+297:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = c0 / math.sqrt((V * (l / A)))
	return tmp

Julia

function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 5e+297)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 5e+297)
		tmp = c0 * sqrt(t_0);
	else
		tmp = c0 / sqrt((V * (l / A)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+297], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 A (*.f64 V l)) < 0.0

   1. Initial program 31.0%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. pow1/2 31.0%

         \[\leadsto c0 \cdot \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{0.5}} \]

      2. clear-num 31.0%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)}}^{0.5} \]

      3. inv-pow 31.0%

         \[\leadsto c0 \cdot {\color{blue}{\left({\left(\frac{V \cdot \ell}{A}\right)}^{-1}\right)}}^{0.5} \]

      4. pow-pow 31.0%

         \[\leadsto c0 \cdot \color{blue}{{\left(\frac{V \cdot \ell}{A}\right)}^{\left(-1 \cdot 0.5\right)}} \]

      5. associate-/l* 46.5%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{V}{\frac{A}{\ell}}\right)}}^{\left(-1 \cdot 0.5\right)} \]

      6. metadata-eval 46.5%

         \[\leadsto c0 \cdot {\left(\frac{V}{\frac{A}{\ell}}\right)}^{\color{blue}{-0.5}} \]

   3. Applied egg-rr 46.5%

      \[\leadsto c0 \cdot \color{blue}{{\left(\frac{V}{\frac{A}{\ell}}\right)}^{-0.5}} \]
   4. Step-by-step derivation

      1. associate-/l* 31.0%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{V \cdot \ell}{A}\right)}}^{-0.5} \]

      2. *-lft-identity 31.0%

         \[\leadsto c0 \cdot {\left(\frac{V \cdot \ell}{\color{blue}{1 \cdot A}}\right)}^{-0.5} \]

      3. times-frac 46.5%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{V}{1} \cdot \frac{\ell}{A}\right)}}^{-0.5} \]

      4. /-rgt-identity 46.5%

         \[\leadsto c0 \cdot {\left(\color{blue}{V} \cdot \frac{\ell}{A}\right)}^{-0.5} \]

   5. Simplified 46.5%

      \[\leadsto c0 \cdot \color{blue}{{\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}} \]

   6. Taylor expanded in c0 around 0 31.0%

      \[\leadsto \color{blue}{\sqrt{\frac{A}{V \cdot \ell}} \cdot c0} \]
   7. Step-by-step derivation

      1. associate-/r* 47.7%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \cdot c0 \]

   8. Simplified 47.7%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0} \]

   if 0.0 < (/.f64 A (*.f64 V l)) < 4.9999999999999998e297

   1. Initial program 98.8%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]

   if 4.9999999999999998e297 < (/.f64 A (*.f64 V l))

   1. Initial program 24.5%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. pow1/2 24.5%

         \[\leadsto c0 \cdot \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{0.5}} \]

      2. clear-num 24.5%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)}}^{0.5} \]

      3. inv-pow 24.5%

         \[\leadsto c0 \cdot {\color{blue}{\left({\left(\frac{V \cdot \ell}{A}\right)}^{-1}\right)}}^{0.5} \]

      4. pow-pow 27.2%

         \[\leadsto c0 \cdot \color{blue}{{\left(\frac{V \cdot \ell}{A}\right)}^{\left(-1 \cdot 0.5\right)}} \]

      5. associate-/l* 44.3%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{V}{\frac{A}{\ell}}\right)}}^{\left(-1 \cdot 0.5\right)} \]

      6. metadata-eval 44.3%

         \[\leadsto c0 \cdot {\left(\frac{V}{\frac{A}{\ell}}\right)}^{\color{blue}{-0.5}} \]

   3. Applied egg-rr 44.3%

      \[\leadsto c0 \cdot \color{blue}{{\left(\frac{V}{\frac{A}{\ell}}\right)}^{-0.5}} \]
   4. Step-by-step derivation

      1. associate-/l* 27.2%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{V \cdot \ell}{A}\right)}}^{-0.5} \]

      2. *-lft-identity 27.2%

         \[\leadsto c0 \cdot {\left(\frac{V \cdot \ell}{\color{blue}{1 \cdot A}}\right)}^{-0.5} \]

      3. times-frac 46.0%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{V}{1} \cdot \frac{\ell}{A}\right)}}^{-0.5} \]

      4. /-rgt-identity 46.0%

         \[\leadsto c0 \cdot {\left(\color{blue}{V} \cdot \frac{\ell}{A}\right)}^{-0.5} \]

   5. Simplified 46.0%

      \[\leadsto c0 \cdot \color{blue}{{\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 46.0%

         \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{{\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}} \cdot \sqrt{{\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}}\right)} \]

      2. sqrt-unprod 44.4%

         \[\leadsto c0 \cdot \color{blue}{\sqrt{{\left(V \cdot \frac{\ell}{A}\right)}^{-0.5} \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}}} \]

      3. pow-prod-up 44.4%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{{\left(V \cdot \frac{\ell}{A}\right)}^{\left(-0.5 + -0.5\right)}}} \]

      4. metadata-eval 44.4%

         \[\leadsto c0 \cdot \sqrt{{\left(V \cdot \frac{\ell}{A}\right)}^{\color{blue}{-1}}} \]

      5. inv-pow 44.4%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{V \cdot \frac{\ell}{A}}}} \]

      6. sqrt-div 46.0%

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{V \cdot \frac{\ell}{A}}}} \]

      7. metadata-eval 46.0%

         \[\leadsto c0 \cdot \frac{\color{blue}{1}}{\sqrt{V \cdot \frac{\ell}{A}}} \]

      8. associate-*r/ 27.1%

         \[\leadsto c0 \cdot \frac{1}{\sqrt{\color{blue}{\frac{V \cdot \ell}{A}}}} \]

      9. sqrt-undiv 22.1%

         \[\leadsto c0 \cdot \frac{1}{\color{blue}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}} \]

      10. div-inv 22.1%

          \[\leadsto \color{blue}{\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}} \]

      11. sqrt-undiv 27.2%

          \[\leadsto \frac{c0}{\color{blue}{\sqrt{\frac{V \cdot \ell}{A}}}} \]

      12. associate-*r/ 46.0%

          \[\leadsto \frac{c0}{\sqrt{\color{blue}{V \cdot \frac{\ell}{A}}}} \]

   7. Applied egg-rr 46.0%

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{+297}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]

Alternative 14: 73.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \end{array} \]

FPCore

(FPCore (c0 A V l) :precision binary64 (* c0 (sqrt (/ A (* V l)))))

C

double code(double c0, double A, double V, double l) {
	return c0 * sqrt((A / (V * l)));
}

Fortran

real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    code = c0 * sqrt((a / (v * l)))
end function

Java

public static double code(double c0, double A, double V, double l) {
	return c0 * Math.sqrt((A / (V * l)));
}

Python

def code(c0, A, V, l):
	return c0 * math.sqrt((A / (V * l)))

Julia

function code(c0, A, V, l)
	return Float64(c0 * sqrt(Float64(A / Float64(V * l))))
end

MATLAB

function tmp = code(c0, A, V, l)
	tmp = c0 * sqrt((A / (V * l)));
end

Wolfram

code[c0_, A_, V_, l_] := N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.5%

   \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]

2. Final simplification 68.5%

   \[\leadsto c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]

Reproduce

herbie shell --seed 2023240 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))