Henrywood and Agarwal, Equation (3)

Percentage Accurate: 72.8% → 89.9%

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: 72.8% 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: 89.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+305}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq -4 \cdot 10^{-315}:\\ \;\;\;\;c0 \cdot \left(\sqrt{-A} \cdot \frac{1}{\sqrt{-V \cdot \ell}}\right)\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-241}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+274}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;A \cdot \frac{c0}{\sqrt{V \cdot \left(A \cdot \ell\right)}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -1e+305)
   (* (/ c0 (sqrt l)) (sqrt (/ A V)))
   (if (<= (* V l) -4e-315)
     (* c0 (* (sqrt (- A)) (/ 1.0 (sqrt (- (* V l))))))
     (if (<= (* V l) 1e-241)
       (/ c0 (sqrt (* l (/ V A))))
       (if (<= (* V l) 5e+274)
         (* c0 (/ (sqrt A) (sqrt (* V l))))
         (* A (/ c0 (sqrt (* V (* A l))))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+305) {
		tmp = (c0 / sqrt(l)) * sqrt((A / V));
	} else if ((V * l) <= -4e-315) {
		tmp = c0 * (sqrt(-A) * (1.0 / sqrt(-(V * l))));
	} else if ((V * l) <= 1e-241) {
		tmp = c0 / sqrt((l * (V / A)));
	} else if ((V * l) <= 5e+274) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = A * (c0 / sqrt((V * (A * l))));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-1d+305)) then
        tmp = (c0 / sqrt(l)) * sqrt((a / v))
    else if ((v * l) <= (-4d-315)) then
        tmp = c0 * (sqrt(-a) * (1.0d0 / sqrt(-(v * l))))
    else if ((v * l) <= 1d-241) then
        tmp = c0 / sqrt((l * (v / a)))
    else if ((v * l) <= 5d+274) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = a * (c0 / sqrt((v * (a * l))))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+305) {
		tmp = (c0 / Math.sqrt(l)) * Math.sqrt((A / V));
	} else if ((V * l) <= -4e-315) {
		tmp = c0 * (Math.sqrt(-A) * (1.0 / Math.sqrt(-(V * l))));
	} else if ((V * l) <= 1e-241) {
		tmp = c0 / Math.sqrt((l * (V / A)));
	} else if ((V * l) <= 5e+274) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = A * (c0 / Math.sqrt((V * (A * l))));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -1e+305:
		tmp = (c0 / math.sqrt(l)) * math.sqrt((A / V))
	elif (V * l) <= -4e-315:
		tmp = c0 * (math.sqrt(-A) * (1.0 / math.sqrt(-(V * l))))
	elif (V * l) <= 1e-241:
		tmp = c0 / math.sqrt((l * (V / A)))
	elif (V * l) <= 5e+274:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = A * (c0 / math.sqrt((V * (A * l))))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -1e+305)
		tmp = Float64(Float64(c0 / sqrt(l)) * sqrt(Float64(A / V)));
	elseif (Float64(V * l) <= -4e-315)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) * Float64(1.0 / sqrt(Float64(-Float64(V * l))))));
	elseif (Float64(V * l) <= 1e-241)
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / A))));
	elseif (Float64(V * l) <= 5e+274)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(A * Float64(c0 / sqrt(Float64(V * Float64(A * l)))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -1e+305)
		tmp = (c0 / sqrt(l)) * sqrt((A / V));
	elseif ((V * l) <= -4e-315)
		tmp = c0 * (sqrt(-A) * (1.0 / sqrt(-(V * l))));
	elseif ((V * l) <= 1e-241)
		tmp = c0 / sqrt((l * (V / A)));
	elseif ((V * l) <= 5e+274)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = A * (c0 / sqrt((V * (A * l))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -1e+305], N[(N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -4e-315], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] * N[(1.0 / N[Sqrt[(-N[(V * l), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e-241], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e+274], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(A * N[(c0 / N[Sqrt[N[(V * N[(A * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -9.9999999999999994e304

   1. Initial program 31.6%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. sqrt-div N/A

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

      3. associate-*r/ N/A

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

      4. associate-*l/ N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. /-lowering-/.f64 40.0

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

   4. Applied egg-rr 40.0%

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

   if -9.9999999999999994e304 < (*.f64 V l) < -3.9999999989e-315

   1. Initial program 82.3%

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

      1. *-commutative N/A

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

      2. /-rgt-identity N/A

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

      3. associate-/r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 82.2

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

   4. Applied egg-rr 82.2%

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

      1. associate-/r/ N/A

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

      2. un-div-inv N/A

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

      3. associate-/l/ N/A

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

      4. frac-2neg N/A

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

      5. sqrt-div N/A

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

      6. div-inv N/A

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

      7. *-lowering-*.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. neg-lowering-neg.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. sqrt-lowering-sqrt.f64 N/A

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

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

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

      13. *-lowering-*.f64 N/A

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

      14. neg-lowering-neg.f64 99.0

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

   6. Applied egg-rr 99.0%

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

   if -3.9999999989e-315 < (*.f64 V l) < 9.9999999999999997e-242

   1. Initial program 54.3%

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

      1. *-commutative N/A

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

      2. /-rgt-identity N/A

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

      3. associate-/r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 54.3

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

   4. Applied egg-rr 54.3%

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

      1. associate-/r/ N/A

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

      2. frac-times N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. sqrt-prod N/A

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

      6. *-inverses N/A

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

      7. sqrt-prod N/A

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

      8. times-frac N/A

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

      9. clear-num N/A

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

      10. sqrt-div N/A

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

      11. metadata-eval N/A

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

      12. un-div-inv N/A

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

      13. /-lowering-/.f64 N/A

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

      14. sqrt-lowering-sqrt.f64 N/A

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

      15. associate-/l* N/A

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

      16. associate-/r* N/A

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

      17. *-inverses N/A

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

      18. div-inv N/A

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

      19. /-lowering-/.f64 N/A

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

   6. Applied egg-rr 54.4%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. metadata-eval N/A

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

      4. div-inv N/A

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

      5. metadata-eval N/A

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

      6. *-inverses N/A

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

      7. clear-num N/A

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

      8. times-frac N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. clear-num N/A

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

      12. times-frac N/A

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

      13. *-inverses N/A

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

      14. *-lft-identity N/A

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

      15. clear-num N/A

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

      16. /-lowering-/.f64 73.9

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

   8. Applied egg-rr 73.9%

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

   if 9.9999999999999997e-242 < (*.f64 V l) < 4.9999999999999998e274

   1. Initial program 88.7%

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

      1. sqrt-div N/A

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

      2. /-lowering-/.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. *-lowering-*.f64 99.4

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

   4. Applied egg-rr 99.4%

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

   if 4.9999999999999998e274 < (*.f64 V l)

   1. Initial program 44.0%

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

      1. *-commutative N/A

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

      2. /-rgt-identity N/A

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

      3. associate-/r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 44.0

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

   4. Applied egg-rr 44.0%

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

      1. associate-/r/ N/A

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

      2. frac-times N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. sqrt-prod N/A

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

      6. *-inverses N/A

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

      7. sqrt-prod N/A

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

      8. times-frac N/A

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

      9. clear-num N/A

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

      10. sqrt-div N/A

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

      11. metadata-eval N/A

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

      12. associate-*r/ N/A

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

      13. sqrt-div N/A

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

      14. associate-*r/ N/A

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

      15. clear-num N/A

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

   6. Applied egg-rr 72.7%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+305}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq -4 \cdot 10^{-315}:\\ \;\;\;\;c0 \cdot \left(\sqrt{-A} \cdot \frac{1}{\sqrt{-V \cdot \ell}}\right)\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-241}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+274}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;A \cdot \frac{c0}{\sqrt{V \cdot \left(A \cdot \ell\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+305}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-307}:\\ \;\;\;\;c0 \cdot \left(\sqrt{-A} \cdot \sqrt{\frac{-1}{V \cdot \ell}}\right)\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-241}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+274}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;A \cdot \frac{c0}{\sqrt{V \cdot \left(A \cdot \ell\right)}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -1e+305)
   (* (/ c0 (sqrt l)) (sqrt (/ A V)))
   (if (<= (* V l) -2e-307)
     (* c0 (* (sqrt (- A)) (sqrt (/ -1.0 (* V l)))))
     (if (<= (* V l) 1e-241)
       (/ c0 (sqrt (* l (/ V A))))
       (if (<= (* V l) 5e+274)
         (* c0 (/ (sqrt A) (sqrt (* V l))))
         (* A (/ c0 (sqrt (* V (* A l))))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+305) {
		tmp = (c0 / sqrt(l)) * sqrt((A / V));
	} else if ((V * l) <= -2e-307) {
		tmp = c0 * (sqrt(-A) * sqrt((-1.0 / (V * l))));
	} else if ((V * l) <= 1e-241) {
		tmp = c0 / sqrt((l * (V / A)));
	} else if ((V * l) <= 5e+274) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = A * (c0 / sqrt((V * (A * l))));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-1d+305)) then
        tmp = (c0 / sqrt(l)) * sqrt((a / v))
    else if ((v * l) <= (-2d-307)) then
        tmp = c0 * (sqrt(-a) * sqrt(((-1.0d0) / (v * l))))
    else if ((v * l) <= 1d-241) then
        tmp = c0 / sqrt((l * (v / a)))
    else if ((v * l) <= 5d+274) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = a * (c0 / sqrt((v * (a * l))))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+305) {
		tmp = (c0 / Math.sqrt(l)) * Math.sqrt((A / V));
	} else if ((V * l) <= -2e-307) {
		tmp = c0 * (Math.sqrt(-A) * Math.sqrt((-1.0 / (V * l))));
	} else if ((V * l) <= 1e-241) {
		tmp = c0 / Math.sqrt((l * (V / A)));
	} else if ((V * l) <= 5e+274) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = A * (c0 / Math.sqrt((V * (A * l))));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -1e+305:
		tmp = (c0 / math.sqrt(l)) * math.sqrt((A / V))
	elif (V * l) <= -2e-307:
		tmp = c0 * (math.sqrt(-A) * math.sqrt((-1.0 / (V * l))))
	elif (V * l) <= 1e-241:
		tmp = c0 / math.sqrt((l * (V / A)))
	elif (V * l) <= 5e+274:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = A * (c0 / math.sqrt((V * (A * l))))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -1e+305)
		tmp = Float64(Float64(c0 / sqrt(l)) * sqrt(Float64(A / V)));
	elseif (Float64(V * l) <= -2e-307)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) * sqrt(Float64(-1.0 / Float64(V * l)))));
	elseif (Float64(V * l) <= 1e-241)
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / A))));
	elseif (Float64(V * l) <= 5e+274)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(A * Float64(c0 / sqrt(Float64(V * Float64(A * l)))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -1e+305)
		tmp = (c0 / sqrt(l)) * sqrt((A / V));
	elseif ((V * l) <= -2e-307)
		tmp = c0 * (sqrt(-A) * sqrt((-1.0 / (V * l))));
	elseif ((V * l) <= 1e-241)
		tmp = c0 / sqrt((l * (V / A)));
	elseif ((V * l) <= 5e+274)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = A * (c0 / sqrt((V * (A * l))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -1e+305], N[(N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-307], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] * N[Sqrt[N[(-1.0 / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e-241], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e+274], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(A * N[(c0 / N[Sqrt[N[(V * N[(A * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -9.9999999999999994e304

   1. Initial program 31.6%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. sqrt-div N/A

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

      3. associate-*r/ N/A

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

      4. associate-*l/ N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. /-lowering-/.f64 40.0

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

   4. Applied egg-rr 40.0%

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

   if -9.9999999999999994e304 < (*.f64 V l) < -1.99999999999999982e-307

   1. Initial program 83.0%

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

      1. *-commutative N/A

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

      2. /-rgt-identity N/A

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

      3. associate-/r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 82.9

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

   4. Applied egg-rr 82.9%

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

      1. frac-2neg N/A

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

      2. frac-2neg N/A

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

      3. associate-/r/ N/A

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

      4. un-div-inv N/A

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

      5. associate-/l/ N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. sqrt-prod N/A

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

      9. *-lowering-*.f64 N/A

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

      10. sqrt-lowering-sqrt.f64 N/A

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

      11. neg-lowering-neg.f64 N/A

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

      12. sqrt-lowering-sqrt.f64 N/A

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

      13. metadata-eval N/A

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

      14. frac-2neg N/A

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

      15. /-lowering-/.f64 N/A

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 99.4

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

   6. Applied egg-rr 99.4%

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

   if -1.99999999999999982e-307 < (*.f64 V l) < 9.9999999999999997e-242

   1. Initial program 53.4%

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

      1. *-commutative N/A

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

      2. /-rgt-identity N/A

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

      3. associate-/r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 53.4

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

   4. Applied egg-rr 53.4%

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

      1. associate-/r/ N/A

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

      2. frac-times N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. sqrt-prod N/A

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

      6. *-inverses N/A

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

      7. sqrt-prod N/A

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

      8. times-frac N/A

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

      9. clear-num N/A

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

      10. sqrt-div N/A

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

      11. metadata-eval N/A

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

      12. un-div-inv N/A

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

      13. /-lowering-/.f64 N/A

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

      14. sqrt-lowering-sqrt.f64 N/A

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

      15. associate-/l* N/A

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

      16. associate-/r* N/A

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

      17. *-inverses N/A

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

      18. div-inv N/A

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

      19. /-lowering-/.f64 N/A

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

   6. Applied egg-rr 53.5%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. metadata-eval N/A

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

      4. div-inv N/A

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

      5. metadata-eval N/A

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

      6. *-inverses N/A

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

      7. clear-num N/A

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

      8. times-frac N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. clear-num N/A

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

      12. times-frac N/A

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

      13. *-inverses N/A

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

      14. *-lft-identity N/A

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

      15. clear-num N/A

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

      16. /-lowering-/.f64 72.6

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

   8. Applied egg-rr 72.6%

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

   if 9.9999999999999997e-242 < (*.f64 V l) < 4.9999999999999998e274

   1. Initial program 88.7%

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

      1. sqrt-div N/A

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

      2. /-lowering-/.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. *-lowering-*.f64 99.4

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

   4. Applied egg-rr 99.4%

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

   if 4.9999999999999998e274 < (*.f64 V l)

   1. Initial program 44.0%

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

      1. *-commutative N/A

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

      2. /-rgt-identity N/A

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

      3. associate-/r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 44.0

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

   4. Applied egg-rr 44.0%

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

      1. associate-/r/ N/A

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

      2. frac-times N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. sqrt-prod N/A

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

      6. *-inverses N/A

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

      7. sqrt-prod N/A

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

      8. times-frac N/A

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

      9. clear-num N/A

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

      10. sqrt-div N/A

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

      11. metadata-eval N/A

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

      12. associate-*r/ N/A

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

      13. sqrt-div N/A

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

      14. associate-*r/ N/A

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

      15. clear-num N/A

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

   6. Applied egg-rr 72.7%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+305}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-307}:\\ \;\;\;\;c0 \cdot \left(\sqrt{-A} \cdot \sqrt{\frac{-1}{V \cdot \ell}}\right)\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-241}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+274}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;A \cdot \frac{c0}{\sqrt{V \cdot \left(A \cdot \ell\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+305}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq -4 \cdot 10^{-315}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{-A}}{\sqrt{-V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-241}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+274}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;A \cdot \frac{c0}{\sqrt{V \cdot \left(A \cdot \ell\right)}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -1e+305)
   (* (/ c0 (sqrt l)) (sqrt (/ A V)))
   (if (<= (* V l) -4e-315)
     (/ (* c0 (sqrt (- A))) (sqrt (- (* V l))))
     (if (<= (* V l) 1e-241)
       (/ c0 (sqrt (* l (/ V A))))
       (if (<= (* V l) 5e+274)
         (* c0 (/ (sqrt A) (sqrt (* V l))))
         (* A (/ c0 (sqrt (* V (* A l))))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+305) {
		tmp = (c0 / sqrt(l)) * sqrt((A / V));
	} else if ((V * l) <= -4e-315) {
		tmp = (c0 * sqrt(-A)) / sqrt(-(V * l));
	} else if ((V * l) <= 1e-241) {
		tmp = c0 / sqrt((l * (V / A)));
	} else if ((V * l) <= 5e+274) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = A * (c0 / sqrt((V * (A * l))));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-1d+305)) then
        tmp = (c0 / sqrt(l)) * sqrt((a / v))
    else if ((v * l) <= (-4d-315)) then
        tmp = (c0 * sqrt(-a)) / sqrt(-(v * l))
    else if ((v * l) <= 1d-241) then
        tmp = c0 / sqrt((l * (v / a)))
    else if ((v * l) <= 5d+274) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = a * (c0 / sqrt((v * (a * l))))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+305) {
		tmp = (c0 / Math.sqrt(l)) * Math.sqrt((A / V));
	} else if ((V * l) <= -4e-315) {
		tmp = (c0 * Math.sqrt(-A)) / Math.sqrt(-(V * l));
	} else if ((V * l) <= 1e-241) {
		tmp = c0 / Math.sqrt((l * (V / A)));
	} else if ((V * l) <= 5e+274) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = A * (c0 / Math.sqrt((V * (A * l))));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -1e+305:
		tmp = (c0 / math.sqrt(l)) * math.sqrt((A / V))
	elif (V * l) <= -4e-315:
		tmp = (c0 * math.sqrt(-A)) / math.sqrt(-(V * l))
	elif (V * l) <= 1e-241:
		tmp = c0 / math.sqrt((l * (V / A)))
	elif (V * l) <= 5e+274:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = A * (c0 / math.sqrt((V * (A * l))))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -1e+305)
		tmp = Float64(Float64(c0 / sqrt(l)) * sqrt(Float64(A / V)));
	elseif (Float64(V * l) <= -4e-315)
		tmp = Float64(Float64(c0 * sqrt(Float64(-A))) / sqrt(Float64(-Float64(V * l))));
	elseif (Float64(V * l) <= 1e-241)
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / A))));
	elseif (Float64(V * l) <= 5e+274)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(A * Float64(c0 / sqrt(Float64(V * Float64(A * l)))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -1e+305)
		tmp = (c0 / sqrt(l)) * sqrt((A / V));
	elseif ((V * l) <= -4e-315)
		tmp = (c0 * sqrt(-A)) / sqrt(-(V * l));
	elseif ((V * l) <= 1e-241)
		tmp = c0 / sqrt((l * (V / A)));
	elseif ((V * l) <= 5e+274)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = A * (c0 / sqrt((V * (A * l))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -1e+305], N[(N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -4e-315], N[(N[(c0 * N[Sqrt[(-A)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-N[(V * l), $MachinePrecision])], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e-241], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e+274], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(A * N[(c0 / N[Sqrt[N[(V * N[(A * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -9.9999999999999994e304

   1. Initial program 31.6%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. sqrt-div N/A

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

      3. associate-*r/ N/A

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

      4. associate-*l/ N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. /-lowering-/.f64 40.0

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

   4. Applied egg-rr 40.0%

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

   if -9.9999999999999994e304 < (*.f64 V l) < -3.9999999989e-315

   1. Initial program 82.3%

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

      1. *-commutative N/A

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

      2. /-rgt-identity N/A

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

      3. associate-/r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 82.2

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

   4. Applied egg-rr 82.2%

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

      1. *-commutative N/A

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

      2. associate-/r/ N/A

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

      3. un-div-inv N/A

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

      4. associate-/l/ N/A

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

      5. frac-2neg N/A

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

      6. sqrt-div N/A

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

      7. associate-*l/ N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. sqrt-lowering-sqrt.f64 N/A

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

      11. neg-lowering-neg.f64 N/A

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

      12. sqrt-lowering-sqrt.f64 N/A

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

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

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

      14. *-lowering-*.f64 N/A

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

      15. neg-lowering-neg.f64 96.9

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

   6. Applied egg-rr 96.9%

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

   if -3.9999999989e-315 < (*.f64 V l) < 9.9999999999999997e-242

   1. Initial program 54.3%

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

      1. *-commutative N/A

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

      2. /-rgt-identity N/A

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

      3. associate-/r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 54.3

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

   4. Applied egg-rr 54.3%

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

      1. associate-/r/ N/A

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

      2. frac-times N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. sqrt-prod N/A

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

      6. *-inverses N/A

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

      7. sqrt-prod N/A

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

      8. times-frac N/A

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

      9. clear-num N/A

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

      10. sqrt-div N/A

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

      11. metadata-eval N/A

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

      12. un-div-inv N/A

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

      13. /-lowering-/.f64 N/A

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

      14. sqrt-lowering-sqrt.f64 N/A

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

      15. associate-/l* N/A

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

      16. associate-/r* N/A

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

      17. *-inverses N/A

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

      18. div-inv N/A

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

      19. /-lowering-/.f64 N/A

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

   6. Applied egg-rr 54.4%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. metadata-eval N/A

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

      4. div-inv N/A

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

      5. metadata-eval N/A

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

      6. *-inverses N/A

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

      7. clear-num N/A

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

      8. times-frac N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. clear-num N/A

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

      12. times-frac N/A

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

      13. *-inverses N/A

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

      14. *-lft-identity N/A

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

      15. clear-num N/A

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

      16. /-lowering-/.f64 73.9

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

   8. Applied egg-rr 73.9%

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

   if 9.9999999999999997e-242 < (*.f64 V l) < 4.9999999999999998e274

   1. Initial program 88.7%

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

      1. sqrt-div N/A

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

      2. /-lowering-/.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. *-lowering-*.f64 99.4

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

   4. Applied egg-rr 99.4%

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

   if 4.9999999999999998e274 < (*.f64 V l)

   1. Initial program 44.0%

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

      1. *-commutative N/A

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

      2. /-rgt-identity N/A

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

      3. associate-/r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 44.0

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

   4. Applied egg-rr 44.0%

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

      1. associate-/r/ N/A

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

      2. frac-times N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. sqrt-prod N/A

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

      6. *-inverses N/A

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

      7. sqrt-prod N/A

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

      8. times-frac N/A

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

      9. clear-num N/A

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

      10. sqrt-div N/A

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

      11. metadata-eval N/A

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

      12. associate-*r/ N/A

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

      13. sqrt-div N/A

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

      14. associate-*r/ N/A

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

      15. clear-num N/A

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

   6. Applied egg-rr 72.7%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+305}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq -4 \cdot 10^{-315}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{-A}}{\sqrt{-V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-241}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+274}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;A \cdot \frac{c0}{\sqrt{V \cdot \left(A \cdot \ell\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+305}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -4 \cdot 10^{-315}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{-A}}{\sqrt{-V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-241}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+274}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;A \cdot \frac{c0}{\sqrt{V \cdot \left(A \cdot \ell\right)}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -1e+305)
   (* c0 (sqrt (/ (/ A V) l)))
   (if (<= (* V l) -4e-315)
     (/ (* c0 (sqrt (- A))) (sqrt (- (* V l))))
     (if (<= (* V l) 1e-241)
       (/ c0 (sqrt (* l (/ V A))))
       (if (<= (* V l) 5e+274)
         (* c0 (/ (sqrt A) (sqrt (* V l))))
         (* A (/ c0 (sqrt (* V (* A l))))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+305) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if ((V * l) <= -4e-315) {
		tmp = (c0 * sqrt(-A)) / sqrt(-(V * l));
	} else if ((V * l) <= 1e-241) {
		tmp = c0 / sqrt((l * (V / A)));
	} else if ((V * l) <= 5e+274) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = A * (c0 / sqrt((V * (A * l))));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-1d+305)) then
        tmp = c0 * sqrt(((a / v) / l))
    else if ((v * l) <= (-4d-315)) then
        tmp = (c0 * sqrt(-a)) / sqrt(-(v * l))
    else if ((v * l) <= 1d-241) then
        tmp = c0 / sqrt((l * (v / a)))
    else if ((v * l) <= 5d+274) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = a * (c0 / sqrt((v * (a * l))))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+305) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if ((V * l) <= -4e-315) {
		tmp = (c0 * Math.sqrt(-A)) / Math.sqrt(-(V * l));
	} else if ((V * l) <= 1e-241) {
		tmp = c0 / Math.sqrt((l * (V / A)));
	} else if ((V * l) <= 5e+274) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = A * (c0 / Math.sqrt((V * (A * l))));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -1e+305:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif (V * l) <= -4e-315:
		tmp = (c0 * math.sqrt(-A)) / math.sqrt(-(V * l))
	elif (V * l) <= 1e-241:
		tmp = c0 / math.sqrt((l * (V / A)))
	elif (V * l) <= 5e+274:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = A * (c0 / math.sqrt((V * (A * l))))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -1e+305)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (Float64(V * l) <= -4e-315)
		tmp = Float64(Float64(c0 * sqrt(Float64(-A))) / sqrt(Float64(-Float64(V * l))));
	elseif (Float64(V * l) <= 1e-241)
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / A))));
	elseif (Float64(V * l) <= 5e+274)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(A * Float64(c0 / sqrt(Float64(V * Float64(A * l)))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -1e+305)
		tmp = c0 * sqrt(((A / V) / l));
	elseif ((V * l) <= -4e-315)
		tmp = (c0 * sqrt(-A)) / sqrt(-(V * l));
	elseif ((V * l) <= 1e-241)
		tmp = c0 / sqrt((l * (V / A)));
	elseif ((V * l) <= 5e+274)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = A * (c0 / sqrt((V * (A * l))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -1e+305], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -4e-315], N[(N[(c0 * N[Sqrt[(-A)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-N[(V * l), $MachinePrecision])], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e-241], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e+274], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(A * N[(c0 / N[Sqrt[N[(V * N[(A * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -9.9999999999999994e304

   1. Initial program 31.6%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 70.2

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

   4. Applied egg-rr 70.2%

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

   if -9.9999999999999994e304 < (*.f64 V l) < -3.9999999989e-315

   1. Initial program 82.3%

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

      1. *-commutative N/A

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

      2. /-rgt-identity N/A

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

      3. associate-/r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 82.2

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

   4. Applied egg-rr 82.2%

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

      1. *-commutative N/A

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

      2. associate-/r/ N/A

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

      3. un-div-inv N/A

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

      4. associate-/l/ N/A

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

      5. frac-2neg N/A

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

      6. sqrt-div N/A

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

      7. associate-*l/ N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. sqrt-lowering-sqrt.f64 N/A

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

      11. neg-lowering-neg.f64 N/A

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

      12. sqrt-lowering-sqrt.f64 N/A

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

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

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

      14. *-lowering-*.f64 N/A

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

      15. neg-lowering-neg.f64 96.9

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

   6. Applied egg-rr 96.9%

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

   if -3.9999999989e-315 < (*.f64 V l) < 9.9999999999999997e-242

   1. Initial program 54.3%

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

      1. *-commutative N/A

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

      2. /-rgt-identity N/A

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

      3. associate-/r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 54.3

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

   4. Applied egg-rr 54.3%

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

      1. associate-/r/ N/A

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

      2. frac-times N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. sqrt-prod N/A

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

      6. *-inverses N/A

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

      7. sqrt-prod N/A

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

      8. times-frac N/A

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

      9. clear-num N/A

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

      10. sqrt-div N/A

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

      11. metadata-eval N/A

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

      12. un-div-inv N/A

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

      13. /-lowering-/.f64 N/A

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

      14. sqrt-lowering-sqrt.f64 N/A

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

      15. associate-/l* N/A

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

      16. associate-/r* N/A

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

      17. *-inverses N/A

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

      18. div-inv N/A

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

      19. /-lowering-/.f64 N/A

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

   6. Applied egg-rr 54.4%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. metadata-eval N/A

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

      4. div-inv N/A

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

      5. metadata-eval N/A

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

      6. *-inverses N/A

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

      7. clear-num N/A

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

      8. times-frac N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. clear-num N/A

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

      12. times-frac N/A

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

      13. *-inverses N/A

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

      14. *-lft-identity N/A

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

      15. clear-num N/A

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

      16. /-lowering-/.f64 73.9

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

   8. Applied egg-rr 73.9%

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

   if 9.9999999999999997e-242 < (*.f64 V l) < 4.9999999999999998e274

   1. Initial program 88.7%

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

      1. sqrt-div N/A

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

      2. /-lowering-/.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. *-lowering-*.f64 99.4

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

   4. Applied egg-rr 99.4%

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

   if 4.9999999999999998e274 < (*.f64 V l)

   1. Initial program 44.0%

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

      1. *-commutative N/A

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

      2. /-rgt-identity N/A

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

      3. associate-/r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 44.0

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

   4. Applied egg-rr 44.0%

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

      1. associate-/r/ N/A

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

      2. frac-times N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. sqrt-prod N/A

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

      6. *-inverses N/A

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

      7. sqrt-prod N/A

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

      8. times-frac N/A

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

      9. clear-num N/A

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

      10. sqrt-div N/A

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

      11. metadata-eval N/A

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

      12. associate-*r/ N/A

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

      13. sqrt-div N/A

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

      14. associate-*r/ N/A

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

      15. clear-num N/A

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

   6. Applied egg-rr 72.7%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+305}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -4 \cdot 10^{-315}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{-A}}{\sqrt{-V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-241}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+274}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;A \cdot \frac{c0}{\sqrt{V \cdot \left(A \cdot \ell\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;A \cdot \frac{c0}{\sqrt{V \cdot \left(A \cdot \ell\right)}}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+291}:\\ \;\;\;\;c0 \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (<= t_0 0.0)
     (* A (/ c0 (sqrt (* V (* A l)))))
     (if (<= t_0 4e+291) (* c0 (sqrt t_0)) (/ c0 (sqrt (* l (/ V A))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = A * (c0 / sqrt((V * (A * l))));
	} else if (t_0 <= 4e+291) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = c0 / sqrt((l * (V / A)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if (t_0 <= 0.0d0) then
        tmp = a * (c0 / sqrt((v * (a * l))))
    else if (t_0 <= 4d+291) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = c0 / sqrt((l * (v / a)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = A * (c0 / Math.sqrt((V * (A * l))));
	} else if (t_0 <= 4e+291) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = c0 / Math.sqrt((l * (V / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 0.0:
		tmp = A * (c0 / math.sqrt((V * (A * l))))
	elif t_0 <= 4e+291:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = c0 / math.sqrt((l * (V / A)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(A * Float64(c0 / sqrt(Float64(V * Float64(A * l)))));
	elseif (t_0 <= 4e+291)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / A))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = A * (c0 / sqrt((V * (A * l))));
	elseif (t_0 <= 4e+291)
		tmp = c0 * sqrt(t_0);
	else
		tmp = c0 / sqrt((l * (V / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(A * N[(c0 / N[Sqrt[N[(V * N[(A * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+291], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 33.0%

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

      1. *-commutative N/A

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

      2. /-rgt-identity N/A

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

      3. associate-/r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 33.0

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

   4. Applied egg-rr 33.0%

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

      1. associate-/r/ N/A

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

      2. frac-times N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. sqrt-prod N/A

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

      6. *-inverses N/A

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

      7. sqrt-prod N/A

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

      8. times-frac N/A

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

      9. clear-num N/A

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

      10. sqrt-div N/A

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

      11. metadata-eval N/A

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

      12. associate-*r/ N/A

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

      13. sqrt-div N/A

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

      14. associate-*r/ N/A

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

      15. clear-num N/A

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

   6. Applied egg-rr 54.1%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 3.9999999999999998e291

   1. Initial program 98.8%

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

   if 3.9999999999999998e291 < (/.f64 A (*.f64 V l))

   1. Initial program 47.3%

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

      1. *-commutative N/A

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

      2. /-rgt-identity N/A

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

      3. associate-/r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 47.3

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

   4. Applied egg-rr 47.3%

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

      1. associate-/r/ N/A

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

      2. frac-times N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. sqrt-prod N/A

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

      6. *-inverses N/A

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

      7. sqrt-prod N/A

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

      8. times-frac N/A

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

      9. clear-num N/A

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

      10. sqrt-div N/A

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

      11. metadata-eval N/A

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

      12. un-div-inv N/A

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

      13. /-lowering-/.f64 N/A

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

      14. sqrt-lowering-sqrt.f64 N/A

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

      15. associate-/l* N/A

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

      16. associate-/r* N/A

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

      17. *-inverses N/A

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

      18. div-inv N/A

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

      19. /-lowering-/.f64 N/A

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

   6. Applied egg-rr 47.3%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. metadata-eval N/A

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

      4. div-inv N/A

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

      5. metadata-eval N/A

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

      6. *-inverses N/A

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

      7. clear-num N/A

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

      8. times-frac N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. clear-num N/A

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

      12. times-frac N/A

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

      13. *-inverses N/A

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

      14. *-lft-identity N/A

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

      15. clear-num N/A

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

      16. /-lowering-/.f64 58.9

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

   8. Applied egg-rr 58.9%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0:\\ \;\;\;\;A \cdot \frac{c0}{\sqrt{V \cdot \left(A \cdot \ell\right)}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 4 \cdot 10^{+291}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;A \cdot \frac{c0}{\sqrt{V \cdot \left(A \cdot \ell\right)}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+296}:\\ \;\;\;\;c0 \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (<= t_0 0.0)
     (* A (/ c0 (sqrt (* V (* A l)))))
     (if (<= t_0 2e+296) (* c0 (sqrt t_0)) (/ c0 (sqrt (* V (/ l A))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = A * (c0 / sqrt((V * (A * l))));
	} else if (t_0 <= 2e+296) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = c0 / sqrt((V * (l / A)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if (t_0 <= 0.0d0) then
        tmp = a * (c0 / sqrt((v * (a * l))))
    else if (t_0 <= 2d+296) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = c0 / sqrt((v * (l / a)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = A * (c0 / Math.sqrt((V * (A * l))));
	} else if (t_0 <= 2e+296) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = c0 / Math.sqrt((V * (l / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 0.0:
		tmp = A * (c0 / math.sqrt((V * (A * l))))
	elif t_0 <= 2e+296:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = c0 / math.sqrt((V * (l / A)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(A * Float64(c0 / sqrt(Float64(V * Float64(A * l)))));
	elseif (t_0 <= 2e+296)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = A * (c0 / sqrt((V * (A * l))));
	elseif (t_0 <= 2e+296)
		tmp = c0 * sqrt(t_0);
	else
		tmp = c0 / sqrt((V * (l / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(A * N[(c0 / N[Sqrt[N[(V * N[(A * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+296], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 33.0%

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

      1. *-commutative N/A

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

      2. /-rgt-identity N/A

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

      3. associate-/r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 33.0

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

   4. Applied egg-rr 33.0%

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

      1. associate-/r/ N/A

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

      2. frac-times N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. sqrt-prod N/A

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

      6. *-inverses N/A

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

      7. sqrt-prod N/A

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

      8. times-frac N/A

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

      9. clear-num N/A

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

      10. sqrt-div N/A

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

      11. metadata-eval N/A

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

      12. associate-*r/ N/A

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

      13. sqrt-div N/A

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

      14. associate-*r/ N/A

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

      15. clear-num N/A

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

   6. Applied egg-rr 54.1%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 1.99999999999999996e296

   1. Initial program 98.8%

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

   if 1.99999999999999996e296 < (/.f64 A (*.f64 V l))

   1. Initial program 46.5%

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

      1. clear-num N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. un-div-inv N/A

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

      5. /-lowering-/.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 59.7

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

   4. Applied egg-rr 59.7%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0:\\ \;\;\;\;A \cdot \frac{c0}{\sqrt{V \cdot \left(A \cdot \ell\right)}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{+296}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;A \cdot \frac{c0}{\sqrt{V \cdot \left(A \cdot \ell\right)}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+296}:\\ \;\;\;\;c0 \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (<= t_0 0.0)
     (* A (/ c0 (sqrt (* V (* A l)))))
     (if (<= t_0 2e+296) (* c0 (sqrt t_0)) (* c0 (sqrt (/ (/ A l) V)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = A * (c0 / sqrt((V * (A * l))));
	} else if (t_0 <= 2e+296) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = c0 * sqrt(((A / l) / V));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if (t_0 <= 0.0d0) then
        tmp = a * (c0 / sqrt((v * (a * l))))
    else if (t_0 <= 2d+296) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = c0 * sqrt(((a / l) / v))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = A * (c0 / Math.sqrt((V * (A * l))));
	} else if (t_0 <= 2e+296) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = c0 * Math.sqrt(((A / l) / V));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 0.0:
		tmp = A * (c0 / math.sqrt((V * (A * l))))
	elif t_0 <= 2e+296:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = c0 * math.sqrt(((A / l) / V))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(A * Float64(c0 / sqrt(Float64(V * Float64(A * l)))));
	elseif (t_0 <= 2e+296)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = A * (c0 / sqrt((V * (A * l))));
	elseif (t_0 <= 2e+296)
		tmp = c0 * sqrt(t_0);
	else
		tmp = c0 * sqrt(((A / l) / V));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(A * N[(c0 / N[Sqrt[N[(V * N[(A * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+296], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 33.0%

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

      1. *-commutative N/A

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

      2. /-rgt-identity N/A

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

      3. associate-/r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 33.0

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

   4. Applied egg-rr 33.0%

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

      1. associate-/r/ N/A

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

      2. frac-times N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. sqrt-prod N/A

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

      6. *-inverses N/A

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

      7. sqrt-prod N/A

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

      8. times-frac N/A

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

      9. clear-num N/A

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

      10. sqrt-div N/A

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

      11. metadata-eval N/A

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

      12. associate-*r/ N/A

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

      13. sqrt-div N/A

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

      14. associate-*r/ N/A

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

      15. clear-num N/A

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

   6. Applied egg-rr 54.1%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 1.99999999999999996e296

   1. Initial program 98.8%

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

   if 1.99999999999999996e296 < (/.f64 A (*.f64 V l))

   1. Initial program 46.5%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 57.9

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

   4. Applied egg-rr 57.9%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0:\\ \;\;\;\;A \cdot \frac{c0}{\sqrt{V \cdot \left(A \cdot \ell\right)}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{+296}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;A \cdot \frac{c0}{\sqrt{V \cdot \left(A \cdot \ell\right)}}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+291}:\\ \;\;\;\;c0 \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (<= t_0 0.0)
     (* A (/ c0 (sqrt (* V (* A l)))))
     (if (<= t_0 4e+291) (* c0 (sqrt t_0)) (* c0 (sqrt (/ (/ A V) l)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = A * (c0 / sqrt((V * (A * l))));
	} else if (t_0 <= 4e+291) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = c0 * sqrt(((A / V) / l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if (t_0 <= 0.0d0) then
        tmp = a * (c0 / sqrt((v * (a * l))))
    else if (t_0 <= 4d+291) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = c0 * sqrt(((a / v) / l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = A * (c0 / Math.sqrt((V * (A * l))));
	} else if (t_0 <= 4e+291) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = c0 * Math.sqrt(((A / V) / l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 0.0:
		tmp = A * (c0 / math.sqrt((V * (A * l))))
	elif t_0 <= 4e+291:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = c0 * math.sqrt(((A / V) / l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(A * Float64(c0 / sqrt(Float64(V * Float64(A * l)))));
	elseif (t_0 <= 4e+291)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = A * (c0 / sqrt((V * (A * l))));
	elseif (t_0 <= 4e+291)
		tmp = c0 * sqrt(t_0);
	else
		tmp = c0 * sqrt(((A / V) / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(A * N[(c0 / N[Sqrt[N[(V * N[(A * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+291], N[(c0 * N[Sqrt[t$95$0], $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 A (*.f64 V l)) < 0.0

   1. Initial program 33.0%

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

      1. *-commutative N/A

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

      2. /-rgt-identity N/A

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

      3. associate-/r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 33.0

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

   4. Applied egg-rr 33.0%

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

      1. associate-/r/ N/A

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

      2. frac-times N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. sqrt-prod N/A

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

      6. *-inverses N/A

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

      7. sqrt-prod N/A

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

      8. times-frac N/A

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

      9. clear-num N/A

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

      10. sqrt-div N/A

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

      11. metadata-eval N/A

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

      12. associate-*r/ N/A

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

      13. sqrt-div N/A

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

      14. associate-*r/ N/A

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

      15. clear-num N/A

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

   6. Applied egg-rr 54.1%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 3.9999999999999998e291

   1. Initial program 98.8%

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

   if 3.9999999999999998e291 < (/.f64 A (*.f64 V l))

   1. Initial program 47.3%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 57.1

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

   4. Applied egg-rr 57.1%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0:\\ \;\;\;\;A \cdot \frac{c0}{\sqrt{V \cdot \left(A \cdot \ell\right)}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 4 \cdot 10^{+291}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -4 \cdot 10^{-315}:\\ \;\;\;\;\sqrt{-A} \cdot \frac{c0}{\sqrt{-V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-241}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+274}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;A \cdot \frac{c0}{\sqrt{V \cdot \left(A \cdot \ell\right)}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -4e-315)
   (* (sqrt (- A)) (/ c0 (sqrt (- (* V l)))))
   (if (<= (* V l) 1e-241)
     (/ c0 (sqrt (* l (/ V A))))
     (if (<= (* V l) 5e+274)
       (* c0 (/ (sqrt A) (sqrt (* V l))))
       (* A (/ c0 (sqrt (* V (* A l)))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -4e-315) {
		tmp = sqrt(-A) * (c0 / sqrt(-(V * l)));
	} else if ((V * l) <= 1e-241) {
		tmp = c0 / sqrt((l * (V / A)));
	} else if ((V * l) <= 5e+274) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = A * (c0 / sqrt((V * (A * l))));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-4d-315)) then
        tmp = sqrt(-a) * (c0 / sqrt(-(v * l)))
    else if ((v * l) <= 1d-241) then
        tmp = c0 / sqrt((l * (v / a)))
    else if ((v * l) <= 5d+274) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = a * (c0 / sqrt((v * (a * l))))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -4e-315) {
		tmp = Math.sqrt(-A) * (c0 / Math.sqrt(-(V * l)));
	} else if ((V * l) <= 1e-241) {
		tmp = c0 / Math.sqrt((l * (V / A)));
	} else if ((V * l) <= 5e+274) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = A * (c0 / Math.sqrt((V * (A * l))));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -4e-315:
		tmp = math.sqrt(-A) * (c0 / math.sqrt(-(V * l)))
	elif (V * l) <= 1e-241:
		tmp = c0 / math.sqrt((l * (V / A)))
	elif (V * l) <= 5e+274:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = A * (c0 / math.sqrt((V * (A * l))))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -4e-315)
		tmp = Float64(sqrt(Float64(-A)) * Float64(c0 / sqrt(Float64(-Float64(V * l)))));
	elseif (Float64(V * l) <= 1e-241)
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / A))));
	elseif (Float64(V * l) <= 5e+274)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(A * Float64(c0 / sqrt(Float64(V * Float64(A * l)))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -4e-315)
		tmp = sqrt(-A) * (c0 / sqrt(-(V * l)));
	elseif ((V * l) <= 1e-241)
		tmp = c0 / sqrt((l * (V / A)));
	elseif ((V * l) <= 5e+274)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = A * (c0 / sqrt((V * (A * l))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -4e-315], N[(N[Sqrt[(-A)], $MachinePrecision] * N[(c0 / N[Sqrt[(-N[(V * l), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e-241], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e+274], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(A * N[(c0 / N[Sqrt[N[(V * N[(A * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -3.9999999989e-315

   1. Initial program 77.9%

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

      1. *-commutative N/A

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

      2. /-rgt-identity N/A

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

      3. associate-/r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 77.8

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

   4. Applied egg-rr 77.8%

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

      1. associate-/r/ N/A

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

      2. frac-times N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. sqrt-prod N/A

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

      6. *-inverses N/A

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

      7. sqrt-prod N/A

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

      8. times-frac N/A

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

      9. clear-num N/A

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

      10. sqrt-div N/A

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

      11. metadata-eval N/A

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

      12. un-div-inv N/A

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

      13. /-lowering-/.f64 N/A

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

      14. sqrt-lowering-sqrt.f64 N/A

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

      15. associate-/l* N/A

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

      16. associate-/r* N/A

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

      17. *-inverses N/A

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

      18. div-inv N/A

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

      19. /-lowering-/.f64 N/A

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

   6. Applied egg-rr 77.3%

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

      1. frac-2neg N/A

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

      2. sqrt-div N/A

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

      3. associate-/r/ N/A

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

      4. un-div-inv N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. sqrt-div N/A

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

      8. frac-2neg N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

   8. Applied egg-rr 89.9%

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

   if -3.9999999989e-315 < (*.f64 V l) < 9.9999999999999997e-242

   1. Initial program 54.3%

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

      1. *-commutative N/A

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

      2. /-rgt-identity N/A

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

      3. associate-/r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 54.3

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

   4. Applied egg-rr 54.3%

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

      1. associate-/r/ N/A

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

      2. frac-times N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. sqrt-prod N/A

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

      6. *-inverses N/A

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

      7. sqrt-prod N/A

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

      8. times-frac N/A

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

      9. clear-num N/A

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

      10. sqrt-div N/A

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

      11. metadata-eval N/A

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

      12. un-div-inv N/A

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

      13. /-lowering-/.f64 N/A

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

      14. sqrt-lowering-sqrt.f64 N/A

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

      15. associate-/l* N/A

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

      16. associate-/r* N/A

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

      17. *-inverses N/A

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

      18. div-inv N/A

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

      19. /-lowering-/.f64 N/A

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

   6. Applied egg-rr 54.4%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. metadata-eval N/A

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

      4. div-inv N/A

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

      5. metadata-eval N/A

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

      6. *-inverses N/A

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

      7. clear-num N/A

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

      8. times-frac N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. clear-num N/A

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

      12. times-frac N/A

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

      13. *-inverses N/A

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

      14. *-lft-identity N/A

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

      15. clear-num N/A

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

      16. /-lowering-/.f64 73.9

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

   8. Applied egg-rr 73.9%

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

   if 9.9999999999999997e-242 < (*.f64 V l) < 4.9999999999999998e274

   1. Initial program 88.7%

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

      1. sqrt-div N/A

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

      2. /-lowering-/.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. *-lowering-*.f64 99.4

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

   4. Applied egg-rr 99.4%

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

   if 4.9999999999999998e274 < (*.f64 V l)

   1. Initial program 44.0%

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

      1. *-commutative N/A

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

      2. /-rgt-identity N/A

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

      3. associate-/r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 44.0

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

   4. Applied egg-rr 44.0%

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

      1. associate-/r/ N/A

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

      2. frac-times N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. sqrt-prod N/A

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

      6. *-inverses N/A

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

      7. sqrt-prod N/A

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

      8. times-frac N/A

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

      9. clear-num N/A

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

      10. sqrt-div N/A

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

      11. metadata-eval N/A

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

      12. associate-*r/ N/A

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

      13. sqrt-div N/A

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

      14. associate-*r/ N/A

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

      15. clear-num N/A

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

   6. Applied egg-rr 72.7%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -4 \cdot 10^{-315}:\\ \;\;\;\;\sqrt{-A} \cdot \frac{c0}{\sqrt{-V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-241}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+274}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;A \cdot \frac{c0}{\sqrt{V \cdot \left(A \cdot \ell\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{-114}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-241}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+274}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;A \cdot \frac{c0}{\sqrt{V \cdot \left(A \cdot \ell\right)}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -1e-114)
   (* c0 (sqrt (/ A (* V l))))
   (if (<= (* V l) 1e-241)
     (/ c0 (sqrt (* l (/ V A))))
     (if (<= (* V l) 5e+274)
       (* c0 (/ (sqrt A) (sqrt (* V l))))
       (* A (/ c0 (sqrt (* V (* A l)))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e-114) {
		tmp = c0 * sqrt((A / (V * l)));
	} else if ((V * l) <= 1e-241) {
		tmp = c0 / sqrt((l * (V / A)));
	} else if ((V * l) <= 5e+274) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = A * (c0 / sqrt((V * (A * l))));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-1d-114)) then
        tmp = c0 * sqrt((a / (v * l)))
    else if ((v * l) <= 1d-241) then
        tmp = c0 / sqrt((l * (v / a)))
    else if ((v * l) <= 5d+274) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = a * (c0 / sqrt((v * (a * l))))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e-114) {
		tmp = c0 * Math.sqrt((A / (V * l)));
	} else if ((V * l) <= 1e-241) {
		tmp = c0 / Math.sqrt((l * (V / A)));
	} else if ((V * l) <= 5e+274) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = A * (c0 / Math.sqrt((V * (A * l))));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -1e-114:
		tmp = c0 * math.sqrt((A / (V * l)))
	elif (V * l) <= 1e-241:
		tmp = c0 / math.sqrt((l * (V / A)))
	elif (V * l) <= 5e+274:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = A * (c0 / math.sqrt((V * (A * l))))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -1e-114)
		tmp = Float64(c0 * sqrt(Float64(A / Float64(V * l))));
	elseif (Float64(V * l) <= 1e-241)
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / A))));
	elseif (Float64(V * l) <= 5e+274)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(A * Float64(c0 / sqrt(Float64(V * Float64(A * l)))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -1e-114)
		tmp = c0 * sqrt((A / (V * l)));
	elseif ((V * l) <= 1e-241)
		tmp = c0 / sqrt((l * (V / A)));
	elseif ((V * l) <= 5e+274)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = A * (c0 / sqrt((V * (A * l))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -1e-114], N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e-241], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e+274], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(A * N[(c0 / N[Sqrt[N[(V * N[(A * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -1.0000000000000001e-114

   1. Initial program 81.8%

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

   if -1.0000000000000001e-114 < (*.f64 V l) < 9.9999999999999997e-242

   1. Initial program 58.4%

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

      1. *-commutative N/A

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

      2. /-rgt-identity N/A

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

      3. associate-/r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 58.3

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

   4. Applied egg-rr 58.3%

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

      1. associate-/r/ N/A

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

      2. frac-times N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. sqrt-prod N/A

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

      6. *-inverses N/A

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

      7. sqrt-prod N/A

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

      8. times-frac N/A

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

      9. clear-num N/A

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

      10. sqrt-div N/A

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

      11. metadata-eval N/A

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

      12. un-div-inv N/A

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

      13. /-lowering-/.f64 N/A

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

      14. sqrt-lowering-sqrt.f64 N/A

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

      15. associate-/l* N/A

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

      16. associate-/r* N/A

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

      17. *-inverses N/A

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

      18. div-inv N/A

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

      19. /-lowering-/.f64 N/A

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

   6. Applied egg-rr 58.5%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. metadata-eval N/A

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

      4. div-inv N/A

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

      5. metadata-eval N/A

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

      6. *-inverses N/A

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

      7. clear-num N/A

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

      8. times-frac N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. clear-num N/A

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

      12. times-frac N/A

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

      13. *-inverses N/A

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

      14. *-lft-identity N/A

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

      15. clear-num N/A

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

      16. /-lowering-/.f64 71.0

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

   8. Applied egg-rr 71.0%

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

   if 9.9999999999999997e-242 < (*.f64 V l) < 4.9999999999999998e274

   1. Initial program 88.7%

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

      1. sqrt-div N/A

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

      2. /-lowering-/.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. *-lowering-*.f64 99.4

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

   4. Applied egg-rr 99.4%

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

   if 4.9999999999999998e274 < (*.f64 V l)

   1. Initial program 44.0%

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

      1. *-commutative N/A

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

      2. /-rgt-identity N/A

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

      3. associate-/r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 44.0

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

   4. Applied egg-rr 44.0%

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

      1. associate-/r/ N/A

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

      2. frac-times N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. sqrt-prod N/A

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

      6. *-inverses N/A

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

      7. sqrt-prod N/A

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

      8. times-frac N/A

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

      9. clear-num N/A

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

      10. sqrt-div N/A

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

      11. metadata-eval N/A

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

      12. associate-*r/ N/A

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

      13. sqrt-div N/A

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

      14. associate-*r/ N/A

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

      15. clear-num N/A

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

   6. Applied egg-rr 72.7%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{-114}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-241}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+274}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;A \cdot \frac{c0}{\sqrt{V \cdot \left(A \cdot \ell\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 90.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -2 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \left(\sqrt{-A} \cdot \frac{1}{\sqrt{-V} \cdot \sqrt{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= A -2e-310)
   (* c0 (* (sqrt (- A)) (/ 1.0 (* (sqrt (- V)) (sqrt l)))))
   (* c0 (/ (sqrt A) (sqrt (* V l))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -2e-310) {
		tmp = c0 * (sqrt(-A) * (1.0 / (sqrt(-V) * sqrt(l))));
	} else {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (a <= (-2d-310)) then
        tmp = c0 * (sqrt(-a) * (1.0d0 / (sqrt(-v) * sqrt(l))))
    else
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -2e-310) {
		tmp = c0 * (Math.sqrt(-A) * (1.0 / (Math.sqrt(-V) * Math.sqrt(l))));
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if A <= -2e-310:
		tmp = c0 * (math.sqrt(-A) * (1.0 / (math.sqrt(-V) * math.sqrt(l))))
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (A <= -2e-310)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) * Float64(1.0 / Float64(sqrt(Float64(-V)) * sqrt(l)))));
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (A <= -2e-310)
		tmp = c0 * (sqrt(-A) * (1.0 / (sqrt(-V) * sqrt(l))));
	else
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[A, -2e-310], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] * N[(1.0 / N[(N[Sqrt[(-V)], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -1.999999999999994e-310

   1. Initial program 72.7%

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

      1. *-commutative N/A

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

      2. /-rgt-identity N/A

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

      3. associate-/r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 72.6

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

   4. Applied egg-rr 72.6%

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

      1. frac-2neg N/A

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

      2. frac-2neg N/A

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

      3. associate-/r/ N/A

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

      4. un-div-inv N/A

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

      5. associate-/l/ N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. sqrt-prod N/A

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

      9. *-lowering-*.f64 N/A

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

      10. sqrt-lowering-sqrt.f64 N/A

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

      11. neg-lowering-neg.f64 N/A

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

      12. sqrt-lowering-sqrt.f64 N/A

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

      13. metadata-eval N/A

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

      14. frac-2neg N/A

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

      15. /-lowering-/.f64 N/A

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 84.6

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

   6. Applied egg-rr 84.6%

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

      1. frac-2neg N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 N/A

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

      7. metadata-eval N/A

         \[\leadsto c0 \cdot \left(\sqrt{\mathsf{neg}\left(A\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\mathsf{neg}\left(\ell \cdot V\right)}}\right) \]

      8. /-rgt-identity N/A

         \[\leadsto c0 \cdot \left(\sqrt{\mathsf{neg}\left(A\right)} \cdot \frac{1}{\sqrt{\color{blue}{\frac{\mathsf{neg}\left(\ell \cdot V\right)}{1}}}}\right) \]

      9. metadata-eval N/A

         \[\leadsto c0 \cdot \left(\sqrt{\mathsf{neg}\left(A\right)} \cdot \frac{1}{\sqrt{\frac{\mathsf{neg}\left(\ell \cdot V\right)}{\color{blue}{\mathsf{neg}\left(-1\right)}}}}\right) \]

      10. frac-2neg N/A

          \[\leadsto c0 \cdot \left(\sqrt{\mathsf{neg}\left(A\right)} \cdot \frac{1}{\sqrt{\color{blue}{\frac{\ell \cdot V}{-1}}}}\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto c0 \cdot \left(\sqrt{\mathsf{neg}\left(A\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{\ell \cdot V}{-1}}}}\right) \]

      12. frac-2neg N/A

          \[\leadsto c0 \cdot \left(\sqrt{\mathsf{neg}\left(A\right)} \cdot \frac{1}{\sqrt{\color{blue}{\frac{\mathsf{neg}\left(\ell \cdot V\right)}{\mathsf{neg}\left(-1\right)}}}}\right) \]

      13. metadata-eval N/A

          \[\leadsto c0 \cdot \left(\sqrt{\mathsf{neg}\left(A\right)} \cdot \frac{1}{\sqrt{\frac{\mathsf{neg}\left(\ell \cdot V\right)}{\color{blue}{1}}}}\right) \]

      14. /-rgt-identity N/A

          \[\leadsto c0 \cdot \left(\sqrt{\mathsf{neg}\left(A\right)} \cdot \frac{1}{\sqrt{\color{blue}{\mathsf{neg}\left(\ell \cdot V\right)}}}\right) \]

      15. distribute-rgt-neg-in N/A

          \[\leadsto c0 \cdot \left(\sqrt{\mathsf{neg}\left(A\right)} \cdot \frac{1}{\sqrt{\color{blue}{\ell \cdot \left(\mathsf{neg}\left(V\right)\right)}}}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto c0 \cdot \left(\sqrt{\mathsf{neg}\left(A\right)} \cdot \frac{1}{\sqrt{\color{blue}{\ell \cdot \left(\mathsf{neg}\left(V\right)\right)}}}\right) \]

      17. neg-lowering-neg.f64 85.2

          \[\leadsto c0 \cdot \left(\sqrt{-A} \cdot \frac{1}{\sqrt{\ell \cdot \color{blue}{\left(-V\right)}}}\right) \]

   8. Applied egg-rr 85.2%

      \[\leadsto c0 \cdot \left(\sqrt{-A} \cdot \color{blue}{\frac{1}{\sqrt{\ell \cdot \left(-V\right)}}}\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto c0 \cdot \left(\sqrt{\mathsf{neg}\left(A\right)} \cdot \frac{1}{\sqrt{\color{blue}{\left(\mathsf{neg}\left(V\right)\right) \cdot \ell}}}\right) \]

      2. sqrt-prod N/A

         \[\leadsto c0 \cdot \left(\sqrt{\mathsf{neg}\left(A\right)} \cdot \frac{1}{\color{blue}{\sqrt{\mathsf{neg}\left(V\right)} \cdot \sqrt{\ell}}}\right) \]

      3. pow1/2 N/A

         \[\leadsto c0 \cdot \left(\sqrt{\mathsf{neg}\left(A\right)} \cdot \frac{1}{\sqrt{\mathsf{neg}\left(V\right)} \cdot \color{blue}{{\ell}^{\frac{1}{2}}}}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto c0 \cdot \left(\sqrt{\mathsf{neg}\left(A\right)} \cdot \frac{1}{\color{blue}{\sqrt{\mathsf{neg}\left(V\right)} \cdot {\ell}^{\frac{1}{2}}}}\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto c0 \cdot \left(\sqrt{\mathsf{neg}\left(A\right)} \cdot \frac{1}{\color{blue}{\sqrt{\mathsf{neg}\left(V\right)}} \cdot {\ell}^{\frac{1}{2}}}\right) \]

      6. neg-lowering-neg.f64 N/A

         \[\leadsto c0 \cdot \left(\sqrt{\mathsf{neg}\left(A\right)} \cdot \frac{1}{\sqrt{\color{blue}{\mathsf{neg}\left(V\right)}} \cdot {\ell}^{\frac{1}{2}}}\right) \]

      7. pow1/2 N/A

         \[\leadsto c0 \cdot \left(\sqrt{\mathsf{neg}\left(A\right)} \cdot \frac{1}{\sqrt{\mathsf{neg}\left(V\right)} \cdot \color{blue}{\sqrt{\ell}}}\right) \]

      8. sqrt-lowering-sqrt.f64 52.8

         \[\leadsto c0 \cdot \left(\sqrt{-A} \cdot \frac{1}{\sqrt{-V} \cdot \color{blue}{\sqrt{\ell}}}\right) \]

   10. Applied egg-rr 52.8%

       \[\leadsto c0 \cdot \left(\sqrt{-A} \cdot \frac{1}{\color{blue}{\sqrt{-V} \cdot \sqrt{\ell}}}\right) \]

   if -1.999999999999994e-310 < A

   1. Initial program 75.9%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sqrt-div N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}} \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{A}}}{\sqrt{V \cdot \ell}} \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\color{blue}{\sqrt{V \cdot \ell}}} \]

      5. *-lowering-*.f64 83.4

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{V \cdot \ell}}} \]

   4. Applied egg-rr 83.4%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 76.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;A \cdot \frac{c0}{\sqrt{V \cdot \left(A \cdot \ell\right)}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (<= t_0 0.0) (* A (/ c0 (sqrt (* V (* A l))))) (* c0 (sqrt t_0)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = A * (c0 / sqrt((V * (A * l))));
	} else {
		tmp = c0 * sqrt(t_0);
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if (t_0 <= 0.0d0) then
        tmp = a * (c0 / sqrt((v * (a * l))))
    else
        tmp = c0 * sqrt(t_0)
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = A * (c0 / Math.sqrt((V * (A * l))));
	} else {
		tmp = c0 * Math.sqrt(t_0);
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 0.0:
		tmp = A * (c0 / math.sqrt((V * (A * l))))
	else:
		tmp = c0 * math.sqrt(t_0)
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(A * Float64(c0 / sqrt(Float64(V * Float64(A * l)))));
	else
		tmp = Float64(c0 * sqrt(t_0));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = A * (c0 / sqrt((V * (A * l))));
	else
		tmp = c0 * sqrt(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(A * N[(c0 / N[Sqrt[N[(V * N[(A * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 0.0

   1. Initial program 33.0%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{\ell \cdot V}}} \]

      2. /-rgt-identity N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{\frac{\ell}{1}} \cdot V}} \]

      3. associate-/r/ N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{\frac{\ell}{\frac{1}{V}}}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{\frac{\ell}{\frac{1}{V}}}}} \]

      5. /-lowering-/.f64 33.0

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\frac{\ell}{\color{blue}{\frac{1}{V}}}}} \]

   4. Applied egg-rr 33.0%

      \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{\frac{\ell}{\frac{1}{V}}}}} \]
   5. Step-by-step derivation

      1. associate-/r/ N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{\ell} \cdot \frac{1}{V}}} \]

      2. frac-times N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A \cdot 1}{\ell \cdot V}}} \]

      3. *-commutative N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot 1}{\color{blue}{V \cdot \ell}}} \]

      4. associate-*l/ N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell} \cdot 1}} \]

      5. sqrt-prod N/A

         \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{\frac{A}{V \cdot \ell}} \cdot \sqrt{1}\right)} \]

      6. *-inverses N/A

         \[\leadsto c0 \cdot \left(\sqrt{\frac{A}{V \cdot \ell}} \cdot \sqrt{\color{blue}{\frac{A}{A}}}\right) \]

      7. sqrt-prod N/A

         \[\leadsto c0 \cdot \color{blue}{\sqrt{\frac{A}{V \cdot \ell} \cdot \frac{A}{A}}} \]

      8. times-frac N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A \cdot A}{\left(V \cdot \ell\right) \cdot A}}} \]

      9. clear-num N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\frac{\left(V \cdot \ell\right) \cdot A}{A \cdot A}}}} \]

      10. sqrt-div N/A

          \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\left(V \cdot \ell\right) \cdot A}{A \cdot A}}}} \]

      11. metadata-eval N/A

          \[\leadsto c0 \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\left(V \cdot \ell\right) \cdot A}{A \cdot A}}} \]

      12. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{c0 \cdot 1}{\sqrt{\frac{\left(V \cdot \ell\right) \cdot A}{A \cdot A}}}} \]

      13. sqrt-div N/A

          \[\leadsto \frac{c0 \cdot 1}{\color{blue}{\frac{\sqrt{\left(V \cdot \ell\right) \cdot A}}{\sqrt{A \cdot A}}}} \]

      14. associate-*r/ N/A

          \[\leadsto \color{blue}{c0 \cdot \frac{1}{\frac{\sqrt{\left(V \cdot \ell\right) \cdot A}}{\sqrt{A \cdot A}}}} \]

      15. clear-num N/A

          \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A \cdot A}}{\sqrt{\left(V \cdot \ell\right) \cdot A}}} \]

   6. Applied egg-rr 54.1%

      \[\leadsto \color{blue}{A \cdot \frac{c0}{\sqrt{V \cdot \left(\ell \cdot A\right)}}} \]

   if 0.0 < (/.f64 A (*.f64 V l))

   1. Initial program 82.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0:\\ \;\;\;\;A \cdot \frac{c0}{\sqrt{V \cdot \left(A \cdot \ell\right)}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 72.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l) :precision binary64 (* c0 (sqrt (/ A (* V l)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	return c0 * sqrt((A / (V * l)));
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    code = c0 * sqrt((a / (v * l)))
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	return c0 * Math.sqrt((A / (V * l)));
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	return c0 * math.sqrt((A / (V * l)))

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	return Float64(c0 * sqrt(Float64(A / Float64(V * l))))
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp = code(c0, A, V, l)
	tmp = c0 * sqrt((A / (V * l)));
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.1%

   \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
2. Add Preprocessing
3. Add Preprocessing

Reproduce

herbie shell --seed 2024198 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))