Henrywood and Agarwal, Equation (3)

Percentage Accurate: 74.2% → 91.5%

Time: 8.3s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -5 \cdot 10^{+274}:\\ \;\;\;\;\sqrt{\frac{A}{-\ell}} \cdot \frac{c0}{\sqrt{-V}}\\ \mathbf{elif}\;\ell \cdot V \leq -1 \cdot 10^{-314}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{-\ell \cdot V}}\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{+287}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{1}{\ell}}{\frac{V}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* l V) -5e+274)
   (* (sqrt (/ A (- l))) (/ c0 (sqrt (- V))))
   (if (<= (* l V) -1e-314)
     (* c0 (/ (sqrt (- A)) (sqrt (- (* l V)))))
     (if (<= (* l V) 0.0)
       (* c0 (sqrt (/ (/ A l) V)))
       (if (<= (* l V) 5e+287)
         (* c0 (/ (sqrt A) (sqrt (* l V))))
         (* c0 (sqrt (/ (/ 1.0 l) (/ V A)))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -5e+274) {
		tmp = sqrt((A / -l)) * (c0 / sqrt(-V));
	} else if ((l * V) <= -1e-314) {
		tmp = c0 * (sqrt(-A) / sqrt(-(l * V)));
	} else if ((l * V) <= 0.0) {
		tmp = c0 * sqrt(((A / l) / V));
	} else if ((l * V) <= 5e+287) {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	} else {
		tmp = c0 * sqrt(((1.0 / l) / (V / A)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((l * v) <= (-5d+274)) then
        tmp = sqrt((a / -l)) * (c0 / sqrt(-v))
    else if ((l * v) <= (-1d-314)) then
        tmp = c0 * (sqrt(-a) / sqrt(-(l * v)))
    else if ((l * v) <= 0.0d0) then
        tmp = c0 * sqrt(((a / l) / v))
    else if ((l * v) <= 5d+287) then
        tmp = c0 * (sqrt(a) / sqrt((l * v)))
    else
        tmp = c0 * sqrt(((1.0d0 / l) / (v / a)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -5e+274) {
		tmp = Math.sqrt((A / -l)) * (c0 / Math.sqrt(-V));
	} else if ((l * V) <= -1e-314) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt(-(l * V)));
	} else if ((l * V) <= 0.0) {
		tmp = c0 * Math.sqrt(((A / l) / V));
	} else if ((l * V) <= 5e+287) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((l * V)));
	} else {
		tmp = c0 * Math.sqrt(((1.0 / l) / (V / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (l * V) <= -5e+274:
		tmp = math.sqrt((A / -l)) * (c0 / math.sqrt(-V))
	elif (l * V) <= -1e-314:
		tmp = c0 * (math.sqrt(-A) / math.sqrt(-(l * V)))
	elif (l * V) <= 0.0:
		tmp = c0 * math.sqrt(((A / l) / V))
	elif (l * V) <= 5e+287:
		tmp = c0 * (math.sqrt(A) / math.sqrt((l * V)))
	else:
		tmp = c0 * math.sqrt(((1.0 / l) / (V / A)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(l * V) <= -5e+274)
		tmp = Float64(sqrt(Float64(A / Float64(-l))) * Float64(c0 / sqrt(Float64(-V))));
	elseif (Float64(l * V) <= -1e-314)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(-Float64(l * V)))));
	elseif (Float64(l * V) <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	elseif (Float64(l * V) <= 5e+287)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(l * V))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(1.0 / l) / Float64(V / A))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((l * V) <= -5e+274)
		tmp = sqrt((A / -l)) * (c0 / sqrt(-V));
	elseif ((l * V) <= -1e-314)
		tmp = c0 * (sqrt(-A) / sqrt(-(l * V)));
	elseif ((l * V) <= 0.0)
		tmp = c0 * sqrt(((A / l) / V));
	elseif ((l * V) <= 5e+287)
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	else
		tmp = c0 * sqrt(((1.0 / l) / (V / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(l * V), $MachinePrecision], -5e+274], N[(N[Sqrt[N[(A / (-l)), $MachinePrecision]], $MachinePrecision] * N[(c0 / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], -1e-314], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[(-N[(l * V), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 0.0], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 5e+287], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -4.9999999999999998e274

   1. Initial program 38.1%

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 71.1

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

   4. Applied egg-rr 71.1%

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

      1. associate-/r* N/A

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

      2. associate-/l/ N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-undiv N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. clear-num N/A

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

      13. sqrt-div N/A

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

      14. metadata-eval N/A

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

      15. frac-times N/A

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

      16. *-commutative N/A

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

      17. *-rgt-identity N/A

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

      18. lift-sqrt.f64 N/A

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

      19. sqrt-prod N/A

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

      20. associate-/r/ N/A

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

      21. lift-/.f64 N/A

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

      22. lower-/.f64 N/A

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

      23. lower-sqrt.f64 N/A

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

   6. Applied egg-rr 38.0%

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

      1. associate-/l* N/A

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

      2. /-rgt-identity N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. clear-num N/A

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

      6. associate-/l/ N/A

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

      7. inv-pow N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. pow-prod-up N/A

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

      12. pow-prod-down N/A

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

      13. remove-double-neg N/A

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

      14. lift-neg.f64 N/A

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

      15. remove-double-neg N/A

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

      16. lift-neg.f64 N/A

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

      17. sqr-neg N/A

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

      18. pow-prod-down N/A

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

      19. pow-prod-up N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. inv-pow N/A

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

      24. *-commutative N/A

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

      25. div-inv N/A

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

      26. lift-/.f64 N/A

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

   8. Applied egg-rr 71.1%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. sqrt-div N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. distribute-neg-frac2 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 65.1

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

   10. Applied egg-rr 65.1%

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

   if -4.9999999999999998e274 < (*.f64 V l) < -9.9999999996e-315

   1. Initial program 84.5%

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 75.6

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

   4. Applied egg-rr 75.6%

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

      1. associate-/l/ N/A

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

      2. lift-*.f64 N/A

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

      3. frac-2neg N/A

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

      4. sqrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 99.2

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

   6. Applied egg-rr 99.2%

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

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

   1. Initial program 42.9%

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 65.0

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

   4. Applied egg-rr 65.0%

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

   if 0.0 < (*.f64 V l) < 5e287

   1. Initial program 86.7%

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

      1. lift-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 98.5

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

   4. Applied egg-rr 98.5%

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

   if 5e287 < (*.f64 V l)

   1. Initial program 48.6%

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

      1. lift-*.f64 N/A

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

      2. div-inv N/A

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

      3. inv-pow N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. pow-pow N/A

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

      7. pow2 N/A

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

      8. remove-double-neg N/A

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

      9. remove-double-neg N/A

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

      10. sqr-neg N/A

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

      11. pow-prod-down N/A

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

      12. pow-prod-up N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. inv-pow N/A

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

      17. div-inv N/A

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

      18. lift-*.f64 N/A

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

      19. distribute-lft-neg-in N/A

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

      20. associate-/r* N/A

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

   4. Applied egg-rr 71.1%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -5 \cdot 10^{+274}:\\ \;\;\;\;\sqrt{\frac{A}{-\ell}} \cdot \frac{c0}{\sqrt{-V}}\\ \mathbf{elif}\;\ell \cdot V \leq -1 \cdot 10^{-314}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{-\ell \cdot V}}\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{+287}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{1}{\ell}}{\frac{V}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \mathbf{if}\;t\_0 \leq 10^{-231}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+246}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

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 (* l V))))))
   (if (<= t_0 1e-231)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 2e+246) 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 / (l * V)));
	double tmp;
	if (t_0 <= 1e-231) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 2e+246) {
		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 / (l * v)))
    if (t_0 <= 1d-231) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 2d+246) 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 / (l * V)));
	double tmp;
	if (t_0 <= 1e-231) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 2e+246) {
		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 / (l * V)))
	tmp = 0
	if t_0 <= 1e-231:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 2e+246:
		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(l * V))))
	tmp = 0.0
	if (t_0 <= 1e-231)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 2e+246)
		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 / (l * V)));
	tmp = 0.0;
	if (t_0 <= 1e-231)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 2e+246)
		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[(l * V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-231], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+246], 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)))) < 9.9999999999999999e-232

   1. Initial program 70.6%

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 72.1

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

   4. Applied egg-rr 72.1%

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

   if 9.9999999999999999e-232 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 2.00000000000000014e246

   1. Initial program 98.4%

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

   if 2.00000000000000014e246 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 47.1%

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 57.1

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

   4. Applied egg-rr 57.1%

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

      1. associate-/r* N/A

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

      2. associate-/l/ N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-undiv N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. clear-num N/A

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

      13. sqrt-div N/A

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

      14. metadata-eval N/A

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

      15. frac-times N/A

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

      16. *-commutative N/A

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

      17. *-rgt-identity N/A

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

      18. lift-sqrt.f64 N/A

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

      19. sqrt-prod N/A

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

      20. associate-/r/ N/A

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

      21. lift-/.f64 N/A

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

      22. lower-/.f64 N/A

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

      23. lower-sqrt.f64 N/A

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

   6. Applied egg-rr 48.4%

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 57.0

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

   8. Applied egg-rr 57.0%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}} \leq 10^{-231}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}} \leq 2 \cdot 10^{+246}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \mathbf{if}\;t\_0 \leq 10^{-231}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+192}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* l V))))))
   (if (<= t_0 1e-231)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 2e+192) t_0 (* c0 (sqrt (/ (/ A l) V)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (l * V)));
	double tmp;
	if (t_0 <= 1e-231) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 2e+192) {
		tmp = t_0;
	} else {
		tmp = c0 * sqrt(((A / l) / V));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (l * v)))
    if (t_0 <= 1d-231) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 2d+192) then
        tmp = t_0
    else
        tmp = c0 * sqrt(((a / l) / v))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (l * V)));
	double tmp;
	if (t_0 <= 1e-231) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 2e+192) {
		tmp = t_0;
	} else {
		tmp = c0 * Math.sqrt(((A / l) / V));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (l * V)))
	tmp = 0
	if t_0 <= 1e-231:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 2e+192:
		tmp = t_0
	else:
		tmp = c0 * math.sqrt(((A / l) / V))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(l * V))))
	tmp = 0.0
	if (t_0 <= 1e-231)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 2e+192)
		tmp = t_0;
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (l * V)));
	tmp = 0.0;
	if (t_0 <= 1e-231)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 2e+192)
		tmp = t_0;
	else
		tmp = c0 * sqrt(((A / l) / V));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(l * V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-231], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+192], t$95$0, N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 9.9999999999999999e-232

   1. Initial program 70.6%

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 72.1

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

   4. Applied egg-rr 72.1%

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

   if 9.9999999999999999e-232 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 2.00000000000000008e192

   1. Initial program 98.4%

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

   if 2.00000000000000008e192 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 50.5%

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 59.9

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

   4. Applied egg-rr 59.9%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}} \leq 10^{-231}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}} \leq 2 \cdot 10^{+192}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ t_1 := c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{if}\;t\_0 \leq 10^{-231}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+277}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* l V))))) (t_1 (* c0 (sqrt (/ (/ A V) l)))))
   (if (<= t_0 1e-231) t_1 (if (<= t_0 5e+277) t_0 t_1))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (l * V)));
	double t_1 = c0 * sqrt(((A / V) / l));
	double tmp;
	if (t_0 <= 1e-231) {
		tmp = t_1;
	} else if (t_0 <= 5e+277) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (l * v)))
    t_1 = c0 * sqrt(((a / v) / l))
    if (t_0 <= 1d-231) then
        tmp = t_1
    else if (t_0 <= 5d+277) then
        tmp = t_0
    else
        tmp = t_1
    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 / (l * V)));
	double t_1 = c0 * Math.sqrt(((A / V) / l));
	double tmp;
	if (t_0 <= 1e-231) {
		tmp = t_1;
	} else if (t_0 <= 5e+277) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (l * V)))
	t_1 = c0 * math.sqrt(((A / V) / l))
	tmp = 0
	if t_0 <= 1e-231:
		tmp = t_1
	elif t_0 <= 5e+277:
		tmp = t_0
	else:
		tmp = t_1
	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(l * V))))
	t_1 = Float64(c0 * sqrt(Float64(Float64(A / V) / l)))
	tmp = 0.0
	if (t_0 <= 1e-231)
		tmp = t_1;
	elseif (t_0 <= 5e+277)
		tmp = t_0;
	else
		tmp = t_1;
	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 / (l * V)));
	t_1 = c0 * sqrt(((A / V) / l));
	tmp = 0.0;
	if (t_0 <= 1e-231)
		tmp = t_1;
	elseif (t_0 <= 5e+277)
		tmp = t_0;
	else
		tmp = t_1;
	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[(l * V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-231], t$95$1, If[LessEqual[t$95$0, 5e+277], t$95$0, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 9.9999999999999999e-232 or 4.99999999999999982e277 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 67.1%

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 70.3

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

   4. Applied egg-rr 70.3%

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

   if 9.9999999999999999e-232 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 4.99999999999999982e277

   1. Initial program 98.5%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}} \leq 10^{-231}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}} \leq 5 \cdot 10^{+277}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -5 \cdot 10^{+274}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;\ell \cdot V \leq -1 \cdot 10^{-314}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{-\ell \cdot V}}\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{+287}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{1}{\ell}}{\frac{V}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* l V) -5e+274)
   (* c0 (/ (sqrt (/ A (- l))) (sqrt (- V))))
   (if (<= (* l V) -1e-314)
     (* c0 (/ (sqrt (- A)) (sqrt (- (* l V)))))
     (if (<= (* l V) 0.0)
       (* c0 (sqrt (/ (/ A l) V)))
       (if (<= (* l V) 5e+287)
         (* c0 (/ (sqrt A) (sqrt (* l V))))
         (* c0 (sqrt (/ (/ 1.0 l) (/ V A)))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -5e+274) {
		tmp = c0 * (sqrt((A / -l)) / sqrt(-V));
	} else if ((l * V) <= -1e-314) {
		tmp = c0 * (sqrt(-A) / sqrt(-(l * V)));
	} else if ((l * V) <= 0.0) {
		tmp = c0 * sqrt(((A / l) / V));
	} else if ((l * V) <= 5e+287) {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	} else {
		tmp = c0 * sqrt(((1.0 / l) / (V / A)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((l * v) <= (-5d+274)) then
        tmp = c0 * (sqrt((a / -l)) / sqrt(-v))
    else if ((l * v) <= (-1d-314)) then
        tmp = c0 * (sqrt(-a) / sqrt(-(l * v)))
    else if ((l * v) <= 0.0d0) then
        tmp = c0 * sqrt(((a / l) / v))
    else if ((l * v) <= 5d+287) then
        tmp = c0 * (sqrt(a) / sqrt((l * v)))
    else
        tmp = c0 * sqrt(((1.0d0 / l) / (v / a)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -5e+274) {
		tmp = c0 * (Math.sqrt((A / -l)) / Math.sqrt(-V));
	} else if ((l * V) <= -1e-314) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt(-(l * V)));
	} else if ((l * V) <= 0.0) {
		tmp = c0 * Math.sqrt(((A / l) / V));
	} else if ((l * V) <= 5e+287) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((l * V)));
	} else {
		tmp = c0 * Math.sqrt(((1.0 / l) / (V / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (l * V) <= -5e+274:
		tmp = c0 * (math.sqrt((A / -l)) / math.sqrt(-V))
	elif (l * V) <= -1e-314:
		tmp = c0 * (math.sqrt(-A) / math.sqrt(-(l * V)))
	elif (l * V) <= 0.0:
		tmp = c0 * math.sqrt(((A / l) / V))
	elif (l * V) <= 5e+287:
		tmp = c0 * (math.sqrt(A) / math.sqrt((l * V)))
	else:
		tmp = c0 * math.sqrt(((1.0 / l) / (V / A)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(l * V) <= -5e+274)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / Float64(-l))) / sqrt(Float64(-V))));
	elseif (Float64(l * V) <= -1e-314)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(-Float64(l * V)))));
	elseif (Float64(l * V) <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	elseif (Float64(l * V) <= 5e+287)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(l * V))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(1.0 / l) / Float64(V / A))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((l * V) <= -5e+274)
		tmp = c0 * (sqrt((A / -l)) / sqrt(-V));
	elseif ((l * V) <= -1e-314)
		tmp = c0 * (sqrt(-A) / sqrt(-(l * V)));
	elseif ((l * V) <= 0.0)
		tmp = c0 * sqrt(((A / l) / V));
	elseif ((l * V) <= 5e+287)
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	else
		tmp = c0 * sqrt(((1.0 / l) / (V / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(l * V), $MachinePrecision], -5e+274], N[(c0 * N[(N[Sqrt[N[(A / (-l)), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], -1e-314], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[(-N[(l * V), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 0.0], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 5e+287], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -4.9999999999999998e274

   1. Initial program 38.1%

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 71.1

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

   4. Applied egg-rr 71.1%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. sqrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-neg.f64 65.0

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

   6. Applied egg-rr 65.0%

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

   if -4.9999999999999998e274 < (*.f64 V l) < -9.9999999996e-315

   1. Initial program 84.5%

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 75.6

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

   4. Applied egg-rr 75.6%

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

      1. associate-/l/ N/A

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

      2. lift-*.f64 N/A

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

      3. frac-2neg N/A

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

      4. sqrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 99.2

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

   6. Applied egg-rr 99.2%

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

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

   1. Initial program 42.9%

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 65.0

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

   4. Applied egg-rr 65.0%

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

   if 0.0 < (*.f64 V l) < 5e287

   1. Initial program 86.7%

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

      1. lift-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 98.5

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

   4. Applied egg-rr 98.5%

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

   if 5e287 < (*.f64 V l)

   1. Initial program 48.6%

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

      1. lift-*.f64 N/A

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

      2. div-inv N/A

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

      3. inv-pow N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. pow-pow N/A

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

      7. pow2 N/A

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

      8. remove-double-neg N/A

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

      9. remove-double-neg N/A

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

      10. sqr-neg N/A

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

      11. pow-prod-down N/A

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

      12. pow-prod-up N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. inv-pow N/A

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

      17. div-inv N/A

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

      18. lift-*.f64 N/A

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

      19. distribute-lft-neg-in N/A

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

      20. associate-/r* N/A

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

   4. Applied egg-rr 71.1%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -5 \cdot 10^{+274}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;\ell \cdot V \leq -1 \cdot 10^{-314}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{-\ell \cdot V}}\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{+287}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{1}{\ell}}{\frac{V}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq -1 \cdot 10^{-314}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{-\ell \cdot V}}\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{+287}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{1}{\ell}}{\frac{V}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* l V) (- INFINITY))
   (* c0 (sqrt (/ (/ A V) l)))
   (if (<= (* l V) -1e-314)
     (* c0 (/ (sqrt (- A)) (sqrt (- (* l V)))))
     (if (<= (* l V) 0.0)
       (* c0 (sqrt (/ (/ A l) V)))
       (if (<= (* l V) 5e+287)
         (* c0 (/ (sqrt A) (sqrt (* l V))))
         (* c0 (sqrt (/ (/ 1.0 l) (/ V A)))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -((double) INFINITY)) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if ((l * V) <= -1e-314) {
		tmp = c0 * (sqrt(-A) / sqrt(-(l * V)));
	} else if ((l * V) <= 0.0) {
		tmp = c0 * sqrt(((A / l) / V));
	} else if ((l * V) <= 5e+287) {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	} else {
		tmp = c0 * sqrt(((1.0 / l) / (V / A)));
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -Double.POSITIVE_INFINITY) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if ((l * V) <= -1e-314) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt(-(l * V)));
	} else if ((l * V) <= 0.0) {
		tmp = c0 * Math.sqrt(((A / l) / V));
	} else if ((l * V) <= 5e+287) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((l * V)));
	} else {
		tmp = c0 * Math.sqrt(((1.0 / l) / (V / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (l * V) <= -math.inf:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif (l * V) <= -1e-314:
		tmp = c0 * (math.sqrt(-A) / math.sqrt(-(l * V)))
	elif (l * V) <= 0.0:
		tmp = c0 * math.sqrt(((A / l) / V))
	elif (l * V) <= 5e+287:
		tmp = c0 * (math.sqrt(A) / math.sqrt((l * V)))
	else:
		tmp = c0 * math.sqrt(((1.0 / l) / (V / A)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(l * V) <= Float64(-Inf))
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (Float64(l * V) <= -1e-314)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(-Float64(l * V)))));
	elseif (Float64(l * V) <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	elseif (Float64(l * V) <= 5e+287)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(l * V))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(1.0 / l) / Float64(V / A))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((l * V) <= -Inf)
		tmp = c0 * sqrt(((A / V) / l));
	elseif ((l * V) <= -1e-314)
		tmp = c0 * (sqrt(-A) / sqrt(-(l * V)));
	elseif ((l * V) <= 0.0)
		tmp = c0 * sqrt(((A / l) / V));
	elseif ((l * V) <= 5e+287)
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	else
		tmp = c0 * sqrt(((1.0 / l) / (V / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(l * V), $MachinePrecision], (-Infinity)], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], -1e-314], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[(-N[(l * V), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 0.0], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 5e+287], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 28.5%

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 70.0

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

   4. Applied egg-rr 70.0%

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

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

   1. Initial program 84.2%

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 75.7

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

   4. Applied egg-rr 75.7%

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

      1. associate-/l/ N/A

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

      2. lift-*.f64 N/A

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

      3. frac-2neg N/A

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

      4. sqrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 99.2

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

   6. Applied egg-rr 99.2%

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

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

   1. Initial program 42.9%

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 65.0

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

   4. Applied egg-rr 65.0%

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

   if 0.0 < (*.f64 V l) < 5e287

   1. Initial program 86.7%

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

      1. lift-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 98.5

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

   4. Applied egg-rr 98.5%

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

   if 5e287 < (*.f64 V l)

   1. Initial program 48.6%

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

      1. lift-*.f64 N/A

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

      2. div-inv N/A

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

      3. inv-pow N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. pow-pow N/A

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

      7. pow2 N/A

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

      8. remove-double-neg N/A

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

      9. remove-double-neg N/A

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

      10. sqr-neg N/A

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

      11. pow-prod-down N/A

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

      12. pow-prod-up N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. inv-pow N/A

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

      17. div-inv N/A

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

      18. lift-*.f64 N/A

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

      19. distribute-lft-neg-in N/A

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

      20. associate-/r* N/A

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

   4. Applied egg-rr 71.1%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq -1 \cdot 10^{-314}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{-\ell \cdot V}}\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{+287}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{1}{\ell}}{\frac{V}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \cdot V \leq -1 \cdot 10^{-314}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{-\ell \cdot V}}\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{+287}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 (<= (* l V) (- INFINITY))
     t_0
     (if (<= (* l V) -1e-314)
       (* c0 (/ (sqrt (- A)) (sqrt (- (* l V)))))
       (if (<= (* l V) 0.0)
         (* c0 (sqrt (/ (/ A l) V)))
         (if (<= (* l V) 5e+287) (* c0 (/ (sqrt A) (sqrt (* l V)))) 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 ((l * V) <= -((double) INFINITY)) {
		tmp = t_0;
	} else if ((l * V) <= -1e-314) {
		tmp = c0 * (sqrt(-A) / sqrt(-(l * V)));
	} else if ((l * V) <= 0.0) {
		tmp = c0 * sqrt(((A / l) / V));
	} else if ((l * V) <= 5e+287) {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt(((A / V) / l));
	double tmp;
	if ((l * V) <= -Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else if ((l * V) <= -1e-314) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt(-(l * V)));
	} else if ((l * V) <= 0.0) {
		tmp = c0 * Math.sqrt(((A / l) / V));
	} else if ((l * V) <= 5e+287) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((l * V)));
	} 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 (l * V) <= -math.inf:
		tmp = t_0
	elif (l * V) <= -1e-314:
		tmp = c0 * (math.sqrt(-A) / math.sqrt(-(l * V)))
	elif (l * V) <= 0.0:
		tmp = c0 * math.sqrt(((A / l) / V))
	elif (l * V) <= 5e+287:
		tmp = c0 * (math.sqrt(A) / math.sqrt((l * V)))
	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(Float64(A / V) / l)))
	tmp = 0.0
	if (Float64(l * V) <= Float64(-Inf))
		tmp = t_0;
	elseif (Float64(l * V) <= -1e-314)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(-Float64(l * V)))));
	elseif (Float64(l * V) <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	elseif (Float64(l * V) <= 5e+287)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(l * V))));
	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 ((l * V) <= -Inf)
		tmp = t_0;
	elseif ((l * V) <= -1e-314)
		tmp = c0 * (sqrt(-A) / sqrt(-(l * V)));
	elseif ((l * V) <= 0.0)
		tmp = c0 * sqrt(((A / l) / V));
	elseif ((l * V) <= 5e+287)
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	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[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(l * V), $MachinePrecision], (-Infinity)], t$95$0, If[LessEqual[N[(l * V), $MachinePrecision], -1e-314], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[(-N[(l * V), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 0.0], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 5e+287], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -inf.0 or 5e287 < (*.f64 V l)

   1. Initial program 40.1%

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 70.6

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

   4. Applied egg-rr 70.6%

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

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

   1. Initial program 84.2%

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 75.7

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

   4. Applied egg-rr 75.7%

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

      1. associate-/l/ N/A

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

      2. lift-*.f64 N/A

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

      3. frac-2neg N/A

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

      4. sqrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 99.2

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

   6. Applied egg-rr 99.2%

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

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

   1. Initial program 42.9%

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 65.0

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

   4. Applied egg-rr 65.0%

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

   if 0.0 < (*.f64 V l) < 5e287

   1. Initial program 86.7%

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

      1. lift-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 98.5

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

   4. Applied egg-rr 98.5%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq -1 \cdot 10^{-314}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{-\ell \cdot V}}\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{+287}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{\sqrt{-A} \cdot \frac{c0}{\sqrt{\ell}}}{\sqrt{-V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= A -1e-309)
   (/ (* (sqrt (- A)) (/ c0 (sqrt l))) (sqrt (- V)))
   (* c0 (/ (sqrt A) (sqrt (* l V))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -1e-309) {
		tmp = (sqrt(-A) * (c0 / sqrt(l))) / sqrt(-V);
	} else {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (a <= (-1d-309)) then
        tmp = (sqrt(-a) * (c0 / sqrt(l))) / sqrt(-v)
    else
        tmp = c0 * (sqrt(a) / sqrt((l * v)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -1e-309) {
		tmp = (Math.sqrt(-A) * (c0 / Math.sqrt(l))) / Math.sqrt(-V);
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((l * V)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if A <= -1e-309:
		tmp = (math.sqrt(-A) * (c0 / math.sqrt(l))) / math.sqrt(-V)
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((l * V)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (A <= -1e-309)
		tmp = Float64(Float64(sqrt(Float64(-A)) * Float64(c0 / sqrt(l))) / sqrt(Float64(-V)));
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(l * V))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (A <= -1e-309)
		tmp = (sqrt(-A) * (c0 / sqrt(l))) / sqrt(-V);
	else
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[A, -1e-309], N[(N[(N[Sqrt[(-A)], $MachinePrecision] * N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -1.000000000000002e-309

   1. Initial program 74.3%

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 74.4

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

   4. Applied egg-rr 74.4%

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

      1. associate-/r* N/A

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

      2. associate-/l/ N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-undiv N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. frac-2neg N/A

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

      13. sqrt-div N/A

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

      14. associate-*l/ N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lower-neg.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. lower-sqrt.f64 N/A

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

      21. lower-neg.f64 42.8

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

   6. Applied egg-rr 42.8%

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

   if -1.000000000000002e-309 < A

   1. Initial program 73.1%

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

      1. lift-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 81.3

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

   4. Applied egg-rr 81.3%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{\sqrt{-A} \cdot \frac{c0}{\sqrt{\ell}}}{\sqrt{-V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{+287}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* l V) 0.0)
   (/ c0 (sqrt (/ V (/ A l))))
   (if (<= (* l V) 5e+287)
     (* c0 (/ (sqrt A) (sqrt (* l V))))
     (* c0 (sqrt (/ (/ A V) l))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= 0.0) {
		tmp = c0 / sqrt((V / (A / l)));
	} else if ((l * V) <= 5e+287) {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	} else {
		tmp = c0 * sqrt(((A / V) / l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((l * v) <= 0.0d0) then
        tmp = c0 / sqrt((v / (a / l)))
    else if ((l * v) <= 5d+287) then
        tmp = c0 * (sqrt(a) / sqrt((l * v)))
    else
        tmp = c0 * sqrt(((a / v) / l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= 0.0) {
		tmp = c0 / Math.sqrt((V / (A / l)));
	} else if ((l * V) <= 5e+287) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((l * V)));
	} else {
		tmp = c0 * Math.sqrt(((A / V) / l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (l * V) <= 0.0:
		tmp = c0 / math.sqrt((V / (A / l)))
	elif (l * V) <= 5e+287:
		tmp = c0 * (math.sqrt(A) / math.sqrt((l * V)))
	else:
		tmp = c0 * math.sqrt(((A / V) / l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(l * V) <= 0.0)
		tmp = Float64(c0 / sqrt(Float64(V / Float64(A / l))));
	elseif (Float64(l * V) <= 5e+287)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(l * V))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((l * V) <= 0.0)
		tmp = c0 / sqrt((V / (A / l)));
	elseif ((l * V) <= 5e+287)
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	else
		tmp = c0 * sqrt(((A / V) / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(l * V), $MachinePrecision], 0.0], N[(c0 / N[Sqrt[N[(V / N[(A / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 5e+287], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 69.5%

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 72.8

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

   4. Applied egg-rr 72.8%

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

      1. associate-/r* N/A

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

      2. associate-/l/ N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-undiv N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. clear-num N/A

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

      13. sqrt-div N/A

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

      14. metadata-eval N/A

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

      15. frac-times N/A

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

      16. *-commutative N/A

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

      17. *-rgt-identity N/A

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

      18. lift-sqrt.f64 N/A

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

      19. sqrt-prod N/A

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

      20. associate-/r/ N/A

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

      21. lift-/.f64 N/A

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

      22. lower-/.f64 N/A

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

      23. lower-sqrt.f64 N/A

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

   6. Applied egg-rr 69.7%

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

      1. associate-/l* N/A

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

      2. /-rgt-identity N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. clear-num N/A

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

      6. associate-/l/ N/A

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

      7. inv-pow N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. pow-prod-up N/A

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

      12. pow-prod-down N/A

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

      13. remove-double-neg N/A

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

      14. lift-neg.f64 N/A

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

      15. remove-double-neg N/A

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

      16. lift-neg.f64 N/A

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

      17. sqr-neg N/A

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

      18. pow-prod-down N/A

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

      19. pow-prod-up N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. inv-pow N/A

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

      24. *-commutative N/A

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

      25. div-inv N/A

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

      26. lift-/.f64 N/A

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

   8. Applied egg-rr 72.9%

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

   if 0.0 < (*.f64 V l) < 5e287

   1. Initial program 86.7%

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

      1. lift-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 98.5

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

   4. Applied egg-rr 98.5%

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

   if 5e287 < (*.f64 V l)

   1. Initial program 48.6%

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 71.0

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

   4. Applied egg-rr 71.0%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{+287}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{+287}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* l V) 0.0)
   (/ c0 (sqrt (* V (/ l A))))
   (if (<= (* l V) 5e+287)
     (* c0 (/ (sqrt A) (sqrt (* l V))))
     (* c0 (sqrt (/ (/ A V) l))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= 0.0) {
		tmp = c0 / sqrt((V * (l / A)));
	} else if ((l * V) <= 5e+287) {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	} else {
		tmp = c0 * sqrt(((A / V) / l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((l * v) <= 0.0d0) then
        tmp = c0 / sqrt((v * (l / a)))
    else if ((l * v) <= 5d+287) then
        tmp = c0 * (sqrt(a) / sqrt((l * v)))
    else
        tmp = c0 * sqrt(((a / v) / l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= 0.0) {
		tmp = c0 / Math.sqrt((V * (l / A)));
	} else if ((l * V) <= 5e+287) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((l * V)));
	} else {
		tmp = c0 * Math.sqrt(((A / V) / l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (l * V) <= 0.0:
		tmp = c0 / math.sqrt((V * (l / A)))
	elif (l * V) <= 5e+287:
		tmp = c0 * (math.sqrt(A) / math.sqrt((l * V)))
	else:
		tmp = c0 * math.sqrt(((A / V) / l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(l * V) <= 0.0)
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	elseif (Float64(l * V) <= 5e+287)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(l * V))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((l * V) <= 0.0)
		tmp = c0 / sqrt((V * (l / A)));
	elseif ((l * V) <= 5e+287)
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	else
		tmp = c0 * sqrt(((A / V) / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(l * V), $MachinePrecision], 0.0], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 5e+287], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 69.5%

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 72.8

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

   4. Applied egg-rr 72.8%

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

      1. associate-/r* N/A

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

      2. associate-/l/ N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-undiv N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. clear-num N/A

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

      13. sqrt-div N/A

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

      14. metadata-eval N/A

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

      15. frac-times N/A

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

      16. *-commutative N/A

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

      17. *-rgt-identity N/A

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

      18. lift-sqrt.f64 N/A

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

      19. sqrt-prod N/A

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

      20. associate-/r/ N/A

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

      21. lift-/.f64 N/A

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

      22. lower-/.f64 N/A

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

      23. lower-sqrt.f64 N/A

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

   6. Applied egg-rr 69.7%

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 72.3

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

   8. Applied egg-rr 72.3%

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

   if 0.0 < (*.f64 V l) < 5e287

   1. Initial program 86.7%

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

      1. lift-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 98.5

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

   4. Applied egg-rr 98.5%

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

   if 5e287 < (*.f64 V l)

   1. Initial program 48.6%

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 71.0

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

   4. Applied egg-rr 71.0%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{+287}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 86.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (V <= -5e-310)
		tmp = Float64(c0 / Float64(sqrt(Float64(-V)) * sqrt(Float64(l / Float64(-A)))));
	else
		tmp = Float64(sqrt(A) * Float64(c0 / Float64(sqrt(l) * sqrt(V))));
	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 <= -5e-310)
		tmp = c0 / (sqrt(-V) * sqrt((l / -A)));
	else
		tmp = sqrt(A) * (c0 / (sqrt(l) * sqrt(V)));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if V < -4.999999999999985e-310

   1. Initial program 71.3%

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 75.0

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

   4. Applied egg-rr 75.0%

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

      1. associate-/r* N/A

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

      2. associate-/l/ N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-undiv N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. clear-num N/A

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

      13. sqrt-div N/A

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

      14. metadata-eval N/A

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

      15. frac-times N/A

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

      16. *-commutative N/A

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

      17. *-rgt-identity N/A

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

      18. lift-sqrt.f64 N/A

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

      19. sqrt-prod N/A

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

      20. associate-/r/ N/A

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

      21. lift-/.f64 N/A

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

      22. lower-/.f64 N/A

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

      23. lower-sqrt.f64 N/A

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

   6. Applied egg-rr 71.4%

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

      1. associate-/l* N/A

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

      2. /-rgt-identity N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. clear-num N/A

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

      6. associate-/l/ N/A

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

      7. inv-pow N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. pow-prod-up N/A

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

      12. pow-prod-down N/A

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

      13. remove-double-neg N/A

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

      14. lift-neg.f64 N/A

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

      15. remove-double-neg N/A

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

      16. lift-neg.f64 N/A

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

      17. sqr-neg N/A

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

      18. pow-prod-down N/A

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

      19. pow-prod-up N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. inv-pow N/A

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

      24. *-commutative N/A

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

      25. div-inv N/A

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

      26. lift-/.f64 N/A

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

   8. Applied egg-rr 75.1%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. sqrt-prod N/A

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

      5. distribute-frac-neg2 N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. clear-num N/A

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

      12. lift-/.f64 N/A

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

      13. distribute-frac-neg2 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. distribute-frac-neg2 N/A

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

      16. lift-/.f64 N/A

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

      17. clear-num N/A

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

      18. distribute-neg-frac N/A

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

      19. lower-/.f64 N/A

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

      20. lower-neg.f64 87.4

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

   10. Applied egg-rr 87.4%

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

   if -4.999999999999985e-310 < V

   1. Initial program 75.9%

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

      1. lift-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 37.3

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

   4. Applied egg-rr 37.3%

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

      1. sqrt-prod N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 44.1

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

   6. Applied egg-rr 44.1%

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

4. Final simplification 65.3%

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

Alternative 12: 86.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (V <= -5e-310)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / Float64(-l))) / sqrt(Float64(-V))));
	else
		tmp = Float64(sqrt(A) * Float64(c0 / Float64(sqrt(l) * sqrt(V))));
	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 <= -5e-310)
		tmp = c0 * (sqrt((A / -l)) / sqrt(-V));
	else
		tmp = sqrt(A) * (c0 / (sqrt(l) * sqrt(V)));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if V < -4.999999999999985e-310

   1. Initial program 71.3%

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 75.0

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

   4. Applied egg-rr 75.0%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. sqrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-neg.f64 87.0

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

   6. Applied egg-rr 87.0%

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

   if -4.999999999999985e-310 < V

   1. Initial program 75.9%

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

      1. lift-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 37.3

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

   4. Applied egg-rr 37.3%

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

      1. sqrt-prod N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 44.1

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

   6. Applied egg-rr 44.1%

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

4. Final simplification 65.0%

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

Alternative 13: 74.2% accurate, 1.0× speedup

Math

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

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 (* l V)))))

C

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

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 / (l * v)))
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 / (l * V)));
}

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 73.7%

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

3. Final simplification 73.7%

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

Reproduce

herbie shell --seed 2024200 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))