Henrywood and Agarwal, Equation (3)

Percentage Accurate: 72.6% → 96.6%

Time: 10.5s

Alternatives: 14

Speedup: 1.0×

• Unsound rule application detected in e-graph. Results may not be sound.

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: 72.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.6% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 72.5%

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

   1. pow1/2 72.5%

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

   2. add-cube-cbrt 72.1%

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

   3. pow3 72.1%

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

   4. pow-pow 72.2%

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

   5. metadata-eval 72.2%

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

3. Applied egg-rr 72.2%

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

   1. associate-/r* 74.6%

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

   2. cbrt-div 83.3%

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

5. Applied egg-rr 83.3%

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

   1. cbrt-div 96.0%

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

7. Applied egg-rr 96.0%

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

8. Final simplification 96.0%

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

Alternative 2: 94.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+306}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-322}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \left(-\ell\right)}}{\sqrt{-A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-302} \lor \neg \left(V \cdot \ell \leq 5 \cdot 10^{+222}\right):\\ \;\;\;\;c0 \cdot {\left(\frac{\sqrt[3]{\frac{A}{V}}}{\sqrt[3]{\ell}}\right)}^{1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -2e+306)
   (* (/ c0 (sqrt l)) (sqrt (/ A V)))
   (if (<= (* V l) -2e-322)
     (/ c0 (/ (sqrt (* V (- l))) (sqrt (- A))))
     (if (or (<= (* V l) 5e-302) (not (<= (* V l) 5e+222)))
       (* c0 (pow (/ (cbrt (/ A V)) (cbrt l)) 1.5))
       (/ c0 (/ (sqrt (* V l)) (sqrt A)))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e+306) {
		tmp = (c0 / sqrt(l)) * sqrt((A / V));
	} else if ((V * l) <= -2e-322) {
		tmp = c0 / (sqrt((V * -l)) / sqrt(-A));
	} else if (((V * l) <= 5e-302) || !((V * l) <= 5e+222)) {
		tmp = c0 * pow((cbrt((A / V)) / cbrt(l)), 1.5);
	} else {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	}
	return tmp;
}

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e+306) {
		tmp = (c0 / Math.sqrt(l)) * Math.sqrt((A / V));
	} else if ((V * l) <= -2e-322) {
		tmp = c0 / (Math.sqrt((V * -l)) / Math.sqrt(-A));
	} else if (((V * l) <= 5e-302) || !((V * l) <= 5e+222)) {
		tmp = c0 * Math.pow((Math.cbrt((A / V)) / Math.cbrt(l)), 1.5);
	} else {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	}
	return tmp;
}

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -2e+306)
		tmp = Float64(Float64(c0 / sqrt(l)) * sqrt(Float64(A / V)));
	elseif (Float64(V * l) <= -2e-322)
		tmp = Float64(c0 / Float64(sqrt(Float64(V * Float64(-l))) / sqrt(Float64(-A))));
	elseif ((Float64(V * l) <= 5e-302) || !(Float64(V * l) <= 5e+222))
		tmp = Float64(c0 * (Float64(cbrt(Float64(A / V)) / cbrt(l)) ^ 1.5));
	else
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	end
	return tmp
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -2e+306], N[(N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-322], N[(c0 / N[(N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-A)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(V * l), $MachinePrecision], 5e-302], N[Not[LessEqual[N[(V * l), $MachinePrecision], 5e+222]], $MachinePrecision]], N[(c0 * N[Power[N[(N[Power[N[(A / V), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision]), $MachinePrecision], N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -2.00000000000000003e306

   1. Initial program 31.7%

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

      1. add-cube-cbrt 31.7%

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

      2. *-commutative 31.7%

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

      3. times-frac 63.3%

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

      4. pow2 63.3%

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

   3. Applied egg-rr 63.3%

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

      1. frac-times 31.7%

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

      2. unpow2 31.7%

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

      3. add-cube-cbrt 31.7%

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

      4. *-commutative 31.7%

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

      5. *-un-lft-identity 31.7%

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

      6. associate-*l/ 31.7%

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

      7. add-sqr-sqrt 30.1%

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

      8. sqrt-unprod 30.3%

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

      9. *-commutative 30.3%

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

      10. *-commutative 30.3%

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

      11. swap-sqr 29.9%

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

   5. Applied egg-rr 29.9%

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

      1. *-commutative 29.9%

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

      2. rem-square-sqrt 29.9%

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

      3. swap-sqr 30.3%

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

      4. associate-*l* 30.3%

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

      5. *-commutative 30.3%

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

      6. associate-*l* 30.3%

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

      7. rem-square-sqrt 30.3%

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

      8. associate-*l/ 30.2%

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

      9. *-commutative 30.2%

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

   7. Simplified 30.2%

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

   8. Taylor expanded in c0 around 0 29.5%

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

      1. times-frac 58.6%

         \[\leadsto \sqrt{\color{blue}{\frac{A}{V} \cdot \frac{{c0}^{2}}{\ell}}} \]

      2. unpow2 58.6%

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

   10. Simplified 58.6%

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

       1. *-commutative 58.6%

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

       2. sqrt-prod 50.4%

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

       3. sqrt-div 45.8%

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

       4. sqrt-prod 40.9%

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

       5. add-sqr-sqrt 59.2%

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

   12. Applied egg-rr 59.2%

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

   if -2.00000000000000003e306 < (*.f64 V l) < -1.97626e-322

   1. Initial program 83.9%

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

      1. frac-2neg 83.9%

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

      2. sqrt-div 98.4%

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

      3. *-commutative 98.4%

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

      4. distribute-rgt-neg-in 98.4%

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

   3. Applied egg-rr 98.4%

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

      1. associate-*r/ 95.2%

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

   5. Applied egg-rr 95.2%

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

      1. associate-/l* 98.5%

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

   7. Simplified 98.5%

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

   if -1.97626e-322 < (*.f64 V l) < 5.00000000000000033e-302 or 5.00000000000000023e222 < (*.f64 V l)

   1. Initial program 43.8%

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

      1. pow1/2 43.8%

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

      2. add-cube-cbrt 43.8%

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

      3. pow3 43.8%

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

      4. pow-pow 43.8%

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

      5. metadata-eval 43.8%

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

   3. Applied egg-rr 43.8%

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

      1. associate-/r* 72.5%

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

      2. cbrt-div 81.5%

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

   5. Applied egg-rr 81.5%

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

   if 5.00000000000000033e-302 < (*.f64 V l) < 5.00000000000000023e222

   1. Initial program 87.8%

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

      1. sqrt-div 99.5%

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

      2. associate-*r/ 96.5%

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

   3. Applied egg-rr 96.5%

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

      1. associate-/l* 99.4%

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

   5. Simplified 99.4%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+306}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-322}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \left(-\ell\right)}}{\sqrt{-A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-302} \lor \neg \left(V \cdot \ell \leq 5 \cdot 10^{+222}\right):\\ \;\;\;\;c0 \cdot {\left(\frac{\sqrt[3]{\frac{A}{V}}}{\sqrt[3]{\ell}}\right)}^{1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \end{array} \]

Alternative 3: 94.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+306}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-322}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \left(-\ell\right)}}{\sqrt{-A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq 2 \cdot 10^{+243}\right):\\ \;\;\;\;c0 \cdot {\left(\frac{\sqrt[3]{\frac{A}{\ell}}}{\sqrt[3]{V}}\right)}^{1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -2e+306)
   (* (/ c0 (sqrt l)) (sqrt (/ A V)))
   (if (<= (* V l) -2e-322)
     (/ c0 (/ (sqrt (* V (- l))) (sqrt (- A))))
     (if (or (<= (* V l) 0.0) (not (<= (* V l) 2e+243)))
       (* c0 (pow (/ (cbrt (/ A l)) (cbrt V)) 1.5))
       (/ c0 (/ (sqrt (* V l)) (sqrt A)))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e+306) {
		tmp = (c0 / sqrt(l)) * sqrt((A / V));
	} else if ((V * l) <= -2e-322) {
		tmp = c0 / (sqrt((V * -l)) / sqrt(-A));
	} else if (((V * l) <= 0.0) || !((V * l) <= 2e+243)) {
		tmp = c0 * pow((cbrt((A / l)) / cbrt(V)), 1.5);
	} else {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	}
	return tmp;
}

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e+306) {
		tmp = (c0 / Math.sqrt(l)) * Math.sqrt((A / V));
	} else if ((V * l) <= -2e-322) {
		tmp = c0 / (Math.sqrt((V * -l)) / Math.sqrt(-A));
	} else if (((V * l) <= 0.0) || !((V * l) <= 2e+243)) {
		tmp = c0 * Math.pow((Math.cbrt((A / l)) / Math.cbrt(V)), 1.5);
	} else {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	}
	return tmp;
}

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -2e+306)
		tmp = Float64(Float64(c0 / sqrt(l)) * sqrt(Float64(A / V)));
	elseif (Float64(V * l) <= -2e-322)
		tmp = Float64(c0 / Float64(sqrt(Float64(V * Float64(-l))) / sqrt(Float64(-A))));
	elseif ((Float64(V * l) <= 0.0) || !(Float64(V * l) <= 2e+243))
		tmp = Float64(c0 * (Float64(cbrt(Float64(A / l)) / cbrt(V)) ^ 1.5));
	else
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	end
	return tmp
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -2e+306], N[(N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-322], N[(c0 / N[(N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-A)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[Not[LessEqual[N[(V * l), $MachinePrecision], 2e+243]], $MachinePrecision]], N[(c0 * N[Power[N[(N[Power[N[(A / l), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[V, 1/3], $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision]), $MachinePrecision], N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -2.00000000000000003e306

   1. Initial program 31.7%

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

      1. add-cube-cbrt 31.7%

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

      2. *-commutative 31.7%

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

      3. times-frac 63.3%

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

      4. pow2 63.3%

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

   3. Applied egg-rr 63.3%

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

      1. frac-times 31.7%

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

      2. unpow2 31.7%

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

      3. add-cube-cbrt 31.7%

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

      4. *-commutative 31.7%

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

      5. *-un-lft-identity 31.7%

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

      6. associate-*l/ 31.7%

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

      7. add-sqr-sqrt 30.1%

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

      8. sqrt-unprod 30.3%

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

      9. *-commutative 30.3%

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

      10. *-commutative 30.3%

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

      11. swap-sqr 29.9%

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

   5. Applied egg-rr 29.9%

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

      1. *-commutative 29.9%

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

      2. rem-square-sqrt 29.9%

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

      3. swap-sqr 30.3%

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

      4. associate-*l* 30.3%

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

      5. *-commutative 30.3%

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

      6. associate-*l* 30.3%

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

      7. rem-square-sqrt 30.3%

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

      8. associate-*l/ 30.2%

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

      9. *-commutative 30.2%

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

   7. Simplified 30.2%

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

   8. Taylor expanded in c0 around 0 29.5%

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

      1. times-frac 58.6%

         \[\leadsto \sqrt{\color{blue}{\frac{A}{V} \cdot \frac{{c0}^{2}}{\ell}}} \]

      2. unpow2 58.6%

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

   10. Simplified 58.6%

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

       1. *-commutative 58.6%

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

       2. sqrt-prod 50.4%

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

       3. sqrt-div 45.8%

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

       4. sqrt-prod 40.9%

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

       5. add-sqr-sqrt 59.2%

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

   12. Applied egg-rr 59.2%

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

   if -2.00000000000000003e306 < (*.f64 V l) < -1.97626e-322

   1. Initial program 83.9%

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

      1. frac-2neg 83.9%

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

      2. sqrt-div 98.4%

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

      3. *-commutative 98.4%

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

      4. distribute-rgt-neg-in 98.4%

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

   3. Applied egg-rr 98.4%

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

      1. associate-*r/ 95.2%

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

   5. Applied egg-rr 95.2%

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

      1. associate-/l* 98.5%

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

   7. Simplified 98.5%

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

   if -1.97626e-322 < (*.f64 V l) < -0.0 or 2.0000000000000001e243 < (*.f64 V l)

   1. Initial program 41.2%

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

      1. pow1/2 41.2%

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

      2. add-cube-cbrt 41.2%

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

      3. pow3 41.2%

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

      4. pow-pow 41.2%

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

      5. metadata-eval 41.2%

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

   3. Applied egg-rr 41.2%

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

      1. add-cube-cbrt 41.2%

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

      2. unpow2 41.2%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\frac{\color{blue}{{\left(\sqrt[3]{A}\right)}^{2}} \cdot \sqrt[3]{A}}{V \cdot \ell}}\right)}^{1.5} \]

      3. *-commutative 41.2%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\frac{{\left(\sqrt[3]{A}\right)}^{2} \cdot \sqrt[3]{A}}{\color{blue}{\ell \cdot V}}}\right)}^{1.5} \]

      4. frac-times 72.2%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\color{blue}{\frac{{\left(\sqrt[3]{A}\right)}^{2}}{\ell} \cdot \frac{\sqrt[3]{A}}{V}}}\right)}^{1.5} \]

      5. associate-*r/ 72.2%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\color{blue}{\frac{\frac{{\left(\sqrt[3]{A}\right)}^{2}}{\ell} \cdot \sqrt[3]{A}}{V}}}\right)}^{1.5} \]

      6. cbrt-div 83.5%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{\sqrt[3]{\frac{{\left(\sqrt[3]{A}\right)}^{2}}{\ell} \cdot \sqrt[3]{A}}}{\sqrt[3]{V}}\right)}}^{1.5} \]

   5. Applied egg-rr 83.5%

      \[\leadsto c0 \cdot {\color{blue}{\left(\frac{\sqrt[3]{\frac{{\left(\sqrt[3]{A}\right)}^{2}}{\ell} \cdot \sqrt[3]{A}}}{\sqrt[3]{V}}\right)}}^{1.5} \]
   6. Step-by-step derivation

      1. associate-*l/ 83.6%

         \[\leadsto c0 \cdot {\left(\frac{\sqrt[3]{\color{blue}{\frac{{\left(\sqrt[3]{A}\right)}^{2} \cdot \sqrt[3]{A}}{\ell}}}}{\sqrt[3]{V}}\right)}^{1.5} \]

      2. unpow2 83.6%

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

      3. rem-3cbrt-lft 83.7%

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

   7. Simplified 83.7%

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

   if -0.0 < (*.f64 V l) < 2.0000000000000001e243

   1. Initial program 87.3%

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

      1. sqrt-div 98.7%

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

      2. associate-*r/ 95.9%

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

   3. Applied egg-rr 95.9%

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

      1. associate-/l* 98.6%

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

   5. Simplified 98.6%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+306}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-322}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \left(-\ell\right)}}{\sqrt{-A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq 2 \cdot 10^{+243}\right):\\ \;\;\;\;c0 \cdot {\left(\frac{\sqrt[3]{\frac{A}{\ell}}}{\sqrt[3]{V}}\right)}^{1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \end{array} \]

Alternative 4: 95.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{-322}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{-V} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq 2 \cdot 10^{+243}\right):\\ \;\;\;\;c0 \cdot {\left(\frac{\sqrt[3]{\frac{A}{\ell}}}{\sqrt[3]{V}}\right)}^{1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -2e-322)
   (* c0 (/ (sqrt (- A)) (* (sqrt (- V)) (sqrt l))))
   (if (or (<= (* V l) 0.0) (not (<= (* V l) 2e+243)))
     (* c0 (pow (/ (cbrt (/ A l)) (cbrt V)) 1.5))
     (/ c0 (/ (sqrt (* V l)) (sqrt A))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e-322) {
		tmp = c0 * (sqrt(-A) / (sqrt(-V) * sqrt(l)));
	} else if (((V * l) <= 0.0) || !((V * l) <= 2e+243)) {
		tmp = c0 * pow((cbrt((A / l)) / cbrt(V)), 1.5);
	} else {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	}
	return tmp;
}

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e-322) {
		tmp = c0 * (Math.sqrt(-A) / (Math.sqrt(-V) * Math.sqrt(l)));
	} else if (((V * l) <= 0.0) || !((V * l) <= 2e+243)) {
		tmp = c0 * Math.pow((Math.cbrt((A / l)) / Math.cbrt(V)), 1.5);
	} else {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	}
	return tmp;
}

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -2e-322)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / Float64(sqrt(Float64(-V)) * sqrt(l))));
	elseif ((Float64(V * l) <= 0.0) || !(Float64(V * l) <= 2e+243))
		tmp = Float64(c0 * (Float64(cbrt(Float64(A / l)) / cbrt(V)) ^ 1.5));
	else
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	end
	return tmp
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -2e-322], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[(N[Sqrt[(-V)], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[Not[LessEqual[N[(V * l), $MachinePrecision], 2e+243]], $MachinePrecision]], N[(c0 * N[Power[N[(N[Power[N[(A / l), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[V, 1/3], $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision]), $MachinePrecision], N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -1.97626e-322

   1. Initial program 73.7%

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

      1. frac-2neg 73.7%

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

      2. sqrt-div 85.9%

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

      3. *-commutative 85.9%

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

      4. distribute-rgt-neg-in 85.9%

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

   3. Applied egg-rr 85.9%

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

      1. pow1/2 85.9%

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

      2. *-commutative 85.9%

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

      3. unpow-prod-down 54.1%

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

      4. pow1/2 54.1%

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

      5. pow1/2 54.1%

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

   5. Applied egg-rr 54.1%

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

   if -1.97626e-322 < (*.f64 V l) < -0.0 or 2.0000000000000001e243 < (*.f64 V l)

   1. Initial program 41.2%

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

      1. pow1/2 41.2%

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

      2. add-cube-cbrt 41.2%

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

      3. pow3 41.2%

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

      4. pow-pow 41.2%

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

      5. metadata-eval 41.2%

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

   3. Applied egg-rr 41.2%

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

      1. add-cube-cbrt 41.2%

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

      2. unpow2 41.2%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\frac{\color{blue}{{\left(\sqrt[3]{A}\right)}^{2}} \cdot \sqrt[3]{A}}{V \cdot \ell}}\right)}^{1.5} \]

      3. *-commutative 41.2%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\frac{{\left(\sqrt[3]{A}\right)}^{2} \cdot \sqrt[3]{A}}{\color{blue}{\ell \cdot V}}}\right)}^{1.5} \]

      4. frac-times 72.2%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\color{blue}{\frac{{\left(\sqrt[3]{A}\right)}^{2}}{\ell} \cdot \frac{\sqrt[3]{A}}{V}}}\right)}^{1.5} \]

      5. associate-*r/ 72.2%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\color{blue}{\frac{\frac{{\left(\sqrt[3]{A}\right)}^{2}}{\ell} \cdot \sqrt[3]{A}}{V}}}\right)}^{1.5} \]

      6. cbrt-div 83.5%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{\sqrt[3]{\frac{{\left(\sqrt[3]{A}\right)}^{2}}{\ell} \cdot \sqrt[3]{A}}}{\sqrt[3]{V}}\right)}}^{1.5} \]

   5. Applied egg-rr 83.5%

      \[\leadsto c0 \cdot {\color{blue}{\left(\frac{\sqrt[3]{\frac{{\left(\sqrt[3]{A}\right)}^{2}}{\ell} \cdot \sqrt[3]{A}}}{\sqrt[3]{V}}\right)}}^{1.5} \]
   6. Step-by-step derivation

      1. associate-*l/ 83.6%

         \[\leadsto c0 \cdot {\left(\frac{\sqrt[3]{\color{blue}{\frac{{\left(\sqrt[3]{A}\right)}^{2} \cdot \sqrt[3]{A}}{\ell}}}}{\sqrt[3]{V}}\right)}^{1.5} \]

      2. unpow2 83.6%

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

      3. rem-3cbrt-lft 83.7%

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

   7. Simplified 83.7%

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

   if -0.0 < (*.f64 V l) < 2.0000000000000001e243

   1. Initial program 87.3%

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

      1. sqrt-div 98.7%

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

      2. associate-*r/ 95.9%

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

   3. Applied egg-rr 95.9%

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

      1. associate-/l* 98.6%

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

   5. Simplified 98.6%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{-322}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{-V} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq 2 \cdot 10^{+243}\right):\\ \;\;\;\;c0 \cdot {\left(\frac{\sqrt[3]{\frac{A}{\ell}}}{\sqrt[3]{V}}\right)}^{1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \end{array} \]

Alternative 5: 90.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-322}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+222}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) (- INFINITY))
   (* (/ c0 (sqrt l)) (sqrt (/ A V)))
   (if (<= (* V l) -2e-322)
     (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
     (if (<= (* V l) 0.0)
       (/ c0 (sqrt (* l (/ V A))))
       (if (<= (* V l) 5e+222)
         (/ c0 (/ (sqrt (* V l)) (sqrt A)))
         (* c0 (sqrt (/ (/ A l) V))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = (c0 / sqrt(l)) * sqrt((A / V));
	} else if ((V * l) <= -2e-322) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 / sqrt((l * (V / A)));
	} else if ((V * l) <= 5e+222) {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	} else {
		tmp = c0 * sqrt(((A / l) / V));
	}
	return tmp;
}

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = (c0 / Math.sqrt(l)) * Math.sqrt((A / V));
	} else if ((V * l) <= -2e-322) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 / Math.sqrt((l * (V / A)));
	} else if ((V * l) <= 5e+222) {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	} else {
		tmp = c0 * Math.sqrt(((A / l) / V));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = (c0 / math.sqrt(l)) * math.sqrt((A / V))
	elif (V * l) <= -2e-322:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 0.0:
		tmp = c0 / math.sqrt((l * (V / A)))
	elif (V * l) <= 5e+222:
		tmp = c0 / (math.sqrt((V * l)) / math.sqrt(A))
	else:
		tmp = c0 * math.sqrt(((A / l) / V))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(Float64(c0 / sqrt(l)) * sqrt(Float64(A / V)));
	elseif (Float64(V * l) <= -2e-322)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / A))));
	elseif (Float64(V * l) <= 5e+222)
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = (c0 / sqrt(l)) * sqrt((A / V));
	elseif ((V * l) <= -2e-322)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 0.0)
		tmp = c0 / sqrt((l * (V / A)));
	elseif ((V * l) <= 5e+222)
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	else
		tmp = c0 * sqrt(((A / l) / V));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-322], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e+222], N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 31.6%

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

      1. add-cube-cbrt 31.6%

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

      2. *-commutative 31.6%

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

      3. times-frac 64.7%

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

      4. pow2 64.7%

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

   3. Applied egg-rr 64.7%

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

      1. frac-times 31.6%

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

      2. unpow2 31.6%

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

      3. add-cube-cbrt 31.6%

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

      4. *-commutative 31.6%

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

      5. *-un-lft-identity 31.6%

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

      6. associate-*l/ 31.6%

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

      7. add-sqr-sqrt 31.6%

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

      8. sqrt-unprod 31.6%

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

      9. *-commutative 31.6%

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

      10. *-commutative 31.6%

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

      11. swap-sqr 31.2%

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

   5. Applied egg-rr 31.2%

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

      1. *-commutative 31.2%

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

      2. rem-square-sqrt 31.2%

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

      3. swap-sqr 31.6%

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

      4. associate-*l* 31.6%

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

      5. *-commutative 31.6%

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

      6. associate-*l* 31.6%

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

      7. rem-square-sqrt 31.6%

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

      8. associate-*l/ 31.5%

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

      9. *-commutative 31.5%

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

   7. Simplified 31.5%

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

   8. Taylor expanded in c0 around 0 30.7%

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

      1. times-frac 61.2%

         \[\leadsto \sqrt{\color{blue}{\frac{A}{V} \cdot \frac{{c0}^{2}}{\ell}}} \]

      2. unpow2 61.2%

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

   10. Simplified 61.2%

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

       1. *-commutative 61.2%

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

       2. sqrt-prod 52.7%

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

       3. sqrt-div 47.9%

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

       4. sqrt-prod 42.8%

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

       5. add-sqr-sqrt 57.3%

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

   12. Applied egg-rr 57.3%

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

   if -inf.0 < (*.f64 V l) < -1.97626e-322

   1. Initial program 83.4%

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

      1. frac-2neg 83.4%

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

      2. sqrt-div 98.4%

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

      3. *-commutative 98.4%

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

      4. distribute-rgt-neg-in 98.4%

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

   3. Applied egg-rr 98.4%

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

   if -1.97626e-322 < (*.f64 V l) < -0.0

   1. Initial program 41.3%

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

      1. add-cube-cbrt 41.3%

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

      2. *-commutative 41.3%

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

      3. times-frac 72.0%

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

      4. pow2 72.0%

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

   3. Applied egg-rr 72.0%

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

      1. frac-times 41.3%

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

      2. unpow2 41.3%

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

      3. add-cube-cbrt 41.3%

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

      4. *-commutative 41.3%

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

      5. *-un-lft-identity 41.3%

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

      6. associate-*l/ 41.3%

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

      7. sqrt-prod 20.6%

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

      8. associate-*r* 20.6%

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

      9. sqrt-div 20.6%

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

      10. metadata-eval 20.6%

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

      11. div-inv 20.6%

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

      12. associate-/r/ 20.6%

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

      13. sqrt-undiv 41.3%

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

      14. associate-*r/ 72.4%

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

   5. Applied egg-rr 72.4%

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

      1. associate-*r/ 41.3%

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

      2. associate-*l/ 72.4%

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

   7. Simplified 72.4%

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

   if -0.0 < (*.f64 V l) < 5.00000000000000023e222

   1. Initial program 87.2%

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

      1. sqrt-div 98.7%

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

      2. associate-*r/ 95.8%

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

   3. Applied egg-rr 95.8%

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

      1. associate-/l* 98.6%

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

   5. Simplified 98.6%

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

   if 5.00000000000000023e222 < (*.f64 V l)

   1. Initial program 44.1%

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

      1. pow1/2 44.1%

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

      2. add-cube-cbrt 44.2%

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

      3. pow3 44.2%

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

      4. pow-pow 44.1%

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

      5. metadata-eval 44.1%

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

   3. Applied egg-rr 44.1%

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

      1. add-cube-cbrt 43.9%

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

      2. unpow2 43.9%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\frac{\color{blue}{{\left(\sqrt[3]{A}\right)}^{2}} \cdot \sqrt[3]{A}}{V \cdot \ell}}\right)}^{1.5} \]

      3. *-commutative 43.9%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\frac{{\left(\sqrt[3]{A}\right)}^{2} \cdot \sqrt[3]{A}}{\color{blue}{\ell \cdot V}}}\right)}^{1.5} \]

      4. frac-times 73.9%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\color{blue}{\frac{{\left(\sqrt[3]{A}\right)}^{2}}{\ell} \cdot \frac{\sqrt[3]{A}}{V}}}\right)}^{1.5} \]

      5. cbrt-prod 83.8%

         \[\leadsto c0 \cdot {\color{blue}{\left(\sqrt[3]{\frac{{\left(\sqrt[3]{A}\right)}^{2}}{\ell}} \cdot \sqrt[3]{\frac{\sqrt[3]{A}}{V}}\right)}}^{1.5} \]

   5. Applied egg-rr 83.8%

      \[\leadsto c0 \cdot {\color{blue}{\left(\sqrt[3]{\frac{{\left(\sqrt[3]{A}\right)}^{2}}{\ell}} \cdot \sqrt[3]{\frac{\sqrt[3]{A}}{V}}\right)}}^{1.5} \]

   6. Taylor expanded in c0 around 0 44.1%

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

      1. *-commutative 44.1%

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

      2. associate-/l/ 74.3%

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

   8. Simplified 74.3%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-322}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+222}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]

Alternative 6: 90.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+306}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-322}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \left(-\ell\right)}}{\sqrt{-A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+222}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -2e+306)
   (* (/ c0 (sqrt l)) (sqrt (/ A V)))
   (if (<= (* V l) -2e-322)
     (/ c0 (/ (sqrt (* V (- l))) (sqrt (- A))))
     (if (<= (* V l) 0.0)
       (/ c0 (sqrt (* l (/ V A))))
       (if (<= (* V l) 5e+222)
         (/ c0 (/ (sqrt (* V l)) (sqrt A)))
         (* c0 (sqrt (/ (/ A l) V))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e+306) {
		tmp = (c0 / sqrt(l)) * sqrt((A / V));
	} else if ((V * l) <= -2e-322) {
		tmp = c0 / (sqrt((V * -l)) / sqrt(-A));
	} else if ((V * l) <= 0.0) {
		tmp = c0 / sqrt((l * (V / A)));
	} else if ((V * l) <= 5e+222) {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	} else {
		tmp = c0 * sqrt(((A / l) / V));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-2d+306)) then
        tmp = (c0 / sqrt(l)) * sqrt((a / v))
    else if ((v * l) <= (-2d-322)) then
        tmp = c0 / (sqrt((v * -l)) / sqrt(-a))
    else if ((v * l) <= 0.0d0) then
        tmp = c0 / sqrt((l * (v / a)))
    else if ((v * l) <= 5d+222) then
        tmp = c0 / (sqrt((v * l)) / sqrt(a))
    else
        tmp = c0 * sqrt(((a / l) / v))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e+306) {
		tmp = (c0 / Math.sqrt(l)) * Math.sqrt((A / V));
	} else if ((V * l) <= -2e-322) {
		tmp = c0 / (Math.sqrt((V * -l)) / Math.sqrt(-A));
	} else if ((V * l) <= 0.0) {
		tmp = c0 / Math.sqrt((l * (V / A)));
	} else if ((V * l) <= 5e+222) {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	} else {
		tmp = c0 * Math.sqrt(((A / l) / V));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -2e+306:
		tmp = (c0 / math.sqrt(l)) * math.sqrt((A / V))
	elif (V * l) <= -2e-322:
		tmp = c0 / (math.sqrt((V * -l)) / math.sqrt(-A))
	elif (V * l) <= 0.0:
		tmp = c0 / math.sqrt((l * (V / A)))
	elif (V * l) <= 5e+222:
		tmp = c0 / (math.sqrt((V * l)) / math.sqrt(A))
	else:
		tmp = c0 * math.sqrt(((A / l) / V))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -2e+306)
		tmp = Float64(Float64(c0 / sqrt(l)) * sqrt(Float64(A / V)));
	elseif (Float64(V * l) <= -2e-322)
		tmp = Float64(c0 / Float64(sqrt(Float64(V * Float64(-l))) / sqrt(Float64(-A))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / A))));
	elseif (Float64(V * l) <= 5e+222)
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -2e+306)
		tmp = (c0 / sqrt(l)) * sqrt((A / V));
	elseif ((V * l) <= -2e-322)
		tmp = c0 / (sqrt((V * -l)) / sqrt(-A));
	elseif ((V * l) <= 0.0)
		tmp = c0 / sqrt((l * (V / A)));
	elseif ((V * l) <= 5e+222)
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	else
		tmp = c0 * sqrt(((A / l) / V));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -2e+306], N[(N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-322], N[(c0 / N[(N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-A)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e+222], N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -2.00000000000000003e306

   1. Initial program 31.7%

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

      1. add-cube-cbrt 31.7%

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

      2. *-commutative 31.7%

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

      3. times-frac 63.3%

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

      4. pow2 63.3%

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

   3. Applied egg-rr 63.3%

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

      1. frac-times 31.7%

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

      2. unpow2 31.7%

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

      3. add-cube-cbrt 31.7%

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

      4. *-commutative 31.7%

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

      5. *-un-lft-identity 31.7%

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

      6. associate-*l/ 31.7%

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

      7. add-sqr-sqrt 30.1%

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

      8. sqrt-unprod 30.3%

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

      9. *-commutative 30.3%

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

      10. *-commutative 30.3%

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

      11. swap-sqr 29.9%

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

   5. Applied egg-rr 29.9%

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

      1. *-commutative 29.9%

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

      2. rem-square-sqrt 29.9%

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

      3. swap-sqr 30.3%

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

      4. associate-*l* 30.3%

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

      5. *-commutative 30.3%

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

      6. associate-*l* 30.3%

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

      7. rem-square-sqrt 30.3%

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

      8. associate-*l/ 30.2%

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

      9. *-commutative 30.2%

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

   7. Simplified 30.2%

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

   8. Taylor expanded in c0 around 0 29.5%

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

      1. times-frac 58.6%

         \[\leadsto \sqrt{\color{blue}{\frac{A}{V} \cdot \frac{{c0}^{2}}{\ell}}} \]

      2. unpow2 58.6%

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

   10. Simplified 58.6%

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

       1. *-commutative 58.6%

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

       2. sqrt-prod 50.4%

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

       3. sqrt-div 45.8%

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

       4. sqrt-prod 40.9%

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

       5. add-sqr-sqrt 59.2%

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

   12. Applied egg-rr 59.2%

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

   if -2.00000000000000003e306 < (*.f64 V l) < -1.97626e-322

   1. Initial program 83.9%

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

      1. frac-2neg 83.9%

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

      2. sqrt-div 98.4%

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

      3. *-commutative 98.4%

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

      4. distribute-rgt-neg-in 98.4%

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

   3. Applied egg-rr 98.4%

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

      1. associate-*r/ 95.2%

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

   5. Applied egg-rr 95.2%

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

      1. associate-/l* 98.5%

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

   7. Simplified 98.5%

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

   if -1.97626e-322 < (*.f64 V l) < -0.0

   1. Initial program 41.3%

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

      1. add-cube-cbrt 41.3%

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

      2. *-commutative 41.3%

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

      3. times-frac 72.0%

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

      4. pow2 72.0%

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

   3. Applied egg-rr 72.0%

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

      1. frac-times 41.3%

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

      2. unpow2 41.3%

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

      3. add-cube-cbrt 41.3%

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

      4. *-commutative 41.3%

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

      5. *-un-lft-identity 41.3%

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

      6. associate-*l/ 41.3%

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

      7. sqrt-prod 20.6%

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

      8. associate-*r* 20.6%

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

      9. sqrt-div 20.6%

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

      10. metadata-eval 20.6%

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

      11. div-inv 20.6%

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

      12. associate-/r/ 20.6%

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

      13. sqrt-undiv 41.3%

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

      14. associate-*r/ 72.4%

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

   5. Applied egg-rr 72.4%

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

      1. associate-*r/ 41.3%

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

      2. associate-*l/ 72.4%

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

   7. Simplified 72.4%

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

   if -0.0 < (*.f64 V l) < 5.00000000000000023e222

   1. Initial program 87.2%

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

      1. sqrt-div 98.7%

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

      2. associate-*r/ 95.8%

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

   3. Applied egg-rr 95.8%

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

      1. associate-/l* 98.6%

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

   5. Simplified 98.6%

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

   if 5.00000000000000023e222 < (*.f64 V l)

   1. Initial program 44.1%

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

      1. pow1/2 44.1%

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

      2. add-cube-cbrt 44.2%

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

      3. pow3 44.2%

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

      4. pow-pow 44.1%

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

      5. metadata-eval 44.1%

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

   3. Applied egg-rr 44.1%

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

      1. add-cube-cbrt 43.9%

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

      2. unpow2 43.9%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\frac{\color{blue}{{\left(\sqrt[3]{A}\right)}^{2}} \cdot \sqrt[3]{A}}{V \cdot \ell}}\right)}^{1.5} \]

      3. *-commutative 43.9%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\frac{{\left(\sqrt[3]{A}\right)}^{2} \cdot \sqrt[3]{A}}{\color{blue}{\ell \cdot V}}}\right)}^{1.5} \]

      4. frac-times 73.9%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\color{blue}{\frac{{\left(\sqrt[3]{A}\right)}^{2}}{\ell} \cdot \frac{\sqrt[3]{A}}{V}}}\right)}^{1.5} \]

      5. cbrt-prod 83.8%

         \[\leadsto c0 \cdot {\color{blue}{\left(\sqrt[3]{\frac{{\left(\sqrt[3]{A}\right)}^{2}}{\ell}} \cdot \sqrt[3]{\frac{\sqrt[3]{A}}{V}}\right)}}^{1.5} \]

   5. Applied egg-rr 83.8%

      \[\leadsto c0 \cdot {\color{blue}{\left(\sqrt[3]{\frac{{\left(\sqrt[3]{A}\right)}^{2}}{\ell}} \cdot \sqrt[3]{\frac{\sqrt[3]{A}}{V}}\right)}}^{1.5} \]

   6. Taylor expanded in c0 around 0 44.1%

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

      1. *-commutative 44.1%

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

      2. associate-/l/ 74.3%

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

   8. Simplified 74.3%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+306}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-322}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \left(-\ell\right)}}{\sqrt{-A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+222}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]

Alternative 7: 81.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= (-5d-310)) then
        tmp = c0 * sqrt(((a / l) / v))
    else
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    end if
    code = tmp
end function

Java

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

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if l <= -5e-310:
		tmp = c0 * math.sqrt(((A / l) / V))
	else:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	return tmp

Julia

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

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (l <= -5e-310)
		tmp = c0 * sqrt(((A / l) / V));
	else
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 72.9%

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

      1. pow1/2 72.9%

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

      2. add-cube-cbrt 72.6%

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

      3. pow3 72.6%

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

      4. pow-pow 72.6%

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

      5. metadata-eval 72.6%

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

   3. Applied egg-rr 72.6%

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

      1. add-cube-cbrt 72.4%

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

      2. unpow2 72.4%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\frac{\color{blue}{{\left(\sqrt[3]{A}\right)}^{2}} \cdot \sqrt[3]{A}}{V \cdot \ell}}\right)}^{1.5} \]

      3. *-commutative 72.4%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\frac{{\left(\sqrt[3]{A}\right)}^{2} \cdot \sqrt[3]{A}}{\color{blue}{\ell \cdot V}}}\right)}^{1.5} \]

      4. frac-times 72.5%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\color{blue}{\frac{{\left(\sqrt[3]{A}\right)}^{2}}{\ell} \cdot \frac{\sqrt[3]{A}}{V}}}\right)}^{1.5} \]

      5. cbrt-prod 82.0%

         \[\leadsto c0 \cdot {\color{blue}{\left(\sqrt[3]{\frac{{\left(\sqrt[3]{A}\right)}^{2}}{\ell}} \cdot \sqrt[3]{\frac{\sqrt[3]{A}}{V}}\right)}}^{1.5} \]

   5. Applied egg-rr 82.0%

      \[\leadsto c0 \cdot {\color{blue}{\left(\sqrt[3]{\frac{{\left(\sqrt[3]{A}\right)}^{2}}{\ell}} \cdot \sqrt[3]{\frac{\sqrt[3]{A}}{V}}\right)}}^{1.5} \]

   6. Taylor expanded in c0 around 0 72.9%

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

      1. *-commutative 72.9%

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

      2. associate-/l/ 72.6%

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

   8. Simplified 72.6%

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

   if -4.999999999999985e-310 < l

   1. Initial program 72.2%

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

      1. associate-/r* 76.5%

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

      2. sqrt-div 86.4%

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

   3. Applied egg-rr 86.4%

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

4. Final simplification 80.2%

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

Alternative 8: 83.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= (-5d-310)) then
        tmp = c0 / (sqrt((v * l)) / sqrt(a))
    else
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    end if
    code = tmp
end function

Java

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

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if l <= -5e-310:
		tmp = c0 / (math.sqrt((V * l)) / math.sqrt(A))
	else:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	return tmp

Julia

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

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (l <= -5e-310)
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	else
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 72.9%

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

      1. sqrt-div 42.8%

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

      2. associate-*r/ 42.7%

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

   3. Applied egg-rr 42.7%

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

      1. associate-/l* 42.8%

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

   5. Simplified 42.8%

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

   if -4.999999999999985e-310 < l

   1. Initial program 72.2%

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

      1. associate-/r* 76.5%

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

      2. sqrt-div 86.4%

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

   3. Applied egg-rr 86.4%

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

4. Final simplification 66.8%

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

Alternative 9: 78.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+250}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{1}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (<= t_0 0.0)
     (* c0 (sqrt (/ (/ A l) V)))
     (if (<= t_0 2e+250)
       (* c0 (sqrt t_0))
       (* c0 (/ 1.0 (sqrt (* V (/ l A)))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * sqrt(((A / l) / V));
	} else if (t_0 <= 2e+250) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = c0 * (1.0 / sqrt((V * (l / A))));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if (t_0 <= 0.0d0) then
        tmp = c0 * sqrt(((a / l) / v))
    else if (t_0 <= 2d+250) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = c0 * (1.0d0 / sqrt((v * (l / a))))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * Math.sqrt(((A / l) / V));
	} else if (t_0 <= 2e+250) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = c0 * (1.0 / Math.sqrt((V * (l / A))));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0 * math.sqrt(((A / l) / V))
	elif t_0 <= 2e+250:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = c0 * (1.0 / math.sqrt((V * (l / A))))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	elseif (t_0 <= 2e+250)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 * Float64(1.0 / sqrt(Float64(V * Float64(l / A)))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0 * sqrt(((A / l) / V));
	elseif (t_0 <= 2e+250)
		tmp = c0 * sqrt(t_0);
	else
		tmp = c0 * (1.0 / sqrt((V * (l / A))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+250], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(1.0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 31.4%

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

      1. pow1/2 31.4%

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

      2. add-cube-cbrt 31.4%

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

      3. pow3 31.4%

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

      4. pow-pow 31.4%

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

      5. metadata-eval 31.4%

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

   3. Applied egg-rr 31.4%

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

      1. add-cube-cbrt 31.4%

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

      2. unpow2 31.4%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\frac{\color{blue}{{\left(\sqrt[3]{A}\right)}^{2}} \cdot \sqrt[3]{A}}{V \cdot \ell}}\right)}^{1.5} \]

      3. *-commutative 31.4%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\frac{{\left(\sqrt[3]{A}\right)}^{2} \cdot \sqrt[3]{A}}{\color{blue}{\ell \cdot V}}}\right)}^{1.5} \]

      4. frac-times 57.1%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\color{blue}{\frac{{\left(\sqrt[3]{A}\right)}^{2}}{\ell} \cdot \frac{\sqrt[3]{A}}{V}}}\right)}^{1.5} \]

      5. cbrt-prod 82.9%

         \[\leadsto c0 \cdot {\color{blue}{\left(\sqrt[3]{\frac{{\left(\sqrt[3]{A}\right)}^{2}}{\ell}} \cdot \sqrt[3]{\frac{\sqrt[3]{A}}{V}}\right)}}^{1.5} \]

   5. Applied egg-rr 82.9%

      \[\leadsto c0 \cdot {\color{blue}{\left(\sqrt[3]{\frac{{\left(\sqrt[3]{A}\right)}^{2}}{\ell}} \cdot \sqrt[3]{\frac{\sqrt[3]{A}}{V}}\right)}}^{1.5} \]

   6. Taylor expanded in c0 around 0 31.4%

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

      1. *-commutative 31.4%

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

      2. associate-/l/ 57.4%

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

   8. Simplified 57.4%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 1.9999999999999998e250

   1. Initial program 98.4%

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

   if 1.9999999999999998e250 < (/.f64 A (*.f64 V l))

   1. Initial program 47.7%

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

      1. clear-num 47.7%

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

      2. sqrt-div 48.9%

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

      3. metadata-eval 48.9%

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

      4. associate-/l* 62.0%

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

   3. Applied egg-rr 62.0%

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

      1. associate-/l* 48.9%

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

      2. *-lft-identity 48.9%

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

      3. times-frac 62.0%

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

      4. /-rgt-identity 62.0%

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

   5. Simplified 62.0%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{+250}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{1}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]

Alternative 10: 79.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0 \lor \neg \left(t_0 \leq 5 \cdot 10^{+288}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (or (<= t_0 0.0) (not (<= t_0 5e+288)))
     (* c0 (sqrt (/ (/ A V) l)))
     (* c0 (sqrt t_0)))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 5e+288)) {
		tmp = c0 * sqrt(((A / V) / l));
	} else {
		tmp = c0 * sqrt(t_0);
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if ((t_0 <= 0.0d0) .or. (.not. (t_0 <= 5d+288))) then
        tmp = c0 * sqrt(((a / v) / l))
    else
        tmp = c0 * sqrt(t_0)
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 5e+288)) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else {
		tmp = c0 * Math.sqrt(t_0);
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 5e+288):
		tmp = c0 * math.sqrt(((A / V) / l))
	else:
		tmp = c0 * math.sqrt(t_0)
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 5e+288))
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	else
		tmp = Float64(c0 * sqrt(t_0));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 5e+288)))
		tmp = c0 * sqrt(((A / V) / l));
	else
		tmp = c0 * sqrt(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 5e+288]], $MachinePrecision]], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 0.0 or 5.0000000000000003e288 < (/.f64 A (*.f64 V l))

   1. Initial program 35.1%

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

      1. add-cube-cbrt 35.1%

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

      2. *-commutative 35.1%

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

      3. times-frac 56.3%

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

      4. pow2 56.3%

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

   3. Applied egg-rr 56.3%

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

      1. frac-times 35.1%

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

      2. unpow2 35.1%

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

      3. add-cube-cbrt 35.1%

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

      4. *-commutative 35.1%

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

      5. associate-/r* 56.4%

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

   5. Applied egg-rr 56.4%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 5.0000000000000003e288

   1. Initial program 98.5%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0 \lor \neg \left(\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{+288}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]

Alternative 11: 78.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0 \lor \neg \left(t_0 \leq 2 \cdot 10^{+250}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (or (<= t_0 0.0) (not (<= t_0 2e+250)))
     (* c0 (sqrt (/ (/ A l) V)))
     (* c0 (sqrt t_0)))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 2e+250)) {
		tmp = c0 * sqrt(((A / l) / V));
	} else {
		tmp = c0 * sqrt(t_0);
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if ((t_0 <= 0.0d0) .or. (.not. (t_0 <= 2d+250))) then
        tmp = c0 * sqrt(((a / l) / v))
    else
        tmp = c0 * sqrt(t_0)
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 2e+250)) {
		tmp = c0 * Math.sqrt(((A / l) / V));
	} else {
		tmp = c0 * Math.sqrt(t_0);
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 2e+250):
		tmp = c0 * math.sqrt(((A / l) / V))
	else:
		tmp = c0 * math.sqrt(t_0)
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 2e+250))
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	else
		tmp = Float64(c0 * sqrt(t_0));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 2e+250)))
		tmp = c0 * sqrt(((A / l) / V));
	else
		tmp = c0 * sqrt(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 2e+250]], $MachinePrecision]], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 0.0 or 1.9999999999999998e250 < (/.f64 A (*.f64 V l))

   1. Initial program 40.8%

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

      1. pow1/2 40.8%

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

      2. add-cube-cbrt 40.7%

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

      3. pow3 40.7%

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

      4. pow-pow 40.7%

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

      5. metadata-eval 40.7%

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

   3. Applied egg-rr 40.7%

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

      1. add-cube-cbrt 40.6%

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

      2. unpow2 40.6%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\frac{\color{blue}{{\left(\sqrt[3]{A}\right)}^{2}} \cdot \sqrt[3]{A}}{V \cdot \ell}}\right)}^{1.5} \]

      3. *-commutative 40.6%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\frac{{\left(\sqrt[3]{A}\right)}^{2} \cdot \sqrt[3]{A}}{\color{blue}{\ell \cdot V}}}\right)}^{1.5} \]

      4. frac-times 58.3%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\color{blue}{\frac{{\left(\sqrt[3]{A}\right)}^{2}}{\ell} \cdot \frac{\sqrt[3]{A}}{V}}}\right)}^{1.5} \]

      5. cbrt-prod 79.8%

         \[\leadsto c0 \cdot {\color{blue}{\left(\sqrt[3]{\frac{{\left(\sqrt[3]{A}\right)}^{2}}{\ell}} \cdot \sqrt[3]{\frac{\sqrt[3]{A}}{V}}\right)}}^{1.5} \]

   5. Applied egg-rr 79.8%

      \[\leadsto c0 \cdot {\color{blue}{\left(\sqrt[3]{\frac{{\left(\sqrt[3]{A}\right)}^{2}}{\ell}} \cdot \sqrt[3]{\frac{\sqrt[3]{A}}{V}}\right)}}^{1.5} \]

   6. Taylor expanded in c0 around 0 40.8%

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

      1. *-commutative 40.8%

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

      2. associate-/l/ 59.3%

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

   8. Simplified 59.3%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 1.9999999999999998e250

   1. Initial program 98.4%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0 \lor \neg \left(\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{+250}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]

Alternative 12: 79.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+275}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (<= t_0 0.0)
     (* c0 (sqrt (/ (/ A l) V)))
     (if (<= t_0 2e+275) (* c0 (sqrt t_0)) (/ c0 (sqrt (* V (/ l A))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * sqrt(((A / l) / V));
	} else if (t_0 <= 2e+275) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = c0 / sqrt((V * (l / A)));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if (t_0 <= 0.0d0) then
        tmp = c0 * sqrt(((a / l) / v))
    else if (t_0 <= 2d+275) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = c0 / sqrt((v * (l / a)))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * Math.sqrt(((A / l) / V));
	} else if (t_0 <= 2e+275) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = c0 / Math.sqrt((V * (l / A)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0 * math.sqrt(((A / l) / V))
	elif t_0 <= 2e+275:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = c0 / math.sqrt((V * (l / A)))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	elseif (t_0 <= 2e+275)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0 * sqrt(((A / l) / V));
	elseif (t_0 <= 2e+275)
		tmp = c0 * sqrt(t_0);
	else
		tmp = c0 / sqrt((V * (l / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+275], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 31.4%

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

      1. pow1/2 31.4%

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

      2. add-cube-cbrt 31.4%

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

      3. pow3 31.4%

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

      4. pow-pow 31.4%

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

      5. metadata-eval 31.4%

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

   3. Applied egg-rr 31.4%

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

      1. add-cube-cbrt 31.4%

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

      2. unpow2 31.4%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\frac{\color{blue}{{\left(\sqrt[3]{A}\right)}^{2}} \cdot \sqrt[3]{A}}{V \cdot \ell}}\right)}^{1.5} \]

      3. *-commutative 31.4%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\frac{{\left(\sqrt[3]{A}\right)}^{2} \cdot \sqrt[3]{A}}{\color{blue}{\ell \cdot V}}}\right)}^{1.5} \]

      4. frac-times 57.1%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\color{blue}{\frac{{\left(\sqrt[3]{A}\right)}^{2}}{\ell} \cdot \frac{\sqrt[3]{A}}{V}}}\right)}^{1.5} \]

      5. cbrt-prod 82.9%

         \[\leadsto c0 \cdot {\color{blue}{\left(\sqrt[3]{\frac{{\left(\sqrt[3]{A}\right)}^{2}}{\ell}} \cdot \sqrt[3]{\frac{\sqrt[3]{A}}{V}}\right)}}^{1.5} \]

   5. Applied egg-rr 82.9%

      \[\leadsto c0 \cdot {\color{blue}{\left(\sqrt[3]{\frac{{\left(\sqrt[3]{A}\right)}^{2}}{\ell}} \cdot \sqrt[3]{\frac{\sqrt[3]{A}}{V}}\right)}}^{1.5} \]

   6. Taylor expanded in c0 around 0 31.4%

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

      1. *-commutative 31.4%

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

      2. associate-/l/ 57.4%

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

   8. Simplified 57.4%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 1.99999999999999992e275

   1. Initial program 98.5%

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

   if 1.99999999999999992e275 < (/.f64 A (*.f64 V l))

   1. Initial program 40.5%

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

      1. add-cube-cbrt 40.5%

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

      2. *-commutative 40.5%

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

      3. times-frac 55.2%

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

      4. pow2 55.2%

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

   3. Applied egg-rr 55.2%

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

      1. frac-times 40.5%

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

      2. unpow2 40.5%

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

      3. add-cube-cbrt 40.5%

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

      4. *-commutative 40.5%

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

      5. *-un-lft-identity 40.5%

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

      6. associate-*l/ 40.5%

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

      7. sqrt-prod 31.6%

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

      8. associate-*r* 31.6%

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

      9. sqrt-div 32.0%

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

      10. metadata-eval 32.0%

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

      11. div-inv 32.0%

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

      12. associate-/r/ 32.0%

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

      13. sqrt-undiv 41.8%

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

      14. associate-*r/ 56.8%

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

   5. Applied egg-rr 56.8%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{+275}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]

Alternative 13: 79.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+288}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (<= t_0 0.0)
     (* c0 (sqrt (/ (/ A l) V)))
     (if (<= t_0 5e+288) (* c0 (sqrt t_0)) (/ c0 (sqrt (* l (/ V A))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * sqrt(((A / l) / V));
	} else if (t_0 <= 5e+288) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = c0 / sqrt((l * (V / A)));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if (t_0 <= 0.0d0) then
        tmp = c0 * sqrt(((a / l) / v))
    else if (t_0 <= 5d+288) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = c0 / sqrt((l * (v / a)))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * Math.sqrt(((A / l) / V));
	} else if (t_0 <= 5e+288) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = c0 / Math.sqrt((l * (V / A)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0 * math.sqrt(((A / l) / V))
	elif t_0 <= 5e+288:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = c0 / math.sqrt((l * (V / A)))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	elseif (t_0 <= 5e+288)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / A))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0 * sqrt(((A / l) / V));
	elseif (t_0 <= 5e+288)
		tmp = c0 * sqrt(t_0);
	else
		tmp = c0 / sqrt((l * (V / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+288], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 31.4%

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

      1. pow1/2 31.4%

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

      2. add-cube-cbrt 31.4%

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

      3. pow3 31.4%

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

      4. pow-pow 31.4%

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

      5. metadata-eval 31.4%

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

   3. Applied egg-rr 31.4%

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

      1. add-cube-cbrt 31.4%

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

      2. unpow2 31.4%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\frac{\color{blue}{{\left(\sqrt[3]{A}\right)}^{2}} \cdot \sqrt[3]{A}}{V \cdot \ell}}\right)}^{1.5} \]

      3. *-commutative 31.4%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\frac{{\left(\sqrt[3]{A}\right)}^{2} \cdot \sqrt[3]{A}}{\color{blue}{\ell \cdot V}}}\right)}^{1.5} \]

      4. frac-times 57.1%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\color{blue}{\frac{{\left(\sqrt[3]{A}\right)}^{2}}{\ell} \cdot \frac{\sqrt[3]{A}}{V}}}\right)}^{1.5} \]

      5. cbrt-prod 82.9%

         \[\leadsto c0 \cdot {\color{blue}{\left(\sqrt[3]{\frac{{\left(\sqrt[3]{A}\right)}^{2}}{\ell}} \cdot \sqrt[3]{\frac{\sqrt[3]{A}}{V}}\right)}}^{1.5} \]

   5. Applied egg-rr 82.9%

      \[\leadsto c0 \cdot {\color{blue}{\left(\sqrt[3]{\frac{{\left(\sqrt[3]{A}\right)}^{2}}{\ell}} \cdot \sqrt[3]{\frac{\sqrt[3]{A}}{V}}\right)}}^{1.5} \]

   6. Taylor expanded in c0 around 0 31.4%

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

      1. *-commutative 31.4%

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

      2. associate-/l/ 57.4%

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

   8. Simplified 57.4%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 5.0000000000000003e288

   1. Initial program 98.5%

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

   if 5.0000000000000003e288 < (/.f64 A (*.f64 V l))

   1. Initial program 38.4%

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

      1. add-cube-cbrt 38.4%

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

      2. *-commutative 38.4%

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

      3. times-frac 55.3%

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

      4. pow2 55.3%

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

   3. Applied egg-rr 55.3%

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

      1. frac-times 38.4%

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

      2. unpow2 38.4%

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

      3. add-cube-cbrt 38.4%

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

      4. *-commutative 38.4%

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

      5. *-un-lft-identity 38.4%

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

      6. associate-*l/ 38.4%

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

      7. sqrt-prod 32.8%

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

      8. associate-*r* 32.8%

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

      9. sqrt-div 33.1%

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

      10. metadata-eval 33.1%

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

      11. div-inv 33.2%

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

      12. associate-/r/ 33.2%

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

      13. sqrt-undiv 39.8%

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

      14. associate-*r/ 57.0%

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

   5. Applied egg-rr 57.0%

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

      1. associate-*r/ 39.8%

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

      2. associate-*l/ 57.0%

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

   7. Simplified 57.0%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{+288}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \]

Alternative 14: 72.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.5%

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

2. Final simplification 72.5%

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

Reproduce

herbie shell --seed 2023174 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))