Henrywood and Agarwal, Equation (3)

Percentage Accurate: 74.9% → 96.8%

Time: 13.2s

Alternatives: 16

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: 74.9% 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: 96.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ c0 \cdot {\left(\frac{\sqrt[3]{A} \cdot \sqrt[3]{\frac{1}{V}}}{\sqrt[3]{\ell}}\right)}^{1.5} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (* c0 (pow (/ (* (cbrt A) (cbrt (/ 1.0 V))) (cbrt l)) 1.5)))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	return c0 * pow(((cbrt(A) * cbrt((1.0 / V))) / cbrt(l)), 1.5);
}

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	return c0 * Math.pow(((Math.cbrt(A) * Math.cbrt((1.0 / V))) / Math.cbrt(l)), 1.5);
}

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	return Float64(c0 * (Float64(Float64(cbrt(A) * cbrt(Float64(1.0 / V))) / cbrt(l)) ^ 1.5))
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := N[(c0 * N[Power[N[(N[(N[Power[A, 1/3], $MachinePrecision] * N[Power[N[(1.0 / V), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.2%

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

   1. pow1/2 73.2%

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

   2. add-cube-cbrt 72.7%

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

   3. pow3 72.7%

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

   4. pow-pow 72.6%

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

   5. metadata-eval 72.6%

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

3. Applied egg-rr 72.6%

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

   1. cbrt-div 83.9%

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

   2. *-rgt-identity 83.9%

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

   3. cbrt-div 72.6%

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

   4. frac-times 74.4%

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

   5. *-commutative 74.4%

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

   6. cbrt-prod 84.2%

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

   7. cbrt-div 84.1%

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

   8. metadata-eval 84.1%

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

5. Applied egg-rr 84.1%

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

   1. associate-*l/ 84.1%

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

   2. *-lft-identity 84.1%

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

7. Simplified 84.1%

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

   1. div-inv 84.1%

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

   2. cbrt-prod 96.7%

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

9. Applied egg-rr 96.7%

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

10. Final simplification 96.7%

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

Alternative 2: 96.8% accurate, 0.3× speedup

Math

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

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (* c0 (pow (/ (/ (cbrt A) (cbrt V)) (cbrt l)) 1.5)))

C

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

Java

assert V < l;
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

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

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := N[(c0 * N[Power[N[(N[(N[Power[A, 1/3], $MachinePrecision] / N[Power[V, 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.2%

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

   1. pow1/2 73.2%

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

   2. add-cube-cbrt 72.7%

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

   3. pow3 72.7%

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

   4. pow-pow 72.6%

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

   5. metadata-eval 72.6%

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

3. Applied egg-rr 72.6%

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

   1. cbrt-div 83.9%

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

   2. *-rgt-identity 83.9%

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

   3. cbrt-div 72.6%

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

   4. frac-times 74.4%

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

   5. *-commutative 74.4%

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

   6. cbrt-prod 84.2%

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

   7. cbrt-div 84.1%

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

   8. metadata-eval 84.1%

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

5. Applied egg-rr 84.1%

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

   1. associate-*l/ 84.1%

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

   2. *-lft-identity 84.1%

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

7. Simplified 84.1%

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

   1. cbrt-div 96.6%

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

   2. div-inv 96.6%

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

9. Applied egg-rr 96.6%

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

    1. associate-*r/ 96.6%

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

    2. *-rgt-identity 96.6%

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

11. Simplified 96.6%

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

12. Final simplification 96.6%

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

Alternative 3: 95.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{\frac{c0}{\sqrt{\frac{V}{A}}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-306}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-301} \lor \neg \left(V \cdot \ell \leq 10^{+298}\right):\\ \;\;\;\;c0 \cdot {\left(\frac{\sqrt[3]{\frac{A}{V}}}{\sqrt[3]{\ell}}\right)}^{1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) (- INFINITY))
   (/ (/ c0 (sqrt (/ V A))) (sqrt l))
   (if (<= (* V l) -1e-306)
     (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
     (if (or (<= (* V l) 2e-301) (not (<= (* V l) 1e+298)))
       (* c0 (pow (/ (cbrt (/ A V)) (cbrt l)) 1.5))
       (/ c0 (/ (sqrt (* V l)) (sqrt A)))))))

C

assert(V < l);
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-306) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if (((V * l) <= 2e-301) || !((V * l) <= 1e+298)) {
		tmp = c0 * pow((cbrt((A / V)) / cbrt(l)), 1.5);
	} else {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	}
	return tmp;
}

Java

assert V < l;
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-306) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if (((V * l) <= 2e-301) || !((V * l) <= 1e+298)) {
		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

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(Float64(c0 / sqrt(Float64(V / A))) / sqrt(l));
	elseif (Float64(V * l) <= -1e-306)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif ((Float64(V * l) <= 2e-301) || !(Float64(V * l) <= 1e+298))
		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

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(N[(c0 / N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-306], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(V * l), $MachinePrecision], 2e-301], N[Not[LessEqual[N[(V * l), $MachinePrecision], 1e+298]], $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 41.1%

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

      1. associate-/r* 76.5%

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

      2. div-inv 76.4%

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

   3. Applied egg-rr 76.4%

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

      1. sqrt-prod 37.8%

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

      2. associate-*r* 37.8%

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

      3. sqrt-div 37.8%

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

      4. metadata-eval 37.8%

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

      5. div-inv 37.9%

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

   5. Applied egg-rr 37.9%

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

      1. sqrt-div 0.0%

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

      2. clear-num 0.0%

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

      3. un-div-inv 0.0%

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

      4. sqrt-undiv 38.0%

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

   7. Applied egg-rr 38.0%

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

   if -inf.0 < (*.f64 V l) < -1.00000000000000003e-306

   1. Initial program 87.5%

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

      1. frac-2neg 87.5%

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

      2. sqrt-div 99.5%

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

      3. *-commutative 99.5%

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

      4. distribute-rgt-neg-in 99.5%

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

   3. Applied egg-rr 99.5%

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

   if -1.00000000000000003e-306 < (*.f64 V l) < 2.00000000000000013e-301 or 9.9999999999999996e297 < (*.f64 V l)

   1. Initial program 34.0%

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

      1. pow1/2 34.0%

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

      2. add-cube-cbrt 34.0%

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

      3. pow3 34.0%

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

      4. pow-pow 34.0%

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

      5. metadata-eval 34.0%

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

   3. Applied egg-rr 34.0%

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

      1. cbrt-div 36.8%

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

      2. *-rgt-identity 36.8%

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

      3. cbrt-div 34.0%

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

      4. frac-times 53.7%

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

      5. *-commutative 53.7%

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

      6. cbrt-prod 75.3%

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

      7. cbrt-div 75.3%

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

      8. metadata-eval 75.3%

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

   5. Applied egg-rr 75.3%

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

      1. associate-*l/ 75.4%

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

      2. *-lft-identity 75.4%

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

   7. Simplified 75.4%

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

   if 2.00000000000000013e-301 < (*.f64 V l) < 9.9999999999999996e297

   1. Initial program 82.6%

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

      1. sqrt-div 99.5%

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

      2. associate-*r/ 96.0%

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

   3. Applied egg-rr 96.0%

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

      1. associate-/l* 99.6%

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

   5. Simplified 99.6%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{\frac{c0}{\sqrt{\frac{V}{A}}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-306}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-301} \lor \neg \left(V \cdot \ell \leq 10^{+298}\right):\\ \;\;\;\;c0 \cdot {\left(\frac{\sqrt[3]{\frac{A}{V}}}{\sqrt[3]{\ell}}\right)}^{1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \end{array} \]

Alternative 4: 95.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{\frac{c0}{\sqrt{\frac{V}{A}}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-306}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-318} \lor \neg \left(V \cdot \ell \leq 10^{+298}\right):\\ \;\;\;\;c0 \cdot {\left(\frac{\sqrt[3]{\frac{A}{\ell}}}{\sqrt[3]{V}}\right)}^{1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) (- INFINITY))
   (/ (/ c0 (sqrt (/ V A))) (sqrt l))
   (if (<= (* V l) -1e-306)
     (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
     (if (or (<= (* V l) 1e-318) (not (<= (* V l) 1e+298)))
       (* c0 (pow (/ (cbrt (/ A l)) (cbrt V)) 1.5))
       (/ c0 (/ (sqrt (* V l)) (sqrt A)))))))

C

assert(V < l);
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-306) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if (((V * l) <= 1e-318) || !((V * l) <= 1e+298)) {
		tmp = c0 * pow((cbrt((A / l)) / cbrt(V)), 1.5);
	} else {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	}
	return tmp;
}

Java

assert V < l;
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-306) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if (((V * l) <= 1e-318) || !((V * l) <= 1e+298)) {
		tmp = c0 * Math.pow((Math.cbrt((A / l)) / Math.cbrt(V)), 1.5);
	} else {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	}
	return tmp;
}

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(Float64(c0 / sqrt(Float64(V / A))) / sqrt(l));
	elseif (Float64(V * l) <= -1e-306)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif ((Float64(V * l) <= 1e-318) || !(Float64(V * l) <= 1e+298))
		tmp = Float64(c0 * (Float64(cbrt(Float64(A / l)) / cbrt(V)) ^ 1.5));
	else
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	end
	return tmp
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(N[(c0 / N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-306], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(V * l), $MachinePrecision], 1e-318], N[Not[LessEqual[N[(V * l), $MachinePrecision], 1e+298]], $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], 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 41.1%

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

      1. associate-/r* 76.5%

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

      2. div-inv 76.4%

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

   3. Applied egg-rr 76.4%

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

      1. sqrt-prod 37.8%

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

      2. associate-*r* 37.8%

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

      3. sqrt-div 37.8%

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

      4. metadata-eval 37.8%

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

      5. div-inv 37.9%

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

   5. Applied egg-rr 37.9%

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

      1. sqrt-div 0.0%

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

      2. clear-num 0.0%

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

      3. un-div-inv 0.0%

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

      4. sqrt-undiv 38.0%

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

   7. Applied egg-rr 38.0%

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

   if -inf.0 < (*.f64 V l) < -1.00000000000000003e-306

   1. Initial program 87.5%

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

      1. frac-2neg 87.5%

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

      2. sqrt-div 99.5%

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

      3. *-commutative 99.5%

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

      4. distribute-rgt-neg-in 99.5%

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

   3. Applied egg-rr 99.5%

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

   if -1.00000000000000003e-306 < (*.f64 V l) < 9.9999875e-319 or 9.9999999999999996e297 < (*.f64 V l)

   1. Initial program 34.6%

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

      1. pow1/2 34.6%

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

      2. add-cube-cbrt 34.6%

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

      3. pow3 34.6%

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

      4. pow-pow 34.6%

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

      5. metadata-eval 34.6%

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

   3. Applied egg-rr 34.6%

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

      1. *-un-lft-identity 34.6%

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

      2. frac-times 54.7%

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

      3. *-commutative 54.7%

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

      4. cbrt-prod 80.0%

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

      5. cbrt-div 80.1%

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

      6. metadata-eval 80.1%

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

   5. Applied egg-rr 80.1%

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

      1. associate-*r/ 80.3%

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

      2. *-rgt-identity 80.3%

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

   7. Simplified 80.3%

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

   if 9.9999875e-319 < (*.f64 V l) < 9.9999999999999996e297

   1. Initial program 81.8%

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

      1. sqrt-div 99.5%

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

      2. associate-*r/ 96.0%

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

   3. Applied egg-rr 96.0%

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

      1. associate-/l* 99.6%

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

   5. Simplified 99.6%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{\frac{c0}{\sqrt{\frac{V}{A}}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-306}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-318} \lor \neg \left(V \cdot \ell \leq 10^{+298}\right):\\ \;\;\;\;c0 \cdot {\left(\frac{\sqrt[3]{\frac{A}{\ell}}}{\sqrt[3]{V}}\right)}^{1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \end{array} \]

Alternative 5: 96.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{-306}:\\ \;\;\;\;c0 \cdot \frac{\frac{\sqrt{-A}}{\sqrt{-V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-318} \lor \neg \left(V \cdot \ell \leq 10^{+298}\right):\\ \;\;\;\;c0 \cdot {\left(\frac{\sqrt[3]{\frac{A}{\ell}}}{\sqrt[3]{V}}\right)}^{1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -1e-306)
   (* c0 (/ (/ (sqrt (- A)) (sqrt (- V))) (sqrt l)))
   (if (or (<= (* V l) 1e-318) (not (<= (* V l) 1e+298)))
     (* c0 (pow (/ (cbrt (/ A l)) (cbrt V)) 1.5))
     (/ c0 (/ (sqrt (* V l)) (sqrt A))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e-306) {
		tmp = c0 * ((sqrt(-A) / sqrt(-V)) / sqrt(l));
	} else if (((V * l) <= 1e-318) || !((V * l) <= 1e+298)) {
		tmp = c0 * pow((cbrt((A / l)) / cbrt(V)), 1.5);
	} else {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	}
	return tmp;
}

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e-306) {
		tmp = c0 * ((Math.sqrt(-A) / Math.sqrt(-V)) / Math.sqrt(l));
	} else if (((V * l) <= 1e-318) || !((V * l) <= 1e+298)) {
		tmp = c0 * Math.pow((Math.cbrt((A / l)) / Math.cbrt(V)), 1.5);
	} else {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	}
	return tmp;
}

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -1e-306)
		tmp = Float64(c0 * Float64(Float64(sqrt(Float64(-A)) / sqrt(Float64(-V))) / sqrt(l)));
	elseif ((Float64(V * l) <= 1e-318) || !(Float64(V * l) <= 1e+298))
		tmp = Float64(c0 * (Float64(cbrt(Float64(A / l)) / cbrt(V)) ^ 1.5));
	else
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	end
	return tmp
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -1e-306], N[(c0 * N[(N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(V * l), $MachinePrecision], 1e-318], N[Not[LessEqual[N[(V * l), $MachinePrecision], 1e+298]], $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], N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -1.00000000000000003e-306

   1. Initial program 81.1%

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

      1. associate-/r* 81.6%

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

      2. sqrt-div 47.2%

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

   3. Applied egg-rr 47.2%

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

      1. frac-2neg 47.2%

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

      2. sqrt-div 51.4%

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

   5. Applied egg-rr 51.4%

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

   if -1.00000000000000003e-306 < (*.f64 V l) < 9.9999875e-319 or 9.9999999999999996e297 < (*.f64 V l)

   1. Initial program 34.6%

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

      1. pow1/2 34.6%

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

      2. add-cube-cbrt 34.6%

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

      3. pow3 34.6%

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

      4. pow-pow 34.6%

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

      5. metadata-eval 34.6%

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

   3. Applied egg-rr 34.6%

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

      1. *-un-lft-identity 34.6%

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

      2. frac-times 54.7%

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

      3. *-commutative 54.7%

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

      4. cbrt-prod 80.0%

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

      5. cbrt-div 80.1%

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

      6. metadata-eval 80.1%

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

   5. Applied egg-rr 80.1%

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

      1. associate-*r/ 80.3%

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

      2. *-rgt-identity 80.3%

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

   7. Simplified 80.3%

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

   if 9.9999875e-319 < (*.f64 V l) < 9.9999999999999996e297

   1. Initial program 81.8%

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

      1. sqrt-div 99.5%

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

      2. associate-*r/ 96.0%

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

   3. Applied egg-rr 96.0%

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

      1. associate-/l* 99.6%

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

   5. Simplified 99.6%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{-306}:\\ \;\;\;\;c0 \cdot \frac{\frac{\sqrt{-A}}{\sqrt{-V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-318} \lor \neg \left(V \cdot \ell \leq 10^{+298}\right):\\ \;\;\;\;c0 \cdot {\left(\frac{\sqrt[3]{\frac{A}{\ell}}}{\sqrt[3]{V}}\right)}^{1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \end{array} \]

Alternative 6: 87.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \leq -5 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot {\left(\frac{\sqrt[3]{A}}{\sqrt[3]{V \cdot \ell}}\right)}^{1.5}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A} \cdot {\ell}^{-0.5}}{\sqrt{V}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= V -5e-310)
   (* c0 (pow (/ (cbrt A) (cbrt (* V l))) 1.5))
   (* c0 (/ (* (sqrt A) (pow l -0.5)) (sqrt V)))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (V <= -5e-310) {
		tmp = c0 * pow((cbrt(A) / cbrt((V * l))), 1.5);
	} else {
		tmp = c0 * ((sqrt(A) * pow(l, -0.5)) / sqrt(V));
	}
	return tmp;
}

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (V <= -5e-310) {
		tmp = c0 * Math.pow((Math.cbrt(A) / Math.cbrt((V * l))), 1.5);
	} else {
		tmp = c0 * ((Math.sqrt(A) * Math.pow(l, -0.5)) / Math.sqrt(V));
	}
	return tmp;
}

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (V <= -5e-310)
		tmp = Float64(c0 * (Float64(cbrt(A) / cbrt(Float64(V * l))) ^ 1.5));
	else
		tmp = Float64(c0 * Float64(Float64(sqrt(A) * (l ^ -0.5)) / sqrt(V)));
	end
	return tmp
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[V, -5e-310], N[(c0 * N[Power[N[(N[Power[A, 1/3], $MachinePrecision] / N[Power[N[(V * l), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[(N[Sqrt[A], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision] / N[Sqrt[V], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if V < -4.999999999999985e-310

   1. Initial program 75.0%

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

      1. pow1/2 75.0%

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

      2. add-cube-cbrt 74.4%

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

      3. pow3 74.4%

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

      4. pow-pow 74.4%

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

      5. metadata-eval 74.4%

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

   3. Applied egg-rr 74.4%

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

      1. cbrt-div 84.2%

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

      2. div-inv 84.1%

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

   5. Applied egg-rr 84.1%

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

      1. associate-*r/ 84.2%

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

      2. *-rgt-identity 84.2%

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

   7. Simplified 84.2%

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

   if -4.999999999999985e-310 < V

   1. Initial program 71.1%

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

      1. pow1/2 71.1%

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

      2. add-cube-cbrt 70.6%

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

      3. pow3 70.6%

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

      4. pow-pow 70.6%

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

      5. metadata-eval 70.6%

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

   3. Applied egg-rr 70.6%

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

      1. cbrt-div 83.5%

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

      2. *-rgt-identity 83.5%

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

      3. cbrt-div 70.6%

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

      4. frac-times 70.3%

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

      5. *-commutative 70.3%

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

      6. cbrt-prod 82.1%

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

      7. cbrt-div 82.0%

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

      8. metadata-eval 82.0%

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

   5. Applied egg-rr 82.0%

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

      1. associate-*l/ 82.0%

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

      2. *-lft-identity 82.0%

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

   7. Simplified 82.0%

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

      1. add-sqr-sqrt 82.0%

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

      2. sqrt-unprod 70.2%

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

      3. pow-prod-up 70.2%

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

      4. metadata-eval 70.2%

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

      5. pow3 70.2%

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

      6. frac-times 70.2%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\sqrt[3]{\frac{A}{V}} \cdot \sqrt[3]{\frac{A}{V}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \frac{\sqrt[3]{\frac{A}{V}}}{\sqrt[3]{\ell}}} \]

      7. times-frac 70.2%

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

      8. add-cube-cbrt 70.4%

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

      9. add-cube-cbrt 70.9%

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

      10. sqrt-div 40.9%

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

      11. div-inv 40.9%

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

      12. sqrt-div 47.5%

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

      13. associate-*l/ 47.5%

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

   9. Applied egg-rr 47.6%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A} \cdot {\ell}^{-0.5}}{\sqrt{V}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \leq -5 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot {\left(\frac{\sqrt[3]{A}}{\sqrt[3]{V \cdot \ell}}\right)}^{1.5}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A} \cdot {\ell}^{-0.5}}{\sqrt{V}}\\ \end{array} \]

Alternative 7: 91.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{\frac{c0}{\sqrt{\frac{V}{A}}}}{\sqrt{\ell}}\\ \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-291}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+298}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ (/ c0 (sqrt (/ V A))) (sqrt l))))
   (if (<= (* V l) (- INFINITY))
     t_0
     (if (<= (* V l) -5e-291)
       (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
       (if (<= (* V l) 0.0)
         t_0
         (if (<= (* V l) 1e+298)
           (/ c0 (/ (sqrt (* V l)) (sqrt A)))
           (* c0 (sqrt (/ (/ A l) V)))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = (c0 / sqrt((V / A))) / sqrt(l);
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = t_0;
	} else if ((V * l) <= -5e-291) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else if ((V * l) <= 1e+298) {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	} else {
		tmp = c0 * sqrt(((A / l) / V));
	}
	return tmp;
}

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = (c0 / Math.sqrt((V / A))) / Math.sqrt(l);
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else if ((V * l) <= -5e-291) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else if ((V * l) <= 1e+298) {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	} else {
		tmp = c0 * Math.sqrt(((A / l) / V));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = (c0 / math.sqrt((V / A))) / math.sqrt(l)
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = t_0
	elif (V * l) <= -5e-291:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 0.0:
		tmp = t_0
	elif (V * l) <= 1e+298:
		tmp = c0 / (math.sqrt((V * l)) / math.sqrt(A))
	else:
		tmp = c0 * math.sqrt(((A / l) / V))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(Float64(c0 / sqrt(Float64(V / A))) / sqrt(l))
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = t_0;
	elseif (Float64(V * l) <= -5e-291)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 0.0)
		tmp = t_0;
	elseif (Float64(V * l) <= 1e+298)
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = (c0 / sqrt((V / A))) / sqrt(l);
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = t_0;
	elseif ((V * l) <= -5e-291)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 0.0)
		tmp = t_0;
	elseif ((V * l) <= 1e+298)
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	else
		tmp = c0 * sqrt(((A / l) / V));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(N[(c0 / N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -5e-291], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], 1e+298], N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -inf.0 or -5.0000000000000003e-291 < (*.f64 V l) < -0.0

   1. Initial program 38.6%

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

      1. associate-/r* 58.3%

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

      2. div-inv 58.3%

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

   3. Applied egg-rr 58.3%

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

      1. sqrt-prod 42.3%

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

      2. associate-*r* 42.3%

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

      3. sqrt-div 42.3%

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

      4. metadata-eval 42.3%

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

      5. div-inv 42.4%

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

   5. Applied egg-rr 42.4%

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

      1. sqrt-div 14.5%

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

      2. clear-num 14.5%

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

      3. un-div-inv 14.5%

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

      4. sqrt-undiv 42.4%

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

   7. Applied egg-rr 42.4%

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

   if -inf.0 < (*.f64 V l) < -5.0000000000000003e-291

   1. Initial program 87.2%

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

      1. frac-2neg 87.2%

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

      2. sqrt-div 99.5%

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

      3. *-commutative 99.5%

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

      4. distribute-rgt-neg-in 99.5%

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

   3. Applied egg-rr 99.5%

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

   if -0.0 < (*.f64 V l) < 9.9999999999999996e297

   1. Initial program 81.4%

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

      1. sqrt-div 98.9%

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

      2. associate-*r/ 95.5%

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

   3. Applied egg-rr 95.5%

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

      1. associate-/l* 99.1%

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

   5. Simplified 99.1%

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

   if 9.9999999999999996e297 < (*.f64 V l)

   1. Initial program 36.8%

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

      1. associate-/r* 71.3%

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

      2. div-inv 71.3%

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

   3. Applied egg-rr 71.3%

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

      1. associate-*l/ 71.3%

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

      2. div-inv 71.3%

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

   5. Applied egg-rr 71.3%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{\frac{c0}{\sqrt{\frac{V}{A}}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-291}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{\frac{c0}{\sqrt{\frac{V}{A}}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+298}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]

Alternative 8: 91.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{\frac{c0}{\sqrt{\frac{V}{A}}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-270}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-301}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+298}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) (- INFINITY))
   (/ (/ c0 (sqrt (/ V A))) (sqrt l))
   (if (<= (* V l) -2e-270)
     (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
     (if (<= (* V l) 2e-301)
       (* c0 (/ (sqrt (/ (- A) l)) (sqrt (- V))))
       (if (<= (* V l) 1e+298)
         (/ c0 (/ (sqrt (* V l)) (sqrt A)))
         (* c0 (sqrt (/ (/ A l) V))))))))

C

assert(V < l);
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) <= -2e-270) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 2e-301) {
		tmp = c0 * (sqrt((-A / l)) / sqrt(-V));
	} else if ((V * l) <= 1e+298) {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	} else {
		tmp = c0 * sqrt(((A / l) / V));
	}
	return tmp;
}

Java

assert V < l;
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) <= -2e-270) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 2e-301) {
		tmp = c0 * (Math.sqrt((-A / l)) / Math.sqrt(-V));
	} else if ((V * l) <= 1e+298) {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	} else {
		tmp = c0 * Math.sqrt(((A / l) / V));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
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) <= -2e-270:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 2e-301:
		tmp = c0 * (math.sqrt((-A / l)) / math.sqrt(-V))
	elif (V * l) <= 1e+298:
		tmp = c0 / (math.sqrt((V * l)) / math.sqrt(A))
	else:
		tmp = c0 * math.sqrt(((A / l) / V))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(Float64(c0 / sqrt(Float64(V / A))) / sqrt(l));
	elseif (Float64(V * l) <= -2e-270)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 2e-301)
		tmp = Float64(c0 * Float64(sqrt(Float64(Float64(-A) / l)) / sqrt(Float64(-V))));
	elseif (Float64(V * l) <= 1e+298)
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
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) <= -2e-270)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 2e-301)
		tmp = c0 * (sqrt((-A / l)) / sqrt(-V));
	elseif ((V * l) <= 1e+298)
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	else
		tmp = c0 * sqrt(((A / l) / V));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(N[(c0 / N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-270], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 2e-301], N[(c0 * N[(N[Sqrt[N[((-A) / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+298], N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 41.1%

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

      1. associate-/r* 76.5%

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

      2. div-inv 76.4%

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

   3. Applied egg-rr 76.4%

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

      1. sqrt-prod 37.8%

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

      2. associate-*r* 37.8%

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

      3. sqrt-div 37.8%

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

      4. metadata-eval 37.8%

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

      5. div-inv 37.9%

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

   5. Applied egg-rr 37.9%

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

      1. sqrt-div 0.0%

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

      2. clear-num 0.0%

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

      3. un-div-inv 0.0%

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

      4. sqrt-undiv 38.0%

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

   7. Applied egg-rr 38.0%

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

   if -inf.0 < (*.f64 V l) < -2.0000000000000001e-270

   1. Initial program 87.8%

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

      1. frac-2neg 87.8%

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

      2. sqrt-div 99.5%

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

      3. *-commutative 99.5%

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

      4. distribute-rgt-neg-in 99.5%

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

   3. Applied egg-rr 99.5%

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

   if -2.0000000000000001e-270 < (*.f64 V l) < 2.00000000000000013e-301

   1. Initial program 39.2%

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

      1. associate-/r* 50.9%

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

      2. div-inv 50.9%

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

   3. Applied egg-rr 50.9%

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

      1. associate-*l/ 50.9%

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

      2. div-inv 50.9%

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

   5. Applied egg-rr 50.9%

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

      1. frac-2neg 50.9%

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

      2. sqrt-div 54.5%

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

   7. Applied egg-rr 54.5%

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

      1. distribute-frac-neg 54.5%

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

   9. Simplified 54.5%

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

   if 2.00000000000000013e-301 < (*.f64 V l) < 9.9999999999999996e297

   1. Initial program 82.6%

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

      1. sqrt-div 99.5%

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

      2. associate-*r/ 96.0%

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

   3. Applied egg-rr 96.0%

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

      1. associate-/l* 99.6%

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

   5. Simplified 99.6%

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

   if 9.9999999999999996e297 < (*.f64 V l)

   1. Initial program 36.8%

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

      1. associate-/r* 71.3%

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

      2. div-inv 71.3%

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

   3. Applied egg-rr 71.3%

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

      1. associate-*l/ 71.3%

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

      2. div-inv 71.3%

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

   5. Applied egg-rr 71.3%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{\ell}}{V}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{\frac{c0}{\sqrt{\frac{V}{A}}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-270}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-301}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+298}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]

Alternative 9: 81.5% accurate, 0.5× speedup

Math

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

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= l -1e-310)
   (* c0 (sqrt (* (/ A V) (/ 1.0 l))))
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= -1e-310) {
		tmp = c0 * sqrt(((A / V) * (1.0 / l)));
	} else {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
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 <= (-1d-310)) then
        tmp = c0 * sqrt(((a / v) * (1.0d0 / l)))
    else
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= -1e-310) {
		tmp = c0 * Math.sqrt(((A / V) * (1.0 / l)));
	} else {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if l <= -1e-310:
		tmp = c0 * math.sqrt(((A / V) * (1.0 / l)))
	else:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (l <= -1e-310)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) * Float64(1.0 / l))));
	else
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (l <= -1e-310)
		tmp = c0 * sqrt(((A / V) * (1.0 / l)));
	else
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[l, -1e-310], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -9.999999999999969e-311

   1. Initial program 74.7%

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

      1. associate-/r* 74.5%

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

      2. div-inv 74.5%

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

   3. Applied egg-rr 74.5%

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

   if -9.999999999999969e-311 < l

   1. Initial program 71.7%

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

      1. associate-/r* 75.3%

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

      2. sqrt-div 85.8%

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

   3. Applied egg-rr 85.8%

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

4. Final simplification 80.3%

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

Alternative 10: 83.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq 2.9 \cdot 10^{-295}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= A 2.9e-295)
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))
   (* c0 (/ (sqrt A) (sqrt (* V l))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= 2.9e-295) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
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 (a <= 2.9d-295) then
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    else
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= 2.9e-295) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if A <= 2.9e-295:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (A <= 2.9e-295)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (A <= 2.9e-295)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	else
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[A, 2.9e-295], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < 2.90000000000000015e-295

   1. Initial program 75.4%

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

      1. associate-/r* 78.0%

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

      2. sqrt-div 48.5%

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

   3. Applied egg-rr 48.5%

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

   if 2.90000000000000015e-295 < A

   1. Initial program 70.6%

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

      1. sqrt-div 84.6%

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

      2. associate-*r/ 81.8%

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

   3. Applied egg-rr 81.8%

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

      1. *-commutative 81.8%

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

      2. associate-*l/ 84.6%

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

   5. Simplified 84.6%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq 2.9 \cdot 10^{-295}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \]

Alternative 11: 83.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq 2.9 \cdot 10^{-295}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= A 2.9e-295)
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))
   (/ c0 (/ (sqrt (* V l)) (sqrt A)))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= 2.9e-295) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
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 (a <= 2.9d-295) then
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    else
        tmp = c0 / (sqrt((v * l)) / sqrt(a))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= 2.9e-295) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if A <= 2.9e-295:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	else:
		tmp = c0 / (math.sqrt((V * l)) / math.sqrt(A))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (A <= 2.9e-295)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	else
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (A <= 2.9e-295)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	else
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[A, 2.9e-295], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < 2.90000000000000015e-295

   1. Initial program 75.4%

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

      1. associate-/r* 78.0%

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

      2. sqrt-div 48.5%

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

   3. Applied egg-rr 48.5%

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

   if 2.90000000000000015e-295 < A

   1. Initial program 70.6%

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

      1. sqrt-div 84.6%

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

      2. associate-*r/ 81.8%

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

   3. Applied egg-rr 81.8%

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

      1. associate-/l* 84.7%

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

   5. Simplified 84.7%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq 2.9 \cdot 10^{-295}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \end{array} \]

Alternative 12: 80.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-317}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t_0 \leq 10^{+280}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(\frac{V}{\frac{A}{\ell}}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (<= t_0 5e-317)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 1e+280)
       (/ c0 (sqrt (/ (* V l) A)))
       (* c0 (pow (/ V (/ A l)) -0.5))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 5e-317) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 1e+280) {
		tmp = c0 / sqrt(((V * l) / A));
	} else {
		tmp = c0 * pow((V / (A / l)), -0.5);
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
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 <= 5d-317) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 1d+280) then
        tmp = c0 / sqrt(((v * l) / a))
    else
        tmp = c0 * ((v / (a / l)) ** (-0.5d0))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 5e-317) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 1e+280) {
		tmp = c0 / Math.sqrt(((V * l) / A));
	} else {
		tmp = c0 * Math.pow((V / (A / l)), -0.5);
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 5e-317:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 1e+280:
		tmp = c0 / math.sqrt(((V * l) / A))
	else:
		tmp = c0 * math.pow((V / (A / l)), -0.5)
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 5e-317)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 1e+280)
		tmp = Float64(c0 / sqrt(Float64(Float64(V * l) / A)));
	else
		tmp = Float64(c0 * (Float64(V / Float64(A / l)) ^ -0.5));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 5e-317)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 1e+280)
		tmp = c0 / sqrt(((V * l) / A));
	else
		tmp = c0 * ((V / (A / l)) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-317], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+280], N[(c0 / N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[Power[N[(V / N[(A / l), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 A (*.f64 V l)) < 5.00000017e-317

   1. Initial program 44.3%

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

      1. associate-/r* 63.6%

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

      2. div-inv 63.6%

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

   3. Applied egg-rr 63.6%

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

      1. un-div-inv 63.6%

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

   5. Applied egg-rr 63.6%

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

   if 5.00000017e-317 < (/.f64 A (*.f64 V l)) < 1e280

   1. Initial program 99.5%

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

      1. pow1/2 99.5%

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

      2. add-cube-cbrt 98.7%

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

      3. pow3 98.7%

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

      4. pow-pow 98.6%

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

      5. metadata-eval 98.6%

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

   3. Applied egg-rr 98.6%

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

      1. cbrt-div 98.4%

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

      2. *-rgt-identity 98.4%

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

      3. cbrt-div 98.6%

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

      4. frac-times 92.7%

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

      5. *-commutative 92.7%

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

      6. cbrt-prod 92.7%

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

      7. cbrt-div 92.5%

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

      8. metadata-eval 92.5%

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

   5. Applied egg-rr 92.5%

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

      1. associate-*l/ 92.5%

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

      2. *-lft-identity 92.5%

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

   7. Simplified 92.5%

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

      1. add-sqr-sqrt 92.5%

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

      2. sqrt-unprod 92.5%

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

      3. pow-prod-up 92.6%

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

      4. metadata-eval 92.6%

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

      5. pow3 92.6%

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

      6. frac-times 92.5%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\sqrt[3]{\frac{A}{V}} \cdot \sqrt[3]{\frac{A}{V}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \frac{\sqrt[3]{\frac{A}{V}}}{\sqrt[3]{\ell}}} \]

      7. times-frac 92.5%

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

      8. add-cube-cbrt 93.0%

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

      9. add-cube-cbrt 93.5%

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

      10. sqrt-div 52.0%

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

      11. associate-*r/ 49.0%

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

      12. associate-/l* 52.0%

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

   9. Applied egg-rr 92.5%

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

       1. expm1-log1p-u 62.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{\sqrt{\ell \cdot \frac{V}{A}}}{c0}}\right)\right)} \]

       2. expm1-udef 29.9%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\frac{\sqrt{\ell \cdot \frac{V}{A}}}{c0}}\right)} - 1} \]

       3. associate-*r/ 31.7%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\frac{\sqrt{\color{blue}{\frac{\ell \cdot V}{A}}}}{c0}}\right)} - 1 \]

       4. *-commutative 31.7%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\frac{\sqrt{\frac{\color{blue}{V \cdot \ell}}{A}}}{c0}}\right)} - 1 \]

       5. sqrt-undiv 16.3%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\frac{\color{blue}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}}{c0}}\right)} - 1 \]

       6. clear-num 16.3%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}}\right)} - 1 \]

       7. sqrt-undiv 31.7%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\color{blue}{\sqrt{\frac{V \cdot \ell}{A}}}}\right)} - 1 \]

       8. *-commutative 31.7%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\frac{\color{blue}{\ell \cdot V}}{A}}}\right)} - 1 \]

       9. associate-*r/ 29.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{\ell \cdot \frac{V}{A}}}}\right)} - 1 \]

   11. Applied egg-rr 29.9%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 63.1%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\right)\right)} \]

       2. expm1-log1p 92.8%

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

       3. *-commutative 92.8%

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

       4. associate-*l/ 99.6%

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

   13. Simplified 99.6%

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

   if 1e280 < (/.f64 A (*.f64 V l))

   1. Initial program 34.0%

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

      1. pow1/2 34.0%

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

      2. clear-num 34.0%

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

      3. inv-pow 34.0%

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

      4. pow-pow 37.1%

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

      5. associate-/l* 44.4%

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

      6. metadata-eval 44.4%

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

   3. Applied egg-rr 44.4%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{-317}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 10^{+280}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(\frac{V}{\frac{A}{\ell}}\right)}^{-0.5}\\ \end{array} \]

Alternative 13: 80.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t_0 \leq 10^{+280}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(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 1e+280) (* c0 (sqrt t_0)) (/ c0 (sqrt (* V (/ l A))))))))

C

assert(V < l);
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 <= 1e+280) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = c0 / sqrt((V * (l / A)));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
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 <= 1d+280) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = c0 / sqrt((v * (l / a)))
    end if
    code = tmp
end function

Java

assert V < l;
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 <= 1e+280) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = c0 / Math.sqrt((V * (l / A)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
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 <= 1e+280:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = c0 / math.sqrt((V * (l / A)))
	return tmp

Julia

V, l = sort([V, l])
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 <= 1e+280)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
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 <= 1e+280)
		tmp = c0 * sqrt(t_0);
	else
		tmp = c0 / sqrt((V * (l / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
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, 1e+280], 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 43.5%

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

      1. associate-/r* 64.9%

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

      2. div-inv 64.8%

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

   3. Applied egg-rr 64.8%

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

      1. un-div-inv 64.9%

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

   5. Applied egg-rr 64.9%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 1e280

   1. Initial program 99.1%

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

   if 1e280 < (/.f64 A (*.f64 V l))

   1. Initial program 34.0%

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

      1. associate-/r* 39.6%

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

      2. div-inv 39.6%

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

   3. Applied egg-rr 39.6%

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

   5. Applied egg-rr 17.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 23.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\right)\right)} \]

      2. expm1-log1p 44.4%

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

   7. Simplified 44.4%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 10^{+280}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]

Alternative 14: 80.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-317}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t_0 \leq 10^{+280}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (<= t_0 5e-317)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 1e+280)
       (/ c0 (sqrt (/ (* V l) A)))
       (/ c0 (sqrt (* V (/ l A))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 5e-317) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 1e+280) {
		tmp = c0 / sqrt(((V * l) / A));
	} else {
		tmp = c0 / sqrt((V * (l / A)));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
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 <= 5d-317) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 1d+280) then
        tmp = c0 / sqrt(((v * l) / a))
    else
        tmp = c0 / sqrt((v * (l / a)))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 5e-317) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 1e+280) {
		tmp = c0 / Math.sqrt(((V * l) / A));
	} else {
		tmp = c0 / Math.sqrt((V * (l / A)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 5e-317:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 1e+280:
		tmp = c0 / math.sqrt(((V * l) / A))
	else:
		tmp = c0 / math.sqrt((V * (l / A)))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 5e-317)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 1e+280)
		tmp = Float64(c0 / sqrt(Float64(Float64(V * l) / A)));
	else
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 5e-317)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 1e+280)
		tmp = c0 / sqrt(((V * l) / A));
	else
		tmp = c0 / sqrt((V * (l / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-317], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+280], N[(c0 / N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $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)) < 5.00000017e-317

   1. Initial program 44.3%

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

      1. associate-/r* 63.6%

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

      2. div-inv 63.6%

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

   3. Applied egg-rr 63.6%

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

      1. un-div-inv 63.6%

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

   5. Applied egg-rr 63.6%

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

   if 5.00000017e-317 < (/.f64 A (*.f64 V l)) < 1e280

   1. Initial program 99.5%

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

      1. pow1/2 99.5%

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

      2. add-cube-cbrt 98.7%

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

      3. pow3 98.7%

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

      4. pow-pow 98.6%

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

      5. metadata-eval 98.6%

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

   3. Applied egg-rr 98.6%

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

      1. cbrt-div 98.4%

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

      2. *-rgt-identity 98.4%

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

      3. cbrt-div 98.6%

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

      4. frac-times 92.7%

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

      5. *-commutative 92.7%

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

      6. cbrt-prod 92.7%

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

      7. cbrt-div 92.5%

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

      8. metadata-eval 92.5%

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

   5. Applied egg-rr 92.5%

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

      1. associate-*l/ 92.5%

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

      2. *-lft-identity 92.5%

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

   7. Simplified 92.5%

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

      1. add-sqr-sqrt 92.5%

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

      2. sqrt-unprod 92.5%

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

      3. pow-prod-up 92.6%

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

      4. metadata-eval 92.6%

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

      5. pow3 92.6%

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

      6. frac-times 92.5%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\sqrt[3]{\frac{A}{V}} \cdot \sqrt[3]{\frac{A}{V}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \frac{\sqrt[3]{\frac{A}{V}}}{\sqrt[3]{\ell}}} \]

      7. times-frac 92.5%

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

      8. add-cube-cbrt 93.0%

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

      9. add-cube-cbrt 93.5%

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

      10. sqrt-div 52.0%

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

      11. associate-*r/ 49.0%

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

      12. associate-/l* 52.0%

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

   9. Applied egg-rr 92.5%

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

       1. expm1-log1p-u 62.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{\sqrt{\ell \cdot \frac{V}{A}}}{c0}}\right)\right)} \]

       2. expm1-udef 29.9%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\frac{\sqrt{\ell \cdot \frac{V}{A}}}{c0}}\right)} - 1} \]

       3. associate-*r/ 31.7%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\frac{\sqrt{\color{blue}{\frac{\ell \cdot V}{A}}}}{c0}}\right)} - 1 \]

       4. *-commutative 31.7%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\frac{\sqrt{\frac{\color{blue}{V \cdot \ell}}{A}}}{c0}}\right)} - 1 \]

       5. sqrt-undiv 16.3%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\frac{\color{blue}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}}{c0}}\right)} - 1 \]

       6. clear-num 16.3%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}}\right)} - 1 \]

       7. sqrt-undiv 31.7%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\color{blue}{\sqrt{\frac{V \cdot \ell}{A}}}}\right)} - 1 \]

       8. *-commutative 31.7%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\frac{\color{blue}{\ell \cdot V}}{A}}}\right)} - 1 \]

       9. associate-*r/ 29.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{\ell \cdot \frac{V}{A}}}}\right)} - 1 \]

   11. Applied egg-rr 29.9%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 63.1%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\right)\right)} \]

       2. expm1-log1p 92.8%

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

       3. *-commutative 92.8%

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

       4. associate-*l/ 99.6%

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

   13. Simplified 99.6%

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

   if 1e280 < (/.f64 A (*.f64 V l))

   1. Initial program 34.0%

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

      1. associate-/r* 39.6%

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

      2. div-inv 39.6%

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

   3. Applied egg-rr 39.6%

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

   5. Applied egg-rr 17.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 23.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\right)\right)} \]

      2. expm1-log1p 44.4%

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

   7. Simplified 44.4%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{-317}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 10^{+280}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]

Alternative 15: 74.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l) :precision binary64 (* c0 (sqrt (/ A (* V l)))))

C

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

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
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

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

Python

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

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	return Float64(c0 * sqrt(Float64(A / Float64(V * l))))
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp = code(c0, A, V, l)
	tmp = c0 * sqrt((A / (V * l)));
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.2%

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

2. Final simplification 73.2%

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

Alternative 16: 74.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l) :precision binary64 (* c0 (sqrt (/ (/ A V) l))))

C

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

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
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

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

Python

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

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	return Float64(c0 * sqrt(Float64(Float64(A / V) / l)))
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp = code(c0, A, V, l)
	tmp = c0 * sqrt(((A / V) / l));
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.2%

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

   1. associate-/r* 74.9%

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

   2. div-inv 74.9%

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

3. Applied egg-rr 74.9%

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

   1. un-div-inv 74.9%

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

5. Applied egg-rr 74.9%

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

6. Final simplification 74.9%

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

Reproduce

herbie shell --seed 2023187 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))