Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.9% → 91.4%

Time: 10.7s

Alternatives: 16

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \ell \cdot \frac{V}{A}\\ \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-299}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-302}:\\ \;\;\;\;c0 \cdot {t\_0}^{-0.5}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+300}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{{t\_0}^{-1}}\\ \end{array} \end{array} \]

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 (* l (/ V A))))
   (if (<= (* V l) (- INFINITY))
     (* c0 (/ (sqrt (/ A (- l))) (sqrt (- V))))
     (if (<= (* V l) -5e-299)
       (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
       (if (<= (* V l) 2e-302)
         (* c0 (pow t_0 -0.5))
         (if (<= (* V l) 1e+300)
           (* c0 (* (sqrt A) (sqrt (/ (/ 1.0 V) l))))
           (* c0 (sqrt (pow t_0 -1.0)))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = l * (V / A);
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = c0 * (sqrt((A / -l)) / sqrt(-V));
	} else if ((V * l) <= -5e-299) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 2e-302) {
		tmp = c0 * pow(t_0, -0.5);
	} else if ((V * l) <= 1e+300) {
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	} else {
		tmp = c0 * sqrt(pow(t_0, -1.0));
	}
	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 = l * (V / A);
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = c0 * (Math.sqrt((A / -l)) / Math.sqrt(-V));
	} else if ((V * l) <= -5e-299) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 2e-302) {
		tmp = c0 * Math.pow(t_0, -0.5);
	} else if ((V * l) <= 1e+300) {
		tmp = c0 * (Math.sqrt(A) * Math.sqrt(((1.0 / V) / l)));
	} else {
		tmp = c0 * Math.sqrt(Math.pow(t_0, -1.0));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = l * (V / A)
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = c0 * (math.sqrt((A / -l)) / math.sqrt(-V))
	elif (V * l) <= -5e-299:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 2e-302:
		tmp = c0 * math.pow(t_0, -0.5)
	elif (V * l) <= 1e+300:
		tmp = c0 * (math.sqrt(A) * math.sqrt(((1.0 / V) / l)))
	else:
		tmp = c0 * math.sqrt(math.pow(t_0, -1.0))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(l * Float64(V / A))
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(c0 * Float64(sqrt(Float64(A / Float64(-l))) / sqrt(Float64(-V))));
	elseif (Float64(V * l) <= -5e-299)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 2e-302)
		tmp = Float64(c0 * (t_0 ^ -0.5));
	elseif (Float64(V * l) <= 1e+300)
		tmp = Float64(c0 * Float64(sqrt(A) * sqrt(Float64(Float64(1.0 / V) / l))));
	else
		tmp = Float64(c0 * sqrt((t_0 ^ -1.0)));
	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 = l * (V / A);
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = c0 * (sqrt((A / -l)) / sqrt(-V));
	elseif ((V * l) <= -5e-299)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 2e-302)
		tmp = c0 * (t_0 ^ -0.5);
	elseif ((V * l) <= 1e+300)
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	else
		tmp = c0 * sqrt((t_0 ^ -1.0));
	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[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(c0 * N[(N[Sqrt[N[(A / (-l)), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -5e-299], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 2e-302], N[(c0 * N[Power[t$95$0, -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+300], 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[Power[t$95$0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 30.1%

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

      1. *-un-lft-identity 30.1%

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

      2. times-frac 72.0%

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

   4. Applied egg-rr 72.0%

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

      1. associate-*l/ 72.1%

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

      2. *-un-lft-identity 72.1%

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

   6. Applied egg-rr 72.1%

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

      1. frac-2neg 72.1%

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

      2. sqrt-div 55.2%

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

      3. distribute-neg-frac 55.2%

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

   8. Applied egg-rr 55.2%

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

   if -inf.0 < (*.f64 V l) < -4.99999999999999956e-299

   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 99.5%

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

      3. distribute-rgt-neg-in 99.5%

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

   4. Applied egg-rr 99.5%

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

   if -4.99999999999999956e-299 < (*.f64 V l) < 1.9999999999999999e-302

   1. Initial program 38.5%

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

      1. *-un-lft-identity 38.5%

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

      2. times-frac 69.7%

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

   4. Applied egg-rr 69.7%

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

      1. associate-*l/ 69.7%

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

      2. *-un-lft-identity 69.7%

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

   6. Applied egg-rr 69.7%

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

      1. clear-num 69.7%

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

      2. sqrt-div 69.5%

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

      3. metadata-eval 69.5%

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

      4. div-inv 69.6%

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

      5. clear-num 69.6%

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

      6. pow1/2 69.6%

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

      7. pow-flip 69.7%

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

      8. associate-*r/ 38.5%

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

      9. *-commutative 38.5%

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

      10. associate-/l* 69.8%

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

      11. metadata-eval 69.8%

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

   8. Applied egg-rr 69.8%

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

   if 1.9999999999999999e-302 < (*.f64 V l) < 1.0000000000000001e300

   1. Initial program 85.8%

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

      1. *-un-lft-identity 85.8%

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

      2. times-frac 74.9%

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

   4. Applied egg-rr 74.9%

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

      1. pow1/2 74.9%

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

      2. associate-*l/ 75.0%

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

      3. *-un-lft-identity 75.0%

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

      4. div-inv 74.9%

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

      5. associate-*l/ 74.5%

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

      6. frac-times 85.8%

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

      7. associate-*r/ 85.7%

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

      8. unpow-prod-down 99.5%

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

      9. pow1/2 99.5%

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

      10. associate-/r* 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. unpow1/2 99.5%

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

   8. Simplified 99.5%

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

   if 1.0000000000000001e300 < (*.f64 V l)

   1. Initial program 22.6%

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

      1. associate-/r* 70.7%

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

      2. div-inv 70.6%

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

      3. clear-num 70.5%

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

      4. frac-times 70.6%

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

      5. metadata-eval 70.6%

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

      6. inv-pow 70.6%

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

   4. Applied egg-rr 70.6%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{{\left(\frac{V}{A} \cdot \ell\right)}^{-1}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-299}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-302}:\\ \;\;\;\;c0 \cdot {\left(\ell \cdot \frac{V}{A}\right)}^{-0.5}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+300}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{{\left(\ell \cdot \frac{V}{A}\right)}^{-1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -2 \cdot 10^{-311}:\\ \;\;\;\;c0 \cdot \frac{\frac{\sqrt{-A}}{\sqrt{-V}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= A -2e-311)
   (* c0 (/ (/ (sqrt (- A)) (sqrt (- V))) (sqrt l)))
   (* c0 (* (sqrt A) (sqrt (/ (/ 1.0 V) l))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -2e-311) {
		tmp = c0 * ((sqrt(-A) / sqrt(-V)) / sqrt(l));
	} else {
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (a <= (-2d-311)) then
        tmp = c0 * ((sqrt(-a) / sqrt(-v)) / sqrt(l))
    else
        tmp = c0 * (sqrt(a) * sqrt(((1.0d0 / v) / l)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -2e-311) {
		tmp = c0 * ((Math.sqrt(-A) / Math.sqrt(-V)) / Math.sqrt(l));
	} else {
		tmp = c0 * (Math.sqrt(A) * Math.sqrt(((1.0 / V) / l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if A <= -2e-311:
		tmp = c0 * ((math.sqrt(-A) / math.sqrt(-V)) / math.sqrt(l))
	else:
		tmp = c0 * (math.sqrt(A) * math.sqrt(((1.0 / V) / l)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (A <= -2e-311)
		tmp = Float64(c0 * Float64(Float64(sqrt(Float64(-A)) / sqrt(Float64(-V))) / sqrt(l)));
	else
		tmp = Float64(c0 * Float64(sqrt(A) * sqrt(Float64(Float64(1.0 / V) / l))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (A <= -2e-311)
		tmp = c0 * ((sqrt(-A) / sqrt(-V)) / sqrt(l));
	else
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[A, -2e-311], N[(c0 * N[(N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Sqrt[N[(N[(1.0 / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -1.9999999999999e-311

   1. Initial program 71.0%

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

      1. associate-/r* 78.2%

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

      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%

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

      1. frac-2neg 46.4%

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

      2. sqrt-div 49.6%

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

   8. Applied egg-rr 49.6%

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

   if -1.9999999999999e-311 < A

   1. Initial program 72.8%

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

      1. *-un-lft-identity 72.8%

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

      2. times-frac 72.5%

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

   4. Applied egg-rr 72.5%

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

      1. pow1/2 72.5%

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

      2. associate-*l/ 72.5%

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

      3. *-un-lft-identity 72.5%

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

      4. div-inv 72.5%

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

      5. associate-*l/ 72.1%

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

      6. frac-times 72.8%

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

      7. associate-*r/ 72.7%

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

      8. unpow-prod-down 83.1%

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

      9. pow1/2 83.1%

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

      10. associate-/r* 83.1%

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

   6. Applied egg-rr 83.1%

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

      1. unpow1/2 83.1%

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

   8. Simplified 83.1%

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 91.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \ell \cdot \frac{V}{A}\\ \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-299}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-302}:\\ \;\;\;\;c0 \cdot {t\_0}^{-0.5}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+300}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{{t\_0}^{-1}}\\ \end{array} \end{array} \]

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 (* l (/ V A))))
   (if (<= (* V l) (- INFINITY))
     (/ c0 (/ (sqrt l) (sqrt (/ A V))))
     (if (<= (* V l) -5e-299)
       (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
       (if (<= (* V l) 2e-302)
         (* c0 (pow t_0 -0.5))
         (if (<= (* V l) 1e+300)
           (* c0 (* (sqrt A) (sqrt (/ (/ 1.0 V) l))))
           (* c0 (sqrt (pow t_0 -1.0)))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = l * (V / A);
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = c0 / (sqrt(l) / sqrt((A / V)));
	} else if ((V * l) <= -5e-299) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 2e-302) {
		tmp = c0 * pow(t_0, -0.5);
	} else if ((V * l) <= 1e+300) {
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	} else {
		tmp = c0 * sqrt(pow(t_0, -1.0));
	}
	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 = l * (V / A);
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = c0 / (Math.sqrt(l) / Math.sqrt((A / V)));
	} else if ((V * l) <= -5e-299) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 2e-302) {
		tmp = c0 * Math.pow(t_0, -0.5);
	} else if ((V * l) <= 1e+300) {
		tmp = c0 * (Math.sqrt(A) * Math.sqrt(((1.0 / V) / l)));
	} else {
		tmp = c0 * Math.sqrt(Math.pow(t_0, -1.0));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = l * (V / A)
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = c0 / (math.sqrt(l) / math.sqrt((A / V)))
	elif (V * l) <= -5e-299:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 2e-302:
		tmp = c0 * math.pow(t_0, -0.5)
	elif (V * l) <= 1e+300:
		tmp = c0 * (math.sqrt(A) * math.sqrt(((1.0 / V) / l)))
	else:
		tmp = c0 * math.sqrt(math.pow(t_0, -1.0))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(l * Float64(V / A))
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(c0 / Float64(sqrt(l) / sqrt(Float64(A / V))));
	elseif (Float64(V * l) <= -5e-299)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 2e-302)
		tmp = Float64(c0 * (t_0 ^ -0.5));
	elseif (Float64(V * l) <= 1e+300)
		tmp = Float64(c0 * Float64(sqrt(A) * sqrt(Float64(Float64(1.0 / V) / l))));
	else
		tmp = Float64(c0 * sqrt((t_0 ^ -1.0)));
	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 = l * (V / A);
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = c0 / (sqrt(l) / sqrt((A / V)));
	elseif ((V * l) <= -5e-299)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 2e-302)
		tmp = c0 * (t_0 ^ -0.5);
	elseif ((V * l) <= 1e+300)
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	else
		tmp = c0 * sqrt((t_0 ^ -1.0));
	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[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(c0 / N[(N[Sqrt[l], $MachinePrecision] / N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -5e-299], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 2e-302], N[(c0 * N[Power[t$95$0, -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+300], 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[Power[t$95$0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 30.1%

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

      1. *-un-lft-identity 30.1%

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

      2. times-frac 72.0%

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

   4. Applied egg-rr 72.0%

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

      1. associate-*l/ 72.1%

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

      2. *-un-lft-identity 72.1%

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

      3. div-inv 72.1%

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

      4. associate-*l/ 72.1%

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

      5. un-div-inv 72.1%

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

      6. associate-/r* 30.1%

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

      7. sqrt-undiv 0.0%

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

      8. clear-num 0.0%

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

      9. un-div-inv 0.0%

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

      10. sqrt-undiv 30.1%

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

      11. associate-/l* 72.0%

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

   6. Applied egg-rr 72.0%

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

      1. *-commutative 72.0%

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

      2. sqrt-prod 34.8%

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

      3. sqrt-div 0.0%

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

      4. associate-/r/ 0.0%

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

      5. sqrt-div 53.9%

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

      6. div-inv 53.9%

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

   8. Applied egg-rr 53.9%

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

      1. associate-*r/ 53.9%

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

      2. *-rgt-identity 53.9%

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

   10. Simplified 53.9%

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

   if -inf.0 < (*.f64 V l) < -4.99999999999999956e-299

   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 99.5%

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

      3. distribute-rgt-neg-in 99.5%

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

   4. Applied egg-rr 99.5%

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

   if -4.99999999999999956e-299 < (*.f64 V l) < 1.9999999999999999e-302

   1. Initial program 38.5%

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

      1. *-un-lft-identity 38.5%

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

      2. times-frac 69.7%

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

   4. Applied egg-rr 69.7%

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

      1. associate-*l/ 69.7%

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

      2. *-un-lft-identity 69.7%

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

   6. Applied egg-rr 69.7%

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

      1. clear-num 69.7%

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

      2. sqrt-div 69.5%

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

      3. metadata-eval 69.5%

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

      4. div-inv 69.6%

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

      5. clear-num 69.6%

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

      6. pow1/2 69.6%

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

      7. pow-flip 69.7%

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

      8. associate-*r/ 38.5%

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

      9. *-commutative 38.5%

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

      10. associate-/l* 69.8%

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

      11. metadata-eval 69.8%

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

   8. Applied egg-rr 69.8%

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

   if 1.9999999999999999e-302 < (*.f64 V l) < 1.0000000000000001e300

   1. Initial program 85.8%

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

      1. *-un-lft-identity 85.8%

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

      2. times-frac 74.9%

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

   4. Applied egg-rr 74.9%

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

      1. pow1/2 74.9%

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

      2. associate-*l/ 75.0%

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

      3. *-un-lft-identity 75.0%

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

      4. div-inv 74.9%

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

      5. associate-*l/ 74.5%

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

      6. frac-times 85.8%

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

      7. associate-*r/ 85.7%

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

      8. unpow-prod-down 99.5%

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

      9. pow1/2 99.5%

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

      10. associate-/r* 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. unpow1/2 99.5%

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

   8. Simplified 99.5%

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

   if 1.0000000000000001e300 < (*.f64 V l)

   1. Initial program 22.6%

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

      1. associate-/r* 70.7%

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

      2. div-inv 70.6%

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

      3. clear-num 70.5%

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

      4. frac-times 70.6%

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

      5. metadata-eval 70.6%

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

      6. inv-pow 70.6%

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

   4. Applied egg-rr 70.6%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{{\left(\frac{V}{A} \cdot \ell\right)}^{-1}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-299}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-302}:\\ \;\;\;\;c0 \cdot {\left(\ell \cdot \frac{V}{A}\right)}^{-0.5}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+300}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{{\left(\ell \cdot \frac{V}{A}\right)}^{-1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{A}{V}} \cdot {\ell}^{-0.5}\right)\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+300}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{{\left(\ell \cdot \frac{V}{A}\right)}^{-1}}\\ \end{array} \end{array} \]

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) 0.0)
   (* c0 (* (sqrt (/ A V)) (pow l -0.5)))
   (if (<= (* V l) 1e+300)
     (* c0 (* (sqrt A) (sqrt (/ (/ 1.0 V) l))))
     (* c0 (sqrt (pow (* l (/ V A)) -1.0))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= 0.0) {
		tmp = c0 * (sqrt((A / V)) * pow(l, -0.5));
	} else if ((V * l) <= 1e+300) {
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	} else {
		tmp = c0 * sqrt(pow((l * (V / A)), -1.0));
	}
	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) :: tmp
    if ((v * l) <= 0.0d0) then
        tmp = c0 * (sqrt((a / v)) * (l ** (-0.5d0)))
    else if ((v * l) <= 1d+300) then
        tmp = c0 * (sqrt(a) * sqrt(((1.0d0 / v) / l)))
    else
        tmp = c0 * sqrt(((l * (v / a)) ** (-1.0d0)))
    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) <= 0.0) {
		tmp = c0 * (Math.sqrt((A / V)) * Math.pow(l, -0.5));
	} else if ((V * l) <= 1e+300) {
		tmp = c0 * (Math.sqrt(A) * Math.sqrt(((1.0 / V) / l)));
	} else {
		tmp = c0 * Math.sqrt(Math.pow((l * (V / A)), -1.0));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= 0.0:
		tmp = c0 * (math.sqrt((A / V)) * math.pow(l, -0.5))
	elif (V * l) <= 1e+300:
		tmp = c0 * (math.sqrt(A) * math.sqrt(((1.0 / V) / l)))
	else:
		tmp = c0 * math.sqrt(math.pow((l * (V / A)), -1.0))
	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) <= 0.0)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) * (l ^ -0.5)));
	elseif (Float64(V * l) <= 1e+300)
		tmp = Float64(c0 * Float64(sqrt(A) * sqrt(Float64(Float64(1.0 / V) / l))));
	else
		tmp = Float64(c0 * sqrt((Float64(l * Float64(V / A)) ^ -1.0)));
	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) <= 0.0)
		tmp = c0 * (sqrt((A / V)) * (l ^ -0.5));
	elseif ((V * l) <= 1e+300)
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	else
		tmp = c0 * sqrt(((l * (V / A)) ^ -1.0));
	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_] := If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+300], 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[Power[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 67.2%

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

      1. *-un-lft-identity 67.2%

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

      2. times-frac 73.2%

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

   4. Applied egg-rr 73.2%

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

      1. associate-*l/ 73.3%

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

      2. *-un-lft-identity 73.3%

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

      3. div-inv 73.3%

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

      4. associate-*l/ 76.2%

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

      5. sqrt-prod 45.2%

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

      6. *-commutative 45.2%

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

      7. inv-pow 45.2%

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

      8. sqrt-pow1 45.3%

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

      9. metadata-eval 45.3%

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

   6. Applied egg-rr 45.3%

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

   if 0.0 < (*.f64 V l) < 1.0000000000000001e300

   1. Initial program 86.2%

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

      1. *-un-lft-identity 86.2%

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

      2. times-frac 75.5%

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

   4. Applied egg-rr 75.5%

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

      1. pow1/2 75.5%

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

      2. associate-*l/ 75.6%

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

      3. *-un-lft-identity 75.6%

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

      4. div-inv 75.5%

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

      5. associate-*l/ 75.0%

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

      6. frac-times 86.2%

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

      7. associate-*r/ 86.0%

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

      8. unpow-prod-down 99.4%

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

      9. pow1/2 99.4%

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

      10. associate-/r* 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. unpow1/2 99.5%

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

   8. Simplified 99.5%

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

   if 1.0000000000000001e300 < (*.f64 V l)

   1. Initial program 22.6%

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

      1. associate-/r* 70.7%

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

      2. div-inv 70.6%

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

      3. clear-num 70.5%

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

      4. frac-times 70.6%

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

      5. metadata-eval 70.6%

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

      6. inv-pow 70.6%

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

   4. Applied egg-rr 70.6%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{A}{V}} \cdot {\ell}^{-0.5}\right)\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+300}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{{\left(\ell \cdot \frac{V}{A}\right)}^{-1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{A}{V}} \cdot {\ell}^{-0.5}\right)\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+295}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) 0.0)
   (* c0 (* (sqrt (/ A V)) (pow l -0.5)))
   (if (<= (* V l) 5e+295)
     (* c0 (* (sqrt A) (pow (* V l) -0.5)))
     (/ c0 (sqrt (* V (/ l A)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= 0.0) {
		tmp = c0 * (sqrt((A / V)) * pow(l, -0.5));
	} else if ((V * l) <= 5e+295) {
		tmp = c0 * (sqrt(A) * pow((V * l), -0.5));
	} 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) :: tmp
    if ((v * l) <= 0.0d0) then
        tmp = c0 * (sqrt((a / v)) * (l ** (-0.5d0)))
    else if ((v * l) <= 5d+295) then
        tmp = c0 * (sqrt(a) * ((v * l) ** (-0.5d0)))
    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 tmp;
	if ((V * l) <= 0.0) {
		tmp = c0 * (Math.sqrt((A / V)) * Math.pow(l, -0.5));
	} else if ((V * l) <= 5e+295) {
		tmp = c0 * (Math.sqrt(A) * Math.pow((V * l), -0.5));
	} 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):
	tmp = 0
	if (V * l) <= 0.0:
		tmp = c0 * (math.sqrt((A / V)) * math.pow(l, -0.5))
	elif (V * l) <= 5e+295:
		tmp = c0 * (math.sqrt(A) * math.pow((V * l), -0.5))
	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)
	tmp = 0.0
	if (Float64(V * l) <= 0.0)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) * (l ^ -0.5)));
	elseif (Float64(V * l) <= 5e+295)
		tmp = Float64(c0 * Float64(sqrt(A) * (Float64(V * l) ^ -0.5)));
	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)
	tmp = 0.0;
	if ((V * l) <= 0.0)
		tmp = c0 * (sqrt((A / V)) * (l ^ -0.5));
	elseif ((V * l) <= 5e+295)
		tmp = c0 * (sqrt(A) * ((V * l) ^ -0.5));
	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_] := If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e+295], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Power[N[(V * l), $MachinePrecision], -0.5], $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 V l) < 0.0

   1. Initial program 67.2%

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

      1. *-un-lft-identity 67.2%

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

      2. times-frac 73.2%

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

   4. Applied egg-rr 73.2%

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

      1. associate-*l/ 73.3%

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

      2. *-un-lft-identity 73.3%

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

      3. div-inv 73.3%

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

      4. associate-*l/ 76.2%

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

      5. sqrt-prod 45.2%

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

      6. *-commutative 45.2%

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

      7. inv-pow 45.2%

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

      8. sqrt-pow1 45.3%

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

      9. metadata-eval 45.3%

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

   6. Applied egg-rr 45.3%

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

   if 0.0 < (*.f64 V l) < 4.99999999999999991e295

   1. Initial program 86.0%

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

      1. *-un-lft-identity 86.0%

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

      2. times-frac 75.2%

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

   4. Applied egg-rr 75.2%

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

      1. sqrt-prod 35.7%

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

      2. sqrt-div 35.7%

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

      3. metadata-eval 35.7%

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

      4. sqrt-div 42.6%

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

      5. times-frac 42.6%

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

      6. sqrt-prod 99.4%

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

      7. *-commutative 99.4%

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

      8. associate-*r/ 99.3%

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

      9. *-commutative 99.3%

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

      10. pow1/2 99.3%

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

      11. pow-flip 99.5%

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

      12. metadata-eval 99.5%

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

   6. Applied egg-rr 99.5%

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

   if 4.99999999999999991e295 < (*.f64 V l)

   1. Initial program 29.5%

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

      1. *-un-lft-identity 29.5%

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

      2. times-frac 73.4%

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

   4. Applied egg-rr 73.4%

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

      1. associate-*l/ 73.0%

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

      2. *-un-lft-identity 73.0%

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

      3. div-inv 73.0%

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

      4. associate-*l/ 73.1%

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

      5. un-div-inv 73.2%

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

      6. associate-/r* 29.5%

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

      7. sqrt-undiv 29.6%

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

      8. clear-num 29.6%

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

      9. un-div-inv 29.6%

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

      10. sqrt-undiv 29.6%

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

      11. associate-/l* 73.4%

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

   6. Applied egg-rr 73.4%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{A}{V}} \cdot {\ell}^{-0.5}\right)\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+295}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{A}{V}} \cdot {\ell}^{-0.5}\right)\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+300}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{{\left(\ell \cdot \frac{V}{A}\right)}^{-1}}\\ \end{array} \end{array} \]

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) 0.0)
   (* c0 (* (sqrt (/ A V)) (pow l -0.5)))
   (if (<= (* V l) 1e+300)
     (/ c0 (/ (sqrt (* V l)) (sqrt A)))
     (* c0 (sqrt (pow (* l (/ V A)) -1.0))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= 0.0) {
		tmp = c0 * (sqrt((A / V)) * pow(l, -0.5));
	} else if ((V * l) <= 1e+300) {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	} else {
		tmp = c0 * sqrt(pow((l * (V / A)), -1.0));
	}
	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) :: tmp
    if ((v * l) <= 0.0d0) then
        tmp = c0 * (sqrt((a / v)) * (l ** (-0.5d0)))
    else if ((v * l) <= 1d+300) then
        tmp = c0 / (sqrt((v * l)) / sqrt(a))
    else
        tmp = c0 * sqrt(((l * (v / a)) ** (-1.0d0)))
    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) <= 0.0) {
		tmp = c0 * (Math.sqrt((A / V)) * Math.pow(l, -0.5));
	} else if ((V * l) <= 1e+300) {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	} else {
		tmp = c0 * Math.sqrt(Math.pow((l * (V / A)), -1.0));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= 0.0:
		tmp = c0 * (math.sqrt((A / V)) * math.pow(l, -0.5))
	elif (V * l) <= 1e+300:
		tmp = c0 / (math.sqrt((V * l)) / math.sqrt(A))
	else:
		tmp = c0 * math.sqrt(math.pow((l * (V / A)), -1.0))
	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) <= 0.0)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) * (l ^ -0.5)));
	elseif (Float64(V * l) <= 1e+300)
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	else
		tmp = Float64(c0 * sqrt((Float64(l * Float64(V / A)) ^ -1.0)));
	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) <= 0.0)
		tmp = c0 * (sqrt((A / V)) * (l ^ -0.5));
	elseif ((V * l) <= 1e+300)
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	else
		tmp = c0 * sqrt(((l * (V / A)) ^ -1.0));
	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_] := If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+300], N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[Power[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 67.2%

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

      1. *-un-lft-identity 67.2%

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

      2. times-frac 73.2%

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

   4. Applied egg-rr 73.2%

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

      1. associate-*l/ 73.3%

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

      2. *-un-lft-identity 73.3%

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

      3. div-inv 73.3%

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

      4. associate-*l/ 76.2%

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

      5. sqrt-prod 45.2%

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

      6. *-commutative 45.2%

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

      7. inv-pow 45.2%

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

      8. sqrt-pow1 45.3%

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

      9. metadata-eval 45.3%

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

   6. Applied egg-rr 45.3%

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

   if 0.0 < (*.f64 V l) < 1.0000000000000001e300

   1. Initial program 86.2%

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

      1. *-un-lft-identity 86.2%

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

      2. times-frac 75.5%

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

   4. Applied egg-rr 75.5%

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

      1. associate-*l/ 75.6%

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

      2. *-un-lft-identity 75.6%

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

      3. div-inv 75.5%

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

      4. associate-*l/ 75.0%

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

      5. un-div-inv 75.1%

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

      6. associate-/r* 86.2%

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

      7. sqrt-undiv 99.4%

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

      8. clear-num 99.3%

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

      9. un-div-inv 99.4%

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

      10. sqrt-undiv 86.1%

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

      11. associate-/l* 74.7%

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

   6. Applied egg-rr 74.7%

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

      1. associate-*r/ 86.1%

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

      2. sqrt-div 99.4%

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

      3. *-commutative 99.4%

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

   8. Applied egg-rr 99.4%

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

   if 1.0000000000000001e300 < (*.f64 V l)

   1. Initial program 22.6%

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

      1. associate-/r* 70.7%

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

      2. div-inv 70.6%

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

      3. clear-num 70.5%

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

      4. frac-times 70.6%

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

      5. metadata-eval 70.6%

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

      6. inv-pow 70.6%

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

   4. Applied egg-rr 70.6%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{A}{V}} \cdot {\ell}^{-0.5}\right)\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+300}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{{\left(\ell \cdot \frac{V}{A}\right)}^{-1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+300}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{{\left(\ell \cdot \frac{V}{A}\right)}^{-1}}\\ \end{array} \end{array} \]

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) 0.0)
   (/ c0 (/ (sqrt l) (sqrt (/ A V))))
   (if (<= (* V l) 1e+300)
     (/ c0 (/ (sqrt (* V l)) (sqrt A)))
     (* c0 (sqrt (pow (* l (/ V A)) -1.0))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= 0.0) {
		tmp = c0 / (sqrt(l) / sqrt((A / V)));
	} else if ((V * l) <= 1e+300) {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	} else {
		tmp = c0 * sqrt(pow((l * (V / A)), -1.0));
	}
	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) :: tmp
    if ((v * l) <= 0.0d0) then
        tmp = c0 / (sqrt(l) / sqrt((a / v)))
    else if ((v * l) <= 1d+300) then
        tmp = c0 / (sqrt((v * l)) / sqrt(a))
    else
        tmp = c0 * sqrt(((l * (v / a)) ** (-1.0d0)))
    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) <= 0.0) {
		tmp = c0 / (Math.sqrt(l) / Math.sqrt((A / V)));
	} else if ((V * l) <= 1e+300) {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	} else {
		tmp = c0 * Math.sqrt(Math.pow((l * (V / A)), -1.0));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= 0.0:
		tmp = c0 / (math.sqrt(l) / math.sqrt((A / V)))
	elif (V * l) <= 1e+300:
		tmp = c0 / (math.sqrt((V * l)) / math.sqrt(A))
	else:
		tmp = c0 * math.sqrt(math.pow((l * (V / A)), -1.0))
	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) <= 0.0)
		tmp = Float64(c0 / Float64(sqrt(l) / sqrt(Float64(A / V))));
	elseif (Float64(V * l) <= 1e+300)
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	else
		tmp = Float64(c0 * sqrt((Float64(l * Float64(V / A)) ^ -1.0)));
	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) <= 0.0)
		tmp = c0 / (sqrt(l) / sqrt((A / V)));
	elseif ((V * l) <= 1e+300)
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	else
		tmp = c0 * sqrt(((l * (V / A)) ^ -1.0));
	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_] := If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 / N[(N[Sqrt[l], $MachinePrecision] / N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+300], N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[Power[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 67.2%

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

      1. *-un-lft-identity 67.2%

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

      2. times-frac 73.2%

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

   4. Applied egg-rr 73.2%

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

      1. associate-*l/ 73.3%

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

      2. *-un-lft-identity 73.3%

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

      3. div-inv 73.3%

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

      4. associate-*l/ 76.2%

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

      5. un-div-inv 76.2%

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

      6. associate-/r* 67.2%

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

      7. sqrt-undiv 2.9%

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

      8. clear-num 2.9%

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

      9. un-div-inv 2.9%

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

      10. sqrt-undiv 67.1%

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

      11. associate-/l* 72.5%

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

   6. Applied egg-rr 72.5%

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

      1. *-commutative 72.5%

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

      2. sqrt-prod 39.4%

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

      3. sqrt-div 3.1%

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

      4. associate-/r/ 3.1%

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

      5. sqrt-div 45.3%

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

      6. div-inv 45.3%

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

   8. Applied egg-rr 45.3%

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

      1. associate-*r/ 45.3%

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

      2. *-rgt-identity 45.3%

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

   10. Simplified 45.3%

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

   if 0.0 < (*.f64 V l) < 1.0000000000000001e300

   1. Initial program 86.2%

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

      1. *-un-lft-identity 86.2%

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

      2. times-frac 75.5%

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

   4. Applied egg-rr 75.5%

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

      1. associate-*l/ 75.6%

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

      2. *-un-lft-identity 75.6%

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

      3. div-inv 75.5%

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

      4. associate-*l/ 75.0%

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

      5. un-div-inv 75.1%

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

      6. associate-/r* 86.2%

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

      7. sqrt-undiv 99.4%

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

      8. clear-num 99.3%

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

      9. un-div-inv 99.4%

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

      10. sqrt-undiv 86.1%

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

      11. associate-/l* 74.7%

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

   6. Applied egg-rr 74.7%

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

      1. associate-*r/ 86.1%

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

      2. sqrt-div 99.4%

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

      3. *-commutative 99.4%

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

   8. Applied egg-rr 99.4%

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

   if 1.0000000000000001e300 < (*.f64 V l)

   1. Initial program 22.6%

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

      1. associate-/r* 70.7%

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

      2. div-inv 70.6%

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

      3. clear-num 70.5%

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

      4. frac-times 70.6%

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

      5. metadata-eval 70.6%

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

      6. inv-pow 70.6%

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

   4. Applied egg-rr 70.6%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+300}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{{\left(\ell \cdot \frac{V}{A}\right)}^{-1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+300}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{1}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) 0.0)
   (/ c0 (/ (sqrt l) (sqrt (/ A V))))
   (if (<= (* V l) 1e+300)
     (/ c0 (/ (sqrt (* V l)) (sqrt A)))
     (* c0 (/ 1.0 (sqrt (* l (/ V A))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= 0.0) {
		tmp = c0 / (sqrt(l) / sqrt((A / V)));
	} else if ((V * l) <= 1e+300) {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	} else {
		tmp = c0 * (1.0 / 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) :: tmp
    if ((v * l) <= 0.0d0) then
        tmp = c0 / (sqrt(l) / sqrt((a / v)))
    else if ((v * l) <= 1d+300) then
        tmp = c0 / (sqrt((v * l)) / sqrt(a))
    else
        tmp = c0 * (1.0d0 / 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 tmp;
	if ((V * l) <= 0.0) {
		tmp = c0 / (Math.sqrt(l) / Math.sqrt((A / V)));
	} else if ((V * l) <= 1e+300) {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	} else {
		tmp = c0 * (1.0 / Math.sqrt((l * (V / A))));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= 0.0:
		tmp = c0 / (math.sqrt(l) / math.sqrt((A / V)))
	elif (V * l) <= 1e+300:
		tmp = c0 / (math.sqrt((V * l)) / math.sqrt(A))
	else:
		tmp = c0 * (1.0 / math.sqrt((l * (V / A))))
	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) <= 0.0)
		tmp = Float64(c0 / Float64(sqrt(l) / sqrt(Float64(A / V))));
	elseif (Float64(V * l) <= 1e+300)
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	else
		tmp = Float64(c0 * Float64(1.0 / 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)
	tmp = 0.0;
	if ((V * l) <= 0.0)
		tmp = c0 / (sqrt(l) / sqrt((A / V)));
	elseif ((V * l) <= 1e+300)
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	else
		tmp = c0 * (1.0 / 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_] := If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 / N[(N[Sqrt[l], $MachinePrecision] / N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+300], N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(1.0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 67.2%

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

      1. *-un-lft-identity 67.2%

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

      2. times-frac 73.2%

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

   4. Applied egg-rr 73.2%

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

      1. associate-*l/ 73.3%

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

      2. *-un-lft-identity 73.3%

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

      3. div-inv 73.3%

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

      4. associate-*l/ 76.2%

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

      5. un-div-inv 76.2%

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

      6. associate-/r* 67.2%

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

      7. sqrt-undiv 2.9%

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

      8. clear-num 2.9%

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

      9. un-div-inv 2.9%

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

      10. sqrt-undiv 67.1%

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

      11. associate-/l* 72.5%

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

   6. Applied egg-rr 72.5%

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

      1. *-commutative 72.5%

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

      2. sqrt-prod 39.4%

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

      3. sqrt-div 3.1%

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

      4. associate-/r/ 3.1%

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

      5. sqrt-div 45.3%

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

      6. div-inv 45.3%

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

   8. Applied egg-rr 45.3%

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

      1. associate-*r/ 45.3%

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

      2. *-rgt-identity 45.3%

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

   10. Simplified 45.3%

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

   if 0.0 < (*.f64 V l) < 1.0000000000000001e300

   1. Initial program 86.2%

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

      1. *-un-lft-identity 86.2%

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

      2. times-frac 75.5%

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

   4. Applied egg-rr 75.5%

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

      1. associate-*l/ 75.6%

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

      2. *-un-lft-identity 75.6%

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

      3. div-inv 75.5%

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

      4. associate-*l/ 75.0%

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

      5. un-div-inv 75.1%

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

      6. associate-/r* 86.2%

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

      7. sqrt-undiv 99.4%

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

      8. clear-num 99.3%

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

      9. un-div-inv 99.4%

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

      10. sqrt-undiv 86.1%

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

      11. associate-/l* 74.7%

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

   6. Applied egg-rr 74.7%

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

      1. associate-*r/ 86.1%

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

      2. sqrt-div 99.4%

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

      3. *-commutative 99.4%

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

   8. Applied egg-rr 99.4%

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

   if 1.0000000000000001e300 < (*.f64 V l)

   1. Initial program 22.6%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. clear-num 70.9%

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

      3. sqrt-div 70.7%

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

      4. metadata-eval 70.7%

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

      5. div-inv 70.7%

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

      6. clear-num 70.7%

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

   4. Applied egg-rr 70.7%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+300}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{1}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 86.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+300}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{1}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) 0.0)
   (/ c0 (/ (sqrt l) (sqrt (/ A V))))
   (if (<= (* V l) 1e+300)
     (* c0 (/ (sqrt A) (sqrt (* V l))))
     (* c0 (/ 1.0 (sqrt (* l (/ V A))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= 0.0) {
		tmp = c0 / (sqrt(l) / sqrt((A / V)));
	} else if ((V * l) <= 1e+300) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0 * (1.0 / 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) :: tmp
    if ((v * l) <= 0.0d0) then
        tmp = c0 / (sqrt(l) / sqrt((a / v)))
    else if ((v * l) <= 1d+300) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = c0 * (1.0d0 / 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 tmp;
	if ((V * l) <= 0.0) {
		tmp = c0 / (Math.sqrt(l) / Math.sqrt((A / V)));
	} else if ((V * l) <= 1e+300) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0 * (1.0 / Math.sqrt((l * (V / A))));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= 0.0:
		tmp = c0 / (math.sqrt(l) / math.sqrt((A / V)))
	elif (V * l) <= 1e+300:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0 * (1.0 / math.sqrt((l * (V / A))))
	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) <= 0.0)
		tmp = Float64(c0 / Float64(sqrt(l) / sqrt(Float64(A / V))));
	elseif (Float64(V * l) <= 1e+300)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 * Float64(1.0 / 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)
	tmp = 0.0;
	if ((V * l) <= 0.0)
		tmp = c0 / (sqrt(l) / sqrt((A / V)));
	elseif ((V * l) <= 1e+300)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0 * (1.0 / 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_] := If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 / N[(N[Sqrt[l], $MachinePrecision] / N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+300], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(1.0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 67.2%

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

      1. *-un-lft-identity 67.2%

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

      2. times-frac 73.2%

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

   4. Applied egg-rr 73.2%

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

      1. associate-*l/ 73.3%

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

      2. *-un-lft-identity 73.3%

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

      3. div-inv 73.3%

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

      4. associate-*l/ 76.2%

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

      5. un-div-inv 76.2%

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

      6. associate-/r* 67.2%

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

      7. sqrt-undiv 2.9%

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

      8. clear-num 2.9%

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

      9. un-div-inv 2.9%

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

      10. sqrt-undiv 67.1%

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

      11. associate-/l* 72.5%

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

   6. Applied egg-rr 72.5%

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

      1. *-commutative 72.5%

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

      2. sqrt-prod 39.4%

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

      3. sqrt-div 3.1%

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

      4. associate-/r/ 3.1%

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

      5. sqrt-div 45.3%

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

      6. div-inv 45.3%

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

   8. Applied egg-rr 45.3%

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

      1. associate-*r/ 45.3%

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

      2. *-rgt-identity 45.3%

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

   10. Simplified 45.3%

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

   if 0.0 < (*.f64 V l) < 1.0000000000000001e300

   1. Initial program 86.2%

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

      1. sqrt-div 99.4%

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

      2. div-inv 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. associate-*r/ 99.4%

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

      2. *-rgt-identity 99.4%

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

   6. Simplified 99.4%

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

   if 1.0000000000000001e300 < (*.f64 V l)

   1. Initial program 22.6%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. clear-num 70.9%

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

      3. sqrt-div 70.7%

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

      4. metadata-eval 70.7%

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

      5. div-inv 70.7%

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

      6. clear-num 70.7%

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

   4. Applied egg-rr 70.7%

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

Alternative 10: 86.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+300}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{1}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) 0.0)
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))
   (if (<= (* V l) 1e+300)
     (* c0 (/ (sqrt A) (sqrt (* V l))))
     (* c0 (/ 1.0 (sqrt (* l (/ V A))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= 0.0) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else if ((V * l) <= 1e+300) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0 * (1.0 / 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) :: tmp
    if ((v * l) <= 0.0d0) then
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    else if ((v * l) <= 1d+300) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = c0 * (1.0d0 / 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 tmp;
	if ((V * l) <= 0.0) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if ((V * l) <= 1e+300) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0 * (1.0 / Math.sqrt((l * (V / A))));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= 0.0:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	elif (V * l) <= 1e+300:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0 * (1.0 / math.sqrt((l * (V / A))))
	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) <= 0.0)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (Float64(V * l) <= 1e+300)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 * Float64(1.0 / 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)
	tmp = 0.0;
	if ((V * l) <= 0.0)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	elseif ((V * l) <= 1e+300)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0 * (1.0 / 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_] := 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+300], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(1.0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 67.2%

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

      1. associate-/r* 76.2%

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

      2. sqrt-div 45.2%

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

      3. div-inv 45.2%

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

   4. Applied egg-rr 45.2%

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

      1. associate-*r/ 45.2%

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

      2. *-rgt-identity 45.2%

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

   6. Simplified 45.2%

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

   if 0.0 < (*.f64 V l) < 1.0000000000000001e300

   1. Initial program 86.2%

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

      1. sqrt-div 99.4%

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

      2. div-inv 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. associate-*r/ 99.4%

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

      2. *-rgt-identity 99.4%

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

   6. Simplified 99.4%

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

   if 1.0000000000000001e300 < (*.f64 V l)

   1. Initial program 22.6%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. clear-num 70.9%

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

      3. sqrt-div 70.7%

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

      4. metadata-eval 70.7%

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

      5. div-inv 70.7%

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

      6. clear-num 70.7%

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

   4. Applied egg-rr 70.7%

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

Alternative 11: 81.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{1}{V} \cdot \frac{A}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+300}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{1}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) 0.0)
   (* c0 (sqrt (* (/ 1.0 V) (/ A l))))
   (if (<= (* V l) 1e+300)
     (* c0 (/ (sqrt A) (sqrt (* V l))))
     (* c0 (/ 1.0 (sqrt (* l (/ V A))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= 0.0) {
		tmp = c0 * sqrt(((1.0 / V) * (A / l)));
	} else if ((V * l) <= 1e+300) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0 * (1.0 / 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) :: tmp
    if ((v * l) <= 0.0d0) then
        tmp = c0 * sqrt(((1.0d0 / v) * (a / l)))
    else if ((v * l) <= 1d+300) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = c0 * (1.0d0 / 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 tmp;
	if ((V * l) <= 0.0) {
		tmp = c0 * Math.sqrt(((1.0 / V) * (A / l)));
	} else if ((V * l) <= 1e+300) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0 * (1.0 / Math.sqrt((l * (V / A))));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= 0.0:
		tmp = c0 * math.sqrt(((1.0 / V) * (A / l)))
	elif (V * l) <= 1e+300:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0 * (1.0 / math.sqrt((l * (V / A))))
	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) <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(1.0 / V) * Float64(A / l))));
	elseif (Float64(V * l) <= 1e+300)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 * Float64(1.0 / 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)
	tmp = 0.0;
	if ((V * l) <= 0.0)
		tmp = c0 * sqrt(((1.0 / V) * (A / l)));
	elseif ((V * l) <= 1e+300)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0 * (1.0 / 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_] := If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 * N[Sqrt[N[(N[(1.0 / V), $MachinePrecision] * N[(A / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+300], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(1.0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 67.2%

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

      1. *-un-lft-identity 67.2%

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

      2. times-frac 73.2%

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

   4. Applied egg-rr 73.2%

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

   if 0.0 < (*.f64 V l) < 1.0000000000000001e300

   1. Initial program 86.2%

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

      1. sqrt-div 99.4%

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

      2. div-inv 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. associate-*r/ 99.4%

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

      2. *-rgt-identity 99.4%

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

   6. Simplified 99.4%

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

   if 1.0000000000000001e300 < (*.f64 V l)

   1. Initial program 22.6%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. clear-num 70.9%

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

      3. sqrt-div 70.7%

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

      4. metadata-eval 70.7%

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

      5. div-inv 70.7%

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

      6. clear-num 70.7%

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

   4. Applied egg-rr 70.7%

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

Alternative 12: 79.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-315} \lor \neg \left(t\_0 \leq 4 \cdot 10^{+304}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{t\_0}\\ \end{array} \end{array} \]

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 5e-315) (not (<= t_0 4e+304)))
     (* 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 <= 5e-315) || !(t_0 <= 4e+304)) {
		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 <= 5d-315) .or. (.not. (t_0 <= 4d+304))) 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 <= 5e-315) || !(t_0 <= 4e+304)) {
		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 <= 5e-315) or not (t_0 <= 4e+304):
		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 <= 5e-315) || !(t_0 <= 4e+304))
		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 <= 5e-315) || ~((t_0 <= 4e+304)))
		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, 5e-315], N[Not[LessEqual[t$95$0, 4e+304]], $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)) < 5.0000000023e-315 or 3.9999999999999998e304 < (/.f64 A (*.f64 V l))

   1. Initial program 33.1%

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

      1. *-commutative 33.1%

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

      2. associate-/l/ 56.8%

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

   3. Simplified 56.8%

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

   if 5.0000000023e-315 < (/.f64 A (*.f64 V l)) < 3.9999999999999998e304

   1. Initial program 99.6%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{-315} \lor \neg \left(\frac{A}{V \cdot \ell} \leq 4 \cdot 10^{+304}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 79.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+290}:\\ \;\;\;\;c0 \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

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 (* l (/ V A))))
     (if (<= t_0 2e+290) (* 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((l * (V / A)));
	} else if (t_0 <= 2e+290) {
		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((l * (v / a)))
    else if (t_0 <= 2d+290) 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((l * (V / A)));
	} else if (t_0 <= 2e+290) {
		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((l * (V / A)))
	elif t_0 <= 2e+290:
		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(l * Float64(V / A))));
	elseif (t_0 <= 2e+290)
		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((l * (V / A)));
	elseif (t_0 <= 2e+290)
		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[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+290], 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 24.9%

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

      1. *-un-lft-identity 24.9%

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

      2. times-frac 54.6%

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

   4. Applied egg-rr 54.6%

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

      1. associate-*l/ 54.6%

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

      2. *-un-lft-identity 54.6%

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

      3. div-inv 54.6%

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

      4. associate-*l/ 54.6%

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

      5. un-div-inv 54.6%

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

      6. associate-/r* 24.9%

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

      7. sqrt-undiv 15.4%

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

      8. clear-num 15.5%

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

      9. un-div-inv 15.4%

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

      10. sqrt-undiv 24.9%

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

      11. associate-/l* 54.5%

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

   6. Applied egg-rr 54.5%

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

      1. associate-*r/ 24.9%

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

      2. associate-*l/ 54.6%

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

      3. *-commutative 54.6%

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

   8. Simplified 54.6%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 2.00000000000000012e290

   1. Initial program 99.4%

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

   if 2.00000000000000012e290 < (/.f64 A (*.f64 V l))

   1. Initial program 42.7%

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

      1. *-un-lft-identity 42.7%

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

      2. times-frac 58.6%

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

   4. Applied egg-rr 58.6%

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

      1. associate-*l/ 58.6%

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

      2. *-un-lft-identity 58.6%

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

      3. div-inv 58.6%

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

      4. associate-*l/ 60.3%

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

      5. un-div-inv 60.3%

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

      6. associate-/r* 42.7%

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

      7. sqrt-undiv 32.0%

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

      8. clear-num 31.9%

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

      9. un-div-inv 32.0%

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

      10. sqrt-undiv 43.5%

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

      11. associate-/l* 58.5%

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

   6. Applied egg-rr 58.5%

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

Alternative 14: 79.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-315}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+290}:\\ \;\;\;\;c0 \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

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 5e-315)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 2e+290) (* 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 <= 5e-315) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 2e+290) {
		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 <= 5d-315) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 2d+290) 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 <= 5e-315) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 2e+290) {
		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 <= 5e-315:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 2e+290:
		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 <= 5e-315)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 2e+290)
		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 <= 5e-315)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 2e+290)
		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, 5e-315], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+290], 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)) < 5.0000000023e-315

   1. Initial program 27.0%

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

      1. *-commutative 27.0%

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

      2. associate-/l/ 55.7%

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

   3. Simplified 55.7%

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

   if 5.0000000023e-315 < (/.f64 A (*.f64 V l)) < 2.00000000000000012e290

   1. Initial program 99.6%

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

   if 2.00000000000000012e290 < (/.f64 A (*.f64 V l))

   1. Initial program 42.7%

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

      1. *-un-lft-identity 42.7%

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

      2. times-frac 58.6%

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

   4. Applied egg-rr 58.6%

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

      1. associate-*l/ 58.6%

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

      2. *-un-lft-identity 58.6%

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

      3. div-inv 58.6%

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

      4. associate-*l/ 60.3%

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

      5. un-div-inv 60.3%

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

      6. associate-/r* 42.7%

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

      7. sqrt-undiv 32.0%

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

      8. clear-num 31.9%

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

      9. un-div-inv 32.0%

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

      10. sqrt-undiv 43.5%

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

      11. associate-/l* 58.5%

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

   6. Applied egg-rr 58.5%

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

Alternative 15: 78.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-315}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+261}:\\ \;\;\;\;c0 \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \end{array} \]

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 (/ A (* V l))))
   (if (<= t_0 5e-315)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 4e+261) (* c0 (sqrt t_0)) (* 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 = A / (V * l);
	double tmp;
	if (t_0 <= 5e-315) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 4e+261) {
		tmp = c0 * sqrt(t_0);
	} 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 = a / (v * l)
    if (t_0 <= 5d-315) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 4d+261) then
        tmp = c0 * sqrt(t_0)
    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 = A / (V * l);
	double tmp;
	if (t_0 <= 5e-315) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 4e+261) {
		tmp = c0 * Math.sqrt(t_0);
	} 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 = A / (V * l)
	tmp = 0
	if t_0 <= 5e-315:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 4e+261:
		tmp = c0 * math.sqrt(t_0)
	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(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 5e-315)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 4e+261)
		tmp = Float64(c0 * sqrt(t_0));
	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 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 5e-315)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 4e+261)
		tmp = c0 * sqrt(t_0);
	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[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-315], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+261], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 A (*.f64 V l)) < 5.0000000023e-315

   1. Initial program 27.0%

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

      1. *-commutative 27.0%

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

      2. associate-/l/ 55.7%

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

   3. Simplified 55.7%

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

   if 5.0000000023e-315 < (/.f64 A (*.f64 V l)) < 3.9999999999999997e261

   1. Initial program 99.6%

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

   if 3.9999999999999997e261 < (/.f64 A (*.f64 V l))

   1. Initial program 48.3%

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

      1. *-un-lft-identity 48.3%

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

      2. times-frac 61.1%

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

   4. Applied egg-rr 61.1%

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

      1. associate-*l/ 61.1%

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

      2. *-un-lft-identity 61.1%

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

   6. Applied egg-rr 61.1%

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

Alternative 16: 73.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    code = c0 * sqrt((a / (v * l)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.8%

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

Reproduce

herbie shell --seed 2024131 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))