Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.6% → 91.9%

Time: 10.2s

Alternatives: 10

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: 73.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{\ell}{\frac{A}{V}}}\\ \mathbf{if}\;\ell \cdot V \leq -2 \cdot 10^{+302}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq -2 \cdot 10^{-313}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{0 - A}}{\sqrt{\ell \cdot \left(0 - V\right)}}\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;\frac{1}{\frac{t\_0}{c0}}\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{+279}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{{\left(\ell \cdot V\right)}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (sqrt (/ l (/ A V)))))
   (if (<= (* l V) -2e+302)
     (* c0 (/ (sqrt (/ A V)) (sqrt l)))
     (if (<= (* l V) -2e-313)
       (* c0 (/ (sqrt (- 0.0 A)) (sqrt (* l (- 0.0 V)))))
       (if (<= (* l V) 0.0)
         (/ 1.0 (/ t_0 c0))
         (if (<= (* l V) 5e+279)
           (* c0 (/ (sqrt A) (pow (* l V) 0.5)))
           (/ c0 t_0)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt((l / (A / V)));
	double tmp;
	if ((l * V) <= -2e+302) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else if ((l * V) <= -2e-313) {
		tmp = c0 * (sqrt((0.0 - A)) / sqrt((l * (0.0 - V))));
	} else if ((l * V) <= 0.0) {
		tmp = 1.0 / (t_0 / c0);
	} else if ((l * V) <= 5e+279) {
		tmp = c0 * (sqrt(A) / pow((l * V), 0.5));
	} else {
		tmp = c0 / t_0;
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((l / (a / v)))
    if ((l * v) <= (-2d+302)) then
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    else if ((l * v) <= (-2d-313)) then
        tmp = c0 * (sqrt((0.0d0 - a)) / sqrt((l * (0.0d0 - v))))
    else if ((l * v) <= 0.0d0) then
        tmp = 1.0d0 / (t_0 / c0)
    else if ((l * v) <= 5d+279) then
        tmp = c0 * (sqrt(a) / ((l * v) ** 0.5d0))
    else
        tmp = c0 / t_0
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = Math.sqrt((l / (A / V)));
	double tmp;
	if ((l * V) <= -2e+302) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if ((l * V) <= -2e-313) {
		tmp = c0 * (Math.sqrt((0.0 - A)) / Math.sqrt((l * (0.0 - V))));
	} else if ((l * V) <= 0.0) {
		tmp = 1.0 / (t_0 / c0);
	} else if ((l * V) <= 5e+279) {
		tmp = c0 * (Math.sqrt(A) / Math.pow((l * V), 0.5));
	} else {
		tmp = c0 / t_0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt((l / (A / V)))
	tmp = 0
	if (l * V) <= -2e+302:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	elif (l * V) <= -2e-313:
		tmp = c0 * (math.sqrt((0.0 - A)) / math.sqrt((l * (0.0 - V))))
	elif (l * V) <= 0.0:
		tmp = 1.0 / (t_0 / c0)
	elif (l * V) <= 5e+279:
		tmp = c0 * (math.sqrt(A) / math.pow((l * V), 0.5))
	else:
		tmp = c0 / t_0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = sqrt(Float64(l / Float64(A / V)))
	tmp = 0.0
	if (Float64(l * V) <= -2e+302)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (Float64(l * V) <= -2e-313)
		tmp = Float64(c0 * Float64(sqrt(Float64(0.0 - A)) / sqrt(Float64(l * Float64(0.0 - V)))));
	elseif (Float64(l * V) <= 0.0)
		tmp = Float64(1.0 / Float64(t_0 / c0));
	elseif (Float64(l * V) <= 5e+279)
		tmp = Float64(c0 * Float64(sqrt(A) / (Float64(l * V) ^ 0.5)));
	else
		tmp = Float64(c0 / t_0);
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = sqrt((l / (A / V)));
	tmp = 0.0;
	if ((l * V) <= -2e+302)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	elseif ((l * V) <= -2e-313)
		tmp = c0 * (sqrt((0.0 - A)) / sqrt((l * (0.0 - V))));
	elseif ((l * V) <= 0.0)
		tmp = 1.0 / (t_0 / c0);
	elseif ((l * V) <= 5e+279)
		tmp = c0 * (sqrt(A) / ((l * V) ^ 0.5));
	else
		tmp = c0 / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[Sqrt[N[(l / N[(A / V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(l * V), $MachinePrecision], -2e+302], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], -2e-313], N[(c0 * N[(N[Sqrt[N[(0.0 - A), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(l * N[(0.0 - V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 0.0], N[(1.0 / N[(t$95$0 / c0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 5e+279], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Power[N[(l * V), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -2.0000000000000002e302

   1. Initial program 25.0%

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

      1. associate-/r* N/A

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

      2. sqrt-div N/A

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

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

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

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

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

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

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

      6. sqrt-lowering-sqrt.f64 44.3%

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

   4. Applied egg-rr 44.3%

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

   if -2.0000000000000002e302 < (*.f64 V l) < -1.99999999998e-313

   1. Initial program 82.9%

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

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

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

      2. associate-/r* N/A

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

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

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

      4. /-lowering-/.f64 74.2%

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

   4. Applied egg-rr 74.2%

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

      1. div-inv N/A

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

      2. frac-2neg N/A

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

      3. frac-times N/A

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

      4. sqrt-div N/A

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

      5. *-rgt-identity N/A

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

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

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

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

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

      8. neg-sub0 N/A

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

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

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

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

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

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

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

      12. neg-sub0 N/A

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

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

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

      14. *-lowering-*.f64 99.2%

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

   6. Applied egg-rr 99.2%

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

   if -1.99999999998e-313 < (*.f64 V l) < -0.0

   1. Initial program 38.3%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. pow1/2 N/A

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

      7. clear-num N/A

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

      8. inv-pow N/A

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

      9. pow-pow N/A

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

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

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

      11. associate-/r* N/A

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

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

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

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

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

      14. metadata-eval 72.8%

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

   4. Applied egg-rr 72.8%

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

   6. Applied egg-rr 72.9%

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

   if -0.0 < (*.f64 V l) < 5.0000000000000002e279

   1. Initial program 80.9%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. pow1/2 N/A

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

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

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

      6. *-lowering-*.f64 98.3%

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

   4. Applied egg-rr 98.3%

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

   if 5.0000000000000002e279 < (*.f64 V l)

   1. Initial program 34.6%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. pow1/2 N/A

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

      7. clear-num N/A

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

      8. inv-pow N/A

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

      9. pow-pow N/A

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

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

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

      11. associate-/r* N/A

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

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

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

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

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

      14. metadata-eval 65.2%

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

   4. Applied egg-rr 65.2%

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

      1. /-lowering-/.f64 N/A

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

      2. associate-/l/ N/A

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

      3. associate-/r* N/A

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

      4. clear-num N/A

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

      5. inv-pow N/A

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

      6. pow-pow N/A

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

      7. metadata-eval N/A

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

      8. pow1/2 N/A

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

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

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

      10. div-inv N/A

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

      11. *-commutative N/A

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

      12. clear-num N/A

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

      13. associate-/r/ N/A

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

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

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

      15. /-lowering-/.f64 65.3%

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

   6. Applied egg-rr 65.3%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -2 \cdot 10^{+302}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq -2 \cdot 10^{-313}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{0 - A}}{\sqrt{\ell \cdot \left(0 - V\right)}}\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\frac{\ell}{\frac{A}{V}}}}{c0}}\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{+279}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{{\left(\ell \cdot V\right)}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ t_1 := \frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+157}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\frac{\ell}{\frac{1}{V}}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* l V))))) (t_1 (/ c0 (sqrt (/ l (/ A V))))))
   (if (<= t_0 0.0)
     t_1
     (if (<= t_0 2e+157) (* c0 (sqrt (/ A (/ l (/ 1.0 V))))) t_1))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (l * V)));
	double t_1 = c0 / sqrt((l / (A / V)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 2e+157) {
		tmp = c0 * sqrt((A / (l / (1.0 / V))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (l * v)))
    t_1 = c0 / sqrt((l / (a / v)))
    if (t_0 <= 0.0d0) then
        tmp = t_1
    else if (t_0 <= 2d+157) then
        tmp = c0 * sqrt((a / (l / (1.0d0 / v))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (l * V)));
	double t_1 = c0 / Math.sqrt((l / (A / V)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 2e+157) {
		tmp = c0 * Math.sqrt((A / (l / (1.0 / V))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (l * V)))
	t_1 = c0 / math.sqrt((l / (A / V)))
	tmp = 0
	if t_0 <= 0.0:
		tmp = t_1
	elif t_0 <= 2e+157:
		tmp = c0 * math.sqrt((A / (l / (1.0 / V))))
	else:
		tmp = t_1
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(l * V))))
	t_1 = Float64(c0 / sqrt(Float64(l / Float64(A / V))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 2e+157)
		tmp = Float64(c0 * sqrt(Float64(A / Float64(l / Float64(1.0 / V)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (l * V)));
	t_1 = c0 / sqrt((l / (A / V)));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 2e+157)
		tmp = c0 * sqrt((A / (l / (1.0 / V))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(l * V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[Sqrt[N[(l / N[(A / V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 2e+157], N[(c0 * N[Sqrt[N[(A / N[(l / N[(1.0 / V), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 0.0 or 1.99999999999999997e157 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 60.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. pow1/2 N/A

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

      7. clear-num N/A

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

      8. inv-pow N/A

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

      9. pow-pow N/A

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

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

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

      11. associate-/r* N/A

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

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

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

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

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

      14. metadata-eval 69.5%

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

   4. Applied egg-rr 69.5%

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

      1. /-lowering-/.f64 N/A

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

      2. associate-/l/ N/A

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

      3. associate-/r* N/A

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

      4. clear-num N/A

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

      5. inv-pow N/A

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

      6. pow-pow N/A

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

      7. metadata-eval N/A

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

      8. pow1/2 N/A

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

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

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

      10. div-inv N/A

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

      11. *-commutative N/A

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

      12. clear-num N/A

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

      13. associate-/r/ N/A

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

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

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

      15. /-lowering-/.f64 68.9%

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

   6. Applied egg-rr 68.9%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 1.99999999999999997e157

   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 99.6%

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

   4. Applied egg-rr 99.6%

      \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{\frac{\ell}{\frac{1}{V}}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}} \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}} \leq 2 \cdot 10^{+157}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\frac{\ell}{\frac{1}{V}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ t_1 := \frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+157}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* l V))))) (t_1 (/ c0 (sqrt (/ l (/ A V))))))
   (if (<= t_0 0.0) t_1 (if (<= t_0 2e+157) t_0 t_1))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (l * V)));
	double t_1 = c0 / sqrt((l / (A / V)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 2e+157) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (l * v)))
    t_1 = c0 / sqrt((l / (a / v)))
    if (t_0 <= 0.0d0) then
        tmp = t_1
    else if (t_0 <= 2d+157) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (l * V)));
	double t_1 = c0 / Math.sqrt((l / (A / V)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 2e+157) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (l * V)))
	t_1 = c0 / math.sqrt((l / (A / V)))
	tmp = 0
	if t_0 <= 0.0:
		tmp = t_1
	elif t_0 <= 2e+157:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(l * V))))
	t_1 = Float64(c0 / sqrt(Float64(l / Float64(A / V))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 2e+157)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (l * V)));
	t_1 = c0 / sqrt((l / (A / V)));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 2e+157)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(l * V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[Sqrt[N[(l / N[(A / V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 2e+157], t$95$0, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 0.0 or 1.99999999999999997e157 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 60.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. pow1/2 N/A

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

      7. clear-num N/A

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

      8. inv-pow N/A

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

      9. pow-pow N/A

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

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

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

      11. associate-/r* N/A

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

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

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

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

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

      14. metadata-eval 69.5%

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

   4. Applied egg-rr 69.5%

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

      1. /-lowering-/.f64 N/A

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

      2. associate-/l/ N/A

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

      3. associate-/r* N/A

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

      4. clear-num N/A

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

      5. inv-pow N/A

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

      6. pow-pow N/A

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

      7. metadata-eval N/A

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

      8. pow1/2 N/A

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

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

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

      10. div-inv N/A

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

      11. *-commutative N/A

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

      12. clear-num N/A

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

      13. associate-/r/ N/A

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

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

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

      15. /-lowering-/.f64 68.9%

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

   6. Applied egg-rr 68.9%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 1.99999999999999997e157

   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 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}} \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}} \leq 2 \cdot 10^{+157}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ t_1 := \frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{if}\;t\_0 \leq 10^{-298}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+226}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* l V))))) (t_1 (/ c0 (sqrt (* l (/ V A))))))
   (if (<= t_0 1e-298) t_1 (if (<= t_0 4e+226) t_0 t_1))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (l * V)));
	double t_1 = c0 / sqrt((l * (V / A)));
	double tmp;
	if (t_0 <= 1e-298) {
		tmp = t_1;
	} else if (t_0 <= 4e+226) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (l * v)))
    t_1 = c0 / sqrt((l * (v / a)))
    if (t_0 <= 1d-298) then
        tmp = t_1
    else if (t_0 <= 4d+226) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (l * V)));
	double t_1 = c0 / Math.sqrt((l * (V / A)));
	double tmp;
	if (t_0 <= 1e-298) {
		tmp = t_1;
	} else if (t_0 <= 4e+226) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (l * V)))
	t_1 = c0 / math.sqrt((l * (V / A)))
	tmp = 0
	if t_0 <= 1e-298:
		tmp = t_1
	elif t_0 <= 4e+226:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(l * V))))
	t_1 = Float64(c0 / sqrt(Float64(l * Float64(V / A))))
	tmp = 0.0
	if (t_0 <= 1e-298)
		tmp = t_1;
	elseif (t_0 <= 4e+226)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (l * V)));
	t_1 = c0 / sqrt((l * (V / A)));
	tmp = 0.0;
	if (t_0 <= 1e-298)
		tmp = t_1;
	elseif (t_0 <= 4e+226)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(l * V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-298], t$95$1, If[LessEqual[t$95$0, 4e+226], t$95$0, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 9.99999999999999912e-299 or 3.99999999999999985e226 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 59.3%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. pow1/2 N/A

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

      7. clear-num N/A

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

      8. inv-pow N/A

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

      9. pow-pow N/A

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

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

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

      11. associate-/r* N/A

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

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

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

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

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

      14. metadata-eval 68.5%

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

   4. Applied egg-rr 68.5%

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

      1. /-lowering-/.f64 N/A

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

      2. associate-/l/ N/A

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

      3. associate-/r* N/A

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

      4. clear-num N/A

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

      5. inv-pow N/A

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

      6. pow-pow N/A

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

      7. metadata-eval N/A

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

      8. pow1/2 N/A

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

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

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

      10. div-inv N/A

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

      11. *-commutative N/A

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

      12. clear-num N/A

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

      13. associate-/r/ N/A

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

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

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

      15. /-lowering-/.f64 67.8%

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

   6. Applied egg-rr 67.8%

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

      1. clear-num N/A

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

      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 67.8%

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

   8. Applied egg-rr 67.8%

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

   if 9.99999999999999912e-299 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 3.99999999999999985e226

   1. Initial program 99.6%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}} \leq 10^{-298}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}} \leq 4 \cdot 10^{+226}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ t_1 := c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+157}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* l V))))) (t_1 (* c0 (sqrt (/ (/ A V) l)))))
   (if (<= t_0 0.0) t_1 (if (<= t_0 2e+157) t_0 t_1))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (l * V)));
	double t_1 = c0 * sqrt(((A / V) / l));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 2e+157) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (l * v)))
    t_1 = c0 * sqrt(((a / v) / l))
    if (t_0 <= 0.0d0) then
        tmp = t_1
    else if (t_0 <= 2d+157) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (l * V)));
	double t_1 = c0 * Math.sqrt(((A / V) / l));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 2e+157) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (l * V)))
	t_1 = c0 * math.sqrt(((A / V) / l))
	tmp = 0
	if t_0 <= 0.0:
		tmp = t_1
	elif t_0 <= 2e+157:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(l * V))))
	t_1 = Float64(c0 * sqrt(Float64(Float64(A / V) / l)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 2e+157)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (l * V)));
	t_1 = c0 * sqrt(((A / V) / l));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 2e+157)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(l * V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 2e+157], t$95$0, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 0.0 or 1.99999999999999997e157 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 60.7%

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

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

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

      2. associate-/r* N/A

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

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

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

      4. /-lowering-/.f64 69.5%

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

   4. Applied egg-rr 69.5%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 1.99999999999999997e157

   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 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}} \leq 2 \cdot 10^{+157}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -2 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{0 - A}}{\sqrt{\ell} \cdot \sqrt{0 - V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{{\left(\ell \cdot V\right)}^{0.5}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= A -2e-310)
   (* c0 (/ (sqrt (- 0.0 A)) (* (sqrt l) (sqrt (- 0.0 V)))))
   (* c0 (/ (sqrt A) (pow (* l V) 0.5)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -2e-310) {
		tmp = c0 * (sqrt((0.0 - A)) / (sqrt(l) * sqrt((0.0 - V))));
	} else {
		tmp = c0 * (sqrt(A) / pow((l * V), 0.5));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (a <= (-2d-310)) then
        tmp = c0 * (sqrt((0.0d0 - a)) / (sqrt(l) * sqrt((0.0d0 - v))))
    else
        tmp = c0 * (sqrt(a) / ((l * v) ** 0.5d0))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -2e-310) {
		tmp = c0 * (Math.sqrt((0.0 - A)) / (Math.sqrt(l) * Math.sqrt((0.0 - V))));
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.pow((l * V), 0.5));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if A <= -2e-310:
		tmp = c0 * (math.sqrt((0.0 - A)) / (math.sqrt(l) * math.sqrt((0.0 - V))))
	else:
		tmp = c0 * (math.sqrt(A) / math.pow((l * V), 0.5))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (A <= -2e-310)
		tmp = Float64(c0 * Float64(sqrt(Float64(0.0 - A)) / Float64(sqrt(l) * sqrt(Float64(0.0 - V)))));
	else
		tmp = Float64(c0 * Float64(sqrt(A) / (Float64(l * V) ^ 0.5)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (A <= -2e-310)
		tmp = c0 * (sqrt((0.0 - A)) / (sqrt(l) * sqrt((0.0 - V))));
	else
		tmp = c0 * (sqrt(A) / ((l * V) ^ 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[A, -2e-310], N[(c0 * N[(N[Sqrt[N[(0.0 - A), $MachinePrecision]], $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[N[(0.0 - V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Power[N[(l * V), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -1.999999999999994e-310

   1. Initial program 68.8%

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

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

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

      2. associate-/r* N/A

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

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

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

      4. /-lowering-/.f64 73.3%

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

   4. Applied egg-rr 73.3%

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

      1. sqrt-div N/A

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

      2. frac-2neg N/A

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

      3. sqrt-div N/A

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

      4. pow1/2 N/A

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

      5. associate-/l/ N/A

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

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

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

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

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

      8. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(0 - A\right)\right), \left({\color{blue}{\ell}}^{\frac{1}{2}} \cdot \sqrt{\mathsf{neg}\left(V\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, A\right)\right), \left({\color{blue}{\ell}}^{\frac{1}{2}} \cdot \sqrt{\mathsf{neg}\left(V\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, A\right)\right), \mathsf{*.f64}\left(\left({\ell}^{\frac{1}{2}}\right), \color{blue}{\left(\sqrt{\mathsf{neg}\left(V\right)}\right)}\right)\right)\right) \]

      11. pow1/2 N/A

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

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

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

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

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

      14. neg-sub0 N/A

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

      15. --lowering--.f64 37.7%

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

   6. Applied egg-rr 37.7%

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

   if -1.999999999999994e-310 < A

   1. Initial program 68.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. pow1/2 N/A

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

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

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

      6. *-lowering-*.f64 80.0%

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

   4. Applied egg-rr 80.0%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{{\left(V \cdot \ell\right)}^{0.5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -2 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{0 - A}}{\sqrt{\ell} \cdot \sqrt{0 - V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{{\left(\ell \cdot V\right)}^{0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \frac{{\left(0 - \frac{A}{V}\right)}^{0.5}}{\sqrt{0 - \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= l -5e-310)
   (* c0 (/ (pow (- 0.0 (/ A V)) 0.5) (sqrt (- 0.0 l))))
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= -5e-310) {
		tmp = c0 * (pow((0.0 - (A / V)), 0.5) / sqrt((0.0 - l)));
	} else {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= (-5d-310)) then
        tmp = c0 * (((0.0d0 - (a / v)) ** 0.5d0) / sqrt((0.0d0 - l)))
    else
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= -5e-310) {
		tmp = c0 * (Math.pow((0.0 - (A / V)), 0.5) / Math.sqrt((0.0 - l)));
	} else {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if l <= -5e-310:
		tmp = c0 * (math.pow((0.0 - (A / V)), 0.5) / math.sqrt((0.0 - l)))
	else:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (l <= -5e-310)
		tmp = Float64(c0 * Float64((Float64(0.0 - Float64(A / V)) ^ 0.5) / sqrt(Float64(0.0 - l))));
	else
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (l <= -5e-310)
		tmp = c0 * (((0.0 - (A / V)) ^ 0.5) / sqrt((0.0 - l)));
	else
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[l, -5e-310], N[(c0 * N[(N[Power[N[(0.0 - N[(A / V), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / N[Sqrt[N[(0.0 - l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 67.8%

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

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

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

      2. associate-/r* N/A

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

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

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

      4. /-lowering-/.f64 69.9%

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

   4. Applied egg-rr 69.9%

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

      1. frac-2neg N/A

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

      2. sqrt-div N/A

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

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

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

      4. pow1/2 N/A

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

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

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

      6. distribute-neg-frac2 N/A

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

      7. div-inv N/A

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

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

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

      9. distribute-frac-neg2 N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

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

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

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

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

      14. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(A, \mathsf{/.f64}\left(-1, V\right)\right), \frac{1}{2}\right), \mathsf{sqrt.f64}\left(\left(0 - \ell\right)\right)\right)\right) \]

      15. --lowering--.f64 83.4%

          \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(A, \mathsf{/.f64}\left(-1, V\right)\right), \frac{1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \ell\right)\right)\right)\right) \]

   6. Applied egg-rr 83.4%

      \[\leadsto c0 \cdot \color{blue}{\frac{{\left(A \cdot \frac{-1}{V}\right)}^{0.5}}{\sqrt{0 - \ell}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{-1}{V} \cdot A\right), \frac{1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\color{blue}{0}, \ell\right)\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{-1 \cdot A}{V}\right), \frac{1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\color{blue}{0}, \ell\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(-1 \cdot \frac{A}{V}\right), \frac{1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\color{blue}{0}, \ell\right)\right)\right)\right) \]

      4. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left(\frac{A}{V}\right)\right), \frac{1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\color{blue}{0}, \ell\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{neg.f64}\left(\left(\frac{A}{V}\right)\right), \frac{1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\color{blue}{0}, \ell\right)\right)\right)\right) \]

      6. /-lowering-/.f64 83.3%

         \[\leadsto \mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(A, V\right)\right), \frac{1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \ell\right)\right)\right)\right) \]

   8. Applied egg-rr 83.3%

      \[\leadsto c0 \cdot \frac{{\color{blue}{\left(-\frac{A}{V}\right)}}^{0.5}}{\sqrt{0 - \ell}} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 83.3%

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

   10. Applied egg-rr 83.3%

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

   if -4.999999999999985e-310 < l

   1. Initial program 69.6%

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

      1. associate-/r* N/A

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

      2. sqrt-div N/A

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

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

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

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

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

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

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

      6. sqrt-lowering-sqrt.f64 87.8%

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

   4. Applied egg-rr 87.8%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \frac{{\left(0 - \frac{A}{V}\right)}^{0.5}}{\sqrt{0 - \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 67.8%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. pow1/2 N/A

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

      7. clear-num N/A

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

      8. inv-pow N/A

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

      9. pow-pow N/A

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

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

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

      11. associate-/r* N/A

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

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

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

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

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

      14. metadata-eval 69.9%

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

   4. Applied egg-rr 69.9%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

      3. clear-num N/A

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

      4. inv-pow N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. pow1/2 N/A

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

      8. associate-/r/ N/A

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

      9. associate-*l/ N/A

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

      10. sqrt-div N/A

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

      11. unpow1/2 N/A

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

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

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

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

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

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

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

      15. unpow1/2 N/A

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

      16. sqrt-lowering-sqrt.f64 30.3%

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

   6. Applied egg-rr 30.3%

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

   if -4.999999999999985e-310 < l

   1. Initial program 69.6%

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

      1. associate-/r* N/A

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

      2. sqrt-div N/A

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

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

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

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

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

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

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

      6. sqrt-lowering-sqrt.f64 87.8%

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

   4. Applied egg-rr 87.8%

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

4. Final simplification 54.8%

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

Alternative 9: 82.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{c0}{{\left(\frac{\frac{A}{V}}{\ell}\right)}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= l -5e-310)
   (/ c0 (pow (/ (/ A V) l) -0.5))
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= -5e-310) {
		tmp = c0 / pow(((A / V) / l), -0.5);
	} else {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= (-5d-310)) then
        tmp = c0 / (((a / v) / l) ** (-0.5d0))
    else
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= -5e-310) {
		tmp = c0 / Math.pow(((A / V) / l), -0.5);
	} else {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if l <= -5e-310:
		tmp = c0 / math.pow(((A / V) / l), -0.5)
	else:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (l <= -5e-310)
		tmp = Float64(c0 / (Float64(Float64(A / V) / l) ^ -0.5));
	else
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (l <= -5e-310)
		tmp = c0 / (((A / V) / l) ^ -0.5);
	else
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[l, -5e-310], N[(c0 / N[Power[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 67.8%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. pow1/2 N/A

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

      7. clear-num N/A

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

      8. inv-pow N/A

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

      9. pow-pow N/A

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

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

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

      11. associate-/r* N/A

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

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

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

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

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

      14. metadata-eval 69.9%

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

   4. Applied egg-rr 69.9%

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

   if -4.999999999999985e-310 < l

   1. Initial program 69.6%

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

      1. associate-/r* N/A

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

      2. sqrt-div N/A

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

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

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

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

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

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

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

      6. sqrt-lowering-sqrt.f64 87.8%

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

   4. Applied egg-rr 87.8%

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

Alternative 10: 73.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.6%

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

3. Final simplification 68.6%

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

Reproduce

herbie shell --seed 2024192 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))