Henrywood and Agarwal, Equation (3)

Percentage Accurate: 74.2% → 89.5%

Time: 12.2s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.2% 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: 89.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+210}:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \left(c0 \cdot {\ell}^{-0.5}\right)\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-308}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell \cdot \left(-V\right)}}{\sqrt{-A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-314}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \end{array} \]

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) -5e+210)
   (* (sqrt (/ A V)) (* c0 (pow l -0.5)))
   (if (<= (* V l) -5e-308)
     (/ c0 (/ (sqrt (* l (- V))) (sqrt (- A))))
     (if (<= (* V l) 2e-314)
       (/ c0 (sqrt (/ V (/ A l))))
       (* c0 (* (sqrt A) (sqrt (/ (/ 1.0 V) l))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -5e+210) {
		tmp = sqrt((A / V)) * (c0 * pow(l, -0.5));
	} else if ((V * l) <= -5e-308) {
		tmp = c0 / (sqrt((l * -V)) / sqrt(-A));
	} else if ((V * l) <= 2e-314) {
		tmp = c0 / sqrt((V / (A / l)));
	} else {
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / 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) :: tmp
    if ((v * l) <= (-5d+210)) then
        tmp = sqrt((a / v)) * (c0 * (l ** (-0.5d0)))
    else if ((v * l) <= (-5d-308)) then
        tmp = c0 / (sqrt((l * -v)) / sqrt(-a))
    else if ((v * l) <= 2d-314) then
        tmp = c0 / sqrt((v / (a / l)))
    else
        tmp = c0 * (sqrt(a) * sqrt(((1.0d0 / v) / 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 tmp;
	if ((V * l) <= -5e+210) {
		tmp = Math.sqrt((A / V)) * (c0 * Math.pow(l, -0.5));
	} else if ((V * l) <= -5e-308) {
		tmp = c0 / (Math.sqrt((l * -V)) / Math.sqrt(-A));
	} else if ((V * l) <= 2e-314) {
		tmp = c0 / Math.sqrt((V / (A / l)));
	} else {
		tmp = c0 * (Math.sqrt(A) * Math.sqrt(((1.0 / V) / l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -5e+210:
		tmp = math.sqrt((A / V)) * (c0 * math.pow(l, -0.5))
	elif (V * l) <= -5e-308:
		tmp = c0 / (math.sqrt((l * -V)) / math.sqrt(-A))
	elif (V * l) <= 2e-314:
		tmp = c0 / math.sqrt((V / (A / l)))
	else:
		tmp = c0 * (math.sqrt(A) * math.sqrt(((1.0 / V) / l)))
	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) <= -5e+210)
		tmp = Float64(sqrt(Float64(A / V)) * Float64(c0 * (l ^ -0.5)));
	elseif (Float64(V * l) <= -5e-308)
		tmp = Float64(c0 / Float64(sqrt(Float64(l * Float64(-V))) / sqrt(Float64(-A))));
	elseif (Float64(V * l) <= 2e-314)
		tmp = Float64(c0 / sqrt(Float64(V / Float64(A / l))));
	else
		tmp = Float64(c0 * Float64(sqrt(A) * sqrt(Float64(Float64(1.0 / V) / l))));
	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) <= -5e+210)
		tmp = sqrt((A / V)) * (c0 * (l ^ -0.5));
	elseif ((V * l) <= -5e-308)
		tmp = c0 / (sqrt((l * -V)) / sqrt(-A));
	elseif ((V * l) <= 2e-314)
		tmp = c0 / sqrt((V / (A / l)));
	else
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / 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_] := If[LessEqual[N[(V * l), $MachinePrecision], -5e+210], N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] * N[(c0 * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -5e-308], N[(c0 / N[(N[Sqrt[N[(l * (-V)), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-A)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 2e-314], N[(c0 / N[Sqrt[N[(V / N[(A / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Sqrt[N[(N[(1.0 / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -4.9999999999999998e210

   1. Initial program 66.4%

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

      1. associate-/r* 85.7%

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

      2. sqrt-div 41.3%

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

      3. associate-*r/ 37.5%

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

   4. Applied egg-rr 37.5%

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

      1. associate-/l* 41.2%

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

   6. Simplified 41.2%

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

      1. div-inv 41.3%

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

      2. clear-num 41.3%

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

      3. sqrt-div 85.7%

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

      4. un-div-inv 85.7%

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

      5. sqrt-prod 41.2%

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

      6. associate-*r* 37.5%

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

      7. inv-pow 37.5%

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

      8. sqrt-pow1 37.5%

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

      9. metadata-eval 37.5%

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

   8. Applied egg-rr 37.5%

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

      1. *-commutative 37.5%

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

      2. associate-*l* 41.3%

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

   10. Simplified 41.3%

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

   if -4.9999999999999998e210 < (*.f64 V l) < -4.99999999999999955e-308

   1. Initial program 87.8%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. div-inv 74.8%

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

      3. add-cube-cbrt 74.2%

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

      4. times-frac 82.5%

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

      5. pow2 82.5%

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

   4. Applied egg-rr 82.5%

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

      1. associate-*l/ 77.8%

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

      2. associate-/l/ 77.8%

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

      3. associate-*r/ 77.8%

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

      4. *-rgt-identity 77.8%

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

      5. *-commutative 77.8%

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

   6. Simplified 77.8%

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

      1. associate-/l/ 87.1%

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

      2. *-commutative 87.1%

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

      3. associate-*r* 87.2%

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

      4. unpow2 87.2%

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

      5. add-cube-cbrt 87.8%

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

      6. *-commutative 87.8%

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

      7. associate-/r* 74.7%

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

      8. sqrt-div 53.2%

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

      9. clear-num 53.1%

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

      10. div-inv 53.2%

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

      11. sqrt-undiv 74.9%

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

      12. div-inv 74.8%

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

      13. clear-num 75.2%

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

   8. Applied egg-rr 75.2%

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

      1. associate-*r/ 88.0%

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

      2. *-commutative 88.0%

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

      3. frac-2neg 88.0%

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

      4. sqrt-div 99.4%

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

      5. *-commutative 99.4%

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

      6. distribute-rgt-neg-in 99.4%

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

   10. Applied egg-rr 99.4%

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

   if -4.99999999999999955e-308 < (*.f64 V l) < 1.9999999999e-314

   1. Initial program 41.7%

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

      1. associate-/r* 75.4%

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

      2. div-inv 75.3%

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

      3. add-cube-cbrt 75.0%

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

      4. times-frac 68.9%

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

      5. pow2 68.9%

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

   4. Applied egg-rr 68.9%

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

      1. associate-*l/ 68.9%

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

      2. associate-/l/ 68.8%

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

      3. associate-*r/ 71.7%

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

      4. *-rgt-identity 71.7%

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

      5. *-commutative 71.7%

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

   6. Simplified 71.7%

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

      1. associate-/l/ 41.7%

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

      2. *-commutative 41.7%

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

      3. associate-*r* 41.7%

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

      4. unpow2 41.7%

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

      5. add-cube-cbrt 41.7%

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

      6. *-commutative 41.7%

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

      7. associate-/r* 75.4%

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

      8. sqrt-div 35.6%

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

      9. clear-num 35.7%

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

      10. div-inv 35.6%

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

      11. sqrt-undiv 78.5%

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

      12. div-inv 78.5%

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

      13. clear-num 78.5%

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

   8. Applied egg-rr 78.5%

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

      1. *-commutative 78.5%

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

      2. associate-/r/ 78.5%

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

   10. Applied egg-rr 78.5%

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

   if 1.9999999999e-314 < (*.f64 V l)

   1. Initial program 79.2%

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

      1. div-inv 79.1%

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

      2. sqrt-prod 84.4%

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

      3. associate-/r* 87.2%

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

   4. Applied egg-rr 87.2%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+210}:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \left(c0 \cdot {\ell}^{-0.5}\right)\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-308}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell \cdot \left(-V\right)}}{\sqrt{-A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-314}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t_0 \leq 0 \lor \neg \left(t_0 \leq 5 \cdot 10^{+263}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

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 (* c0 (sqrt (/ A (* V l))))))
   (if (or (<= t_0 0.0) (not (<= t_0 5e+263)))
     (* c0 (sqrt (/ (/ A V) l)))
     t_0)))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 5e+263)) {
		tmp = c0 * sqrt(((A / V) / l));
	} else {
		tmp = 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 = c0 * sqrt((a / (v * l)))
    if ((t_0 <= 0.0d0) .or. (.not. (t_0 <= 5d+263))) then
        tmp = c0 * sqrt(((a / v) / l))
    else
        tmp = 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 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 5e+263)) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 5e+263):
		tmp = c0 * math.sqrt(((A / V) / l))
	else:
		tmp = t_0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 5e+263))
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	else
		tmp = 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 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 5e+263)))
		tmp = c0 * sqrt(((A / V) / l));
	else
		tmp = 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[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 5e+263]], $MachinePrecision]], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 0.0 or 5.00000000000000022e263 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 68.6%

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

      1. *-commutative 68.6%

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

      2. associate-/l/ 72.6%

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

   3. Simplified 72.6%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 5.00000000000000022e263

   1. Initial program 97.7%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 0 \lor \neg \left(c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 5 \cdot 10^{+263}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-268} \lor \neg \left(t_0 \leq 10^{+239}\right):\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

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 (* c0 (sqrt (/ A (* V l))))))
   (if (or (<= t_0 2e-268) (not (<= t_0 1e+239)))
     (/ c0 (sqrt (* l (/ V A))))
     t_0)))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if ((t_0 <= 2e-268) || !(t_0 <= 1e+239)) {
		tmp = c0 / sqrt((l * (V / A)));
	} else {
		tmp = 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 = c0 * sqrt((a / (v * l)))
    if ((t_0 <= 2d-268) .or. (.not. (t_0 <= 1d+239))) then
        tmp = c0 / sqrt((l * (v / a)))
    else
        tmp = 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 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if ((t_0 <= 2e-268) || !(t_0 <= 1e+239)) {
		tmp = c0 / Math.sqrt((l * (V / A)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if (t_0 <= 2e-268) or not (t_0 <= 1e+239):
		tmp = c0 / math.sqrt((l * (V / A)))
	else:
		tmp = t_0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if ((t_0 <= 2e-268) || !(t_0 <= 1e+239))
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / A))));
	else
		tmp = 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 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if ((t_0 <= 2e-268) || ~((t_0 <= 1e+239)))
		tmp = c0 / sqrt((l * (V / A)));
	else
		tmp = 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[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 2e-268], N[Not[LessEqual[t$95$0, 1e+239]], $MachinePrecision]], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 1.99999999999999992e-268 or 9.99999999999999991e238 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 69.7%

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

      1. associate-/r* 73.6%

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

      2. div-inv 73.6%

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

      3. add-cube-cbrt 73.2%

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

      4. times-frac 76.1%

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

      5. pow2 76.1%

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

   4. Applied egg-rr 76.1%

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

      1. associate-*l/ 73.7%

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

      2. associate-/l/ 73.7%

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

      3. associate-*r/ 74.2%

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

      4. *-rgt-identity 74.2%

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

      5. *-commutative 74.2%

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

   6. Simplified 74.2%

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

      1. associate-/l/ 69.3%

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

      2. *-commutative 69.3%

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

      3. associate-*r* 69.4%

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

      4. unpow2 69.4%

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

      5. add-cube-cbrt 69.7%

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

      6. *-commutative 69.7%

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

      7. associate-/r* 73.6%

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

      8. sqrt-div 43.4%

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

      9. clear-num 43.4%

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

      10. div-inv 43.4%

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

      11. sqrt-undiv 74.6%

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

      12. div-inv 74.6%

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

      13. clear-num 74.6%

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

   8. Applied egg-rr 74.6%

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

   if 1.99999999999999992e-268 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 9.99999999999999991e238

   1. Initial program 97.5%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 2 \cdot 10^{-268} \lor \neg \left(c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 10^{+239}\right):\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-230}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \mathbf{elif}\;t_0 \leq 10^{+239}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \end{array} \]

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 (* c0 (sqrt (/ A (* V l))))))
   (if (<= t_0 2e-230)
     (/ c0 (sqrt (/ V (/ A l))))
     (if (<= t_0 1e+239) t_0 (/ c0 (sqrt (* l (/ V A))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 2e-230) {
		tmp = c0 / sqrt((V / (A / l)));
	} else if (t_0 <= 1e+239) {
		tmp = t_0;
	} else {
		tmp = c0 / sqrt((l * (V / 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 = c0 * sqrt((a / (v * l)))
    if (t_0 <= 2d-230) then
        tmp = c0 / sqrt((v / (a / l)))
    else if (t_0 <= 1d+239) then
        tmp = t_0
    else
        tmp = c0 / sqrt((l * (v / 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 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 2e-230) {
		tmp = c0 / Math.sqrt((V / (A / l)));
	} else if (t_0 <= 1e+239) {
		tmp = t_0;
	} else {
		tmp = c0 / Math.sqrt((l * (V / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 2e-230:
		tmp = c0 / math.sqrt((V / (A / l)))
	elif t_0 <= 1e+239:
		tmp = t_0
	else:
		tmp = c0 / math.sqrt((l * (V / A)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if (t_0 <= 2e-230)
		tmp = Float64(c0 / sqrt(Float64(V / Float64(A / l))));
	elseif (t_0 <= 1e+239)
		tmp = t_0;
	else
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / 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 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= 2e-230)
		tmp = c0 / sqrt((V / (A / l)));
	elseif (t_0 <= 1e+239)
		tmp = t_0;
	else
		tmp = c0 / sqrt((l * (V / 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[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-230], N[(c0 / N[Sqrt[N[(V / N[(A / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+239], t$95$0, N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 2.00000000000000009e-230

   1. Initial program 71.4%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 76.6%

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

      5. pow2 76.6%

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

   4. Applied egg-rr 76.6%

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

      1. associate-*l/ 73.8%

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

      2. associate-/l/ 73.8%

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

      3. associate-*r/ 74.1%

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

      4. *-rgt-identity 74.1%

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

      5. *-commutative 74.1%

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

   6. Simplified 74.1%

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

      1. associate-/l/ 71.0%

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

      2. *-commutative 71.0%

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

      3. associate-*r* 71.0%

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

      4. unpow2 71.0%

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

      5. add-cube-cbrt 71.4%

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

      6. *-commutative 71.4%

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

      7. associate-/r* 73.0%

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

      8. sqrt-div 42.8%

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

      9. clear-num 42.7%

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

      10. div-inv 42.7%

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

      11. sqrt-undiv 74.2%

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

      12. div-inv 74.2%

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

      13. 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-/r/ 78.2%

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

   10. Applied egg-rr 78.2%

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

   if 2.00000000000000009e-230 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 9.99999999999999991e238

   1. Initial program 97.5%

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

   if 9.99999999999999991e238 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 59.7%

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

      1. associate-/r* 78.0%

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

      2. div-inv 77.9%

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

      3. add-cube-cbrt 77.7%

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

      4. times-frac 70.5%

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

      5. pow2 70.5%

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

   4. Applied egg-rr 70.5%

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

      1. associate-*l/ 74.0%

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

      2. associate-/l/ 73.8%

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

      3. associate-*r/ 75.2%

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

      4. *-rgt-identity 75.2%

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

      5. *-commutative 75.2%

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

   6. Simplified 75.2%

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

      1. associate-/l/ 59.5%

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

      2. *-commutative 59.5%

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

      3. associate-*r* 59.5%

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

      4. unpow2 59.5%

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

      5. add-cube-cbrt 59.7%

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

      6. *-commutative 59.7%

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

      7. associate-/r* 78.0%

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

      8. sqrt-div 46.0%

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

      9. clear-num 46.0%

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

      10. div-inv 46.1%

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

      11. sqrt-undiv 78.2%

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

      12. div-inv 78.2%

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

      13. clear-num 78.1%

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

   8. Applied egg-rr 78.1%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 2 \cdot 10^{-230}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 10^{+239}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-230}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \mathbf{elif}\;t_0 \leq 4 \cdot 10^{+257}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \end{array} \]

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 (/ A (* V l))))))
   (if (<= t_0 2e-230)
     (/ c0 (sqrt (/ V (/ A l))))
     (if (<= t_0 4e+257) t_0 (/ c0 (sqrt (/ l (/ A V))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 2e-230) {
		tmp = c0 / sqrt((V / (A / l)));
	} else if (t_0 <= 4e+257) {
		tmp = t_0;
	} else {
		tmp = c0 / sqrt((l / (A / 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 = c0 * sqrt((a / (v * l)))
    if (t_0 <= 2d-230) then
        tmp = c0 / sqrt((v / (a / l)))
    else if (t_0 <= 4d+257) then
        tmp = t_0
    else
        tmp = c0 / sqrt((l / (a / 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 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 2e-230) {
		tmp = c0 / Math.sqrt((V / (A / l)));
	} else if (t_0 <= 4e+257) {
		tmp = t_0;
	} else {
		tmp = c0 / Math.sqrt((l / (A / V)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 2e-230:
		tmp = c0 / math.sqrt((V / (A / l)))
	elif t_0 <= 4e+257:
		tmp = t_0
	else:
		tmp = c0 / math.sqrt((l / (A / V)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if (t_0 <= 2e-230)
		tmp = Float64(c0 / sqrt(Float64(V / Float64(A / l))));
	elseif (t_0 <= 4e+257)
		tmp = t_0;
	else
		tmp = Float64(c0 / sqrt(Float64(l / Float64(A / 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((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= 2e-230)
		tmp = c0 / sqrt((V / (A / l)));
	elseif (t_0 <= 4e+257)
		tmp = t_0;
	else
		tmp = c0 / sqrt((l / (A / 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[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-230], N[(c0 / N[Sqrt[N[(V / N[(A / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+257], t$95$0, N[(c0 / N[Sqrt[N[(l / N[(A / V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 2.00000000000000009e-230

   1. Initial program 71.4%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 76.6%

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

      5. pow2 76.6%

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

   4. Applied egg-rr 76.6%

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

      1. associate-*l/ 73.8%

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

      2. associate-/l/ 73.8%

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

      3. associate-*r/ 74.1%

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

      4. *-rgt-identity 74.1%

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

      5. *-commutative 74.1%

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

   6. Simplified 74.1%

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

      1. associate-/l/ 71.0%

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

      2. *-commutative 71.0%

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

      3. associate-*r* 71.0%

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

      4. unpow2 71.0%

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

      5. add-cube-cbrt 71.4%

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

      6. *-commutative 71.4%

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

      7. associate-/r* 73.0%

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

      8. sqrt-div 42.8%

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

      9. clear-num 42.7%

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

      10. div-inv 42.7%

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

      11. sqrt-undiv 74.2%

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

      12. div-inv 74.2%

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

      13. 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-/r/ 78.2%

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

   10. Applied egg-rr 78.2%

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

   if 2.00000000000000009e-230 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 4.00000000000000012e257

   1. Initial program 97.6%

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

   if 4.00000000000000012e257 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 56.4%

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

      1. associate-/r* 76.2%

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

      2. div-inv 76.1%

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

      3. add-cube-cbrt 76.0%

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

      4. times-frac 68.2%

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

      5. pow2 68.2%

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

   4. Applied egg-rr 68.2%

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

      1. associate-*l/ 72.0%

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

      2. associate-/l/ 71.8%

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

      3. associate-*r/ 73.3%

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

      4. *-rgt-identity 73.3%

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

      5. *-commutative 73.3%

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

   6. Simplified 73.3%

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

      1. associate-/l/ 56.3%

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

      2. *-commutative 56.3%

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

      3. associate-*r* 56.3%

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

      4. unpow2 56.3%

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

      5. add-cube-cbrt 56.4%

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

      6. *-commutative 56.4%

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

      7. associate-/r* 76.2%

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

      8. sqrt-div 41.6%

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

      9. clear-num 41.6%

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

      10. div-inv 41.6%

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

      11. sqrt-undiv 76.3%

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

      12. div-inv 76.3%

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

      13. clear-num 76.3%

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

   8. Applied egg-rr 76.3%

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

      1. clear-num 76.3%

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

      2. un-div-inv 76.3%

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

   10. Applied egg-rr 76.3%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 2 \cdot 10^{-230}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 4 \cdot 10^{+257}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+118}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-205}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+267}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \end{array} \]

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 (* (sqrt (/ A V)) (/ c0 (sqrt l)))))
   (if (<= (* V l) -1e+118)
     t_0
     (if (<= (* V l) -1e-205)
       (/ c0 (sqrt (/ (* V l) A)))
       (if (<= (* V l) 0.0)
         t_0
         (if (<= (* V l) 2e+267)
           (* c0 (/ (sqrt A) (sqrt (* V l))))
           (/ c0 (sqrt (* l (/ V A))))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt((A / V)) * (c0 / sqrt(l));
	double tmp;
	if ((V * l) <= -1e+118) {
		tmp = t_0;
	} else if ((V * l) <= -1e-205) {
		tmp = c0 / sqrt(((V * l) / A));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else if ((V * l) <= 2e+267) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0 / sqrt((l * (V / 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 = sqrt((a / v)) * (c0 / sqrt(l))
    if ((v * l) <= (-1d+118)) then
        tmp = t_0
    else if ((v * l) <= (-1d-205)) then
        tmp = c0 / sqrt(((v * l) / a))
    else if ((v * l) <= 0.0d0) then
        tmp = t_0
    else if ((v * l) <= 2d+267) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = c0 / sqrt((l * (v / 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 = Math.sqrt((A / V)) * (c0 / Math.sqrt(l));
	double tmp;
	if ((V * l) <= -1e+118) {
		tmp = t_0;
	} else if ((V * l) <= -1e-205) {
		tmp = c0 / Math.sqrt(((V * l) / A));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else if ((V * l) <= 2e+267) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0 / Math.sqrt((l * (V / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt((A / V)) * (c0 / math.sqrt(l))
	tmp = 0
	if (V * l) <= -1e+118:
		tmp = t_0
	elif (V * l) <= -1e-205:
		tmp = c0 / math.sqrt(((V * l) / A))
	elif (V * l) <= 0.0:
		tmp = t_0
	elif (V * l) <= 2e+267:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0 / math.sqrt((l * (V / A)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(sqrt(Float64(A / V)) * Float64(c0 / sqrt(l)))
	tmp = 0.0
	if (Float64(V * l) <= -1e+118)
		tmp = t_0;
	elseif (Float64(V * l) <= -1e-205)
		tmp = Float64(c0 / sqrt(Float64(Float64(V * l) / A)));
	elseif (Float64(V * l) <= 0.0)
		tmp = t_0;
	elseif (Float64(V * l) <= 2e+267)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / 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 = sqrt((A / V)) * (c0 / sqrt(l));
	tmp = 0.0;
	if ((V * l) <= -1e+118)
		tmp = t_0;
	elseif ((V * l) <= -1e-205)
		tmp = c0 / sqrt(((V * l) / A));
	elseif ((V * l) <= 0.0)
		tmp = t_0;
	elseif ((V * l) <= 2e+267)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0 / sqrt((l * (V / 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[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] * N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -1e+118], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -1e-205], N[(c0 / N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], 2e+267], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -9.99999999999999967e117 or -1e-205 < (*.f64 V l) < -0.0

   1. Initial program 57.0%

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

      1. associate-/r* 75.5%

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

      2. sqrt-div 46.2%

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

      3. associate-*r/ 44.9%

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

   4. Applied egg-rr 44.9%

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

      1. associate-/l* 46.2%

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

   6. Simplified 46.2%

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

      1. associate-/r/ 45.1%

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

   8. Applied egg-rr 45.1%

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

   if -9.99999999999999967e117 < (*.f64 V l) < -1e-205

   1. Initial program 97.8%

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

      1. associate-/r* 78.5%

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

      2. div-inv 78.5%

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

      3. add-cube-cbrt 78.0%

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

      4. times-frac 90.2%

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

      5. pow2 90.2%

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

   4. Applied egg-rr 90.2%

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

      1. associate-*l/ 83.2%

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

      2. associate-/l/ 83.2%

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

      3. associate-*r/ 83.2%

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

      4. *-rgt-identity 83.2%

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

      5. *-commutative 83.2%

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

   6. Simplified 83.2%

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

      1. associate-/l/ 97.0%

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

      2. *-commutative 97.0%

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

      3. associate-*r* 97.1%

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

      4. unpow2 97.1%

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

      5. add-cube-cbrt 97.8%

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

      6. *-commutative 97.8%

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

      7. associate-/r* 78.5%

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

      8. sqrt-div 49.1%

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

      9. clear-num 49.1%

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

      10. div-inv 49.1%

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

      11. sqrt-undiv 78.7%

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

      12. div-inv 78.6%

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

      13. clear-num 79.2%

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

   8. Applied egg-rr 79.2%

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

      1. *-commutative 79.2%

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

      2. associate-*l/ 98.0%

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

   10. Simplified 98.0%

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

   if -0.0 < (*.f64 V l) < 1.9999999999999999e267

   1. Initial program 92.3%

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

      1. sqrt-div 99.3%

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

      2. associate-*r/ 93.6%

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

   4. Applied egg-rr 93.6%

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

      1. *-commutative 93.6%

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

      2. associate-/l* 95.8%

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

      3. associate-/r/ 99.3%

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

   6. Simplified 99.3%

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

   if 1.9999999999999999e267 < (*.f64 V l)

   1. Initial program 35.6%

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

      1. associate-/r* 56.1%

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

      2. div-inv 56.1%

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

      3. add-cube-cbrt 55.9%

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

      4. times-frac 52.7%

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

      5. pow2 52.7%

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

   4. Applied egg-rr 52.7%

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

      1. associate-*l/ 52.8%

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

      2. associate-/l/ 52.7%

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

      3. associate-*r/ 52.7%

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

      4. *-rgt-identity 52.7%

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

      5. *-commutative 52.7%

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

   6. Simplified 52.7%

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

      1. associate-/l/ 35.6%

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

      2. *-commutative 35.6%

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

      3. associate-*r* 35.6%

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

      4. unpow2 35.6%

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

      5. add-cube-cbrt 35.6%

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

      6. *-commutative 35.6%

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

      7. associate-/r* 56.1%

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

      8. sqrt-div 32.5%

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

      9. clear-num 32.4%

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

      10. div-inv 32.5%

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

      11. sqrt-undiv 56.2%

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

      12. div-inv 56.3%

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

      13. clear-num 56.3%

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

   8. Applied egg-rr 56.3%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+118}:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-205}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+267}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{V}}\\ \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+118}:\\ \;\;\;\;t_0 \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-205}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{t_0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+267}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \end{array} \]

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 (sqrt (/ A V))))
   (if (<= (* V l) -1e+118)
     (* t_0 (/ c0 (sqrt l)))
     (if (<= (* V l) -1e-205)
       (/ c0 (sqrt (/ (* V l) A)))
       (if (<= (* V l) 0.0)
         (* c0 (/ t_0 (sqrt l)))
         (if (<= (* V l) 2e+267)
           (* c0 (/ (sqrt A) (sqrt (* V l))))
           (/ c0 (sqrt (* l (/ V A))))))))))

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) <= -1e+118) {
		tmp = t_0 * (c0 / sqrt(l));
	} else if ((V * l) <= -1e-205) {
		tmp = c0 / sqrt(((V * l) / A));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * (t_0 / sqrt(l));
	} else if ((V * l) <= 2e+267) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0 / sqrt((l * (V / 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 = sqrt((a / v))
    if ((v * l) <= (-1d+118)) then
        tmp = t_0 * (c0 / sqrt(l))
    else if ((v * l) <= (-1d-205)) then
        tmp = c0 / sqrt(((v * l) / a))
    else if ((v * l) <= 0.0d0) then
        tmp = c0 * (t_0 / sqrt(l))
    else if ((v * l) <= 2d+267) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = c0 / sqrt((l * (v / 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 = Math.sqrt((A / V));
	double tmp;
	if ((V * l) <= -1e+118) {
		tmp = t_0 * (c0 / Math.sqrt(l));
	} else if ((V * l) <= -1e-205) {
		tmp = c0 / Math.sqrt(((V * l) / A));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * (t_0 / Math.sqrt(l));
	} else if ((V * l) <= 2e+267) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0 / Math.sqrt((l * (V / A)));
	}
	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) <= -1e+118:
		tmp = t_0 * (c0 / math.sqrt(l))
	elif (V * l) <= -1e-205:
		tmp = c0 / math.sqrt(((V * l) / A))
	elif (V * l) <= 0.0:
		tmp = c0 * (t_0 / math.sqrt(l))
	elif (V * l) <= 2e+267:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0 / math.sqrt((l * (V / A)))
	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) <= -1e+118)
		tmp = Float64(t_0 * Float64(c0 / sqrt(l)));
	elseif (Float64(V * l) <= -1e-205)
		tmp = Float64(c0 / sqrt(Float64(Float64(V * l) / A)));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 * Float64(t_0 / sqrt(l)));
	elseif (Float64(V * l) <= 2e+267)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / 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 = sqrt((A / V));
	tmp = 0.0;
	if ((V * l) <= -1e+118)
		tmp = t_0 * (c0 / sqrt(l));
	elseif ((V * l) <= -1e-205)
		tmp = c0 / sqrt(((V * l) / A));
	elseif ((V * l) <= 0.0)
		tmp = c0 * (t_0 / sqrt(l));
	elseif ((V * l) <= 2e+267)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0 / sqrt((l * (V / 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[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -1e+118], N[(t$95$0 * N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-205], N[(c0 / N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 * N[(t$95$0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 2e+267], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -9.99999999999999967e117

   1. Initial program 65.5%

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

      1. associate-/r* 77.2%

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

      2. sqrt-div 47.3%

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

      3. associate-*r/ 45.0%

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

   4. Applied egg-rr 45.0%

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

      1. associate-/l* 47.3%

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

   6. Simplified 47.3%

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

      1. associate-/r/ 47.4%

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

   8. Applied egg-rr 47.4%

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

   if -9.99999999999999967e117 < (*.f64 V l) < -1e-205

   1. Initial program 97.8%

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

      1. associate-/r* 78.5%

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

      2. div-inv 78.5%

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

      3. add-cube-cbrt 78.0%

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

      4. times-frac 90.2%

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

      5. pow2 90.2%

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

   4. Applied egg-rr 90.2%

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

      1. associate-*l/ 83.2%

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

      2. associate-/l/ 83.2%

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

      3. associate-*r/ 83.2%

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

      4. *-rgt-identity 83.2%

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

      5. *-commutative 83.2%

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

   6. Simplified 83.2%

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

      1. associate-/l/ 97.0%

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

      2. *-commutative 97.0%

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

      3. associate-*r* 97.1%

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

      4. unpow2 97.1%

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

      5. add-cube-cbrt 97.8%

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

      6. *-commutative 97.8%

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

      7. associate-/r* 78.5%

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

      8. sqrt-div 49.1%

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

      9. clear-num 49.1%

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

      10. div-inv 49.1%

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

      11. sqrt-undiv 78.7%

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

      12. div-inv 78.6%

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

      13. clear-num 79.2%

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

   8. Applied egg-rr 79.2%

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

      1. *-commutative 79.2%

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

      2. associate-*l/ 98.0%

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

   10. Simplified 98.0%

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

   if -1e-205 < (*.f64 V l) < -0.0

   1. Initial program 48.2%

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

      1. associate-/r* 73.6%

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

      2. sqrt-div 45.0%

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

      3. associate-*r/ 44.9%

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

   4. Applied egg-rr 44.9%

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

      1. *-commutative 44.9%

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

      2. associate-/l* 44.7%

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

      3. associate-/r/ 45.0%

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

   6. Simplified 45.0%

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

   if -0.0 < (*.f64 V l) < 1.9999999999999999e267

   1. Initial program 92.3%

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

      1. sqrt-div 99.3%

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

      2. associate-*r/ 93.6%

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

   4. Applied egg-rr 93.6%

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

      1. *-commutative 93.6%

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

      2. associate-/l* 95.8%

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

      3. associate-/r/ 99.3%

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

   6. Simplified 99.3%

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

   if 1.9999999999999999e267 < (*.f64 V l)

   1. Initial program 35.6%

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

      1. associate-/r* 56.1%

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

      2. div-inv 56.1%

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

      3. add-cube-cbrt 55.9%

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

      4. times-frac 52.7%

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

      5. pow2 52.7%

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

   4. Applied egg-rr 52.7%

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

      1. associate-*l/ 52.8%

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

      2. associate-/l/ 52.7%

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

      3. associate-*r/ 52.7%

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

      4. *-rgt-identity 52.7%

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

      5. *-commutative 52.7%

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

   6. Simplified 52.7%

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

      1. associate-/l/ 35.6%

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

      2. *-commutative 35.6%

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

      3. associate-*r* 35.6%

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

      4. unpow2 35.6%

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

      5. add-cube-cbrt 35.6%

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

      6. *-commutative 35.6%

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

      7. associate-/r* 56.1%

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

      8. sqrt-div 32.5%

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

      9. clear-num 32.4%

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

      10. div-inv 32.5%

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

      11. sqrt-undiv 56.2%

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

      12. div-inv 56.3%

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

      13. clear-num 56.3%

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

   8. Applied egg-rr 56.3%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+118}:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-205}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+267}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+118}:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-205}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-314}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \end{array} \]

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+118)
   (* (sqrt (/ A V)) (/ c0 (sqrt l)))
   (if (<= (* V l) -1e-205)
     (/ c0 (sqrt (/ (* V l) A)))
     (if (<= (* V l) 2e-314)
       (/ c0 (sqrt (/ l (/ A V))))
       (* c0 (* (sqrt A) (sqrt (/ (/ 1.0 V) l))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+118) {
		tmp = sqrt((A / V)) * (c0 / sqrt(l));
	} else if ((V * l) <= -1e-205) {
		tmp = c0 / sqrt(((V * l) / A));
	} else if ((V * l) <= 2e-314) {
		tmp = c0 / sqrt((l / (A / V)));
	} else {
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / 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) :: tmp
    if ((v * l) <= (-1d+118)) then
        tmp = sqrt((a / v)) * (c0 / sqrt(l))
    else if ((v * l) <= (-1d-205)) then
        tmp = c0 / sqrt(((v * l) / a))
    else if ((v * l) <= 2d-314) then
        tmp = c0 / sqrt((l / (a / v)))
    else
        tmp = c0 * (sqrt(a) * sqrt(((1.0d0 / v) / 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 tmp;
	if ((V * l) <= -1e+118) {
		tmp = Math.sqrt((A / V)) * (c0 / Math.sqrt(l));
	} else if ((V * l) <= -1e-205) {
		tmp = c0 / Math.sqrt(((V * l) / A));
	} else if ((V * l) <= 2e-314) {
		tmp = c0 / Math.sqrt((l / (A / V)));
	} else {
		tmp = c0 * (Math.sqrt(A) * Math.sqrt(((1.0 / V) / l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -1e+118:
		tmp = math.sqrt((A / V)) * (c0 / math.sqrt(l))
	elif (V * l) <= -1e-205:
		tmp = c0 / math.sqrt(((V * l) / A))
	elif (V * l) <= 2e-314:
		tmp = c0 / math.sqrt((l / (A / V)))
	else:
		tmp = c0 * (math.sqrt(A) * math.sqrt(((1.0 / V) / l)))
	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+118)
		tmp = Float64(sqrt(Float64(A / V)) * Float64(c0 / sqrt(l)));
	elseif (Float64(V * l) <= -1e-205)
		tmp = Float64(c0 / sqrt(Float64(Float64(V * l) / A)));
	elseif (Float64(V * l) <= 2e-314)
		tmp = Float64(c0 / sqrt(Float64(l / Float64(A / V))));
	else
		tmp = Float64(c0 * Float64(sqrt(A) * sqrt(Float64(Float64(1.0 / V) / l))));
	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+118)
		tmp = sqrt((A / V)) * (c0 / sqrt(l));
	elseif ((V * l) <= -1e-205)
		tmp = c0 / sqrt(((V * l) / A));
	elseif ((V * l) <= 2e-314)
		tmp = c0 / sqrt((l / (A / V)));
	else
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / 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_] := If[LessEqual[N[(V * l), $MachinePrecision], -1e+118], N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] * N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-205], N[(c0 / N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 2e-314], N[(c0 / N[Sqrt[N[(l / N[(A / V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Sqrt[N[(N[(1.0 / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -9.99999999999999967e117

   1. Initial program 65.5%

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

      1. associate-/r* 77.2%

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

      2. sqrt-div 47.3%

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

      3. associate-*r/ 45.0%

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

   4. Applied egg-rr 45.0%

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

      1. associate-/l* 47.3%

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

   6. Simplified 47.3%

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

      1. associate-/r/ 47.4%

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

   8. Applied egg-rr 47.4%

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

   if -9.99999999999999967e117 < (*.f64 V l) < -1e-205

   1. Initial program 97.8%

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

      1. associate-/r* 78.5%

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

      2. div-inv 78.5%

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

      3. add-cube-cbrt 78.0%

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

      4. times-frac 90.2%

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

      5. pow2 90.2%

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

   4. Applied egg-rr 90.2%

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

      1. associate-*l/ 83.2%

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

      2. associate-/l/ 83.2%

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

      3. associate-*r/ 83.2%

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

      4. *-rgt-identity 83.2%

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

      5. *-commutative 83.2%

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

   6. Simplified 83.2%

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

      1. associate-/l/ 97.0%

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

      2. *-commutative 97.0%

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

      3. associate-*r* 97.1%

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

      4. unpow2 97.1%

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

      5. add-cube-cbrt 97.8%

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

      6. *-commutative 97.8%

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

      7. associate-/r* 78.5%

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

      8. sqrt-div 49.1%

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

      9. clear-num 49.1%

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

      10. div-inv 49.1%

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

      11. sqrt-undiv 78.7%

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

      12. div-inv 78.6%

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

      13. clear-num 79.2%

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

   8. Applied egg-rr 79.2%

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

      1. *-commutative 79.2%

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

      2. associate-*l/ 98.0%

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

   10. Simplified 98.0%

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

   if -1e-205 < (*.f64 V l) < 1.9999999999e-314

   1. Initial program 48.8%

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

      1. associate-/r* 74.3%

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

      2. div-inv 74.2%

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

      3. add-cube-cbrt 73.9%

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

      4. times-frac 69.2%

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

      5. pow2 69.2%

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

   4. Applied egg-rr 69.2%

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

      1. associate-*l/ 69.2%

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

      2. associate-/l/ 69.2%

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

      3. associate-*r/ 71.4%

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

      4. *-rgt-identity 71.4%

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

      5. *-commutative 71.4%

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

   6. Simplified 71.4%

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

      1. associate-/l/ 48.6%

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

      2. *-commutative 48.6%

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

      3. associate-*r* 48.6%

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

      4. unpow2 48.6%

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

      5. add-cube-cbrt 48.8%

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

      6. *-commutative 48.8%

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

      7. associate-/r* 74.3%

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

      8. sqrt-div 43.9%

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

      9. clear-num 43.9%

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

      10. div-inv 43.8%

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

      11. sqrt-undiv 76.6%

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

      12. div-inv 76.7%

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

      13. clear-num 76.6%

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

   8. Applied egg-rr 76.6%

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

      1. clear-num 76.7%

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

      2. un-div-inv 76.6%

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

   10. Applied egg-rr 76.6%

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

   if 1.9999999999e-314 < (*.f64 V l)

   1. Initial program 79.2%

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

      1. div-inv 79.1%

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

      2. sqrt-prod 84.4%

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

      3. associate-/r* 87.2%

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

   4. Applied egg-rr 87.2%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+118}:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-205}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-314}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 89.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+210}:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \left(c0 \cdot {\ell}^{-0.5}\right)\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-308}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-314}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \end{array} \]

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) -5e+210)
   (* (sqrt (/ A V)) (* c0 (pow l -0.5)))
   (if (<= (* V l) -5e-308)
     (* c0 (/ (sqrt (- A)) (sqrt (* l (- V)))))
     (if (<= (* V l) 2e-314)
       (/ c0 (sqrt (/ V (/ A l))))
       (* c0 (* (sqrt A) (sqrt (/ (/ 1.0 V) l))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -5e+210) {
		tmp = sqrt((A / V)) * (c0 * pow(l, -0.5));
	} else if ((V * l) <= -5e-308) {
		tmp = c0 * (sqrt(-A) / sqrt((l * -V)));
	} else if ((V * l) <= 2e-314) {
		tmp = c0 / sqrt((V / (A / l)));
	} else {
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / 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) :: tmp
    if ((v * l) <= (-5d+210)) then
        tmp = sqrt((a / v)) * (c0 * (l ** (-0.5d0)))
    else if ((v * l) <= (-5d-308)) then
        tmp = c0 * (sqrt(-a) / sqrt((l * -v)))
    else if ((v * l) <= 2d-314) then
        tmp = c0 / sqrt((v / (a / l)))
    else
        tmp = c0 * (sqrt(a) * sqrt(((1.0d0 / v) / 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 tmp;
	if ((V * l) <= -5e+210) {
		tmp = Math.sqrt((A / V)) * (c0 * Math.pow(l, -0.5));
	} else if ((V * l) <= -5e-308) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((l * -V)));
	} else if ((V * l) <= 2e-314) {
		tmp = c0 / Math.sqrt((V / (A / l)));
	} else {
		tmp = c0 * (Math.sqrt(A) * Math.sqrt(((1.0 / V) / l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -5e+210:
		tmp = math.sqrt((A / V)) * (c0 * math.pow(l, -0.5))
	elif (V * l) <= -5e-308:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((l * -V)))
	elif (V * l) <= 2e-314:
		tmp = c0 / math.sqrt((V / (A / l)))
	else:
		tmp = c0 * (math.sqrt(A) * math.sqrt(((1.0 / V) / l)))
	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) <= -5e+210)
		tmp = Float64(sqrt(Float64(A / V)) * Float64(c0 * (l ^ -0.5)));
	elseif (Float64(V * l) <= -5e-308)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(l * Float64(-V)))));
	elseif (Float64(V * l) <= 2e-314)
		tmp = Float64(c0 / sqrt(Float64(V / Float64(A / l))));
	else
		tmp = Float64(c0 * Float64(sqrt(A) * sqrt(Float64(Float64(1.0 / V) / l))));
	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) <= -5e+210)
		tmp = sqrt((A / V)) * (c0 * (l ^ -0.5));
	elseif ((V * l) <= -5e-308)
		tmp = c0 * (sqrt(-A) / sqrt((l * -V)));
	elseif ((V * l) <= 2e-314)
		tmp = c0 / sqrt((V / (A / l)));
	else
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / 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_] := If[LessEqual[N[(V * l), $MachinePrecision], -5e+210], N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] * N[(c0 * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -5e-308], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(l * (-V)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 2e-314], N[(c0 / N[Sqrt[N[(V / N[(A / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Sqrt[N[(N[(1.0 / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -4.9999999999999998e210

   1. Initial program 66.4%

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

      1. associate-/r* 85.7%

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

      2. sqrt-div 41.3%

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

      3. associate-*r/ 37.5%

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

   4. Applied egg-rr 37.5%

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

      1. associate-/l* 41.2%

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

   6. Simplified 41.2%

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

      1. div-inv 41.3%

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

      2. clear-num 41.3%

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

      3. sqrt-div 85.7%

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

      4. un-div-inv 85.7%

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

      5. sqrt-prod 41.2%

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

      6. associate-*r* 37.5%

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

      7. inv-pow 37.5%

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

      8. sqrt-pow1 37.5%

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

      9. metadata-eval 37.5%

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

   8. Applied egg-rr 37.5%

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

      1. *-commutative 37.5%

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

      2. associate-*l* 41.3%

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

   10. Simplified 41.3%

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

   if -4.9999999999999998e210 < (*.f64 V l) < -4.99999999999999955e-308

   1. Initial program 87.8%

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

      1. frac-2neg 87.8%

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

      2. sqrt-div 99.2%

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

      3. distribute-rgt-neg-in 99.2%

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

   4. Applied egg-rr 99.2%

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

      1. distribute-rgt-neg-out 99.2%

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

      2. *-commutative 99.2%

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

      3. distribute-rgt-neg-in 99.2%

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

   6. Simplified 99.2%

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

   if -4.99999999999999955e-308 < (*.f64 V l) < 1.9999999999e-314

   1. Initial program 41.7%

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

      1. associate-/r* 75.4%

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

      2. div-inv 75.3%

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

      3. add-cube-cbrt 75.0%

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

      4. times-frac 68.9%

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

      5. pow2 68.9%

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

   4. Applied egg-rr 68.9%

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

      1. associate-*l/ 68.9%

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

      2. associate-/l/ 68.8%

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

      3. associate-*r/ 71.7%

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

      4. *-rgt-identity 71.7%

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

      5. *-commutative 71.7%

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

   6. Simplified 71.7%

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

      1. associate-/l/ 41.7%

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

      2. *-commutative 41.7%

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

      3. associate-*r* 41.7%

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

      4. unpow2 41.7%

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

      5. add-cube-cbrt 41.7%

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

      6. *-commutative 41.7%

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

      7. associate-/r* 75.4%

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

      8. sqrt-div 35.6%

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

      9. clear-num 35.7%

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

      10. div-inv 35.6%

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

      11. sqrt-undiv 78.5%

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

      12. div-inv 78.5%

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

      13. clear-num 78.5%

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

   8. Applied egg-rr 78.5%

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

      1. *-commutative 78.5%

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

      2. associate-/r/ 78.5%

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

   10. Applied egg-rr 78.5%

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

   if 1.9999999999e-314 < (*.f64 V l)

   1. Initial program 79.2%

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

      1. div-inv 79.1%

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

      2. sqrt-prod 84.4%

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

      3. associate-/r* 87.2%

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

   4. Applied egg-rr 87.2%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+210}:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \left(c0 \cdot {\ell}^{-0.5}\right)\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-308}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-314}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 80.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 10^{-312}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t_0 \leq 10^{+300}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \end{array} \]

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 (/ A (* V l))))
   (if (<= t_0 1e-312)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 1e+300)
       (/ c0 (sqrt (/ (* V l) A)))
       (/ c0 (sqrt (/ l (/ A V))))))))

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 <= 1e-312) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 1e+300) {
		tmp = c0 / sqrt(((V * l) / A));
	} else {
		tmp = c0 / sqrt((l / (A / 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 = a / (v * l)
    if (t_0 <= 1d-312) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 1d+300) then
        tmp = c0 / sqrt(((v * l) / a))
    else
        tmp = c0 / sqrt((l / (a / 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 = A / (V * l);
	double tmp;
	if (t_0 <= 1e-312) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 1e+300) {
		tmp = c0 / Math.sqrt(((V * l) / A));
	} else {
		tmp = c0 / Math.sqrt((l / (A / V)));
	}
	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 <= 1e-312:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 1e+300:
		tmp = c0 / math.sqrt(((V * l) / A))
	else:
		tmp = c0 / math.sqrt((l / (A / V)))
	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 <= 1e-312)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 1e+300)
		tmp = Float64(c0 / sqrt(Float64(Float64(V * l) / A)));
	else
		tmp = Float64(c0 / sqrt(Float64(l / Float64(A / 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 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 1e-312)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 1e+300)
		tmp = c0 / sqrt(((V * l) / A));
	else
		tmp = c0 / sqrt((l / (A / 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[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-312], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+300], N[(c0 / N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(l / N[(A / V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 A (*.f64 V l)) < 9.9999999999847e-313

   1. Initial program 42.3%

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

      1. *-commutative 42.3%

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

      2. associate-/l/ 58.6%

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

   3. Simplified 58.6%

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

   if 9.9999999999847e-313 < (/.f64 A (*.f64 V l)) < 1.0000000000000001e300

   1. Initial program 98.8%

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

      1. associate-/r* 85.5%

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

      2. div-inv 85.5%

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

      3. add-cube-cbrt 84.8%

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

      4. times-frac 93.3%

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

      5. pow2 93.3%

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

   4. Applied egg-rr 93.3%

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

      1. associate-*l/ 89.4%

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

      2. associate-/l/ 89.4%

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

      3. associate-*r/ 89.3%

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

      4. *-rgt-identity 89.3%

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

      5. *-commutative 89.3%

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

   6. Simplified 89.3%

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

      1. associate-/l/ 98.0%

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

      2. *-commutative 98.0%

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

      3. associate-*r* 98.0%

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

      4. unpow2 98.0%

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

      5. add-cube-cbrt 98.8%

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

      6. *-commutative 98.8%

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

      7. associate-/r* 85.5%

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

      8. sqrt-div 45.9%

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

      9. clear-num 45.9%

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

      10. div-inv 45.9%

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

      11. sqrt-undiv 85.6%

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

      12. div-inv 85.3%

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

      13. clear-num 85.5%

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

   8. Applied egg-rr 85.5%

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

      1. *-commutative 85.5%

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

      2. associate-*l/ 99.0%

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

   10. Simplified 99.0%

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

   if 1.0000000000000001e300 < (/.f64 A (*.f64 V l))

   1. Initial program 48.3%

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

      1. associate-/r* 68.1%

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

      2. div-inv 68.0%

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

      3. add-cube-cbrt 67.9%

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

      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/ 63.9%

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

      3. associate-*r/ 65.8%

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

      4. *-rgt-identity 65.8%

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

      5. *-commutative 65.8%

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

   6. Simplified 65.8%

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

      1. associate-/l/ 48.3%

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

      2. *-commutative 48.3%

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

      3. associate-*r* 48.3%

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

      4. unpow2 48.3%

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

      5. add-cube-cbrt 48.3%

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

      6. *-commutative 48.3%

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

      7. associate-/r* 68.1%

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

      8. sqrt-div 43.8%

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

      9. clear-num 43.8%

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

      10. div-inv 43.8%

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

      11. sqrt-undiv 72.1%

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

      12. div-inv 72.1%

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

      13. clear-num 72.1%

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

   8. Applied egg-rr 72.1%

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

      1. clear-num 72.1%

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

      2. un-div-inv 72.1%

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

   10. Applied egg-rr 72.1%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 10^{-312}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 10^{+300}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 74.2% 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 76.1%

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

3. Final simplification 76.1%

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

Reproduce

herbie shell --seed 2024014 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))