Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.9% → 88.8%

Time: 9.4s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-300}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+228}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{V} \cdot \frac{\sqrt{\ell}}{c0}}\\ \end{array} \end{array} \]

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 (/ (- A) l)) (sqrt (- V))))
   (if (<= (* V l) -1e-300)
     (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
     (if (<= (* V l) 0.0)
       (* c0 (/ (sqrt (/ A V)) (sqrt l)))
       (if (<= (* V l) 1e+228)
         (* c0 (* (sqrt A) (pow (* V l) -0.5)))
         (/ (sqrt A) (* (sqrt V) (/ (sqrt l) c0))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = c0 * (sqrt((-A / l)) / sqrt(-V));
	} else if ((V * l) <= -1e-300) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else if ((V * l) <= 1e+228) {
		tmp = c0 * (sqrt(A) * pow((V * l), -0.5));
	} else {
		tmp = sqrt(A) / (sqrt(V) * (sqrt(l) / c0));
	}
	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((-A / l)) / Math.sqrt(-V));
	} else if ((V * l) <= -1e-300) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if ((V * l) <= 1e+228) {
		tmp = c0 * (Math.sqrt(A) * Math.pow((V * l), -0.5));
	} else {
		tmp = Math.sqrt(A) / (Math.sqrt(V) * (Math.sqrt(l) / c0));
	}
	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((-A / l)) / math.sqrt(-V))
	elif (V * l) <= -1e-300:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 0.0:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	elif (V * l) <= 1e+228:
		tmp = c0 * (math.sqrt(A) * math.pow((V * l), -0.5))
	else:
		tmp = math.sqrt(A) / (math.sqrt(V) * (math.sqrt(l) / c0))
	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(c0 * Float64(sqrt(Float64(Float64(-A) / l)) / sqrt(Float64(-V))));
	elseif (Float64(V * l) <= -1e-300)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (Float64(V * l) <= 1e+228)
		tmp = Float64(c0 * Float64(sqrt(A) * (Float64(V * l) ^ -0.5)));
	else
		tmp = Float64(sqrt(A) / Float64(sqrt(V) * Float64(sqrt(l) / c0)));
	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((-A / l)) / sqrt(-V));
	elseif ((V * l) <= -1e-300)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 0.0)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	elseif ((V * l) <= 1e+228)
		tmp = c0 * (sqrt(A) * ((V * l) ^ -0.5));
	else
		tmp = sqrt(A) / (sqrt(V) * (sqrt(l) / c0));
	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[(c0 * N[(N[Sqrt[N[((-A) / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-300], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+228], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Power[N[(V * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[A], $MachinePrecision] / N[(N[Sqrt[V], $MachinePrecision] * N[(N[Sqrt[l], $MachinePrecision] / c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 58.4%

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

      1. associate-/r* 70.4%

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

      2. div-inv 70.4%

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

   3. Applied egg-rr 70.4%

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

      1. un-div-inv 70.4%

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

   5. Applied egg-rr 70.4%

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

      1. associate-/l/ 58.4%

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

      2. *-commutative 58.4%

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

      3. associate-/l/ 70.3%

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

      4. frac-2neg 70.3%

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

      5. sqrt-div 50.3%

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

      6. distribute-neg-frac 50.3%

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

   7. Applied egg-rr 50.3%

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

   if -inf.0 < (*.f64 V l) < -1.00000000000000003e-300

   1. Initial program 83.8%

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

      1. frac-2neg 83.8%

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

      2. sqrt-div 99.5%

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

      3. *-commutative 99.5%

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

      4. distribute-rgt-neg-in 99.5%

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

   3. Applied egg-rr 99.5%

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

   if -1.00000000000000003e-300 < (*.f64 V l) < 0.0

   1. Initial program 41.4%

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

      1. associate-/r* 67.3%

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

      2. sqrt-div 55.2%

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

   3. Applied egg-rr 55.2%

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

   if 0.0 < (*.f64 V l) < 9.9999999999999992e227

   1. Initial program 89.4%

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

      1. div-inv 89.0%

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

      2. sqrt-prod 98.5%

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

      3. pow1/2 98.5%

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

      4. inv-pow 98.5%

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

      5. pow-pow 98.8%

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

      6. metadata-eval 98.8%

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

   3. Applied egg-rr 98.8%

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

   if 9.9999999999999992e227 < (*.f64 V l)

   1. Initial program 40.1%

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

      1. associate-/r* 45.4%

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

      2. div-inv 45.3%

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

   3. Applied egg-rr 45.3%

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

      1. frac-times 40.1%

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

      2. div-inv 40.1%

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

      3. *-rgt-identity 40.1%

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

      4. inv-pow 40.1%

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

      5. metadata-eval 40.1%

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

      6. pow-prod-up 40.1%

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

      7. sqrt-prod 48.1%

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

      8. sqrt-unprod 48.2%

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

      9. add-sqr-sqrt 48.1%

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

      10. associate-*r* 48.1%

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

      11. metadata-eval 48.1%

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

      12. pow-flip 48.1%

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

      13. pow1/2 48.1%

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

      14. div-inv 48.2%

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

      15. associate-*l/ 48.2%

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

      16. *-commutative 48.2%

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

      17. clear-num 48.2%

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

      18. un-div-inv 48.3%

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

   5. Applied egg-rr 48.3%

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

      1. div-inv 48.2%

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

      2. sqrt-prod 78.7%

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

      3. associate-*l* 78.9%

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

   7. Applied egg-rr 78.9%

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

      1. associate-*r/ 78.7%

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

      2. *-rgt-identity 78.7%

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

   9. Simplified 78.7%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-300}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+228}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{V} \cdot \frac{\sqrt{\ell}}{c0}}\\ \end{array} \]

Alternative 2: 90.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-300}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+279}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A} \cdot \frac{\frac{c0}{\sqrt{V}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) (- INFINITY))
   (* c0 (/ (sqrt (/ (- A) l)) (sqrt (- V))))
   (if (<= (* V l) -1e-300)
     (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
     (if (<= (* V l) 0.0)
       (* c0 (/ (sqrt (/ A V)) (sqrt l)))
       (if (<= (* V l) 5e+279)
         (* c0 (* (sqrt A) (pow (* V l) -0.5)))
         (* (sqrt A) (/ (/ c0 (sqrt V)) (sqrt l))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = c0 * (sqrt((-A / l)) / sqrt(-V));
	} else if ((V * l) <= -1e-300) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else if ((V * l) <= 5e+279) {
		tmp = c0 * (sqrt(A) * pow((V * l), -0.5));
	} else {
		tmp = sqrt(A) * ((c0 / sqrt(V)) / sqrt(l));
	}
	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((-A / l)) / Math.sqrt(-V));
	} else if ((V * l) <= -1e-300) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if ((V * l) <= 5e+279) {
		tmp = c0 * (Math.sqrt(A) * Math.pow((V * l), -0.5));
	} else {
		tmp = Math.sqrt(A) * ((c0 / Math.sqrt(V)) / Math.sqrt(l));
	}
	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((-A / l)) / math.sqrt(-V))
	elif (V * l) <= -1e-300:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 0.0:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	elif (V * l) <= 5e+279:
		tmp = c0 * (math.sqrt(A) * math.pow((V * l), -0.5))
	else:
		tmp = math.sqrt(A) * ((c0 / math.sqrt(V)) / math.sqrt(l))
	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(c0 * Float64(sqrt(Float64(Float64(-A) / l)) / sqrt(Float64(-V))));
	elseif (Float64(V * l) <= -1e-300)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (Float64(V * l) <= 5e+279)
		tmp = Float64(c0 * Float64(sqrt(A) * (Float64(V * l) ^ -0.5)));
	else
		tmp = Float64(sqrt(A) * Float64(Float64(c0 / sqrt(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 ((V * l) <= -Inf)
		tmp = c0 * (sqrt((-A / l)) / sqrt(-V));
	elseif ((V * l) <= -1e-300)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 0.0)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	elseif ((V * l) <= 5e+279)
		tmp = c0 * (sqrt(A) * ((V * l) ^ -0.5));
	else
		tmp = sqrt(A) * ((c0 / sqrt(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[N[(V * l), $MachinePrecision], (-Infinity)], N[(c0 * N[(N[Sqrt[N[((-A) / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-300], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e+279], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Power[N[(V * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[A], $MachinePrecision] * N[(N[(c0 / N[Sqrt[V], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 58.4%

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

      1. associate-/r* 70.4%

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

      2. div-inv 70.4%

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

   3. Applied egg-rr 70.4%

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

      1. un-div-inv 70.4%

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

   5. Applied egg-rr 70.4%

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

      1. associate-/l/ 58.4%

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

      2. *-commutative 58.4%

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

      3. associate-/l/ 70.3%

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

      4. frac-2neg 70.3%

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

      5. sqrt-div 50.3%

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

      6. distribute-neg-frac 50.3%

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

   7. Applied egg-rr 50.3%

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

   if -inf.0 < (*.f64 V l) < -1.00000000000000003e-300

   1. Initial program 83.8%

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

      1. frac-2neg 83.8%

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

      2. sqrt-div 99.5%

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

      3. *-commutative 99.5%

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

      4. distribute-rgt-neg-in 99.5%

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

   3. Applied egg-rr 99.5%

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

   if -1.00000000000000003e-300 < (*.f64 V l) < 0.0

   1. Initial program 41.4%

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

      1. associate-/r* 67.3%

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

      2. sqrt-div 55.2%

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

   3. Applied egg-rr 55.2%

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

   if 0.0 < (*.f64 V l) < 5.0000000000000002e279

   1. Initial program 89.0%

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

      1. div-inv 88.6%

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

      2. sqrt-prod 98.5%

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

      3. pow1/2 98.5%

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

      4. inv-pow 98.5%

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

      5. pow-pow 98.9%

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

      6. metadata-eval 98.9%

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

   3. Applied egg-rr 98.9%

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

   if 5.0000000000000002e279 < (*.f64 V l)

   1. Initial program 29.6%

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

      1. sqrt-div 34.7%

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

      2. associate-*r/ 34.6%

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

   3. Applied egg-rr 34.6%

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

      1. associate-*l/ 34.7%

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

   5. Simplified 34.7%

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

      1. *-un-lft-identity 34.7%

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

      2. sqrt-prod 73.3%

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

      3. times-frac 73.2%

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

   7. Applied egg-rr 73.2%

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

      1. associate-*r/ 73.1%

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

      2. associate-*l/ 73.3%

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

      3. *-lft-identity 73.3%

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

   9. Simplified 73.3%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-300}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+279}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A} \cdot \frac{\frac{c0}{\sqrt{V}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 3: 90.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-300}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+279}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot \frac{\frac{c0}{V}}{\frac{\ell}{c0}}}\\ \end{array} \end{array} \]

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 (/ (- A) l)) (sqrt (- V))))
   (if (<= (* V l) -1e-300)
     (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
     (if (<= (* V l) 0.0)
       (* c0 (/ (sqrt (/ A V)) (sqrt l)))
       (if (<= (* V l) 5e+279)
         (* c0 (* (sqrt A) (pow (* V l) -0.5)))
         (sqrt (* A (/ (/ c0 V) (/ l c0)))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = c0 * (sqrt((-A / l)) / sqrt(-V));
	} else if ((V * l) <= -1e-300) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else if ((V * l) <= 5e+279) {
		tmp = c0 * (sqrt(A) * pow((V * l), -0.5));
	} else {
		tmp = sqrt((A * ((c0 / V) / (l / c0))));
	}
	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((-A / l)) / Math.sqrt(-V));
	} else if ((V * l) <= -1e-300) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if ((V * l) <= 5e+279) {
		tmp = c0 * (Math.sqrt(A) * Math.pow((V * l), -0.5));
	} else {
		tmp = Math.sqrt((A * ((c0 / V) / (l / c0))));
	}
	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((-A / l)) / math.sqrt(-V))
	elif (V * l) <= -1e-300:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 0.0:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	elif (V * l) <= 5e+279:
		tmp = c0 * (math.sqrt(A) * math.pow((V * l), -0.5))
	else:
		tmp = math.sqrt((A * ((c0 / V) / (l / c0))))
	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(c0 * Float64(sqrt(Float64(Float64(-A) / l)) / sqrt(Float64(-V))));
	elseif (Float64(V * l) <= -1e-300)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (Float64(V * l) <= 5e+279)
		tmp = Float64(c0 * Float64(sqrt(A) * (Float64(V * l) ^ -0.5)));
	else
		tmp = sqrt(Float64(A * Float64(Float64(c0 / V) / Float64(l / c0))));
	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((-A / l)) / sqrt(-V));
	elseif ((V * l) <= -1e-300)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 0.0)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	elseif ((V * l) <= 5e+279)
		tmp = c0 * (sqrt(A) * ((V * l) ^ -0.5));
	else
		tmp = sqrt((A * ((c0 / V) / (l / c0))));
	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[(c0 * N[(N[Sqrt[N[((-A) / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-300], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e+279], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Power[N[(V * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(A * N[(N[(c0 / V), $MachinePrecision] / N[(l / c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 58.4%

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

      1. associate-/r* 70.4%

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

      2. div-inv 70.4%

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

   3. Applied egg-rr 70.4%

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

      1. un-div-inv 70.4%

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

   5. Applied egg-rr 70.4%

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

      1. associate-/l/ 58.4%

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

      2. *-commutative 58.4%

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

      3. associate-/l/ 70.3%

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

      4. frac-2neg 70.3%

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

      5. sqrt-div 50.3%

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

      6. distribute-neg-frac 50.3%

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

   7. Applied egg-rr 50.3%

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

   if -inf.0 < (*.f64 V l) < -1.00000000000000003e-300

   1. Initial program 83.8%

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

      1. frac-2neg 83.8%

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

      2. sqrt-div 99.5%

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

      3. *-commutative 99.5%

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

      4. distribute-rgt-neg-in 99.5%

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

   3. Applied egg-rr 99.5%

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

   if -1.00000000000000003e-300 < (*.f64 V l) < 0.0

   1. Initial program 41.4%

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

      1. associate-/r* 67.3%

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

      2. sqrt-div 55.2%

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

   3. Applied egg-rr 55.2%

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

   if 0.0 < (*.f64 V l) < 5.0000000000000002e279

   1. Initial program 89.0%

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

      1. div-inv 88.6%

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

      2. sqrt-prod 98.5%

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

      3. pow1/2 98.5%

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

      4. inv-pow 98.5%

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

      5. pow-pow 98.9%

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

      6. metadata-eval 98.9%

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

   3. Applied egg-rr 98.9%

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

   if 5.0000000000000002e279 < (*.f64 V l)

   1. Initial program 29.6%

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

      1. associate-/r* 36.2%

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

      2. div-inv 36.2%

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

   3. Applied egg-rr 36.2%

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

      1. un-div-inv 36.2%

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

   5. Applied egg-rr 36.2%

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

      1. add-sqr-sqrt 12.1%

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

      2. sqrt-prod 28.1%

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

      3. pow1/2 28.1%

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

      4. pow1/2 28.1%

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

      5. associate-/l/ 28.0%

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

      6. *-commutative 28.0%

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

      7. associate-/l/ 28.1%

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

      8. unpow-prod-down 28.1%

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

      9. clear-num 28.2%

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

      10. associate-/l* 28.0%

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

      11. div-inv 28.0%

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

      12. associate-/r/ 28.1%

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

      13. frac-times 59.6%

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

      14. *-commutative 59.6%

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

      15. associate-*l* 59.4%

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

   7. Applied egg-rr 59.4%

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

      1. unpow1/2 59.4%

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

      2. associate-*r* 59.6%

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

      3. *-commutative 59.6%

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

      4. *-commutative 59.6%

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

      5. associate-*r/ 54.5%

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

      6. associate-/l* 59.7%

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

   9. Simplified 59.7%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-300}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+279}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot \frac{\frac{c0}{V}}{\frac{\ell}{c0}}}\\ \end{array} \]

Alternative 4: 87.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{-300}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+279}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot \frac{\frac{c0}{V}}{\frac{\ell}{c0}}}\\ \end{array} \end{array} \]

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) -1e-300)
   (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
   (if (<= (* V l) 0.0)
     (* c0 (/ (sqrt (/ A V)) (sqrt l)))
     (if (<= (* V l) 5e+279)
       (* c0 (* (sqrt A) (pow (* V l) -0.5)))
       (sqrt (* A (/ (/ c0 V) (/ l c0))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e-300) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else if ((V * l) <= 5e+279) {
		tmp = c0 * (sqrt(A) * pow((V * l), -0.5));
	} else {
		tmp = sqrt((A * ((c0 / V) / (l / c0))));
	}
	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) <= (-1d-300)) then
        tmp = c0 * (sqrt(-a) / sqrt((v * -l)))
    else if ((v * l) <= 0.0d0) then
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    else if ((v * l) <= 5d+279) then
        tmp = c0 * (sqrt(a) * ((v * l) ** (-0.5d0)))
    else
        tmp = sqrt((a * ((c0 / v) / (l / c0))))
    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) <= -1e-300) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if ((V * l) <= 5e+279) {
		tmp = c0 * (Math.sqrt(A) * Math.pow((V * l), -0.5));
	} else {
		tmp = Math.sqrt((A * ((c0 / V) / (l / c0))));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -1e-300:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 0.0:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	elif (V * l) <= 5e+279:
		tmp = c0 * (math.sqrt(A) * math.pow((V * l), -0.5))
	else:
		tmp = math.sqrt((A * ((c0 / V) / (l / c0))))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -1e-300)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (Float64(V * l) <= 5e+279)
		tmp = Float64(c0 * Float64(sqrt(A) * (Float64(V * l) ^ -0.5)));
	else
		tmp = sqrt(Float64(A * Float64(Float64(c0 / V) / Float64(l / c0))));
	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) <= -1e-300)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 0.0)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	elseif ((V * l) <= 5e+279)
		tmp = c0 * (sqrt(A) * ((V * l) ^ -0.5));
	else
		tmp = sqrt((A * ((c0 / V) / (l / c0))));
	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], -1e-300], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e+279], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Power[N[(V * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(A * N[(N[(c0 / V), $MachinePrecision] / N[(l / c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -1.00000000000000003e-300

   1. Initial program 79.9%

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

      1. frac-2neg 79.9%

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

      2. sqrt-div 93.2%

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

      3. *-commutative 93.2%

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

      4. distribute-rgt-neg-in 93.2%

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

   3. Applied egg-rr 93.2%

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

   if -1.00000000000000003e-300 < (*.f64 V l) < 0.0

   1. Initial program 41.4%

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

      1. associate-/r* 67.3%

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

      2. sqrt-div 55.2%

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

   3. Applied egg-rr 55.2%

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

   if 0.0 < (*.f64 V l) < 5.0000000000000002e279

   1. Initial program 89.0%

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

      1. div-inv 88.6%

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

      2. sqrt-prod 98.5%

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

      3. pow1/2 98.5%

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

      4. inv-pow 98.5%

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

      5. pow-pow 98.9%

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

      6. metadata-eval 98.9%

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

   3. Applied egg-rr 98.9%

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

   if 5.0000000000000002e279 < (*.f64 V l)

   1. Initial program 29.6%

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

      1. associate-/r* 36.2%

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

      2. div-inv 36.2%

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

   3. Applied egg-rr 36.2%

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

      1. un-div-inv 36.2%

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

   5. Applied egg-rr 36.2%

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

      1. add-sqr-sqrt 12.1%

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

      2. sqrt-prod 28.1%

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

      3. pow1/2 28.1%

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

      4. pow1/2 28.1%

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

      5. associate-/l/ 28.0%

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

      6. *-commutative 28.0%

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

      7. associate-/l/ 28.1%

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

      8. unpow-prod-down 28.1%

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

      9. clear-num 28.2%

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

      10. associate-/l* 28.0%

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

      11. div-inv 28.0%

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

      12. associate-/r/ 28.1%

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

      13. frac-times 59.6%

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

      14. *-commutative 59.6%

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

      15. associate-*l* 59.4%

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

   7. Applied egg-rr 59.4%

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

      1. unpow1/2 59.4%

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

      2. associate-*r* 59.6%

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

      3. *-commutative 59.6%

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

      4. *-commutative 59.6%

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

      5. associate-*r/ 54.5%

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

      6. associate-/l* 59.7%

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

   9. Simplified 59.7%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{-300}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+279}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot \frac{\frac{c0}{V}}{\frac{\ell}{c0}}}\\ \end{array} \]

Alternative 5: 84.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+279}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot \frac{\frac{c0}{V}}{\frac{\ell}{c0}}}\\ \end{array} \end{array} \]

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) 0.0)
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))
   (if (<= (* V l) 5e+279)
     (* c0 (* (sqrt A) (pow (* V l) -0.5)))
     (sqrt (* A (/ (/ c0 V) (/ l c0)))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= 0.0) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else if ((V * l) <= 5e+279) {
		tmp = c0 * (sqrt(A) * pow((V * l), -0.5));
	} else {
		tmp = sqrt((A * ((c0 / V) / (l / c0))));
	}
	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) <= 0.0d0) then
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    else if ((v * l) <= 5d+279) then
        tmp = c0 * (sqrt(a) * ((v * l) ** (-0.5d0)))
    else
        tmp = sqrt((a * ((c0 / v) / (l / c0))))
    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) <= 0.0) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if ((V * l) <= 5e+279) {
		tmp = c0 * (Math.sqrt(A) * Math.pow((V * l), -0.5));
	} else {
		tmp = Math.sqrt((A * ((c0 / V) / (l / c0))));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= 0.0:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	elif (V * l) <= 5e+279:
		tmp = c0 * (math.sqrt(A) * math.pow((V * l), -0.5))
	else:
		tmp = math.sqrt((A * ((c0 / V) / (l / c0))))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= 0.0)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (Float64(V * l) <= 5e+279)
		tmp = Float64(c0 * Float64(sqrt(A) * (Float64(V * l) ^ -0.5)));
	else
		tmp = sqrt(Float64(A * Float64(Float64(c0 / V) / Float64(l / c0))));
	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) <= 0.0)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	elseif ((V * l) <= 5e+279)
		tmp = c0 * (sqrt(A) * ((V * l) ^ -0.5));
	else
		tmp = sqrt((A * ((c0 / V) / (l / c0))));
	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], 0.0], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e+279], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Power[N[(V * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(A * N[(N[(c0 / V), $MachinePrecision] / N[(l / c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 71.6%

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

      1. associate-/r* 76.5%

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

      2. sqrt-div 44.1%

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

   3. Applied egg-rr 44.1%

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

   if 0.0 < (*.f64 V l) < 5.0000000000000002e279

   1. Initial program 89.0%

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

      1. div-inv 88.6%

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

      2. sqrt-prod 98.5%

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

      3. pow1/2 98.5%

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

      4. inv-pow 98.5%

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

      5. pow-pow 98.9%

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

      6. metadata-eval 98.9%

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

   3. Applied egg-rr 98.9%

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

   if 5.0000000000000002e279 < (*.f64 V l)

   1. Initial program 29.6%

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

      1. associate-/r* 36.2%

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

      2. div-inv 36.2%

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

   3. Applied egg-rr 36.2%

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

      1. un-div-inv 36.2%

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

   5. Applied egg-rr 36.2%

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

      1. add-sqr-sqrt 12.1%

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

      2. sqrt-prod 28.1%

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

      3. pow1/2 28.1%

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

      4. pow1/2 28.1%

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

      5. associate-/l/ 28.0%

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

      6. *-commutative 28.0%

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

      7. associate-/l/ 28.1%

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

      8. unpow-prod-down 28.1%

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

      9. clear-num 28.2%

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

      10. associate-/l* 28.0%

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

      11. div-inv 28.0%

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

      12. associate-/r/ 28.1%

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

      13. frac-times 59.6%

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

      14. *-commutative 59.6%

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

      15. associate-*l* 59.4%

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

   7. Applied egg-rr 59.4%

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

      1. unpow1/2 59.4%

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

      2. associate-*r* 59.6%

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

      3. *-commutative 59.6%

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

      4. *-commutative 59.6%

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

      5. associate-*r/ 54.5%

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

      6. associate-/l* 59.7%

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

   9. Simplified 59.7%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+279}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot \frac{\frac{c0}{V}}{\frac{\ell}{c0}}}\\ \end{array} \]

Alternative 6: 84.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+279}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot \frac{\frac{c0}{V}}{\frac{\ell}{c0}}}\\ \end{array} \end{array} \]

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) 0.0)
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))
   (if (<= (* V l) 5e+279)
     (* c0 (/ (sqrt A) (sqrt (* V l))))
     (sqrt (* A (/ (/ c0 V) (/ l c0)))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= 0.0) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else if ((V * l) <= 5e+279) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = sqrt((A * ((c0 / V) / (l / c0))));
	}
	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) <= 0.0d0) then
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    else if ((v * l) <= 5d+279) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = sqrt((a * ((c0 / v) / (l / c0))))
    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) <= 0.0) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if ((V * l) <= 5e+279) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = Math.sqrt((A * ((c0 / V) / (l / c0))));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= 0.0:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	elif (V * l) <= 5e+279:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = math.sqrt((A * ((c0 / V) / (l / c0))))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= 0.0)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (Float64(V * l) <= 5e+279)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = sqrt(Float64(A * Float64(Float64(c0 / V) / Float64(l / c0))));
	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) <= 0.0)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	elseif ((V * l) <= 5e+279)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = sqrt((A * ((c0 / V) / (l / c0))));
	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], 0.0], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e+279], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(A * N[(N[(c0 / V), $MachinePrecision] / N[(l / c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 71.6%

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

      1. associate-/r* 76.5%

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

      2. sqrt-div 44.1%

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

   3. Applied egg-rr 44.1%

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

   if 0.0 < (*.f64 V l) < 5.0000000000000002e279

   1. Initial program 89.0%

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

      1. sqrt-div 98.8%

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

      2. associate-*r/ 96.2%

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

   3. Applied egg-rr 96.2%

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

      1. *-commutative 96.2%

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

      2. associate-*l/ 98.8%

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

   5. Simplified 98.8%

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

   if 5.0000000000000002e279 < (*.f64 V l)

   1. Initial program 29.6%

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

      1. associate-/r* 36.2%

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

      2. div-inv 36.2%

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

   3. Applied egg-rr 36.2%

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

      1. un-div-inv 36.2%

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

   5. Applied egg-rr 36.2%

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

      1. add-sqr-sqrt 12.1%

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

      2. sqrt-prod 28.1%

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

      3. pow1/2 28.1%

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

      4. pow1/2 28.1%

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

      5. associate-/l/ 28.0%

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

      6. *-commutative 28.0%

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

      7. associate-/l/ 28.1%

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

      8. unpow-prod-down 28.1%

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

      9. clear-num 28.2%

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

      10. associate-/l* 28.0%

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

      11. div-inv 28.0%

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

      12. associate-/r/ 28.1%

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

      13. frac-times 59.6%

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

      14. *-commutative 59.6%

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

      15. associate-*l* 59.4%

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

   7. Applied egg-rr 59.4%

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

      1. unpow1/2 59.4%

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

      2. associate-*r* 59.6%

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

      3. *-commutative 59.6%

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

      4. *-commutative 59.6%

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

      5. associate-*r/ 54.5%

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

      6. associate-/l* 59.7%

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

   9. Simplified 59.7%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+279}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot \frac{\frac{c0}{V}}{\frac{\ell}{c0}}}\\ \end{array} \]

Alternative 7: 81.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

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 (pow (/ (* V l) A) -0.5))
   (* 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 * pow(((V * l) / A), -0.5);
	} 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 * (((v * l) / a) ** (-0.5d0))
    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.pow(((V * l) / A), -0.5);
	} 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.pow(((V * l) / A), -0.5)
	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(Float64(V * l) / A) ^ -0.5));
	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 * (((V * l) / A) ^ -0.5);
	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[Power[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision], -0.5], $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 80.9%

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

      1. pow1/2 80.9%

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

      2. clear-num 80.9%

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

      3. inv-pow 80.9%

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

      4. pow-pow 80.9%

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

      5. associate-/l* 82.8%

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

      6. metadata-eval 82.8%

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

   3. Applied egg-rr 82.8%

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

      1. associate-/l* 80.9%

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

   5. Simplified 80.9%

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

   if -4.999999999999985e-310 < l

   1. Initial program 70.8%

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

      1. associate-/r* 70.6%

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

      2. sqrt-div 84.3%

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

   3. Applied egg-rr 84.3%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 8: 80.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;\sqrt{A \cdot \frac{\frac{c0}{V}}{\frac{\ell}{c0}}}\\ \mathbf{elif}\;t_0 \leq 10^{+304}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\ \end{array} \end{array} \]

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)
     (sqrt (* A (/ (/ c0 V) (/ l c0))))
     (if (<= t_0 1e+304) (* c0 (sqrt t_0)) (* c0 (pow (* V (/ l A)) -0.5))))))

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 = sqrt((A * ((c0 / V) / (l / c0))));
	} else if (t_0 <= 1e+304) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = c0 * pow((V * (l / A)), -0.5);
	}
	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 = sqrt((a * ((c0 / v) / (l / c0))))
    else if (t_0 <= 1d+304) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = c0 * ((v * (l / a)) ** (-0.5d0))
    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 = Math.sqrt((A * ((c0 / V) / (l / c0))));
	} else if (t_0 <= 1e+304) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = c0 * Math.pow((V * (l / A)), -0.5);
	}
	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 = math.sqrt((A * ((c0 / V) / (l / c0))))
	elif t_0 <= 1e+304:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = c0 * math.pow((V * (l / A)), -0.5)
	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 = sqrt(Float64(A * Float64(Float64(c0 / V) / Float64(l / c0))));
	elseif (t_0 <= 1e+304)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 * (Float64(V * Float64(l / A)) ^ -0.5));
	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 = sqrt((A * ((c0 / V) / (l / c0))));
	elseif (t_0 <= 1e+304)
		tmp = c0 * sqrt(t_0);
	else
		tmp = c0 * ((V * (l / A)) ^ -0.5);
	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[Sqrt[N[(A * N[(N[(c0 / V), $MachinePrecision] / N[(l / c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 1e+304], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 * N[Power[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 37.6%

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

      1. associate-/r* 43.7%

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

      2. div-inv 43.7%

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

   3. Applied egg-rr 43.7%

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

      1. un-div-inv 43.7%

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

   5. Applied egg-rr 43.7%

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

      1. add-sqr-sqrt 20.6%

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

      2. sqrt-prod 38.7%

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

      3. pow1/2 38.7%

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

      4. pow1/2 38.7%

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

      5. associate-/l/ 36.8%

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

      6. *-commutative 36.8%

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

      7. associate-/l/ 38.6%

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

      8. unpow-prod-down 36.8%

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

      9. clear-num 36.9%

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

      10. associate-/l* 36.8%

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

      11. div-inv 36.8%

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

      12. associate-/r/ 42.4%

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

      13. frac-times 55.7%

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

      14. *-commutative 55.7%

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

      15. associate-*l* 58.5%

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

   7. Applied egg-rr 58.5%

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

      1. unpow1/2 58.5%

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

      2. associate-*r* 55.7%

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

      3. *-commutative 55.7%

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

      4. *-commutative 55.7%

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

      5. associate-*r/ 52.0%

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

      6. associate-/l* 55.8%

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

   9. Simplified 55.8%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 9.9999999999999994e303

   1. Initial program 98.2%

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

   if 9.9999999999999994e303 < (/.f64 A (*.f64 V l))

   1. Initial program 44.9%

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

      1. pow1/2 44.9%

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

      2. clear-num 44.9%

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

      3. inv-pow 44.9%

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

      4. pow-pow 46.7%

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

      5. associate-/l* 58.4%

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

      6. metadata-eval 58.4%

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

   3. Applied egg-rr 58.4%

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

      1. associate-/l* 46.7%

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

      2. *-lft-identity 46.7%

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

      3. times-frac 58.4%

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

      4. /-rgt-identity 58.4%

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

   5. Simplified 58.4%

      \[\leadsto c0 \cdot \color{blue}{{\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0:\\ \;\;\;\;\sqrt{A \cdot \frac{\frac{c0}{V}}{\frac{\ell}{c0}}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 10^{+304}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\ \end{array} \]

Alternative 9: 80.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;\sqrt{\frac{c0}{V} \cdot \left(A \cdot \frac{c0}{\ell}\right)}\\ \mathbf{elif}\;t_0 \leq 10^{+304}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\ \end{array} \end{array} \]

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)
     (sqrt (* (/ c0 V) (* A (/ c0 l))))
     (if (<= t_0 1e+304) (* c0 (sqrt t_0)) (* c0 (pow (* V (/ l A)) -0.5))))))

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 = sqrt(((c0 / V) * (A * (c0 / l))));
	} else if (t_0 <= 1e+304) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = c0 * pow((V * (l / A)), -0.5);
	}
	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 = sqrt(((c0 / v) * (a * (c0 / l))))
    else if (t_0 <= 1d+304) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = c0 * ((v * (l / a)) ** (-0.5d0))
    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 = Math.sqrt(((c0 / V) * (A * (c0 / l))));
	} else if (t_0 <= 1e+304) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = c0 * Math.pow((V * (l / A)), -0.5);
	}
	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 = math.sqrt(((c0 / V) * (A * (c0 / l))))
	elif t_0 <= 1e+304:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = c0 * math.pow((V * (l / A)), -0.5)
	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 = sqrt(Float64(Float64(c0 / V) * Float64(A * Float64(c0 / l))));
	elseif (t_0 <= 1e+304)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 * (Float64(V * Float64(l / A)) ^ -0.5));
	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 = sqrt(((c0 / V) * (A * (c0 / l))));
	elseif (t_0 <= 1e+304)
		tmp = c0 * sqrt(t_0);
	else
		tmp = c0 * ((V * (l / A)) ^ -0.5);
	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[Sqrt[N[(N[(c0 / V), $MachinePrecision] * N[(A * N[(c0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 1e+304], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 * N[Power[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 37.6%

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

      1. add-sqr-sqrt 37.6%

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

      2. sqrt-unprod 37.6%

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

      3. pow1/2 37.6%

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

      4. *-commutative 37.6%

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

      5. *-commutative 37.6%

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

      6. swap-sqr 36.8%

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

      7. add-sqr-sqrt 36.8%

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

   3. Applied egg-rr 36.8%

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

      1. unpow1/2 36.8%

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

   5. Simplified 36.8%

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

      1. *-commutative 36.8%

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

      2. clear-num 36.8%

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

      3. associate-*r/ 36.9%

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

      4. un-div-inv 36.9%

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

      5. associate-*r/ 36.8%

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

   7. Applied egg-rr 36.8%

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

      1. associate-/r/ 42.4%

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

      2. frac-times 55.7%

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

      3. associate-*l* 56.3%

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

   9. Applied egg-rr 56.3%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 9.9999999999999994e303

   1. Initial program 98.2%

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

   if 9.9999999999999994e303 < (/.f64 A (*.f64 V l))

   1. Initial program 44.9%

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

      1. pow1/2 44.9%

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

      2. clear-num 44.9%

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

      3. inv-pow 44.9%

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

      4. pow-pow 46.7%

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

      5. associate-/l* 58.4%

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

      6. metadata-eval 58.4%

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

   3. Applied egg-rr 58.4%

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

      1. associate-/l* 46.7%

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

      2. *-lft-identity 46.7%

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

      3. times-frac 58.4%

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

      4. /-rgt-identity 58.4%

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

   5. Simplified 58.4%

      \[\leadsto c0 \cdot \color{blue}{{\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0:\\ \;\;\;\;\sqrt{\frac{c0}{V} \cdot \left(A \cdot \frac{c0}{\ell}\right)}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 10^{+304}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\ \end{array} \]

Alternative 10: 75.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 10^{-264}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) 1e-264)
   (* c0 (sqrt (/ (/ A V) l)))
   (* c0 (sqrt (/ A (* V l))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= 1e-264) {
		tmp = c0 * sqrt(((A / V) / l));
	} else {
		tmp = c0 * sqrt((A / (V * 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 ((v * l) <= 1d-264) then
        tmp = c0 * sqrt(((a / v) / l))
    else
        tmp = c0 * sqrt((a / (v * 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 ((V * l) <= 1e-264) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else {
		tmp = c0 * Math.sqrt((A / (V * l)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= 1e-264:
		tmp = c0 * math.sqrt(((A / V) / l))
	else:
		tmp = c0 * math.sqrt((A / (V * l)))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= 1e-264)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	else
		tmp = Float64(c0 * sqrt(Float64(A / Float64(V * l))));
	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) <= 1e-264)
		tmp = c0 * sqrt(((A / V) / l));
	else
		tmp = c0 * sqrt((A / (V * 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[N[(V * l), $MachinePrecision], 1e-264], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 V l) < 1e-264

   1. Initial program 72.4%

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

      1. associate-/r* 77.5%

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

      2. div-inv 77.3%

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

   3. Applied egg-rr 77.3%

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

      1. un-div-inv 77.5%

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

   5. Applied egg-rr 77.5%

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

   if 1e-264 < (*.f64 V l)

   1. Initial program 79.2%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 10^{-264}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]

Alternative 11: 73.9% 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 75.5%

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

2. Final simplification 75.5%

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

Reproduce

herbie shell --seed 2023249 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))