Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.3% → 91.7%

Time: 10.9s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.3% 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: 91.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -5 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \left(\frac{\sqrt{-A}}{\sqrt{-V}} \cdot \frac{1}{\sqrt{\ell}}\right)\\ \mathbf{elif}\;A \leq 4.5 \cdot 10^{-146}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \mathbf{elif}\;A \leq 3.1 \cdot 10^{+187}:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{\frac{A}{V}}{\sqrt[3]{\ell}}} \cdot \sqrt{{\left(\sqrt[3]{\ell}\right)}^{-2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= A -5e-310)
   (* c0 (* (/ (sqrt (- A)) (sqrt (- V))) (/ 1.0 (sqrt l))))
   (if (<= A 4.5e-146)
     (* c0 (* (sqrt A) (sqrt (/ (/ 1.0 V) l))))
     (if (<= A 3.1e+187)
       (* c0 (* (sqrt (/ (/ A V) (cbrt l))) (sqrt (pow (cbrt l) -2.0))))
       (/ c0 (/ (sqrt (* V l)) (sqrt A)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -5e-310) {
		tmp = c0 * ((sqrt(-A) / sqrt(-V)) * (1.0 / sqrt(l)));
	} else if (A <= 4.5e-146) {
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	} else if (A <= 3.1e+187) {
		tmp = c0 * (sqrt(((A / V) / cbrt(l))) * sqrt(pow(cbrt(l), -2.0)));
	} else {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -5e-310) {
		tmp = c0 * ((Math.sqrt(-A) / Math.sqrt(-V)) * (1.0 / Math.sqrt(l)));
	} else if (A <= 4.5e-146) {
		tmp = c0 * (Math.sqrt(A) * Math.sqrt(((1.0 / V) / l)));
	} else if (A <= 3.1e+187) {
		tmp = c0 * (Math.sqrt(((A / V) / Math.cbrt(l))) * Math.sqrt(Math.pow(Math.cbrt(l), -2.0)));
	} else {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	}
	return tmp;
}

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (A <= -5e-310)
		tmp = Float64(c0 * Float64(Float64(sqrt(Float64(-A)) / sqrt(Float64(-V))) * Float64(1.0 / sqrt(l))));
	elseif (A <= 4.5e-146)
		tmp = Float64(c0 * Float64(sqrt(A) * sqrt(Float64(Float64(1.0 / V) / l))));
	elseif (A <= 3.1e+187)
		tmp = Float64(c0 * Float64(sqrt(Float64(Float64(A / V) / cbrt(l))) * sqrt((cbrt(l) ^ -2.0))));
	else
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	end
	return tmp
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[A, -5e-310], N[(c0 * N[(N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[A, 4.5e-146], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Sqrt[N[(N[(1.0 / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[A, 3.1e+187], N[(c0 * N[(N[Sqrt[N[(N[(A / V), $MachinePrecision] / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]], $MachinePrecision]), $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 A < -4.999999999999985e-310

   1. Initial program 69.5%

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

      1. associate-/r* 70.9%

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

      2. sqrt-div 43.9%

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

      3. div-inv 43.8%

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

   4. Applied egg-rr 43.8%

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

      1. frac-2neg 43.8%

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

      2. sqrt-div 50.9%

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

   6. Applied egg-rr 50.9%

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

   if -4.999999999999985e-310 < A < 4.5000000000000001e-146

   1. Initial program 66.4%

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

      1. pow1/2 66.4%

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

      2. div-inv 66.4%

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

      3. unpow-prod-down 83.0%

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

      4. pow1/2 83.0%

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

      5. associate-/r* 84.3%

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

   4. Applied egg-rr 84.3%

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{A} \cdot {\left(\frac{\frac{1}{V}}{\ell}\right)}^{0.5}\right)} \]
   5. Step-by-step derivation

      1. unpow1/2 84.4%

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

   6. Simplified 84.4%

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

   if 4.5000000000000001e-146 < A < 3.10000000000000012e187

   1. Initial program 80.2%

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

      1. associate-/r* 83.1%

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

      2. div-inv 83.1%

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

      3. add-cube-cbrt 82.7%

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

      4. times-frac 88.5%

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

      5. pow2 88.5%

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}} \]

   4. Applied egg-rr 88.5%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]
   5. Step-by-step derivation

      1. associate-*l/ 86.7%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

      2. associate-/l/ 85.1%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \color{blue}{\frac{1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      3. associate-*r/ 85.2%

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A \cdot 1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      4. *-rgt-identity 85.2%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{\color{blue}{A}}{\sqrt[3]{\ell} \cdot V}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      5. *-commutative 85.2%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{A}{\color{blue}{V \cdot \sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

   6. Simplified 85.2%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V \cdot \sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]
   7. Step-by-step derivation

      1. pow1/2 85.2%

         \[\leadsto c0 \cdot \color{blue}{{\left(\frac{\frac{A}{V \cdot \sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{0.5}} \]

      2. div-inv 85.2%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{A}{V \cdot \sqrt[3]{\ell}} \cdot \frac{1}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}}^{0.5} \]

      3. unpow-prod-down 90.5%

         \[\leadsto c0 \cdot \color{blue}{\left({\left(\frac{A}{V \cdot \sqrt[3]{\ell}}\right)}^{0.5} \cdot {\left(\frac{1}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{0.5}\right)} \]

      4. pow1/2 90.5%

         \[\leadsto c0 \cdot \left(\color{blue}{\sqrt{\frac{A}{V \cdot \sqrt[3]{\ell}}}} \cdot {\left(\frac{1}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{0.5}\right) \]

      5. associate-/r* 89.6%

         \[\leadsto c0 \cdot \left(\sqrt{\color{blue}{\frac{\frac{A}{V}}{\sqrt[3]{\ell}}}} \cdot {\left(\frac{1}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{0.5}\right) \]

      6. pow-flip 89.6%

         \[\leadsto c0 \cdot \left(\sqrt{\frac{\frac{A}{V}}{\sqrt[3]{\ell}}} \cdot {\color{blue}{\left({\left(\sqrt[3]{\ell}\right)}^{\left(-2\right)}\right)}}^{0.5}\right) \]

      7. metadata-eval 89.6%

         \[\leadsto c0 \cdot \left(\sqrt{\frac{\frac{A}{V}}{\sqrt[3]{\ell}}} \cdot {\left({\left(\sqrt[3]{\ell}\right)}^{\color{blue}{-2}}\right)}^{0.5}\right) \]

   8. Applied egg-rr 89.6%

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{\frac{\frac{A}{V}}{\sqrt[3]{\ell}}} \cdot {\left({\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{0.5}\right)} \]
   9. Step-by-step derivation

      1. unpow1/2 89.6%

         \[\leadsto c0 \cdot \left(\sqrt{\frac{\frac{A}{V}}{\sqrt[3]{\ell}}} \cdot \color{blue}{\sqrt{{\left(\sqrt[3]{\ell}\right)}^{-2}}}\right) \]

   10. Simplified 89.6%

       \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{\frac{\frac{A}{V}}{\sqrt[3]{\ell}}} \cdot \sqrt{{\left(\sqrt[3]{\ell}\right)}^{-2}}\right)} \]

   if 3.10000000000000012e187 < A

   1. Initial program 73.9%

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

      1. associate-/r* 70.7%

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

      2. div-inv 70.8%

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

      3. add-cube-cbrt 70.6%

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

      4. times-frac 73.6%

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

      5. pow2 73.6%

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}} \]

   4. Applied egg-rr 73.6%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]
   5. Step-by-step derivation

      1. associate-*l/ 76.4%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

      2. associate-/l/ 76.4%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \color{blue}{\frac{1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      3. associate-*r/ 76.5%

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A \cdot 1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      4. *-rgt-identity 76.5%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{\color{blue}{A}}{\sqrt[3]{\ell} \cdot V}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      5. *-commutative 76.5%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{A}{\color{blue}{V \cdot \sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

   6. Simplified 76.5%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V \cdot \sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]
   7. Step-by-step derivation

      1. associate-/r* 70.6%

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{\frac{A}{V}}{\sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      2. associate-/l/ 70.5%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{\ell}}}} \]

      3. unpow2 70.5%

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

      4. add-cube-cbrt 70.7%

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

      5. sqrt-undiv 46.0%

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

      6. clear-num 46.0%

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

      7. un-div-inv 46.2%

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

      8. sqrt-undiv 73.4%

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

      9. div-inv 73.4%

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

      10. clear-num 74.5%

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

   8. Applied egg-rr 74.5%

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

      1. *-commutative 74.5%

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

      2. associate-*l/ 76.6%

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

      3. associate-*r/ 71.0%

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

   10. Simplified 71.0%

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

       1. associate-*r/ 76.6%

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

       2. sqrt-div 90.8%

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

   12. Applied egg-rr 90.8%

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

Alternative 2: 90.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{-159}:\\ \;\;\;\;c0 \cdot \frac{\frac{\sqrt{-A}}{\sqrt{-V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-315}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+298}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -1e-159)
   (* c0 (/ (/ (sqrt (- A)) (sqrt (- V))) (sqrt l)))
   (if (<= (* V l) 1e-315)
     (* (/ c0 (sqrt l)) (sqrt (/ A V)))
     (if (<= (* V l) 1e+298)
       (/ c0 (/ (sqrt (* V l)) (sqrt A)))
       (* c0 (sqrt (/ (/ A l) V)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e-159) {
		tmp = c0 * ((sqrt(-A) / sqrt(-V)) / sqrt(l));
	} else if ((V * l) <= 1e-315) {
		tmp = (c0 / sqrt(l)) * sqrt((A / V));
	} else if ((V * l) <= 1e+298) {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	} else {
		tmp = c0 * sqrt(((A / l) / V));
	}
	return tmp;
}

Fortran

NOTE: c0, A, 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 ((v * l) <= (-1d-159)) then
        tmp = c0 * ((sqrt(-a) / sqrt(-v)) / sqrt(l))
    else if ((v * l) <= 1d-315) then
        tmp = (c0 / sqrt(l)) * sqrt((a / v))
    else if ((v * l) <= 1d+298) then
        tmp = c0 / (sqrt((v * l)) / sqrt(a))
    else
        tmp = c0 * sqrt(((a / l) / v))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e-159) {
		tmp = c0 * ((Math.sqrt(-A) / Math.sqrt(-V)) / Math.sqrt(l));
	} else if ((V * l) <= 1e-315) {
		tmp = (c0 / Math.sqrt(l)) * Math.sqrt((A / 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

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -1e-159:
		tmp = c0 * ((math.sqrt(-A) / math.sqrt(-V)) / math.sqrt(l))
	elif (V * l) <= 1e-315:
		tmp = (c0 / math.sqrt(l)) * math.sqrt((A / 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

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -1e-159)
		tmp = Float64(c0 * Float64(Float64(sqrt(Float64(-A)) / sqrt(Float64(-V))) / sqrt(l)));
	elseif (Float64(V * l) <= 1e-315)
		tmp = Float64(Float64(c0 / sqrt(l)) * sqrt(Float64(A / 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

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -1e-159)
		tmp = c0 * ((sqrt(-A) / sqrt(-V)) / sqrt(l));
	elseif ((V * l) <= 1e-315)
		tmp = (c0 / sqrt(l)) * sqrt((A / 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: c0, A, 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-159], N[(c0 * N[(N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e-315], N[(N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(A / 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 4 regimes

2. if (*.f64 V l) < -9.99999999999999989e-160

   1. Initial program 78.5%

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

      1. associate-/r* 74.7%

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

      2. sqrt-div 45.2%

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

      3. div-inv 45.1%

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

   4. Applied egg-rr 45.1%

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

      1. associate-*r/ 45.2%

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

      2. *-rgt-identity 45.2%

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

   6. Simplified 45.2%

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

      1. frac-2neg 45.1%

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

      2. sqrt-div 54.5%

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

   8. Applied egg-rr 54.6%

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

   if -9.99999999999999989e-160 < (*.f64 V l) < 9.999999985e-316

   1. Initial program 52.3%

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

      1. associate-/r* 66.3%

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

      2. div-inv 66.1%

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

      3. add-cube-cbrt 65.7%

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

      4. times-frac 63.9%

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

      5. pow2 63.9%

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}} \]

   4. Applied egg-rr 63.9%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]
   5. Step-by-step derivation

      1. associate-*l/ 63.9%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

      2. associate-/l/ 64.0%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \color{blue}{\frac{1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      3. associate-*r/ 64.0%

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A \cdot 1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      4. *-rgt-identity 64.0%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{\color{blue}{A}}{\sqrt[3]{\ell} \cdot V}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      5. *-commutative 64.0%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{A}{\color{blue}{V \cdot \sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

   6. Simplified 64.0%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V \cdot \sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]
   7. Step-by-step derivation

      1. associate-/r* 65.9%

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{\frac{A}{V}}{\sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      2. associate-/l/ 65.7%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{\ell}}}} \]

      3. unpow2 65.7%

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

      4. add-cube-cbrt 66.3%

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

      5. sqrt-undiv 47.3%

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

      6. clear-num 47.3%

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

      7. un-div-inv 47.3%

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

      8. sqrt-undiv 66.1%

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

      9. div-inv 66.1%

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

      10. clear-num 66.1%

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

   8. Applied egg-rr 66.1%

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

      1. *-commutative 66.1%

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

      2. associate-*l/ 52.2%

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

      3. associate-*r/ 66.1%

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

   10. Simplified 66.1%

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

       1. clear-num 66.1%

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

       2. associate-*r/ 52.2%

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

       3. *-commutative 52.2%

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

       4. associate-*r/ 66.1%

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

       5. sqrt-prod 47.2%

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

       6. associate-/l* 47.2%

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

       7. clear-num 47.2%

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

       8. sqrt-div 47.1%

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

       9. metadata-eval 47.1%

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

       10. associate-/r* 47.2%

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

       11. *-commutative 47.2%

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

       12. div-inv 47.2%

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

       13. associate-/r* 47.3%

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

       14. associate-/r/ 47.3%

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

       15. clear-num 47.3%

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

   12. Applied egg-rr 47.3%

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

   if 9.999999985e-316 < (*.f64 V l) < 9.9999999999999996e297

   1. Initial program 85.3%

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

      1. associate-/r* 73.0%

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

      2. div-inv 73.0%

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

      3. add-cube-cbrt 72.6%

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

      4. times-frac 81.1%

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

      5. pow2 81.1%

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}} \]

   4. Applied egg-rr 81.1%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]
   5. Step-by-step derivation

      1. associate-*l/ 79.1%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

      2. associate-/l/ 79.1%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \color{blue}{\frac{1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      3. associate-*r/ 79.1%

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A \cdot 1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      4. *-rgt-identity 79.1%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{\color{blue}{A}}{\sqrt[3]{\ell} \cdot V}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      5. *-commutative 79.1%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{A}{\color{blue}{V \cdot \sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

   6. Simplified 79.1%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V \cdot \sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]
   7. Step-by-step derivation

      1. associate-/r* 72.6%

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{\frac{A}{V}}{\sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      2. associate-/l/ 72.6%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{\ell}}}} \]

      3. unpow2 72.6%

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

      4. add-cube-cbrt 73.0%

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

      5. sqrt-undiv 38.1%

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

      6. clear-num 38.1%

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

      7. un-div-inv 38.1%

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

      8. sqrt-undiv 74.2%

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

      9. div-inv 73.2%

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

      10. clear-num 74.2%

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

   8. Applied egg-rr 74.2%

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

      1. *-commutative 74.2%

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

      2. associate-*l/ 85.7%

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

      3. associate-*r/ 79.1%

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

   10. Simplified 79.1%

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

       1. associate-*r/ 85.7%

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

       2. sqrt-div 99.4%

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

   12. Applied egg-rr 99.4%

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

   if 9.9999999999999996e297 < (*.f64 V l)

   1. Initial program 33.3%

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

      1. *-un-lft-identity 33.3%

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

      2. times-frac 61.6%

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

   4. Applied egg-rr 61.6%

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

      1. associate-*l/ 61.6%

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

      2. *-un-lft-identity 61.6%

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

   6. Applied egg-rr 61.6%

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

Alternative 3: 90.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-159}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-315}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+298}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, 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 l)) (sqrt (/ A V)))))
   (if (<= (* V l) (- INFINITY))
     t_0
     (if (<= (* V l) -1e-159)
       (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
       (if (<= (* V l) 1e-315)
         t_0
         (if (<= (* V l) 1e+298)
           (/ c0 (/ (sqrt (* V l)) (sqrt A)))
           (* c0 (sqrt (/ (/ A l) V)))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = (c0 / sqrt(l)) * sqrt((A / V));
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = t_0;
	} else if ((V * l) <= -1e-159) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 1e-315) {
		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 c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = (c0 / Math.sqrt(l)) * Math.sqrt((A / V));
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else if ((V * l) <= -1e-159) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 1e-315) {
		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

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = (c0 / math.sqrt(l)) * math.sqrt((A / V))
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = t_0
	elif (V * l) <= -1e-159:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 1e-315:
		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

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(Float64(c0 / sqrt(l)) * sqrt(Float64(A / V)))
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = t_0;
	elseif (Float64(V * l) <= -1e-159)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 1e-315)
		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

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = (c0 / sqrt(l)) * sqrt((A / V));
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = t_0;
	elseif ((V * l) <= -1e-159)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 1e-315)
		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: c0, A, 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[l], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -1e-159], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e-315], 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 -9.99999999999999989e-160 < (*.f64 V l) < 9.999999985e-316

   1. Initial program 47.5%

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

      1. associate-/r* 65.1%

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

      2. div-inv 65.0%

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

      3. add-cube-cbrt 64.5%

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

      4. times-frac 61.3%

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

      5. pow2 61.3%

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}} \]

   4. Applied egg-rr 61.3%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]
   5. Step-by-step derivation

      1. associate-*l/ 61.3%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

      2. associate-/l/ 60.6%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \color{blue}{\frac{1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      3. associate-*r/ 60.6%

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A \cdot 1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      4. *-rgt-identity 60.6%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{\color{blue}{A}}{\sqrt[3]{\ell} \cdot V}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      5. *-commutative 60.6%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{A}{\color{blue}{V \cdot \sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

   6. Simplified 60.6%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V \cdot \sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]
   7. Step-by-step derivation

      1. associate-/r* 64.7%

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{\frac{A}{V}}{\sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      2. associate-/l/ 64.5%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{\ell}}}} \]

      3. unpow2 64.5%

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

      4. add-cube-cbrt 65.1%

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

      5. sqrt-undiv 47.9%

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

      6. clear-num 47.9%

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

      7. un-div-inv 47.9%

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

      8. sqrt-undiv 65.0%

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

      9. div-inv 65.0%

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

      10. clear-num 65.0%

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

   8. Applied egg-rr 65.0%

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

      1. *-commutative 65.0%

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

      2. associate-*l/ 47.4%

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

      3. associate-*r/ 65.0%

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

   10. Simplified 65.0%

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

       1. clear-num 64.9%

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

       2. associate-*r/ 47.4%

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

       3. *-commutative 47.4%

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

       4. associate-*r/ 64.9%

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

       5. sqrt-prod 47.8%

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

       6. associate-/l* 47.8%

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

       7. clear-num 47.8%

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

       8. sqrt-div 47.8%

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

       9. metadata-eval 47.8%

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

       10. associate-/r* 47.8%

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

       11. *-commutative 47.8%

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

       12. div-inv 47.8%

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

       13. associate-/r* 47.9%

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

       14. associate-/r/ 47.9%

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

       15. clear-num 47.9%

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

   12. Applied egg-rr 47.9%

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

   if -inf.0 < (*.f64 V l) < -9.99999999999999989e-160

   1. Initial program 89.8%

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

      1. frac-2neg 89.8%

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

      2. sqrt-div 99.5%

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

      3. distribute-rgt-neg-in 99.5%

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

   4. Applied egg-rr 99.5%

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

      1. distribute-rgt-neg-out 99.5%

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

      2. *-commutative 99.5%

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

      3. distribute-rgt-neg-in 99.5%

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

   6. Simplified 99.5%

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

   if 9.999999985e-316 < (*.f64 V l) < 9.9999999999999996e297

   1. Initial program 85.3%

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

      1. associate-/r* 73.0%

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

      2. div-inv 73.0%

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

      3. add-cube-cbrt 72.6%

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

      4. times-frac 81.1%

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

      5. pow2 81.1%

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}} \]

   4. Applied egg-rr 81.1%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]
   5. Step-by-step derivation

      1. associate-*l/ 79.1%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

      2. associate-/l/ 79.1%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \color{blue}{\frac{1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      3. associate-*r/ 79.1%

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A \cdot 1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      4. *-rgt-identity 79.1%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{\color{blue}{A}}{\sqrt[3]{\ell} \cdot V}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      5. *-commutative 79.1%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{A}{\color{blue}{V \cdot \sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

   6. Simplified 79.1%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V \cdot \sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]
   7. Step-by-step derivation

      1. associate-/r* 72.6%

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{\frac{A}{V}}{\sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      2. associate-/l/ 72.6%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{\ell}}}} \]

      3. unpow2 72.6%

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

      4. add-cube-cbrt 73.0%

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

      5. sqrt-undiv 38.1%

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

      6. clear-num 38.1%

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

      7. un-div-inv 38.1%

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

      8. sqrt-undiv 74.2%

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

      9. div-inv 73.2%

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

      10. clear-num 74.2%

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

   8. Applied egg-rr 74.2%

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

      1. *-commutative 74.2%

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

      2. associate-*l/ 85.7%

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

      3. associate-*r/ 79.1%

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

   10. Simplified 79.1%

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

       1. associate-*r/ 85.7%

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

       2. sqrt-div 99.4%

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

   12. Applied egg-rr 99.4%

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

   if 9.9999999999999996e297 < (*.f64 V l)

   1. Initial program 33.3%

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

      1. *-un-lft-identity 33.3%

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

      2. times-frac 61.6%

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

   4. Applied egg-rr 61.6%

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

      1. associate-*l/ 61.6%

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

      2. *-un-lft-identity 61.6%

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

   6. Applied egg-rr 61.6%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-159}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-315}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+298}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{V}}\\ \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+193}:\\ \;\;\;\;c0 \cdot \frac{t\_0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-154}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-315}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot t\_0\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+298}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (sqrt (/ A V))))
   (if (<= (* V l) -5e+193)
     (* c0 (/ t_0 (sqrt l)))
     (if (<= (* V l) -5e-154)
       (/ c0 (sqrt (/ (* V l) A)))
       (if (<= (* V l) 1e-315)
         (* (/ c0 (sqrt l)) t_0)
         (if (<= (* V l) 1e+298)
           (/ c0 (/ (sqrt (* V l)) (sqrt A)))
           (* c0 (sqrt (/ (/ A l) V)))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt((A / V));
	double tmp;
	if ((V * l) <= -5e+193) {
		tmp = c0 * (t_0 / sqrt(l));
	} else if ((V * l) <= -5e-154) {
		tmp = c0 / sqrt(((V * l) / A));
	} else if ((V * l) <= 1e-315) {
		tmp = (c0 / sqrt(l)) * t_0;
	} else if ((V * l) <= 1e+298) {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	} else {
		tmp = c0 * sqrt(((A / l) / V));
	}
	return tmp;
}

Fortran

NOTE: c0, A, 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 = sqrt((a / v))
    if ((v * l) <= (-5d+193)) then
        tmp = c0 * (t_0 / sqrt(l))
    else if ((v * l) <= (-5d-154)) then
        tmp = c0 / sqrt(((v * l) / a))
    else if ((v * l) <= 1d-315) then
        tmp = (c0 / sqrt(l)) * t_0
    else if ((v * l) <= 1d+298) then
        tmp = c0 / (sqrt((v * l)) / sqrt(a))
    else
        tmp = c0 * sqrt(((a / l) / v))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = Math.sqrt((A / V));
	double tmp;
	if ((V * l) <= -5e+193) {
		tmp = c0 * (t_0 / Math.sqrt(l));
	} else if ((V * l) <= -5e-154) {
		tmp = c0 / Math.sqrt(((V * l) / A));
	} else if ((V * l) <= 1e-315) {
		tmp = (c0 / Math.sqrt(l)) * 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

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt((A / V))
	tmp = 0
	if (V * l) <= -5e+193:
		tmp = c0 * (t_0 / math.sqrt(l))
	elif (V * l) <= -5e-154:
		tmp = c0 / math.sqrt(((V * l) / A))
	elif (V * l) <= 1e-315:
		tmp = (c0 / math.sqrt(l)) * 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

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = sqrt(Float64(A / V))
	tmp = 0.0
	if (Float64(V * l) <= -5e+193)
		tmp = Float64(c0 * Float64(t_0 / sqrt(l)));
	elseif (Float64(V * l) <= -5e-154)
		tmp = Float64(c0 / sqrt(Float64(Float64(V * l) / A)));
	elseif (Float64(V * l) <= 1e-315)
		tmp = Float64(Float64(c0 / sqrt(l)) * 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

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = sqrt((A / V));
	tmp = 0.0;
	if ((V * l) <= -5e+193)
		tmp = c0 * (t_0 / sqrt(l));
	elseif ((V * l) <= -5e-154)
		tmp = c0 / sqrt(((V * l) / A));
	elseif ((V * l) <= 1e-315)
		tmp = (c0 / sqrt(l)) * 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: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -5e+193], N[(c0 * N[(t$95$0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -5e-154], N[(c0 / N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e-315], N[(N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * t$95$0), $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) < -4.99999999999999972e193

   1. Initial program 42.9%

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

      1. associate-/r* 59.1%

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

      2. sqrt-div 41.6%

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

      3. div-inv 41.4%

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

   4. Applied egg-rr 41.4%

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

      1. associate-*r/ 41.6%

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

      2. *-rgt-identity 41.6%

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

   6. Simplified 41.6%

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

   if -4.99999999999999972e193 < (*.f64 V l) < -5.0000000000000002e-154

   1. Initial program 97.9%

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

      1. associate-/r* 82.9%

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

      2. div-inv 82.9%

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

      3. add-cube-cbrt 82.3%

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

      4. times-frac 91.5%

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

      5. pow2 91.5%

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}} \]

   4. Applied egg-rr 91.5%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]
   5. Step-by-step derivation

      1. associate-*l/ 91.7%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

      2. associate-/l/ 91.7%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \color{blue}{\frac{1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      3. associate-*r/ 91.7%

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A \cdot 1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      4. *-rgt-identity 91.7%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{\color{blue}{A}}{\sqrt[3]{\ell} \cdot V}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      5. *-commutative 91.7%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{A}{\color{blue}{V \cdot \sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

   6. Simplified 91.7%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V \cdot \sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]
   7. Step-by-step derivation

      1. associate-/r* 82.2%

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{\frac{A}{V}}{\sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      2. associate-/l/ 82.3%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{\ell}}}} \]

      3. unpow2 82.3%

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

      4. add-cube-cbrt 82.9%

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

      5. sqrt-undiv 46.2%

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

      6. clear-num 46.2%

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

      7. un-div-inv 46.1%

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

      8. sqrt-undiv 83.0%

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

      9. div-inv 81.6%

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

      10. clear-num 81.5%

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

   8. Applied egg-rr 81.5%

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

      1. *-commutative 81.5%

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

      2. associate-*l/ 98.0%

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

      3. associate-*r/ 87.6%

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

   10. Simplified 87.6%

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

   11. Taylor expanded in V around 0 98.0%

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

   if -5.0000000000000002e-154 < (*.f64 V l) < 9.999999985e-316

   1. Initial program 53.3%

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

      1. associate-/r* 66.9%

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

      2. div-inv 66.8%

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

      3. add-cube-cbrt 66.3%

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

      4. times-frac 64.6%

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

      5. pow2 64.6%

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}} \]

   4. Applied egg-rr 64.6%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]
   5. Step-by-step derivation

      1. associate-*l/ 64.6%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

      2. associate-/l/ 64.6%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \color{blue}{\frac{1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      3. associate-*r/ 64.6%

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A \cdot 1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      4. *-rgt-identity 64.6%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{\color{blue}{A}}{\sqrt[3]{\ell} \cdot V}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      5. *-commutative 64.6%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{A}{\color{blue}{V \cdot \sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

   6. Simplified 64.6%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V \cdot \sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]
   7. Step-by-step derivation

      1. associate-/r* 66.5%

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{\frac{A}{V}}{\sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      2. associate-/l/ 66.3%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{\ell}}}} \]

      3. unpow2 66.3%

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

      4. add-cube-cbrt 66.9%

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

      5. sqrt-undiv 48.3%

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

      6. clear-num 48.2%

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

      7. un-div-inv 48.3%

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

      8. sqrt-undiv 66.8%

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

      9. div-inv 66.7%

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

      10. clear-num 66.8%

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

   8. Applied egg-rr 66.8%

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

      1. *-commutative 66.8%

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

      2. associate-*l/ 53.1%

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

      3. associate-*r/ 66.8%

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

   10. Simplified 66.8%

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

       1. clear-num 66.7%

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

       2. associate-*r/ 53.2%

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

       3. *-commutative 53.2%

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

       4. associate-*r/ 66.7%

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

       5. sqrt-prod 48.2%

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

       6. associate-/l* 48.2%

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

       7. clear-num 48.2%

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

       8. sqrt-div 48.1%

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

       9. metadata-eval 48.1%

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

       10. associate-/r* 48.2%

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

       11. *-commutative 48.2%

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

       12. div-inv 48.2%

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

       13. associate-/r* 48.3%

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

       14. associate-/r/ 48.2%

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

       15. clear-num 48.2%

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

   12. Applied egg-rr 48.2%

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

   if 9.999999985e-316 < (*.f64 V l) < 9.9999999999999996e297

   1. Initial program 85.3%

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

      1. associate-/r* 73.0%

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

      2. div-inv 73.0%

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

      3. add-cube-cbrt 72.6%

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

      4. times-frac 81.1%

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

      5. pow2 81.1%

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}} \]

   4. Applied egg-rr 81.1%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]
   5. Step-by-step derivation

      1. associate-*l/ 79.1%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

      2. associate-/l/ 79.1%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \color{blue}{\frac{1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      3. associate-*r/ 79.1%

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A \cdot 1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      4. *-rgt-identity 79.1%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{\color{blue}{A}}{\sqrt[3]{\ell} \cdot V}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      5. *-commutative 79.1%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{A}{\color{blue}{V \cdot \sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

   6. Simplified 79.1%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V \cdot \sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]
   7. Step-by-step derivation

      1. associate-/r* 72.6%

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{\frac{A}{V}}{\sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      2. associate-/l/ 72.6%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{\ell}}}} \]

      3. unpow2 72.6%

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

      4. add-cube-cbrt 73.0%

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

      5. sqrt-undiv 38.1%

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

      6. clear-num 38.1%

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

      7. un-div-inv 38.1%

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

      8. sqrt-undiv 74.2%

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

      9. div-inv 73.2%

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

      10. clear-num 74.2%

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

   8. Applied egg-rr 74.2%

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

      1. *-commutative 74.2%

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

      2. associate-*l/ 85.7%

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

      3. associate-*r/ 79.1%

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

   10. Simplified 79.1%

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

       1. associate-*r/ 85.7%

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

       2. sqrt-div 99.4%

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

   12. Applied egg-rr 99.4%

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

   if 9.9999999999999996e297 < (*.f64 V l)

   1. Initial program 33.3%

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

      1. *-un-lft-identity 33.3%

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

      2. times-frac 61.6%

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

   4. Applied egg-rr 61.6%

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

      1. associate-*l/ 61.6%

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

      2. *-un-lft-identity 61.6%

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

   6. Applied egg-rr 61.6%

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

Alternative 5: 78.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-309}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t\_0 \leq 10^{+253}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, 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 4e-309)
     (* c0 (sqrt (/ (/ A l) V)))
     (if (<= t_0 1e+253)
       (/ c0 (sqrt (/ (* V l) A)))
       (* c0 (/ (sqrt (/ A V)) (sqrt l)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 4e-309) {
		tmp = c0 * sqrt(((A / l) / V));
	} else if (t_0 <= 1e+253) {
		tmp = c0 / sqrt(((V * l) / A));
	} else {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, 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 <= 4d-309) then
        tmp = c0 * sqrt(((a / l) / v))
    else if (t_0 <= 1d+253) then
        tmp = c0 / sqrt(((v * l) / a))
    else
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && 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 <= 4e-309) {
		tmp = c0 * Math.sqrt(((A / l) / V));
	} else if (t_0 <= 1e+253) {
		tmp = c0 / Math.sqrt(((V * l) / A));
	} else {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 4e-309:
		tmp = c0 * math.sqrt(((A / l) / V))
	elif t_0 <= 1e+253:
		tmp = c0 / math.sqrt(((V * l) / A))
	else:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 4e-309)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	elseif (t_0 <= 1e+253)
		tmp = Float64(c0 / sqrt(Float64(Float64(V * l) / A)));
	else
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 4e-309)
		tmp = c0 * sqrt(((A / l) / V));
	elseif (t_0 <= 1e+253)
		tmp = c0 / sqrt(((V * l) / A));
	else
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, 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, 4e-309], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+253], N[(c0 / N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 A (*.f64 V l)) < 3.9999999999999977e-309

   1. Initial program 32.6%

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

      1. *-un-lft-identity 32.6%

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

      2. times-frac 51.2%

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

   4. Applied egg-rr 51.2%

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

      1. associate-*l/ 51.2%

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

      2. *-un-lft-identity 51.2%

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

   6. Applied egg-rr 51.2%

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

   if 3.9999999999999977e-309 < (/.f64 A (*.f64 V l)) < 9.9999999999999994e252

   1. Initial program 99.6%

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

      1. associate-/r* 86.6%

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

      2. div-inv 86.6%

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

      3. add-cube-cbrt 86.0%

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

      4. times-frac 94.1%

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

      5. pow2 94.1%

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}} \]

   4. Applied egg-rr 94.1%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]
   5. Step-by-step derivation

      1. associate-*l/ 94.1%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

      2. associate-/l/ 94.1%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \color{blue}{\frac{1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      3. associate-*r/ 94.1%

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A \cdot 1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      4. *-rgt-identity 94.1%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{\color{blue}{A}}{\sqrt[3]{\ell} \cdot V}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      5. *-commutative 94.1%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{A}{\color{blue}{V \cdot \sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

   6. Simplified 94.1%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V \cdot \sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]
   7. Step-by-step derivation

      1. associate-/r* 85.9%

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{\frac{A}{V}}{\sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      2. associate-/l/ 86.0%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{\ell}}}} \]

      3. unpow2 86.0%

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

      4. add-cube-cbrt 86.6%

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

      5. sqrt-undiv 42.7%

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

      6. clear-num 42.7%

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

      7. un-div-inv 42.7%

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

      8. sqrt-undiv 86.6%

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

      9. div-inv 85.4%

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

      10. clear-num 86.1%

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

   8. Applied egg-rr 86.1%

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

      1. *-commutative 86.1%

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

      2. associate-*l/ 99.7%

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

      3. associate-*r/ 91.7%

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

   10. Simplified 91.7%

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

   11. Taylor expanded in V around 0 99.7%

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

   if 9.9999999999999994e252 < (/.f64 A (*.f64 V l))

   1. Initial program 44.7%

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

      1. associate-/r* 55.6%

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

      2. sqrt-div 52.2%

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

      3. div-inv 52.2%

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

   4. Applied egg-rr 52.2%

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

      1. associate-*r/ 52.2%

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

      2. *-rgt-identity 52.2%

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

   6. Simplified 52.2%

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

Alternative 6: 79.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 6 \cdot 10^{+260}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, 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 (or (<= t_0 0.0) (not (<= t_0 6e+260)))
     (* c0 (sqrt (/ (/ A l) V)))
     (* c0 (sqrt t_0)))))

C

assert(c0 < A && A < V && 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) || !(t_0 <= 6e+260)) {
		tmp = c0 * sqrt(((A / l) / V));
	} else {
		tmp = c0 * sqrt(t_0);
	}
	return tmp;
}

Fortran

NOTE: c0, A, 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) .or. (.not. (t_0 <= 6d+260))) then
        tmp = c0 * sqrt(((a / l) / v))
    else
        tmp = c0 * sqrt(t_0)
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && 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) || !(t_0 <= 6e+260)) {
		tmp = c0 * Math.sqrt(((A / l) / V));
	} else {
		tmp = c0 * Math.sqrt(t_0);
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 6e+260):
		tmp = c0 * math.sqrt(((A / l) / V))
	else:
		tmp = c0 * math.sqrt(t_0)
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 6e+260))
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	else
		tmp = Float64(c0 * sqrt(t_0));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 6e+260)))
		tmp = c0 * sqrt(((A / l) / V));
	else
		tmp = c0 * sqrt(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, 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[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 6e+260]], $MachinePrecision]], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 0.0 or 5.9999999999999996e260 < (/.f64 A (*.f64 V l))

   1. Initial program 36.3%

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

      1. *-un-lft-identity 36.3%

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

      2. times-frac 52.5%

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

   4. Applied egg-rr 52.5%

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

      1. associate-*l/ 52.5%

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

      2. *-un-lft-identity 52.5%

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

   6. Applied egg-rr 52.5%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 5.9999999999999996e260

   1. Initial program 99.6%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0 \lor \neg \left(\frac{A}{V \cdot \ell} \leq 6 \cdot 10^{+260}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 10^{+253}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, 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 (or (<= t_0 0.0) (not (<= t_0 1e+253)))
     (* c0 (sqrt (/ (/ A V) l)))
     (* c0 (sqrt t_0)))))

C

assert(c0 < A && A < V && 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) || !(t_0 <= 1e+253)) {
		tmp = c0 * sqrt(((A / V) / l));
	} else {
		tmp = c0 * sqrt(t_0);
	}
	return tmp;
}

Fortran

NOTE: c0, A, 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) .or. (.not. (t_0 <= 1d+253))) then
        tmp = c0 * sqrt(((a / v) / l))
    else
        tmp = c0 * sqrt(t_0)
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && 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) || !(t_0 <= 1e+253)) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else {
		tmp = c0 * Math.sqrt(t_0);
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 1e+253):
		tmp = c0 * math.sqrt(((A / V) / l))
	else:
		tmp = c0 * math.sqrt(t_0)
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 1e+253))
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	else
		tmp = Float64(c0 * sqrt(t_0));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 1e+253)))
		tmp = c0 * sqrt(((A / V) / l));
	else
		tmp = c0 * sqrt(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, 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[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 1e+253]], $MachinePrecision]], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 0.0 or 9.9999999999999994e252 < (/.f64 A (*.f64 V l))

   1. Initial program 37.9%

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

      1. associate-/r* 53.0%

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

   3. Simplified 53.0%

      \[\leadsto \color{blue}{c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}} \]
   4. Add Preprocessing

   if 0.0 < (/.f64 A (*.f64 V l)) < 9.9999999999999994e252

   1. Initial program 99.6%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0 \lor \neg \left(\frac{A}{V \cdot \ell} \leq 10^{+253}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-309}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t\_0 \leq 6 \cdot 10^{+260}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, 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 4e-309)
     (* c0 (sqrt (/ (/ A l) V)))
     (if (<= t_0 6e+260)
       (/ c0 (sqrt (/ (* V l) A)))
       (/ c0 (sqrt (* V (/ l A))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 4e-309) {
		tmp = c0 * sqrt(((A / l) / V));
	} else if (t_0 <= 6e+260) {
		tmp = c0 / sqrt(((V * l) / A));
	} else {
		tmp = c0 / sqrt((V * (l / A)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, 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 <= 4d-309) then
        tmp = c0 * sqrt(((a / l) / v))
    else if (t_0 <= 6d+260) then
        tmp = c0 / sqrt(((v * l) / a))
    else
        tmp = c0 / sqrt((v * (l / a)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && 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 <= 4e-309) {
		tmp = c0 * Math.sqrt(((A / l) / V));
	} else if (t_0 <= 6e+260) {
		tmp = c0 / Math.sqrt(((V * l) / A));
	} else {
		tmp = c0 / Math.sqrt((V * (l / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 4e-309:
		tmp = c0 * math.sqrt(((A / l) / V))
	elif t_0 <= 6e+260:
		tmp = c0 / math.sqrt(((V * l) / A))
	else:
		tmp = c0 / math.sqrt((V * (l / A)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 4e-309)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	elseif (t_0 <= 6e+260)
		tmp = Float64(c0 / sqrt(Float64(Float64(V * l) / A)));
	else
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 4e-309)
		tmp = c0 * sqrt(((A / l) / V));
	elseif (t_0 <= 6e+260)
		tmp = c0 / sqrt(((V * l) / A));
	else
		tmp = c0 / sqrt((V * (l / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, 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, 4e-309], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 6e+260], 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)) < 3.9999999999999977e-309

   1. Initial program 32.6%

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

      1. *-un-lft-identity 32.6%

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

      2. times-frac 51.2%

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

   4. Applied egg-rr 51.2%

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

      1. associate-*l/ 51.2%

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

      2. *-un-lft-identity 51.2%

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

   6. Applied egg-rr 51.2%

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

   if 3.9999999999999977e-309 < (/.f64 A (*.f64 V l)) < 5.9999999999999996e260

   1. Initial program 99.6%

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

      1. associate-/r* 86.2%

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

      2. div-inv 86.2%

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

      3. add-cube-cbrt 85.6%

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

      4. times-frac 94.2%

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

      5. pow2 94.2%

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}} \]

   4. Applied egg-rr 94.2%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]
   5. Step-by-step derivation

      1. associate-*l/ 93.5%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

      2. associate-/l/ 93.5%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \color{blue}{\frac{1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      3. associate-*r/ 93.5%

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A \cdot 1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      4. *-rgt-identity 93.5%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{\color{blue}{A}}{\sqrt[3]{\ell} \cdot V}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      5. *-commutative 93.5%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{A}{\color{blue}{V \cdot \sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

   6. Simplified 93.5%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V \cdot \sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]
   7. Step-by-step derivation

      1. associate-/r* 85.5%

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{\frac{A}{V}}{\sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      2. associate-/l/ 85.6%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{\ell}}}} \]

      3. unpow2 85.6%

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

      4. add-cube-cbrt 86.2%

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

      5. sqrt-undiv 43.2%

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

      6. clear-num 43.2%

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

      7. un-div-inv 43.2%

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

      8. sqrt-undiv 86.2%

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

      9. div-inv 85.0%

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

      10. clear-num 85.7%

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

   8. Applied egg-rr 85.7%

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

      1. *-commutative 85.7%

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

      2. associate-*l/ 99.7%

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

      3. associate-*r/ 91.9%

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

   10. Simplified 91.9%

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

   11. Taylor expanded in V around 0 99.7%

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

   if 5.9999999999999996e260 < (/.f64 A (*.f64 V l))

   1. Initial program 41.6%

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

      1. associate-/r* 54.9%

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

      2. div-inv 54.8%

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

      3. add-cube-cbrt 54.7%

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

      4. times-frac 52.9%

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

      5. pow2 52.9%

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}} \]

   4. Applied egg-rr 52.9%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]
   5. Step-by-step derivation

      1. associate-*l/ 53.0%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

      2. associate-/l/ 53.0%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \color{blue}{\frac{1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      3. associate-*r/ 53.0%

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A \cdot 1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      4. *-rgt-identity 53.0%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{\color{blue}{A}}{\sqrt[3]{\ell} \cdot V}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      5. *-commutative 53.0%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{A}{\color{blue}{V \cdot \sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

   6. Simplified 53.0%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V \cdot \sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]
   7. Step-by-step derivation

      1. associate-/r* 54.8%

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{\frac{A}{V}}{\sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      2. associate-/l/ 54.7%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{\ell}}}} \]

      3. unpow2 54.7%

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

      4. add-cube-cbrt 54.9%

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

      5. sqrt-undiv 51.5%

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

      6. clear-num 51.4%

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

      7. un-div-inv 51.5%

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

      8. sqrt-undiv 57.2%

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

      9. div-inv 57.2%

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

      10. clear-num 57.2%

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

   8. Applied egg-rr 57.2%

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

      1. *-commutative 57.2%

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

      2. associate-*l/ 43.9%

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

      3. associate-*r/ 55.7%

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

   10. Simplified 55.7%

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

Alternative 9: 79.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t\_0 \leq 10^{+261}:\\ \;\;\;\;c0 \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, 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 l) V)))
     (if (<= t_0 1e+261) (* c0 (sqrt t_0)) (/ c0 (sqrt (* V (/ l A))))))))

C

assert(c0 < A && A < V && 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 / l) / V));
	} else if (t_0 <= 1e+261) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = c0 / sqrt((V * (l / A)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, 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 / l) / v))
    else if (t_0 <= 1d+261) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = c0 / sqrt((v * (l / a)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && 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 / l) / V));
	} else if (t_0 <= 1e+261) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = c0 / Math.sqrt((V * (l / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, 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 / l) / V))
	elif t_0 <= 1e+261:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = c0 / math.sqrt((V * (l / A)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, 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 / l) / V)));
	elseif (t_0 <= 1e+261)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, 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 / l) / V));
	elseif (t_0 <= 1e+261)
		tmp = c0 * sqrt(t_0);
	else
		tmp = c0 / sqrt((V * (l / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, 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 / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+261], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 31.5%

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

      1. *-un-lft-identity 31.5%

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

      2. times-frac 50.4%

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

   4. Applied egg-rr 50.4%

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

      1. associate-*l/ 50.4%

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

      2. *-un-lft-identity 50.4%

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

   6. Applied egg-rr 50.4%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 9.9999999999999993e260

   1. Initial program 99.6%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing

   if 9.9999999999999993e260 < (/.f64 A (*.f64 V l))

   1. Initial program 40.5%

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

      1. associate-/r* 54.1%

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

      2. div-inv 54.0%

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

      3. add-cube-cbrt 53.8%

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

      4. times-frac 52.0%

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

      5. pow2 52.0%

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}} \]

   4. Applied egg-rr 52.0%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]
   5. Step-by-step derivation

      1. associate-*l/ 52.1%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

      2. associate-/l/ 52.1%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \color{blue}{\frac{1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      3. associate-*r/ 52.1%

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A \cdot 1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      4. *-rgt-identity 52.1%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{\color{blue}{A}}{\sqrt[3]{\ell} \cdot V}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      5. *-commutative 52.1%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{A}{\color{blue}{V \cdot \sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

   6. Simplified 52.1%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V \cdot \sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]
   7. Step-by-step derivation

      1. associate-/r* 54.0%

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{\frac{A}{V}}{\sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      2. associate-/l/ 53.9%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{\ell}}}} \]

      3. unpow2 53.9%

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

      4. add-cube-cbrt 54.1%

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

      5. sqrt-undiv 52.4%

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

      6. clear-num 52.4%

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

      7. un-div-inv 52.5%

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

      8. sqrt-undiv 56.4%

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

      9. div-inv 56.4%

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

      10. clear-num 56.4%

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

   8. Applied egg-rr 56.4%

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

      1. *-commutative 56.4%

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

      2. associate-*l/ 42.8%

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

      3. associate-*r/ 54.8%

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

   10. Simplified 54.8%

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

Alternative 10: 73.3% accurate, 1.0× speedup

Math

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

FPCore

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

Fortran

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: c0, A, 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 71.9%

   \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
2. Add Preprocessing
3. Add Preprocessing

Reproduce

herbie shell --seed 2024090 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))