Henrywood and Agarwal, Equation (3)

Percentage Accurate: 74.2% → 92.0%

Time: 9.3s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-314}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;{\ell}^{-0.5} \cdot \frac{c0}{\sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+300}:\\ \;\;\;\;c0 \cdot \left({\left(V \cdot \ell\right)}^{-0.5} \cdot \sqrt{A}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) (- INFINITY))
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))
   (if (<= (* V l) -1e-314)
     (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
     (if (<= (* V l) 0.0)
       (* (pow l -0.5) (/ c0 (sqrt (/ V A))))
       (if (<= (* V l) 1e+300)
         (* c0 (* (pow (* V l) -0.5) (sqrt A)))
         (* c0 (sqrt (/ (/ A V) l))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else if ((V * l) <= -1e-314) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = pow(l, -0.5) * (c0 / sqrt((V / A)));
	} else if ((V * l) <= 1e+300) {
		tmp = c0 * (pow((V * l), -0.5) * sqrt(A));
	} else {
		tmp = c0 * sqrt(((A / V) / l));
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if ((V * l) <= -1e-314) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = Math.pow(l, -0.5) * (c0 / Math.sqrt((V / A)));
	} else if ((V * l) <= 1e+300) {
		tmp = c0 * (Math.pow((V * l), -0.5) * Math.sqrt(A));
	} else {
		tmp = c0 * Math.sqrt(((A / V) / l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	elif (V * l) <= -1e-314:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 0.0:
		tmp = math.pow(l, -0.5) * (c0 / math.sqrt((V / A)))
	elif (V * l) <= 1e+300:
		tmp = c0 * (math.pow((V * l), -0.5) * math.sqrt(A))
	else:
		tmp = c0 * math.sqrt(((A / V) / l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (Float64(V * l) <= -1e-314)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64((l ^ -0.5) * Float64(c0 / sqrt(Float64(V / A))));
	elseif (Float64(V * l) <= 1e+300)
		tmp = Float64(c0 * Float64((Float64(V * l) ^ -0.5) * sqrt(A)));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	elseif ((V * l) <= -1e-314)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 0.0)
		tmp = (l ^ -0.5) * (c0 / sqrt((V / A)));
	elseif ((V * l) <= 1e+300)
		tmp = c0 * (((V * l) ^ -0.5) * sqrt(A));
	else
		tmp = c0 * sqrt(((A / V) / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-314], 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[(N[Power[l, -0.5], $MachinePrecision] * N[(c0 / N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+300], N[(c0 * N[(N[Power[N[(V * l), $MachinePrecision], -0.5], $MachinePrecision] * N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 38.5%

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

      1. associate-/r* 73.5%

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

      2. sqrt-div 37.0%

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

      3. div-inv 37.0%

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

   4. Applied egg-rr 37.0%

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

      1. associate-*r/ 37.0%

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

      2. *-rgt-identity 37.0%

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

   6. Simplified 37.0%

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

   if -inf.0 < (*.f64 V l) < -9.9999999996e-315

   1. Initial program 84.0%

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

      1. frac-2neg 84.0%

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

      2. sqrt-div 98.9%

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

      3. distribute-rgt-neg-in 98.9%

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

   4. Applied egg-rr 98.9%

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

      1. distribute-rgt-neg-out 98.9%

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

      2. *-commutative 98.9%

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

      3. distribute-rgt-neg-in 98.9%

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

   6. Simplified 98.9%

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

   if -9.9999999996e-315 < (*.f64 V l) < -0.0

   1. Initial program 43.3%

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

      1. *-un-lft-identity 43.3%

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

      2. times-frac 59.9%

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

   4. Applied egg-rr 59.9%

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

      1. associate-*r/ 59.9%

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

      2. sqrt-div 51.4%

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

      3. associate-*l/ 51.4%

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

      4. *-un-lft-identity 51.4%

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

      5. clear-num 51.6%

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

      6. un-div-inv 51.7%

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

      7. clear-num 51.5%

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

      8. sqrt-undiv 60.0%

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

      9. associate-/r* 43.3%

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

      10. frac-2neg 43.3%

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

      11. *-commutative 43.3%

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

      12. distribute-rgt-neg-out 43.3%

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

      13. sqrt-undiv 24.5%

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

      14. clear-num 24.5%

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

      15. sqrt-undiv 43.3%

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

      16. distribute-rgt-neg-out 43.3%

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

      17. *-commutative 43.3%

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

      18. frac-2neg 43.3%

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

   6. Applied egg-rr 62.3%

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

      1. *-un-lft-identity 62.3%

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

      2. sqrt-prod 51.6%

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

      3. times-frac 51.5%

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

      4. pow1/2 51.5%

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

      5. pow-flip 51.6%

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

      6. metadata-eval 51.6%

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

   8. Applied egg-rr 51.6%

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

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

   1. Initial program 80.8%

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

      1. *-un-lft-identity 80.8%

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

      2. times-frac 66.7%

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

   4. Applied egg-rr 66.7%

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

      1. associate-*r/ 76.7%

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

      2. associate-*l/ 79.0%

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

      3. *-commutative 79.0%

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

      4. associate-*r/ 76.7%

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

      5. div-inv 76.8%

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

      6. frac-2neg 76.8%

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

      7. associate-/l/ 80.8%

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

      8. sqrt-undiv 0.0%

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

      9. clear-num 0.0%

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

      10. associate-/r/ 0.0%

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

      11. pow1/2 0.0%

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

      12. pow-flip 0.0%

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

      13. *-commutative 0.0%

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

      14. add-sqr-sqrt 0.0%

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

      15. sqrt-unprod 0.0%

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

      16. sqr-neg 0.0%

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

      17. sqrt-unprod 0.0%

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

      18. add-sqr-sqrt 0.0%

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

      19. metadata-eval 0.0%

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

      20. add-sqr-sqrt 0.0%

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

      21. sqrt-unprod 63.5%

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

      22. sqr-neg 63.5%

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

      23. sqrt-unprod 98.5%

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

   6. Applied egg-rr 98.5%

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

   if 1.0000000000000001e300 < (*.f64 V l)

   1. Initial program 31.6%

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

      1. associate-/r* 76.7%

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

   3. Simplified 76.7%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-314}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;{\ell}^{-0.5} \cdot \frac{c0}{\sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+300}:\\ \;\;\;\;c0 \cdot \left({\left(V \cdot \ell\right)}^{-0.5} \cdot \sqrt{A}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+178}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-236}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;{\ell}^{-0.5} \cdot \frac{c0}{\sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+300}:\\ \;\;\;\;c0 \cdot \left({\left(V \cdot \ell\right)}^{-0.5} \cdot \sqrt{A}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -5e+178)
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))
   (if (<= (* V l) -5e-236)
     (/ c0 (sqrt (/ (* V l) A)))
     (if (<= (* V l) 0.0)
       (* (pow l -0.5) (/ c0 (sqrt (/ V A))))
       (if (<= (* V l) 1e+300)
         (* c0 (* (pow (* V l) -0.5) (sqrt A)))
         (* c0 (sqrt (/ (/ A V) l))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -5e+178) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else if ((V * l) <= -5e-236) {
		tmp = c0 / sqrt(((V * l) / A));
	} else if ((V * l) <= 0.0) {
		tmp = pow(l, -0.5) * (c0 / sqrt((V / A)));
	} else if ((V * l) <= 1e+300) {
		tmp = c0 * (pow((V * l), -0.5) * sqrt(A));
	} else {
		tmp = c0 * sqrt(((A / V) / l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-5d+178)) then
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    else if ((v * l) <= (-5d-236)) then
        tmp = c0 / sqrt(((v * l) / a))
    else if ((v * l) <= 0.0d0) then
        tmp = (l ** (-0.5d0)) * (c0 / sqrt((v / a)))
    else if ((v * l) <= 1d+300) then
        tmp = c0 * (((v * l) ** (-0.5d0)) * sqrt(a))
    else
        tmp = c0 * sqrt(((a / v) / l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -5e+178) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if ((V * l) <= -5e-236) {
		tmp = c0 / Math.sqrt(((V * l) / A));
	} else if ((V * l) <= 0.0) {
		tmp = Math.pow(l, -0.5) * (c0 / Math.sqrt((V / A)));
	} else if ((V * l) <= 1e+300) {
		tmp = c0 * (Math.pow((V * l), -0.5) * Math.sqrt(A));
	} else {
		tmp = c0 * Math.sqrt(((A / V) / l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -5e+178:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	elif (V * l) <= -5e-236:
		tmp = c0 / math.sqrt(((V * l) / A))
	elif (V * l) <= 0.0:
		tmp = math.pow(l, -0.5) * (c0 / math.sqrt((V / A)))
	elif (V * l) <= 1e+300:
		tmp = c0 * (math.pow((V * l), -0.5) * math.sqrt(A))
	else:
		tmp = c0 * math.sqrt(((A / V) / l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -5e+178)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (Float64(V * l) <= -5e-236)
		tmp = Float64(c0 / sqrt(Float64(Float64(V * l) / A)));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64((l ^ -0.5) * Float64(c0 / sqrt(Float64(V / A))));
	elseif (Float64(V * l) <= 1e+300)
		tmp = Float64(c0 * Float64((Float64(V * l) ^ -0.5) * sqrt(A)));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -5e+178)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	elseif ((V * l) <= -5e-236)
		tmp = c0 / sqrt(((V * l) / A));
	elseif ((V * l) <= 0.0)
		tmp = (l ^ -0.5) * (c0 / sqrt((V / A)));
	elseif ((V * l) <= 1e+300)
		tmp = c0 * (((V * l) ^ -0.5) * sqrt(A));
	else
		tmp = c0 * sqrt(((A / V) / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -5e+178], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -5e-236], N[(c0 / N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(N[Power[l, -0.5], $MachinePrecision] * N[(c0 / N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+300], N[(c0 * N[(N[Power[N[(V * l), $MachinePrecision], -0.5], $MachinePrecision] * N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -4.9999999999999999e178

   1. Initial program 53.2%

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

      1. associate-/r* 68.1%

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

      2. sqrt-div 37.2%

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

      3. div-inv 37.1%

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

   4. Applied egg-rr 37.1%

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

      1. associate-*r/ 37.2%

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

      2. *-rgt-identity 37.2%

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

   6. Simplified 37.2%

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

   if -4.9999999999999999e178 < (*.f64 V l) < -4.9999999999999998e-236

   1. Initial program 92.0%

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

      1. *-un-lft-identity 92.0%

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

      2. times-frac 83.3%

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

   4. Applied egg-rr 83.3%

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

      1. associate-*r/ 79.3%

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

      2. sqrt-div 39.1%

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

      3. associate-*l/ 39.1%

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

      4. *-un-lft-identity 39.1%

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

      5. clear-num 39.2%

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

      6. un-div-inv 39.2%

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

      7. clear-num 39.0%

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

      8. sqrt-undiv 79.4%

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

      9. associate-/r* 92.0%

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

      10. frac-2neg 92.0%

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

      11. *-commutative 92.0%

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

      12. distribute-rgt-neg-out 92.0%

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

      13. sqrt-undiv 99.3%

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

      14. clear-num 99.4%

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

      15. sqrt-undiv 92.4%

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

      16. distribute-rgt-neg-out 92.4%

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

      17. *-commutative 92.4%

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

      18. frac-2neg 92.4%

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

   6. Applied egg-rr 77.8%

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

      1. associate-*r/ 92.4%

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

   8. Applied egg-rr 92.4%

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

   if -4.9999999999999998e-236 < (*.f64 V l) < -0.0

   1. Initial program 46.3%

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

      1. *-un-lft-identity 46.3%

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

      2. times-frac 59.1%

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

   4. Applied egg-rr 59.1%

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

      1. associate-*r/ 59.1%

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

      2. sqrt-div 45.2%

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

      3. associate-*l/ 45.2%

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

      4. *-un-lft-identity 45.2%

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

      5. clear-num 45.4%

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

      6. un-div-inv 45.5%

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

      7. clear-num 45.3%

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

      8. sqrt-undiv 59.1%

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

      9. associate-/r* 46.3%

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

      10. frac-2neg 46.3%

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

      11. *-commutative 46.3%

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

      12. distribute-rgt-neg-out 46.3%

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

      13. sqrt-undiv 42.9%

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

      14. clear-num 43.0%

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

      15. sqrt-undiv 46.3%

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

      16. distribute-rgt-neg-out 46.3%

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

      17. *-commutative 46.3%

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

      18. frac-2neg 46.3%

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

   6. Applied egg-rr 60.8%

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

      1. *-un-lft-identity 60.8%

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

      2. sqrt-prod 45.4%

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

      3. times-frac 45.4%

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

      4. pow1/2 45.4%

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

      5. pow-flip 45.4%

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

      6. metadata-eval 45.4%

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

   8. Applied egg-rr 45.4%

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

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

   1. Initial program 80.8%

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

      1. *-un-lft-identity 80.8%

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

      2. times-frac 66.7%

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

   4. Applied egg-rr 66.7%

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

      1. associate-*r/ 76.7%

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

      2. associate-*l/ 79.0%

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

      3. *-commutative 79.0%

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

      4. associate-*r/ 76.7%

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

      5. div-inv 76.8%

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

      6. frac-2neg 76.8%

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

      7. associate-/l/ 80.8%

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

      8. sqrt-undiv 0.0%

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

      9. clear-num 0.0%

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

      10. associate-/r/ 0.0%

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

      11. pow1/2 0.0%

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

      12. pow-flip 0.0%

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

      13. *-commutative 0.0%

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

      14. add-sqr-sqrt 0.0%

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

      15. sqrt-unprod 0.0%

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

      16. sqr-neg 0.0%

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

      17. sqrt-unprod 0.0%

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

      18. add-sqr-sqrt 0.0%

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

      19. metadata-eval 0.0%

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

      20. add-sqr-sqrt 0.0%

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

      21. sqrt-unprod 63.5%

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

      22. sqr-neg 63.5%

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

      23. sqrt-unprod 98.5%

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

   6. Applied egg-rr 98.5%

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

   if 1.0000000000000001e300 < (*.f64 V l)

   1. Initial program 31.6%

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

      1. associate-/r* 76.7%

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

   3. Simplified 76.7%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+178}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-236}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;{\ell}^{-0.5} \cdot \frac{c0}{\sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+300}:\\ \;\;\;\;c0 \cdot \left({\left(V \cdot \ell\right)}^{-0.5} \cdot \sqrt{A}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+178}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-236}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+300}:\\ \;\;\;\;c0 \cdot \left({\left(V \cdot \ell\right)}^{-0.5} \cdot \sqrt{A}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -5e+178)
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))
   (if (<= (* V l) -5e-236)
     (/ c0 (sqrt (/ (* V l) A)))
     (if (<= (* V l) 0.0)
       (/ c0 (* (sqrt l) (sqrt (/ V A))))
       (if (<= (* V l) 1e+300)
         (* c0 (* (pow (* V l) -0.5) (sqrt A)))
         (* c0 (sqrt (/ (/ A V) l))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -5e+178) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else if ((V * l) <= -5e-236) {
		tmp = c0 / sqrt(((V * l) / A));
	} else if ((V * l) <= 0.0) {
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	} else if ((V * l) <= 1e+300) {
		tmp = c0 * (pow((V * l), -0.5) * sqrt(A));
	} else {
		tmp = c0 * sqrt(((A / V) / l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-5d+178)) then
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    else if ((v * l) <= (-5d-236)) then
        tmp = c0 / sqrt(((v * l) / a))
    else if ((v * l) <= 0.0d0) then
        tmp = c0 / (sqrt(l) * sqrt((v / a)))
    else if ((v * l) <= 1d+300) then
        tmp = c0 * (((v * l) ** (-0.5d0)) * sqrt(a))
    else
        tmp = c0 * sqrt(((a / v) / l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -5e+178) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if ((V * l) <= -5e-236) {
		tmp = c0 / Math.sqrt(((V * l) / A));
	} else if ((V * l) <= 0.0) {
		tmp = c0 / (Math.sqrt(l) * Math.sqrt((V / A)));
	} else if ((V * l) <= 1e+300) {
		tmp = c0 * (Math.pow((V * l), -0.5) * Math.sqrt(A));
	} else {
		tmp = c0 * Math.sqrt(((A / V) / l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -5e+178:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	elif (V * l) <= -5e-236:
		tmp = c0 / math.sqrt(((V * l) / A))
	elif (V * l) <= 0.0:
		tmp = c0 / (math.sqrt(l) * math.sqrt((V / A)))
	elif (V * l) <= 1e+300:
		tmp = c0 * (math.pow((V * l), -0.5) * math.sqrt(A))
	else:
		tmp = c0 * math.sqrt(((A / V) / l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -5e+178)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (Float64(V * l) <= -5e-236)
		tmp = Float64(c0 / sqrt(Float64(Float64(V * l) / A)));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 / Float64(sqrt(l) * sqrt(Float64(V / A))));
	elseif (Float64(V * l) <= 1e+300)
		tmp = Float64(c0 * Float64((Float64(V * l) ^ -0.5) * sqrt(A)));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -5e+178)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	elseif ((V * l) <= -5e-236)
		tmp = c0 / sqrt(((V * l) / A));
	elseif ((V * l) <= 0.0)
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	elseif ((V * l) <= 1e+300)
		tmp = c0 * (((V * l) ^ -0.5) * sqrt(A));
	else
		tmp = c0 * sqrt(((A / V) / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -5e+178], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -5e-236], N[(c0 / N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+300], N[(c0 * N[(N[Power[N[(V * l), $MachinePrecision], -0.5], $MachinePrecision] * N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -4.9999999999999999e178

   1. Initial program 53.2%

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

      1. associate-/r* 68.1%

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

      2. sqrt-div 37.2%

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

      3. div-inv 37.1%

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

   4. Applied egg-rr 37.1%

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

      1. associate-*r/ 37.2%

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

      2. *-rgt-identity 37.2%

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

   6. Simplified 37.2%

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

   if -4.9999999999999999e178 < (*.f64 V l) < -4.9999999999999998e-236

   1. Initial program 92.0%

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

      1. *-un-lft-identity 92.0%

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

      2. times-frac 83.3%

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

   4. Applied egg-rr 83.3%

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

      1. associate-*r/ 79.3%

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

      2. sqrt-div 39.1%

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

      3. associate-*l/ 39.1%

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

      4. *-un-lft-identity 39.1%

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

      5. clear-num 39.2%

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

      6. un-div-inv 39.2%

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

      7. clear-num 39.0%

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

      8. sqrt-undiv 79.4%

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

      9. associate-/r* 92.0%

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

      10. frac-2neg 92.0%

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

      11. *-commutative 92.0%

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

      12. distribute-rgt-neg-out 92.0%

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

      13. sqrt-undiv 99.3%

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

      14. clear-num 99.4%

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

      15. sqrt-undiv 92.4%

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

      16. distribute-rgt-neg-out 92.4%

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

      17. *-commutative 92.4%

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

      18. frac-2neg 92.4%

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

   6. Applied egg-rr 77.8%

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

      1. associate-*r/ 92.4%

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

   8. Applied egg-rr 92.4%

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

   if -4.9999999999999998e-236 < (*.f64 V l) < -0.0

   1. Initial program 46.3%

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

      1. *-un-lft-identity 46.3%

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

      2. times-frac 59.1%

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

   4. Applied egg-rr 59.1%

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

      1. associate-*r/ 59.1%

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

      2. sqrt-div 45.2%

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

      3. associate-*l/ 45.2%

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

      4. *-un-lft-identity 45.2%

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

      5. clear-num 45.4%

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

      6. un-div-inv 45.5%

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

      7. clear-num 45.3%

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

      8. sqrt-undiv 59.1%

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

      9. associate-/r* 46.3%

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

      10. frac-2neg 46.3%

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

      11. *-commutative 46.3%

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

      12. distribute-rgt-neg-out 46.3%

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

      13. sqrt-undiv 42.9%

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

      14. clear-num 43.0%

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

      15. sqrt-undiv 46.3%

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

      16. distribute-rgt-neg-out 46.3%

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

      17. *-commutative 46.3%

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

      18. frac-2neg 46.3%

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

   6. Applied egg-rr 60.8%

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

      1. *-commutative 60.8%

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

      2. sqrt-prod 45.4%

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

   8. Applied egg-rr 45.4%

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

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

   1. Initial program 80.8%

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

      1. *-un-lft-identity 80.8%

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

      2. times-frac 66.7%

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

   4. Applied egg-rr 66.7%

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

      1. associate-*r/ 76.7%

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

      2. associate-*l/ 79.0%

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

      3. *-commutative 79.0%

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

      4. associate-*r/ 76.7%

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

      5. div-inv 76.8%

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

      6. frac-2neg 76.8%

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

      7. associate-/l/ 80.8%

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

      8. sqrt-undiv 0.0%

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

      9. clear-num 0.0%

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

      10. associate-/r/ 0.0%

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

      11. pow1/2 0.0%

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

      12. pow-flip 0.0%

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

      13. *-commutative 0.0%

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

      14. add-sqr-sqrt 0.0%

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

      15. sqrt-unprod 0.0%

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

      16. sqr-neg 0.0%

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

      17. sqrt-unprod 0.0%

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

      18. add-sqr-sqrt 0.0%

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

      19. metadata-eval 0.0%

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

      20. add-sqr-sqrt 0.0%

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

      21. sqrt-unprod 63.5%

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

      22. sqr-neg 63.5%

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

      23. sqrt-unprod 98.5%

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

   6. Applied egg-rr 98.5%

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

   if 1.0000000000000001e300 < (*.f64 V l)

   1. Initial program 31.6%

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

      1. associate-/r* 76.7%

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

   3. Simplified 76.7%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+178}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-236}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+300}:\\ \;\;\;\;c0 \cdot \left({\left(V \cdot \ell\right)}^{-0.5} \cdot \sqrt{A}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+178}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-236}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+300}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -5e+178)
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))
   (if (<= (* V l) -5e-236)
     (/ c0 (sqrt (/ (* V l) A)))
     (if (<= (* V l) 0.0)
       (/ c0 (* (sqrt l) (sqrt (/ V A))))
       (if (<= (* V l) 1e+300)
         (/ c0 (/ (sqrt (* V l)) (sqrt A)))
         (* c0 (sqrt (/ (/ A V) l))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -5e+178) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else if ((V * l) <= -5e-236) {
		tmp = c0 / sqrt(((V * l) / A));
	} else if ((V * l) <= 0.0) {
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	} else if ((V * l) <= 1e+300) {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	} else {
		tmp = c0 * sqrt(((A / V) / l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-5d+178)) then
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    else if ((v * l) <= (-5d-236)) then
        tmp = c0 / sqrt(((v * l) / a))
    else if ((v * l) <= 0.0d0) then
        tmp = c0 / (sqrt(l) * sqrt((v / a)))
    else if ((v * l) <= 1d+300) then
        tmp = c0 / (sqrt((v * l)) / sqrt(a))
    else
        tmp = c0 * sqrt(((a / v) / l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -5e+178) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if ((V * l) <= -5e-236) {
		tmp = c0 / Math.sqrt(((V * l) / A));
	} else if ((V * l) <= 0.0) {
		tmp = c0 / (Math.sqrt(l) * Math.sqrt((V / A)));
	} else if ((V * l) <= 1e+300) {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	} else {
		tmp = c0 * Math.sqrt(((A / V) / l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -5e+178:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	elif (V * l) <= -5e-236:
		tmp = c0 / math.sqrt(((V * l) / A))
	elif (V * l) <= 0.0:
		tmp = c0 / (math.sqrt(l) * math.sqrt((V / A)))
	elif (V * l) <= 1e+300:
		tmp = c0 / (math.sqrt((V * l)) / math.sqrt(A))
	else:
		tmp = c0 * math.sqrt(((A / V) / l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -5e+178)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (Float64(V * l) <= -5e-236)
		tmp = Float64(c0 / sqrt(Float64(Float64(V * l) / A)));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 / Float64(sqrt(l) * sqrt(Float64(V / A))));
	elseif (Float64(V * l) <= 1e+300)
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -5e+178)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	elseif ((V * l) <= -5e-236)
		tmp = c0 / sqrt(((V * l) / A));
	elseif ((V * l) <= 0.0)
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	elseif ((V * l) <= 1e+300)
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	else
		tmp = c0 * sqrt(((A / V) / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -5e+178], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -5e-236], N[(c0 / N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+300], N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -4.9999999999999999e178

   1. Initial program 53.2%

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

      1. associate-/r* 68.1%

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

      2. sqrt-div 37.2%

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

      3. div-inv 37.1%

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

   4. Applied egg-rr 37.1%

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

      1. associate-*r/ 37.2%

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

      2. *-rgt-identity 37.2%

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

   6. Simplified 37.2%

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

   if -4.9999999999999999e178 < (*.f64 V l) < -4.9999999999999998e-236

   1. Initial program 92.0%

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

      1. *-un-lft-identity 92.0%

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

      2. times-frac 83.3%

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

   4. Applied egg-rr 83.3%

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

      1. associate-*r/ 79.3%

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

      2. sqrt-div 39.1%

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

      3. associate-*l/ 39.1%

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

      4. *-un-lft-identity 39.1%

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

      5. clear-num 39.2%

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

      6. un-div-inv 39.2%

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

      7. clear-num 39.0%

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

      8. sqrt-undiv 79.4%

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

      9. associate-/r* 92.0%

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

      10. frac-2neg 92.0%

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

      11. *-commutative 92.0%

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

      12. distribute-rgt-neg-out 92.0%

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

      13. sqrt-undiv 99.3%

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

      14. clear-num 99.4%

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

      15. sqrt-undiv 92.4%

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

      16. distribute-rgt-neg-out 92.4%

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

      17. *-commutative 92.4%

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

      18. frac-2neg 92.4%

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

   6. Applied egg-rr 77.8%

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

      1. associate-*r/ 92.4%

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

   8. Applied egg-rr 92.4%

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

   if -4.9999999999999998e-236 < (*.f64 V l) < -0.0

   1. Initial program 46.3%

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

      1. *-un-lft-identity 46.3%

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

      2. times-frac 59.1%

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

   4. Applied egg-rr 59.1%

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

      1. associate-*r/ 59.1%

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

      2. sqrt-div 45.2%

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

      3. associate-*l/ 45.2%

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

      4. *-un-lft-identity 45.2%

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

      5. clear-num 45.4%

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

      6. un-div-inv 45.5%

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

      7. clear-num 45.3%

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

      8. sqrt-undiv 59.1%

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

      9. associate-/r* 46.3%

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

      10. frac-2neg 46.3%

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

      11. *-commutative 46.3%

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

      12. distribute-rgt-neg-out 46.3%

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

      13. sqrt-undiv 42.9%

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

      14. clear-num 43.0%

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

      15. sqrt-undiv 46.3%

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

      16. distribute-rgt-neg-out 46.3%

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

      17. *-commutative 46.3%

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

      18. frac-2neg 46.3%

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

   6. Applied egg-rr 60.8%

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

      1. *-commutative 60.8%

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

      2. sqrt-prod 45.4%

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

   8. Applied egg-rr 45.4%

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

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

   1. Initial program 80.8%

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

      1. *-un-lft-identity 80.8%

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

      2. times-frac 66.7%

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

   4. Applied egg-rr 66.7%

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

      1. associate-*r/ 76.7%

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

      2. sqrt-div 44.8%

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

      3. associate-*l/ 44.8%

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

      4. *-un-lft-identity 44.8%

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

      5. clear-num 44.8%

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

      6. un-div-inv 44.8%

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

      7. clear-num 44.7%

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

      8. sqrt-undiv 76.7%

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

      9. associate-/r* 80.7%

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

      10. frac-2neg 80.7%

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

      11. *-commutative 80.7%

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

      12. distribute-rgt-neg-out 80.7%

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

      13. sqrt-undiv 0.0%

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

      14. clear-num 0.0%

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

      15. sqrt-undiv 80.5%

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

      16. distribute-rgt-neg-out 80.5%

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

      17. *-commutative 80.5%

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

      18. frac-2neg 80.5%

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

   6. Applied egg-rr 75.6%

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

      1. associate-*r/ 80.5%

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

      2. *-commutative 80.5%

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

      3. sqrt-div 98.5%

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

      4. *-commutative 98.5%

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

   8. Applied egg-rr 98.5%

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

   if 1.0000000000000001e300 < (*.f64 V l)

   1. Initial program 31.6%

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

      1. associate-/r* 76.7%

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

   3. Simplified 76.7%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+178}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-236}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+300}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+178}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-236}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+300}:\\ \;\;\;\;\sqrt{A} \cdot \frac{c0}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -5e+178)
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))
   (if (<= (* V l) -5e-236)
     (/ c0 (sqrt (/ (* V l) A)))
     (if (<= (* V l) 0.0)
       (/ c0 (* (sqrt l) (sqrt (/ V A))))
       (if (<= (* V l) 1e+300)
         (* (sqrt A) (/ c0 (sqrt (* V l))))
         (* c0 (sqrt (/ (/ A V) l))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -5e+178) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else if ((V * l) <= -5e-236) {
		tmp = c0 / sqrt(((V * l) / A));
	} else if ((V * l) <= 0.0) {
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	} else if ((V * l) <= 1e+300) {
		tmp = sqrt(A) * (c0 / sqrt((V * l)));
	} else {
		tmp = c0 * sqrt(((A / V) / l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-5d+178)) then
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    else if ((v * l) <= (-5d-236)) then
        tmp = c0 / sqrt(((v * l) / a))
    else if ((v * l) <= 0.0d0) then
        tmp = c0 / (sqrt(l) * sqrt((v / a)))
    else if ((v * l) <= 1d+300) then
        tmp = sqrt(a) * (c0 / sqrt((v * l)))
    else
        tmp = c0 * sqrt(((a / v) / l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -5e+178) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if ((V * l) <= -5e-236) {
		tmp = c0 / Math.sqrt(((V * l) / A));
	} else if ((V * l) <= 0.0) {
		tmp = c0 / (Math.sqrt(l) * Math.sqrt((V / A)));
	} else if ((V * l) <= 1e+300) {
		tmp = Math.sqrt(A) * (c0 / Math.sqrt((V * l)));
	} else {
		tmp = c0 * Math.sqrt(((A / V) / l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -5e+178:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	elif (V * l) <= -5e-236:
		tmp = c0 / math.sqrt(((V * l) / A))
	elif (V * l) <= 0.0:
		tmp = c0 / (math.sqrt(l) * math.sqrt((V / A)))
	elif (V * l) <= 1e+300:
		tmp = math.sqrt(A) * (c0 / math.sqrt((V * l)))
	else:
		tmp = c0 * math.sqrt(((A / V) / l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -5e+178)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (Float64(V * l) <= -5e-236)
		tmp = Float64(c0 / sqrt(Float64(Float64(V * l) / A)));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 / Float64(sqrt(l) * sqrt(Float64(V / A))));
	elseif (Float64(V * l) <= 1e+300)
		tmp = Float64(sqrt(A) * Float64(c0 / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -5e+178)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	elseif ((V * l) <= -5e-236)
		tmp = c0 / sqrt(((V * l) / A));
	elseif ((V * l) <= 0.0)
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	elseif ((V * l) <= 1e+300)
		tmp = sqrt(A) * (c0 / sqrt((V * l)));
	else
		tmp = c0 * sqrt(((A / V) / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -5e+178], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -5e-236], N[(c0 / N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+300], N[(N[Sqrt[A], $MachinePrecision] * N[(c0 / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -4.9999999999999999e178

   1. Initial program 53.2%

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

      1. associate-/r* 68.1%

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

      2. sqrt-div 37.2%

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

      3. div-inv 37.1%

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

   4. Applied egg-rr 37.1%

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

      1. associate-*r/ 37.2%

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

      2. *-rgt-identity 37.2%

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

   6. Simplified 37.2%

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

   if -4.9999999999999999e178 < (*.f64 V l) < -4.9999999999999998e-236

   1. Initial program 92.0%

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

      1. *-un-lft-identity 92.0%

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

      2. times-frac 83.3%

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

   4. Applied egg-rr 83.3%

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

      1. associate-*r/ 79.3%

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

      2. sqrt-div 39.1%

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

      3. associate-*l/ 39.1%

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

      4. *-un-lft-identity 39.1%

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

      5. clear-num 39.2%

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

      6. un-div-inv 39.2%

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

      7. clear-num 39.0%

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

      8. sqrt-undiv 79.4%

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

      9. associate-/r* 92.0%

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

      10. frac-2neg 92.0%

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

      11. *-commutative 92.0%

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

      12. distribute-rgt-neg-out 92.0%

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

      13. sqrt-undiv 99.3%

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

      14. clear-num 99.4%

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

      15. sqrt-undiv 92.4%

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

      16. distribute-rgt-neg-out 92.4%

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

      17. *-commutative 92.4%

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

      18. frac-2neg 92.4%

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

   6. Applied egg-rr 77.8%

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

      1. associate-*r/ 92.4%

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

   8. Applied egg-rr 92.4%

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

   if -4.9999999999999998e-236 < (*.f64 V l) < -0.0

   1. Initial program 46.3%

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

      1. *-un-lft-identity 46.3%

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

      2. times-frac 59.1%

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

   4. Applied egg-rr 59.1%

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

      1. associate-*r/ 59.1%

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

      2. sqrt-div 45.2%

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

      3. associate-*l/ 45.2%

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

      4. *-un-lft-identity 45.2%

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

      5. clear-num 45.4%

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

      6. un-div-inv 45.5%

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

      7. clear-num 45.3%

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

      8. sqrt-undiv 59.1%

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

      9. associate-/r* 46.3%

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

      10. frac-2neg 46.3%

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

      11. *-commutative 46.3%

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

      12. distribute-rgt-neg-out 46.3%

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

      13. sqrt-undiv 42.9%

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

      14. clear-num 43.0%

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

      15. sqrt-undiv 46.3%

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

      16. distribute-rgt-neg-out 46.3%

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

      17. *-commutative 46.3%

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

      18. frac-2neg 46.3%

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

   6. Applied egg-rr 60.8%

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

      1. *-commutative 60.8%

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

      2. sqrt-prod 45.4%

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

   8. Applied egg-rr 45.4%

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

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

   1. Initial program 80.8%

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

      1. *-un-lft-identity 80.8%

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

      2. times-frac 66.7%

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

   4. Applied egg-rr 66.7%

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

      1. associate-*r/ 76.7%

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

      2. sqrt-div 44.8%

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

      3. associate-*l/ 44.8%

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

      4. *-un-lft-identity 44.8%

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

      5. clear-num 44.8%

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

      6. un-div-inv 44.8%

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

      7. clear-num 44.7%

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

      8. sqrt-undiv 76.7%

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

      9. associate-/r* 80.7%

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

      10. frac-2neg 80.7%

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

      11. *-commutative 80.7%

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

      12. distribute-rgt-neg-out 80.7%

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

      13. sqrt-undiv 0.0%

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

      14. clear-num 0.0%

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

      15. sqrt-undiv 80.5%

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

      16. distribute-rgt-neg-out 80.5%

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

      17. *-commutative 80.5%

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

      18. frac-2neg 80.5%

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

   6. Applied egg-rr 75.6%

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

      1. clear-num 73.9%

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

      2. associate-*r/ 78.8%

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

      3. *-commutative 78.8%

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

      4. sqrt-div 96.8%

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

      5. associate-/r* 94.8%

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

      6. *-commutative 94.8%

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

      7. clear-num 95.8%

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

      8. *-commutative 95.8%

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

      9. associate-/l* 94.8%

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

      10. *-commutative 94.8%

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

   8. Applied egg-rr 94.8%

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

   if 1.0000000000000001e300 < (*.f64 V l)

   1. Initial program 31.6%

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

      1. associate-/r* 76.7%

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

   3. Simplified 76.7%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+178}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-236}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+300}:\\ \;\;\;\;\sqrt{A} \cdot \frac{c0}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+178}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-84}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+300}:\\ \;\;\;\;\sqrt{A} \cdot \frac{c0}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (/ (sqrt (/ A V)) (sqrt l)))))
   (if (<= (* V l) -5e+178)
     t_0
     (if (<= (* V l) -1e-84)
       (/ c0 (sqrt (/ (* V l) A)))
       (if (<= (* V l) 0.0)
         t_0
         (if (<= (* V l) 1e+300)
           (* (sqrt A) (/ c0 (sqrt (* V l))))
           (* c0 (sqrt (/ (/ A V) l)))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * (sqrt((A / V)) / sqrt(l));
	double tmp;
	if ((V * l) <= -5e+178) {
		tmp = t_0;
	} else if ((V * l) <= -1e-84) {
		tmp = c0 / sqrt(((V * l) / A));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else if ((V * l) <= 1e+300) {
		tmp = sqrt(A) * (c0 / sqrt((V * l)));
	} else {
		tmp = c0 * sqrt(((A / V) / l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * (sqrt((a / v)) / sqrt(l))
    if ((v * l) <= (-5d+178)) then
        tmp = t_0
    else if ((v * l) <= (-1d-84)) then
        tmp = c0 / sqrt(((v * l) / a))
    else if ((v * l) <= 0.0d0) then
        tmp = t_0
    else if ((v * l) <= 1d+300) then
        tmp = sqrt(a) * (c0 / sqrt((v * l)))
    else
        tmp = c0 * sqrt(((a / v) / l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	double tmp;
	if ((V * l) <= -5e+178) {
		tmp = t_0;
	} else if ((V * l) <= -1e-84) {
		tmp = c0 / Math.sqrt(((V * l) / A));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else if ((V * l) <= 1e+300) {
		tmp = Math.sqrt(A) * (c0 / Math.sqrt((V * l)));
	} else {
		tmp = c0 * Math.sqrt(((A / V) / l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	tmp = 0
	if (V * l) <= -5e+178:
		tmp = t_0
	elif (V * l) <= -1e-84:
		tmp = c0 / math.sqrt(((V * l) / A))
	elif (V * l) <= 0.0:
		tmp = t_0
	elif (V * l) <= 1e+300:
		tmp = math.sqrt(A) * (c0 / math.sqrt((V * l)))
	else:
		tmp = c0 * math.sqrt(((A / V) / l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)))
	tmp = 0.0
	if (Float64(V * l) <= -5e+178)
		tmp = t_0;
	elseif (Float64(V * l) <= -1e-84)
		tmp = Float64(c0 / sqrt(Float64(Float64(V * l) / A)));
	elseif (Float64(V * l) <= 0.0)
		tmp = t_0;
	elseif (Float64(V * l) <= 1e+300)
		tmp = Float64(sqrt(A) * Float64(c0 / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * (sqrt((A / V)) / sqrt(l));
	tmp = 0.0;
	if ((V * l) <= -5e+178)
		tmp = t_0;
	elseif ((V * l) <= -1e-84)
		tmp = c0 / sqrt(((V * l) / A));
	elseif ((V * l) <= 0.0)
		tmp = t_0;
	elseif ((V * l) <= 1e+300)
		tmp = sqrt(A) * (c0 / sqrt((V * l)));
	else
		tmp = c0 * sqrt(((A / V) / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -5e+178], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -1e-84], N[(c0 / N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], 1e+300], N[(N[Sqrt[A], $MachinePrecision] * N[(c0 / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -4.9999999999999999e178 or -1e-84 < (*.f64 V l) < -0.0

   1. Initial program 58.7%

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

      1. associate-/r* 68.9%

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

      2. sqrt-div 42.0%

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

      3. div-inv 41.9%

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

   4. Applied egg-rr 41.9%

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

      1. associate-*r/ 42.0%

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

      2. *-rgt-identity 42.0%

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

   6. Simplified 42.0%

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

   if -4.9999999999999999e178 < (*.f64 V l) < -1e-84

   1. Initial program 92.5%

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

      1. *-un-lft-identity 92.5%

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

      2. times-frac 81.5%

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

   4. Applied egg-rr 81.5%

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

      1. associate-*r/ 75.6%

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

      2. sqrt-div 38.1%

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

      3. associate-*l/ 38.1%

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

      4. *-un-lft-identity 38.1%

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

      5. clear-num 38.2%

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

      6. un-div-inv 38.3%

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

      7. clear-num 38.1%

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

      8. sqrt-undiv 75.7%

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

      9. associate-/r* 92.5%

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

      10. frac-2neg 92.5%

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

      11. *-commutative 92.5%

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

      12. distribute-rgt-neg-out 92.5%

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

      13. sqrt-undiv 99.3%

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

      14. clear-num 99.5%

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

      15. sqrt-undiv 93.1%

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

      16. distribute-rgt-neg-out 93.1%

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

      17. *-commutative 93.1%

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

      18. frac-2neg 93.1%

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

   6. Applied egg-rr 74.8%

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

      1. associate-*r/ 93.1%

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

   8. Applied egg-rr 93.1%

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

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

   1. Initial program 80.8%

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

      1. *-un-lft-identity 80.8%

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

      2. times-frac 66.7%

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

   4. Applied egg-rr 66.7%

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

      1. associate-*r/ 76.7%

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

      2. sqrt-div 44.8%

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

      3. associate-*l/ 44.8%

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

      4. *-un-lft-identity 44.8%

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

      5. clear-num 44.8%

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

      6. un-div-inv 44.8%

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

      7. clear-num 44.7%

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

      8. sqrt-undiv 76.7%

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

      9. associate-/r* 80.7%

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

      10. frac-2neg 80.7%

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

      11. *-commutative 80.7%

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

      12. distribute-rgt-neg-out 80.7%

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

      13. sqrt-undiv 0.0%

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

      14. clear-num 0.0%

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

      15. sqrt-undiv 80.5%

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

      16. distribute-rgt-neg-out 80.5%

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

      17. *-commutative 80.5%

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

      18. frac-2neg 80.5%

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

   6. Applied egg-rr 75.6%

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

      1. clear-num 73.9%

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

      2. associate-*r/ 78.8%

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

      3. *-commutative 78.8%

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

      4. sqrt-div 96.8%

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

      5. associate-/r* 94.8%

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

      6. *-commutative 94.8%

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

      7. clear-num 95.8%

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

      8. *-commutative 95.8%

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

      9. associate-/l* 94.8%

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

      10. *-commutative 94.8%

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

   8. Applied egg-rr 94.8%

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

   if 1.0000000000000001e300 < (*.f64 V l)

   1. Initial program 31.6%

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

      1. associate-/r* 76.7%

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

   3. Simplified 76.7%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+178}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-84}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+300}:\\ \;\;\;\;\sqrt{A} \cdot \frac{c0}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.55 \cdot 10^{-305}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= l 1.55e-305)
   (* c0 (sqrt (/ (/ A V) l)))
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= 1.55e-305) {
		tmp = c0 * sqrt(((A / V) / l));
	} else {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= 1.55d-305) then
        tmp = c0 * sqrt(((a / v) / l))
    else
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= 1.55e-305) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if l <= 1.55e-305:
		tmp = c0 * math.sqrt(((A / V) / l))
	else:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (l <= 1.55e-305)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	else
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (l <= 1.55e-305)
		tmp = c0 * sqrt(((A / V) / l));
	else
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[l, 1.55e-305], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.5499999999999999e-305

   1. Initial program 71.7%

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

      1. associate-/r* 74.1%

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

   3. Simplified 74.1%

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

   if 1.5499999999999999e-305 < l

   1. Initial program 74.2%

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

      1. associate-/r* 73.3%

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

      2. sqrt-div 85.3%

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

      3. div-inv 85.2%

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

   4. Applied egg-rr 85.2%

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

      1. associate-*r/ 85.3%

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

      2. *-rgt-identity 85.3%

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

   6. Simplified 85.3%

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

Alternative 8: 79.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-324} \lor \neg \left(t\_0 \leq 2 \cdot 10^{+273}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (or (<= t_0 5e-324) (not (<= t_0 2e+273)))
     (* c0 (sqrt (/ (/ A V) l)))
     (* c0 (sqrt t_0)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 5e-324) || !(t_0 <= 2e+273)) {
		tmp = c0 * sqrt(((A / V) / l));
	} else {
		tmp = c0 * sqrt(t_0);
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if ((t_0 <= 5d-324) .or. (.not. (t_0 <= 2d+273))) then
        tmp = c0 * sqrt(((a / v) / l))
    else
        tmp = c0 * sqrt(t_0)
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 5e-324) || !(t_0 <= 2e+273)) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else {
		tmp = c0 * Math.sqrt(t_0);
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if (t_0 <= 5e-324) or not (t_0 <= 2e+273):
		tmp = c0 * math.sqrt(((A / V) / l))
	else:
		tmp = c0 * math.sqrt(t_0)
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if ((t_0 <= 5e-324) || !(t_0 <= 2e+273))
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	else
		tmp = Float64(c0 * sqrt(t_0));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if ((t_0 <= 5e-324) || ~((t_0 <= 2e+273)))
		tmp = c0 * sqrt(((A / V) / l));
	else
		tmp = c0 * sqrt(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 5e-324], N[Not[LessEqual[t$95$0, 2e+273]], $MachinePrecision]], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 4.94066e-324 or 1.99999999999999989e273 < (/.f64 A (*.f64 V l))

   1. Initial program 37.1%

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

      1. associate-/r* 51.5%

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

   3. Simplified 51.5%

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

   if 4.94066e-324 < (/.f64 A (*.f64 V l)) < 1.99999999999999989e273

   1. Initial program 99.0%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{-324} \lor \neg \left(\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{+273}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-324}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+273}:\\ \;\;\;\;c0 \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (<= t_0 5e-324)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 2e+273) (* c0 (sqrt t_0)) (/ c0 (sqrt (* l (/ V A))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 5e-324) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 2e+273) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = c0 / sqrt((l * (V / A)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if (t_0 <= 5d-324) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 2d+273) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = c0 / sqrt((l * (v / a)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 5e-324) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 2e+273) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = c0 / Math.sqrt((l * (V / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 5e-324:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 2e+273:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = c0 / math.sqrt((l * (V / A)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 5e-324)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 2e+273)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / A))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 5e-324)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 2e+273)
		tmp = c0 * sqrt(t_0);
	else
		tmp = c0 / sqrt((l * (V / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-324], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+273], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 A (*.f64 V l)) < 4.94066e-324

   1. Initial program 31.8%

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

      1. associate-/r* 53.8%

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

   3. Simplified 53.8%

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

   if 4.94066e-324 < (/.f64 A (*.f64 V l)) < 1.99999999999999989e273

   1. Initial program 99.0%

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

   if 1.99999999999999989e273 < (/.f64 A (*.f64 V l))

   1. Initial program 41.3%

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

      1. *-un-lft-identity 41.3%

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

      2. times-frac 49.7%

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

   4. Applied egg-rr 49.7%

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

      1. associate-*r/ 49.8%

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

      2. sqrt-div 44.3%

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

      3. associate-*l/ 44.3%

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

      4. *-un-lft-identity 44.3%

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

      5. clear-num 44.5%

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

      6. un-div-inv 44.6%

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

      7. clear-num 44.4%

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

      8. sqrt-undiv 49.8%

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

      9. associate-/r* 41.3%

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

      10. frac-2neg 41.3%

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

      11. *-commutative 41.3%

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

      12. distribute-rgt-neg-out 41.3%

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

      13. sqrt-undiv 33.1%

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

      14. clear-num 33.1%

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

      15. sqrt-undiv 43.8%

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

      16. distribute-rgt-neg-out 43.8%

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

      17. *-commutative 43.8%

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

      18. frac-2neg 43.8%

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

   6. Applied egg-rr 51.9%

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

Alternative 10: 80.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-324}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+266}:\\ \;\;\;\;c0 \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (<= t_0 5e-324)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 5e+266) (* c0 (sqrt t_0)) (/ c0 (sqrt (* V (/ l A))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 5e-324) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 5e+266) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = c0 / sqrt((V * (l / A)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if (t_0 <= 5d-324) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 5d+266) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = c0 / sqrt((v * (l / a)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 5e-324) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 5e+266) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = c0 / Math.sqrt((V * (l / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 5e-324:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 5e+266:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = c0 / math.sqrt((V * (l / A)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 5e-324)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 5e+266)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 5e-324)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 5e+266)
		tmp = c0 * sqrt(t_0);
	else
		tmp = c0 / sqrt((V * (l / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-324], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+266], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 A (*.f64 V l)) < 4.94066e-324

   1. Initial program 31.8%

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

      1. associate-/r* 53.8%

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

   3. Simplified 53.8%

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

   if 4.94066e-324 < (/.f64 A (*.f64 V l)) < 4.9999999999999999e266

   1. Initial program 99.0%

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

   if 4.9999999999999999e266 < (/.f64 A (*.f64 V l))

   1. Initial program 43.1%

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

      1. *-un-lft-identity 43.1%

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

      2. times-frac 51.3%

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

   4. Applied egg-rr 51.3%

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

      1. associate-*r/ 51.3%

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

      2. sqrt-div 44.5%

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

      3. associate-*l/ 44.5%

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

      4. *-un-lft-identity 44.5%

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

      5. clear-num 44.6%

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

      6. un-div-inv 44.7%

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

      7. clear-num 44.5%

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

      8. sqrt-undiv 51.3%

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

      9. associate-/r* 43.1%

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

      10. frac-2neg 43.1%

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

      11. *-commutative 43.1%

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

      12. distribute-rgt-neg-out 43.1%

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

      13. sqrt-undiv 35.2%

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

      14. clear-num 35.2%

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

      15. sqrt-undiv 45.6%

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

      16. distribute-rgt-neg-out 45.6%

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

      17. *-commutative 45.6%

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

      18. frac-2neg 45.6%

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

   6. Applied egg-rr 53.4%

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

      1. associate-*r/ 45.6%

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

      2. associate-*l/ 54.9%

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

      3. *-commutative 54.9%

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

   8. Simplified 54.9%

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

Alternative 11: 79.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-324}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+273}:\\ \;\;\;\;c0 \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (<= t_0 5e-324)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 2e+273) (* c0 (sqrt t_0)) (* c0 (sqrt (/ (/ A l) V)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 5e-324) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 2e+273) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = c0 * sqrt(((A / l) / V));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if (t_0 <= 5d-324) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 2d+273) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = c0 * sqrt(((a / l) / v))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 5e-324) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 2e+273) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = c0 * Math.sqrt(((A / l) / V));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 5e-324:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 2e+273:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = c0 * math.sqrt(((A / l) / V))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 5e-324)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 2e+273)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 5e-324)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 2e+273)
		tmp = c0 * sqrt(t_0);
	else
		tmp = c0 * sqrt(((A / l) / V));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-324], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+273], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 A (*.f64 V l)) < 4.94066e-324

   1. Initial program 31.8%

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

      1. associate-/r* 53.8%

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

   3. Simplified 53.8%

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

   if 4.94066e-324 < (/.f64 A (*.f64 V l)) < 1.99999999999999989e273

   1. Initial program 99.0%

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

   if 1.99999999999999989e273 < (/.f64 A (*.f64 V l))

   1. Initial program 41.3%

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

      1. *-un-lft-identity 41.3%

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

      2. times-frac 49.7%

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

   4. Applied egg-rr 49.7%

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

   5. Taylor expanded in V around 0 41.3%

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

      1. associate-/l/ 49.7%

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

   7. Simplified 49.7%

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

Alternative 12: 74.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.9%

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

Reproduce

herbie shell --seed 2024097 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))