Henrywood and Agarwal, Equation (3)

Percentage Accurate: 74.5% → 92.2%

Time: 10.0s

Alternatives: 19

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{0 - A}\\ \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\left(\sqrt{\frac{-1}{V}} \cdot t\_0\right) \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-253}:\\ \;\;\;\;c0 \cdot \left(t\_0 \cdot \sqrt{\frac{\frac{-1}{V}}{\ell}}\right)\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-294}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\frac{V}{\frac{A}{\ell}}}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+285}:\\ \;\;\;\;\frac{c0}{{A}^{-0.5} \cdot \sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (sqrt (- 0.0 A))))
   (if (<= (* V l) (- INFINITY))
     (* (* (sqrt (/ -1.0 V)) t_0) (/ c0 (sqrt l)))
     (if (<= (* V l) -1e-253)
       (* c0 (* t_0 (sqrt (/ (/ -1.0 V) l))))
       (if (<= (* V l) 1e-294)
         (/ 1.0 (/ (sqrt (/ V (/ A l))) c0))
         (if (<= (* V l) 1e+285)
           (/ c0 (* (pow A -0.5) (sqrt (* V l))))
           (* c0 (sqrt (/ (/ A V) l)))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt((0.0 - A));
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = (sqrt((-1.0 / V)) * t_0) * (c0 / sqrt(l));
	} else if ((V * l) <= -1e-253) {
		tmp = c0 * (t_0 * sqrt(((-1.0 / V) / l)));
	} else if ((V * l) <= 1e-294) {
		tmp = 1.0 / (sqrt((V / (A / l))) / c0);
	} else if ((V * l) <= 1e+285) {
		tmp = c0 / (pow(A, -0.5) * sqrt((V * l)));
	} else {
		tmp = c0 * sqrt(((A / V) / l));
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = Math.sqrt((0.0 - A));
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = (Math.sqrt((-1.0 / V)) * t_0) * (c0 / Math.sqrt(l));
	} else if ((V * l) <= -1e-253) {
		tmp = c0 * (t_0 * Math.sqrt(((-1.0 / V) / l)));
	} else if ((V * l) <= 1e-294) {
		tmp = 1.0 / (Math.sqrt((V / (A / l))) / c0);
	} else if ((V * l) <= 1e+285) {
		tmp = c0 / (Math.pow(A, -0.5) * Math.sqrt((V * l)));
	} else {
		tmp = c0 * Math.sqrt(((A / V) / l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt((0.0 - A))
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = (math.sqrt((-1.0 / V)) * t_0) * (c0 / math.sqrt(l))
	elif (V * l) <= -1e-253:
		tmp = c0 * (t_0 * math.sqrt(((-1.0 / V) / l)))
	elif (V * l) <= 1e-294:
		tmp = 1.0 / (math.sqrt((V / (A / l))) / c0)
	elif (V * l) <= 1e+285:
		tmp = c0 / (math.pow(A, -0.5) * math.sqrt((V * l)))
	else:
		tmp = c0 * math.sqrt(((A / V) / l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = sqrt(Float64(0.0 - A))
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(Float64(-1.0 / V)) * t_0) * Float64(c0 / sqrt(l)));
	elseif (Float64(V * l) <= -1e-253)
		tmp = Float64(c0 * Float64(t_0 * sqrt(Float64(Float64(-1.0 / V) / l))));
	elseif (Float64(V * l) <= 1e-294)
		tmp = Float64(1.0 / Float64(sqrt(Float64(V / Float64(A / l))) / c0));
	elseif (Float64(V * l) <= 1e+285)
		tmp = Float64(c0 / Float64((A ^ -0.5) * sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = sqrt((0.0 - A));
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = (sqrt((-1.0 / V)) * t_0) * (c0 / sqrt(l));
	elseif ((V * l) <= -1e-253)
		tmp = c0 * (t_0 * sqrt(((-1.0 / V) / l)));
	elseif ((V * l) <= 1e-294)
		tmp = 1.0 / (sqrt((V / (A / l))) / c0);
	elseif ((V * l) <= 1e+285)
		tmp = c0 / ((A ^ -0.5) * sqrt((V * l)));
	else
		tmp = c0 * sqrt(((A / V) / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[Sqrt[N[(0.0 - A), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(N[(N[Sqrt[N[(-1.0 / V), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] * N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-253], N[(c0 * N[(t$95$0 * N[Sqrt[N[(N[(-1.0 / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e-294], N[(1.0 / N[(N[Sqrt[N[(V / N[(A / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / c0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+285], N[(c0 / N[(N[Power[A, -0.5], $MachinePrecision] * N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 45.9%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. sqrt-div N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{A}{V}}\right), \color{blue}{\left(\frac{c0}{\sqrt{\ell}}\right)}\right) \]

      7. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{A}{V}\right)}^{\frac{1}{2}}\right), \left(\frac{\color{blue}{c0}}{\sqrt{\ell}}\right)\right) \]

      8. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{1}{\frac{V}{A}}\right)}^{\frac{1}{2}}\right), \left(\frac{c0}{\sqrt{\ell}}\right)\right) \]

      9. inv-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left({\left(\frac{V}{A}\right)}^{-1}\right)}^{\frac{1}{2}}\right), \left(\frac{c0}{\sqrt{\ell}}\right)\right) \]

      10. pow-pow N/A

          \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{V}{A}\right)}^{\left(-1 \cdot \frac{1}{2}\right)}\right), \left(\frac{\color{blue}{c0}}{\sqrt{\ell}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\frac{V}{A}\right), \left(-1 \cdot \frac{1}{2}\right)\right), \left(\frac{\color{blue}{c0}}{\sqrt{\ell}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(V, A\right), \left(-1 \cdot \frac{1}{2}\right)\right), \left(\frac{c0}{\sqrt{\ell}}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(V, A\right), \frac{-1}{2}\right), \left(\frac{c0}{\sqrt{\ell}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(V, A\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(c0, \color{blue}{\left(\sqrt{\ell}\right)}\right)\right) \]

      15. sqrt-lowering-sqrt.f64 39.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(V, A\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\ell\right)\right)\right) \]

   4. Applied egg-rr 39.5%

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

      1. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{V}{A}\right)}^{\left(\frac{-1}{2}\right)}\right), \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\ell\right)\right)\right) \]

      2. sqrt-pow1 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{{\left(\frac{V}{A}\right)}^{-1}}\right), \mathsf{/.f64}\left(\color{blue}{c0}, \mathsf{sqrt.f64}\left(\ell\right)\right)\right) \]

      3. inv-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{\frac{V}{A}}}\right), \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\ell\right)\right)\right) \]

      4. frac-2neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{\frac{\mathsf{neg}\left(V\right)}{\mathsf{neg}\left(A\right)}}}\right), \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\ell\right)\right)\right) \]

      5. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{\mathsf{neg}\left(V\right)} \cdot \left(\mathsf{neg}\left(A\right)\right)}\right), \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\ell\right)\right)\right) \]

      6. sqrt-prod N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{\mathsf{neg}\left(V\right)}} \cdot \sqrt{\mathsf{neg}\left(A\right)}\right), \mathsf{/.f64}\left(\color{blue}{c0}, \mathsf{sqrt.f64}\left(\ell\right)\right)\right) \]

      7. sub0-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{\mathsf{neg}\left(V\right)}} \cdot \sqrt{0 - A}\right), \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\ell\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{1}{\mathsf{neg}\left(V\right)}}\right), \left(\sqrt{0 - A}\right)\right), \mathsf{/.f64}\left(\color{blue}{c0}, \mathsf{sqrt.f64}\left(\ell\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{\mathsf{neg}\left(V\right)}\right)\right), \left(\sqrt{0 - A}\right)\right), \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\ell\right)\right)\right) \]

      10. distribute-frac-neg2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\frac{1}{V}\right)\right)\right), \left(\sqrt{0 - A}\right)\right), \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\ell\right)\right)\right) \]

      11. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\mathsf{neg}\left(1\right)}{V}\right)\right), \left(\sqrt{0 - A}\right)\right), \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\ell\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{-1}{V}\right)\right), \left(\sqrt{0 - A}\right)\right), \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\ell\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(-1, V\right)\right), \left(\sqrt{0 - A}\right)\right), \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\ell\right)\right)\right) \]

      14. sub0-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(-1, V\right)\right), \left(\sqrt{\mathsf{neg}\left(A\right)}\right)\right), \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\ell\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(-1, V\right)\right), \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(A\right)\right)\right)\right), \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\ell\right)\right)\right) \]

      16. sub0-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(-1, V\right)\right), \mathsf{sqrt.f64}\left(\left(0 - A\right)\right)\right), \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\ell\right)\right)\right) \]

      17. --lowering--.f64 47.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(-1, V\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, A\right)\right)\right), \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\ell\right)\right)\right) \]

   6. Applied egg-rr 47.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{-1}{V}} \cdot \sqrt{0 - A}\right)} \cdot \frac{c0}{\sqrt{\ell}} \]

   if -inf.0 < (*.f64 V l) < -1.0000000000000001e-253

   1. Initial program 84.9%

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

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\ell \cdot V}{A}}\right)\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\ell \cdot \frac{V}{A}}\right)\right)\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{\ell}}{\frac{V}{A}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\ell}\right), \left(\frac{V}{A}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \left(\frac{V}{A}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 79.9%

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \mathsf{/.f64}\left(V, A\right)\right)\right)\right) \]

   4. Applied egg-rr 79.9%

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

      1. frac-2neg N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \left(\sqrt{\frac{\frac{1}{\ell}}{\frac{\mathsf{neg}\left(V\right)}{\mathsf{neg}\left(A\right)}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \left(\sqrt{\frac{\frac{1}{\ell}}{\mathsf{neg}\left(V\right)} \cdot \left(\mathsf{neg}\left(A\right)\right)}\right)\right) \]

      3. sqrt-prod N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \left(\sqrt{\frac{\frac{1}{\ell}}{\mathsf{neg}\left(V\right)}} \cdot \color{blue}{\sqrt{\mathsf{neg}\left(A\right)}}\right)\right) \]

      4. sub0-neg N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \left(\sqrt{\frac{\frac{1}{\ell}}{\mathsf{neg}\left(V\right)}} \cdot \sqrt{0 - A}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\left(\sqrt{\frac{\frac{1}{\ell}}{\mathsf{neg}\left(V\right)}}\right), \color{blue}{\left(\sqrt{0 - A}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{\ell}}{\mathsf{neg}\left(V\right)}\right)\right), \left(\sqrt{\color{blue}{0 - A}}\right)\right)\right) \]

      7. frac-2neg N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\mathsf{neg}\left(\frac{1}{\ell}\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(V\right)\right)\right)}\right)\right), \left(\sqrt{\color{blue}{0} - A}\right)\right)\right) \]

      8. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\mathsf{neg}\left(\frac{1}{\ell}\right)}{V}\right)\right), \left(\sqrt{0 - A}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{\ell}\right)\right), V\right)\right), \left(\sqrt{\color{blue}{0} - A}\right)\right)\right) \]

      10. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(1\right)}{\ell}\right), V\right)\right), \left(\sqrt{0 - A}\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{\ell}\right), V\right)\right), \left(\sqrt{0 - A}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, \ell\right), V\right)\right), \left(\sqrt{0 - A}\right)\right)\right) \]

      13. sub0-neg N/A

          \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, \ell\right), V\right)\right), \left(\sqrt{\mathsf{neg}\left(A\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, \ell\right), V\right)\right), \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(A\right)\right)\right)\right)\right) \]

      15. sub0-neg N/A

          \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, \ell\right), V\right)\right), \mathsf{sqrt.f64}\left(\left(0 - A\right)\right)\right)\right) \]

      16. --lowering--.f64 99.4%

          \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, \ell\right), V\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, A\right)\right)\right)\right) \]

   6. Applied egg-rr 99.4%

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{\frac{\frac{-1}{\ell}}{V}} \cdot \sqrt{0 - A}\right)} \]
   7. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{-1}{V \cdot \ell}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\color{blue}{0}, A\right)\right)\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\frac{-1}{V}}{\ell}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\color{blue}{0}, A\right)\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(V\right)}}{\ell}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, A\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{\mathsf{neg}\left(V\right)}}{\ell}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, A\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\mathsf{neg}\left(V\right)}\right), \ell\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\color{blue}{0}, A\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(V\right)}\right), \ell\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, A\right)\right)\right)\right) \]

      7. frac-2neg N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{V}\right), \ell\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, A\right)\right)\right)\right) \]

      8. /-lowering-/.f64 99.3%

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, V\right), \ell\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, A\right)\right)\right)\right) \]

   8. Applied egg-rr 99.3%

      \[\leadsto c0 \cdot \left(\sqrt{\color{blue}{\frac{\frac{-1}{V}}{\ell}}} \cdot \sqrt{0 - A}\right) \]

   if -1.0000000000000001e-253 < (*.f64 V l) < 1.00000000000000002e-294

   1. Initial program 52.8%

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

      1. *-commutative N/A

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

      2. sqrt-div N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\sqrt{V \cdot \ell}}{\sqrt{A} \cdot c0}\right)}\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}{\color{blue}{c0}}\right)\right) \]

      7. sqrt-div N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\sqrt{\frac{V \cdot \ell}{A}}}{c0}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\sqrt{\frac{V \cdot \ell}{A}}\right), \color{blue}{c0}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{V \cdot \ell}{A}\right)\right), c0\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{V}{A} \cdot \ell\right)\right), c0\right)\right) \]

      11. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{V}{\frac{A}{\ell}}\right)\right), c0\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(V, \left(\frac{A}{\ell}\right)\right)\right), c0\right)\right) \]

      13. /-lowering-/.f64 68.0%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(V, \mathsf{/.f64}\left(A, \ell\right)\right)\right), c0\right)\right) \]

   4. Applied egg-rr 68.0%

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

   if 1.00000000000000002e-294 < (*.f64 V l) < 9.9999999999999998e284

   1. Initial program 81.6%

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

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\ell \cdot V}{A}}\right)\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\ell \cdot \frac{V}{A}}\right)\right)\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{\ell}}{\frac{V}{A}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\ell}\right), \left(\frac{V}{A}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \left(\frac{V}{A}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 77.0%

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \mathsf{/.f64}\left(V, A\right)\right)\right)\right) \]

   4. Applied egg-rr 77.0%

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

      1. associate-/l/ N/A

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

      2. associate-/r/ N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(c0, \color{blue}{\left(\sqrt{\frac{V}{\frac{A}{\ell}}}\right)}\right) \]

      7. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{V}{A} \cdot \ell}\right)\right) \]

      8. sqrt-prod N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{V}{A}} \cdot \color{blue}{\sqrt{\ell}}\right)\right) \]

      9. /-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{1}} \cdot \sqrt{\ell}\right)\right) \]

      10. sqrt-prod N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{1} \cdot \ell}\right)\right) \]

      11. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{\frac{1}{\ell}}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{V}{A}}{\frac{1}{\ell}}\right)\right)\right) \]

      13. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{\frac{A}{V}}}{\frac{1}{\ell}}\right)\right)\right) \]

      14. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{A}{V} \cdot \frac{1}{\ell}}\right)\right)\right) \]

      15. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\frac{A}{V}}{\ell}}\right)\right)\right) \]

      16. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\ell}{\frac{A}{V}}\right)\right)\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, \left(\frac{A}{V}\right)\right)\right)\right) \]

      18. /-lowering-/.f64 77.2%

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(A, V\right)\right)\right)\right) \]

   6. Applied egg-rr 77.2%

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

      1. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{1}{\frac{\frac{A}{V}}{\ell}}}\right)\right) \]

      2. associate-/l/ N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{1}{\frac{A}{\ell \cdot V}}}\right)\right) \]

      3. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{1}{A} \cdot \left(\ell \cdot V\right)}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{1}{A} \cdot \left(V \cdot \ell\right)}\right)\right) \]

      5. sqrt-prod N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{1}{A}} \cdot \color{blue}{\sqrt{V \cdot \ell}}\right)\right) \]

      6. pow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left({\left(\frac{1}{A}\right)}^{\frac{1}{2}} \cdot \sqrt{\color{blue}{V \cdot \ell}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\left({\left(\frac{1}{A}\right)}^{\frac{1}{2}}\right), \color{blue}{\left(\sqrt{V \cdot \ell}\right)}\right)\right) \]

      8. inv-pow N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\left({\left({A}^{-1}\right)}^{\frac{1}{2}}\right), \left(\sqrt{\color{blue}{V} \cdot \ell}\right)\right)\right) \]

      9. pow-pow N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\left({A}^{\left(-1 \cdot \frac{1}{2}\right)}\right), \left(\sqrt{\color{blue}{V \cdot \ell}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\left({A}^{\frac{-1}{2}}\right), \left(\sqrt{V \cdot \color{blue}{\ell}}\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\left({A}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \left(\sqrt{V \cdot \color{blue}{\ell}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(A, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \left(\sqrt{\color{blue}{V \cdot \ell}}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(A, \frac{-1}{2}\right), \left(\sqrt{V \cdot \color{blue}{\ell}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(A, \frac{-1}{2}\right), \mathsf{sqrt.f64}\left(\left(V \cdot \ell\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(A, \frac{-1}{2}\right), \mathsf{sqrt.f64}\left(\left(\ell \cdot V\right)\right)\right)\right) \]

      16. *-lowering-*.f64 99.5%

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(A, \frac{-1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\ell, V\right)\right)\right)\right) \]

   8. Applied egg-rr 99.5%

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

   if 9.9999999999999998e284 < (*.f64 V l)

   1. Initial program 38.2%

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

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{A}{V}}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{A}{V}\right), \ell\right)\right)\right) \]

      3. /-lowering-/.f64 85.9%

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(A, V\right), \ell\right)\right)\right) \]

   4. Applied egg-rr 85.9%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\left(\sqrt{\frac{-1}{V}} \cdot \sqrt{0 - A}\right) \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-253}:\\ \;\;\;\;c0 \cdot \left(\sqrt{0 - A} \cdot \sqrt{\frac{\frac{-1}{V}}{\ell}}\right)\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-294}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\frac{V}{\frac{A}{\ell}}}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+285}:\\ \;\;\;\;\frac{c0}{{A}^{-0.5} \cdot \sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 10^{-142}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{1}{\ell}}{\frac{V}{A}}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+134}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\frac{\ell}{\frac{A}{V}}}}{c0}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (<= t_0 1e-142)
     (* c0 (sqrt (/ (/ 1.0 l) (/ V A))))
     (if (<= t_0 2e+134) t_0 (/ 1.0 (/ (sqrt (/ l (/ A V))) c0))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 1e-142) {
		tmp = c0 * sqrt(((1.0 / l) / (V / A)));
	} else if (t_0 <= 2e+134) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (sqrt((l / (A / V))) / c0);
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    if (t_0 <= 1d-142) then
        tmp = c0 * sqrt(((1.0d0 / l) / (v / a)))
    else if (t_0 <= 2d+134) then
        tmp = t_0
    else
        tmp = 1.0d0 / (sqrt((l / (a / v))) / c0)
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 1e-142) {
		tmp = c0 * Math.sqrt(((1.0 / l) / (V / A)));
	} else if (t_0 <= 2e+134) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (Math.sqrt((l / (A / V))) / c0);
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 1e-142:
		tmp = c0 * math.sqrt(((1.0 / l) / (V / A)))
	elif t_0 <= 2e+134:
		tmp = t_0
	else:
		tmp = 1.0 / (math.sqrt((l / (A / V))) / c0)
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if (t_0 <= 1e-142)
		tmp = Float64(c0 * sqrt(Float64(Float64(1.0 / l) / Float64(V / A))));
	elseif (t_0 <= 2e+134)
		tmp = t_0;
	else
		tmp = Float64(1.0 / Float64(sqrt(Float64(l / Float64(A / V))) / c0));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= 1e-142)
		tmp = c0 * sqrt(((1.0 / l) / (V / A)));
	elseif (t_0 <= 2e+134)
		tmp = t_0;
	else
		tmp = 1.0 / (sqrt((l / (A / V))) / c0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-142], N[(c0 * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+134], t$95$0, N[(1.0 / N[(N[Sqrt[N[(l / N[(A / V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / c0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 1e-142

   1. Initial program 69.8%

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

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\ell \cdot V}{A}}\right)\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\ell \cdot \frac{V}{A}}\right)\right)\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{\ell}}{\frac{V}{A}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\ell}\right), \left(\frac{V}{A}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \left(\frac{V}{A}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 76.9%

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \mathsf{/.f64}\left(V, A\right)\right)\right)\right) \]

   4. Applied egg-rr 76.9%

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

   if 1e-142 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 1.99999999999999984e134

   1. Initial program 99.5%

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

   if 1.99999999999999984e134 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 66.4%

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

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\ell \cdot V}{A}}\right)\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\ell \cdot \frac{V}{A}}\right)\right)\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{\ell}}{\frac{V}{A}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\ell}\right), \left(\frac{V}{A}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \left(\frac{V}{A}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 71.0%

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \mathsf{/.f64}\left(V, A\right)\right)\right)\right) \]

   4. Applied egg-rr 71.0%

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

      1. associate-/l/ N/A

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

      2. associate-/r/ N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(c0, \color{blue}{\left(\sqrt{\frac{V}{\frac{A}{\ell}}}\right)}\right) \]

      7. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{V}{A} \cdot \ell}\right)\right) \]

      8. sqrt-prod N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{V}{A}} \cdot \color{blue}{\sqrt{\ell}}\right)\right) \]

      9. /-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{1}} \cdot \sqrt{\ell}\right)\right) \]

      10. sqrt-prod N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{1} \cdot \ell}\right)\right) \]

      11. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{\frac{1}{\ell}}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{V}{A}}{\frac{1}{\ell}}\right)\right)\right) \]

      13. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{\frac{A}{V}}}{\frac{1}{\ell}}\right)\right)\right) \]

      14. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{A}{V} \cdot \frac{1}{\ell}}\right)\right)\right) \]

      15. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\frac{A}{V}}{\ell}}\right)\right)\right) \]

      16. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\ell}{\frac{A}{V}}\right)\right)\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, \left(\frac{A}{V}\right)\right)\right)\right) \]

      18. /-lowering-/.f64 72.7%

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(A, V\right)\right)\right)\right) \]

   6. Applied egg-rr 72.7%

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

      1. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\frac{A}{V}}{\ell}}\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{A}{V}} \cdot \ell\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{V}{A} \cdot \ell\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{V}{A}\right), \ell\right)\right)\right) \]

      5. /-lowering-/.f64 72.7%

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(V, A\right), \ell\right)\right)\right) \]

   8. Applied egg-rr 72.7%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\sqrt{\frac{V}{\frac{A}{\ell}}}}{c0}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\sqrt{\frac{V}{\frac{A}{\ell}}}\right), \color{blue}{c0}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{V}{\frac{A}{\ell}}\right)\right), c0\right)\right) \]

      6. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{V}{A} \cdot \ell\right)\right), c0\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\ell \cdot \frac{V}{A}\right)\right), c0\right)\right) \]

      8. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\ell \cdot \frac{1}{\frac{A}{V}}\right)\right), c0\right)\right) \]

      9. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\ell}{\frac{A}{V}}\right)\right), c0\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, \left(\frac{A}{V}\right)\right)\right), c0\right)\right) \]

      11. /-lowering-/.f64 72.7%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(A, V\right)\right)\right), c0\right)\right) \]

   10. Applied egg-rr 72.7%

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

Alternative 3: 76.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 10^{-209}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+134}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\frac{\ell}{\frac{1}{V}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (<= t_0 1e-209)
     (/ c0 (sqrt (/ l (/ A V))))
     (if (<= t_0 2e+134)
       (* c0 (sqrt (/ A (/ l (/ 1.0 V)))))
       (/ c0 (sqrt (* l (/ V A))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 1e-209) {
		tmp = c0 / sqrt((l / (A / V)));
	} else if (t_0 <= 2e+134) {
		tmp = c0 * sqrt((A / (l / (1.0 / V))));
	} else {
		tmp = c0 / sqrt((l * (V / A)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    if (t_0 <= 1d-209) then
        tmp = c0 / sqrt((l / (a / v)))
    else if (t_0 <= 2d+134) then
        tmp = c0 * sqrt((a / (l / (1.0d0 / v))))
    else
        tmp = c0 / sqrt((l * (v / a)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 1e-209) {
		tmp = c0 / Math.sqrt((l / (A / V)));
	} else if (t_0 <= 2e+134) {
		tmp = c0 * Math.sqrt((A / (l / (1.0 / V))));
	} else {
		tmp = c0 / Math.sqrt((l * (V / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 1e-209:
		tmp = c0 / math.sqrt((l / (A / V)))
	elif t_0 <= 2e+134:
		tmp = c0 * math.sqrt((A / (l / (1.0 / V))))
	else:
		tmp = c0 / math.sqrt((l * (V / A)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if (t_0 <= 1e-209)
		tmp = Float64(c0 / sqrt(Float64(l / Float64(A / V))));
	elseif (t_0 <= 2e+134)
		tmp = Float64(c0 * sqrt(Float64(A / Float64(l / Float64(1.0 / V)))));
	else
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / A))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= 1e-209)
		tmp = c0 / sqrt((l / (A / V)));
	elseif (t_0 <= 2e+134)
		tmp = c0 * sqrt((A / (l / (1.0 / V))));
	else
		tmp = c0 / sqrt((l * (V / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-209], N[(c0 / N[Sqrt[N[(l / N[(A / V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+134], N[(c0 * N[Sqrt[N[(A / N[(l / N[(1.0 / V), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 1e-209

   1. Initial program 68.8%

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

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\ell \cdot V}{A}}\right)\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\ell \cdot \frac{V}{A}}\right)\right)\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{\ell}}{\frac{V}{A}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\ell}\right), \left(\frac{V}{A}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \left(\frac{V}{A}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 76.1%

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \mathsf{/.f64}\left(V, A\right)\right)\right)\right) \]

   4. Applied egg-rr 76.1%

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

      1. associate-/l/ N/A

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

      2. associate-/r/ N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(c0, \color{blue}{\left(\sqrt{\frac{V}{\frac{A}{\ell}}}\right)}\right) \]

      7. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{V}{A} \cdot \ell}\right)\right) \]

      8. sqrt-prod N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{V}{A}} \cdot \color{blue}{\sqrt{\ell}}\right)\right) \]

      9. /-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{1}} \cdot \sqrt{\ell}\right)\right) \]

      10. sqrt-prod N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{1} \cdot \ell}\right)\right) \]

      11. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{\frac{1}{\ell}}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{V}{A}}{\frac{1}{\ell}}\right)\right)\right) \]

      13. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{\frac{A}{V}}}{\frac{1}{\ell}}\right)\right)\right) \]

      14. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{A}{V} \cdot \frac{1}{\ell}}\right)\right)\right) \]

      15. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\frac{A}{V}}{\ell}}\right)\right)\right) \]

      16. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\ell}{\frac{A}{V}}\right)\right)\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, \left(\frac{A}{V}\right)\right)\right)\right) \]

      18. /-lowering-/.f64 75.2%

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(A, V\right)\right)\right)\right) \]

   6. Applied egg-rr 75.2%

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

   if 1e-209 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 1.99999999999999984e134

   1. Initial program 99.5%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(A, \left(\ell \cdot V\right)\right)\right)\right) \]

      2. /-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(A, \left(\frac{\ell}{1} \cdot V\right)\right)\right)\right) \]

      3. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(A, \left(\frac{\ell}{\frac{1}{V}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(A, \mathsf{/.f64}\left(\ell, \left(\frac{1}{V}\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 99.4%

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(A, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(1, V\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.4%

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

   if 1.99999999999999984e134 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 66.4%

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

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\ell \cdot V}{A}}\right)\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\ell \cdot \frac{V}{A}}\right)\right)\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{\ell}}{\frac{V}{A}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\ell}\right), \left(\frac{V}{A}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \left(\frac{V}{A}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 71.0%

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \mathsf{/.f64}\left(V, A\right)\right)\right)\right) \]

   4. Applied egg-rr 71.0%

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

      1. associate-/l/ N/A

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

      2. associate-/r/ N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(c0, \color{blue}{\left(\sqrt{\frac{V}{\frac{A}{\ell}}}\right)}\right) \]

      7. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{V}{A} \cdot \ell}\right)\right) \]

      8. sqrt-prod N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{V}{A}} \cdot \color{blue}{\sqrt{\ell}}\right)\right) \]

      9. /-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{1}} \cdot \sqrt{\ell}\right)\right) \]

      10. sqrt-prod N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{1} \cdot \ell}\right)\right) \]

      11. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{\frac{1}{\ell}}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{V}{A}}{\frac{1}{\ell}}\right)\right)\right) \]

      13. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{\frac{A}{V}}}{\frac{1}{\ell}}\right)\right)\right) \]

      14. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{A}{V} \cdot \frac{1}{\ell}}\right)\right)\right) \]

      15. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\frac{A}{V}}{\ell}}\right)\right)\right) \]

      16. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\ell}{\frac{A}{V}}\right)\right)\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, \left(\frac{A}{V}\right)\right)\right)\right) \]

      18. /-lowering-/.f64 72.7%

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(A, V\right)\right)\right)\right) \]

   6. Applied egg-rr 72.7%

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

      1. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\frac{A}{V}}{\ell}}\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{A}{V}} \cdot \ell\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{V}{A} \cdot \ell\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{V}{A}\right), \ell\right)\right)\right) \]

      5. /-lowering-/.f64 72.7%

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(V, A\right), \ell\right)\right)\right) \]

   8. Applied egg-rr 72.7%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 10^{-209}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 2 \cdot 10^{+134}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\frac{\ell}{\frac{1}{V}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 10^{-142}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{1}{\ell}}{\frac{V}{A}}}\\ \mathbf{elif}\;t\_0 \leq 10^{+145}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (<= t_0 1e-142)
     (* c0 (sqrt (/ (/ 1.0 l) (/ V A))))
     (if (<= t_0 1e+145) t_0 (/ c0 (sqrt (* l (/ V A))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 1e-142) {
		tmp = c0 * sqrt(((1.0 / l) / (V / A)));
	} else if (t_0 <= 1e+145) {
		tmp = t_0;
	} else {
		tmp = c0 / sqrt((l * (V / A)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    if (t_0 <= 1d-142) then
        tmp = c0 * sqrt(((1.0d0 / l) / (v / a)))
    else if (t_0 <= 1d+145) then
        tmp = t_0
    else
        tmp = c0 / sqrt((l * (v / a)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 1e-142) {
		tmp = c0 * Math.sqrt(((1.0 / l) / (V / A)));
	} else if (t_0 <= 1e+145) {
		tmp = t_0;
	} else {
		tmp = c0 / Math.sqrt((l * (V / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 1e-142:
		tmp = c0 * math.sqrt(((1.0 / l) / (V / A)))
	elif t_0 <= 1e+145:
		tmp = t_0
	else:
		tmp = c0 / math.sqrt((l * (V / A)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if (t_0 <= 1e-142)
		tmp = Float64(c0 * sqrt(Float64(Float64(1.0 / l) / Float64(V / A))));
	elseif (t_0 <= 1e+145)
		tmp = t_0;
	else
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / A))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= 1e-142)
		tmp = c0 * sqrt(((1.0 / l) / (V / A)));
	elseif (t_0 <= 1e+145)
		tmp = t_0;
	else
		tmp = c0 / sqrt((l * (V / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-142], N[(c0 * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+145], t$95$0, N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 1e-142

   1. Initial program 69.8%

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

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\ell \cdot V}{A}}\right)\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\ell \cdot \frac{V}{A}}\right)\right)\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{\ell}}{\frac{V}{A}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\ell}\right), \left(\frac{V}{A}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \left(\frac{V}{A}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 76.9%

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \mathsf{/.f64}\left(V, A\right)\right)\right)\right) \]

   4. Applied egg-rr 76.9%

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

   if 1e-142 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 9.9999999999999999e144

   1. Initial program 99.5%

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

   if 9.9999999999999999e144 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 66.4%

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

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\ell \cdot V}{A}}\right)\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\ell \cdot \frac{V}{A}}\right)\right)\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{\ell}}{\frac{V}{A}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\ell}\right), \left(\frac{V}{A}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \left(\frac{V}{A}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 71.0%

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \mathsf{/.f64}\left(V, A\right)\right)\right)\right) \]

   4. Applied egg-rr 71.0%

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

      1. associate-/l/ N/A

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

      2. associate-/r/ N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(c0, \color{blue}{\left(\sqrt{\frac{V}{\frac{A}{\ell}}}\right)}\right) \]

      7. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{V}{A} \cdot \ell}\right)\right) \]

      8. sqrt-prod N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{V}{A}} \cdot \color{blue}{\sqrt{\ell}}\right)\right) \]

      9. /-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{1}} \cdot \sqrt{\ell}\right)\right) \]

      10. sqrt-prod N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{1} \cdot \ell}\right)\right) \]

      11. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{\frac{1}{\ell}}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{V}{A}}{\frac{1}{\ell}}\right)\right)\right) \]

      13. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{\frac{A}{V}}}{\frac{1}{\ell}}\right)\right)\right) \]

      14. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{A}{V} \cdot \frac{1}{\ell}}\right)\right)\right) \]

      15. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\frac{A}{V}}{\ell}}\right)\right)\right) \]

      16. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\ell}{\frac{A}{V}}\right)\right)\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, \left(\frac{A}{V}\right)\right)\right)\right) \]

      18. /-lowering-/.f64 72.7%

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(A, V\right)\right)\right)\right) \]

   6. Applied egg-rr 72.7%

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

      1. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\frac{A}{V}}{\ell}}\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{A}{V}} \cdot \ell\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{V}{A} \cdot \ell\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{V}{A}\right), \ell\right)\right)\right) \]

      5. /-lowering-/.f64 72.7%

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(V, A\right), \ell\right)\right)\right) \]

   8. Applied egg-rr 72.7%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 10^{-142}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{1}{\ell}}{\frac{V}{A}}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 10^{+145}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-252}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \mathbf{elif}\;t\_0 \leq 10^{+145}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (<= t_0 5e-252)
     (/ c0 (sqrt (/ l (/ A V))))
     (if (<= t_0 1e+145) t_0 (/ c0 (sqrt (* l (/ V A))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 5e-252) {
		tmp = c0 / sqrt((l / (A / V)));
	} else if (t_0 <= 1e+145) {
		tmp = t_0;
	} else {
		tmp = c0 / sqrt((l * (V / A)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    if (t_0 <= 5d-252) then
        tmp = c0 / sqrt((l / (a / v)))
    else if (t_0 <= 1d+145) then
        tmp = t_0
    else
        tmp = c0 / sqrt((l * (v / a)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 5e-252) {
		tmp = c0 / Math.sqrt((l / (A / V)));
	} else if (t_0 <= 1e+145) {
		tmp = t_0;
	} else {
		tmp = c0 / Math.sqrt((l * (V / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 5e-252:
		tmp = c0 / math.sqrt((l / (A / V)))
	elif t_0 <= 1e+145:
		tmp = t_0
	else:
		tmp = c0 / math.sqrt((l * (V / A)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if (t_0 <= 5e-252)
		tmp = Float64(c0 / sqrt(Float64(l / Float64(A / V))));
	elseif (t_0 <= 1e+145)
		tmp = t_0;
	else
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / A))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= 5e-252)
		tmp = c0 / sqrt((l / (A / V)));
	elseif (t_0 <= 1e+145)
		tmp = t_0;
	else
		tmp = c0 / sqrt((l * (V / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-252], N[(c0 / N[Sqrt[N[(l / N[(A / V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+145], t$95$0, N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 5.00000000000000008e-252

   1. Initial program 68.5%

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

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\ell \cdot V}{A}}\right)\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\ell \cdot \frac{V}{A}}\right)\right)\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{\ell}}{\frac{V}{A}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\ell}\right), \left(\frac{V}{A}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \left(\frac{V}{A}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 75.8%

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \mathsf{/.f64}\left(V, A\right)\right)\right)\right) \]

   4. Applied egg-rr 75.8%

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

      1. associate-/l/ N/A

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

      2. associate-/r/ N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(c0, \color{blue}{\left(\sqrt{\frac{V}{\frac{A}{\ell}}}\right)}\right) \]

      7. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{V}{A} \cdot \ell}\right)\right) \]

      8. sqrt-prod N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{V}{A}} \cdot \color{blue}{\sqrt{\ell}}\right)\right) \]

      9. /-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{1}} \cdot \sqrt{\ell}\right)\right) \]

      10. sqrt-prod N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{1} \cdot \ell}\right)\right) \]

      11. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{\frac{1}{\ell}}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{V}{A}}{\frac{1}{\ell}}\right)\right)\right) \]

      13. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{\frac{A}{V}}}{\frac{1}{\ell}}\right)\right)\right) \]

      14. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{A}{V} \cdot \frac{1}{\ell}}\right)\right)\right) \]

      15. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\frac{A}{V}}{\ell}}\right)\right)\right) \]

      16. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\ell}{\frac{A}{V}}\right)\right)\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, \left(\frac{A}{V}\right)\right)\right)\right) \]

      18. /-lowering-/.f64 74.9%

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(A, V\right)\right)\right)\right) \]

   6. Applied egg-rr 74.9%

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

   if 5.00000000000000008e-252 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 9.9999999999999999e144

   1. Initial program 99.5%

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

   if 9.9999999999999999e144 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 66.4%

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

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\ell \cdot V}{A}}\right)\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\ell \cdot \frac{V}{A}}\right)\right)\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{\ell}}{\frac{V}{A}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\ell}\right), \left(\frac{V}{A}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \left(\frac{V}{A}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 71.0%

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \mathsf{/.f64}\left(V, A\right)\right)\right)\right) \]

   4. Applied egg-rr 71.0%

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

      1. associate-/l/ N/A

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

      2. associate-/r/ N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(c0, \color{blue}{\left(\sqrt{\frac{V}{\frac{A}{\ell}}}\right)}\right) \]

      7. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{V}{A} \cdot \ell}\right)\right) \]

      8. sqrt-prod N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{V}{A}} \cdot \color{blue}{\sqrt{\ell}}\right)\right) \]

      9. /-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{1}} \cdot \sqrt{\ell}\right)\right) \]

      10. sqrt-prod N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{1} \cdot \ell}\right)\right) \]

      11. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{\frac{1}{\ell}}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{V}{A}}{\frac{1}{\ell}}\right)\right)\right) \]

      13. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{\frac{A}{V}}}{\frac{1}{\ell}}\right)\right)\right) \]

      14. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{A}{V} \cdot \frac{1}{\ell}}\right)\right)\right) \]

      15. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\frac{A}{V}}{\ell}}\right)\right)\right) \]

      16. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\ell}{\frac{A}{V}}\right)\right)\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, \left(\frac{A}{V}\right)\right)\right)\right) \]

      18. /-lowering-/.f64 72.7%

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(A, V\right)\right)\right)\right) \]

   6. Applied egg-rr 72.7%

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

      1. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\frac{A}{V}}{\ell}}\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{A}{V}} \cdot \ell\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{V}{A} \cdot \ell\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{V}{A}\right), \ell\right)\right)\right) \]

      5. /-lowering-/.f64 72.7%

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(V, A\right), \ell\right)\right)\right) \]

   8. Applied egg-rr 72.7%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 5 \cdot 10^{-252}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 10^{+145}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-252}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \mathbf{elif}\;t\_0 \leq 10^{+145}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (<= t_0 5e-252)
     (/ c0 (sqrt (/ V (/ A l))))
     (if (<= t_0 1e+145) t_0 (/ c0 (sqrt (* l (/ V A))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 5e-252) {
		tmp = c0 / sqrt((V / (A / l)));
	} else if (t_0 <= 1e+145) {
		tmp = t_0;
	} else {
		tmp = c0 / sqrt((l * (V / A)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    if (t_0 <= 5d-252) then
        tmp = c0 / sqrt((v / (a / l)))
    else if (t_0 <= 1d+145) then
        tmp = t_0
    else
        tmp = c0 / sqrt((l * (v / a)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 5e-252) {
		tmp = c0 / Math.sqrt((V / (A / l)));
	} else if (t_0 <= 1e+145) {
		tmp = t_0;
	} else {
		tmp = c0 / Math.sqrt((l * (V / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 5e-252:
		tmp = c0 / math.sqrt((V / (A / l)))
	elif t_0 <= 1e+145:
		tmp = t_0
	else:
		tmp = c0 / math.sqrt((l * (V / A)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if (t_0 <= 5e-252)
		tmp = Float64(c0 / sqrt(Float64(V / Float64(A / l))));
	elseif (t_0 <= 1e+145)
		tmp = t_0;
	else
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / A))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= 5e-252)
		tmp = c0 / sqrt((V / (A / l)));
	elseif (t_0 <= 1e+145)
		tmp = t_0;
	else
		tmp = c0 / sqrt((l * (V / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-252], N[(c0 / N[Sqrt[N[(V / N[(A / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+145], t$95$0, N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 5.00000000000000008e-252

   1. Initial program 68.5%

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

      1. clear-num N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. un-div-inv N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(c0, \color{blue}{\left(\sqrt{\frac{V \cdot \ell}{A}}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{V \cdot \ell}{A}\right)\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{V}{A} \cdot \ell\right)\right)\right) \]

      8. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{V}{\frac{A}{\ell}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(V, \left(\frac{A}{\ell}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 72.7%

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(V, \mathsf{/.f64}\left(A, \ell\right)\right)\right)\right) \]

   4. Applied egg-rr 72.7%

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

   if 5.00000000000000008e-252 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 9.9999999999999999e144

   1. Initial program 99.5%

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

   if 9.9999999999999999e144 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 66.4%

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

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\ell \cdot V}{A}}\right)\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\ell \cdot \frac{V}{A}}\right)\right)\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{\ell}}{\frac{V}{A}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\ell}\right), \left(\frac{V}{A}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \left(\frac{V}{A}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 71.0%

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \mathsf{/.f64}\left(V, A\right)\right)\right)\right) \]

   4. Applied egg-rr 71.0%

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

      1. associate-/l/ N/A

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

      2. associate-/r/ N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(c0, \color{blue}{\left(\sqrt{\frac{V}{\frac{A}{\ell}}}\right)}\right) \]

      7. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{V}{A} \cdot \ell}\right)\right) \]

      8. sqrt-prod N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{V}{A}} \cdot \color{blue}{\sqrt{\ell}}\right)\right) \]

      9. /-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{1}} \cdot \sqrt{\ell}\right)\right) \]

      10. sqrt-prod N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{1} \cdot \ell}\right)\right) \]

      11. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{\frac{1}{\ell}}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{V}{A}}{\frac{1}{\ell}}\right)\right)\right) \]

      13. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{\frac{A}{V}}}{\frac{1}{\ell}}\right)\right)\right) \]

      14. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{A}{V} \cdot \frac{1}{\ell}}\right)\right)\right) \]

      15. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\frac{A}{V}}{\ell}}\right)\right)\right) \]

      16. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\ell}{\frac{A}{V}}\right)\right)\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, \left(\frac{A}{V}\right)\right)\right)\right) \]

      18. /-lowering-/.f64 72.7%

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(A, V\right)\right)\right)\right) \]

   6. Applied egg-rr 72.7%

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

      1. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\frac{A}{V}}{\ell}}\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{A}{V}} \cdot \ell\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{V}{A} \cdot \ell\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{V}{A}\right), \ell\right)\right)\right) \]

      5. /-lowering-/.f64 72.7%

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(V, A\right), \ell\right)\right)\right) \]

   8. Applied egg-rr 72.7%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 5 \cdot 10^{-252}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 10^{+145}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ t_1 := \frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{if}\;t\_0 \leq 10^{-154}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+145}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))) (t_1 (/ c0 (sqrt (* l (/ V A))))))
   (if (<= t_0 1e-154) t_1 (if (<= t_0 1e+145) t_0 t_1))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double t_1 = c0 / sqrt((l * (V / A)));
	double tmp;
	if (t_0 <= 1e-154) {
		tmp = t_1;
	} else if (t_0 <= 1e+145) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    t_1 = c0 / sqrt((l * (v / a)))
    if (t_0 <= 1d-154) then
        tmp = t_1
    else if (t_0 <= 1d+145) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double t_1 = c0 / Math.sqrt((l * (V / A)));
	double tmp;
	if (t_0 <= 1e-154) {
		tmp = t_1;
	} else if (t_0 <= 1e+145) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	t_1 = c0 / math.sqrt((l * (V / A)))
	tmp = 0
	if t_0 <= 1e-154:
		tmp = t_1
	elif t_0 <= 1e+145:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	t_1 = Float64(c0 / sqrt(Float64(l * Float64(V / A))))
	tmp = 0.0
	if (t_0 <= 1e-154)
		tmp = t_1;
	elseif (t_0 <= 1e+145)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	t_1 = c0 / sqrt((l * (V / A)));
	tmp = 0.0;
	if (t_0 <= 1e-154)
		tmp = t_1;
	elseif (t_0 <= 1e+145)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-154], t$95$1, If[LessEqual[t$95$0, 1e+145], t$95$0, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 9.9999999999999997e-155 or 9.9999999999999999e144 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 68.9%

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

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\ell \cdot V}{A}}\right)\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\ell \cdot \frac{V}{A}}\right)\right)\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{\ell}}{\frac{V}{A}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\ell}\right), \left(\frac{V}{A}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \left(\frac{V}{A}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 75.6%

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \mathsf{/.f64}\left(V, A\right)\right)\right)\right) \]

   4. Applied egg-rr 75.6%

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

      1. associate-/l/ N/A

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

      2. associate-/r/ N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(c0, \color{blue}{\left(\sqrt{\frac{V}{\frac{A}{\ell}}}\right)}\right) \]

      7. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{V}{A} \cdot \ell}\right)\right) \]

      8. sqrt-prod N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{V}{A}} \cdot \color{blue}{\sqrt{\ell}}\right)\right) \]

      9. /-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{1}} \cdot \sqrt{\ell}\right)\right) \]

      10. sqrt-prod N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{1} \cdot \ell}\right)\right) \]

      11. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{\frac{1}{\ell}}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{V}{A}}{\frac{1}{\ell}}\right)\right)\right) \]

      13. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{\frac{A}{V}}}{\frac{1}{\ell}}\right)\right)\right) \]

      14. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{A}{V} \cdot \frac{1}{\ell}}\right)\right)\right) \]

      15. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\frac{A}{V}}{\ell}}\right)\right)\right) \]

      16. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\ell}{\frac{A}{V}}\right)\right)\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, \left(\frac{A}{V}\right)\right)\right)\right) \]

      18. /-lowering-/.f64 75.2%

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(A, V\right)\right)\right)\right) \]

   6. Applied egg-rr 75.2%

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

      1. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\frac{A}{V}}{\ell}}\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{A}{V}} \cdot \ell\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{V}{A} \cdot \ell\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{V}{A}\right), \ell\right)\right)\right) \]

      5. /-lowering-/.f64 75.6%

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(V, A\right), \ell\right)\right)\right) \]

   8. Applied egg-rr 75.6%

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

   if 9.9999999999999997e-155 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 9.9999999999999999e144

   1. Initial program 99.5%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 10^{-154}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 10^{+145}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+134}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \end{array} \]

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 (* c0 (sqrt (/ A (* V l))))))
   (if (<= t_0 0.0)
     (* c0 (sqrt (/ (/ A l) V)))
     (if (<= t_0 2e+134) 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 = c0 * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * sqrt(((A / l) / V));
	} else if (t_0 <= 2e+134) {
		tmp = 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 = c0 * sqrt((a / (v * l)))
    if (t_0 <= 0.0d0) then
        tmp = c0 * sqrt(((a / l) / v))
    else if (t_0 <= 2d+134) then
        tmp = 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 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * Math.sqrt(((A / l) / V));
	} else if (t_0 <= 2e+134) {
		tmp = 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 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0 * math.sqrt(((A / l) / V))
	elif t_0 <= 2e+134:
		tmp = 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(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	elseif (t_0 <= 2e+134)
		tmp = 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 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0 * sqrt(((A / l) / V));
	elseif (t_0 <= 2e+134)
		tmp = 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[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+134], t$95$0, N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 0.0

   1. Initial program 68.3%

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

      1. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{A}{\ell}}{V}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{A}{\ell}\right), V\right)\right)\right) \]

      3. /-lowering-/.f64 72.5%

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(A, \ell\right), V\right)\right)\right) \]

   4. Applied egg-rr 72.5%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 1.99999999999999984e134

   1. Initial program 99.5%

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

   if 1.99999999999999984e134 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 66.4%

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

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{A}{V}}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{A}{V}\right), \ell\right)\right)\right) \]

      3. /-lowering-/.f64 71.1%

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(A, V\right), \ell\right)\right)\right) \]

   4. Applied egg-rr 71.1%

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

Alternative 9: 76.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ t_1 := c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-252}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+134}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))) (t_1 (* c0 (sqrt (/ (/ A V) l)))))
   (if (<= t_0 5e-252) t_1 (if (<= t_0 2e+134) t_0 t_1))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double t_1 = c0 * sqrt(((A / V) / l));
	double tmp;
	if (t_0 <= 5e-252) {
		tmp = t_1;
	} else if (t_0 <= 2e+134) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    t_1 = c0 * sqrt(((a / v) / l))
    if (t_0 <= 5d-252) then
        tmp = t_1
    else if (t_0 <= 2d+134) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double t_1 = c0 * Math.sqrt(((A / V) / l));
	double tmp;
	if (t_0 <= 5e-252) {
		tmp = t_1;
	} else if (t_0 <= 2e+134) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	t_1 = c0 * math.sqrt(((A / V) / l))
	tmp = 0
	if t_0 <= 5e-252:
		tmp = t_1
	elif t_0 <= 2e+134:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	t_1 = Float64(c0 * sqrt(Float64(Float64(A / V) / l)))
	tmp = 0.0
	if (t_0 <= 5e-252)
		tmp = t_1;
	elseif (t_0 <= 2e+134)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	t_1 = c0 * sqrt(((A / V) / l));
	tmp = 0.0;
	if (t_0 <= 5e-252)
		tmp = t_1;
	elseif (t_0 <= 2e+134)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-252], t$95$1, If[LessEqual[t$95$0, 2e+134], t$95$0, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 5.00000000000000008e-252 or 1.99999999999999984e134 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 68.1%

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

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{A}{V}}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{A}{V}\right), \ell\right)\right)\right) \]

      3. /-lowering-/.f64 74.5%

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(A, V\right), \ell\right)\right)\right) \]

   4. Applied egg-rr 74.5%

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

   if 5.00000000000000008e-252 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 1.99999999999999984e134

   1. Initial program 99.5%

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

Alternative 10: 86.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{0 - A}} \cdot \sqrt{0 - V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c0 \cdot \sqrt{A}}{\sqrt{\ell}}}{\sqrt{V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= V -5e-310)
   (/ c0 (* (sqrt (/ l (- 0.0 A))) (sqrt (- 0.0 V))))
   (/ (/ (* c0 (sqrt A)) (sqrt l)) (sqrt V))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (V <= -5e-310) {
		tmp = c0 / (sqrt((l / (0.0 - A))) * sqrt((0.0 - V)));
	} else {
		tmp = ((c0 * sqrt(A)) / sqrt(l)) / sqrt(V);
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (v <= (-5d-310)) then
        tmp = c0 / (sqrt((l / (0.0d0 - a))) * sqrt((0.0d0 - v)))
    else
        tmp = ((c0 * sqrt(a)) / sqrt(l)) / sqrt(v)
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (V <= -5e-310) {
		tmp = c0 / (Math.sqrt((l / (0.0 - A))) * Math.sqrt((0.0 - V)));
	} else {
		tmp = ((c0 * Math.sqrt(A)) / Math.sqrt(l)) / Math.sqrt(V);
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if V <= -5e-310:
		tmp = c0 / (math.sqrt((l / (0.0 - A))) * math.sqrt((0.0 - V)))
	else:
		tmp = ((c0 * math.sqrt(A)) / math.sqrt(l)) / math.sqrt(V)
	return tmp

Julia

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

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (V <= -5e-310)
		tmp = c0 / (sqrt((l / (0.0 - A))) * sqrt((0.0 - V)));
	else
		tmp = ((c0 * sqrt(A)) / sqrt(l)) / sqrt(V);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[V, -5e-310], N[(c0 / N[(N[Sqrt[N[(l / N[(0.0 - A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(0.0 - V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c0 * N[Sqrt[A], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[V], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if V < -4.999999999999985e-310

   1. Initial program 72.6%

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

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\ell \cdot V}{A}}\right)\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\ell \cdot \frac{V}{A}}\right)\right)\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{\ell}}{\frac{V}{A}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\ell}\right), \left(\frac{V}{A}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \left(\frac{V}{A}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 76.4%

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \mathsf{/.f64}\left(V, A\right)\right)\right)\right) \]

   4. Applied egg-rr 76.4%

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

      1. associate-/l/ N/A

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

      2. associate-/r/ N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(c0, \color{blue}{\left(\sqrt{\frac{V}{\frac{A}{\ell}}}\right)}\right) \]

      7. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{V}{A} \cdot \ell}\right)\right) \]

      8. sqrt-prod N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{V}{A}} \cdot \color{blue}{\sqrt{\ell}}\right)\right) \]

      9. /-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{1}} \cdot \sqrt{\ell}\right)\right) \]

      10. sqrt-prod N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{1} \cdot \ell}\right)\right) \]

      11. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{\frac{1}{\ell}}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{V}{A}}{\frac{1}{\ell}}\right)\right)\right) \]

      13. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{\frac{A}{V}}}{\frac{1}{\ell}}\right)\right)\right) \]

      14. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{A}{V} \cdot \frac{1}{\ell}}\right)\right)\right) \]

      15. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\frac{A}{V}}{\ell}}\right)\right)\right) \]

      16. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\ell}{\frac{A}{V}}\right)\right)\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, \left(\frac{A}{V}\right)\right)\right)\right) \]

      18. /-lowering-/.f64 76.1%

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(A, V\right)\right)\right)\right) \]

   6. Applied egg-rr 76.1%

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

      1. frac-2neg N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\ell}{\frac{\mathsf{neg}\left(A\right)}{\mathsf{neg}\left(V\right)}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\ell}{\mathsf{neg}\left(A\right)} \cdot \left(\mathsf{neg}\left(V\right)\right)}\right)\right) \]

      3. sqrt-prod N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\ell}{\mathsf{neg}\left(A\right)}} \cdot \color{blue}{\sqrt{\mathsf{neg}\left(V\right)}}\right)\right) \]

      4. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\mathsf{neg}\left(\frac{\ell}{A}\right)} \cdot \sqrt{\mathsf{neg}\left(\color{blue}{V}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\left(\sqrt{\mathsf{neg}\left(\frac{\ell}{A}\right)}\right), \color{blue}{\left(\sqrt{\mathsf{neg}\left(V\right)}\right)}\right)\right) \]

      6. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\left(\sqrt{\frac{\ell}{\mathsf{neg}\left(A\right)}}\right), \left(\sqrt{\mathsf{neg}\left(\color{blue}{V}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\ell}{\mathsf{neg}\left(A\right)}\right)\right), \left(\sqrt{\color{blue}{\mathsf{neg}\left(V\right)}}\right)\right)\right) \]

      8. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\frac{\ell}{A}\right)\right)\right), \left(\sqrt{\mathsf{neg}\left(\color{blue}{V}\right)}\right)\right)\right) \]

      9. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(0 - \frac{\ell}{A}\right)\right), \left(\sqrt{\mathsf{neg}\left(\color{blue}{V}\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \left(\frac{\ell}{A}\right)\right)\right), \left(\sqrt{\mathsf{neg}\left(\color{blue}{V}\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\ell, A\right)\right)\right), \left(\sqrt{\mathsf{neg}\left(V\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\ell, A\right)\right)\right), \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(V\right)\right)\right)\right)\right) \]

      13. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\ell, A\right)\right)\right), \mathsf{sqrt.f64}\left(\left(0 - V\right)\right)\right)\right) \]

      14. --lowering--.f64 86.5%

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\ell, A\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, V\right)\right)\right)\right) \]

   8. Applied egg-rr 86.5%

      \[\leadsto \frac{c0}{\color{blue}{\sqrt{0 - \frac{\ell}{A}} \cdot \sqrt{0 - V}}} \]

   if -4.999999999999985e-310 < V

   1. Initial program 72.9%

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

      1. *-commutative N/A

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

      2. associate-/l/ N/A

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

      3. sqrt-div N/A

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

      4. pow1/2 N/A

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

      5. associate-*l/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{A}{\ell}\right)}^{\frac{1}{2}} \cdot c0\right), \color{blue}{\left(\sqrt{V}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\left(\frac{A}{\ell}\right)}^{\frac{1}{2}}\right), c0\right), \left(\sqrt{\color{blue}{V}}\right)\right) \]

      8. pow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{A}{\ell}}\right), c0\right), \left(\sqrt{V}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{A}{\ell}\right)\right), c0\right), \left(\sqrt{V}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(A, \ell\right)\right), c0\right), \left(\sqrt{V}\right)\right) \]

      11. sqrt-lowering-sqrt.f64 83.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(A, \ell\right)\right), c0\right), \mathsf{sqrt.f64}\left(V\right)\right) \]

   4. Applied egg-rr 83.0%

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

      1. sqrt-div N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\sqrt{A}}{\sqrt{\ell}} \cdot c0\right), \mathsf{sqrt.f64}\left(V\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\sqrt{A} \cdot c0}{\sqrt{\ell}}\right), \mathsf{sqrt.f64}\left(\color{blue}{V}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{A} \cdot c0\right), \left(\sqrt{\ell}\right)\right), \mathsf{sqrt.f64}\left(\color{blue}{V}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(c0 \cdot \sqrt{A}\right), \left(\sqrt{\ell}\right)\right), \mathsf{sqrt.f64}\left(V\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\sqrt{A}\right)\right), \left(\sqrt{\ell}\right)\right), \mathsf{sqrt.f64}\left(V\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(A\right)\right), \left(\sqrt{\ell}\right)\right), \mathsf{sqrt.f64}\left(V\right)\right) \]

      7. sqrt-lowering-sqrt.f64 50.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(A\right)\right), \mathsf{sqrt.f64}\left(\ell\right)\right), \mathsf{sqrt.f64}\left(V\right)\right) \]

   6. Applied egg-rr 50.0%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{0 - A}} \cdot \sqrt{0 - V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c0 \cdot \sqrt{A}}{\sqrt{\ell}}}{\sqrt{V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 91.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{-1}{V}} \cdot \sqrt{0 - \frac{A}{\ell}}\right)\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-253}:\\ \;\;\;\;c0 \cdot \left(\sqrt{0 - A} \cdot \sqrt{\frac{\frac{-1}{V}}{\ell}}\right)\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-294}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\frac{V}{\frac{A}{\ell}}}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+285}:\\ \;\;\;\;\frac{c0}{{A}^{-0.5} \cdot \sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) (- INFINITY))
   (* c0 (* (sqrt (/ -1.0 V)) (sqrt (- 0.0 (/ A l)))))
   (if (<= (* V l) -1e-253)
     (* c0 (* (sqrt (- 0.0 A)) (sqrt (/ (/ -1.0 V) l))))
     (if (<= (* V l) 1e-294)
       (/ 1.0 (/ (sqrt (/ V (/ A l))) c0))
       (if (<= (* V l) 1e+285)
         (/ c0 (* (pow A -0.5) (sqrt (* V l))))
         (* c0 (sqrt (/ (/ A V) l))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = c0 * (sqrt((-1.0 / V)) * sqrt((0.0 - (A / l))));
	} else if ((V * l) <= -1e-253) {
		tmp = c0 * (sqrt((0.0 - A)) * sqrt(((-1.0 / V) / l)));
	} else if ((V * l) <= 1e-294) {
		tmp = 1.0 / (sqrt((V / (A / l))) / c0);
	} else if ((V * l) <= 1e+285) {
		tmp = c0 / (pow(A, -0.5) * sqrt((V * l)));
	} else {
		tmp = c0 * sqrt(((A / V) / l));
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = c0 * (Math.sqrt((-1.0 / V)) * Math.sqrt((0.0 - (A / l))));
	} else if ((V * l) <= -1e-253) {
		tmp = c0 * (Math.sqrt((0.0 - A)) * Math.sqrt(((-1.0 / V) / l)));
	} else if ((V * l) <= 1e-294) {
		tmp = 1.0 / (Math.sqrt((V / (A / l))) / c0);
	} else if ((V * l) <= 1e+285) {
		tmp = c0 / (Math.pow(A, -0.5) * Math.sqrt((V * l)));
	} else {
		tmp = c0 * Math.sqrt(((A / V) / l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = c0 * (math.sqrt((-1.0 / V)) * math.sqrt((0.0 - (A / l))))
	elif (V * l) <= -1e-253:
		tmp = c0 * (math.sqrt((0.0 - A)) * math.sqrt(((-1.0 / V) / l)))
	elif (V * l) <= 1e-294:
		tmp = 1.0 / (math.sqrt((V / (A / l))) / c0)
	elif (V * l) <= 1e+285:
		tmp = c0 / (math.pow(A, -0.5) * math.sqrt((V * l)))
	else:
		tmp = c0 * math.sqrt(((A / V) / l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(c0 * Float64(sqrt(Float64(-1.0 / V)) * sqrt(Float64(0.0 - Float64(A / l)))));
	elseif (Float64(V * l) <= -1e-253)
		tmp = Float64(c0 * Float64(sqrt(Float64(0.0 - A)) * sqrt(Float64(Float64(-1.0 / V) / l))));
	elseif (Float64(V * l) <= 1e-294)
		tmp = Float64(1.0 / Float64(sqrt(Float64(V / Float64(A / l))) / c0));
	elseif (Float64(V * l) <= 1e+285)
		tmp = Float64(c0 / Float64((A ^ -0.5) * sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = c0 * (sqrt((-1.0 / V)) * sqrt((0.0 - (A / l))));
	elseif ((V * l) <= -1e-253)
		tmp = c0 * (sqrt((0.0 - A)) * sqrt(((-1.0 / V) / l)));
	elseif ((V * l) <= 1e-294)
		tmp = 1.0 / (sqrt((V / (A / l))) / c0);
	elseif ((V * l) <= 1e+285)
		tmp = c0 / ((A ^ -0.5) * sqrt((V * l)));
	else
		tmp = c0 * sqrt(((A / V) / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(c0 * N[(N[Sqrt[N[(-1.0 / V), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(0.0 - N[(A / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-253], N[(c0 * N[(N[Sqrt[N[(0.0 - A), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(-1.0 / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e-294], N[(1.0 / N[(N[Sqrt[N[(V / N[(A / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / c0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+285], N[(c0 / N[(N[Power[A, -0.5], $MachinePrecision] * N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 45.9%

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

      1. frac-2neg N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \left(\sqrt{\frac{\mathsf{neg}\left(A\right)}{\mathsf{neg}\left(V \cdot \ell\right)}}\right)\right) \]

      2. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \left(\sqrt{\frac{-1 \cdot A}{\mathsf{neg}\left(V \cdot \ell\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \left(\sqrt{\frac{-1 \cdot A}{V \cdot \left(\mathsf{neg}\left(\ell\right)\right)}}\right)\right) \]

      4. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \left(\sqrt{\frac{-1}{V} \cdot \frac{A}{\mathsf{neg}\left(\ell\right)}}\right)\right) \]

      5. sqrt-prod N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \left(\sqrt{\frac{-1}{V}} \cdot \color{blue}{\sqrt{\frac{A}{\mathsf{neg}\left(\ell\right)}}}\right)\right) \]

      6. frac-2neg N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \left(\sqrt{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(V\right)}} \cdot \sqrt{\frac{\color{blue}{A}}{\mathsf{neg}\left(\ell\right)}}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \left(\sqrt{\frac{1}{\mathsf{neg}\left(V\right)}} \cdot \sqrt{\frac{A}{\mathsf{neg}\left(\ell\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{\mathsf{neg}\left(V\right)}}\right), \color{blue}{\left(\sqrt{\frac{A}{\mathsf{neg}\left(\ell\right)}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{\mathsf{neg}\left(V\right)}\right)\right), \left(\sqrt{\color{blue}{\frac{A}{\mathsf{neg}\left(\ell\right)}}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(V\right)}\right)\right), \left(\sqrt{\frac{A}{\mathsf{neg}\left(\ell\right)}}\right)\right)\right) \]

      11. frac-2neg N/A

          \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{-1}{V}\right)\right), \left(\sqrt{\frac{\color{blue}{A}}{\mathsf{neg}\left(\ell\right)}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(-1, V\right)\right), \left(\sqrt{\frac{\color{blue}{A}}{\mathsf{neg}\left(\ell\right)}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(-1, V\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{A}{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(-1, V\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(A\right)\right)\right)}{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right) \]

      15. frac-2neg N/A

          \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(-1, V\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{\mathsf{neg}\left(A\right)}{\ell}\right)\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(-1, V\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(A\right)\right), \ell\right)\right)\right)\right) \]

      17. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(-1, V\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(0 - A\right), \ell\right)\right)\right)\right) \]

      18. --lowering--.f64 43.6%

          \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(-1, V\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, A\right), \ell\right)\right)\right)\right) \]

   4. Applied egg-rr 43.6%

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{\frac{-1}{V}} \cdot \sqrt{\frac{0 - A}{\ell}}\right)} \]

   if -inf.0 < (*.f64 V l) < -1.0000000000000001e-253

   1. Initial program 84.9%

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

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\ell \cdot V}{A}}\right)\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\ell \cdot \frac{V}{A}}\right)\right)\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{\ell}}{\frac{V}{A}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\ell}\right), \left(\frac{V}{A}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \left(\frac{V}{A}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 79.9%

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \mathsf{/.f64}\left(V, A\right)\right)\right)\right) \]

   4. Applied egg-rr 79.9%

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

      1. frac-2neg N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \left(\sqrt{\frac{\frac{1}{\ell}}{\frac{\mathsf{neg}\left(V\right)}{\mathsf{neg}\left(A\right)}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \left(\sqrt{\frac{\frac{1}{\ell}}{\mathsf{neg}\left(V\right)} \cdot \left(\mathsf{neg}\left(A\right)\right)}\right)\right) \]

      3. sqrt-prod N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \left(\sqrt{\frac{\frac{1}{\ell}}{\mathsf{neg}\left(V\right)}} \cdot \color{blue}{\sqrt{\mathsf{neg}\left(A\right)}}\right)\right) \]

      4. sub0-neg N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \left(\sqrt{\frac{\frac{1}{\ell}}{\mathsf{neg}\left(V\right)}} \cdot \sqrt{0 - A}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\left(\sqrt{\frac{\frac{1}{\ell}}{\mathsf{neg}\left(V\right)}}\right), \color{blue}{\left(\sqrt{0 - A}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{\ell}}{\mathsf{neg}\left(V\right)}\right)\right), \left(\sqrt{\color{blue}{0 - A}}\right)\right)\right) \]

      7. frac-2neg N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\mathsf{neg}\left(\frac{1}{\ell}\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(V\right)\right)\right)}\right)\right), \left(\sqrt{\color{blue}{0} - A}\right)\right)\right) \]

      8. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\mathsf{neg}\left(\frac{1}{\ell}\right)}{V}\right)\right), \left(\sqrt{0 - A}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{\ell}\right)\right), V\right)\right), \left(\sqrt{\color{blue}{0} - A}\right)\right)\right) \]

      10. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(1\right)}{\ell}\right), V\right)\right), \left(\sqrt{0 - A}\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{\ell}\right), V\right)\right), \left(\sqrt{0 - A}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, \ell\right), V\right)\right), \left(\sqrt{0 - A}\right)\right)\right) \]

      13. sub0-neg N/A

          \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, \ell\right), V\right)\right), \left(\sqrt{\mathsf{neg}\left(A\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, \ell\right), V\right)\right), \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(A\right)\right)\right)\right)\right) \]

      15. sub0-neg N/A

          \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, \ell\right), V\right)\right), \mathsf{sqrt.f64}\left(\left(0 - A\right)\right)\right)\right) \]

      16. --lowering--.f64 99.4%

          \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, \ell\right), V\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, A\right)\right)\right)\right) \]

   6. Applied egg-rr 99.4%

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{\frac{\frac{-1}{\ell}}{V}} \cdot \sqrt{0 - A}\right)} \]
   7. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{-1}{V \cdot \ell}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\color{blue}{0}, A\right)\right)\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\frac{-1}{V}}{\ell}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\color{blue}{0}, A\right)\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(V\right)}}{\ell}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, A\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{\mathsf{neg}\left(V\right)}}{\ell}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, A\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\mathsf{neg}\left(V\right)}\right), \ell\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\color{blue}{0}, A\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(V\right)}\right), \ell\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, A\right)\right)\right)\right) \]

      7. frac-2neg N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{V}\right), \ell\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, A\right)\right)\right)\right) \]

      8. /-lowering-/.f64 99.3%

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, V\right), \ell\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, A\right)\right)\right)\right) \]

   8. Applied egg-rr 99.3%

      \[\leadsto c0 \cdot \left(\sqrt{\color{blue}{\frac{\frac{-1}{V}}{\ell}}} \cdot \sqrt{0 - A}\right) \]

   if -1.0000000000000001e-253 < (*.f64 V l) < 1.00000000000000002e-294

   1. Initial program 52.8%

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

      1. *-commutative N/A

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

      2. sqrt-div N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\sqrt{V \cdot \ell}}{\sqrt{A} \cdot c0}\right)}\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}{\color{blue}{c0}}\right)\right) \]

      7. sqrt-div N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\sqrt{\frac{V \cdot \ell}{A}}}{c0}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\sqrt{\frac{V \cdot \ell}{A}}\right), \color{blue}{c0}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{V \cdot \ell}{A}\right)\right), c0\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{V}{A} \cdot \ell\right)\right), c0\right)\right) \]

      11. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{V}{\frac{A}{\ell}}\right)\right), c0\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(V, \left(\frac{A}{\ell}\right)\right)\right), c0\right)\right) \]

      13. /-lowering-/.f64 68.0%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(V, \mathsf{/.f64}\left(A, \ell\right)\right)\right), c0\right)\right) \]

   4. Applied egg-rr 68.0%

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

   if 1.00000000000000002e-294 < (*.f64 V l) < 9.9999999999999998e284

   1. Initial program 81.6%

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

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\ell \cdot V}{A}}\right)\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\ell \cdot \frac{V}{A}}\right)\right)\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{\ell}}{\frac{V}{A}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\ell}\right), \left(\frac{V}{A}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \left(\frac{V}{A}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 77.0%

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \mathsf{/.f64}\left(V, A\right)\right)\right)\right) \]

   4. Applied egg-rr 77.0%

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

      1. associate-/l/ N/A

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

      2. associate-/r/ N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(c0, \color{blue}{\left(\sqrt{\frac{V}{\frac{A}{\ell}}}\right)}\right) \]

      7. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{V}{A} \cdot \ell}\right)\right) \]

      8. sqrt-prod N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{V}{A}} \cdot \color{blue}{\sqrt{\ell}}\right)\right) \]

      9. /-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{1}} \cdot \sqrt{\ell}\right)\right) \]

      10. sqrt-prod N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{1} \cdot \ell}\right)\right) \]

      11. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{\frac{1}{\ell}}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{V}{A}}{\frac{1}{\ell}}\right)\right)\right) \]

      13. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{\frac{A}{V}}}{\frac{1}{\ell}}\right)\right)\right) \]

      14. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{A}{V} \cdot \frac{1}{\ell}}\right)\right)\right) \]

      15. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\frac{A}{V}}{\ell}}\right)\right)\right) \]

      16. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\ell}{\frac{A}{V}}\right)\right)\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, \left(\frac{A}{V}\right)\right)\right)\right) \]

      18. /-lowering-/.f64 77.2%

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(A, V\right)\right)\right)\right) \]

   6. Applied egg-rr 77.2%

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

      1. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{1}{\frac{\frac{A}{V}}{\ell}}}\right)\right) \]

      2. associate-/l/ N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{1}{\frac{A}{\ell \cdot V}}}\right)\right) \]

      3. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{1}{A} \cdot \left(\ell \cdot V\right)}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{1}{A} \cdot \left(V \cdot \ell\right)}\right)\right) \]

      5. sqrt-prod N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{1}{A}} \cdot \color{blue}{\sqrt{V \cdot \ell}}\right)\right) \]

      6. pow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left({\left(\frac{1}{A}\right)}^{\frac{1}{2}} \cdot \sqrt{\color{blue}{V \cdot \ell}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\left({\left(\frac{1}{A}\right)}^{\frac{1}{2}}\right), \color{blue}{\left(\sqrt{V \cdot \ell}\right)}\right)\right) \]

      8. inv-pow N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\left({\left({A}^{-1}\right)}^{\frac{1}{2}}\right), \left(\sqrt{\color{blue}{V} \cdot \ell}\right)\right)\right) \]

      9. pow-pow N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\left({A}^{\left(-1 \cdot \frac{1}{2}\right)}\right), \left(\sqrt{\color{blue}{V \cdot \ell}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\left({A}^{\frac{-1}{2}}\right), \left(\sqrt{V \cdot \color{blue}{\ell}}\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\left({A}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \left(\sqrt{V \cdot \color{blue}{\ell}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(A, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \left(\sqrt{\color{blue}{V \cdot \ell}}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(A, \frac{-1}{2}\right), \left(\sqrt{V \cdot \color{blue}{\ell}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(A, \frac{-1}{2}\right), \mathsf{sqrt.f64}\left(\left(V \cdot \ell\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(A, \frac{-1}{2}\right), \mathsf{sqrt.f64}\left(\left(\ell \cdot V\right)\right)\right)\right) \]

      16. *-lowering-*.f64 99.5%

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(A, \frac{-1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\ell, V\right)\right)\right)\right) \]

   8. Applied egg-rr 99.5%

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

   if 9.9999999999999998e284 < (*.f64 V l)

   1. Initial program 38.2%

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

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{A}{V}}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{A}{V}\right), \ell\right)\right)\right) \]

      3. /-lowering-/.f64 85.9%

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(A, V\right), \ell\right)\right)\right) \]

   4. Applied egg-rr 85.9%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{-1}{V}} \cdot \sqrt{0 - \frac{A}{\ell}}\right)\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-253}:\\ \;\;\;\;c0 \cdot \left(\sqrt{0 - A} \cdot \sqrt{\frac{\frac{-1}{V}}{\ell}}\right)\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-294}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\frac{V}{\frac{A}{\ell}}}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+285}:\\ \;\;\;\;\frac{c0}{{A}^{-0.5} \cdot \sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 90.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+293}:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{-1}{V}} \cdot \sqrt{0 - \frac{A}{\ell}}\right)\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-272}:\\ \;\;\;\;\sqrt{0 - A} \cdot \frac{c0}{\sqrt{0 - V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-315}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\ell \cdot \frac{V}{A}}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+285}:\\ \;\;\;\;\frac{c0}{{A}^{-0.5} \cdot \sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -1e+293)
   (* c0 (* (sqrt (/ -1.0 V)) (sqrt (- 0.0 (/ A l)))))
   (if (<= (* V l) -1e-272)
     (* (sqrt (- 0.0 A)) (/ c0 (sqrt (- 0.0 (* V l)))))
     (if (<= (* V l) 5e-315)
       (/ 1.0 (/ (sqrt (* l (/ V A))) c0))
       (if (<= (* V l) 1e+285)
         (/ c0 (* (pow A -0.5) (sqrt (* V l))))
         (* c0 (sqrt (/ (/ A V) l))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+293) {
		tmp = c0 * (sqrt((-1.0 / V)) * sqrt((0.0 - (A / l))));
	} else if ((V * l) <= -1e-272) {
		tmp = sqrt((0.0 - A)) * (c0 / sqrt((0.0 - (V * l))));
	} else if ((V * l) <= 5e-315) {
		tmp = 1.0 / (sqrt((l * (V / A))) / c0);
	} else if ((V * l) <= 1e+285) {
		tmp = c0 / (pow(A, -0.5) * sqrt((V * l)));
	} else {
		tmp = c0 * sqrt(((A / V) / l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-1d+293)) then
        tmp = c0 * (sqrt(((-1.0d0) / v)) * sqrt((0.0d0 - (a / l))))
    else if ((v * l) <= (-1d-272)) then
        tmp = sqrt((0.0d0 - a)) * (c0 / sqrt((0.0d0 - (v * l))))
    else if ((v * l) <= 5d-315) then
        tmp = 1.0d0 / (sqrt((l * (v / a))) / c0)
    else if ((v * l) <= 1d+285) then
        tmp = c0 / ((a ** (-0.5d0)) * sqrt((v * l)))
    else
        tmp = c0 * sqrt(((a / v) / l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+293) {
		tmp = c0 * (Math.sqrt((-1.0 / V)) * Math.sqrt((0.0 - (A / l))));
	} else if ((V * l) <= -1e-272) {
		tmp = Math.sqrt((0.0 - A)) * (c0 / Math.sqrt((0.0 - (V * l))));
	} else if ((V * l) <= 5e-315) {
		tmp = 1.0 / (Math.sqrt((l * (V / A))) / c0);
	} else if ((V * l) <= 1e+285) {
		tmp = c0 / (Math.pow(A, -0.5) * Math.sqrt((V * l)));
	} else {
		tmp = c0 * Math.sqrt(((A / V) / l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -1e+293:
		tmp = c0 * (math.sqrt((-1.0 / V)) * math.sqrt((0.0 - (A / l))))
	elif (V * l) <= -1e-272:
		tmp = math.sqrt((0.0 - A)) * (c0 / math.sqrt((0.0 - (V * l))))
	elif (V * l) <= 5e-315:
		tmp = 1.0 / (math.sqrt((l * (V / A))) / c0)
	elif (V * l) <= 1e+285:
		tmp = c0 / (math.pow(A, -0.5) * math.sqrt((V * l)))
	else:
		tmp = c0 * math.sqrt(((A / V) / l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -1e+293)
		tmp = Float64(c0 * Float64(sqrt(Float64(-1.0 / V)) * sqrt(Float64(0.0 - Float64(A / l)))));
	elseif (Float64(V * l) <= -1e-272)
		tmp = Float64(sqrt(Float64(0.0 - A)) * Float64(c0 / sqrt(Float64(0.0 - Float64(V * l)))));
	elseif (Float64(V * l) <= 5e-315)
		tmp = Float64(1.0 / Float64(sqrt(Float64(l * Float64(V / A))) / c0));
	elseif (Float64(V * l) <= 1e+285)
		tmp = Float64(c0 / Float64((A ^ -0.5) * sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -1e+293)
		tmp = c0 * (sqrt((-1.0 / V)) * sqrt((0.0 - (A / l))));
	elseif ((V * l) <= -1e-272)
		tmp = sqrt((0.0 - A)) * (c0 / sqrt((0.0 - (V * l))));
	elseif ((V * l) <= 5e-315)
		tmp = 1.0 / (sqrt((l * (V / A))) / c0);
	elseif ((V * l) <= 1e+285)
		tmp = c0 / ((A ^ -0.5) * sqrt((V * l)));
	else
		tmp = c0 * sqrt(((A / V) / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -1e+293], N[(c0 * N[(N[Sqrt[N[(-1.0 / V), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(0.0 - N[(A / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-272], N[(N[Sqrt[N[(0.0 - A), $MachinePrecision]], $MachinePrecision] * N[(c0 / N[Sqrt[N[(0.0 - N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e-315], N[(1.0 / N[(N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / c0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+285], N[(c0 / N[(N[Power[A, -0.5], $MachinePrecision] * N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -9.9999999999999992e292

   1. Initial program 48.1%

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

      1. frac-2neg N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \left(\sqrt{\frac{\mathsf{neg}\left(A\right)}{\mathsf{neg}\left(V \cdot \ell\right)}}\right)\right) \]

      2. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \left(\sqrt{\frac{-1 \cdot A}{\mathsf{neg}\left(V \cdot \ell\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \left(\sqrt{\frac{-1 \cdot A}{V \cdot \left(\mathsf{neg}\left(\ell\right)\right)}}\right)\right) \]

      4. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \left(\sqrt{\frac{-1}{V} \cdot \frac{A}{\mathsf{neg}\left(\ell\right)}}\right)\right) \]

      5. sqrt-prod N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \left(\sqrt{\frac{-1}{V}} \cdot \color{blue}{\sqrt{\frac{A}{\mathsf{neg}\left(\ell\right)}}}\right)\right) \]

      6. frac-2neg N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \left(\sqrt{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(V\right)}} \cdot \sqrt{\frac{\color{blue}{A}}{\mathsf{neg}\left(\ell\right)}}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \left(\sqrt{\frac{1}{\mathsf{neg}\left(V\right)}} \cdot \sqrt{\frac{A}{\mathsf{neg}\left(\ell\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{\mathsf{neg}\left(V\right)}}\right), \color{blue}{\left(\sqrt{\frac{A}{\mathsf{neg}\left(\ell\right)}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{\mathsf{neg}\left(V\right)}\right)\right), \left(\sqrt{\color{blue}{\frac{A}{\mathsf{neg}\left(\ell\right)}}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(V\right)}\right)\right), \left(\sqrt{\frac{A}{\mathsf{neg}\left(\ell\right)}}\right)\right)\right) \]

      11. frac-2neg N/A

          \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{-1}{V}\right)\right), \left(\sqrt{\frac{\color{blue}{A}}{\mathsf{neg}\left(\ell\right)}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(-1, V\right)\right), \left(\sqrt{\frac{\color{blue}{A}}{\mathsf{neg}\left(\ell\right)}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(-1, V\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{A}{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(-1, V\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(A\right)\right)\right)}{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right) \]

      15. frac-2neg N/A

          \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(-1, V\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{\mathsf{neg}\left(A\right)}{\ell}\right)\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(-1, V\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(A\right)\right), \ell\right)\right)\right)\right) \]

      17. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(-1, V\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(0 - A\right), \ell\right)\right)\right)\right) \]

      18. --lowering--.f64 45.9%

          \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(-1, V\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, A\right), \ell\right)\right)\right)\right) \]

   4. Applied egg-rr 45.9%

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{\frac{-1}{V}} \cdot \sqrt{\frac{0 - A}{\ell}}\right)} \]

   if -9.9999999999999992e292 < (*.f64 V l) < -9.9999999999999993e-273

   1. Initial program 84.9%

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

      1. clear-num N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. un-div-inv N/A

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

      5. frac-2neg N/A

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

      6. sqrt-div N/A

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

      7. associate-/r/ N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{c0}{\sqrt{\mathsf{neg}\left(V \cdot \ell\right)}}\right), \color{blue}{\left(\sqrt{\mathsf{neg}\left(A\right)}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \left(\sqrt{\mathsf{neg}\left(V \cdot \ell\right)}\right)\right), \left(\sqrt{\color{blue}{\mathsf{neg}\left(A\right)}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(V \cdot \ell\right)\right)\right)\right), \left(\sqrt{\mathsf{neg}\left(A\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(V \cdot \left(\mathsf{neg}\left(\ell\right)\right)\right)\right)\right), \left(\sqrt{\mathsf{neg}\left(A\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(V, \left(\mathsf{neg}\left(\ell\right)\right)\right)\right)\right), \left(\sqrt{\mathsf{neg}\left(A\right)}\right)\right) \]

      13. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(V, \left(0 - \ell\right)\right)\right)\right), \left(\sqrt{\mathsf{neg}\left(A\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(V, \mathsf{\_.f64}\left(0, \ell\right)\right)\right)\right), \left(\sqrt{\mathsf{neg}\left(A\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(V, \mathsf{\_.f64}\left(0, \ell\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(A\right)\right)\right)\right) \]

      16. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(V, \mathsf{\_.f64}\left(0, \ell\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\left(0 - A\right)\right)\right) \]

      17. --lowering--.f64 98.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(V, \mathsf{\_.f64}\left(0, \ell\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, A\right)\right)\right) \]

   4. Applied egg-rr 98.4%

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{V \cdot \left(0 - \ell\right)}} \cdot \sqrt{0 - A}} \]
   5. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(V \cdot \left(\mathsf{neg}\left(\ell\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, A\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot V\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, A\right)\right)\right) \]

      3. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\ell \cdot V\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, A\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{neg.f64}\left(\left(\ell \cdot V\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, A\right)\right)\right) \]

      5. *-lowering-*.f64 98.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\ell, V\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, A\right)\right)\right) \]

   6. Applied egg-rr 98.4%

      \[\leadsto \frac{c0}{\sqrt{\color{blue}{-\ell \cdot V}}} \cdot \sqrt{0 - A} \]

   if -9.9999999999999993e-273 < (*.f64 V l) < 5.0000000023e-315

   1. Initial program 48.3%

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

      1. *-commutative N/A

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

      2. sqrt-div N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\sqrt{V \cdot \ell}}{\sqrt{A} \cdot c0}\right)}\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}{\color{blue}{c0}}\right)\right) \]

      7. sqrt-div N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\sqrt{\frac{V \cdot \ell}{A}}}{c0}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\sqrt{\frac{V \cdot \ell}{A}}\right), \color{blue}{c0}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{V \cdot \ell}{A}\right)\right), c0\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{V}{A} \cdot \ell\right)\right), c0\right)\right) \]

      11. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{V}{\frac{A}{\ell}}\right)\right), c0\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(V, \left(\frac{A}{\ell}\right)\right)\right), c0\right)\right) \]

      13. /-lowering-/.f64 68.0%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(V, \mathsf{/.f64}\left(A, \ell\right)\right)\right), c0\right)\right) \]

   4. Applied egg-rr 68.0%

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

      1. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{V}{A} \cdot \ell\right)\right), c0\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{V}{A}\right), \ell\right)\right), c0\right)\right) \]

      3. /-lowering-/.f64 68.1%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(V, A\right), \ell\right)\right), c0\right)\right) \]

   6. Applied egg-rr 68.1%

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

   if 5.0000000023e-315 < (*.f64 V l) < 9.9999999999999998e284

   1. Initial program 82.0%

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

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\ell \cdot V}{A}}\right)\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\ell \cdot \frac{V}{A}}\right)\right)\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{\ell}}{\frac{V}{A}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\ell}\right), \left(\frac{V}{A}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \left(\frac{V}{A}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 77.5%

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \mathsf{/.f64}\left(V, A\right)\right)\right)\right) \]

   4. Applied egg-rr 77.5%

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

      1. associate-/l/ N/A

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

      2. associate-/r/ N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(c0, \color{blue}{\left(\sqrt{\frac{V}{\frac{A}{\ell}}}\right)}\right) \]

      7. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{V}{A} \cdot \ell}\right)\right) \]

      8. sqrt-prod N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{V}{A}} \cdot \color{blue}{\sqrt{\ell}}\right)\right) \]

      9. /-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{1}} \cdot \sqrt{\ell}\right)\right) \]

      10. sqrt-prod N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{1} \cdot \ell}\right)\right) \]

      11. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{\frac{1}{\ell}}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{V}{A}}{\frac{1}{\ell}}\right)\right)\right) \]

      13. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{\frac{A}{V}}}{\frac{1}{\ell}}\right)\right)\right) \]

      14. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{A}{V} \cdot \frac{1}{\ell}}\right)\right)\right) \]

      15. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\frac{A}{V}}{\ell}}\right)\right)\right) \]

      16. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\ell}{\frac{A}{V}}\right)\right)\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, \left(\frac{A}{V}\right)\right)\right)\right) \]

      18. /-lowering-/.f64 77.8%

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(A, V\right)\right)\right)\right) \]

   6. Applied egg-rr 77.8%

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

      1. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{1}{\frac{\frac{A}{V}}{\ell}}}\right)\right) \]

      2. associate-/l/ N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{1}{\frac{A}{\ell \cdot V}}}\right)\right) \]

      3. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{1}{A} \cdot \left(\ell \cdot V\right)}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{1}{A} \cdot \left(V \cdot \ell\right)}\right)\right) \]

      5. sqrt-prod N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{1}{A}} \cdot \color{blue}{\sqrt{V \cdot \ell}}\right)\right) \]

      6. pow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left({\left(\frac{1}{A}\right)}^{\frac{1}{2}} \cdot \sqrt{\color{blue}{V \cdot \ell}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\left({\left(\frac{1}{A}\right)}^{\frac{1}{2}}\right), \color{blue}{\left(\sqrt{V \cdot \ell}\right)}\right)\right) \]

      8. inv-pow N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\left({\left({A}^{-1}\right)}^{\frac{1}{2}}\right), \left(\sqrt{\color{blue}{V} \cdot \ell}\right)\right)\right) \]

      9. pow-pow N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\left({A}^{\left(-1 \cdot \frac{1}{2}\right)}\right), \left(\sqrt{\color{blue}{V \cdot \ell}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\left({A}^{\frac{-1}{2}}\right), \left(\sqrt{V \cdot \color{blue}{\ell}}\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\left({A}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \left(\sqrt{V \cdot \color{blue}{\ell}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(A, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \left(\sqrt{\color{blue}{V \cdot \ell}}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(A, \frac{-1}{2}\right), \left(\sqrt{V \cdot \color{blue}{\ell}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(A, \frac{-1}{2}\right), \mathsf{sqrt.f64}\left(\left(V \cdot \ell\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(A, \frac{-1}{2}\right), \mathsf{sqrt.f64}\left(\left(\ell \cdot V\right)\right)\right)\right) \]

      16. *-lowering-*.f64 99.5%

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(A, \frac{-1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\ell, V\right)\right)\right)\right) \]

   8. Applied egg-rr 99.5%

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

   if 9.9999999999999998e284 < (*.f64 V l)

   1. Initial program 38.2%

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

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{A}{V}}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{A}{V}\right), \ell\right)\right)\right) \]

      3. /-lowering-/.f64 85.9%

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(A, V\right), \ell\right)\right)\right) \]

   4. Applied egg-rr 85.9%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+293}:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{-1}{V}} \cdot \sqrt{0 - \frac{A}{\ell}}\right)\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-272}:\\ \;\;\;\;\sqrt{0 - A} \cdot \frac{c0}{\sqrt{0 - V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-315}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\ell \cdot \frac{V}{A}}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+285}:\\ \;\;\;\;\frac{c0}{{A}^{-0.5} \cdot \sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 90.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+293}:\\ \;\;\;\;\frac{1}{\frac{\frac{\sqrt{\ell}}{{\left(\frac{A}{V}\right)}^{0.5}}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-272}:\\ \;\;\;\;\sqrt{0 - A} \cdot \frac{c0}{\sqrt{0 - V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-315}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\ell \cdot \frac{V}{A}}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+285}:\\ \;\;\;\;\frac{c0}{{A}^{-0.5} \cdot \sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -1e+293)
   (/ 1.0 (/ (/ (sqrt l) (pow (/ A V) 0.5)) c0))
   (if (<= (* V l) -1e-272)
     (* (sqrt (- 0.0 A)) (/ c0 (sqrt (- 0.0 (* V l)))))
     (if (<= (* V l) 5e-315)
       (/ 1.0 (/ (sqrt (* l (/ V A))) c0))
       (if (<= (* V l) 1e+285)
         (/ c0 (* (pow A -0.5) (sqrt (* V l))))
         (* c0 (sqrt (/ (/ A V) l))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+293) {
		tmp = 1.0 / ((sqrt(l) / pow((A / V), 0.5)) / c0);
	} else if ((V * l) <= -1e-272) {
		tmp = sqrt((0.0 - A)) * (c0 / sqrt((0.0 - (V * l))));
	} else if ((V * l) <= 5e-315) {
		tmp = 1.0 / (sqrt((l * (V / A))) / c0);
	} else if ((V * l) <= 1e+285) {
		tmp = c0 / (pow(A, -0.5) * sqrt((V * l)));
	} else {
		tmp = c0 * sqrt(((A / V) / l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-1d+293)) then
        tmp = 1.0d0 / ((sqrt(l) / ((a / v) ** 0.5d0)) / c0)
    else if ((v * l) <= (-1d-272)) then
        tmp = sqrt((0.0d0 - a)) * (c0 / sqrt((0.0d0 - (v * l))))
    else if ((v * l) <= 5d-315) then
        tmp = 1.0d0 / (sqrt((l * (v / a))) / c0)
    else if ((v * l) <= 1d+285) then
        tmp = c0 / ((a ** (-0.5d0)) * sqrt((v * l)))
    else
        tmp = c0 * sqrt(((a / v) / l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+293) {
		tmp = 1.0 / ((Math.sqrt(l) / Math.pow((A / V), 0.5)) / c0);
	} else if ((V * l) <= -1e-272) {
		tmp = Math.sqrt((0.0 - A)) * (c0 / Math.sqrt((0.0 - (V * l))));
	} else if ((V * l) <= 5e-315) {
		tmp = 1.0 / (Math.sqrt((l * (V / A))) / c0);
	} else if ((V * l) <= 1e+285) {
		tmp = c0 / (Math.pow(A, -0.5) * Math.sqrt((V * l)));
	} else {
		tmp = c0 * Math.sqrt(((A / V) / l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -1e+293:
		tmp = 1.0 / ((math.sqrt(l) / math.pow((A / V), 0.5)) / c0)
	elif (V * l) <= -1e-272:
		tmp = math.sqrt((0.0 - A)) * (c0 / math.sqrt((0.0 - (V * l))))
	elif (V * l) <= 5e-315:
		tmp = 1.0 / (math.sqrt((l * (V / A))) / c0)
	elif (V * l) <= 1e+285:
		tmp = c0 / (math.pow(A, -0.5) * math.sqrt((V * l)))
	else:
		tmp = c0 * math.sqrt(((A / V) / l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -1e+293)
		tmp = Float64(1.0 / Float64(Float64(sqrt(l) / (Float64(A / V) ^ 0.5)) / c0));
	elseif (Float64(V * l) <= -1e-272)
		tmp = Float64(sqrt(Float64(0.0 - A)) * Float64(c0 / sqrt(Float64(0.0 - Float64(V * l)))));
	elseif (Float64(V * l) <= 5e-315)
		tmp = Float64(1.0 / Float64(sqrt(Float64(l * Float64(V / A))) / c0));
	elseif (Float64(V * l) <= 1e+285)
		tmp = Float64(c0 / Float64((A ^ -0.5) * sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -1e+293)
		tmp = 1.0 / ((sqrt(l) / ((A / V) ^ 0.5)) / c0);
	elseif ((V * l) <= -1e-272)
		tmp = sqrt((0.0 - A)) * (c0 / sqrt((0.0 - (V * l))));
	elseif ((V * l) <= 5e-315)
		tmp = 1.0 / (sqrt((l * (V / A))) / c0);
	elseif ((V * l) <= 1e+285)
		tmp = c0 / ((A ^ -0.5) * sqrt((V * l)));
	else
		tmp = c0 * sqrt(((A / V) / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -1e+293], N[(1.0 / N[(N[(N[Sqrt[l], $MachinePrecision] / N[Power[N[(A / V), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] / c0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-272], N[(N[Sqrt[N[(0.0 - A), $MachinePrecision]], $MachinePrecision] * N[(c0 / N[Sqrt[N[(0.0 - N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e-315], N[(1.0 / N[(N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / c0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+285], N[(c0 / N[(N[Power[A, -0.5], $MachinePrecision] * N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -9.9999999999999992e292

   1. Initial program 48.1%

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

      1. *-commutative N/A

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

      2. sqrt-div N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\sqrt{V \cdot \ell}}{\sqrt{A} \cdot c0}\right)}\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}{\color{blue}{c0}}\right)\right) \]

      7. sqrt-div N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\sqrt{\frac{V \cdot \ell}{A}}}{c0}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\sqrt{\frac{V \cdot \ell}{A}}\right), \color{blue}{c0}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{V \cdot \ell}{A}\right)\right), c0\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{V}{A} \cdot \ell\right)\right), c0\right)\right) \]

      11. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{V}{\frac{A}{\ell}}\right)\right), c0\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(V, \left(\frac{A}{\ell}\right)\right)\right), c0\right)\right) \]

      13. /-lowering-/.f64 76.1%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(V, \mathsf{/.f64}\left(A, \ell\right)\right)\right), c0\right)\right) \]

   4. Applied egg-rr 76.1%

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

      1. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\sqrt{\frac{V}{A} \cdot \ell}\right), c0\right)\right) \]

      2. sqrt-prod N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}\right), c0\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{\frac{A}{V}}} \cdot \sqrt{\ell}\right), c0\right)\right) \]

      4. sqrt-div N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{\sqrt{1}}{\sqrt{\frac{A}{V}}} \cdot \sqrt{\ell}\right), c0\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{\sqrt{\frac{A}{V}}} \cdot \sqrt{\ell}\right), c0\right)\right) \]

      6. pow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{{\left(\frac{A}{V}\right)}^{\frac{1}{2}}} \cdot \sqrt{\ell}\right), c0\right)\right) \]

      7. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{\frac{{\left(\frac{A}{V}\right)}^{\frac{1}{2}}}{\sqrt{\ell}}}\right), c0\right)\right) \]

      8. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{\sqrt{\ell}}{{\left(\frac{A}{V}\right)}^{\frac{1}{2}}}\right), c0\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{\ell}\right), \left({\left(\frac{A}{V}\right)}^{\frac{1}{2}}\right)\right), c0\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \left({\left(\frac{A}{V}\right)}^{\frac{1}{2}}\right)\right), c0\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \mathsf{pow.f64}\left(\left(\frac{A}{V}\right), \frac{1}{2}\right)\right), c0\right)\right) \]

      12. /-lowering-/.f64 42.0%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(A, V\right), \frac{1}{2}\right)\right), c0\right)\right) \]

   6. Applied egg-rr 42.0%

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

   if -9.9999999999999992e292 < (*.f64 V l) < -9.9999999999999993e-273

   1. Initial program 84.9%

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

      1. clear-num N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. un-div-inv N/A

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

      5. frac-2neg N/A

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

      6. sqrt-div N/A

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

      7. associate-/r/ N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{c0}{\sqrt{\mathsf{neg}\left(V \cdot \ell\right)}}\right), \color{blue}{\left(\sqrt{\mathsf{neg}\left(A\right)}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \left(\sqrt{\mathsf{neg}\left(V \cdot \ell\right)}\right)\right), \left(\sqrt{\color{blue}{\mathsf{neg}\left(A\right)}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(V \cdot \ell\right)\right)\right)\right), \left(\sqrt{\mathsf{neg}\left(A\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(V \cdot \left(\mathsf{neg}\left(\ell\right)\right)\right)\right)\right), \left(\sqrt{\mathsf{neg}\left(A\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(V, \left(\mathsf{neg}\left(\ell\right)\right)\right)\right)\right), \left(\sqrt{\mathsf{neg}\left(A\right)}\right)\right) \]

      13. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(V, \left(0 - \ell\right)\right)\right)\right), \left(\sqrt{\mathsf{neg}\left(A\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(V, \mathsf{\_.f64}\left(0, \ell\right)\right)\right)\right), \left(\sqrt{\mathsf{neg}\left(A\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(V, \mathsf{\_.f64}\left(0, \ell\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(A\right)\right)\right)\right) \]

      16. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(V, \mathsf{\_.f64}\left(0, \ell\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\left(0 - A\right)\right)\right) \]

      17. --lowering--.f64 98.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(V, \mathsf{\_.f64}\left(0, \ell\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, A\right)\right)\right) \]

   4. Applied egg-rr 98.4%

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{V \cdot \left(0 - \ell\right)}} \cdot \sqrt{0 - A}} \]
   5. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(V \cdot \left(\mathsf{neg}\left(\ell\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, A\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot V\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, A\right)\right)\right) \]

      3. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\ell \cdot V\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, A\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{neg.f64}\left(\left(\ell \cdot V\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, A\right)\right)\right) \]

      5. *-lowering-*.f64 98.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\ell, V\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, A\right)\right)\right) \]

   6. Applied egg-rr 98.4%

      \[\leadsto \frac{c0}{\sqrt{\color{blue}{-\ell \cdot V}}} \cdot \sqrt{0 - A} \]

   if -9.9999999999999993e-273 < (*.f64 V l) < 5.0000000023e-315

   1. Initial program 48.3%

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

      1. *-commutative N/A

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

      2. sqrt-div N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\sqrt{V \cdot \ell}}{\sqrt{A} \cdot c0}\right)}\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}{\color{blue}{c0}}\right)\right) \]

      7. sqrt-div N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\sqrt{\frac{V \cdot \ell}{A}}}{c0}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\sqrt{\frac{V \cdot \ell}{A}}\right), \color{blue}{c0}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{V \cdot \ell}{A}\right)\right), c0\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{V}{A} \cdot \ell\right)\right), c0\right)\right) \]

      11. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{V}{\frac{A}{\ell}}\right)\right), c0\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(V, \left(\frac{A}{\ell}\right)\right)\right), c0\right)\right) \]

      13. /-lowering-/.f64 68.0%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(V, \mathsf{/.f64}\left(A, \ell\right)\right)\right), c0\right)\right) \]

   4. Applied egg-rr 68.0%

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

      1. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{V}{A} \cdot \ell\right)\right), c0\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{V}{A}\right), \ell\right)\right), c0\right)\right) \]

      3. /-lowering-/.f64 68.1%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(V, A\right), \ell\right)\right), c0\right)\right) \]

   6. Applied egg-rr 68.1%

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

   if 5.0000000023e-315 < (*.f64 V l) < 9.9999999999999998e284

   1. Initial program 82.0%

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

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\ell \cdot V}{A}}\right)\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\ell \cdot \frac{V}{A}}\right)\right)\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{\ell}}{\frac{V}{A}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\ell}\right), \left(\frac{V}{A}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \left(\frac{V}{A}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 77.5%

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \mathsf{/.f64}\left(V, A\right)\right)\right)\right) \]

   4. Applied egg-rr 77.5%

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

      1. associate-/l/ N/A

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

      2. associate-/r/ N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(c0, \color{blue}{\left(\sqrt{\frac{V}{\frac{A}{\ell}}}\right)}\right) \]

      7. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{V}{A} \cdot \ell}\right)\right) \]

      8. sqrt-prod N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{V}{A}} \cdot \color{blue}{\sqrt{\ell}}\right)\right) \]

      9. /-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{1}} \cdot \sqrt{\ell}\right)\right) \]

      10. sqrt-prod N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{1} \cdot \ell}\right)\right) \]

      11. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{\frac{1}{\ell}}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{V}{A}}{\frac{1}{\ell}}\right)\right)\right) \]

      13. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{\frac{A}{V}}}{\frac{1}{\ell}}\right)\right)\right) \]

      14. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{A}{V} \cdot \frac{1}{\ell}}\right)\right)\right) \]

      15. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\frac{A}{V}}{\ell}}\right)\right)\right) \]

      16. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\ell}{\frac{A}{V}}\right)\right)\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, \left(\frac{A}{V}\right)\right)\right)\right) \]

      18. /-lowering-/.f64 77.8%

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(A, V\right)\right)\right)\right) \]

   6. Applied egg-rr 77.8%

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

      1. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{1}{\frac{\frac{A}{V}}{\ell}}}\right)\right) \]

      2. associate-/l/ N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{1}{\frac{A}{\ell \cdot V}}}\right)\right) \]

      3. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{1}{A} \cdot \left(\ell \cdot V\right)}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{1}{A} \cdot \left(V \cdot \ell\right)}\right)\right) \]

      5. sqrt-prod N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{1}{A}} \cdot \color{blue}{\sqrt{V \cdot \ell}}\right)\right) \]

      6. pow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left({\left(\frac{1}{A}\right)}^{\frac{1}{2}} \cdot \sqrt{\color{blue}{V \cdot \ell}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\left({\left(\frac{1}{A}\right)}^{\frac{1}{2}}\right), \color{blue}{\left(\sqrt{V \cdot \ell}\right)}\right)\right) \]

      8. inv-pow N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\left({\left({A}^{-1}\right)}^{\frac{1}{2}}\right), \left(\sqrt{\color{blue}{V} \cdot \ell}\right)\right)\right) \]

      9. pow-pow N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\left({A}^{\left(-1 \cdot \frac{1}{2}\right)}\right), \left(\sqrt{\color{blue}{V \cdot \ell}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\left({A}^{\frac{-1}{2}}\right), \left(\sqrt{V \cdot \color{blue}{\ell}}\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\left({A}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \left(\sqrt{V \cdot \color{blue}{\ell}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(A, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \left(\sqrt{\color{blue}{V \cdot \ell}}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(A, \frac{-1}{2}\right), \left(\sqrt{V \cdot \color{blue}{\ell}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(A, \frac{-1}{2}\right), \mathsf{sqrt.f64}\left(\left(V \cdot \ell\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(A, \frac{-1}{2}\right), \mathsf{sqrt.f64}\left(\left(\ell \cdot V\right)\right)\right)\right) \]

      16. *-lowering-*.f64 99.5%

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(A, \frac{-1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\ell, V\right)\right)\right)\right) \]

   8. Applied egg-rr 99.5%

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

   if 9.9999999999999998e284 < (*.f64 V l)

   1. Initial program 38.2%

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

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{A}{V}}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{A}{V}\right), \ell\right)\right)\right) \]

      3. /-lowering-/.f64 85.9%

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(A, V\right), \ell\right)\right)\right) \]

   4. Applied egg-rr 85.9%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+293}:\\ \;\;\;\;\frac{1}{\frac{\frac{\sqrt{\ell}}{{\left(\frac{A}{V}\right)}^{0.5}}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-272}:\\ \;\;\;\;\sqrt{0 - A} \cdot \frac{c0}{\sqrt{0 - V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-315}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\ell \cdot \frac{V}{A}}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+285}:\\ \;\;\;\;\frac{c0}{{A}^{-0.5} \cdot \sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 90.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+293}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-272}:\\ \;\;\;\;\sqrt{0 - A} \cdot \frac{c0}{\sqrt{0 - V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-315}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\ell \cdot \frac{V}{A}}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+285}:\\ \;\;\;\;\frac{c0}{{A}^{-0.5} \cdot \sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -1e+293)
   (/ c0 (/ (sqrt l) (sqrt (/ A V))))
   (if (<= (* V l) -1e-272)
     (* (sqrt (- 0.0 A)) (/ c0 (sqrt (- 0.0 (* V l)))))
     (if (<= (* V l) 5e-315)
       (/ 1.0 (/ (sqrt (* l (/ V A))) c0))
       (if (<= (* V l) 1e+285)
         (/ c0 (* (pow A -0.5) (sqrt (* V l))))
         (* c0 (sqrt (/ (/ A V) l))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+293) {
		tmp = c0 / (sqrt(l) / sqrt((A / V)));
	} else if ((V * l) <= -1e-272) {
		tmp = sqrt((0.0 - A)) * (c0 / sqrt((0.0 - (V * l))));
	} else if ((V * l) <= 5e-315) {
		tmp = 1.0 / (sqrt((l * (V / A))) / c0);
	} else if ((V * l) <= 1e+285) {
		tmp = c0 / (pow(A, -0.5) * sqrt((V * l)));
	} else {
		tmp = c0 * sqrt(((A / V) / l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-1d+293)) then
        tmp = c0 / (sqrt(l) / sqrt((a / v)))
    else if ((v * l) <= (-1d-272)) then
        tmp = sqrt((0.0d0 - a)) * (c0 / sqrt((0.0d0 - (v * l))))
    else if ((v * l) <= 5d-315) then
        tmp = 1.0d0 / (sqrt((l * (v / a))) / c0)
    else if ((v * l) <= 1d+285) then
        tmp = c0 / ((a ** (-0.5d0)) * sqrt((v * l)))
    else
        tmp = c0 * sqrt(((a / v) / l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+293) {
		tmp = c0 / (Math.sqrt(l) / Math.sqrt((A / V)));
	} else if ((V * l) <= -1e-272) {
		tmp = Math.sqrt((0.0 - A)) * (c0 / Math.sqrt((0.0 - (V * l))));
	} else if ((V * l) <= 5e-315) {
		tmp = 1.0 / (Math.sqrt((l * (V / A))) / c0);
	} else if ((V * l) <= 1e+285) {
		tmp = c0 / (Math.pow(A, -0.5) * Math.sqrt((V * l)));
	} else {
		tmp = c0 * Math.sqrt(((A / V) / l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -1e+293:
		tmp = c0 / (math.sqrt(l) / math.sqrt((A / V)))
	elif (V * l) <= -1e-272:
		tmp = math.sqrt((0.0 - A)) * (c0 / math.sqrt((0.0 - (V * l))))
	elif (V * l) <= 5e-315:
		tmp = 1.0 / (math.sqrt((l * (V / A))) / c0)
	elif (V * l) <= 1e+285:
		tmp = c0 / (math.pow(A, -0.5) * math.sqrt((V * l)))
	else:
		tmp = c0 * math.sqrt(((A / V) / l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -1e+293)
		tmp = Float64(c0 / Float64(sqrt(l) / sqrt(Float64(A / V))));
	elseif (Float64(V * l) <= -1e-272)
		tmp = Float64(sqrt(Float64(0.0 - A)) * Float64(c0 / sqrt(Float64(0.0 - Float64(V * l)))));
	elseif (Float64(V * l) <= 5e-315)
		tmp = Float64(1.0 / Float64(sqrt(Float64(l * Float64(V / A))) / c0));
	elseif (Float64(V * l) <= 1e+285)
		tmp = Float64(c0 / Float64((A ^ -0.5) * sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -1e+293)
		tmp = c0 / (sqrt(l) / sqrt((A / V)));
	elseif ((V * l) <= -1e-272)
		tmp = sqrt((0.0 - A)) * (c0 / sqrt((0.0 - (V * l))));
	elseif ((V * l) <= 5e-315)
		tmp = 1.0 / (sqrt((l * (V / A))) / c0);
	elseif ((V * l) <= 1e+285)
		tmp = c0 / ((A ^ -0.5) * sqrt((V * l)));
	else
		tmp = c0 * sqrt(((A / V) / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -1e+293], N[(c0 / N[(N[Sqrt[l], $MachinePrecision] / N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-272], N[(N[Sqrt[N[(0.0 - A), $MachinePrecision]], $MachinePrecision] * N[(c0 / N[Sqrt[N[(0.0 - N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e-315], N[(1.0 / N[(N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / c0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+285], N[(c0 / N[(N[Power[A, -0.5], $MachinePrecision] * N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -9.9999999999999992e292

   1. Initial program 48.1%

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

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\ell \cdot V}{A}}\right)\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\ell \cdot \frac{V}{A}}\right)\right)\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{\ell}}{\frac{V}{A}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\ell}\right), \left(\frac{V}{A}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \left(\frac{V}{A}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 76.0%

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \mathsf{/.f64}\left(V, A\right)\right)\right)\right) \]

   4. Applied egg-rr 76.0%

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

      1. associate-/l/ N/A

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

      2. associate-/r/ N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(c0, \color{blue}{\left(\sqrt{\frac{V}{\frac{A}{\ell}}}\right)}\right) \]

      7. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{V}{A} \cdot \ell}\right)\right) \]

      8. sqrt-prod N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{V}{A}} \cdot \color{blue}{\sqrt{\ell}}\right)\right) \]

      9. /-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{1}} \cdot \sqrt{\ell}\right)\right) \]

      10. sqrt-prod N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{1} \cdot \ell}\right)\right) \]

      11. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{\frac{1}{\ell}}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{V}{A}}{\frac{1}{\ell}}\right)\right)\right) \]

      13. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{\frac{A}{V}}}{\frac{1}{\ell}}\right)\right)\right) \]

      14. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{A}{V} \cdot \frac{1}{\ell}}\right)\right)\right) \]

      15. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\frac{A}{V}}{\ell}}\right)\right)\right) \]

      16. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\ell}{\frac{A}{V}}\right)\right)\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, \left(\frac{A}{V}\right)\right)\right)\right) \]

      18. /-lowering-/.f64 76.1%

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(A, V\right)\right)\right)\right) \]

   6. Applied egg-rr 76.1%

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

      1. sqrt-div N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\frac{\sqrt{\ell}}{\color{blue}{\sqrt{\frac{A}{V}}}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\left(\sqrt{\ell}\right), \color{blue}{\left(\sqrt{\frac{A}{V}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \left(\sqrt{\color{blue}{\frac{A}{V}}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \mathsf{sqrt.f64}\left(\left(\frac{A}{V}\right)\right)\right)\right) \]

      5. /-lowering-/.f64 42.0%

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(A, V\right)\right)\right)\right) \]

   8. Applied egg-rr 42.0%

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

   if -9.9999999999999992e292 < (*.f64 V l) < -9.9999999999999993e-273

   1. Initial program 84.9%

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

      1. clear-num N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. un-div-inv N/A

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

      5. frac-2neg N/A

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

      6. sqrt-div N/A

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

      7. associate-/r/ N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{c0}{\sqrt{\mathsf{neg}\left(V \cdot \ell\right)}}\right), \color{blue}{\left(\sqrt{\mathsf{neg}\left(A\right)}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \left(\sqrt{\mathsf{neg}\left(V \cdot \ell\right)}\right)\right), \left(\sqrt{\color{blue}{\mathsf{neg}\left(A\right)}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(V \cdot \ell\right)\right)\right)\right), \left(\sqrt{\mathsf{neg}\left(A\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(V \cdot \left(\mathsf{neg}\left(\ell\right)\right)\right)\right)\right), \left(\sqrt{\mathsf{neg}\left(A\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(V, \left(\mathsf{neg}\left(\ell\right)\right)\right)\right)\right), \left(\sqrt{\mathsf{neg}\left(A\right)}\right)\right) \]

      13. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(V, \left(0 - \ell\right)\right)\right)\right), \left(\sqrt{\mathsf{neg}\left(A\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(V, \mathsf{\_.f64}\left(0, \ell\right)\right)\right)\right), \left(\sqrt{\mathsf{neg}\left(A\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(V, \mathsf{\_.f64}\left(0, \ell\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(A\right)\right)\right)\right) \]

      16. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(V, \mathsf{\_.f64}\left(0, \ell\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\left(0 - A\right)\right)\right) \]

      17. --lowering--.f64 98.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(V, \mathsf{\_.f64}\left(0, \ell\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, A\right)\right)\right) \]

   4. Applied egg-rr 98.4%

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{V \cdot \left(0 - \ell\right)}} \cdot \sqrt{0 - A}} \]
   5. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(V \cdot \left(\mathsf{neg}\left(\ell\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, A\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot V\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, A\right)\right)\right) \]

      3. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\ell \cdot V\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, A\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{neg.f64}\left(\left(\ell \cdot V\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, A\right)\right)\right) \]

      5. *-lowering-*.f64 98.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\ell, V\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, A\right)\right)\right) \]

   6. Applied egg-rr 98.4%

      \[\leadsto \frac{c0}{\sqrt{\color{blue}{-\ell \cdot V}}} \cdot \sqrt{0 - A} \]

   if -9.9999999999999993e-273 < (*.f64 V l) < 5.0000000023e-315

   1. Initial program 48.3%

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

      1. *-commutative N/A

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

      2. sqrt-div N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\sqrt{V \cdot \ell}}{\sqrt{A} \cdot c0}\right)}\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}{\color{blue}{c0}}\right)\right) \]

      7. sqrt-div N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\sqrt{\frac{V \cdot \ell}{A}}}{c0}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\sqrt{\frac{V \cdot \ell}{A}}\right), \color{blue}{c0}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{V \cdot \ell}{A}\right)\right), c0\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{V}{A} \cdot \ell\right)\right), c0\right)\right) \]

      11. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{V}{\frac{A}{\ell}}\right)\right), c0\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(V, \left(\frac{A}{\ell}\right)\right)\right), c0\right)\right) \]

      13. /-lowering-/.f64 68.0%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(V, \mathsf{/.f64}\left(A, \ell\right)\right)\right), c0\right)\right) \]

   4. Applied egg-rr 68.0%

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

      1. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{V}{A} \cdot \ell\right)\right), c0\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{V}{A}\right), \ell\right)\right), c0\right)\right) \]

      3. /-lowering-/.f64 68.1%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(V, A\right), \ell\right)\right), c0\right)\right) \]

   6. Applied egg-rr 68.1%

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

   if 5.0000000023e-315 < (*.f64 V l) < 9.9999999999999998e284

   1. Initial program 82.0%

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

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\ell \cdot V}{A}}\right)\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\ell \cdot \frac{V}{A}}\right)\right)\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{\ell}}{\frac{V}{A}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\ell}\right), \left(\frac{V}{A}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \left(\frac{V}{A}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 77.5%

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \mathsf{/.f64}\left(V, A\right)\right)\right)\right) \]

   4. Applied egg-rr 77.5%

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

      1. associate-/l/ N/A

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

      2. associate-/r/ N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(c0, \color{blue}{\left(\sqrt{\frac{V}{\frac{A}{\ell}}}\right)}\right) \]

      7. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{V}{A} \cdot \ell}\right)\right) \]

      8. sqrt-prod N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{V}{A}} \cdot \color{blue}{\sqrt{\ell}}\right)\right) \]

      9. /-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{1}} \cdot \sqrt{\ell}\right)\right) \]

      10. sqrt-prod N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{1} \cdot \ell}\right)\right) \]

      11. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{\frac{1}{\ell}}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{V}{A}}{\frac{1}{\ell}}\right)\right)\right) \]

      13. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{\frac{A}{V}}}{\frac{1}{\ell}}\right)\right)\right) \]

      14. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{A}{V} \cdot \frac{1}{\ell}}\right)\right)\right) \]

      15. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\frac{A}{V}}{\ell}}\right)\right)\right) \]

      16. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\ell}{\frac{A}{V}}\right)\right)\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, \left(\frac{A}{V}\right)\right)\right)\right) \]

      18. /-lowering-/.f64 77.8%

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(A, V\right)\right)\right)\right) \]

   6. Applied egg-rr 77.8%

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

      1. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{1}{\frac{\frac{A}{V}}{\ell}}}\right)\right) \]

      2. associate-/l/ N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{1}{\frac{A}{\ell \cdot V}}}\right)\right) \]

      3. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{1}{A} \cdot \left(\ell \cdot V\right)}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{1}{A} \cdot \left(V \cdot \ell\right)}\right)\right) \]

      5. sqrt-prod N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{1}{A}} \cdot \color{blue}{\sqrt{V \cdot \ell}}\right)\right) \]

      6. pow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left({\left(\frac{1}{A}\right)}^{\frac{1}{2}} \cdot \sqrt{\color{blue}{V \cdot \ell}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\left({\left(\frac{1}{A}\right)}^{\frac{1}{2}}\right), \color{blue}{\left(\sqrt{V \cdot \ell}\right)}\right)\right) \]

      8. inv-pow N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\left({\left({A}^{-1}\right)}^{\frac{1}{2}}\right), \left(\sqrt{\color{blue}{V} \cdot \ell}\right)\right)\right) \]

      9. pow-pow N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\left({A}^{\left(-1 \cdot \frac{1}{2}\right)}\right), \left(\sqrt{\color{blue}{V \cdot \ell}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\left({A}^{\frac{-1}{2}}\right), \left(\sqrt{V \cdot \color{blue}{\ell}}\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\left({A}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \left(\sqrt{V \cdot \color{blue}{\ell}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(A, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \left(\sqrt{\color{blue}{V \cdot \ell}}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(A, \frac{-1}{2}\right), \left(\sqrt{V \cdot \color{blue}{\ell}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(A, \frac{-1}{2}\right), \mathsf{sqrt.f64}\left(\left(V \cdot \ell\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(A, \frac{-1}{2}\right), \mathsf{sqrt.f64}\left(\left(\ell \cdot V\right)\right)\right)\right) \]

      16. *-lowering-*.f64 99.5%

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(A, \frac{-1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\ell, V\right)\right)\right)\right) \]

   8. Applied egg-rr 99.5%

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

   if 9.9999999999999998e284 < (*.f64 V l)

   1. Initial program 38.2%

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

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{A}{V}}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{A}{V}\right), \ell\right)\right)\right) \]

      3. /-lowering-/.f64 85.9%

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(A, V\right), \ell\right)\right)\right) \]

   4. Applied egg-rr 85.9%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+293}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-272}:\\ \;\;\;\;\sqrt{0 - A} \cdot \frac{c0}{\sqrt{0 - V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-315}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\ell \cdot \frac{V}{A}}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+285}:\\ \;\;\;\;\frac{c0}{{A}^{-0.5} \cdot \sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 86.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+220}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-176}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-294}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\frac{V}{\frac{A}{\ell}}}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+283}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -5e+220)
   (* (/ c0 (sqrt l)) (sqrt (/ A V)))
   (if (<= (* V l) -1e-176)
     (* c0 (sqrt (/ A (* V l))))
     (if (<= (* V l) 1e-294)
       (/ 1.0 (/ (sqrt (/ V (/ A l))) c0))
       (if (<= (* V l) 4e+283)
         (* c0 (/ (sqrt A) (sqrt (* V l))))
         (/ c0 (sqrt (/ l (/ A V)))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -5e+220) {
		tmp = (c0 / sqrt(l)) * sqrt((A / V));
	} else if ((V * l) <= -1e-176) {
		tmp = c0 * sqrt((A / (V * l)));
	} else if ((V * l) <= 1e-294) {
		tmp = 1.0 / (sqrt((V / (A / l))) / c0);
	} else if ((V * l) <= 4e+283) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0 / sqrt((l / (A / V)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-5d+220)) then
        tmp = (c0 / sqrt(l)) * sqrt((a / v))
    else if ((v * l) <= (-1d-176)) then
        tmp = c0 * sqrt((a / (v * l)))
    else if ((v * l) <= 1d-294) then
        tmp = 1.0d0 / (sqrt((v / (a / l))) / c0)
    else if ((v * l) <= 4d+283) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = c0 / sqrt((l / (a / v)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -5e+220) {
		tmp = (c0 / Math.sqrt(l)) * Math.sqrt((A / V));
	} else if ((V * l) <= -1e-176) {
		tmp = c0 * Math.sqrt((A / (V * l)));
	} else if ((V * l) <= 1e-294) {
		tmp = 1.0 / (Math.sqrt((V / (A / l))) / c0);
	} else if ((V * l) <= 4e+283) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0 / Math.sqrt((l / (A / V)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -5e+220:
		tmp = (c0 / math.sqrt(l)) * math.sqrt((A / V))
	elif (V * l) <= -1e-176:
		tmp = c0 * math.sqrt((A / (V * l)))
	elif (V * l) <= 1e-294:
		tmp = 1.0 / (math.sqrt((V / (A / l))) / c0)
	elif (V * l) <= 4e+283:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0 / math.sqrt((l / (A / V)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -5e+220)
		tmp = Float64(Float64(c0 / sqrt(l)) * sqrt(Float64(A / V)));
	elseif (Float64(V * l) <= -1e-176)
		tmp = Float64(c0 * sqrt(Float64(A / Float64(V * l))));
	elseif (Float64(V * l) <= 1e-294)
		tmp = Float64(1.0 / Float64(sqrt(Float64(V / Float64(A / l))) / c0));
	elseif (Float64(V * l) <= 4e+283)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 / sqrt(Float64(l / Float64(A / V))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -5e+220)
		tmp = (c0 / sqrt(l)) * sqrt((A / V));
	elseif ((V * l) <= -1e-176)
		tmp = c0 * sqrt((A / (V * l)));
	elseif ((V * l) <= 1e-294)
		tmp = 1.0 / (sqrt((V / (A / l))) / c0);
	elseif ((V * l) <= 4e+283)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0 / sqrt((l / (A / V)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -5e+220], N[(N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-176], N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e-294], N[(1.0 / N[(N[Sqrt[N[(V / N[(A / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / c0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 4e+283], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(l / N[(A / V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -5.0000000000000002e220

   1. Initial program 54.8%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. sqrt-div N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{A}{V}}\right), \color{blue}{\left(\frac{c0}{\sqrt{\ell}}\right)}\right) \]

      7. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{A}{V}\right)}^{\frac{1}{2}}\right), \left(\frac{\color{blue}{c0}}{\sqrt{\ell}}\right)\right) \]

      8. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{1}{\frac{V}{A}}\right)}^{\frac{1}{2}}\right), \left(\frac{c0}{\sqrt{\ell}}\right)\right) \]

      9. inv-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left({\left(\frac{V}{A}\right)}^{-1}\right)}^{\frac{1}{2}}\right), \left(\frac{c0}{\sqrt{\ell}}\right)\right) \]

      10. pow-pow N/A

          \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{V}{A}\right)}^{\left(-1 \cdot \frac{1}{2}\right)}\right), \left(\frac{\color{blue}{c0}}{\sqrt{\ell}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\frac{V}{A}\right), \left(-1 \cdot \frac{1}{2}\right)\right), \left(\frac{\color{blue}{c0}}{\sqrt{\ell}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(V, A\right), \left(-1 \cdot \frac{1}{2}\right)\right), \left(\frac{c0}{\sqrt{\ell}}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(V, A\right), \frac{-1}{2}\right), \left(\frac{c0}{\sqrt{\ell}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(V, A\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(c0, \color{blue}{\left(\sqrt{\ell}\right)}\right)\right) \]

      15. sqrt-lowering-sqrt.f64 41.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(V, A\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\ell\right)\right)\right) \]

   4. Applied egg-rr 41.6%

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

      1. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{V}{A}\right)}^{\left(\frac{-1}{2}\right)}\right), \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\ell\right)\right)\right) \]

      2. sqrt-pow1 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{{\left(\frac{V}{A}\right)}^{-1}}\right), \mathsf{/.f64}\left(\color{blue}{c0}, \mathsf{sqrt.f64}\left(\ell\right)\right)\right) \]

      3. inv-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{\frac{V}{A}}}\right), \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\ell\right)\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{A}{V}}\right), \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\ell\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{A}{V}\right)\right), \mathsf{/.f64}\left(\color{blue}{c0}, \mathsf{sqrt.f64}\left(\ell\right)\right)\right) \]

      6. /-lowering-/.f64 41.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(A, V\right)\right), \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\ell\right)\right)\right) \]

   6. Applied egg-rr 41.6%

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

   if -5.0000000000000002e220 < (*.f64 V l) < -1e-176

   1. Initial program 87.7%

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

   if -1e-176 < (*.f64 V l) < 1.00000000000000002e-294

   1. Initial program 60.6%

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

      1. *-commutative N/A

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

      2. sqrt-div N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\sqrt{V \cdot \ell}}{\sqrt{A} \cdot c0}\right)}\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}{\color{blue}{c0}}\right)\right) \]

      7. sqrt-div N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\sqrt{\frac{V \cdot \ell}{A}}}{c0}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\sqrt{\frac{V \cdot \ell}{A}}\right), \color{blue}{c0}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{V \cdot \ell}{A}\right)\right), c0\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{V}{A} \cdot \ell\right)\right), c0\right)\right) \]

      11. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{V}{\frac{A}{\ell}}\right)\right), c0\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(V, \left(\frac{A}{\ell}\right)\right)\right), c0\right)\right) \]

      13. /-lowering-/.f64 71.7%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(V, \mathsf{/.f64}\left(A, \ell\right)\right)\right), c0\right)\right) \]

   4. Applied egg-rr 71.7%

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

   if 1.00000000000000002e-294 < (*.f64 V l) < 3.99999999999999982e283

   1. Initial program 81.4%

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

      1. sqrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \left(\frac{\sqrt{A}}{\color{blue}{\sqrt{V \cdot \ell}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\left(\sqrt{A}\right), \color{blue}{\left(\sqrt{V \cdot \ell}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(A\right), \left(\sqrt{\color{blue}{V \cdot \ell}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(A\right), \mathsf{sqrt.f64}\left(\left(V \cdot \ell\right)\right)\right)\right) \]

      5. *-lowering-*.f64 99.5%

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(A\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(V, \ell\right)\right)\right)\right) \]

   4. Applied egg-rr 99.5%

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

   if 3.99999999999999982e283 < (*.f64 V l)

   1. Initial program 41.4%

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

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\ell \cdot V}{A}}\right)\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\ell \cdot \frac{V}{A}}\right)\right)\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{\ell}}{\frac{V}{A}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\ell}\right), \left(\frac{V}{A}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \left(\frac{V}{A}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 86.6%

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \mathsf{/.f64}\left(V, A\right)\right)\right)\right) \]

   4. Applied egg-rr 86.6%

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

      1. associate-/l/ N/A

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

      2. associate-/r/ N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(c0, \color{blue}{\left(\sqrt{\frac{V}{\frac{A}{\ell}}}\right)}\right) \]

      7. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{V}{A} \cdot \ell}\right)\right) \]

      8. sqrt-prod N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{V}{A}} \cdot \color{blue}{\sqrt{\ell}}\right)\right) \]

      9. /-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{1}} \cdot \sqrt{\ell}\right)\right) \]

      10. sqrt-prod N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{1} \cdot \ell}\right)\right) \]

      11. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{\frac{1}{\ell}}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{V}{A}}{\frac{1}{\ell}}\right)\right)\right) \]

      13. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{\frac{A}{V}}}{\frac{1}{\ell}}\right)\right)\right) \]

      14. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{A}{V} \cdot \frac{1}{\ell}}\right)\right)\right) \]

      15. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\frac{A}{V}}{\ell}}\right)\right)\right) \]

      16. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\ell}{\frac{A}{V}}\right)\right)\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, \left(\frac{A}{V}\right)\right)\right)\right) \]

      18. /-lowering-/.f64 86.7%

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(A, V\right)\right)\right)\right) \]

   6. Applied egg-rr 86.7%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+220}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-176}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-294}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\frac{V}{\frac{A}{\ell}}}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+283}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 82.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 10^{-294}:\\ \;\;\;\;c0 \cdot {\left(\frac{V}{\frac{A}{\ell}}\right)}^{-0.5}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+283}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) 1e-294)
   (* c0 (pow (/ V (/ A l)) -0.5))
   (if (<= (* V l) 4e+283)
     (* c0 (/ (sqrt A) (sqrt (* V l))))
     (/ c0 (sqrt (/ l (/ A V)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= 1e-294) {
		tmp = c0 * pow((V / (A / l)), -0.5);
	} else if ((V * l) <= 4e+283) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0 / sqrt((l / (A / V)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= 1d-294) then
        tmp = c0 * ((v / (a / l)) ** (-0.5d0))
    else if ((v * l) <= 4d+283) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = c0 / sqrt((l / (a / v)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= 1e-294) {
		tmp = c0 * Math.pow((V / (A / l)), -0.5);
	} else if ((V * l) <= 4e+283) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0 / Math.sqrt((l / (A / V)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= 1e-294:
		tmp = c0 * math.pow((V / (A / l)), -0.5)
	elif (V * l) <= 4e+283:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0 / math.sqrt((l / (A / V)))
	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-294)
		tmp = Float64(c0 * (Float64(V / Float64(A / l)) ^ -0.5));
	elseif (Float64(V * l) <= 4e+283)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 / sqrt(Float64(l / Float64(A / V))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= 1e-294)
		tmp = c0 * ((V / (A / l)) ^ -0.5);
	elseif ((V * l) <= 4e+283)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0 / sqrt((l / (A / V)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], 1e-294], N[(c0 * N[Power[N[(V / N[(A / l), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 4e+283], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(l / N[(A / V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < 1.00000000000000002e-294

   1. Initial program 72.0%

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

      1. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{A}{V \cdot \ell}}\right), \color{blue}{c0}\right) \]

      3. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{A}{V \cdot \ell}\right)}^{\frac{1}{2}}\right), c0\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)}^{\frac{1}{2}}\right), c0\right) \]

      5. inv-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left({\left(\frac{V \cdot \ell}{A}\right)}^{-1}\right)}^{\frac{1}{2}}\right), c0\right) \]

      6. pow-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{V \cdot \ell}{A}\right)}^{\left(-1 \cdot \frac{1}{2}\right)}\right), c0\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\frac{V \cdot \ell}{A}\right), \left(-1 \cdot \frac{1}{2}\right)\right), c0\right) \]

      8. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\frac{V}{A} \cdot \ell\right), \left(-1 \cdot \frac{1}{2}\right)\right), c0\right) \]

      9. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\frac{V}{\frac{A}{\ell}}\right), \left(-1 \cdot \frac{1}{2}\right)\right), c0\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(V, \left(\frac{A}{\ell}\right)\right), \left(-1 \cdot \frac{1}{2}\right)\right), c0\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(V, \mathsf{/.f64}\left(A, \ell\right)\right), \left(-1 \cdot \frac{1}{2}\right)\right), c0\right) \]

      12. metadata-eval 74.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(V, \mathsf{/.f64}\left(A, \ell\right)\right), \frac{-1}{2}\right), c0\right) \]

   4. Applied egg-rr 74.4%

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

   if 1.00000000000000002e-294 < (*.f64 V l) < 3.99999999999999982e283

   1. Initial program 81.4%

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

      1. sqrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \left(\frac{\sqrt{A}}{\color{blue}{\sqrt{V \cdot \ell}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\left(\sqrt{A}\right), \color{blue}{\left(\sqrt{V \cdot \ell}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(A\right), \left(\sqrt{\color{blue}{V \cdot \ell}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(A\right), \mathsf{sqrt.f64}\left(\left(V \cdot \ell\right)\right)\right)\right) \]

      5. *-lowering-*.f64 99.5%

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(A\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(V, \ell\right)\right)\right)\right) \]

   4. Applied egg-rr 99.5%

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

   if 3.99999999999999982e283 < (*.f64 V l)

   1. Initial program 41.4%

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

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\ell \cdot V}{A}}\right)\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\ell \cdot \frac{V}{A}}\right)\right)\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{\ell}}{\frac{V}{A}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\ell}\right), \left(\frac{V}{A}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \left(\frac{V}{A}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 86.6%

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \mathsf{/.f64}\left(V, A\right)\right)\right)\right) \]

   4. Applied egg-rr 86.6%

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

      1. associate-/l/ N/A

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

      2. associate-/r/ N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(c0, \color{blue}{\left(\sqrt{\frac{V}{\frac{A}{\ell}}}\right)}\right) \]

      7. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{V}{A} \cdot \ell}\right)\right) \]

      8. sqrt-prod N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{V}{A}} \cdot \color{blue}{\sqrt{\ell}}\right)\right) \]

      9. /-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{1}} \cdot \sqrt{\ell}\right)\right) \]

      10. sqrt-prod N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{1} \cdot \ell}\right)\right) \]

      11. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{\frac{1}{\ell}}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{V}{A}}{\frac{1}{\ell}}\right)\right)\right) \]

      13. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{\frac{A}{V}}}{\frac{1}{\ell}}\right)\right)\right) \]

      14. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{A}{V} \cdot \frac{1}{\ell}}\right)\right)\right) \]

      15. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\frac{A}{V}}{\ell}}\right)\right)\right) \]

      16. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\ell}{\frac{A}{V}}\right)\right)\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, \left(\frac{A}{V}\right)\right)\right)\right) \]

      18. /-lowering-/.f64 86.7%

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(A, V\right)\right)\right)\right) \]

   6. Applied egg-rr 86.7%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 10^{-294}:\\ \;\;\;\;c0 \cdot {\left(\frac{V}{\frac{A}{\ell}}\right)}^{-0.5}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+283}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 84.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{c0}{{A}^{-0.5} \cdot \sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= l -2e-310)
   (/ c0 (* (pow A -0.5) (sqrt (* V l))))
   (/ c0 (/ (sqrt l) (sqrt (/ A V))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= -2e-310) {
		tmp = c0 / (pow(A, -0.5) * sqrt((V * l)));
	} else {
		tmp = c0 / (sqrt(l) / sqrt((A / V)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= (-2d-310)) then
        tmp = c0 / ((a ** (-0.5d0)) * sqrt((v * l)))
    else
        tmp = c0 / (sqrt(l) / sqrt((a / v)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= -2e-310) {
		tmp = c0 / (Math.pow(A, -0.5) * Math.sqrt((V * l)));
	} else {
		tmp = c0 / (Math.sqrt(l) / Math.sqrt((A / V)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if l <= -2e-310:
		tmp = c0 / (math.pow(A, -0.5) * math.sqrt((V * l)))
	else:
		tmp = c0 / (math.sqrt(l) / math.sqrt((A / V)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (l <= -2e-310)
		tmp = Float64(c0 / Float64((A ^ -0.5) * sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 / Float64(sqrt(l) / sqrt(Float64(A / V))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (l <= -2e-310)
		tmp = c0 / ((A ^ -0.5) * sqrt((V * l)));
	else
		tmp = c0 / (sqrt(l) / sqrt((A / V)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[l, -2e-310], N[(c0 / N[(N[Power[A, -0.5], $MachinePrecision] * N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[(N[Sqrt[l], $MachinePrecision] / N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.999999999999994e-310

   1. Initial program 73.4%

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

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\ell \cdot V}{A}}\right)\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\ell \cdot \frac{V}{A}}\right)\right)\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{\ell}}{\frac{V}{A}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\ell}\right), \left(\frac{V}{A}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \left(\frac{V}{A}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 79.7%

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \mathsf{/.f64}\left(V, A\right)\right)\right)\right) \]

   4. Applied egg-rr 79.7%

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

      1. associate-/l/ N/A

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

      2. associate-/r/ N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(c0, \color{blue}{\left(\sqrt{\frac{V}{\frac{A}{\ell}}}\right)}\right) \]

      7. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{V}{A} \cdot \ell}\right)\right) \]

      8. sqrt-prod N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{V}{A}} \cdot \color{blue}{\sqrt{\ell}}\right)\right) \]

      9. /-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{1}} \cdot \sqrt{\ell}\right)\right) \]

      10. sqrt-prod N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{1} \cdot \ell}\right)\right) \]

      11. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{\frac{1}{\ell}}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{V}{A}}{\frac{1}{\ell}}\right)\right)\right) \]

      13. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{\frac{A}{V}}}{\frac{1}{\ell}}\right)\right)\right) \]

      14. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{A}{V} \cdot \frac{1}{\ell}}\right)\right)\right) \]

      15. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\frac{A}{V}}{\ell}}\right)\right)\right) \]

      16. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\ell}{\frac{A}{V}}\right)\right)\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, \left(\frac{A}{V}\right)\right)\right)\right) \]

      18. /-lowering-/.f64 79.3%

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(A, V\right)\right)\right)\right) \]

   6. Applied egg-rr 79.3%

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

      1. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{1}{\frac{\frac{A}{V}}{\ell}}}\right)\right) \]

      2. associate-/l/ N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{1}{\frac{A}{\ell \cdot V}}}\right)\right) \]

      3. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{1}{A} \cdot \left(\ell \cdot V\right)}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{1}{A} \cdot \left(V \cdot \ell\right)}\right)\right) \]

      5. sqrt-prod N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{1}{A}} \cdot \color{blue}{\sqrt{V \cdot \ell}}\right)\right) \]

      6. pow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left({\left(\frac{1}{A}\right)}^{\frac{1}{2}} \cdot \sqrt{\color{blue}{V \cdot \ell}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\left({\left(\frac{1}{A}\right)}^{\frac{1}{2}}\right), \color{blue}{\left(\sqrt{V \cdot \ell}\right)}\right)\right) \]

      8. inv-pow N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\left({\left({A}^{-1}\right)}^{\frac{1}{2}}\right), \left(\sqrt{\color{blue}{V} \cdot \ell}\right)\right)\right) \]

      9. pow-pow N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\left({A}^{\left(-1 \cdot \frac{1}{2}\right)}\right), \left(\sqrt{\color{blue}{V \cdot \ell}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\left({A}^{\frac{-1}{2}}\right), \left(\sqrt{V \cdot \color{blue}{\ell}}\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\left({A}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \left(\sqrt{V \cdot \color{blue}{\ell}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(A, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \left(\sqrt{\color{blue}{V \cdot \ell}}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(A, \frac{-1}{2}\right), \left(\sqrt{V \cdot \color{blue}{\ell}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(A, \frac{-1}{2}\right), \mathsf{sqrt.f64}\left(\left(V \cdot \ell\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(A, \frac{-1}{2}\right), \mathsf{sqrt.f64}\left(\left(\ell \cdot V\right)\right)\right)\right) \]

      16. *-lowering-*.f64 40.4%

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(A, \frac{-1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\ell, V\right)\right)\right)\right) \]

   8. Applied egg-rr 40.4%

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

   if -1.999999999999994e-310 < l

   1. Initial program 72.2%

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

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\ell \cdot V}{A}}\right)\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\ell \cdot \frac{V}{A}}\right)\right)\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{\ell}}{\frac{V}{A}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\ell}\right), \left(\frac{V}{A}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \left(\frac{V}{A}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 75.4%

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \mathsf{/.f64}\left(V, A\right)\right)\right)\right) \]

   4. Applied egg-rr 75.4%

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

      1. associate-/l/ N/A

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

      2. associate-/r/ N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(c0, \color{blue}{\left(\sqrt{\frac{V}{\frac{A}{\ell}}}\right)}\right) \]

      7. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{V}{A} \cdot \ell}\right)\right) \]

      8. sqrt-prod N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{V}{A}} \cdot \color{blue}{\sqrt{\ell}}\right)\right) \]

      9. /-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{1}} \cdot \sqrt{\ell}\right)\right) \]

      10. sqrt-prod N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{1} \cdot \ell}\right)\right) \]

      11. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{\frac{1}{\ell}}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{V}{A}}{\frac{1}{\ell}}\right)\right)\right) \]

      13. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{\frac{A}{V}}}{\frac{1}{\ell}}\right)\right)\right) \]

      14. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{A}{V} \cdot \frac{1}{\ell}}\right)\right)\right) \]

      15. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\frac{A}{V}}{\ell}}\right)\right)\right) \]

      16. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\ell}{\frac{A}{V}}\right)\right)\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, \left(\frac{A}{V}\right)\right)\right)\right) \]

      18. /-lowering-/.f64 75.2%

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(A, V\right)\right)\right)\right) \]

   6. Applied egg-rr 75.2%

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

      1. sqrt-div N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\frac{\sqrt{\ell}}{\color{blue}{\sqrt{\frac{A}{V}}}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\left(\sqrt{\ell}\right), \color{blue}{\left(\sqrt{\frac{A}{V}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \left(\sqrt{\color{blue}{\frac{A}{V}}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \mathsf{sqrt.f64}\left(\left(\frac{A}{V}\right)\right)\right)\right) \]

      5. /-lowering-/.f64 87.8%

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(A, V\right)\right)\right)\right) \]

   8. Applied egg-rr 87.8%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{c0}{{A}^{-0.5} \cdot \sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 84.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= l -2e-310)
   (* c0 (/ (sqrt A) (sqrt (* V l))))
   (/ c0 (/ (sqrt l) (sqrt (/ A V))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= -2e-310) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0 / (sqrt(l) / sqrt((A / V)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= (-2d-310)) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = c0 / (sqrt(l) / sqrt((a / v)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= -2e-310) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0 / (Math.sqrt(l) / Math.sqrt((A / V)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if l <= -2e-310:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0 / (math.sqrt(l) / math.sqrt((A / V)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (l <= -2e-310)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 / Float64(sqrt(l) / sqrt(Float64(A / V))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (l <= -2e-310)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0 / (sqrt(l) / sqrt((A / V)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[l, -2e-310], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[(N[Sqrt[l], $MachinePrecision] / N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.999999999999994e-310

   1. Initial program 73.4%

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

      1. sqrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \left(\frac{\sqrt{A}}{\color{blue}{\sqrt{V \cdot \ell}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\left(\sqrt{A}\right), \color{blue}{\left(\sqrt{V \cdot \ell}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(A\right), \left(\sqrt{\color{blue}{V \cdot \ell}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(A\right), \mathsf{sqrt.f64}\left(\left(V \cdot \ell\right)\right)\right)\right) \]

      5. *-lowering-*.f64 40.4%

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(A\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(V, \ell\right)\right)\right)\right) \]

   4. Applied egg-rr 40.4%

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

   if -1.999999999999994e-310 < l

   1. Initial program 72.2%

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

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\ell \cdot V}{A}}\right)\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\ell \cdot \frac{V}{A}}\right)\right)\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{\ell}}{\frac{V}{A}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\ell}\right), \left(\frac{V}{A}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \left(\frac{V}{A}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 75.4%

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \mathsf{/.f64}\left(V, A\right)\right)\right)\right) \]

   4. Applied egg-rr 75.4%

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

      1. associate-/l/ N/A

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

      2. associate-/r/ N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(c0, \color{blue}{\left(\sqrt{\frac{V}{\frac{A}{\ell}}}\right)}\right) \]

      7. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{V}{A} \cdot \ell}\right)\right) \]

      8. sqrt-prod N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{V}{A}} \cdot \color{blue}{\sqrt{\ell}}\right)\right) \]

      9. /-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{1}} \cdot \sqrt{\ell}\right)\right) \]

      10. sqrt-prod N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{1} \cdot \ell}\right)\right) \]

      11. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \left(\sqrt{\frac{\frac{V}{A}}{\frac{1}{\ell}}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{V}{A}}{\frac{1}{\ell}}\right)\right)\right) \]

      13. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{\frac{A}{V}}}{\frac{1}{\ell}}\right)\right)\right) \]

      14. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{A}{V} \cdot \frac{1}{\ell}}\right)\right)\right) \]

      15. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\frac{A}{V}}{\ell}}\right)\right)\right) \]

      16. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\left(\frac{\ell}{\frac{A}{V}}\right)\right)\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, \left(\frac{A}{V}\right)\right)\right)\right) \]

      18. /-lowering-/.f64 75.2%

          \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(A, V\right)\right)\right)\right) \]

   6. Applied egg-rr 75.2%

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

      1. sqrt-div N/A

         \[\leadsto \mathsf{/.f64}\left(c0, \left(\frac{\sqrt{\ell}}{\color{blue}{\sqrt{\frac{A}{V}}}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\left(\sqrt{\ell}\right), \color{blue}{\left(\sqrt{\frac{A}{V}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \left(\sqrt{\color{blue}{\frac{A}{V}}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \mathsf{sqrt.f64}\left(\left(\frac{A}{V}\right)\right)\right)\right) \]

      5. /-lowering-/.f64 87.8%

         \[\leadsto \mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(A, V\right)\right)\right)\right) \]

   8. Applied egg-rr 87.8%

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

Alternative 19: 74.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.8%

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

Reproduce

herbie shell --seed 2024150 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))