Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.5% → 91.1%

Time: 10.8s

Alternatives: 14

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-308}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \left(-\ell\right)}}{\sqrt{-A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+274}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (/ (sqrt (/ A V)) (sqrt l)))))
   (if (<= (* V l) (- INFINITY))
     t_0
     (if (<= (* V l) -2e-308)
       (/ c0 (/ (sqrt (* V (- l))) (sqrt (- A))))
       (if (<= (* V l) 0.0)
         t_0
         (if (<= (* V l) 1e+274)
           (* c0 (* (sqrt A) (sqrt (/ (/ 1.0 V) l))))
           (* c0 (sqrt (/ (/ A l) V)))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * (sqrt((A / V)) / sqrt(l));
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = t_0;
	} else if ((V * l) <= -2e-308) {
		tmp = c0 / (sqrt((V * -l)) / sqrt(-A));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else if ((V * l) <= 1e+274) {
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	} else {
		tmp = c0 * sqrt(((A / l) / V));
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else if ((V * l) <= -2e-308) {
		tmp = c0 / (Math.sqrt((V * -l)) / Math.sqrt(-A));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else if ((V * l) <= 1e+274) {
		tmp = c0 * (Math.sqrt(A) * Math.sqrt(((1.0 / V) / l)));
	} else {
		tmp = c0 * Math.sqrt(((A / l) / V));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = t_0
	elif (V * l) <= -2e-308:
		tmp = c0 / (math.sqrt((V * -l)) / math.sqrt(-A))
	elif (V * l) <= 0.0:
		tmp = t_0
	elif (V * l) <= 1e+274:
		tmp = c0 * (math.sqrt(A) * math.sqrt(((1.0 / V) / l)))
	else:
		tmp = c0 * math.sqrt(((A / l) / V))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)))
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = t_0;
	elseif (Float64(V * l) <= -2e-308)
		tmp = Float64(c0 / Float64(sqrt(Float64(V * Float64(-l))) / sqrt(Float64(-A))));
	elseif (Float64(V * l) <= 0.0)
		tmp = t_0;
	elseif (Float64(V * l) <= 1e+274)
		tmp = Float64(c0 * Float64(sqrt(A) * sqrt(Float64(Float64(1.0 / V) / l))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * (sqrt((A / V)) / sqrt(l));
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = t_0;
	elseif ((V * l) <= -2e-308)
		tmp = c0 / (sqrt((V * -l)) / sqrt(-A));
	elseif ((V * l) <= 0.0)
		tmp = t_0;
	elseif ((V * l) <= 1e+274)
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	else
		tmp = c0 * sqrt(((A / l) / V));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -2e-308], N[(c0 / N[(N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-A)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], 1e+274], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Sqrt[N[(N[(1.0 / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -inf.0 or -1.9999999999999998e-308 < (*.f64 V l) < 0.0

   1. Initial program 33.7%

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

      1. associate-/r* 68.4%

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

      2. sqrt-div 61.4%

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

      3. div-inv 61.3%

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

   4. Applied egg-rr 61.3%

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

      1. associate-*r/ 61.4%

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

      2. *-rgt-identity 61.4%

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

   6. Simplified 61.4%

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

   if -inf.0 < (*.f64 V l) < -1.9999999999999998e-308

   1. Initial program 87.1%

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

      1. associate-/r* 75.8%

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

      2. clear-num 75.8%

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

      3. sqrt-div 75.8%

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

      4. metadata-eval 75.8%

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

      5. div-inv 75.7%

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

      6. clear-num 75.8%

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

   4. Applied egg-rr 75.8%

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

      1. *-commutative 75.8%

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

      2. associate-*l/ 87.0%

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

      3. associate-/l* 77.5%

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

   6. Simplified 77.5%

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

      1. metadata-eval 77.5%

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

      2. sqrt-div 77.1%

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

      3. associate-*r/ 86.5%

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

      4. clear-num 87.1%

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

      5. associate-/r* 75.8%

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

      6. sqrt-undiv 35.7%

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

      7. clear-num 35.7%

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

      8. un-div-inv 35.8%

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

      9. sqrt-div 0.0%

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

      10. associate-/r/ 0.0%

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

      11. sqrt-div 45.8%

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

      12. sqrt-prod 77.6%

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

      13. *-commutative 77.6%

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

      14. clear-num 75.9%

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

      15. un-div-inv 76.9%

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

   8. Applied egg-rr 76.9%

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

      1. clear-num 76.4%

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

      2. associate-/r/ 75.9%

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

      3. clear-num 77.6%

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

   10. Applied egg-rr 77.6%

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

       1. *-commutative 77.6%

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

       2. associate-/l* 87.1%

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

       3. frac-2neg 87.1%

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

       4. sqrt-div 99.3%

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

       5. *-commutative 99.3%

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

       6. distribute-rgt-neg-in 99.3%

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

   12. Applied egg-rr 99.3%

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

   if 0.0 < (*.f64 V l) < 9.99999999999999921e273

   1. Initial program 85.1%

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

      1. pow1/2 85.1%

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

      2. div-inv 85.1%

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

      3. unpow-prod-down 99.5%

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

      4. pow1/2 99.5%

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

      5. associate-/r* 99.5%

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

   4. Applied egg-rr 99.5%

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

      1. unpow1/2 99.5%

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

   6. Simplified 99.5%

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

   if 9.99999999999999921e273 < (*.f64 V l)

   1. Initial program 52.7%

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

      1. associate-/r* 83.1%

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

      2. clear-num 83.1%

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

      3. sqrt-div 82.8%

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

      4. metadata-eval 82.8%

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

      5. div-inv 82.9%

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

      6. clear-num 82.9%

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

   4. Applied egg-rr 82.9%

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

      1. *-commutative 82.9%

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

      2. associate-*l/ 52.7%

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

      3. associate-/l* 83.0%

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

   6. Simplified 83.0%

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

      1. metadata-eval 83.0%

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

      2. sqrt-div 83.0%

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

      3. associate-*r/ 52.7%

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

      4. clear-num 52.7%

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

      5. *-commutative 52.7%

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

      6. associate-/r* 83.0%

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

   8. Applied egg-rr 83.0%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-308}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \left(-\ell\right)}}{\sqrt{-A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+274}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+244}:\\ \;\;\;\;c0 \cdot \frac{\frac{1}{\sqrt{\frac{V}{A}}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-308}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+274}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -1e+244)
   (* c0 (/ (/ 1.0 (sqrt (/ V A))) (sqrt l)))
   (if (<= (* V l) -2e-308)
     (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
     (if (<= (* V l) 0.0)
       (* c0 (/ (sqrt (/ A V)) (sqrt l)))
       (if (<= (* V l) 1e+274)
         (* c0 (* (sqrt A) (sqrt (/ (/ 1.0 V) l))))
         (* c0 (sqrt (/ (/ A l) V))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+244) {
		tmp = c0 * ((1.0 / sqrt((V / A))) / sqrt(l));
	} else if ((V * l) <= -2e-308) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else if ((V * l) <= 1e+274) {
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	} else {
		tmp = c0 * sqrt(((A / l) / V));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-1d+244)) then
        tmp = c0 * ((1.0d0 / sqrt((v / a))) / sqrt(l))
    else if ((v * l) <= (-2d-308)) then
        tmp = c0 * (sqrt(-a) / sqrt((v * -l)))
    else if ((v * l) <= 0.0d0) then
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    else if ((v * l) <= 1d+274) then
        tmp = c0 * (sqrt(a) * sqrt(((1.0d0 / v) / l)))
    else
        tmp = c0 * sqrt(((a / l) / v))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+244) {
		tmp = c0 * ((1.0 / Math.sqrt((V / A))) / Math.sqrt(l));
	} else if ((V * l) <= -2e-308) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if ((V * l) <= 1e+274) {
		tmp = c0 * (Math.sqrt(A) * Math.sqrt(((1.0 / V) / l)));
	} else {
		tmp = c0 * Math.sqrt(((A / l) / V));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -1e+244:
		tmp = c0 * ((1.0 / math.sqrt((V / A))) / math.sqrt(l))
	elif (V * l) <= -2e-308:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 0.0:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	elif (V * l) <= 1e+274:
		tmp = c0 * (math.sqrt(A) * math.sqrt(((1.0 / V) / l)))
	else:
		tmp = c0 * math.sqrt(((A / l) / V))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -1e+244)
		tmp = Float64(c0 * Float64(Float64(1.0 / sqrt(Float64(V / A))) / sqrt(l)));
	elseif (Float64(V * l) <= -2e-308)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (Float64(V * l) <= 1e+274)
		tmp = Float64(c0 * Float64(sqrt(A) * sqrt(Float64(Float64(1.0 / V) / l))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -1e+244)
		tmp = c0 * ((1.0 / sqrt((V / A))) / sqrt(l));
	elseif ((V * l) <= -2e-308)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 0.0)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	elseif ((V * l) <= 1e+274)
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	else
		tmp = c0 * sqrt(((A / l) / V));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -1e+244], N[(c0 * N[(N[(1.0 / N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-308], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+274], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Sqrt[N[(N[(1.0 / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -1.00000000000000007e244

   1. Initial program 59.2%

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

      1. associate-/r* 78.3%

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

      2. sqrt-div 72.9%

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

      3. div-inv 72.8%

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

   4. Applied egg-rr 72.8%

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

      1. associate-*r/ 72.9%

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

      2. *-rgt-identity 72.9%

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

   6. Simplified 72.9%

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

      1. clear-num 72.9%

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

      2. sqrt-div 73.1%

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

      3. metadata-eval 73.1%

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

   8. Applied egg-rr 73.1%

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

   if -1.00000000000000007e244 < (*.f64 V l) < -1.9999999999999998e-308

   1. Initial program 85.4%

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

      1. frac-2neg 85.4%

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

      2. sqrt-div 99.3%

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

      3. distribute-rgt-neg-in 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. distribute-rgt-neg-out 99.3%

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

      2. *-commutative 99.3%

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

      3. distribute-rgt-neg-in 99.3%

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

   6. Simplified 99.3%

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

   if -1.9999999999999998e-308 < (*.f64 V l) < 0.0

   1. Initial program 36.3%

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

      1. associate-/r* 72.0%

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

      2. sqrt-div 53.9%

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

      3. div-inv 53.9%

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

   4. Applied egg-rr 53.9%

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

      1. associate-*r/ 53.9%

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

      2. *-rgt-identity 53.9%

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

   6. Simplified 53.9%

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

   if 0.0 < (*.f64 V l) < 9.99999999999999921e273

   1. Initial program 85.1%

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

      1. pow1/2 85.1%

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

      2. div-inv 85.1%

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

      3. unpow-prod-down 99.5%

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

      4. pow1/2 99.5%

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

      5. associate-/r* 99.5%

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

   4. Applied egg-rr 99.5%

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

      1. unpow1/2 99.5%

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

   6. Simplified 99.5%

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

   if 9.99999999999999921e273 < (*.f64 V l)

   1. Initial program 52.7%

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

      1. associate-/r* 83.1%

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

      2. clear-num 83.1%

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

      3. sqrt-div 82.8%

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

      4. metadata-eval 82.8%

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

      5. div-inv 82.9%

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

      6. clear-num 82.9%

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

   4. Applied egg-rr 82.9%

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

      1. *-commutative 82.9%

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

      2. associate-*l/ 52.7%

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

      3. associate-/l* 83.0%

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

   6. Simplified 83.0%

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

      1. metadata-eval 83.0%

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

      2. sqrt-div 83.0%

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

      3. associate-*r/ 52.7%

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

      4. clear-num 52.7%

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

      5. *-commutative 52.7%

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

      6. associate-/r* 83.0%

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

   8. Applied egg-rr 83.0%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+244}:\\ \;\;\;\;c0 \cdot \frac{\frac{1}{\sqrt{\frac{V}{A}}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-308}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+274}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+244}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-308}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+274}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -1e+244)
   (/ c0 (* (sqrt l) (sqrt (/ V A))))
   (if (<= (* V l) -2e-308)
     (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
     (if (<= (* V l) 0.0)
       (* c0 (/ (sqrt (/ A V)) (sqrt l)))
       (if (<= (* V l) 1e+274)
         (* c0 (* (sqrt A) (sqrt (/ (/ 1.0 V) l))))
         (* c0 (sqrt (/ (/ A l) V))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+244) {
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	} else if ((V * l) <= -2e-308) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else if ((V * l) <= 1e+274) {
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	} else {
		tmp = c0 * sqrt(((A / l) / V));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-1d+244)) then
        tmp = c0 / (sqrt(l) * sqrt((v / a)))
    else if ((v * l) <= (-2d-308)) then
        tmp = c0 * (sqrt(-a) / sqrt((v * -l)))
    else if ((v * l) <= 0.0d0) then
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    else if ((v * l) <= 1d+274) then
        tmp = c0 * (sqrt(a) * sqrt(((1.0d0 / v) / l)))
    else
        tmp = c0 * sqrt(((a / l) / v))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+244) {
		tmp = c0 / (Math.sqrt(l) * Math.sqrt((V / A)));
	} else if ((V * l) <= -2e-308) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if ((V * l) <= 1e+274) {
		tmp = c0 * (Math.sqrt(A) * Math.sqrt(((1.0 / V) / l)));
	} else {
		tmp = c0 * Math.sqrt(((A / l) / V));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -1e+244:
		tmp = c0 / (math.sqrt(l) * math.sqrt((V / A)))
	elif (V * l) <= -2e-308:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 0.0:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	elif (V * l) <= 1e+274:
		tmp = c0 * (math.sqrt(A) * math.sqrt(((1.0 / V) / l)))
	else:
		tmp = c0 * math.sqrt(((A / l) / V))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -1e+244)
		tmp = Float64(c0 / Float64(sqrt(l) * sqrt(Float64(V / A))));
	elseif (Float64(V * l) <= -2e-308)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (Float64(V * l) <= 1e+274)
		tmp = Float64(c0 * Float64(sqrt(A) * sqrt(Float64(Float64(1.0 / V) / l))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -1e+244)
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	elseif ((V * l) <= -2e-308)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 0.0)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	elseif ((V * l) <= 1e+274)
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	else
		tmp = c0 * sqrt(((A / l) / V));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -1e+244], N[(c0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-308], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+274], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Sqrt[N[(N[(1.0 / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -1.00000000000000007e244

   1. Initial program 59.2%

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

      1. associate-/r* 78.3%

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

      2. clear-num 77.5%

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

      3. sqrt-div 77.4%

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

      4. metadata-eval 77.4%

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

      5. div-inv 77.3%

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

      6. clear-num 77.3%

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

   4. Applied egg-rr 77.3%

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

      1. *-commutative 77.3%

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

      2. associate-*l/ 59.2%

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

      3. associate-/l* 77.4%

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

   6. Simplified 77.4%

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

      1. metadata-eval 77.4%

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

      2. sqrt-div 77.4%

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

      3. associate-*r/ 59.2%

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

      4. clear-num 59.2%

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

      5. associate-/r* 78.3%

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

      6. sqrt-undiv 72.9%

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

      7. clear-num 72.9%

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

      8. un-div-inv 72.9%

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

      9. sqrt-div 0.0%

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

      10. associate-/r/ 0.0%

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

      11. sqrt-div 23.0%

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

      12. sqrt-prod 77.6%

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

      13. *-commutative 77.6%

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

      14. clear-num 77.6%

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

      15. un-div-inv 77.6%

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

   8. Applied egg-rr 77.6%

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

      1. associate-/r/ 77.5%

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

      2. sqrt-prod 72.9%

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

   10. Applied egg-rr 72.9%

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

   if -1.00000000000000007e244 < (*.f64 V l) < -1.9999999999999998e-308

   1. Initial program 85.4%

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

      1. frac-2neg 85.4%

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

      2. sqrt-div 99.3%

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

      3. distribute-rgt-neg-in 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. distribute-rgt-neg-out 99.3%

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

      2. *-commutative 99.3%

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

      3. distribute-rgt-neg-in 99.3%

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

   6. Simplified 99.3%

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

   if -1.9999999999999998e-308 < (*.f64 V l) < 0.0

   1. Initial program 36.3%

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

      1. associate-/r* 72.0%

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

      2. sqrt-div 53.9%

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

      3. div-inv 53.9%

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

   4. Applied egg-rr 53.9%

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

      1. associate-*r/ 53.9%

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

      2. *-rgt-identity 53.9%

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

   6. Simplified 53.9%

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

   if 0.0 < (*.f64 V l) < 9.99999999999999921e273

   1. Initial program 85.1%

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

      1. pow1/2 85.1%

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

      2. div-inv 85.1%

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

      3. unpow-prod-down 99.5%

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

      4. pow1/2 99.5%

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

      5. associate-/r* 99.5%

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

   4. Applied egg-rr 99.5%

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

      1. unpow1/2 99.5%

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

   6. Simplified 99.5%

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

   if 9.99999999999999921e273 < (*.f64 V l)

   1. Initial program 52.7%

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

      1. associate-/r* 83.1%

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

      2. clear-num 83.1%

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

      3. sqrt-div 82.8%

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

      4. metadata-eval 82.8%

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

      5. div-inv 82.9%

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

      6. clear-num 82.9%

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

   4. Applied egg-rr 82.9%

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

      1. *-commutative 82.9%

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

      2. associate-*l/ 52.7%

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

      3. associate-/l* 83.0%

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

   6. Simplified 83.0%

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

      1. metadata-eval 83.0%

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

      2. sqrt-div 83.0%

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

      3. associate-*r/ 52.7%

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

      4. clear-num 52.7%

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

      5. *-commutative 52.7%

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

      6. associate-/r* 83.0%

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

   8. Applied egg-rr 83.0%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+244}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-308}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+274}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+197}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-240}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+274}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -2e+197)
   (/ c0 (* (sqrt l) (sqrt (/ V A))))
   (if (<= (* V l) -2e-240)
     (* c0 (sqrt (/ A (* V l))))
     (if (<= (* V l) 0.0)
       (* c0 (/ (sqrt (/ A V)) (sqrt l)))
       (if (<= (* V l) 1e+274)
         (* c0 (* (sqrt A) (sqrt (/ (/ 1.0 V) l))))
         (* c0 (sqrt (/ (/ A l) V))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e+197) {
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	} else if ((V * l) <= -2e-240) {
		tmp = c0 * sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else if ((V * l) <= 1e+274) {
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	} else {
		tmp = c0 * sqrt(((A / l) / V));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-2d+197)) then
        tmp = c0 / (sqrt(l) * sqrt((v / a)))
    else if ((v * l) <= (-2d-240)) then
        tmp = c0 * sqrt((a / (v * l)))
    else if ((v * l) <= 0.0d0) then
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    else if ((v * l) <= 1d+274) then
        tmp = c0 * (sqrt(a) * sqrt(((1.0d0 / v) / l)))
    else
        tmp = c0 * sqrt(((a / l) / v))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e+197) {
		tmp = c0 / (Math.sqrt(l) * Math.sqrt((V / A)));
	} else if ((V * l) <= -2e-240) {
		tmp = c0 * Math.sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if ((V * l) <= 1e+274) {
		tmp = c0 * (Math.sqrt(A) * Math.sqrt(((1.0 / V) / l)));
	} else {
		tmp = c0 * Math.sqrt(((A / l) / V));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -2e+197:
		tmp = c0 / (math.sqrt(l) * math.sqrt((V / A)))
	elif (V * l) <= -2e-240:
		tmp = c0 * math.sqrt((A / (V * l)))
	elif (V * l) <= 0.0:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	elif (V * l) <= 1e+274:
		tmp = c0 * (math.sqrt(A) * math.sqrt(((1.0 / V) / l)))
	else:
		tmp = c0 * math.sqrt(((A / l) / V))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -2e+197)
		tmp = Float64(c0 / Float64(sqrt(l) * sqrt(Float64(V / A))));
	elseif (Float64(V * l) <= -2e-240)
		tmp = Float64(c0 * sqrt(Float64(A / Float64(V * l))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (Float64(V * l) <= 1e+274)
		tmp = Float64(c0 * Float64(sqrt(A) * sqrt(Float64(Float64(1.0 / V) / l))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -2e+197)
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	elseif ((V * l) <= -2e-240)
		tmp = c0 * sqrt((A / (V * l)));
	elseif ((V * l) <= 0.0)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	elseif ((V * l) <= 1e+274)
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	else
		tmp = c0 * sqrt(((A / l) / V));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -2e+197], N[(c0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-240], N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+274], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Sqrt[N[(N[(1.0 / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -1.9999999999999999e197

   1. Initial program 57.5%

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

      1. associate-/r* 72.1%

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

      2. clear-num 71.4%

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

      3. sqrt-div 71.4%

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

      4. metadata-eval 71.4%

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

      5. div-inv 71.3%

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

      6. clear-num 71.3%

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

   4. Applied egg-rr 71.3%

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

      1. *-commutative 71.3%

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

      2. associate-*l/ 57.5%

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

      3. associate-/l* 71.4%

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

   6. Simplified 71.4%

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

      1. metadata-eval 71.4%

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

      2. sqrt-div 71.4%

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

      3. associate-*r/ 57.4%

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

      4. clear-num 57.5%

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

      5. associate-/r* 72.1%

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

      6. sqrt-undiv 70.5%

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

      7. clear-num 70.6%

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

      8. un-div-inv 70.6%

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

      9. sqrt-div 0.0%

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

      10. associate-/r/ 0.0%

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

      11. sqrt-div 20.6%

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

      12. sqrt-prod 71.6%

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

      13. *-commutative 71.6%

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

      14. clear-num 71.6%

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

      15. un-div-inv 71.6%

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

   8. Applied egg-rr 71.6%

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

      1. associate-/r/ 71.6%

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

      2. sqrt-prod 70.6%

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

   10. Applied egg-rr 70.6%

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

   if -1.9999999999999999e197 < (*.f64 V l) < -1.9999999999999999e-240

   1. Initial program 91.0%

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

   if -1.9999999999999999e-240 < (*.f64 V l) < 0.0

   1. Initial program 38.5%

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

      1. associate-/r* 69.1%

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

      2. sqrt-div 46.4%

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

      3. div-inv 46.4%

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

   4. Applied egg-rr 46.4%

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

      1. associate-*r/ 46.4%

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

      2. *-rgt-identity 46.4%

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

   6. Simplified 46.4%

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

   if 0.0 < (*.f64 V l) < 9.99999999999999921e273

   1. Initial program 85.1%

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

      1. pow1/2 85.1%

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

      2. div-inv 85.1%

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

      3. unpow-prod-down 99.5%

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

      4. pow1/2 99.5%

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

      5. associate-/r* 99.5%

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

   4. Applied egg-rr 99.5%

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

      1. unpow1/2 99.5%

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

   6. Simplified 99.5%

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

   if 9.99999999999999921e273 < (*.f64 V l)

   1. Initial program 52.7%

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

      1. associate-/r* 83.1%

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

      2. clear-num 83.1%

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

      3. sqrt-div 82.8%

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

      4. metadata-eval 82.8%

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

      5. div-inv 82.9%

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

      6. clear-num 82.9%

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

   4. Applied egg-rr 82.9%

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

      1. *-commutative 82.9%

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

      2. associate-*l/ 52.7%

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

      3. associate-/l* 83.0%

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

   6. Simplified 83.0%

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

      1. metadata-eval 83.0%

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

      2. sqrt-div 83.0%

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

      3. associate-*r/ 52.7%

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

      4. clear-num 52.7%

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

      5. *-commutative 52.7%

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

      6. associate-/r* 83.0%

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

   8. Applied egg-rr 83.0%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+197}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-240}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+274}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+197}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-240}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+274}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -2e+197)
   (/ c0 (* (sqrt l) (sqrt (/ V A))))
   (if (<= (* V l) -2e-240)
     (* c0 (sqrt (/ A (* V l))))
     (if (<= (* V l) 0.0)
       (* c0 (/ (sqrt (/ A V)) (sqrt l)))
       (if (<= (* V l) 1e+274)
         (* c0 (* (sqrt A) (pow (* V l) -0.5)))
         (* c0 (sqrt (/ (/ A l) V))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e+197) {
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	} else if ((V * l) <= -2e-240) {
		tmp = c0 * sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else if ((V * l) <= 1e+274) {
		tmp = c0 * (sqrt(A) * pow((V * l), -0.5));
	} else {
		tmp = c0 * sqrt(((A / l) / V));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-2d+197)) then
        tmp = c0 / (sqrt(l) * sqrt((v / a)))
    else if ((v * l) <= (-2d-240)) then
        tmp = c0 * sqrt((a / (v * l)))
    else if ((v * l) <= 0.0d0) then
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    else if ((v * l) <= 1d+274) then
        tmp = c0 * (sqrt(a) * ((v * l) ** (-0.5d0)))
    else
        tmp = c0 * sqrt(((a / l) / v))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e+197) {
		tmp = c0 / (Math.sqrt(l) * Math.sqrt((V / A)));
	} else if ((V * l) <= -2e-240) {
		tmp = c0 * Math.sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if ((V * l) <= 1e+274) {
		tmp = c0 * (Math.sqrt(A) * Math.pow((V * l), -0.5));
	} else {
		tmp = c0 * Math.sqrt(((A / l) / V));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -2e+197:
		tmp = c0 / (math.sqrt(l) * math.sqrt((V / A)))
	elif (V * l) <= -2e-240:
		tmp = c0 * math.sqrt((A / (V * l)))
	elif (V * l) <= 0.0:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	elif (V * l) <= 1e+274:
		tmp = c0 * (math.sqrt(A) * math.pow((V * l), -0.5))
	else:
		tmp = c0 * math.sqrt(((A / l) / V))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -2e+197)
		tmp = Float64(c0 / Float64(sqrt(l) * sqrt(Float64(V / A))));
	elseif (Float64(V * l) <= -2e-240)
		tmp = Float64(c0 * sqrt(Float64(A / Float64(V * l))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (Float64(V * l) <= 1e+274)
		tmp = Float64(c0 * Float64(sqrt(A) * (Float64(V * l) ^ -0.5)));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -2e+197)
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	elseif ((V * l) <= -2e-240)
		tmp = c0 * sqrt((A / (V * l)));
	elseif ((V * l) <= 0.0)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	elseif ((V * l) <= 1e+274)
		tmp = c0 * (sqrt(A) * ((V * l) ^ -0.5));
	else
		tmp = c0 * sqrt(((A / l) / V));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -2e+197], N[(c0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-240], N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+274], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Power[N[(V * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -1.9999999999999999e197

   1. Initial program 57.5%

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

      1. associate-/r* 72.1%

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

      2. clear-num 71.4%

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

      3. sqrt-div 71.4%

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

      4. metadata-eval 71.4%

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

      5. div-inv 71.3%

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

      6. clear-num 71.3%

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

   4. Applied egg-rr 71.3%

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

      1. *-commutative 71.3%

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

      2. associate-*l/ 57.5%

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

      3. associate-/l* 71.4%

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

   6. Simplified 71.4%

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

      1. metadata-eval 71.4%

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

      2. sqrt-div 71.4%

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

      3. associate-*r/ 57.4%

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

      4. clear-num 57.5%

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

      5. associate-/r* 72.1%

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

      6. sqrt-undiv 70.5%

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

      7. clear-num 70.6%

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

      8. un-div-inv 70.6%

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

      9. sqrt-div 0.0%

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

      10. associate-/r/ 0.0%

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

      11. sqrt-div 20.6%

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

      12. sqrt-prod 71.6%

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

      13. *-commutative 71.6%

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

      14. clear-num 71.6%

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

      15. un-div-inv 71.6%

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

   8. Applied egg-rr 71.6%

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

      1. associate-/r/ 71.6%

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

      2. sqrt-prod 70.6%

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

   10. Applied egg-rr 70.6%

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

   if -1.9999999999999999e197 < (*.f64 V l) < -1.9999999999999999e-240

   1. Initial program 91.0%

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

   if -1.9999999999999999e-240 < (*.f64 V l) < 0.0

   1. Initial program 38.5%

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

      1. associate-/r* 69.1%

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

      2. sqrt-div 46.4%

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

      3. div-inv 46.4%

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

   4. Applied egg-rr 46.4%

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

      1. associate-*r/ 46.4%

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

      2. *-rgt-identity 46.4%

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

   6. Simplified 46.4%

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

   if 0.0 < (*.f64 V l) < 9.99999999999999921e273

   1. Initial program 85.1%

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

      1. associate-/r* 73.5%

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

      2. clear-num 73.5%

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

      3. sqrt-div 74.1%

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

      4. metadata-eval 74.1%

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

      5. div-inv 74.1%

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

      6. clear-num 74.1%

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

   4. Applied egg-rr 74.1%

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

      1. *-commutative 74.1%

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

      2. associate-*l/ 87.5%

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

      3. associate-/l* 75.3%

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

   6. Simplified 75.3%

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

      1. metadata-eval 75.3%

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

      2. sqrt-div 72.9%

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

      3. associate-*r/ 85.1%

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

      4. clear-num 85.1%

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

      5. frac-2neg 85.1%

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

      6. *-commutative 85.1%

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

      7. distribute-rgt-neg-out 85.1%

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

      8. sqrt-undiv 0.0%

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

      9. div-inv 0.0%

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

      10. *-commutative 0.0%

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

      11. pow1/2 0.0%

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

      12. pow-flip 0.0%

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

      13. *-commutative 0.0%

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

      14. add-sqr-sqrt 0.0%

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

      15. sqrt-unprod 0.0%

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

      16. sqr-neg 0.0%

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

      17. sqrt-unprod 0.0%

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

      18. add-sqr-sqrt 0.0%

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

      19. metadata-eval 0.0%

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

      20. add-sqr-sqrt 0.0%

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

      21. sqrt-unprod 59.2%

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

      22. sqr-neg 59.2%

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

   8. Applied egg-rr 99.5%

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

   if 9.99999999999999921e273 < (*.f64 V l)

   1. Initial program 52.7%

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

      1. associate-/r* 83.1%

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

      2. clear-num 83.1%

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

      3. sqrt-div 82.8%

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

      4. metadata-eval 82.8%

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

      5. div-inv 82.9%

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

      6. clear-num 82.9%

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

   4. Applied egg-rr 82.9%

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

      1. *-commutative 82.9%

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

      2. associate-*l/ 52.7%

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

      3. associate-/l* 83.0%

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

   6. Simplified 83.0%

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

      1. metadata-eval 83.0%

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

      2. sqrt-div 83.0%

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

      3. associate-*r/ 52.7%

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

      4. clear-num 52.7%

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

      5. *-commutative 52.7%

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

      6. associate-/r* 83.0%

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

   8. Applied egg-rr 83.0%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+197}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-240}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+274}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+197}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-240}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+274}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -2e+197)
   (/ c0 (* (sqrt l) (sqrt (/ V A))))
   (if (<= (* V l) -2e-240)
     (* c0 (sqrt (/ A (* V l))))
     (if (<= (* V l) 0.0)
       (* c0 (/ (sqrt (/ A V)) (sqrt l)))
       (if (<= (* V l) 1e+274)
         (* c0 (/ (sqrt A) (sqrt (* V l))))
         (* c0 (sqrt (/ (/ A l) V))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e+197) {
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	} else if ((V * l) <= -2e-240) {
		tmp = c0 * sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else if ((V * l) <= 1e+274) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0 * sqrt(((A / l) / V));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-2d+197)) then
        tmp = c0 / (sqrt(l) * sqrt((v / a)))
    else if ((v * l) <= (-2d-240)) then
        tmp = c0 * sqrt((a / (v * l)))
    else if ((v * l) <= 0.0d0) then
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    else if ((v * l) <= 1d+274) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = c0 * sqrt(((a / l) / v))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e+197) {
		tmp = c0 / (Math.sqrt(l) * Math.sqrt((V / A)));
	} else if ((V * l) <= -2e-240) {
		tmp = c0 * Math.sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if ((V * l) <= 1e+274) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0 * Math.sqrt(((A / l) / V));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -2e+197:
		tmp = c0 / (math.sqrt(l) * math.sqrt((V / A)))
	elif (V * l) <= -2e-240:
		tmp = c0 * math.sqrt((A / (V * l)))
	elif (V * l) <= 0.0:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	elif (V * l) <= 1e+274:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0 * math.sqrt(((A / l) / V))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -2e+197)
		tmp = Float64(c0 / Float64(sqrt(l) * sqrt(Float64(V / A))));
	elseif (Float64(V * l) <= -2e-240)
		tmp = Float64(c0 * sqrt(Float64(A / Float64(V * l))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (Float64(V * l) <= 1e+274)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -2e+197)
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	elseif ((V * l) <= -2e-240)
		tmp = c0 * sqrt((A / (V * l)));
	elseif ((V * l) <= 0.0)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	elseif ((V * l) <= 1e+274)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0 * sqrt(((A / l) / V));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -2e+197], N[(c0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-240], N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+274], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -1.9999999999999999e197

   1. Initial program 57.5%

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

      1. associate-/r* 72.1%

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

      2. clear-num 71.4%

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

      3. sqrt-div 71.4%

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

      4. metadata-eval 71.4%

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

      5. div-inv 71.3%

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

      6. clear-num 71.3%

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

   4. Applied egg-rr 71.3%

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

      1. *-commutative 71.3%

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

      2. associate-*l/ 57.5%

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

      3. associate-/l* 71.4%

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

   6. Simplified 71.4%

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

      1. metadata-eval 71.4%

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

      2. sqrt-div 71.4%

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

      3. associate-*r/ 57.4%

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

      4. clear-num 57.5%

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

      5. associate-/r* 72.1%

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

      6. sqrt-undiv 70.5%

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

      7. clear-num 70.6%

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

      8. un-div-inv 70.6%

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

      9. sqrt-div 0.0%

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

      10. associate-/r/ 0.0%

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

      11. sqrt-div 20.6%

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

      12. sqrt-prod 71.6%

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

      13. *-commutative 71.6%

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

      14. clear-num 71.6%

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

      15. un-div-inv 71.6%

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

   8. Applied egg-rr 71.6%

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

      1. associate-/r/ 71.6%

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

      2. sqrt-prod 70.6%

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

   10. Applied egg-rr 70.6%

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

   if -1.9999999999999999e197 < (*.f64 V l) < -1.9999999999999999e-240

   1. Initial program 91.0%

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

   if -1.9999999999999999e-240 < (*.f64 V l) < 0.0

   1. Initial program 38.5%

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

      1. associate-/r* 69.1%

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

      2. sqrt-div 46.4%

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

      3. div-inv 46.4%

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

   4. Applied egg-rr 46.4%

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

      1. associate-*r/ 46.4%

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

      2. *-rgt-identity 46.4%

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

   6. Simplified 46.4%

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

   if 0.0 < (*.f64 V l) < 9.99999999999999921e273

   1. Initial program 85.1%

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

      1. sqrt-div 99.5%

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

      2. div-inv 99.4%

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

   4. Applied egg-rr 99.4%

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

      1. associate-*r/ 99.5%

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

      2. *-rgt-identity 99.5%

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

   6. Simplified 99.5%

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

   if 9.99999999999999921e273 < (*.f64 V l)

   1. Initial program 52.7%

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

      1. associate-/r* 83.1%

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

      2. clear-num 83.1%

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

      3. sqrt-div 82.8%

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

      4. metadata-eval 82.8%

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

      5. div-inv 82.9%

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

      6. clear-num 82.9%

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

   4. Applied egg-rr 82.9%

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

      1. *-commutative 82.9%

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

      2. associate-*l/ 52.7%

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

      3. associate-/l* 83.0%

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

   6. Simplified 83.0%

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

      1. metadata-eval 83.0%

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

      2. sqrt-div 83.0%

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

      3. associate-*r/ 52.7%

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

      4. clear-num 52.7%

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

      5. *-commutative 52.7%

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

      6. associate-/r* 83.0%

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

   8. Applied egg-rr 83.0%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+197}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-240}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+274}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+197}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-240}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+274}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (/ (sqrt (/ A V)) (sqrt l)))))
   (if (<= (* V l) -2e+197)
     t_0
     (if (<= (* V l) -2e-240)
       (* c0 (sqrt (/ A (* V l))))
       (if (<= (* V l) 0.0)
         t_0
         (if (<= (* V l) 1e+274)
           (* c0 (/ (sqrt A) (sqrt (* V l))))
           (* c0 (sqrt (/ (/ A l) V)))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * (sqrt((A / V)) / sqrt(l));
	double tmp;
	if ((V * l) <= -2e+197) {
		tmp = t_0;
	} else if ((V * l) <= -2e-240) {
		tmp = c0 * sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else if ((V * l) <= 1e+274) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0 * sqrt(((A / l) / V));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * (sqrt((a / v)) / sqrt(l))
    if ((v * l) <= (-2d+197)) then
        tmp = t_0
    else if ((v * l) <= (-2d-240)) then
        tmp = c0 * sqrt((a / (v * l)))
    else if ((v * l) <= 0.0d0) then
        tmp = t_0
    else if ((v * l) <= 1d+274) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = c0 * sqrt(((a / l) / v))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	double tmp;
	if ((V * l) <= -2e+197) {
		tmp = t_0;
	} else if ((V * l) <= -2e-240) {
		tmp = c0 * Math.sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else if ((V * l) <= 1e+274) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0 * Math.sqrt(((A / l) / V));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	tmp = 0
	if (V * l) <= -2e+197:
		tmp = t_0
	elif (V * l) <= -2e-240:
		tmp = c0 * math.sqrt((A / (V * l)))
	elif (V * l) <= 0.0:
		tmp = t_0
	elif (V * l) <= 1e+274:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0 * math.sqrt(((A / l) / V))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)))
	tmp = 0.0
	if (Float64(V * l) <= -2e+197)
		tmp = t_0;
	elseif (Float64(V * l) <= -2e-240)
		tmp = Float64(c0 * sqrt(Float64(A / Float64(V * l))));
	elseif (Float64(V * l) <= 0.0)
		tmp = t_0;
	elseif (Float64(V * l) <= 1e+274)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * (sqrt((A / V)) / sqrt(l));
	tmp = 0.0;
	if ((V * l) <= -2e+197)
		tmp = t_0;
	elseif ((V * l) <= -2e-240)
		tmp = c0 * sqrt((A / (V * l)));
	elseif ((V * l) <= 0.0)
		tmp = t_0;
	elseif ((V * l) <= 1e+274)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0 * sqrt(((A / l) / V));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -2e+197], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -2e-240], N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], 1e+274], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -1.9999999999999999e197 or -1.9999999999999999e-240 < (*.f64 V l) < 0.0

   1. Initial program 48.9%

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

      1. associate-/r* 70.7%

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

      2. sqrt-div 59.6%

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

      3. div-inv 59.6%

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

   4. Applied egg-rr 59.6%

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

      1. associate-*r/ 59.6%

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

      2. *-rgt-identity 59.6%

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

   6. Simplified 59.6%

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

   if -1.9999999999999999e197 < (*.f64 V l) < -1.9999999999999999e-240

   1. Initial program 91.0%

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

   if 0.0 < (*.f64 V l) < 9.99999999999999921e273

   1. Initial program 85.1%

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

      1. sqrt-div 99.5%

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

      2. div-inv 99.4%

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

   4. Applied egg-rr 99.4%

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

      1. associate-*r/ 99.5%

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

      2. *-rgt-identity 99.5%

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

   6. Simplified 99.5%

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

   if 9.99999999999999921e273 < (*.f64 V l)

   1. Initial program 52.7%

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

      1. associate-/r* 83.1%

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

      2. clear-num 83.1%

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

      3. sqrt-div 82.8%

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

      4. metadata-eval 82.8%

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

      5. div-inv 82.9%

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

      6. clear-num 82.9%

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

   4. Applied egg-rr 82.9%

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

      1. *-commutative 82.9%

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

      2. associate-*l/ 52.7%

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

      3. associate-/l* 83.0%

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

   6. Simplified 83.0%

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

      1. metadata-eval 83.0%

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

      2. sqrt-div 83.0%

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

      3. associate-*r/ 52.7%

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

      4. clear-num 52.7%

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

      5. *-commutative 52.7%

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

      6. associate-/r* 83.0%

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

   8. Applied egg-rr 83.0%

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

Alternative 8: 84.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+262}:\\ \;\;\;\;\frac{\sqrt{\frac{\frac{A}{V}}{\ell}}}{\frac{1}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq -4 \cdot 10^{-295}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \frac{\frac{1}{V}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-247}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+274}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -2e+262)
   (/ (sqrt (/ (/ A V) l)) (/ 1.0 c0))
   (if (<= (* V l) -4e-295)
     (* c0 (sqrt (* A (/ (/ 1.0 V) l))))
     (if (<= (* V l) 2e-247)
       (/ c0 (sqrt (* V (/ l A))))
       (if (<= (* V l) 1e+274)
         (* c0 (/ (sqrt A) (sqrt (* V l))))
         (* c0 (sqrt (/ (/ A l) V))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e+262) {
		tmp = sqrt(((A / V) / l)) / (1.0 / c0);
	} else if ((V * l) <= -4e-295) {
		tmp = c0 * sqrt((A * ((1.0 / V) / l)));
	} else if ((V * l) <= 2e-247) {
		tmp = c0 / sqrt((V * (l / A)));
	} else if ((V * l) <= 1e+274) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0 * sqrt(((A / l) / V));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-2d+262)) then
        tmp = sqrt(((a / v) / l)) / (1.0d0 / c0)
    else if ((v * l) <= (-4d-295)) then
        tmp = c0 * sqrt((a * ((1.0d0 / v) / l)))
    else if ((v * l) <= 2d-247) then
        tmp = c0 / sqrt((v * (l / a)))
    else if ((v * l) <= 1d+274) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = c0 * sqrt(((a / l) / v))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e+262) {
		tmp = Math.sqrt(((A / V) / l)) / (1.0 / c0);
	} else if ((V * l) <= -4e-295) {
		tmp = c0 * Math.sqrt((A * ((1.0 / V) / l)));
	} else if ((V * l) <= 2e-247) {
		tmp = c0 / Math.sqrt((V * (l / A)));
	} else if ((V * l) <= 1e+274) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0 * Math.sqrt(((A / l) / V));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -2e+262:
		tmp = math.sqrt(((A / V) / l)) / (1.0 / c0)
	elif (V * l) <= -4e-295:
		tmp = c0 * math.sqrt((A * ((1.0 / V) / l)))
	elif (V * l) <= 2e-247:
		tmp = c0 / math.sqrt((V * (l / A)))
	elif (V * l) <= 1e+274:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0 * math.sqrt(((A / l) / V))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -2e+262)
		tmp = Float64(sqrt(Float64(Float64(A / V) / l)) / Float64(1.0 / c0));
	elseif (Float64(V * l) <= -4e-295)
		tmp = Float64(c0 * sqrt(Float64(A * Float64(Float64(1.0 / V) / l))));
	elseif (Float64(V * l) <= 2e-247)
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	elseif (Float64(V * l) <= 1e+274)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -2e+262)
		tmp = sqrt(((A / V) / l)) / (1.0 / c0);
	elseif ((V * l) <= -4e-295)
		tmp = c0 * sqrt((A * ((1.0 / V) / l)));
	elseif ((V * l) <= 2e-247)
		tmp = c0 / sqrt((V * (l / A)));
	elseif ((V * l) <= 1e+274)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0 * sqrt(((A / l) / V));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -2e+262], N[(N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] / N[(1.0 / c0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -4e-295], N[(c0 * N[Sqrt[N[(A * N[(N[(1.0 / V), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 2e-247], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+274], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -2e262

   1. Initial program 51.9%

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

      1. associate-/r* 74.5%

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

      2. clear-num 73.5%

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

      3. sqrt-div 73.5%

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

      4. metadata-eval 73.5%

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

      5. div-inv 73.4%

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

      6. clear-num 73.4%

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

   4. Applied egg-rr 73.4%

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

      1. *-commutative 73.4%

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

      2. associate-*l/ 51.9%

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

      3. associate-/l* 73.5%

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

   6. Simplified 73.5%

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

      1. metadata-eval 73.5%

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

      2. sqrt-div 73.5%

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

      3. associate-*r/ 51.9%

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

      4. clear-num 51.9%

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

      5. *-commutative 51.9%

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

      6. associate-/r* 74.4%

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

   8. Applied egg-rr 74.4%

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

      1. clear-num 73.5%

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

      2. sqrt-div 73.4%

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

      3. metadata-eval 73.4%

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

      4. div-inv 73.5%

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

      5. clear-num 71.8%

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

      6. div-inv 71.8%

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

      7. associate-/r* 73.5%

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

      8. metadata-eval 73.5%

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

      9. sqrt-div 73.5%

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

      10. clear-num 74.4%

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

      11. associate-/l/ 51.9%

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

   10. Applied egg-rr 51.9%

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

   11. Taylor expanded in A around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. unpow2 0.0%

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

       3. rem-square-sqrt 51.9%

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

       4. associate-*r* 51.9%

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

       5. metadata-eval 51.9%

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

       6. *-lft-identity 51.9%

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

       7. associate-/r* 74.4%

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

   13. Simplified 74.4%

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

   if -2e262 < (*.f64 V l) < -4.00000000000000024e-295

   1. Initial program 85.9%

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

      1. clear-num 85.4%

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

      2. associate-/r/ 85.9%

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

      3. associate-/r* 85.9%

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

   4. Applied egg-rr 85.9%

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

   if -4.00000000000000024e-295 < (*.f64 V l) < 2e-247

   1. Initial program 48.9%

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

      1. associate-/r* 77.5%

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

      2. clear-num 77.5%

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

      3. sqrt-div 77.5%

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

      4. metadata-eval 77.5%

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

      5. div-inv 77.5%

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

      6. clear-num 77.5%

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

   4. Applied egg-rr 77.5%

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

      1. *-commutative 77.5%

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

      2. associate-*l/ 48.9%

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

      3. associate-/l* 77.5%

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

   6. Simplified 77.5%

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

      1. metadata-eval 77.5%

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

      2. sqrt-div 77.5%

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

      3. associate-*r/ 48.9%

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

      4. clear-num 48.9%

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

      5. associate-/r* 77.5%

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

      6. sqrt-undiv 46.5%

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

      7. clear-num 46.3%

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

      8. un-div-inv 46.4%

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

      9. sqrt-div 26.5%

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

      10. associate-/r/ 26.6%

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

      11. sqrt-div 46.6%

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

      12. sqrt-prod 77.7%

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

      13. *-commutative 77.7%

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

      14. clear-num 77.7%

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

      15. un-div-inv 77.7%

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

   8. Applied egg-rr 77.7%

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

      1. clear-num 77.6%

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

      2. associate-/r/ 77.7%

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

      3. clear-num 77.7%

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

   10. Applied egg-rr 77.7%

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

   if 2e-247 < (*.f64 V l) < 9.99999999999999921e273

   1. Initial program 84.3%

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

      1. sqrt-div 99.5%

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

      2. div-inv 99.4%

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

   4. Applied egg-rr 99.4%

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

      1. associate-*r/ 99.5%

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

      2. *-rgt-identity 99.5%

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

   6. Simplified 99.5%

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

   if 9.99999999999999921e273 < (*.f64 V l)

   1. Initial program 52.7%

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

      1. associate-/r* 83.1%

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

      2. clear-num 83.1%

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

      3. sqrt-div 82.8%

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

      4. metadata-eval 82.8%

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

      5. div-inv 82.9%

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

      6. clear-num 82.9%

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

   4. Applied egg-rr 82.9%

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

      1. *-commutative 82.9%

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

      2. associate-*l/ 52.7%

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

      3. associate-/l* 83.0%

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

   6. Simplified 83.0%

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

      1. metadata-eval 83.0%

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

      2. sqrt-div 83.0%

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

      3. associate-*r/ 52.7%

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

      4. clear-num 52.7%

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

      5. *-commutative 52.7%

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

      6. associate-/r* 83.0%

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

   8. Applied egg-rr 83.0%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+262}:\\ \;\;\;\;\frac{\sqrt{\frac{\frac{A}{V}}{\ell}}}{\frac{1}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq -4 \cdot 10^{-295}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \frac{\frac{1}{V}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-247}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+274}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 2 \cdot 10^{+297}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (or (<= t_0 0.0) (not (<= t_0 2e+297)))
     (* c0 (sqrt (/ (/ A l) V)))
     (* c0 (sqrt t_0)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 2e+297)) {
		tmp = c0 * sqrt(((A / l) / V));
	} else {
		tmp = c0 * sqrt(t_0);
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if ((t_0 <= 0.0d0) .or. (.not. (t_0 <= 2d+297))) then
        tmp = c0 * sqrt(((a / l) / v))
    else
        tmp = c0 * sqrt(t_0)
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 2e+297)) {
		tmp = c0 * Math.sqrt(((A / l) / V));
	} else {
		tmp = c0 * Math.sqrt(t_0);
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 2e+297):
		tmp = c0 * math.sqrt(((A / l) / V))
	else:
		tmp = c0 * math.sqrt(t_0)
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 2e+297))
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	else
		tmp = Float64(c0 * sqrt(t_0));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 2e+297)))
		tmp = c0 * sqrt(((A / l) / V));
	else
		tmp = c0 * sqrt(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 2e+297]], $MachinePrecision]], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 0.0 or 2e297 < (/.f64 A (*.f64 V l))

   1. Initial program 36.9%

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

      1. associate-/r* 57.6%

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

      2. clear-num 57.4%

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

      3. sqrt-div 58.0%

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

      4. metadata-eval 58.0%

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

      5. div-inv 58.0%

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

      6. clear-num 58.0%

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

   4. Applied egg-rr 58.0%

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

      1. *-commutative 58.0%

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

      2. associate-*l/ 39.7%

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

      3. associate-/l* 59.2%

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

   6. Simplified 59.2%

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

      1. metadata-eval 59.2%

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

      2. sqrt-div 56.4%

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

      3. associate-*r/ 36.9%

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

      4. clear-num 36.9%

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

      5. *-commutative 36.9%

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

      6. associate-/r* 56.6%

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

   8. Applied egg-rr 56.6%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 2e297

   1. Initial program 99.2%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0 \lor \neg \left(\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{+297}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 5 \cdot 10^{+306}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (or (<= t_0 0.0) (not (<= t_0 5e+306)))
     (* c0 (sqrt (/ (/ A V) l)))
     (* c0 (sqrt t_0)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 5e+306)) {
		tmp = c0 * sqrt(((A / V) / l));
	} else {
		tmp = c0 * sqrt(t_0);
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if ((t_0 <= 0.0d0) .or. (.not. (t_0 <= 5d+306))) then
        tmp = c0 * sqrt(((a / v) / l))
    else
        tmp = c0 * sqrt(t_0)
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 5e+306)) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else {
		tmp = c0 * Math.sqrt(t_0);
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 5e+306):
		tmp = c0 * math.sqrt(((A / V) / l))
	else:
		tmp = c0 * math.sqrt(t_0)
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 5e+306))
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	else
		tmp = Float64(c0 * sqrt(t_0));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 5e+306)))
		tmp = c0 * sqrt(((A / V) / l));
	else
		tmp = c0 * sqrt(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 5e+306]], $MachinePrecision]], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 0.0 or 4.99999999999999993e306 < (/.f64 A (*.f64 V l))

   1. Initial program 36.2%

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

      1. associate-/r* 57.2%

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

   3. Simplified 57.2%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 4.99999999999999993e306

   1. Initial program 99.2%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0 \lor \neg \left(\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{+306}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-319}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+297}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (<= t_0 5e-319)
     (* c0 (sqrt (/ (/ A l) V)))
     (if (<= t_0 2e+297)
       (/ c0 (sqrt (/ (* V l) A)))
       (/ c0 (sqrt (* V (/ l A))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 5e-319) {
		tmp = c0 * sqrt(((A / l) / V));
	} else if (t_0 <= 2e+297) {
		tmp = c0 / sqrt(((V * l) / A));
	} else {
		tmp = c0 / sqrt((V * (l / A)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if (t_0 <= 5d-319) then
        tmp = c0 * sqrt(((a / l) / v))
    else if (t_0 <= 2d+297) then
        tmp = c0 / sqrt(((v * l) / a))
    else
        tmp = c0 / sqrt((v * (l / a)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 5e-319) {
		tmp = c0 * Math.sqrt(((A / l) / V));
	} else if (t_0 <= 2e+297) {
		tmp = c0 / Math.sqrt(((V * l) / A));
	} else {
		tmp = c0 / Math.sqrt((V * (l / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 5e-319:
		tmp = c0 * math.sqrt(((A / l) / V))
	elif t_0 <= 2e+297:
		tmp = c0 / math.sqrt(((V * l) / A))
	else:
		tmp = c0 / math.sqrt((V * (l / A)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 5e-319)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	elseif (t_0 <= 2e+297)
		tmp = Float64(c0 / sqrt(Float64(Float64(V * l) / A)));
	else
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 5e-319)
		tmp = c0 * sqrt(((A / l) / V));
	elseif (t_0 <= 2e+297)
		tmp = c0 / sqrt(((V * l) / A));
	else
		tmp = c0 / sqrt((V * (l / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-319], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+297], N[(c0 / N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 A (*.f64 V l)) < 4.9999937e-319

   1. Initial program 41.3%

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

      1. associate-/r* 61.5%

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

      2. clear-num 61.1%

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

      3. sqrt-div 61.0%

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

      4. metadata-eval 61.0%

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

      5. div-inv 61.0%

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

      6. clear-num 61.0%

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

   4. Applied egg-rr 61.0%

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

      1. *-commutative 61.0%

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

      2. associate-*l/ 40.5%

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

      3. associate-/l* 61.1%

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

   6. Simplified 61.1%

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

      1. metadata-eval 61.1%

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

      2. sqrt-div 61.1%

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

      3. associate-*r/ 40.5%

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

      4. clear-num 41.3%

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

      5. *-commutative 41.3%

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

      6. associate-/r* 62.2%

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

   8. Applied egg-rr 62.2%

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

   if 4.9999937e-319 < (/.f64 A (*.f64 V l)) < 2e297

   1. Initial program 99.5%

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

      1. associate-/r* 85.4%

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

      2. clear-num 85.4%

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

      3. sqrt-div 85.4%

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

      4. metadata-eval 85.4%

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

      5. div-inv 85.3%

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

      6. clear-num 85.4%

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

   4. Applied egg-rr 85.4%

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

      1. *-commutative 85.4%

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

      2. associate-*l/ 99.5%

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

      3. associate-/l* 86.5%

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

   6. Simplified 86.5%

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

      1. metadata-eval 86.5%

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

      2. sqrt-div 86.5%

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

      3. associate-*r/ 99.5%

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

      4. clear-num 99.5%

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

      5. associate-/r* 85.4%

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

      6. sqrt-undiv 35.3%

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

      7. clear-num 35.3%

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

      8. un-div-inv 35.3%

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

      9. sqrt-div 21.9%

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

      10. associate-/r/ 21.9%

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

      11. sqrt-div 44.7%

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

      12. sqrt-prod 86.6%

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

      13. *-commutative 86.6%

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

      14. clear-num 84.6%

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

      15. un-div-inv 85.2%

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

   8. Applied egg-rr 85.2%

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

   9. Taylor expanded in V around 0 99.6%

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

   if 2e297 < (/.f64 A (*.f64 V l))

   1. Initial program 34.3%

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

      1. associate-/r* 53.3%

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

      2. clear-num 53.4%

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

      3. sqrt-div 54.8%

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

      4. metadata-eval 54.8%

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

      5. div-inv 54.8%

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

      6. clear-num 54.8%

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

   4. Applied egg-rr 54.8%

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

      1. *-commutative 54.8%

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

      2. associate-*l/ 40.5%

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

      3. associate-/l* 57.4%

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

   6. Simplified 57.4%

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

      1. metadata-eval 57.4%

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

      2. sqrt-div 51.2%

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

      3. associate-*r/ 34.3%

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

      4. clear-num 34.3%

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

      5. associate-/r* 53.3%

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

      6. sqrt-undiv 45.0%

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

      7. clear-num 44.8%

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

      8. un-div-inv 44.9%

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

      9. sqrt-div 35.4%

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

      10. associate-/r/ 35.4%

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

      11. sqrt-div 47.3%

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

      12. sqrt-prod 57.5%

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

      13. *-commutative 57.5%

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

      14. clear-num 57.4%

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

      15. un-div-inv 57.4%

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

   8. Applied egg-rr 57.4%

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

      1. clear-num 51.2%

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

      2. associate-/r/ 57.4%

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

      3. clear-num 57.5%

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

   10. Applied egg-rr 57.5%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{-319}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{+297}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 80.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+297}:\\ \;\;\;\;c0 \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (<= t_0 0.0)
     (* c0 (sqrt (/ (/ A l) V)))
     (if (<= t_0 2e+297) (* c0 (sqrt t_0)) (/ c0 (sqrt (* V (/ l A))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * sqrt(((A / l) / V));
	} else if (t_0 <= 2e+297) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = c0 / sqrt((V * (l / A)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if (t_0 <= 0.0d0) then
        tmp = c0 * sqrt(((a / l) / v))
    else if (t_0 <= 2d+297) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = c0 / sqrt((v * (l / a)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * Math.sqrt(((A / l) / V));
	} else if (t_0 <= 2e+297) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = c0 / Math.sqrt((V * (l / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0 * math.sqrt(((A / l) / V))
	elif t_0 <= 2e+297:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = c0 / math.sqrt((V * (l / A)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	elseif (t_0 <= 2e+297)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0 * sqrt(((A / l) / V));
	elseif (t_0 <= 2e+297)
		tmp = c0 * sqrt(t_0);
	else
		tmp = c0 / sqrt((V * (l / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+297], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 39.0%

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

      1. associate-/r* 61.1%

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

      2. clear-num 60.8%

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

      3. sqrt-div 60.6%

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

      4. metadata-eval 60.6%

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

      5. div-inv 60.7%

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

      6. clear-num 60.7%

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

   4. Applied egg-rr 60.7%

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

      1. *-commutative 60.7%

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

      2. associate-*l/ 39.0%

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

      3. associate-/l* 60.7%

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

   6. Simplified 60.7%

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

      1. metadata-eval 60.7%

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

      2. sqrt-div 60.7%

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

      3. associate-*r/ 39.0%

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

      4. clear-num 39.0%

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

      5. *-commutative 39.0%

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

      6. associate-/r* 61.1%

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

   8. Applied egg-rr 61.1%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 2e297

   1. Initial program 99.2%

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

   if 2e297 < (/.f64 A (*.f64 V l))

   1. Initial program 34.3%

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

      1. associate-/r* 53.3%

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

      2. clear-num 53.4%

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

      3. sqrt-div 54.8%

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

      4. metadata-eval 54.8%

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

      5. div-inv 54.8%

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

      6. clear-num 54.8%

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

   4. Applied egg-rr 54.8%

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

      1. *-commutative 54.8%

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

      2. associate-*l/ 40.5%

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

      3. associate-/l* 57.4%

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

   6. Simplified 57.4%

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

      1. metadata-eval 57.4%

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

      2. sqrt-div 51.2%

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

      3. associate-*r/ 34.3%

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

      4. clear-num 34.3%

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

      5. associate-/r* 53.3%

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

      6. sqrt-undiv 45.0%

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

      7. clear-num 44.8%

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

      8. un-div-inv 44.9%

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

      9. sqrt-div 35.4%

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

      10. associate-/r/ 35.4%

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

      11. sqrt-div 47.3%

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

      12. sqrt-prod 57.5%

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

      13. *-commutative 57.5%

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

      14. clear-num 57.4%

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

      15. un-div-inv 57.4%

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

   8. Applied egg-rr 57.4%

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

      1. clear-num 51.2%

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

      2. associate-/r/ 57.4%

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

      3. clear-num 57.5%

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

   10. Applied egg-rr 57.5%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{+297}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 80.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;c0 \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \end{array} \]

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 (/ A (* V l))))
   (if (<= t_0 0.0)
     (* c0 (sqrt (/ (/ A l) V)))
     (if (<= t_0 5e+306) (* c0 (sqrt 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 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * sqrt(((A / l) / V));
	} else if (t_0 <= 5e+306) {
		tmp = c0 * sqrt(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 = a / (v * l)
    if (t_0 <= 0.0d0) then
        tmp = c0 * sqrt(((a / l) / v))
    else if (t_0 <= 5d+306) then
        tmp = c0 * sqrt(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 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * Math.sqrt(((A / l) / V));
	} else if (t_0 <= 5e+306) {
		tmp = c0 * Math.sqrt(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 = A / (V * l)
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0 * math.sqrt(((A / l) / V))
	elif t_0 <= 5e+306:
		tmp = c0 * math.sqrt(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(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	elseif (t_0 <= 5e+306)
		tmp = Float64(c0 * sqrt(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 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0 * sqrt(((A / l) / V));
	elseif (t_0 <= 5e+306)
		tmp = c0 * sqrt(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[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+306], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 39.0%

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

      1. associate-/r* 61.1%

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

      2. clear-num 60.8%

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

      3. sqrt-div 60.6%

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

      4. metadata-eval 60.6%

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

      5. div-inv 60.7%

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

      6. clear-num 60.7%

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

   4. Applied egg-rr 60.7%

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

      1. *-commutative 60.7%

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

      2. associate-*l/ 39.0%

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

      3. associate-/l* 60.7%

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

   6. Simplified 60.7%

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

      1. metadata-eval 60.7%

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

      2. sqrt-div 60.7%

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

      3. associate-*r/ 39.0%

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

      4. clear-num 39.0%

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

      5. *-commutative 39.0%

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

      6. associate-/r* 61.1%

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

   8. Applied egg-rr 61.1%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 4.99999999999999993e306

   1. Initial program 99.2%

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

   if 4.99999999999999993e306 < (/.f64 A (*.f64 V l))

   1. Initial program 32.8%

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

      1. associate-/r* 52.3%

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

      2. clear-num 52.3%

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

      3. sqrt-div 53.8%

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

      4. metadata-eval 53.8%

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

      5. div-inv 53.8%

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

      6. clear-num 53.8%

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

   4. Applied egg-rr 53.8%

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

      1. *-commutative 53.8%

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

      2. associate-*l/ 39.1%

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

      3. associate-/l* 58.6%

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

   6. Simplified 58.6%

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

      1. metadata-eval 58.6%

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

      2. sqrt-div 52.3%

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

      3. associate-*r/ 32.8%

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

      4. clear-num 32.8%

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

      5. associate-/r* 52.3%

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

      6. sqrt-undiv 46.0%

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

      7. clear-num 45.8%

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

      8. un-div-inv 45.9%

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

      9. sqrt-div 36.2%

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

      10. associate-/r/ 36.2%

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

      11. sqrt-div 48.3%

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

      12. sqrt-prod 58.7%

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

      13. *-commutative 58.7%

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

      14. clear-num 58.7%

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

      15. un-div-inv 58.7%

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

   8. Applied egg-rr 58.7%

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

      1. associate-/r/ 53.8%

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

      2. associate-*l/ 39.1%

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

      3. associate-*l/ 53.8%

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{V}{A} \cdot \ell}}} \]

      4. *-commutative 53.8%

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{\ell \cdot \frac{V}{A}}}} \]

   10. Simplified 53.8%

       \[\leadsto \color{blue}{\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 73.5% 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 75.1%

   \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
2. Add Preprocessing
3. Add Preprocessing

Reproduce

herbie shell --seed 2024092 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))