Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.5% → 90.3%

Time: 13.3s

Alternatives: 13

Speedup: 1.0×

• could not determine a ground truth

  1. c0 = 1.6060098356936378e-90
  2. A = -2.998878508298789e+254
  3. V = -1.4423375282003516e-100
  4. l = 2.2846448242722777e-260

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+208}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{-V}}{\sqrt{\frac{A}{-\ell}}}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-304}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-298}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+299}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -1e+208)
   (/ c0 (/ (sqrt (- V)) (sqrt (/ A (- l)))))
   (if (<= (* V l) -2e-304)
     (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
     (if (<= (* V l) 5e-298)
       (* (/ c0 (sqrt l)) (sqrt (/ A V)))
       (if (<= (* V l) 1e+299)
         (/ c0 (/ (sqrt (* V l)) (sqrt A)))
         (* c0 (pow (* V (/ l A)) -0.5)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+208) {
		tmp = c0 / (sqrt(-V) / sqrt((A / -l)));
	} else if ((V * l) <= -2e-304) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 5e-298) {
		tmp = (c0 / sqrt(l)) * sqrt((A / V));
	} else if ((V * l) <= 1e+299) {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	} else {
		tmp = c0 * pow((V * (l / A)), -0.5);
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-1d+208)) then
        tmp = c0 / (sqrt(-v) / sqrt((a / -l)))
    else if ((v * l) <= (-2d-304)) then
        tmp = c0 * (sqrt(-a) / sqrt((v * -l)))
    else if ((v * l) <= 5d-298) then
        tmp = (c0 / sqrt(l)) * sqrt((a / v))
    else if ((v * l) <= 1d+299) then
        tmp = c0 / (sqrt((v * l)) / sqrt(a))
    else
        tmp = c0 * ((v * (l / a)) ** (-0.5d0))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+208) {
		tmp = c0 / (Math.sqrt(-V) / Math.sqrt((A / -l)));
	} else if ((V * l) <= -2e-304) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 5e-298) {
		tmp = (c0 / Math.sqrt(l)) * Math.sqrt((A / V));
	} else if ((V * l) <= 1e+299) {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	} else {
		tmp = c0 * Math.pow((V * (l / A)), -0.5);
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -1e+208:
		tmp = c0 / (math.sqrt(-V) / math.sqrt((A / -l)))
	elif (V * l) <= -2e-304:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 5e-298:
		tmp = (c0 / math.sqrt(l)) * math.sqrt((A / V))
	elif (V * l) <= 1e+299:
		tmp = c0 / (math.sqrt((V * l)) / math.sqrt(A))
	else:
		tmp = c0 * math.pow((V * (l / A)), -0.5)
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -1e+208)
		tmp = Float64(c0 / Float64(sqrt(Float64(-V)) / sqrt(Float64(A / Float64(-l)))));
	elseif (Float64(V * l) <= -2e-304)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 5e-298)
		tmp = Float64(Float64(c0 / sqrt(l)) * sqrt(Float64(A / V)));
	elseif (Float64(V * l) <= 1e+299)
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	else
		tmp = Float64(c0 * (Float64(V * Float64(l / A)) ^ -0.5));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -1e+208)
		tmp = c0 / (sqrt(-V) / sqrt((A / -l)));
	elseif ((V * l) <= -2e-304)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 5e-298)
		tmp = (c0 / sqrt(l)) * sqrt((A / V));
	elseif ((V * l) <= 1e+299)
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	else
		tmp = c0 * ((V * (l / A)) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -1e+208], N[(c0 / N[(N[Sqrt[(-V)], $MachinePrecision] / N[Sqrt[N[(A / (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-304], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e-298], N[(N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+299], N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Power[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -9.9999999999999998e207

   1. Initial program 52.1%

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

      1. associate-/r* 70.6%

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

      2. clear-num 67.0%

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

      3. sqrt-div 67.0%

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

      4. metadata-eval 67.0%

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

      5. div-inv 67.0%

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

      6. clear-num 67.0%

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

   4. Applied egg-rr 67.0%

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

      1. *-commutative 67.0%

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

      2. associate-*l/ 52.1%

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

      3. associate-/l* 67.1%

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

   6. Simplified 67.1%

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

      1. un-div-inv 67.2%

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

      2. clear-num 66.9%

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

   8. Applied egg-rr 66.9%

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

      1. associate-/r/ 67.1%

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

      2. associate-*l/ 67.2%

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

      3. *-lft-identity 67.2%

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

      4. associate-*r/ 52.2%

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

      5. associate-*l/ 67.0%

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

      6. associate-/r/ 67.1%

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

   10. Simplified 67.1%

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

       1. frac-2neg 67.1%

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

       2. sqrt-div 50.1%

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

       3. distribute-neg-frac2 50.1%

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

   12. Applied egg-rr 50.1%

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

   if -9.9999999999999998e207 < (*.f64 V l) < -1.99999999999999994e-304

   1. Initial program 85.5%

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

      1. frac-2neg 85.5%

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

      2. sqrt-div 99.4%

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

      3. distribute-rgt-neg-in 99.4%

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

   4. Applied egg-rr 99.4%

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

      1. distribute-rgt-neg-out 99.4%

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

      2. *-commutative 99.4%

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

      3. distribute-rgt-neg-in 99.4%

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

   6. Simplified 99.4%

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

   if -1.99999999999999994e-304 < (*.f64 V l) < 5.0000000000000002e-298

   1. Initial program 46.1%

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

      1. add-sqr-sqrt 31.8%

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

      2. sqrt-unprod 28.0%

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

      3. *-commutative 28.0%

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

      4. *-commutative 28.0%

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

      5. swap-sqr 27.6%

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

      6. add-sqr-sqrt 27.6%

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

      7. pow2 27.6%

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

   4. Applied egg-rr 27.6%

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

      1. sqrt-prod 31.7%

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

      2. div-inv 31.7%

         \[\leadsto \sqrt{\color{blue}{A \cdot \frac{1}{V \cdot \ell}}} \cdot \sqrt{{c0}^{2}} \]

      3. associate-/r* 31.7%

         \[\leadsto \sqrt{A \cdot \color{blue}{\frac{\frac{1}{V}}{\ell}}} \cdot \sqrt{{c0}^{2}} \]

      4. *-commutative 31.7%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{V}}{\ell} \cdot A}} \cdot \sqrt{{c0}^{2}} \]

      5. *-commutative 31.7%

         \[\leadsto \sqrt{\color{blue}{A \cdot \frac{\frac{1}{V}}{\ell}}} \cdot \sqrt{{c0}^{2}} \]

      6. associate-*r/ 44.6%

         \[\leadsto \sqrt{\color{blue}{\frac{A \cdot \frac{1}{V}}{\ell}}} \cdot \sqrt{{c0}^{2}} \]

      7. div-inv 44.7%

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

      8. frac-2neg 44.7%

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

      9. associate-/r* 31.7%

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

      10. *-commutative 31.7%

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

      11. sqrt-undiv 17.9%

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

      12. sqrt-pow1 25.2%

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

      13. metadata-eval 25.2%

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

      14. pow1 25.2%

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

      15. *-commutative 25.2%

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

      16. associate-*r/ 25.1%

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

      17. sqrt-prod 30.3%

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

      18. times-frac 30.3%

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

      19. sqrt-div 60.6%

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

   6. Applied egg-rr 60.6%

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

   if 5.0000000000000002e-298 < (*.f64 V l) < 1.0000000000000001e299

   1. Initial program 85.4%

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

      1. associate-/r* 78.2%

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

      2. clear-num 77.3%

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

      3. sqrt-div 77.3%

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

      4. metadata-eval 77.3%

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

      5. div-inv 76.4%

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

      6. clear-num 76.7%

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

   4. Applied egg-rr 76.7%

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

      1. *-commutative 76.7%

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

      2. associate-*l/ 84.9%

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

      3. associate-/l* 75.2%

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

   6. Simplified 75.2%

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

      1. un-div-inv 75.3%

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

   8. Applied egg-rr 75.3%

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

      1. associate-*r/ 85.1%

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

      2. frac-2neg 85.1%

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

      3. *-commutative 85.1%

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

      4. distribute-rgt-neg-out 85.1%

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

      5. sqrt-undiv 0.0%

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

      6. div-inv 0.0%

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

      7. *-commutative 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 0.0%

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

      10. sqr-neg 0.0%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.0%

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

      13. add-sqr-sqrt 0.0%

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

      14. sqrt-unprod 63.2%

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

      15. sqr-neg 63.2%

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

      16. sqrt-unprod 99.3%

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

      17. add-sqr-sqrt 99.3%

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

   10. Applied egg-rr 99.3%

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

       1. associate-*r/ 99.5%

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

       2. *-rgt-identity 99.5%

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

   12. Simplified 99.5%

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

   if 1.0000000000000001e299 < (*.f64 V l)

   1. Initial program 12.2%

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

      1. associate-/r* 65.0%

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

      2. clear-num 64.9%

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

      3. sqrt-div 64.9%

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

      4. metadata-eval 64.9%

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

      5. div-inv 64.9%

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

      6. clear-num 64.9%

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

   4. Applied egg-rr 64.9%

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

      1. *-commutative 64.9%

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

      2. associate-*l/ 12.2%

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

      3. associate-/l* 64.9%

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

   6. Simplified 64.9%

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

      1. inv-pow 64.9%

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

      2. sqrt-pow2 65.0%

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

      3. metadata-eval 65.0%

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

   8. Applied egg-rr 65.0%

      \[\leadsto c0 \cdot \color{blue}{{\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+208}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{-V}}{\sqrt{\frac{A}{-\ell}}}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-304}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-298}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+299}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-298} \lor \neg \left(t\_0 \leq 2 \cdot 10^{+231}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 (* c0 (sqrt (/ A (* V l))))))
   (if (or (<= t_0 5e-298) (not (<= t_0 2e+231)))
     (* c0 (sqrt (/ (/ A V) l)))
     t_0)))

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 * l)));
	double tmp;
	if ((t_0 <= 5e-298) || !(t_0 <= 2e+231)) {
		tmp = c0 * sqrt(((A / V) / l));
	} else {
		tmp = 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 = c0 * sqrt((a / (v * l)))
    if ((t_0 <= 5d-298) .or. (.not. (t_0 <= 2d+231))) then
        tmp = c0 * sqrt(((a / v) / l))
    else
        tmp = 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 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if ((t_0 <= 5e-298) || !(t_0 <= 2e+231)) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else {
		tmp = t_0;
	}
	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 * l)))
	tmp = 0
	if (t_0 <= 5e-298) or not (t_0 <= 2e+231):
		tmp = c0 * math.sqrt(((A / V) / l))
	else:
		tmp = t_0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if ((t_0 <= 5e-298) || !(t_0 <= 2e+231))
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	else
		tmp = 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 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if ((t_0 <= 5e-298) || ~((t_0 <= 2e+231)))
		tmp = c0 * sqrt(((A / V) / l));
	else
		tmp = 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[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 5e-298], N[Not[LessEqual[t$95$0, 2e+231]], $MachinePrecision]], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 5.0000000000000002e-298 or 2.0000000000000001e231 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 66.8%

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

      1. associate-/r* 73.4%

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

   3. Simplified 73.4%

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

   if 5.0000000000000002e-298 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 2.0000000000000001e231

   1. Initial program 99.2%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 5 \cdot 10^{-298} \lor \neg \left(c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 2 \cdot 10^{+231}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-298}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+304}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

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 (* c0 (sqrt (/ A (* V l))))))
   (if (<= t_0 5e-298)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 5e+304) 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 = c0 * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 5e-298) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 5e+304) {
		tmp = 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 = c0 * sqrt((a / (v * l)))
    if (t_0 <= 5d-298) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 5d+304) then
        tmp = 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 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 5e-298) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 5e+304) {
		tmp = 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 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 5e-298:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 5e+304:
		tmp = 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(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if (t_0 <= 5e-298)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 5e+304)
		tmp = 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 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= 5e-298)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 5e+304)
		tmp = 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[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-298], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+304], t$95$0, N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 5.0000000000000002e-298

   1. Initial program 66.9%

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

      1. associate-/r* 72.8%

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

   3. Simplified 72.8%

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

   if 5.0000000000000002e-298 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 4.9999999999999997e304

   1. Initial program 99.2%

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

   if 4.9999999999999997e304 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 60.1%

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

      1. associate-/r* 73.5%

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

      2. clear-num 73.4%

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

      3. sqrt-div 76.5%

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

      4. metadata-eval 76.5%

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

      5. div-inv 76.5%

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

      6. clear-num 76.5%

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

   4. Applied egg-rr 76.5%

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

      1. *-commutative 76.5%

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

      2. associate-*l/ 61.3%

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

      3. associate-/l* 76.0%

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

   6. Simplified 76.0%

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

      1. un-div-inv 76.0%

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

   8. Applied egg-rr 76.0%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 5 \cdot 10^{-298}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 5 \cdot 10^{+304}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-298}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+231}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \end{array} \]

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 (* c0 (sqrt (/ A (* V l))))))
   (if (<= t_0 5e-298)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 2e+231) 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 = c0 * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 5e-298) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 2e+231) {
		tmp = 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 = c0 * sqrt((a / (v * l)))
    if (t_0 <= 5d-298) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 2d+231) then
        tmp = 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 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 5e-298) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 2e+231) {
		tmp = 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 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 5e-298:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 2e+231:
		tmp = 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(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if (t_0 <= 5e-298)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 2e+231)
		tmp = 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 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= 5e-298)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 2e+231)
		tmp = 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[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-298], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+231], t$95$0, N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 5.0000000000000002e-298

   1. Initial program 66.9%

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

      1. associate-/r* 72.8%

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

   3. Simplified 72.8%

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

   if 5.0000000000000002e-298 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 2.0000000000000001e231

   1. Initial program 99.2%

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

   if 2.0000000000000001e231 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 66.4%

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

      1. associate-/r* 75.7%

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

      2. clear-num 75.6%

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

      3. sqrt-div 78.2%

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

      4. metadata-eval 78.2%

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

      5. div-inv 78.2%

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

      6. clear-num 78.2%

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

   4. Applied egg-rr 78.2%

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

      1. *-commutative 78.2%

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

      2. associate-*l/ 67.5%

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

      3. associate-/l* 78.6%

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

   6. Simplified 78.6%

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

      1. un-div-inv 78.7%

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

      2. clear-num 78.6%

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

   8. Applied egg-rr 78.6%

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

      1. associate-/r/ 78.6%

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

      2. associate-*l/ 78.7%

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

      3. *-lft-identity 78.7%

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

      4. associate-*r/ 67.5%

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

      5. associate-*l/ 78.3%

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

      6. associate-/r/ 77.7%

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

   10. Simplified 77.7%

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

       1. associate-/r/ 78.3%

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

   12. Applied egg-rr 78.3%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 5 \cdot 10^{-298}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 2 \cdot 10^{+231}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+62}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-170}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq 10^{+299}\right):\\ \;\;\;\;c0 \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

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) -2e+62)
   (* c0 (sqrt (/ (/ A l) V)))
   (if (<= (* V l) -5e-170)
     (/ c0 (sqrt (/ (* V l) A)))
     (if (or (<= (* V l) 0.0) (not (<= (* V l) 1e+299)))
       (* c0 (pow (* V (/ l A)) -0.5))
       (* c0 (/ (sqrt A) (sqrt (* 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) <= -2e+62) {
		tmp = c0 * sqrt(((A / l) / V));
	} else if ((V * l) <= -5e-170) {
		tmp = c0 / sqrt(((V * l) / A));
	} else if (((V * l) <= 0.0) || !((V * l) <= 1e+299)) {
		tmp = c0 * pow((V * (l / A)), -0.5);
	} else {
		tmp = c0 * (sqrt(A) / sqrt((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) <= (-2d+62)) then
        tmp = c0 * sqrt(((a / l) / v))
    else if ((v * l) <= (-5d-170)) then
        tmp = c0 / sqrt(((v * l) / a))
    else if (((v * l) <= 0.0d0) .or. (.not. ((v * l) <= 1d+299))) then
        tmp = c0 * ((v * (l / a)) ** (-0.5d0))
    else
        tmp = c0 * (sqrt(a) / sqrt((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) <= -2e+62) {
		tmp = c0 * Math.sqrt(((A / l) / V));
	} else if ((V * l) <= -5e-170) {
		tmp = c0 / Math.sqrt(((V * l) / A));
	} else if (((V * l) <= 0.0) || !((V * l) <= 1e+299)) {
		tmp = c0 * Math.pow((V * (l / A)), -0.5);
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((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) <= -2e+62:
		tmp = c0 * math.sqrt(((A / l) / V))
	elif (V * l) <= -5e-170:
		tmp = c0 / math.sqrt(((V * l) / A))
	elif ((V * l) <= 0.0) or not ((V * l) <= 1e+299):
		tmp = c0 * math.pow((V * (l / A)), -0.5)
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((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) <= -2e+62)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	elseif (Float64(V * l) <= -5e-170)
		tmp = Float64(c0 / sqrt(Float64(Float64(V * l) / A)));
	elseif ((Float64(V * l) <= 0.0) || !(Float64(V * l) <= 1e+299))
		tmp = Float64(c0 * (Float64(V * Float64(l / A)) ^ -0.5));
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(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) <= -2e+62)
		tmp = c0 * sqrt(((A / l) / V));
	elseif ((V * l) <= -5e-170)
		tmp = c0 / sqrt(((V * l) / A));
	elseif (((V * l) <= 0.0) || ~(((V * l) <= 1e+299)))
		tmp = c0 * ((V * (l / A)) ^ -0.5);
	else
		tmp = c0 * (sqrt(A) / sqrt((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], -2e+62], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -5e-170], N[(c0 / N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[Not[LessEqual[N[(V * l), $MachinePrecision], 1e+299]], $MachinePrecision]], N[(c0 * N[Power[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -2.00000000000000007e62

   1. Initial program 66.3%

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

      1. clear-num 66.4%

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

      2. associate-/r/ 66.3%

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

      3. associate-/r* 66.3%

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

   4. Applied egg-rr 66.3%

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

      1. *-commutative 66.3%

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

      2. associate-/r* 66.3%

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

      3. div-inv 66.3%

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

      4. *-commutative 66.3%

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

      5. associate-/r* 77.3%

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

   6. Applied egg-rr 77.3%

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

   if -2.00000000000000007e62 < (*.f64 V l) < -5.0000000000000001e-170

   1. Initial program 88.2%

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

      1. clear-num 88.2%

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

      2. associate-/r/ 88.3%

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

      3. associate-/r* 88.3%

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

   4. Applied egg-rr 88.3%

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

      1. *-commutative 88.3%

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

      2. associate-/r* 88.3%

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

      3. div-inv 88.2%

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

      4. *-commutative 88.2%

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

      5. associate-/r* 75.6%

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

   6. Applied egg-rr 75.6%

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

      1. clear-num 75.6%

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

      2. sqrt-div 75.8%

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

      3. metadata-eval 75.8%

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

      4. div-inv 75.2%

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

      5. clear-num 76.1%

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

      6. div-inv 76.2%

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

      7. *-commutative 76.2%

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

      8. associate-*l/ 90.3%

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

   8. Applied egg-rr 90.3%

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

   if -5.0000000000000001e-170 < (*.f64 V l) < -0.0 or 1.0000000000000001e299 < (*.f64 V l)

   1. Initial program 46.1%

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

      1. associate-/r* 72.2%

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

      2. clear-num 72.1%

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

      3. sqrt-div 73.8%

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

      4. metadata-eval 73.8%

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

      5. div-inv 73.8%

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

      6. clear-num 73.8%

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

   4. Applied egg-rr 73.8%

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

      1. *-commutative 73.8%

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

      2. associate-*l/ 46.0%

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

      3. associate-/l* 73.8%

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

   6. Simplified 73.8%

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

      1. inv-pow 73.8%

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

      2. sqrt-pow2 73.9%

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

      3. metadata-eval 73.9%

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

   8. Applied egg-rr 73.9%

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

   if -0.0 < (*.f64 V l) < 1.0000000000000001e299

   1. Initial program 85.6%

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

      1. sqrt-div 99.4%

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

      2. div-inv 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. associate-*r/ 99.4%

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

      2. *-rgt-identity 99.4%

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

   6. Simplified 99.4%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+62}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-170}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq 10^{+299}\right):\\ \;\;\;\;c0 \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+95}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-69}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-298}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+299}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\ \end{array} \end{array} \]

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+95)
     t_0
     (if (<= (* V l) -2e-69)
       (/ c0 (sqrt (/ (* V l) A)))
       (if (<= (* V l) 5e-298)
         t_0
         (if (<= (* V l) 1e+299)
           (* c0 (/ (sqrt A) (sqrt (* V l))))
           (* c0 (pow (* V (/ l A)) -0.5))))))))

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+95) {
		tmp = t_0;
	} else if ((V * l) <= -2e-69) {
		tmp = c0 / sqrt(((V * l) / A));
	} else if ((V * l) <= 5e-298) {
		tmp = t_0;
	} else if ((V * l) <= 1e+299) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0 * pow((V * (l / A)), -0.5);
	}
	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+95)) then
        tmp = t_0
    else if ((v * l) <= (-2d-69)) then
        tmp = c0 / sqrt(((v * l) / a))
    else if ((v * l) <= 5d-298) then
        tmp = t_0
    else if ((v * l) <= 1d+299) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = c0 * ((v * (l / a)) ** (-0.5d0))
    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+95) {
		tmp = t_0;
	} else if ((V * l) <= -2e-69) {
		tmp = c0 / Math.sqrt(((V * l) / A));
	} else if ((V * l) <= 5e-298) {
		tmp = t_0;
	} else if ((V * l) <= 1e+299) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0 * Math.pow((V * (l / A)), -0.5);
	}
	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+95:
		tmp = t_0
	elif (V * l) <= -2e-69:
		tmp = c0 / math.sqrt(((V * l) / A))
	elif (V * l) <= 5e-298:
		tmp = t_0
	elif (V * l) <= 1e+299:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0 * math.pow((V * (l / A)), -0.5)
	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+95)
		tmp = t_0;
	elseif (Float64(V * l) <= -2e-69)
		tmp = Float64(c0 / sqrt(Float64(Float64(V * l) / A)));
	elseif (Float64(V * l) <= 5e-298)
		tmp = t_0;
	elseif (Float64(V * l) <= 1e+299)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 * (Float64(V * Float64(l / A)) ^ -0.5));
	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+95)
		tmp = t_0;
	elseif ((V * l) <= -2e-69)
		tmp = c0 / sqrt(((V * l) / A));
	elseif ((V * l) <= 5e-298)
		tmp = t_0;
	elseif ((V * l) <= 1e+299)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0 * ((V * (l / A)) ^ -0.5);
	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+95], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -2e-69], N[(c0 / N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e-298], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], 1e+299], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Power[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -5.00000000000000025e95 or -1.9999999999999999e-69 < (*.f64 V l) < 5.0000000000000002e-298

   1. Initial program 65.5%

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

      1. associate-/r* 75.2%

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

      2. sqrt-div 47.2%

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

      3. div-inv 47.3%

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

   4. Applied egg-rr 47.3%

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

      1. associate-*r/ 47.2%

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

      2. *-rgt-identity 47.2%

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

   6. Simplified 47.2%

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

   if -5.00000000000000025e95 < (*.f64 V l) < -1.9999999999999999e-69

   1. Initial program 93.4%

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

      1. clear-num 93.3%

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

      2. associate-/r/ 93.4%

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

      3. associate-/r* 93.4%

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

   4. Applied egg-rr 93.4%

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

      1. *-commutative 93.4%

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

      2. associate-/r* 93.4%

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

      3. div-inv 93.4%

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

      4. *-commutative 93.4%

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

      5. associate-/r* 73.0%

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

   6. Applied egg-rr 73.0%

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

      1. clear-num 73.0%

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

      2. sqrt-div 73.0%

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

      3. metadata-eval 73.0%

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

      4. div-inv 72.0%

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

      5. clear-num 73.4%

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

      6. div-inv 73.5%

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

      7. *-commutative 73.5%

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

      8. associate-*l/ 95.1%

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

   8. Applied egg-rr 95.1%

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

   if 5.0000000000000002e-298 < (*.f64 V l) < 1.0000000000000001e299

   1. Initial program 85.4%

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

      1. sqrt-div 99.4%

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

      2. div-inv 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. associate-*r/ 99.4%

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

      2. *-rgt-identity 99.4%

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

   6. Simplified 99.4%

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

   if 1.0000000000000001e299 < (*.f64 V l)

   1. Initial program 12.2%

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

      1. associate-/r* 65.0%

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

      2. clear-num 64.9%

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

      3. sqrt-div 64.9%

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

      4. metadata-eval 64.9%

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

      5. div-inv 64.9%

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

      6. clear-num 64.9%

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

   4. Applied egg-rr 64.9%

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

      1. *-commutative 64.9%

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

      2. associate-*l/ 12.2%

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

      3. associate-/l* 64.9%

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

   6. Simplified 64.9%

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

      1. inv-pow 64.9%

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

      2. sqrt-pow2 65.0%

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

      3. metadata-eval 65.0%

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

   8. Applied egg-rr 65.0%

      \[\leadsto c0 \cdot \color{blue}{{\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+95}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-69}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-298}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+299}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{V}}\\ \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+165}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot t\_0\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-69}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-298}:\\ \;\;\;\;c0 \cdot \frac{t\_0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+299}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\ \end{array} \end{array} \]

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 (sqrt (/ A V))))
   (if (<= (* V l) -5e+165)
     (* (/ c0 (sqrt l)) t_0)
     (if (<= (* V l) -2e-69)
       (/ c0 (sqrt (/ (* V l) A)))
       (if (<= (* V l) 5e-298)
         (* c0 (/ t_0 (sqrt l)))
         (if (<= (* V l) 1e+299)
           (* c0 (/ (sqrt A) (sqrt (* V l))))
           (* c0 (pow (* V (/ l A)) -0.5))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt((A / V));
	double tmp;
	if ((V * l) <= -5e+165) {
		tmp = (c0 / sqrt(l)) * t_0;
	} else if ((V * l) <= -2e-69) {
		tmp = c0 / sqrt(((V * l) / A));
	} else if ((V * l) <= 5e-298) {
		tmp = c0 * (t_0 / sqrt(l));
	} else if ((V * l) <= 1e+299) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0 * pow((V * (l / A)), -0.5);
	}
	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 = sqrt((a / v))
    if ((v * l) <= (-5d+165)) then
        tmp = (c0 / sqrt(l)) * t_0
    else if ((v * l) <= (-2d-69)) then
        tmp = c0 / sqrt(((v * l) / a))
    else if ((v * l) <= 5d-298) then
        tmp = c0 * (t_0 / sqrt(l))
    else if ((v * l) <= 1d+299) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = c0 * ((v * (l / a)) ** (-0.5d0))
    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 = Math.sqrt((A / V));
	double tmp;
	if ((V * l) <= -5e+165) {
		tmp = (c0 / Math.sqrt(l)) * t_0;
	} else if ((V * l) <= -2e-69) {
		tmp = c0 / Math.sqrt(((V * l) / A));
	} else if ((V * l) <= 5e-298) {
		tmp = c0 * (t_0 / Math.sqrt(l));
	} else if ((V * l) <= 1e+299) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0 * Math.pow((V * (l / A)), -0.5);
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt((A / V))
	tmp = 0
	if (V * l) <= -5e+165:
		tmp = (c0 / math.sqrt(l)) * t_0
	elif (V * l) <= -2e-69:
		tmp = c0 / math.sqrt(((V * l) / A))
	elif (V * l) <= 5e-298:
		tmp = c0 * (t_0 / math.sqrt(l))
	elif (V * l) <= 1e+299:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0 * math.pow((V * (l / A)), -0.5)
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = sqrt(Float64(A / V))
	tmp = 0.0
	if (Float64(V * l) <= -5e+165)
		tmp = Float64(Float64(c0 / sqrt(l)) * t_0);
	elseif (Float64(V * l) <= -2e-69)
		tmp = Float64(c0 / sqrt(Float64(Float64(V * l) / A)));
	elseif (Float64(V * l) <= 5e-298)
		tmp = Float64(c0 * Float64(t_0 / sqrt(l)));
	elseif (Float64(V * l) <= 1e+299)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 * (Float64(V * Float64(l / A)) ^ -0.5));
	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 = sqrt((A / V));
	tmp = 0.0;
	if ((V * l) <= -5e+165)
		tmp = (c0 / sqrt(l)) * t_0;
	elseif ((V * l) <= -2e-69)
		tmp = c0 / sqrt(((V * l) / A));
	elseif ((V * l) <= 5e-298)
		tmp = c0 * (t_0 / sqrt(l));
	elseif ((V * l) <= 1e+299)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0 * ((V * (l / A)) ^ -0.5);
	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[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -5e+165], N[(N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-69], N[(c0 / N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e-298], N[(c0 * N[(t$95$0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+299], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Power[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -4.9999999999999997e165

   1. Initial program 54.7%

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

      1. add-sqr-sqrt 39.0%

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

      2. sqrt-unprod 39.7%

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

      3. *-commutative 39.7%

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

      4. *-commutative 39.7%

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

      5. swap-sqr 34.1%

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

      6. add-sqr-sqrt 34.1%

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

      7. pow2 34.1%

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

   4. Applied egg-rr 34.1%

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

      1. sqrt-prod 34.0%

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

      2. div-inv 34.0%

         \[\leadsto \sqrt{\color{blue}{A \cdot \frac{1}{V \cdot \ell}}} \cdot \sqrt{{c0}^{2}} \]

      3. associate-/r* 34.0%

         \[\leadsto \sqrt{A \cdot \color{blue}{\frac{\frac{1}{V}}{\ell}}} \cdot \sqrt{{c0}^{2}} \]

      4. *-commutative 34.0%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{V}}{\ell} \cdot A}} \cdot \sqrt{{c0}^{2}} \]

      5. *-commutative 34.0%

         \[\leadsto \sqrt{\color{blue}{A \cdot \frac{\frac{1}{V}}{\ell}}} \cdot \sqrt{{c0}^{2}} \]

      6. associate-*r/ 34.1%

         \[\leadsto \sqrt{\color{blue}{\frac{A \cdot \frac{1}{V}}{\ell}}} \cdot \sqrt{{c0}^{2}} \]

      7. div-inv 34.1%

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

      8. frac-2neg 34.1%

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

      9. associate-/r* 34.0%

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

      10. *-commutative 34.0%

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

      11. sqrt-undiv 36.4%

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

      12. sqrt-pow1 62.0%

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

      13. metadata-eval 62.0%

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

      14. pow1 62.0%

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

      15. *-commutative 62.0%

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

      16. associate-*r/ 61.7%

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

      17. sqrt-prod 55.2%

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

      18. times-frac 63.0%

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

      19. sqrt-div 56.0%

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

   6. Applied egg-rr 56.0%

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

   if -4.9999999999999997e165 < (*.f64 V l) < -1.9999999999999999e-69

   1. Initial program 95.1%

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

      1. clear-num 95.0%

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

      2. associate-/r/ 95.1%

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

      3. associate-/r* 95.1%

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

   4. Applied egg-rr 95.1%

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

      1. *-commutative 95.1%

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

      2. associate-/r* 95.1%

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

      3. div-inv 95.1%

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

      4. *-commutative 95.1%

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

      5. associate-/r* 80.0%

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

   6. Applied egg-rr 80.0%

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

      1. clear-num 80.0%

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

      2. sqrt-div 80.0%

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

      3. metadata-eval 80.0%

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

      4. div-inv 79.3%

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

      5. clear-num 80.3%

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

      6. div-inv 80.3%

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

      7. *-commutative 80.3%

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

      8. associate-*l/ 96.3%

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

   8. Applied egg-rr 96.3%

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

   if -1.9999999999999999e-69 < (*.f64 V l) < 5.0000000000000002e-298

   1. Initial program 66.1%

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

      1. associate-/r* 73.9%

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

      2. sqrt-div 45.4%

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

      3. div-inv 45.5%

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

   4. Applied egg-rr 45.5%

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

      1. associate-*r/ 45.4%

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

      2. *-rgt-identity 45.4%

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

   6. Simplified 45.4%

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

   if 5.0000000000000002e-298 < (*.f64 V l) < 1.0000000000000001e299

   1. Initial program 85.4%

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

      1. sqrt-div 99.4%

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

      2. div-inv 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. associate-*r/ 99.4%

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

      2. *-rgt-identity 99.4%

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

   6. Simplified 99.4%

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

   if 1.0000000000000001e299 < (*.f64 V l)

   1. Initial program 12.2%

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

      1. associate-/r* 65.0%

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

      2. clear-num 64.9%

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

      3. sqrt-div 64.9%

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

      4. metadata-eval 64.9%

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

      5. div-inv 64.9%

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

      6. clear-num 64.9%

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

   4. Applied egg-rr 64.9%

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

      1. *-commutative 64.9%

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

      2. associate-*l/ 12.2%

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

      3. associate-/l* 64.9%

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

   6. Simplified 64.9%

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

      1. inv-pow 64.9%

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

      2. sqrt-pow2 65.0%

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

      3. metadata-eval 65.0%

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

   8. Applied egg-rr 65.0%

      \[\leadsto c0 \cdot \color{blue}{{\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+165}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-69}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-298}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+299}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{V}}\\ \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+165}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot t\_0\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-69}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-298}:\\ \;\;\;\;c0 \cdot \frac{t\_0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+299}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\ \end{array} \end{array} \]

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 (sqrt (/ A V))))
   (if (<= (* V l) -5e+165)
     (* (/ c0 (sqrt l)) t_0)
     (if (<= (* V l) -2e-69)
       (/ c0 (sqrt (/ (* V l) A)))
       (if (<= (* V l) 5e-298)
         (* c0 (/ t_0 (sqrt l)))
         (if (<= (* V l) 1e+299)
           (/ c0 (/ (sqrt (* V l)) (sqrt A)))
           (* c0 (pow (* V (/ l A)) -0.5))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt((A / V));
	double tmp;
	if ((V * l) <= -5e+165) {
		tmp = (c0 / sqrt(l)) * t_0;
	} else if ((V * l) <= -2e-69) {
		tmp = c0 / sqrt(((V * l) / A));
	} else if ((V * l) <= 5e-298) {
		tmp = c0 * (t_0 / sqrt(l));
	} else if ((V * l) <= 1e+299) {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	} else {
		tmp = c0 * pow((V * (l / A)), -0.5);
	}
	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 = sqrt((a / v))
    if ((v * l) <= (-5d+165)) then
        tmp = (c0 / sqrt(l)) * t_0
    else if ((v * l) <= (-2d-69)) then
        tmp = c0 / sqrt(((v * l) / a))
    else if ((v * l) <= 5d-298) then
        tmp = c0 * (t_0 / sqrt(l))
    else if ((v * l) <= 1d+299) then
        tmp = c0 / (sqrt((v * l)) / sqrt(a))
    else
        tmp = c0 * ((v * (l / a)) ** (-0.5d0))
    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 = Math.sqrt((A / V));
	double tmp;
	if ((V * l) <= -5e+165) {
		tmp = (c0 / Math.sqrt(l)) * t_0;
	} else if ((V * l) <= -2e-69) {
		tmp = c0 / Math.sqrt(((V * l) / A));
	} else if ((V * l) <= 5e-298) {
		tmp = c0 * (t_0 / Math.sqrt(l));
	} else if ((V * l) <= 1e+299) {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	} else {
		tmp = c0 * Math.pow((V * (l / A)), -0.5);
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt((A / V))
	tmp = 0
	if (V * l) <= -5e+165:
		tmp = (c0 / math.sqrt(l)) * t_0
	elif (V * l) <= -2e-69:
		tmp = c0 / math.sqrt(((V * l) / A))
	elif (V * l) <= 5e-298:
		tmp = c0 * (t_0 / math.sqrt(l))
	elif (V * l) <= 1e+299:
		tmp = c0 / (math.sqrt((V * l)) / math.sqrt(A))
	else:
		tmp = c0 * math.pow((V * (l / A)), -0.5)
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = sqrt(Float64(A / V))
	tmp = 0.0
	if (Float64(V * l) <= -5e+165)
		tmp = Float64(Float64(c0 / sqrt(l)) * t_0);
	elseif (Float64(V * l) <= -2e-69)
		tmp = Float64(c0 / sqrt(Float64(Float64(V * l) / A)));
	elseif (Float64(V * l) <= 5e-298)
		tmp = Float64(c0 * Float64(t_0 / sqrt(l)));
	elseif (Float64(V * l) <= 1e+299)
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	else
		tmp = Float64(c0 * (Float64(V * Float64(l / A)) ^ -0.5));
	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 = sqrt((A / V));
	tmp = 0.0;
	if ((V * l) <= -5e+165)
		tmp = (c0 / sqrt(l)) * t_0;
	elseif ((V * l) <= -2e-69)
		tmp = c0 / sqrt(((V * l) / A));
	elseif ((V * l) <= 5e-298)
		tmp = c0 * (t_0 / sqrt(l));
	elseif ((V * l) <= 1e+299)
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	else
		tmp = c0 * ((V * (l / A)) ^ -0.5);
	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[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -5e+165], N[(N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-69], N[(c0 / N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e-298], N[(c0 * N[(t$95$0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+299], N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Power[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -4.9999999999999997e165

   1. Initial program 54.7%

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

      1. add-sqr-sqrt 39.0%

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

      2. sqrt-unprod 39.7%

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

      3. *-commutative 39.7%

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

      4. *-commutative 39.7%

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

      5. swap-sqr 34.1%

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

      6. add-sqr-sqrt 34.1%

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

      7. pow2 34.1%

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

   4. Applied egg-rr 34.1%

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

      1. sqrt-prod 34.0%

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

      2. div-inv 34.0%

         \[\leadsto \sqrt{\color{blue}{A \cdot \frac{1}{V \cdot \ell}}} \cdot \sqrt{{c0}^{2}} \]

      3. associate-/r* 34.0%

         \[\leadsto \sqrt{A \cdot \color{blue}{\frac{\frac{1}{V}}{\ell}}} \cdot \sqrt{{c0}^{2}} \]

      4. *-commutative 34.0%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{V}}{\ell} \cdot A}} \cdot \sqrt{{c0}^{2}} \]

      5. *-commutative 34.0%

         \[\leadsto \sqrt{\color{blue}{A \cdot \frac{\frac{1}{V}}{\ell}}} \cdot \sqrt{{c0}^{2}} \]

      6. associate-*r/ 34.1%

         \[\leadsto \sqrt{\color{blue}{\frac{A \cdot \frac{1}{V}}{\ell}}} \cdot \sqrt{{c0}^{2}} \]

      7. div-inv 34.1%

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

      8. frac-2neg 34.1%

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

      9. associate-/r* 34.0%

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

      10. *-commutative 34.0%

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

      11. sqrt-undiv 36.4%

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

      12. sqrt-pow1 62.0%

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

      13. metadata-eval 62.0%

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

      14. pow1 62.0%

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

      15. *-commutative 62.0%

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

      16. associate-*r/ 61.7%

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

      17. sqrt-prod 55.2%

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

      18. times-frac 63.0%

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

      19. sqrt-div 56.0%

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

   6. Applied egg-rr 56.0%

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

   if -4.9999999999999997e165 < (*.f64 V l) < -1.9999999999999999e-69

   1. Initial program 95.1%

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

      1. clear-num 95.0%

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

      2. associate-/r/ 95.1%

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

      3. associate-/r* 95.1%

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

   4. Applied egg-rr 95.1%

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

      1. *-commutative 95.1%

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

      2. associate-/r* 95.1%

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

      3. div-inv 95.1%

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

      4. *-commutative 95.1%

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

      5. associate-/r* 80.0%

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

   6. Applied egg-rr 80.0%

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

      1. clear-num 80.0%

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

      2. sqrt-div 80.0%

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

      3. metadata-eval 80.0%

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

      4. div-inv 79.3%

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

      5. clear-num 80.3%

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

      6. div-inv 80.3%

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

      7. *-commutative 80.3%

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

      8. associate-*l/ 96.3%

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

   8. Applied egg-rr 96.3%

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

   if -1.9999999999999999e-69 < (*.f64 V l) < 5.0000000000000002e-298

   1. Initial program 66.1%

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

      1. associate-/r* 73.9%

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

      2. sqrt-div 45.4%

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

      3. div-inv 45.5%

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

   4. Applied egg-rr 45.5%

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

      1. associate-*r/ 45.4%

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

      2. *-rgt-identity 45.4%

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

   6. Simplified 45.4%

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

   if 5.0000000000000002e-298 < (*.f64 V l) < 1.0000000000000001e299

   1. Initial program 85.4%

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

      1. associate-/r* 78.2%

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

      2. clear-num 77.3%

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

      3. sqrt-div 77.3%

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

      4. metadata-eval 77.3%

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

      5. div-inv 76.4%

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

      6. clear-num 76.7%

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

   4. Applied egg-rr 76.7%

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

      1. *-commutative 76.7%

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

      2. associate-*l/ 84.9%

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

      3. associate-/l* 75.2%

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

   6. Simplified 75.2%

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

      1. un-div-inv 75.3%

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

   8. Applied egg-rr 75.3%

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

      1. associate-*r/ 85.1%

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

      2. frac-2neg 85.1%

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

      3. *-commutative 85.1%

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

      4. distribute-rgt-neg-out 85.1%

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

      5. sqrt-undiv 0.0%

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

      6. div-inv 0.0%

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

      7. *-commutative 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 0.0%

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

      10. sqr-neg 0.0%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.0%

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

      13. add-sqr-sqrt 0.0%

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

      14. sqrt-unprod 63.2%

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

      15. sqr-neg 63.2%

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

      16. sqrt-unprod 99.3%

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

      17. add-sqr-sqrt 99.3%

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

   10. Applied egg-rr 99.3%

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

       1. associate-*r/ 99.5%

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

       2. *-rgt-identity 99.5%

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

   12. Simplified 99.5%

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

   if 1.0000000000000001e299 < (*.f64 V l)

   1. Initial program 12.2%

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

      1. associate-/r* 65.0%

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

      2. clear-num 64.9%

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

      3. sqrt-div 64.9%

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

      4. metadata-eval 64.9%

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

      5. div-inv 64.9%

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

      6. clear-num 64.9%

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

   4. Applied egg-rr 64.9%

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

      1. *-commutative 64.9%

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

      2. associate-*l/ 12.2%

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

      3. associate-/l* 64.9%

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

   6. Simplified 64.9%

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

      1. inv-pow 64.9%

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

      2. sqrt-pow2 65.0%

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

      3. metadata-eval 65.0%

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

   8. Applied egg-rr 65.0%

      \[\leadsto c0 \cdot \color{blue}{{\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+165}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-69}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-298}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+299}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-304}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-298}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+299}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\ \end{array} \end{array} \]

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 l)) (sqrt (/ A V)))))
   (if (<= (* V l) (- INFINITY))
     t_0
     (if (<= (* V l) -2e-304)
       (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
       (if (<= (* V l) 5e-298)
         t_0
         (if (<= (* V l) 1e+299)
           (/ c0 (/ (sqrt (* V l)) (sqrt A)))
           (* c0 (pow (* V (/ l A)) -0.5))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = (c0 / sqrt(l)) * sqrt((A / V));
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = t_0;
	} else if ((V * l) <= -2e-304) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 5e-298) {
		tmp = t_0;
	} else if ((V * l) <= 1e+299) {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	} else {
		tmp = c0 * pow((V * (l / A)), -0.5);
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = (c0 / Math.sqrt(l)) * Math.sqrt((A / V));
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else if ((V * l) <= -2e-304) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 5e-298) {
		tmp = t_0;
	} else if ((V * l) <= 1e+299) {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	} else {
		tmp = c0 * Math.pow((V * (l / A)), -0.5);
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = (c0 / math.sqrt(l)) * math.sqrt((A / V))
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = t_0
	elif (V * l) <= -2e-304:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 5e-298:
		tmp = t_0
	elif (V * l) <= 1e+299:
		tmp = c0 / (math.sqrt((V * l)) / math.sqrt(A))
	else:
		tmp = c0 * math.pow((V * (l / A)), -0.5)
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(Float64(c0 / sqrt(l)) * sqrt(Float64(A / V)))
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = t_0;
	elseif (Float64(V * l) <= -2e-304)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 5e-298)
		tmp = t_0;
	elseif (Float64(V * l) <= 1e+299)
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	else
		tmp = Float64(c0 * (Float64(V * Float64(l / A)) ^ -0.5));
	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(l)) * sqrt((A / V));
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = t_0;
	elseif ((V * l) <= -2e-304)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 5e-298)
		tmp = t_0;
	elseif ((V * l) <= 1e+299)
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	else
		tmp = c0 * ((V * (l / A)) ^ -0.5);
	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[(N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -2e-304], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e-298], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], 1e+299], N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Power[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -inf.0 or -1.99999999999999994e-304 < (*.f64 V l) < 5.0000000000000002e-298

   1. Initial program 37.9%

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

      1. add-sqr-sqrt 30.2%

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

      2. sqrt-unprod 28.2%

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

      3. *-commutative 28.2%

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

      4. *-commutative 28.2%

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

      5. swap-sqr 27.5%

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

      6. add-sqr-sqrt 27.5%

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

      7. pow2 27.5%

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

   4. Applied egg-rr 27.5%

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

      1. sqrt-prod 29.7%

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

      2. div-inv 29.7%

         \[\leadsto \sqrt{\color{blue}{A \cdot \frac{1}{V \cdot \ell}}} \cdot \sqrt{{c0}^{2}} \]

      3. associate-/r* 29.7%

         \[\leadsto \sqrt{A \cdot \color{blue}{\frac{\frac{1}{V}}{\ell}}} \cdot \sqrt{{c0}^{2}} \]

      4. *-commutative 29.7%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{V}}{\ell} \cdot A}} \cdot \sqrt{{c0}^{2}} \]

      5. *-commutative 29.7%

         \[\leadsto \sqrt{\color{blue}{A \cdot \frac{\frac{1}{V}}{\ell}}} \cdot \sqrt{{c0}^{2}} \]

      6. associate-*r/ 36.6%

         \[\leadsto \sqrt{\color{blue}{\frac{A \cdot \frac{1}{V}}{\ell}}} \cdot \sqrt{{c0}^{2}} \]

      7. div-inv 36.7%

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

      8. frac-2neg 36.7%

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

      9. associate-/r* 29.7%

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

      10. *-commutative 29.7%

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

      11. sqrt-undiv 22.3%

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

      12. sqrt-pow1 26.7%

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

      13. metadata-eval 26.7%

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

      14. pow1 26.7%

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

      15. *-commutative 26.7%

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

      16. associate-*r/ 26.3%

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

      17. sqrt-prod 32.6%

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

      18. times-frac 39.4%

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

      19. sqrt-div 51.8%

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

   6. Applied egg-rr 51.8%

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

   if -inf.0 < (*.f64 V l) < -1.99999999999999994e-304

   1. Initial program 86.3%

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

      1. frac-2neg 86.3%

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

      2. sqrt-div 99.4%

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

      3. distribute-rgt-neg-in 99.4%

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

   4. Applied egg-rr 99.4%

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

      1. distribute-rgt-neg-out 99.4%

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

      2. *-commutative 99.4%

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

      3. distribute-rgt-neg-in 99.4%

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

   6. Simplified 99.4%

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

   if 5.0000000000000002e-298 < (*.f64 V l) < 1.0000000000000001e299

   1. Initial program 85.4%

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

      1. associate-/r* 78.2%

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

      2. clear-num 77.3%

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

      3. sqrt-div 77.3%

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

      4. metadata-eval 77.3%

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

      5. div-inv 76.4%

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

      6. clear-num 76.7%

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

   4. Applied egg-rr 76.7%

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

      1. *-commutative 76.7%

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

      2. associate-*l/ 84.9%

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

      3. associate-/l* 75.2%

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

   6. Simplified 75.2%

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

      1. un-div-inv 75.3%

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

   8. Applied egg-rr 75.3%

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

      1. associate-*r/ 85.1%

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

      2. frac-2neg 85.1%

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

      3. *-commutative 85.1%

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

      4. distribute-rgt-neg-out 85.1%

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

      5. sqrt-undiv 0.0%

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

      6. div-inv 0.0%

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

      7. *-commutative 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 0.0%

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

      10. sqr-neg 0.0%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.0%

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

      13. add-sqr-sqrt 0.0%

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

      14. sqrt-unprod 63.2%

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

      15. sqr-neg 63.2%

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

      16. sqrt-unprod 99.3%

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

      17. add-sqr-sqrt 99.3%

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

   10. Applied egg-rr 99.3%

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

       1. associate-*r/ 99.5%

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

       2. *-rgt-identity 99.5%

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

   12. Simplified 99.5%

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

   if 1.0000000000000001e299 < (*.f64 V l)

   1. Initial program 12.2%

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

      1. associate-/r* 65.0%

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

      2. clear-num 64.9%

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

      3. sqrt-div 64.9%

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

      4. metadata-eval 64.9%

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

      5. div-inv 64.9%

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

      6. clear-num 64.9%

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

   4. Applied egg-rr 64.9%

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

      1. *-commutative 64.9%

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

      2. associate-*l/ 12.2%

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

      3. associate-/l* 64.9%

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

   6. Simplified 64.9%

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

      1. inv-pow 64.9%

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

      2. sqrt-pow2 65.0%

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

      3. metadata-eval 65.0%

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

   8. Applied egg-rr 65.0%

      \[\leadsto c0 \cdot \color{blue}{{\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-304}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-298}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+299}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 90.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+208}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-304}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-298}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+299}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -1e+208)
   (* c0 (/ (sqrt (/ A (- l))) (sqrt (- V))))
   (if (<= (* V l) -2e-304)
     (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
     (if (<= (* V l) 5e-298)
       (* (/ c0 (sqrt l)) (sqrt (/ A V)))
       (if (<= (* V l) 1e+299)
         (/ c0 (/ (sqrt (* V l)) (sqrt A)))
         (* c0 (pow (* V (/ l A)) -0.5)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+208) {
		tmp = c0 * (sqrt((A / -l)) / sqrt(-V));
	} else if ((V * l) <= -2e-304) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 5e-298) {
		tmp = (c0 / sqrt(l)) * sqrt((A / V));
	} else if ((V * l) <= 1e+299) {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	} else {
		tmp = c0 * pow((V * (l / A)), -0.5);
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-1d+208)) then
        tmp = c0 * (sqrt((a / -l)) / sqrt(-v))
    else if ((v * l) <= (-2d-304)) then
        tmp = c0 * (sqrt(-a) / sqrt((v * -l)))
    else if ((v * l) <= 5d-298) then
        tmp = (c0 / sqrt(l)) * sqrt((a / v))
    else if ((v * l) <= 1d+299) then
        tmp = c0 / (sqrt((v * l)) / sqrt(a))
    else
        tmp = c0 * ((v * (l / a)) ** (-0.5d0))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+208) {
		tmp = c0 * (Math.sqrt((A / -l)) / Math.sqrt(-V));
	} else if ((V * l) <= -2e-304) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 5e-298) {
		tmp = (c0 / Math.sqrt(l)) * Math.sqrt((A / V));
	} else if ((V * l) <= 1e+299) {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	} else {
		tmp = c0 * Math.pow((V * (l / A)), -0.5);
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -1e+208:
		tmp = c0 * (math.sqrt((A / -l)) / math.sqrt(-V))
	elif (V * l) <= -2e-304:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 5e-298:
		tmp = (c0 / math.sqrt(l)) * math.sqrt((A / V))
	elif (V * l) <= 1e+299:
		tmp = c0 / (math.sqrt((V * l)) / math.sqrt(A))
	else:
		tmp = c0 * math.pow((V * (l / A)), -0.5)
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -1e+208)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / Float64(-l))) / sqrt(Float64(-V))));
	elseif (Float64(V * l) <= -2e-304)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 5e-298)
		tmp = Float64(Float64(c0 / sqrt(l)) * sqrt(Float64(A / V)));
	elseif (Float64(V * l) <= 1e+299)
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	else
		tmp = Float64(c0 * (Float64(V * Float64(l / A)) ^ -0.5));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -1e+208)
		tmp = c0 * (sqrt((A / -l)) / sqrt(-V));
	elseif ((V * l) <= -2e-304)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 5e-298)
		tmp = (c0 / sqrt(l)) * sqrt((A / V));
	elseif ((V * l) <= 1e+299)
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	else
		tmp = c0 * ((V * (l / A)) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -1e+208], N[(c0 * N[(N[Sqrt[N[(A / (-l)), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-304], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e-298], N[(N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+299], N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Power[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -9.9999999999999998e207

   1. Initial program 52.1%

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

      1. clear-num 52.2%

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

      2. associate-/r/ 52.1%

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

      3. associate-/r* 52.1%

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

   4. Applied egg-rr 52.1%

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

      1. *-commutative 52.1%

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

      2. associate-/r* 52.1%

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

      3. div-inv 52.1%

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

      4. *-commutative 52.1%

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

      5. associate-/r* 70.6%

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

   6. Applied egg-rr 70.6%

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

      1. frac-2neg 70.6%

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

      2. sqrt-div 50.1%

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

      3. distribute-neg-frac2 50.1%

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

   8. Applied egg-rr 50.1%

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

      1. distribute-frac-neg2 50.1%

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

      2. distribute-neg-frac 50.1%

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

   10. Simplified 50.1%

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

   if -9.9999999999999998e207 < (*.f64 V l) < -1.99999999999999994e-304

   1. Initial program 85.5%

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

      1. frac-2neg 85.5%

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

      2. sqrt-div 99.4%

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

      3. distribute-rgt-neg-in 99.4%

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

   4. Applied egg-rr 99.4%

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

      1. distribute-rgt-neg-out 99.4%

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

      2. *-commutative 99.4%

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

      3. distribute-rgt-neg-in 99.4%

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

   6. Simplified 99.4%

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

   if -1.99999999999999994e-304 < (*.f64 V l) < 5.0000000000000002e-298

   1. Initial program 46.1%

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

      1. add-sqr-sqrt 31.8%

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

      2. sqrt-unprod 28.0%

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

      3. *-commutative 28.0%

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

      4. *-commutative 28.0%

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

      5. swap-sqr 27.6%

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

      6. add-sqr-sqrt 27.6%

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

      7. pow2 27.6%

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

   4. Applied egg-rr 27.6%

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

      1. sqrt-prod 31.7%

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

      2. div-inv 31.7%

         \[\leadsto \sqrt{\color{blue}{A \cdot \frac{1}{V \cdot \ell}}} \cdot \sqrt{{c0}^{2}} \]

      3. associate-/r* 31.7%

         \[\leadsto \sqrt{A \cdot \color{blue}{\frac{\frac{1}{V}}{\ell}}} \cdot \sqrt{{c0}^{2}} \]

      4. *-commutative 31.7%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{V}}{\ell} \cdot A}} \cdot \sqrt{{c0}^{2}} \]

      5. *-commutative 31.7%

         \[\leadsto \sqrt{\color{blue}{A \cdot \frac{\frac{1}{V}}{\ell}}} \cdot \sqrt{{c0}^{2}} \]

      6. associate-*r/ 44.6%

         \[\leadsto \sqrt{\color{blue}{\frac{A \cdot \frac{1}{V}}{\ell}}} \cdot \sqrt{{c0}^{2}} \]

      7. div-inv 44.7%

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

      8. frac-2neg 44.7%

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

      9. associate-/r* 31.7%

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

      10. *-commutative 31.7%

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

      11. sqrt-undiv 17.9%

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

      12. sqrt-pow1 25.2%

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

      13. metadata-eval 25.2%

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

      14. pow1 25.2%

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

      15. *-commutative 25.2%

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

      16. associate-*r/ 25.1%

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

      17. sqrt-prod 30.3%

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

      18. times-frac 30.3%

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

      19. sqrt-div 60.6%

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

   6. Applied egg-rr 60.6%

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

   if 5.0000000000000002e-298 < (*.f64 V l) < 1.0000000000000001e299

   1. Initial program 85.4%

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

      1. associate-/r* 78.2%

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

      2. clear-num 77.3%

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

      3. sqrt-div 77.3%

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

      4. metadata-eval 77.3%

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

      5. div-inv 76.4%

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

      6. clear-num 76.7%

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

   4. Applied egg-rr 76.7%

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

      1. *-commutative 76.7%

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

      2. associate-*l/ 84.9%

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

      3. associate-/l* 75.2%

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

   6. Simplified 75.2%

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

      1. un-div-inv 75.3%

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

   8. Applied egg-rr 75.3%

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

      1. associate-*r/ 85.1%

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

      2. frac-2neg 85.1%

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

      3. *-commutative 85.1%

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

      4. distribute-rgt-neg-out 85.1%

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

      5. sqrt-undiv 0.0%

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

      6. div-inv 0.0%

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

      7. *-commutative 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 0.0%

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

      10. sqr-neg 0.0%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.0%

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

      13. add-sqr-sqrt 0.0%

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

      14. sqrt-unprod 63.2%

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

      15. sqr-neg 63.2%

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

      16. sqrt-unprod 99.3%

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

      17. add-sqr-sqrt 99.3%

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

   10. Applied egg-rr 99.3%

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

       1. associate-*r/ 99.5%

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

       2. *-rgt-identity 99.5%

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

   12. Simplified 99.5%

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

   if 1.0000000000000001e299 < (*.f64 V l)

   1. Initial program 12.2%

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

      1. associate-/r* 65.0%

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

      2. clear-num 64.9%

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

      3. sqrt-div 64.9%

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

      4. metadata-eval 64.9%

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

      5. div-inv 64.9%

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

      6. clear-num 64.9%

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

   4. Applied egg-rr 64.9%

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

      1. *-commutative 64.9%

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

      2. associate-*l/ 12.2%

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

      3. associate-/l* 64.9%

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

   6. Simplified 64.9%

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

      1. inv-pow 64.9%

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

      2. sqrt-pow2 65.0%

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

      3. metadata-eval 65.0%

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

   8. Applied egg-rr 65.0%

      \[\leadsto c0 \cdot \color{blue}{{\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+208}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-304}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-298}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+299}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-312}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+292}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{1}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (<= t_0 4e-312)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 2e+292)
       (/ c0 (sqrt (/ (* V l) A)))
       (* c0 (/ 1.0 (sqrt (* l (/ V A)))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 4e-312) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 2e+292) {
		tmp = c0 / sqrt(((V * l) / A));
	} else {
		tmp = c0 * (1.0 / sqrt((l * (V / A))));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if (t_0 <= 4d-312) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 2d+292) then
        tmp = c0 / sqrt(((v * l) / a))
    else
        tmp = c0 * (1.0d0 / sqrt((l * (v / a))))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 4e-312) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 2e+292) {
		tmp = c0 / Math.sqrt(((V * l) / A));
	} else {
		tmp = c0 * (1.0 / Math.sqrt((l * (V / A))));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 4e-312:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 2e+292:
		tmp = c0 / math.sqrt(((V * l) / A))
	else:
		tmp = c0 * (1.0 / math.sqrt((l * (V / A))))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 4e-312)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 2e+292)
		tmp = Float64(c0 / sqrt(Float64(Float64(V * l) / A)));
	else
		tmp = Float64(c0 * Float64(1.0 / sqrt(Float64(l * Float64(V / A)))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 4e-312)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 2e+292)
		tmp = c0 / sqrt(((V * l) / A));
	else
		tmp = c0 * (1.0 / sqrt((l * (V / A))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 4e-312], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+292], N[(c0 / N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(1.0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 A (*.f64 V l)) < 3.9999999999988e-312

   1. Initial program 34.6%

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

      1. associate-/r* 56.8%

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

   3. Simplified 56.8%

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

   if 3.9999999999988e-312 < (/.f64 A (*.f64 V l)) < 2e292

   1. Initial program 99.5%

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

      1. clear-num 99.5%

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

      2. associate-/r/ 99.5%

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

      3. associate-/r* 99.5%

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

   4. Applied egg-rr 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-/r* 99.5%

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

      3. div-inv 99.5%

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

      4. *-commutative 99.5%

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

      5. associate-/r* 89.2%

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

   6. Applied egg-rr 89.2%

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

      1. clear-num 89.2%

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

      2. sqrt-div 89.1%

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

      3. metadata-eval 89.1%

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

      4. div-inv 88.2%

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

      5. clear-num 88.8%

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

      6. div-inv 88.9%

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

      7. *-commutative 88.9%

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

      8. associate-*l/ 99.6%

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

   8. Applied egg-rr 99.6%

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

   if 2e292 < (/.f64 A (*.f64 V l))

   1. Initial program 42.1%

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

      1. associate-/r* 55.3%

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

      2. clear-num 55.3%

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

      3. sqrt-div 58.5%

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

      4. metadata-eval 58.5%

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

      5. div-inv 58.5%

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

      6. clear-num 59.0%

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

   4. Applied egg-rr 59.0%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 4 \cdot 10^{-312}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{+292}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{1}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 79.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-312}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+292}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \end{array} \]

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 4e-312)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 2e+292)
       (/ c0 (sqrt (/ (* V l) A)))
       (/ 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 <= 4e-312) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 2e+292) {
		tmp = c0 / sqrt(((V * l) / A));
	} 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 <= 4d-312) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 2d+292) then
        tmp = c0 / sqrt(((v * l) / a))
    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 <= 4e-312) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 2e+292) {
		tmp = c0 / Math.sqrt(((V * l) / A));
	} 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 <= 4e-312:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 2e+292:
		tmp = c0 / math.sqrt(((V * l) / A))
	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 <= 4e-312)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 2e+292)
		tmp = Float64(c0 / sqrt(Float64(Float64(V * l) / A)));
	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 <= 4e-312)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 2e+292)
		tmp = c0 / sqrt(((V * l) / A));
	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, 4e-312], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+292], N[(c0 / N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $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)) < 3.9999999999988e-312

   1. Initial program 34.6%

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

      1. associate-/r* 56.8%

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

   3. Simplified 56.8%

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

   if 3.9999999999988e-312 < (/.f64 A (*.f64 V l)) < 2e292

   1. Initial program 99.5%

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

      1. clear-num 99.5%

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

      2. associate-/r/ 99.5%

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

      3. associate-/r* 99.5%

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

   4. Applied egg-rr 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-/r* 99.5%

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

      3. div-inv 99.5%

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

      4. *-commutative 99.5%

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

      5. associate-/r* 89.2%

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

   6. Applied egg-rr 89.2%

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

      1. clear-num 89.2%

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

      2. sqrt-div 89.1%

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

      3. metadata-eval 89.1%

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

      4. div-inv 88.2%

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

      5. clear-num 88.8%

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

      6. div-inv 88.9%

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

      7. *-commutative 88.9%

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

      8. associate-*l/ 99.6%

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

   8. Applied egg-rr 99.6%

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

   if 2e292 < (/.f64 A (*.f64 V l))

   1. Initial program 42.1%

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

      1. associate-/r* 55.3%

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

      2. clear-num 55.3%

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

      3. sqrt-div 58.5%

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

      4. metadata-eval 58.5%

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

      5. div-inv 58.5%

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

      6. clear-num 59.0%

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

   4. Applied egg-rr 59.0%

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

      1. *-commutative 59.0%

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

      2. associate-*l/ 44.5%

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

      3. associate-/l* 57.7%

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

   6. Simplified 57.7%

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

      1. un-div-inv 57.6%

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

      2. clear-num 57.6%

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

   8. Applied egg-rr 57.6%

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

      1. associate-/r/ 57.7%

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

      2. associate-*l/ 57.6%

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

      3. *-lft-identity 57.6%

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

      4. associate-*r/ 44.5%

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

      5. associate-*l/ 59.0%

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

      6. associate-/r/ 57.6%

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

   10. Simplified 57.6%

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

       1. associate-/r/ 59.0%

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

   12. Applied egg-rr 59.0%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 4 \cdot 10^{-312}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{+292}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 73.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.6%

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

3. Final simplification 74.6%

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

Reproduce

herbie shell --seed 2024073 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))