Henrywood and Agarwal, Equation (3)

Percentage Accurate: 74.5% → 91.3%

Time: 7.5s

Alternatives: 14

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{-A}\\ \mathbf{if}\;\ell \cdot V \leq -2 \cdot 10^{+273}:\\ \;\;\;\;\frac{c0 \cdot t\_0}{\sqrt{\ell} \cdot \sqrt{-V}}\\ \mathbf{elif}\;\ell \cdot V \leq -5 \cdot 10^{-250}:\\ \;\;\;\;\frac{t\_0}{\sqrt{\left(-\ell\right) \cdot V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{-292}:\\ \;\;\;\;\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{+291}:\\ \;\;\;\;\left(\sqrt{A} \cdot \sqrt{\frac{1}{\ell \cdot V}}\right) \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{-1}{V} \cdot A}{-\ell}} \cdot c0\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (sqrt (- A))))
   (if (<= (* l V) -2e+273)
     (/ (* c0 t_0) (* (sqrt l) (sqrt (- V))))
     (if (<= (* l V) -5e-250)
       (* (/ t_0 (sqrt (* (- l) V))) c0)
       (if (<= (* l V) 5e-292)
         (* (/ (sqrt (/ A V)) (sqrt l)) c0)
         (if (<= (* l V) 5e+291)
           (* (* (sqrt A) (sqrt (/ 1.0 (* l V)))) c0)
           (* (sqrt (/ (* (/ -1.0 V) A) (- l))) c0)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt(-A);
	double tmp;
	if ((l * V) <= -2e+273) {
		tmp = (c0 * t_0) / (sqrt(l) * sqrt(-V));
	} else if ((l * V) <= -5e-250) {
		tmp = (t_0 / sqrt((-l * V))) * c0;
	} else if ((l * V) <= 5e-292) {
		tmp = (sqrt((A / V)) / sqrt(l)) * c0;
	} else if ((l * V) <= 5e+291) {
		tmp = (sqrt(A) * sqrt((1.0 / (l * V)))) * c0;
	} else {
		tmp = sqrt((((-1.0 / V) * A) / -l)) * c0;
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(-a)
    if ((l * v) <= (-2d+273)) then
        tmp = (c0 * t_0) / (sqrt(l) * sqrt(-v))
    else if ((l * v) <= (-5d-250)) then
        tmp = (t_0 / sqrt((-l * v))) * c0
    else if ((l * v) <= 5d-292) then
        tmp = (sqrt((a / v)) / sqrt(l)) * c0
    else if ((l * v) <= 5d+291) then
        tmp = (sqrt(a) * sqrt((1.0d0 / (l * v)))) * c0
    else
        tmp = sqrt(((((-1.0d0) / v) * a) / -l)) * c0
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = Math.sqrt(-A);
	double tmp;
	if ((l * V) <= -2e+273) {
		tmp = (c0 * t_0) / (Math.sqrt(l) * Math.sqrt(-V));
	} else if ((l * V) <= -5e-250) {
		tmp = (t_0 / Math.sqrt((-l * V))) * c0;
	} else if ((l * V) <= 5e-292) {
		tmp = (Math.sqrt((A / V)) / Math.sqrt(l)) * c0;
	} else if ((l * V) <= 5e+291) {
		tmp = (Math.sqrt(A) * Math.sqrt((1.0 / (l * V)))) * c0;
	} else {
		tmp = Math.sqrt((((-1.0 / V) * A) / -l)) * c0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt(-A)
	tmp = 0
	if (l * V) <= -2e+273:
		tmp = (c0 * t_0) / (math.sqrt(l) * math.sqrt(-V))
	elif (l * V) <= -5e-250:
		tmp = (t_0 / math.sqrt((-l * V))) * c0
	elif (l * V) <= 5e-292:
		tmp = (math.sqrt((A / V)) / math.sqrt(l)) * c0
	elif (l * V) <= 5e+291:
		tmp = (math.sqrt(A) * math.sqrt((1.0 / (l * V)))) * c0
	else:
		tmp = math.sqrt((((-1.0 / V) * A) / -l)) * c0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = sqrt(Float64(-A))
	tmp = 0.0
	if (Float64(l * V) <= -2e+273)
		tmp = Float64(Float64(c0 * t_0) / Float64(sqrt(l) * sqrt(Float64(-V))));
	elseif (Float64(l * V) <= -5e-250)
		tmp = Float64(Float64(t_0 / sqrt(Float64(Float64(-l) * V))) * c0);
	elseif (Float64(l * V) <= 5e-292)
		tmp = Float64(Float64(sqrt(Float64(A / V)) / sqrt(l)) * c0);
	elseif (Float64(l * V) <= 5e+291)
		tmp = Float64(Float64(sqrt(A) * sqrt(Float64(1.0 / Float64(l * V)))) * c0);
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(-1.0 / V) * A) / Float64(-l))) * c0);
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = sqrt(-A);
	tmp = 0.0;
	if ((l * V) <= -2e+273)
		tmp = (c0 * t_0) / (sqrt(l) * sqrt(-V));
	elseif ((l * V) <= -5e-250)
		tmp = (t_0 / sqrt((-l * V))) * c0;
	elseif ((l * V) <= 5e-292)
		tmp = (sqrt((A / V)) / sqrt(l)) * c0;
	elseif ((l * V) <= 5e+291)
		tmp = (sqrt(A) * sqrt((1.0 / (l * V)))) * c0;
	else
		tmp = sqrt((((-1.0 / V) * A) / -l)) * c0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[Sqrt[(-A)], $MachinePrecision]}, If[LessEqual[N[(l * V), $MachinePrecision], -2e+273], N[(N[(c0 * t$95$0), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], -5e-250], N[(N[(t$95$0 / N[Sqrt[N[((-l) * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 5e-292], N[(N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 5e+291], N[(N[(N[Sqrt[A], $MachinePrecision] * N[Sqrt[N[(1.0 / N[(l * V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(-1.0 / V), $MachinePrecision] * A), $MachinePrecision] / (-l)), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -1.99999999999999989e273

   1. Initial program 32.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 57.5

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

   4. Applied rewrites 57.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-neg.f64 57.5

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

   6. Applied rewrites 57.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. sqrt-div N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. sqrt-div N/A

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

      10. neg-mul-1 N/A

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

      11. lift-neg.f64 N/A

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

      12. remove-double-neg N/A

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

      13. sqrt-div N/A

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

      14. lift-/.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. associate-*l/ N/A

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

      18. associate-*r/ N/A

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

   8. Applied rewrites 47.0%

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

   if -1.99999999999999989e273 < (*.f64 V l) < -5.00000000000000027e-250

   1. Initial program 83.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 74.9

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

   4. Applied rewrites 74.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. div-inv N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-2neg N/A

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

      10. sqrt-div N/A

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

      11. frac-times N/A

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

      12. metadata-eval N/A

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

      13. sqrt-prod N/A

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

      14. *-rgt-identity N/A

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

      15. lower-/.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. lift-sqrt.f64 N/A

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

      19. sqrt-prod N/A

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

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

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

      21. lower-sqrt.f64 N/A

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

      22. *-commutative N/A

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

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

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

      24. lower-*.f64 N/A

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

      25. lower-neg.f64 99.5

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

   6. Applied rewrites 99.5%

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

   if -5.00000000000000027e-250 < (*.f64 V l) < 4.99999999999999981e-292

   1. Initial program 43.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. sqrt-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 53.5

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

   4. Applied rewrites 53.5%

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

   if 4.99999999999999981e-292 < (*.f64 V l) < 5.0000000000000001e291

   1. Initial program 83.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 74.9

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

   4. Applied rewrites 74.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-neg.f64 74.9

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

   6. Applied rewrites 74.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. neg-mul-1 N/A

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

      10. lift-neg.f64 N/A

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

      11. remove-double-neg N/A

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

      12. associate-/r/ N/A

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

      13. associate-*l/ N/A

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

      14. sqrt-div N/A

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

      15. associate-/r/ N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-sqrt.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-sqrt.f64 99.3

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

   8. Applied rewrites 99.3%

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

   9. Taylor expanded in V around 0

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

       1. lower-sqrt.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 99.3

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

   11. Applied rewrites 99.3%

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

   if 5.0000000000000001e291 < (*.f64 V l)

   1. Initial program 44.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 69.0

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

   4. Applied rewrites 69.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-neg.f64 69.1

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

   6. Applied rewrites 69.1%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -2 \cdot 10^{+273}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{-A}}{\sqrt{\ell} \cdot \sqrt{-V}}\\ \mathbf{elif}\;\ell \cdot V \leq -5 \cdot 10^{-250}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\left(-\ell\right) \cdot V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{-292}:\\ \;\;\;\;\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{+291}:\\ \;\;\;\;\left(\sqrt{A} \cdot \sqrt{\frac{1}{\ell \cdot V}}\right) \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{-1}{V} \cdot A}{-\ell}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\frac{-A}{\ell}} \cdot c0}{\sqrt{-V}}\\ \mathbf{elif}\;\ell \cdot V \leq -5 \cdot 10^{-250}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\left(-\ell\right) \cdot V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{-292}:\\ \;\;\;\;\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{+291}:\\ \;\;\;\;\left(\sqrt{A} \cdot \sqrt{\frac{1}{\ell \cdot V}}\right) \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{-1}{V} \cdot A}{-\ell}} \cdot c0\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* l V) (- INFINITY))
   (/ (* (sqrt (/ (- A) l)) c0) (sqrt (- V)))
   (if (<= (* l V) -5e-250)
     (* (/ (sqrt (- A)) (sqrt (* (- l) V))) c0)
     (if (<= (* l V) 5e-292)
       (* (/ (sqrt (/ A V)) (sqrt l)) c0)
       (if (<= (* l V) 5e+291)
         (* (* (sqrt A) (sqrt (/ 1.0 (* l V)))) c0)
         (* (sqrt (/ (* (/ -1.0 V) A) (- l))) c0))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -((double) INFINITY)) {
		tmp = (sqrt((-A / l)) * c0) / sqrt(-V);
	} else if ((l * V) <= -5e-250) {
		tmp = (sqrt(-A) / sqrt((-l * V))) * c0;
	} else if ((l * V) <= 5e-292) {
		tmp = (sqrt((A / V)) / sqrt(l)) * c0;
	} else if ((l * V) <= 5e+291) {
		tmp = (sqrt(A) * sqrt((1.0 / (l * V)))) * c0;
	} else {
		tmp = sqrt((((-1.0 / V) * A) / -l)) * c0;
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -Double.POSITIVE_INFINITY) {
		tmp = (Math.sqrt((-A / l)) * c0) / Math.sqrt(-V);
	} else if ((l * V) <= -5e-250) {
		tmp = (Math.sqrt(-A) / Math.sqrt((-l * V))) * c0;
	} else if ((l * V) <= 5e-292) {
		tmp = (Math.sqrt((A / V)) / Math.sqrt(l)) * c0;
	} else if ((l * V) <= 5e+291) {
		tmp = (Math.sqrt(A) * Math.sqrt((1.0 / (l * V)))) * c0;
	} else {
		tmp = Math.sqrt((((-1.0 / V) * A) / -l)) * c0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (l * V) <= -math.inf:
		tmp = (math.sqrt((-A / l)) * c0) / math.sqrt(-V)
	elif (l * V) <= -5e-250:
		tmp = (math.sqrt(-A) / math.sqrt((-l * V))) * c0
	elif (l * V) <= 5e-292:
		tmp = (math.sqrt((A / V)) / math.sqrt(l)) * c0
	elif (l * V) <= 5e+291:
		tmp = (math.sqrt(A) * math.sqrt((1.0 / (l * V)))) * c0
	else:
		tmp = math.sqrt((((-1.0 / V) * A) / -l)) * c0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(l * V) <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(Float64(Float64(-A) / l)) * c0) / sqrt(Float64(-V)));
	elseif (Float64(l * V) <= -5e-250)
		tmp = Float64(Float64(sqrt(Float64(-A)) / sqrt(Float64(Float64(-l) * V))) * c0);
	elseif (Float64(l * V) <= 5e-292)
		tmp = Float64(Float64(sqrt(Float64(A / V)) / sqrt(l)) * c0);
	elseif (Float64(l * V) <= 5e+291)
		tmp = Float64(Float64(sqrt(A) * sqrt(Float64(1.0 / Float64(l * V)))) * c0);
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(-1.0 / V) * A) / Float64(-l))) * c0);
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((l * V) <= -Inf)
		tmp = (sqrt((-A / l)) * c0) / sqrt(-V);
	elseif ((l * V) <= -5e-250)
		tmp = (sqrt(-A) / sqrt((-l * V))) * c0;
	elseif ((l * V) <= 5e-292)
		tmp = (sqrt((A / V)) / sqrt(l)) * c0;
	elseif ((l * V) <= 5e+291)
		tmp = (sqrt(A) * sqrt((1.0 / (l * V)))) * c0;
	else
		tmp = sqrt((((-1.0 / V) * A) / -l)) * c0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(l * V), $MachinePrecision], (-Infinity)], N[(N[(N[Sqrt[N[((-A) / l), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], -5e-250], N[(N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[((-l) * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 5e-292], N[(N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 5e+291], N[(N[(N[Sqrt[A], $MachinePrecision] * N[Sqrt[N[(1.0 / N[(l * V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(-1.0 / V), $MachinePrecision] * A), $MachinePrecision] / (-l)), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 27.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 57.5

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

   4. Applied rewrites 57.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. sqrt-div N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. div-inv N/A

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

      9. div-inv N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. sqrt-div N/A

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

      13. div-inv N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. div-inv N/A

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

      17. lift-/.f64 N/A

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

      18. frac-2neg N/A

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

      19. sqrt-div N/A

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

      20. associate-*l/ N/A

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

      21. lower-/.f64 N/A

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

   6. Applied rewrites 43.9%

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

   if -inf.0 < (*.f64 V l) < -5.00000000000000027e-250

   1. Initial program 82.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 74.3

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

   4. Applied rewrites 74.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. div-inv N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-2neg N/A

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

      10. sqrt-div N/A

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

      11. frac-times N/A

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

      12. metadata-eval N/A

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

      13. sqrt-prod N/A

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

      14. *-rgt-identity N/A

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

      15. lower-/.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. lift-sqrt.f64 N/A

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

      19. sqrt-prod N/A

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

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

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

      21. lower-sqrt.f64 N/A

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

      22. *-commutative N/A

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

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

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

      24. lower-*.f64 N/A

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

      25. lower-neg.f64 99.4

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

   6. Applied rewrites 99.4%

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

   if -5.00000000000000027e-250 < (*.f64 V l) < 4.99999999999999981e-292

   1. Initial program 43.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. sqrt-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 53.5

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

   4. Applied rewrites 53.5%

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

   if 4.99999999999999981e-292 < (*.f64 V l) < 5.0000000000000001e291

   1. Initial program 83.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 74.9

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

   4. Applied rewrites 74.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-neg.f64 74.9

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

   6. Applied rewrites 74.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. neg-mul-1 N/A

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

      10. lift-neg.f64 N/A

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

      11. remove-double-neg N/A

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

      12. associate-/r/ N/A

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

      13. associate-*l/ N/A

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

      14. sqrt-div N/A

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

      15. associate-/r/ N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-sqrt.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-sqrt.f64 99.3

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

   8. Applied rewrites 99.3%

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

   9. Taylor expanded in V around 0

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

       1. lower-sqrt.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 99.3

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

   11. Applied rewrites 99.3%

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

   if 5.0000000000000001e291 < (*.f64 V l)

   1. Initial program 44.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 69.0

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

   4. Applied rewrites 69.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-neg.f64 69.1

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

   6. Applied rewrites 69.1%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\frac{-A}{\ell}} \cdot c0}{\sqrt{-V}}\\ \mathbf{elif}\;\ell \cdot V \leq -5 \cdot 10^{-250}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\left(-\ell\right) \cdot V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{-292}:\\ \;\;\;\;\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{+291}:\\ \;\;\;\;\left(\sqrt{A} \cdot \sqrt{\frac{1}{\ell \cdot V}}\right) \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{-1}{V} \cdot A}{-\ell}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -5 \cdot 10^{+300}:\\ \;\;\;\;\frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq -5 \cdot 10^{-250}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\left(-\ell\right) \cdot V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{-292}:\\ \;\;\;\;\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{+291}:\\ \;\;\;\;\left(\sqrt{A} \cdot \sqrt{\frac{1}{\ell \cdot V}}\right) \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{-1}{V} \cdot A}{-\ell}} \cdot c0\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* l V) -5e+300)
   (* (/ (sqrt (/ (- A) l)) (sqrt (- V))) c0)
   (if (<= (* l V) -5e-250)
     (* (/ (sqrt (- A)) (sqrt (* (- l) V))) c0)
     (if (<= (* l V) 5e-292)
       (* (/ (sqrt (/ A V)) (sqrt l)) c0)
       (if (<= (* l V) 5e+291)
         (* (* (sqrt A) (sqrt (/ 1.0 (* l V)))) c0)
         (* (sqrt (/ (* (/ -1.0 V) A) (- l))) c0))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -5e+300) {
		tmp = (sqrt((-A / l)) / sqrt(-V)) * c0;
	} else if ((l * V) <= -5e-250) {
		tmp = (sqrt(-A) / sqrt((-l * V))) * c0;
	} else if ((l * V) <= 5e-292) {
		tmp = (sqrt((A / V)) / sqrt(l)) * c0;
	} else if ((l * V) <= 5e+291) {
		tmp = (sqrt(A) * sqrt((1.0 / (l * V)))) * c0;
	} else {
		tmp = sqrt((((-1.0 / V) * A) / -l)) * c0;
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((l * v) <= (-5d+300)) then
        tmp = (sqrt((-a / l)) / sqrt(-v)) * c0
    else if ((l * v) <= (-5d-250)) then
        tmp = (sqrt(-a) / sqrt((-l * v))) * c0
    else if ((l * v) <= 5d-292) then
        tmp = (sqrt((a / v)) / sqrt(l)) * c0
    else if ((l * v) <= 5d+291) then
        tmp = (sqrt(a) * sqrt((1.0d0 / (l * v)))) * c0
    else
        tmp = sqrt(((((-1.0d0) / v) * a) / -l)) * c0
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -5e+300) {
		tmp = (Math.sqrt((-A / l)) / Math.sqrt(-V)) * c0;
	} else if ((l * V) <= -5e-250) {
		tmp = (Math.sqrt(-A) / Math.sqrt((-l * V))) * c0;
	} else if ((l * V) <= 5e-292) {
		tmp = (Math.sqrt((A / V)) / Math.sqrt(l)) * c0;
	} else if ((l * V) <= 5e+291) {
		tmp = (Math.sqrt(A) * Math.sqrt((1.0 / (l * V)))) * c0;
	} else {
		tmp = Math.sqrt((((-1.0 / V) * A) / -l)) * c0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (l * V) <= -5e+300:
		tmp = (math.sqrt((-A / l)) / math.sqrt(-V)) * c0
	elif (l * V) <= -5e-250:
		tmp = (math.sqrt(-A) / math.sqrt((-l * V))) * c0
	elif (l * V) <= 5e-292:
		tmp = (math.sqrt((A / V)) / math.sqrt(l)) * c0
	elif (l * V) <= 5e+291:
		tmp = (math.sqrt(A) * math.sqrt((1.0 / (l * V)))) * c0
	else:
		tmp = math.sqrt((((-1.0 / V) * A) / -l)) * c0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(l * V) <= -5e+300)
		tmp = Float64(Float64(sqrt(Float64(Float64(-A) / l)) / sqrt(Float64(-V))) * c0);
	elseif (Float64(l * V) <= -5e-250)
		tmp = Float64(Float64(sqrt(Float64(-A)) / sqrt(Float64(Float64(-l) * V))) * c0);
	elseif (Float64(l * V) <= 5e-292)
		tmp = Float64(Float64(sqrt(Float64(A / V)) / sqrt(l)) * c0);
	elseif (Float64(l * V) <= 5e+291)
		tmp = Float64(Float64(sqrt(A) * sqrt(Float64(1.0 / Float64(l * V)))) * c0);
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(-1.0 / V) * A) / Float64(-l))) * c0);
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((l * V) <= -5e+300)
		tmp = (sqrt((-A / l)) / sqrt(-V)) * c0;
	elseif ((l * V) <= -5e-250)
		tmp = (sqrt(-A) / sqrt((-l * V))) * c0;
	elseif ((l * V) <= 5e-292)
		tmp = (sqrt((A / V)) / sqrt(l)) * c0;
	elseif ((l * V) <= 5e+291)
		tmp = (sqrt(A) * sqrt((1.0 / (l * V)))) * c0;
	else
		tmp = sqrt((((-1.0 / V) * A) / -l)) * c0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(l * V), $MachinePrecision], -5e+300], N[(N[(N[Sqrt[N[((-A) / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], -5e-250], N[(N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[((-l) * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 5e-292], N[(N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 5e+291], N[(N[(N[Sqrt[A], $MachinePrecision] * N[Sqrt[N[(1.0 / N[(l * V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(-1.0 / V), $MachinePrecision] * A), $MachinePrecision] / (-l)), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -5.00000000000000026e300

   1. Initial program 30.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 58.2

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

   4. Applied rewrites 58.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-/l/ N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. sqrt-div N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. distribute-neg-frac N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-neg.f64 46.9

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

   6. Applied rewrites 46.9%

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

   if -5.00000000000000026e300 < (*.f64 V l) < -5.00000000000000027e-250

   1. Initial program 83.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 74.4

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

   4. Applied rewrites 74.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. div-inv N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-2neg N/A

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

      10. sqrt-div N/A

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

      11. frac-times N/A

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

      12. metadata-eval N/A

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

      13. sqrt-prod N/A

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

      14. *-rgt-identity N/A

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

      15. lower-/.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. lift-sqrt.f64 N/A

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

      19. sqrt-prod N/A

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

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

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

      21. lower-sqrt.f64 N/A

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

      22. *-commutative N/A

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

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

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

      24. lower-*.f64 N/A

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

      25. lower-neg.f64 99.5

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

   6. Applied rewrites 99.5%

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

   if -5.00000000000000027e-250 < (*.f64 V l) < 4.99999999999999981e-292

   1. Initial program 43.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. sqrt-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 53.5

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

   4. Applied rewrites 53.5%

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

   if 4.99999999999999981e-292 < (*.f64 V l) < 5.0000000000000001e291

   1. Initial program 83.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 74.9

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

   4. Applied rewrites 74.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-neg.f64 74.9

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

   6. Applied rewrites 74.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. neg-mul-1 N/A

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

      10. lift-neg.f64 N/A

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

      11. remove-double-neg N/A

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

      12. associate-/r/ N/A

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

      13. associate-*l/ N/A

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

      14. sqrt-div N/A

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

      15. associate-/r/ N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-sqrt.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-sqrt.f64 99.3

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

   8. Applied rewrites 99.3%

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

   9. Taylor expanded in V around 0

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

       1. lower-sqrt.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 99.3

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

   11. Applied rewrites 99.3%

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

   if 5.0000000000000001e291 < (*.f64 V l)

   1. Initial program 44.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 69.0

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

   4. Applied rewrites 69.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-neg.f64 69.1

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

   6. Applied rewrites 69.1%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -5 \cdot 10^{+300}:\\ \;\;\;\;\frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq -5 \cdot 10^{-250}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\left(-\ell\right) \cdot V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{-292}:\\ \;\;\;\;\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{+291}:\\ \;\;\;\;\left(\sqrt{A} \cdot \sqrt{\frac{1}{\ell \cdot V}}\right) \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{-1}{V} \cdot A}{-\ell}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{V}}\\ \mathbf{if}\;\ell \cdot V \leq -5 \cdot 10^{+300}:\\ \;\;\;\;\frac{t\_0}{\sqrt{\ell}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq -5 \cdot 10^{-303}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\left(-\ell\right) \cdot V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 10^{-293}:\\ \;\;\;\;\frac{t\_0 \cdot c0}{\sqrt{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{+291}:\\ \;\;\;\;\left(\sqrt{A} \cdot \sqrt{\frac{1}{\ell \cdot V}}\right) \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{-1}{V} \cdot A}{-\ell}} \cdot c0\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (sqrt (/ A V))))
   (if (<= (* l V) -5e+300)
     (* (/ t_0 (sqrt l)) c0)
     (if (<= (* l V) -5e-303)
       (* (/ (sqrt (- A)) (sqrt (* (- l) V))) c0)
       (if (<= (* l V) 1e-293)
         (/ (* t_0 c0) (sqrt l))
         (if (<= (* l V) 5e+291)
           (* (* (sqrt A) (sqrt (/ 1.0 (* l V)))) c0)
           (* (sqrt (/ (* (/ -1.0 V) A) (- l))) c0)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt((A / V));
	double tmp;
	if ((l * V) <= -5e+300) {
		tmp = (t_0 / sqrt(l)) * c0;
	} else if ((l * V) <= -5e-303) {
		tmp = (sqrt(-A) / sqrt((-l * V))) * c0;
	} else if ((l * V) <= 1e-293) {
		tmp = (t_0 * c0) / sqrt(l);
	} else if ((l * V) <= 5e+291) {
		tmp = (sqrt(A) * sqrt((1.0 / (l * V)))) * c0;
	} else {
		tmp = sqrt((((-1.0 / V) * A) / -l)) * c0;
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((a / v))
    if ((l * v) <= (-5d+300)) then
        tmp = (t_0 / sqrt(l)) * c0
    else if ((l * v) <= (-5d-303)) then
        tmp = (sqrt(-a) / sqrt((-l * v))) * c0
    else if ((l * v) <= 1d-293) then
        tmp = (t_0 * c0) / sqrt(l)
    else if ((l * v) <= 5d+291) then
        tmp = (sqrt(a) * sqrt((1.0d0 / (l * v)))) * c0
    else
        tmp = sqrt(((((-1.0d0) / v) * a) / -l)) * c0
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = Math.sqrt((A / V));
	double tmp;
	if ((l * V) <= -5e+300) {
		tmp = (t_0 / Math.sqrt(l)) * c0;
	} else if ((l * V) <= -5e-303) {
		tmp = (Math.sqrt(-A) / Math.sqrt((-l * V))) * c0;
	} else if ((l * V) <= 1e-293) {
		tmp = (t_0 * c0) / Math.sqrt(l);
	} else if ((l * V) <= 5e+291) {
		tmp = (Math.sqrt(A) * Math.sqrt((1.0 / (l * V)))) * c0;
	} else {
		tmp = Math.sqrt((((-1.0 / V) * A) / -l)) * c0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt((A / V))
	tmp = 0
	if (l * V) <= -5e+300:
		tmp = (t_0 / math.sqrt(l)) * c0
	elif (l * V) <= -5e-303:
		tmp = (math.sqrt(-A) / math.sqrt((-l * V))) * c0
	elif (l * V) <= 1e-293:
		tmp = (t_0 * c0) / math.sqrt(l)
	elif (l * V) <= 5e+291:
		tmp = (math.sqrt(A) * math.sqrt((1.0 / (l * V)))) * c0
	else:
		tmp = math.sqrt((((-1.0 / V) * A) / -l)) * c0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = sqrt(Float64(A / V))
	tmp = 0.0
	if (Float64(l * V) <= -5e+300)
		tmp = Float64(Float64(t_0 / sqrt(l)) * c0);
	elseif (Float64(l * V) <= -5e-303)
		tmp = Float64(Float64(sqrt(Float64(-A)) / sqrt(Float64(Float64(-l) * V))) * c0);
	elseif (Float64(l * V) <= 1e-293)
		tmp = Float64(Float64(t_0 * c0) / sqrt(l));
	elseif (Float64(l * V) <= 5e+291)
		tmp = Float64(Float64(sqrt(A) * sqrt(Float64(1.0 / Float64(l * V)))) * c0);
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(-1.0 / V) * A) / Float64(-l))) * c0);
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = sqrt((A / V));
	tmp = 0.0;
	if ((l * V) <= -5e+300)
		tmp = (t_0 / sqrt(l)) * c0;
	elseif ((l * V) <= -5e-303)
		tmp = (sqrt(-A) / sqrt((-l * V))) * c0;
	elseif ((l * V) <= 1e-293)
		tmp = (t_0 * c0) / sqrt(l);
	elseif ((l * V) <= 5e+291)
		tmp = (sqrt(A) * sqrt((1.0 / (l * V)))) * c0;
	else
		tmp = sqrt((((-1.0 / V) * A) / -l)) * c0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(l * V), $MachinePrecision], -5e+300], N[(N[(t$95$0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], -5e-303], N[(N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[((-l) * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 1e-293], N[(N[(t$95$0 * c0), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 5e+291], N[(N[(N[Sqrt[A], $MachinePrecision] * N[Sqrt[N[(1.0 / N[(l * V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(-1.0 / V), $MachinePrecision] * A), $MachinePrecision] / (-l)), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -5.00000000000000026e300

   1. Initial program 30.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. sqrt-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 41.4

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

   4. Applied rewrites 41.4%

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

   if -5.00000000000000026e300 < (*.f64 V l) < -4.9999999999999998e-303

   1. Initial program 83.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 75.0

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

   4. Applied rewrites 75.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. div-inv N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-2neg N/A

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

      10. sqrt-div N/A

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

      11. frac-times N/A

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

      12. metadata-eval N/A

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

      13. sqrt-prod N/A

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

      14. *-rgt-identity N/A

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

      15. lower-/.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. lift-sqrt.f64 N/A

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

      19. sqrt-prod N/A

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

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

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

      21. lower-sqrt.f64 N/A

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

      22. *-commutative N/A

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

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

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

      24. lower-*.f64 N/A

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

      25. lower-neg.f64 99.5

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

   6. Applied rewrites 99.5%

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

   if -4.9999999999999998e-303 < (*.f64 V l) < 1.0000000000000001e-293

   1. Initial program 31.8%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. sqrt-div N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sqrt.f64 56.5

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

   4. Applied rewrites 56.5%

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

   if 1.0000000000000001e-293 < (*.f64 V l) < 5.0000000000000001e291

   1. Initial program 83.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 75.2

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

   4. Applied rewrites 75.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-neg.f64 75.2

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

   6. Applied rewrites 75.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. neg-mul-1 N/A

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

      10. lift-neg.f64 N/A

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

      11. remove-double-neg N/A

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

      12. associate-/r/ N/A

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

      13. associate-*l/ N/A

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

      14. sqrt-div N/A

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

      15. associate-/r/ N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-sqrt.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-sqrt.f64 99.3

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

   8. Applied rewrites 99.3%

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

   9. Taylor expanded in V around 0

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

       1. lower-sqrt.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 99.3

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

   11. Applied rewrites 99.3%

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

   if 5.0000000000000001e291 < (*.f64 V l)

   1. Initial program 44.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 69.0

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

   4. Applied rewrites 69.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-neg.f64 69.1

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

   6. Applied rewrites 69.1%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -5 \cdot 10^{+300}:\\ \;\;\;\;\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq -5 \cdot 10^{-303}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\left(-\ell\right) \cdot V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 10^{-293}:\\ \;\;\;\;\frac{\sqrt{\frac{A}{V}} \cdot c0}{\sqrt{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{+291}:\\ \;\;\;\;\left(\sqrt{A} \cdot \sqrt{\frac{1}{\ell \cdot V}}\right) \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{-1}{V} \cdot A}{-\ell}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}} \cdot c0\\ \mathbf{if}\;\ell \cdot V \leq -5 \cdot 10^{+300}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \cdot V \leq -5 \cdot 10^{-250}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\left(-\ell\right) \cdot V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{-292}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{+291}:\\ \;\;\;\;\left(\sqrt{A} \cdot \sqrt{\frac{1}{\ell \cdot V}}\right) \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{-1}{V} \cdot A}{-\ell}} \cdot c0\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* (/ (sqrt (/ A V)) (sqrt l)) c0)))
   (if (<= (* l V) -5e+300)
     t_0
     (if (<= (* l V) -5e-250)
       (* (/ (sqrt (- A)) (sqrt (* (- l) V))) c0)
       (if (<= (* l V) 5e-292)
         t_0
         (if (<= (* l V) 5e+291)
           (* (* (sqrt A) (sqrt (/ 1.0 (* l V)))) c0)
           (* (sqrt (/ (* (/ -1.0 V) A) (- l))) c0)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = (sqrt((A / V)) / sqrt(l)) * c0;
	double tmp;
	if ((l * V) <= -5e+300) {
		tmp = t_0;
	} else if ((l * V) <= -5e-250) {
		tmp = (sqrt(-A) / sqrt((-l * V))) * c0;
	} else if ((l * V) <= 5e-292) {
		tmp = t_0;
	} else if ((l * V) <= 5e+291) {
		tmp = (sqrt(A) * sqrt((1.0 / (l * V)))) * c0;
	} else {
		tmp = sqrt((((-1.0 / V) * A) / -l)) * c0;
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt((a / v)) / sqrt(l)) * c0
    if ((l * v) <= (-5d+300)) then
        tmp = t_0
    else if ((l * v) <= (-5d-250)) then
        tmp = (sqrt(-a) / sqrt((-l * v))) * c0
    else if ((l * v) <= 5d-292) then
        tmp = t_0
    else if ((l * v) <= 5d+291) then
        tmp = (sqrt(a) * sqrt((1.0d0 / (l * v)))) * c0
    else
        tmp = sqrt(((((-1.0d0) / v) * a) / -l)) * c0
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = (Math.sqrt((A / V)) / Math.sqrt(l)) * c0;
	double tmp;
	if ((l * V) <= -5e+300) {
		tmp = t_0;
	} else if ((l * V) <= -5e-250) {
		tmp = (Math.sqrt(-A) / Math.sqrt((-l * V))) * c0;
	} else if ((l * V) <= 5e-292) {
		tmp = t_0;
	} else if ((l * V) <= 5e+291) {
		tmp = (Math.sqrt(A) * Math.sqrt((1.0 / (l * V)))) * c0;
	} else {
		tmp = Math.sqrt((((-1.0 / V) * A) / -l)) * c0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = (math.sqrt((A / V)) / math.sqrt(l)) * c0
	tmp = 0
	if (l * V) <= -5e+300:
		tmp = t_0
	elif (l * V) <= -5e-250:
		tmp = (math.sqrt(-A) / math.sqrt((-l * V))) * c0
	elif (l * V) <= 5e-292:
		tmp = t_0
	elif (l * V) <= 5e+291:
		tmp = (math.sqrt(A) * math.sqrt((1.0 / (l * V)))) * c0
	else:
		tmp = math.sqrt((((-1.0 / V) * A) / -l)) * c0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(Float64(sqrt(Float64(A / V)) / sqrt(l)) * c0)
	tmp = 0.0
	if (Float64(l * V) <= -5e+300)
		tmp = t_0;
	elseif (Float64(l * V) <= -5e-250)
		tmp = Float64(Float64(sqrt(Float64(-A)) / sqrt(Float64(Float64(-l) * V))) * c0);
	elseif (Float64(l * V) <= 5e-292)
		tmp = t_0;
	elseif (Float64(l * V) <= 5e+291)
		tmp = Float64(Float64(sqrt(A) * sqrt(Float64(1.0 / Float64(l * V)))) * c0);
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(-1.0 / V) * A) / Float64(-l))) * c0);
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = (sqrt((A / V)) / sqrt(l)) * c0;
	tmp = 0.0;
	if ((l * V) <= -5e+300)
		tmp = t_0;
	elseif ((l * V) <= -5e-250)
		tmp = (sqrt(-A) / sqrt((-l * V))) * c0;
	elseif ((l * V) <= 5e-292)
		tmp = t_0;
	elseif ((l * V) <= 5e+291)
		tmp = (sqrt(A) * sqrt((1.0 / (l * V)))) * c0;
	else
		tmp = sqrt((((-1.0 / V) * A) / -l)) * c0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision]}, If[LessEqual[N[(l * V), $MachinePrecision], -5e+300], t$95$0, If[LessEqual[N[(l * V), $MachinePrecision], -5e-250], N[(N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[((-l) * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 5e-292], t$95$0, If[LessEqual[N[(l * V), $MachinePrecision], 5e+291], N[(N[(N[Sqrt[A], $MachinePrecision] * N[Sqrt[N[(1.0 / N[(l * V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(-1.0 / V), $MachinePrecision] * A), $MachinePrecision] / (-l)), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -5.00000000000000026e300 or -5.00000000000000027e-250 < (*.f64 V l) < 4.99999999999999981e-292

   1. Initial program 38.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. sqrt-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 49.3

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

   4. Applied rewrites 49.3%

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

   if -5.00000000000000026e300 < (*.f64 V l) < -5.00000000000000027e-250

   1. Initial program 83.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 74.4

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

   4. Applied rewrites 74.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. div-inv N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-2neg N/A

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

      10. sqrt-div N/A

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

      11. frac-times N/A

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

      12. metadata-eval N/A

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

      13. sqrt-prod N/A

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

      14. *-rgt-identity N/A

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

      15. lower-/.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. lift-sqrt.f64 N/A

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

      19. sqrt-prod N/A

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

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

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

      21. lower-sqrt.f64 N/A

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

      22. *-commutative N/A

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

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

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

      24. lower-*.f64 N/A

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

      25. lower-neg.f64 99.5

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

   6. Applied rewrites 99.5%

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

   if 4.99999999999999981e-292 < (*.f64 V l) < 5.0000000000000001e291

   1. Initial program 83.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 74.9

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

   4. Applied rewrites 74.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-neg.f64 74.9

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

   6. Applied rewrites 74.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. neg-mul-1 N/A

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

      10. lift-neg.f64 N/A

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

      11. remove-double-neg N/A

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

      12. associate-/r/ N/A

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

      13. associate-*l/ N/A

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

      14. sqrt-div N/A

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

      15. associate-/r/ N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-sqrt.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-sqrt.f64 99.3

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

   8. Applied rewrites 99.3%

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

   9. Taylor expanded in V around 0

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

       1. lower-sqrt.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 99.3

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

   11. Applied rewrites 99.3%

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

   if 5.0000000000000001e291 < (*.f64 V l)

   1. Initial program 44.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 69.0

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

   4. Applied rewrites 69.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-neg.f64 69.1

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

   6. Applied rewrites 69.1%

      \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{-1}{V} \cdot \left(-A\right)}}{\ell}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -5 \cdot 10^{+300}:\\ \;\;\;\;\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq -5 \cdot 10^{-250}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\left(-\ell\right) \cdot V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{-292}:\\ \;\;\;\;\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{+291}:\\ \;\;\;\;\left(\sqrt{A} \cdot \sqrt{\frac{1}{\ell \cdot V}}\right) \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{-1}{V} \cdot A}{-\ell}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \cdot V \leq -5 \cdot 10^{-303}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\left(-\ell\right) \cdot V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 10^{-256}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{+291}:\\ \;\;\;\;\left(\frac{1}{\sqrt{\ell \cdot V}} \cdot \sqrt{A}\right) \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{-1}{V} \cdot A}{-\ell}} \cdot c0\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ c0 (sqrt (* (/ l A) V)))))
   (if (<= (* l V) (- INFINITY))
     t_0
     (if (<= (* l V) -5e-303)
       (* (/ (sqrt (- A)) (sqrt (* (- l) V))) c0)
       (if (<= (* l V) 1e-256)
         t_0
         (if (<= (* l V) 5e+291)
           (* (* (/ 1.0 (sqrt (* l V))) (sqrt A)) c0)
           (* (sqrt (/ (* (/ -1.0 V) A) (- l))) c0)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 / sqrt(((l / A) * V));
	double tmp;
	if ((l * V) <= -((double) INFINITY)) {
		tmp = t_0;
	} else if ((l * V) <= -5e-303) {
		tmp = (sqrt(-A) / sqrt((-l * V))) * c0;
	} else if ((l * V) <= 1e-256) {
		tmp = t_0;
	} else if ((l * V) <= 5e+291) {
		tmp = ((1.0 / sqrt((l * V))) * sqrt(A)) * c0;
	} else {
		tmp = sqrt((((-1.0 / V) * A) / -l)) * c0;
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 / Math.sqrt(((l / A) * V));
	double tmp;
	if ((l * V) <= -Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else if ((l * V) <= -5e-303) {
		tmp = (Math.sqrt(-A) / Math.sqrt((-l * V))) * c0;
	} else if ((l * V) <= 1e-256) {
		tmp = t_0;
	} else if ((l * V) <= 5e+291) {
		tmp = ((1.0 / Math.sqrt((l * V))) * Math.sqrt(A)) * c0;
	} else {
		tmp = Math.sqrt((((-1.0 / V) * A) / -l)) * c0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 / math.sqrt(((l / A) * V))
	tmp = 0
	if (l * V) <= -math.inf:
		tmp = t_0
	elif (l * V) <= -5e-303:
		tmp = (math.sqrt(-A) / math.sqrt((-l * V))) * c0
	elif (l * V) <= 1e-256:
		tmp = t_0
	elif (l * V) <= 5e+291:
		tmp = ((1.0 / math.sqrt((l * V))) * math.sqrt(A)) * c0
	else:
		tmp = math.sqrt((((-1.0 / V) * A) / -l)) * c0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 / sqrt(Float64(Float64(l / A) * V)))
	tmp = 0.0
	if (Float64(l * V) <= Float64(-Inf))
		tmp = t_0;
	elseif (Float64(l * V) <= -5e-303)
		tmp = Float64(Float64(sqrt(Float64(-A)) / sqrt(Float64(Float64(-l) * V))) * c0);
	elseif (Float64(l * V) <= 1e-256)
		tmp = t_0;
	elseif (Float64(l * V) <= 5e+291)
		tmp = Float64(Float64(Float64(1.0 / sqrt(Float64(l * V))) * sqrt(A)) * c0);
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(-1.0 / V) * A) / Float64(-l))) * c0);
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 / sqrt(((l / A) * V));
	tmp = 0.0;
	if ((l * V) <= -Inf)
		tmp = t_0;
	elseif ((l * V) <= -5e-303)
		tmp = (sqrt(-A) / sqrt((-l * V))) * c0;
	elseif ((l * V) <= 1e-256)
		tmp = t_0;
	elseif ((l * V) <= 5e+291)
		tmp = ((1.0 / sqrt((l * V))) * sqrt(A)) * c0;
	else
		tmp = sqrt((((-1.0 / V) * A) / -l)) * c0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 / N[Sqrt[N[(N[(l / A), $MachinePrecision] * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(l * V), $MachinePrecision], (-Infinity)], t$95$0, If[LessEqual[N[(l * V), $MachinePrecision], -5e-303], N[(N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[((-l) * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 1e-256], t$95$0, If[LessEqual[N[(l * V), $MachinePrecision], 5e+291], N[(N[(N[(1.0 / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[A], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(-1.0 / V), $MachinePrecision] * A), $MachinePrecision] / (-l)), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -inf.0 or -4.9999999999999998e-303 < (*.f64 V l) < 9.99999999999999977e-257

   1. Initial program 40.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 67.8

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

   4. Applied rewrites 67.8%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. sqrt-undiv N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-/r/ N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 67.9

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

   6. Applied rewrites 67.9%

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

   if -inf.0 < (*.f64 V l) < -4.9999999999999998e-303

   1. Initial program 82.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 74.9

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

   4. Applied rewrites 74.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. div-inv N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-2neg N/A

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

      10. sqrt-div N/A

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

      11. frac-times N/A

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

      12. metadata-eval N/A

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

      13. sqrt-prod N/A

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

      14. *-rgt-identity N/A

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

      15. lower-/.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. lift-sqrt.f64 N/A

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

      19. sqrt-prod N/A

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

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

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

      21. lower-sqrt.f64 N/A

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

      22. *-commutative N/A

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

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

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

      24. lower-*.f64 N/A

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

      25. lower-neg.f64 99.5

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

   6. Applied rewrites 99.5%

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

   if 9.99999999999999977e-257 < (*.f64 V l) < 5.0000000000000001e291

   1. Initial program 82.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 73.4

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

   4. Applied rewrites 73.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-neg.f64 73.3

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

   6. Applied rewrites 73.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. neg-mul-1 N/A

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

      10. lift-neg.f64 N/A

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

      11. remove-double-neg N/A

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

      12. associate-/r/ N/A

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

      13. associate-*l/ N/A

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

      14. sqrt-div N/A

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

      15. associate-/r/ N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-sqrt.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-sqrt.f64 99.3

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

   8. Applied rewrites 99.3%

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

   if 5.0000000000000001e291 < (*.f64 V l)

   1. Initial program 44.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 69.0

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

   4. Applied rewrites 69.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-neg.f64 69.1

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

   6. Applied rewrites 69.1%

      \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{-1}{V} \cdot \left(-A\right)}}{\ell}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \mathbf{elif}\;\ell \cdot V \leq -5 \cdot 10^{-303}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\left(-\ell\right) \cdot V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 10^{-256}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{+291}:\\ \;\;\;\;\left(\frac{1}{\sqrt{\ell \cdot V}} \cdot \sqrt{A}\right) \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{-1}{V} \cdot A}{-\ell}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \cdot V \leq -5 \cdot 10^{-303}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\left(-\ell\right) \cdot V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 10^{-256}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{+291}:\\ \;\;\;\;\left(\sqrt{A} \cdot \sqrt{\frac{1}{\ell \cdot V}}\right) \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{-1}{V} \cdot A}{-\ell}} \cdot c0\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ c0 (sqrt (* (/ l A) V)))))
   (if (<= (* l V) (- INFINITY))
     t_0
     (if (<= (* l V) -5e-303)
       (* (/ (sqrt (- A)) (sqrt (* (- l) V))) c0)
       (if (<= (* l V) 1e-256)
         t_0
         (if (<= (* l V) 5e+291)
           (* (* (sqrt A) (sqrt (/ 1.0 (* l V)))) c0)
           (* (sqrt (/ (* (/ -1.0 V) A) (- l))) c0)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 / sqrt(((l / A) * V));
	double tmp;
	if ((l * V) <= -((double) INFINITY)) {
		tmp = t_0;
	} else if ((l * V) <= -5e-303) {
		tmp = (sqrt(-A) / sqrt((-l * V))) * c0;
	} else if ((l * V) <= 1e-256) {
		tmp = t_0;
	} else if ((l * V) <= 5e+291) {
		tmp = (sqrt(A) * sqrt((1.0 / (l * V)))) * c0;
	} else {
		tmp = sqrt((((-1.0 / V) * A) / -l)) * c0;
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 / Math.sqrt(((l / A) * V));
	double tmp;
	if ((l * V) <= -Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else if ((l * V) <= -5e-303) {
		tmp = (Math.sqrt(-A) / Math.sqrt((-l * V))) * c0;
	} else if ((l * V) <= 1e-256) {
		tmp = t_0;
	} else if ((l * V) <= 5e+291) {
		tmp = (Math.sqrt(A) * Math.sqrt((1.0 / (l * V)))) * c0;
	} else {
		tmp = Math.sqrt((((-1.0 / V) * A) / -l)) * c0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 / math.sqrt(((l / A) * V))
	tmp = 0
	if (l * V) <= -math.inf:
		tmp = t_0
	elif (l * V) <= -5e-303:
		tmp = (math.sqrt(-A) / math.sqrt((-l * V))) * c0
	elif (l * V) <= 1e-256:
		tmp = t_0
	elif (l * V) <= 5e+291:
		tmp = (math.sqrt(A) * math.sqrt((1.0 / (l * V)))) * c0
	else:
		tmp = math.sqrt((((-1.0 / V) * A) / -l)) * c0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 / sqrt(Float64(Float64(l / A) * V)))
	tmp = 0.0
	if (Float64(l * V) <= Float64(-Inf))
		tmp = t_0;
	elseif (Float64(l * V) <= -5e-303)
		tmp = Float64(Float64(sqrt(Float64(-A)) / sqrt(Float64(Float64(-l) * V))) * c0);
	elseif (Float64(l * V) <= 1e-256)
		tmp = t_0;
	elseif (Float64(l * V) <= 5e+291)
		tmp = Float64(Float64(sqrt(A) * sqrt(Float64(1.0 / Float64(l * V)))) * c0);
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(-1.0 / V) * A) / Float64(-l))) * c0);
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 / sqrt(((l / A) * V));
	tmp = 0.0;
	if ((l * V) <= -Inf)
		tmp = t_0;
	elseif ((l * V) <= -5e-303)
		tmp = (sqrt(-A) / sqrt((-l * V))) * c0;
	elseif ((l * V) <= 1e-256)
		tmp = t_0;
	elseif ((l * V) <= 5e+291)
		tmp = (sqrt(A) * sqrt((1.0 / (l * V)))) * c0;
	else
		tmp = sqrt((((-1.0 / V) * A) / -l)) * c0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 / N[Sqrt[N[(N[(l / A), $MachinePrecision] * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(l * V), $MachinePrecision], (-Infinity)], t$95$0, If[LessEqual[N[(l * V), $MachinePrecision], -5e-303], N[(N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[((-l) * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 1e-256], t$95$0, If[LessEqual[N[(l * V), $MachinePrecision], 5e+291], N[(N[(N[Sqrt[A], $MachinePrecision] * N[Sqrt[N[(1.0 / N[(l * V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(-1.0 / V), $MachinePrecision] * A), $MachinePrecision] / (-l)), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -inf.0 or -4.9999999999999998e-303 < (*.f64 V l) < 9.99999999999999977e-257

   1. Initial program 40.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 67.8

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

   4. Applied rewrites 67.8%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. sqrt-undiv N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-/r/ N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 67.9

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

   6. Applied rewrites 67.9%

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

   if -inf.0 < (*.f64 V l) < -4.9999999999999998e-303

   1. Initial program 82.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      5. lower-/.f64 74.9

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{V}}}{\ell}} \]

   4. Applied rewrites 74.9%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\sqrt{\frac{\frac{A}{V}}{\ell}}} \]

      2. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      3. sqrt-div N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{\frac{A}{V}}}}{\sqrt{\ell}} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\color{blue}{\sqrt{\ell}}} \]

      6. div-inv N/A

         \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{\frac{A}{V}} \cdot \frac{1}{\sqrt{\ell}}\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \left(\color{blue}{\sqrt{\frac{A}{V}}} \cdot \frac{1}{\sqrt{\ell}}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto c0 \cdot \left(\sqrt{\color{blue}{\frac{A}{V}}} \cdot \frac{1}{\sqrt{\ell}}\right) \]

      9. frac-2neg N/A

         \[\leadsto c0 \cdot \left(\sqrt{\color{blue}{\frac{\mathsf{neg}\left(A\right)}{\mathsf{neg}\left(V\right)}}} \cdot \frac{1}{\sqrt{\ell}}\right) \]

      10. sqrt-div N/A

          \[\leadsto c0 \cdot \left(\color{blue}{\frac{\sqrt{\mathsf{neg}\left(A\right)}}{\sqrt{\mathsf{neg}\left(V\right)}}} \cdot \frac{1}{\sqrt{\ell}}\right) \]

      11. frac-times N/A

          \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\mathsf{neg}\left(A\right)} \cdot 1}{\sqrt{\mathsf{neg}\left(V\right)} \cdot \sqrt{\ell}}} \]

      12. metadata-eval N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\mathsf{neg}\left(A\right)} \cdot \color{blue}{\sqrt{1}}}{\sqrt{\mathsf{neg}\left(V\right)} \cdot \sqrt{\ell}} \]

      13. sqrt-prod N/A

          \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{\left(\mathsf{neg}\left(A\right)\right) \cdot 1}}}{\sqrt{\mathsf{neg}\left(V\right)} \cdot \sqrt{\ell}} \]

      14. *-rgt-identity N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{\mathsf{neg}\left(A\right)}}}{\sqrt{\mathsf{neg}\left(V\right)} \cdot \sqrt{\ell}} \]

      15. lower-/.f64 N/A

          \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\mathsf{neg}\left(A\right)}}{\sqrt{\mathsf{neg}\left(V\right)} \cdot \sqrt{\ell}}} \]

      16. lower-sqrt.f64 N/A

          \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{\mathsf{neg}\left(A\right)}}}{\sqrt{\mathsf{neg}\left(V\right)} \cdot \sqrt{\ell}} \]

      17. lower-neg.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{-A}}}{\sqrt{\mathsf{neg}\left(V\right)} \cdot \sqrt{\ell}} \]

      18. lift-sqrt.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{-A}}{\sqrt{\mathsf{neg}\left(V\right)} \cdot \color{blue}{\sqrt{\ell}}} \]

      19. sqrt-prod N/A

          \[\leadsto c0 \cdot \frac{\sqrt{-A}}{\color{blue}{\sqrt{\left(\mathsf{neg}\left(V\right)\right) \cdot \ell}}} \]

      20. distribute-lft-neg-in N/A

          \[\leadsto c0 \cdot \frac{\sqrt{-A}}{\sqrt{\color{blue}{\mathsf{neg}\left(V \cdot \ell\right)}}} \]

      21. lower-sqrt.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{-A}}{\color{blue}{\sqrt{\mathsf{neg}\left(V \cdot \ell\right)}}} \]

      22. *-commutative N/A

          \[\leadsto c0 \cdot \frac{\sqrt{-A}}{\sqrt{\mathsf{neg}\left(\color{blue}{\ell \cdot V}\right)}} \]

      23. distribute-lft-neg-in N/A

          \[\leadsto c0 \cdot \frac{\sqrt{-A}}{\sqrt{\color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot V}}} \]

      24. lower-*.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{-A}}{\sqrt{\color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot V}}} \]

      25. lower-neg.f64 99.5

          \[\leadsto c0 \cdot \frac{\sqrt{-A}}{\sqrt{\color{blue}{\left(-\ell\right)} \cdot V}} \]

   6. Applied rewrites 99.5%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{-A}}{\sqrt{\left(-\ell\right) \cdot V}}} \]

   if 9.99999999999999977e-257 < (*.f64 V l) < 5.0000000000000001e291

   1. Initial program 82.1%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      3. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      4. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      5. lower-/.f64 73.4

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{V}}}{\ell}} \]

   4. Applied rewrites 73.4%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{V}}}{\ell}} \]

      2. clear-num N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{1}{\frac{V}{A}}}}{\ell}} \]

      3. frac-2neg N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{1}{\color{blue}{\frac{\mathsf{neg}\left(V\right)}{\mathsf{neg}\left(A\right)}}}}{\ell}} \]

      4. associate-/r/ N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{1}{\mathsf{neg}\left(V\right)} \cdot \left(\mathsf{neg}\left(A\right)\right)}}{\ell}} \]

      5. lower-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{1}{\mathsf{neg}\left(V\right)} \cdot \left(\mathsf{neg}\left(A\right)\right)}}{\ell}} \]

      6. metadata-eval N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\mathsf{neg}\left(V\right)} \cdot \left(\mathsf{neg}\left(A\right)\right)}{\ell}} \]

      7. frac-2neg N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{-1}{V}} \cdot \left(\mathsf{neg}\left(A\right)\right)}{\ell}} \]

      8. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{-1}{V}} \cdot \left(\mathsf{neg}\left(A\right)\right)}{\ell}} \]

      9. lower-neg.f64 73.3

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{-1}{V} \cdot \color{blue}{\left(-A\right)}}{\ell}} \]

   6. Applied rewrites 73.3%

      \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{-1}{V} \cdot \left(-A\right)}}{\ell}} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\sqrt{\frac{\frac{-1}{V} \cdot \left(-A\right)}{\ell}}} \]

      2. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{-1}{V} \cdot \left(-A\right)}{\ell}}} \]

      3. clear-num N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{\frac{-1}{V} \cdot \left(-A\right)}}}} \]

      4. sqrt-div N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{\frac{-1}{V} \cdot \left(-A\right)}}}} \]

      5. metadata-eval N/A

         \[\leadsto c0 \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{\frac{-1}{V} \cdot \left(-A\right)}}} \]

      6. lift-*.f64 N/A

         \[\leadsto c0 \cdot \frac{1}{\sqrt{\frac{\ell}{\color{blue}{\frac{-1}{V} \cdot \left(-A\right)}}}} \]

      7. lift-/.f64 N/A

         \[\leadsto c0 \cdot \frac{1}{\sqrt{\frac{\ell}{\color{blue}{\frac{-1}{V}} \cdot \left(-A\right)}}} \]

      8. associate-*l/ N/A

         \[\leadsto c0 \cdot \frac{1}{\sqrt{\frac{\ell}{\color{blue}{\frac{-1 \cdot \left(-A\right)}{V}}}}} \]

      9. neg-mul-1 N/A

         \[\leadsto c0 \cdot \frac{1}{\sqrt{\frac{\ell}{\frac{\color{blue}{\mathsf{neg}\left(\left(-A\right)\right)}}{V}}}} \]

      10. lift-neg.f64 N/A

          \[\leadsto c0 \cdot \frac{1}{\sqrt{\frac{\ell}{\frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(A\right)\right)}\right)}{V}}}} \]

      11. remove-double-neg N/A

          \[\leadsto c0 \cdot \frac{1}{\sqrt{\frac{\ell}{\frac{\color{blue}{A}}{V}}}} \]

      12. associate-/r/ N/A

          \[\leadsto c0 \cdot \frac{1}{\sqrt{\color{blue}{\frac{\ell}{A} \cdot V}}} \]

      13. associate-*l/ N/A

          \[\leadsto c0 \cdot \frac{1}{\sqrt{\color{blue}{\frac{\ell \cdot V}{A}}}} \]

      14. sqrt-div N/A

          \[\leadsto c0 \cdot \frac{1}{\color{blue}{\frac{\sqrt{\ell \cdot V}}{\sqrt{A}}}} \]

      15. associate-/r/ N/A

          \[\leadsto c0 \cdot \color{blue}{\left(\frac{1}{\sqrt{\ell \cdot V}} \cdot \sqrt{A}\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto c0 \cdot \color{blue}{\left(\frac{1}{\sqrt{\ell \cdot V}} \cdot \sqrt{A}\right)} \]

      17. lower-/.f64 N/A

          \[\leadsto c0 \cdot \left(\color{blue}{\frac{1}{\sqrt{\ell \cdot V}}} \cdot \sqrt{A}\right) \]

      18. lower-sqrt.f64 N/A

          \[\leadsto c0 \cdot \left(\frac{1}{\color{blue}{\sqrt{\ell \cdot V}}} \cdot \sqrt{A}\right) \]

      19. lower-*.f64 N/A

          \[\leadsto c0 \cdot \left(\frac{1}{\sqrt{\color{blue}{\ell \cdot V}}} \cdot \sqrt{A}\right) \]

      20. lower-sqrt.f64 99.3

          \[\leadsto c0 \cdot \left(\frac{1}{\sqrt{\ell \cdot V}} \cdot \color{blue}{\sqrt{A}}\right) \]

   8. Applied rewrites 99.3%

      \[\leadsto c0 \cdot \color{blue}{\left(\frac{1}{\sqrt{\ell \cdot V}} \cdot \sqrt{A}\right)} \]

   9. Taylor expanded in V around 0

      \[\leadsto c0 \cdot \left(\color{blue}{\sqrt{\frac{1}{V \cdot \ell}}} \cdot \sqrt{A}\right) \]
   10. Step-by-step derivation

       1. lower-sqrt.f64 N/A

          \[\leadsto c0 \cdot \left(\color{blue}{\sqrt{\frac{1}{V \cdot \ell}}} \cdot \sqrt{A}\right) \]

       2. lower-/.f64 N/A

          \[\leadsto c0 \cdot \left(\sqrt{\color{blue}{\frac{1}{V \cdot \ell}}} \cdot \sqrt{A}\right) \]

       3. *-commutative N/A

          \[\leadsto c0 \cdot \left(\sqrt{\frac{1}{\color{blue}{\ell \cdot V}}} \cdot \sqrt{A}\right) \]

       4. lower-*.f64 99.3

          \[\leadsto c0 \cdot \left(\sqrt{\frac{1}{\color{blue}{\ell \cdot V}}} \cdot \sqrt{A}\right) \]

   11. Applied rewrites 99.3%

       \[\leadsto c0 \cdot \left(\color{blue}{\sqrt{\frac{1}{\ell \cdot V}}} \cdot \sqrt{A}\right) \]

   if 5.0000000000000001e291 < (*.f64 V l)

   1. Initial program 44.4%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      3. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      4. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      5. lower-/.f64 69.0

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{V}}}{\ell}} \]

   4. Applied rewrites 69.0%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{V}}}{\ell}} \]

      2. clear-num N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{1}{\frac{V}{A}}}}{\ell}} \]

      3. frac-2neg N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{1}{\color{blue}{\frac{\mathsf{neg}\left(V\right)}{\mathsf{neg}\left(A\right)}}}}{\ell}} \]

      4. associate-/r/ N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{1}{\mathsf{neg}\left(V\right)} \cdot \left(\mathsf{neg}\left(A\right)\right)}}{\ell}} \]

      5. lower-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{1}{\mathsf{neg}\left(V\right)} \cdot \left(\mathsf{neg}\left(A\right)\right)}}{\ell}} \]

      6. metadata-eval N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\mathsf{neg}\left(V\right)} \cdot \left(\mathsf{neg}\left(A\right)\right)}{\ell}} \]

      7. frac-2neg N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{-1}{V}} \cdot \left(\mathsf{neg}\left(A\right)\right)}{\ell}} \]

      8. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{-1}{V}} \cdot \left(\mathsf{neg}\left(A\right)\right)}{\ell}} \]

      9. lower-neg.f64 69.1

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{-1}{V} \cdot \color{blue}{\left(-A\right)}}{\ell}} \]

   6. Applied rewrites 69.1%

      \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{-1}{V} \cdot \left(-A\right)}}{\ell}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \mathbf{elif}\;\ell \cdot V \leq -5 \cdot 10^{-303}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\left(-\ell\right) \cdot V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 10^{-256}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{+291}:\\ \;\;\;\;\left(\sqrt{A} \cdot \sqrt{\frac{1}{\ell \cdot V}}\right) \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{-1}{V} \cdot A}{-\ell}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \cdot V \leq -5 \cdot 10^{-303}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\left(-\ell\right) \cdot V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{-292}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{+291}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot V}} \cdot \sqrt{A}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{-1}{V} \cdot A}{-\ell}} \cdot c0\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ c0 (sqrt (* (/ l A) V)))))
   (if (<= (* l V) (- INFINITY))
     t_0
     (if (<= (* l V) -5e-303)
       (* (/ (sqrt (- A)) (sqrt (* (- l) V))) c0)
       (if (<= (* l V) 5e-292)
         t_0
         (if (<= (* l V) 5e+291)
           (* (/ c0 (sqrt (* l V))) (sqrt A))
           (* (sqrt (/ (* (/ -1.0 V) A) (- l))) c0)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 / sqrt(((l / A) * V));
	double tmp;
	if ((l * V) <= -((double) INFINITY)) {
		tmp = t_0;
	} else if ((l * V) <= -5e-303) {
		tmp = (sqrt(-A) / sqrt((-l * V))) * c0;
	} else if ((l * V) <= 5e-292) {
		tmp = t_0;
	} else if ((l * V) <= 5e+291) {
		tmp = (c0 / sqrt((l * V))) * sqrt(A);
	} else {
		tmp = sqrt((((-1.0 / V) * A) / -l)) * c0;
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 / Math.sqrt(((l / A) * V));
	double tmp;
	if ((l * V) <= -Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else if ((l * V) <= -5e-303) {
		tmp = (Math.sqrt(-A) / Math.sqrt((-l * V))) * c0;
	} else if ((l * V) <= 5e-292) {
		tmp = t_0;
	} else if ((l * V) <= 5e+291) {
		tmp = (c0 / Math.sqrt((l * V))) * Math.sqrt(A);
	} else {
		tmp = Math.sqrt((((-1.0 / V) * A) / -l)) * c0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 / math.sqrt(((l / A) * V))
	tmp = 0
	if (l * V) <= -math.inf:
		tmp = t_0
	elif (l * V) <= -5e-303:
		tmp = (math.sqrt(-A) / math.sqrt((-l * V))) * c0
	elif (l * V) <= 5e-292:
		tmp = t_0
	elif (l * V) <= 5e+291:
		tmp = (c0 / math.sqrt((l * V))) * math.sqrt(A)
	else:
		tmp = math.sqrt((((-1.0 / V) * A) / -l)) * c0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 / sqrt(Float64(Float64(l / A) * V)))
	tmp = 0.0
	if (Float64(l * V) <= Float64(-Inf))
		tmp = t_0;
	elseif (Float64(l * V) <= -5e-303)
		tmp = Float64(Float64(sqrt(Float64(-A)) / sqrt(Float64(Float64(-l) * V))) * c0);
	elseif (Float64(l * V) <= 5e-292)
		tmp = t_0;
	elseif (Float64(l * V) <= 5e+291)
		tmp = Float64(Float64(c0 / sqrt(Float64(l * V))) * sqrt(A));
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(-1.0 / V) * A) / Float64(-l))) * c0);
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 / sqrt(((l / A) * V));
	tmp = 0.0;
	if ((l * V) <= -Inf)
		tmp = t_0;
	elseif ((l * V) <= -5e-303)
		tmp = (sqrt(-A) / sqrt((-l * V))) * c0;
	elseif ((l * V) <= 5e-292)
		tmp = t_0;
	elseif ((l * V) <= 5e+291)
		tmp = (c0 / sqrt((l * V))) * sqrt(A);
	else
		tmp = sqrt((((-1.0 / V) * A) / -l)) * c0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 / N[Sqrt[N[(N[(l / A), $MachinePrecision] * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(l * V), $MachinePrecision], (-Infinity)], t$95$0, If[LessEqual[N[(l * V), $MachinePrecision], -5e-303], N[(N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[((-l) * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 5e-292], t$95$0, If[LessEqual[N[(l * V), $MachinePrecision], 5e+291], N[(N[(c0 / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[A], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(-1.0 / V), $MachinePrecision] * A), $MachinePrecision] / (-l)), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -inf.0 or -4.9999999999999998e-303 < (*.f64 V l) < 4.99999999999999981e-292

   1. Initial program 31.8%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      3. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      4. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      5. lower-/.f64 63.2

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{V}}}{\ell}} \]

   4. Applied rewrites 63.2%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\sqrt{\frac{\frac{A}{V}}{\ell}}} \]

      3. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      4. sqrt-div N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{\frac{A}{V}}}}{\sqrt{\ell}} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\color{blue}{\sqrt{\ell}}} \]

      7. clear-num N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{1}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}} \]

      8. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}} \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \frac{c0}{\frac{\color{blue}{\sqrt{\ell}}}{\sqrt{\frac{A}{V}}}} \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \frac{c0}{\frac{\sqrt{\ell}}{\color{blue}{\sqrt{\frac{A}{V}}}}} \]

      12. sqrt-undiv N/A

          \[\leadsto \frac{c0}{\color{blue}{\sqrt{\frac{\ell}{\frac{A}{V}}}}} \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \frac{c0}{\color{blue}{\sqrt{\frac{\ell}{\frac{A}{V}}}}} \]

      14. lift-/.f64 N/A

          \[\leadsto \frac{c0}{\sqrt{\frac{\ell}{\color{blue}{\frac{A}{V}}}}} \]

      15. associate-/r/ N/A

          \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{\ell}{A} \cdot V}}} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{\ell}{A} \cdot V}}} \]

      17. lower-/.f64 63.3

          \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{\ell}{A}} \cdot V}} \]

   6. Applied rewrites 63.3%

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}} \]

   if -inf.0 < (*.f64 V l) < -4.9999999999999998e-303

   1. Initial program 82.9%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      3. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      4. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      5. lower-/.f64 74.9

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{V}}}{\ell}} \]

   4. Applied rewrites 74.9%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\sqrt{\frac{\frac{A}{V}}{\ell}}} \]

      2. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      3. sqrt-div N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{\frac{A}{V}}}}{\sqrt{\ell}} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\color{blue}{\sqrt{\ell}}} \]

      6. div-inv N/A

         \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{\frac{A}{V}} \cdot \frac{1}{\sqrt{\ell}}\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \left(\color{blue}{\sqrt{\frac{A}{V}}} \cdot \frac{1}{\sqrt{\ell}}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto c0 \cdot \left(\sqrt{\color{blue}{\frac{A}{V}}} \cdot \frac{1}{\sqrt{\ell}}\right) \]

      9. frac-2neg N/A

         \[\leadsto c0 \cdot \left(\sqrt{\color{blue}{\frac{\mathsf{neg}\left(A\right)}{\mathsf{neg}\left(V\right)}}} \cdot \frac{1}{\sqrt{\ell}}\right) \]

      10. sqrt-div N/A

          \[\leadsto c0 \cdot \left(\color{blue}{\frac{\sqrt{\mathsf{neg}\left(A\right)}}{\sqrt{\mathsf{neg}\left(V\right)}}} \cdot \frac{1}{\sqrt{\ell}}\right) \]

      11. frac-times N/A

          \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\mathsf{neg}\left(A\right)} \cdot 1}{\sqrt{\mathsf{neg}\left(V\right)} \cdot \sqrt{\ell}}} \]

      12. metadata-eval N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\mathsf{neg}\left(A\right)} \cdot \color{blue}{\sqrt{1}}}{\sqrt{\mathsf{neg}\left(V\right)} \cdot \sqrt{\ell}} \]

      13. sqrt-prod N/A

          \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{\left(\mathsf{neg}\left(A\right)\right) \cdot 1}}}{\sqrt{\mathsf{neg}\left(V\right)} \cdot \sqrt{\ell}} \]

      14. *-rgt-identity N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{\mathsf{neg}\left(A\right)}}}{\sqrt{\mathsf{neg}\left(V\right)} \cdot \sqrt{\ell}} \]

      15. lower-/.f64 N/A

          \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\mathsf{neg}\left(A\right)}}{\sqrt{\mathsf{neg}\left(V\right)} \cdot \sqrt{\ell}}} \]

      16. lower-sqrt.f64 N/A

          \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{\mathsf{neg}\left(A\right)}}}{\sqrt{\mathsf{neg}\left(V\right)} \cdot \sqrt{\ell}} \]

      17. lower-neg.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{-A}}}{\sqrt{\mathsf{neg}\left(V\right)} \cdot \sqrt{\ell}} \]

      18. lift-sqrt.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{-A}}{\sqrt{\mathsf{neg}\left(V\right)} \cdot \color{blue}{\sqrt{\ell}}} \]

      19. sqrt-prod N/A

          \[\leadsto c0 \cdot \frac{\sqrt{-A}}{\color{blue}{\sqrt{\left(\mathsf{neg}\left(V\right)\right) \cdot \ell}}} \]

      20. distribute-lft-neg-in N/A

          \[\leadsto c0 \cdot \frac{\sqrt{-A}}{\sqrt{\color{blue}{\mathsf{neg}\left(V \cdot \ell\right)}}} \]

      21. lower-sqrt.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{-A}}{\color{blue}{\sqrt{\mathsf{neg}\left(V \cdot \ell\right)}}} \]

      22. *-commutative N/A

          \[\leadsto c0 \cdot \frac{\sqrt{-A}}{\sqrt{\mathsf{neg}\left(\color{blue}{\ell \cdot V}\right)}} \]

      23. distribute-lft-neg-in N/A

          \[\leadsto c0 \cdot \frac{\sqrt{-A}}{\sqrt{\color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot V}}} \]

      24. lower-*.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{-A}}{\sqrt{\color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot V}}} \]

      25. lower-neg.f64 99.5

          \[\leadsto c0 \cdot \frac{\sqrt{-A}}{\sqrt{\color{blue}{\left(-\ell\right)} \cdot V}} \]

   6. Applied rewrites 99.5%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{-A}}{\sqrt{\left(-\ell\right) \cdot V}}} \]

   if 4.99999999999999981e-292 < (*.f64 V l) < 5.0000000000000001e291

   1. Initial program 83.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{A}{V \cdot \ell}} \cdot c0} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{A}{V \cdot \ell}}} \cdot c0 \]

      4. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \cdot c0 \]

      5. sqrt-div N/A

         \[\leadsto \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}} \cdot c0 \]

      6. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sqrt{A} \cdot c0}{\sqrt{V \cdot \ell}}} \]

      7. associate-/l* N/A

         \[\leadsto \color{blue}{\sqrt{A} \cdot \frac{c0}{\sqrt{V \cdot \ell}}} \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\frac{c0}{\sqrt{V \cdot \ell}} \cdot \sqrt{A}} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{c0}{\sqrt{V \cdot \ell}} \cdot \sqrt{A}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{c0}{\sqrt{V \cdot \ell}}} \cdot \sqrt{A} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \frac{c0}{\color{blue}{\sqrt{V \cdot \ell}}} \cdot \sqrt{A} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{c0}{\sqrt{\color{blue}{V \cdot \ell}}} \cdot \sqrt{A} \]

      13. *-commutative N/A

          \[\leadsto \frac{c0}{\sqrt{\color{blue}{\ell \cdot V}}} \cdot \sqrt{A} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{c0}{\sqrt{\color{blue}{\ell \cdot V}}} \cdot \sqrt{A} \]

      15. lower-sqrt.f64 98.4

          \[\leadsto \frac{c0}{\sqrt{\ell \cdot V}} \cdot \color{blue}{\sqrt{A}} \]

   4. Applied rewrites 98.4%

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{\ell \cdot V}} \cdot \sqrt{A}} \]

   if 5.0000000000000001e291 < (*.f64 V l)

   1. Initial program 44.4%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      3. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      4. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      5. lower-/.f64 69.0

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{V}}}{\ell}} \]

   4. Applied rewrites 69.0%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{V}}}{\ell}} \]

      2. clear-num N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{1}{\frac{V}{A}}}}{\ell}} \]

      3. frac-2neg N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{1}{\color{blue}{\frac{\mathsf{neg}\left(V\right)}{\mathsf{neg}\left(A\right)}}}}{\ell}} \]

      4. associate-/r/ N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{1}{\mathsf{neg}\left(V\right)} \cdot \left(\mathsf{neg}\left(A\right)\right)}}{\ell}} \]

      5. lower-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{1}{\mathsf{neg}\left(V\right)} \cdot \left(\mathsf{neg}\left(A\right)\right)}}{\ell}} \]

      6. metadata-eval N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\mathsf{neg}\left(V\right)} \cdot \left(\mathsf{neg}\left(A\right)\right)}{\ell}} \]

      7. frac-2neg N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{-1}{V}} \cdot \left(\mathsf{neg}\left(A\right)\right)}{\ell}} \]

      8. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{-1}{V}} \cdot \left(\mathsf{neg}\left(A\right)\right)}{\ell}} \]

      9. lower-neg.f64 69.1

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{-1}{V} \cdot \color{blue}{\left(-A\right)}}{\ell}} \]

   6. Applied rewrites 69.1%

      \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{-1}{V} \cdot \left(-A\right)}}{\ell}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \mathbf{elif}\;\ell \cdot V \leq -5 \cdot 10^{-303}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\left(-\ell\right) \cdot V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{-292}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{+291}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot V}} \cdot \sqrt{A}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{-1}{V} \cdot A}{-\ell}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{\ell \cdot V}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-295}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{\ell}}{V}} \cdot c0\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+303}:\\ \;\;\;\;\sqrt{t\_0} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* l V))))
   (if (<= t_0 4e-295)
     (* (sqrt (/ (/ A l) V)) c0)
     (if (<= t_0 2e+303) (* (sqrt t_0) c0) (/ c0 (sqrt (* (/ l A) V)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (l * V);
	double tmp;
	if (t_0 <= 4e-295) {
		tmp = sqrt(((A / l) / V)) * c0;
	} else if (t_0 <= 2e+303) {
		tmp = sqrt(t_0) * c0;
	} else {
		tmp = c0 / sqrt(((l / A) * V));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (l * v)
    if (t_0 <= 4d-295) then
        tmp = sqrt(((a / l) / v)) * c0
    else if (t_0 <= 2d+303) then
        tmp = sqrt(t_0) * c0
    else
        tmp = c0 / sqrt(((l / a) * v))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (l * V);
	double tmp;
	if (t_0 <= 4e-295) {
		tmp = Math.sqrt(((A / l) / V)) * c0;
	} else if (t_0 <= 2e+303) {
		tmp = Math.sqrt(t_0) * c0;
	} else {
		tmp = c0 / Math.sqrt(((l / A) * V));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (l * V)
	tmp = 0
	if t_0 <= 4e-295:
		tmp = math.sqrt(((A / l) / V)) * c0
	elif t_0 <= 2e+303:
		tmp = math.sqrt(t_0) * c0
	else:
		tmp = c0 / math.sqrt(((l / A) * V))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(l * V))
	tmp = 0.0
	if (t_0 <= 4e-295)
		tmp = Float64(sqrt(Float64(Float64(A / l) / V)) * c0);
	elseif (t_0 <= 2e+303)
		tmp = Float64(sqrt(t_0) * c0);
	else
		tmp = Float64(c0 / sqrt(Float64(Float64(l / A) * V)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (l * V);
	tmp = 0.0;
	if (t_0 <= 4e-295)
		tmp = sqrt(((A / l) / V)) * c0;
	elseif (t_0 <= 2e+303)
		tmp = sqrt(t_0) * c0;
	else
		tmp = c0 / sqrt(((l / A) * V));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(l * V), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 4e-295], N[(N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[t$95$0, 2e+303], N[(N[Sqrt[t$95$0], $MachinePrecision] * c0), $MachinePrecision], N[(c0 / N[Sqrt[N[(N[(l / A), $MachinePrecision] * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 A (*.f64 V l)) < 4.00000000000000024e-295

   1. Initial program 36.1%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      3. associate-/l/ N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{\ell}}{V}}} \]

      4. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{\ell}}{V}}} \]

      5. lower-/.f64 49.3

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{\ell}}}{V}} \]

   4. Applied rewrites 49.3%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{\ell}}{V}}} \]

   if 4.00000000000000024e-295 < (/.f64 A (*.f64 V l)) < 2e303

   1. Initial program 99.6%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing

   if 2e303 < (/.f64 A (*.f64 V l))

   1. Initial program 32.3%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      3. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      4. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      5. lower-/.f64 50.9

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{V}}}{\ell}} \]

   4. Applied rewrites 50.9%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\sqrt{\frac{\frac{A}{V}}{\ell}}} \]

      3. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      4. sqrt-div N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{\frac{A}{V}}}}{\sqrt{\ell}} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\color{blue}{\sqrt{\ell}}} \]

      7. clear-num N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{1}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}} \]

      8. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}} \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \frac{c0}{\frac{\color{blue}{\sqrt{\ell}}}{\sqrt{\frac{A}{V}}}} \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \frac{c0}{\frac{\sqrt{\ell}}{\color{blue}{\sqrt{\frac{A}{V}}}}} \]

      12. sqrt-undiv N/A

          \[\leadsto \frac{c0}{\color{blue}{\sqrt{\frac{\ell}{\frac{A}{V}}}}} \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \frac{c0}{\color{blue}{\sqrt{\frac{\ell}{\frac{A}{V}}}}} \]

      14. lift-/.f64 N/A

          \[\leadsto \frac{c0}{\sqrt{\frac{\ell}{\color{blue}{\frac{A}{V}}}}} \]

      15. associate-/r/ N/A

          \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{\ell}{A} \cdot V}}} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{\ell}{A} \cdot V}}} \]

      17. lower-/.f64 53.9

          \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{\ell}{A}} \cdot V}} \]

   6. Applied rewrites 53.9%

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{\ell \cdot V} \leq 4 \cdot 10^{-295}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{\ell}}{V}} \cdot c0\\ \mathbf{elif}\;\frac{A}{\ell \cdot V} \leq 2 \cdot 10^{+303}:\\ \;\;\;\;\sqrt{\frac{A}{\ell \cdot V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 80.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{\ell \cdot V}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-295}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{\ell}}{V}} \cdot c0\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+266}:\\ \;\;\;\;\sqrt{t\_0} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* l V))))
   (if (<= t_0 4e-295)
     (* (sqrt (/ (/ A l) V)) c0)
     (if (<= t_0 5e+266) (* (sqrt t_0) c0) (* (sqrt (/ (/ A V) l)) c0)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (l * V);
	double tmp;
	if (t_0 <= 4e-295) {
		tmp = sqrt(((A / l) / V)) * c0;
	} else if (t_0 <= 5e+266) {
		tmp = sqrt(t_0) * c0;
	} else {
		tmp = sqrt(((A / V) / l)) * c0;
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (l * v)
    if (t_0 <= 4d-295) then
        tmp = sqrt(((a / l) / v)) * c0
    else if (t_0 <= 5d+266) then
        tmp = sqrt(t_0) * c0
    else
        tmp = sqrt(((a / v) / l)) * c0
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (l * V);
	double tmp;
	if (t_0 <= 4e-295) {
		tmp = Math.sqrt(((A / l) / V)) * c0;
	} else if (t_0 <= 5e+266) {
		tmp = Math.sqrt(t_0) * c0;
	} else {
		tmp = Math.sqrt(((A / V) / l)) * c0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (l * V)
	tmp = 0
	if t_0 <= 4e-295:
		tmp = math.sqrt(((A / l) / V)) * c0
	elif t_0 <= 5e+266:
		tmp = math.sqrt(t_0) * c0
	else:
		tmp = math.sqrt(((A / V) / l)) * c0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(l * V))
	tmp = 0.0
	if (t_0 <= 4e-295)
		tmp = Float64(sqrt(Float64(Float64(A / l) / V)) * c0);
	elseif (t_0 <= 5e+266)
		tmp = Float64(sqrt(t_0) * c0);
	else
		tmp = Float64(sqrt(Float64(Float64(A / V) / l)) * c0);
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (l * V);
	tmp = 0.0;
	if (t_0 <= 4e-295)
		tmp = sqrt(((A / l) / V)) * c0;
	elseif (t_0 <= 5e+266)
		tmp = sqrt(t_0) * c0;
	else
		tmp = sqrt(((A / V) / l)) * c0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(l * V), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 4e-295], N[(N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[t$95$0, 5e+266], N[(N[Sqrt[t$95$0], $MachinePrecision] * c0), $MachinePrecision], N[(N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 A (*.f64 V l)) < 4.00000000000000024e-295

   1. Initial program 36.1%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      3. associate-/l/ N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{\ell}}{V}}} \]

      4. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{\ell}}{V}}} \]

      5. lower-/.f64 49.3

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{\ell}}}{V}} \]

   4. Applied rewrites 49.3%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{\ell}}{V}}} \]

   if 4.00000000000000024e-295 < (/.f64 A (*.f64 V l)) < 4.9999999999999999e266

   1. Initial program 99.6%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing

   if 4.9999999999999999e266 < (/.f64 A (*.f64 V l))

   1. Initial program 42.5%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      3. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      4. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      5. lower-/.f64 54.8

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{V}}}{\ell}} \]

   4. Applied rewrites 54.8%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{\ell \cdot V} \leq 4 \cdot 10^{-295}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{\ell}}{V}} \cdot c0\\ \mathbf{elif}\;\frac{A}{\ell \cdot V} \leq 5 \cdot 10^{+266}:\\ \;\;\;\;\sqrt{\frac{A}{\ell \cdot V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 80.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{\ell \cdot V}\\ t_1 := \sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+266}:\\ \;\;\;\;\sqrt{t\_0} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (/ A (* l V))) (t_1 (* (sqrt (/ (/ A V) l)) c0)))
   (if (<= t_0 0.0) t_1 (if (<= t_0 5e+266) (* (sqrt t_0) c0) t_1))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (l * V);
	double t_1 = sqrt(((A / V) / l)) * c0;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 5e+266) {
		tmp = sqrt(t_0) * c0;
	} 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 = a / (l * v)
    t_1 = sqrt(((a / v) / l)) * c0
    if (t_0 <= 0.0d0) then
        tmp = t_1
    else if (t_0 <= 5d+266) then
        tmp = sqrt(t_0) * c0
    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 = A / (l * V);
	double t_1 = Math.sqrt(((A / V) / l)) * c0;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 5e+266) {
		tmp = Math.sqrt(t_0) * c0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (l * V)
	t_1 = math.sqrt(((A / V) / l)) * c0
	tmp = 0
	if t_0 <= 0.0:
		tmp = t_1
	elif t_0 <= 5e+266:
		tmp = math.sqrt(t_0) * c0
	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(A / Float64(l * V))
	t_1 = Float64(sqrt(Float64(Float64(A / V) / l)) * c0)
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 5e+266)
		tmp = Float64(sqrt(t_0) * c0);
	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 = A / (l * V);
	t_1 = sqrt(((A / V) / l)) * c0;
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 5e+266)
		tmp = sqrt(t_0) * c0;
	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[(A / N[(l * V), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 5e+266], N[(N[Sqrt[t$95$0], $MachinePrecision] * c0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 0.0 or 4.9999999999999999e266 < (/.f64 A (*.f64 V l))

   1. Initial program 36.3%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      3. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      4. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      5. lower-/.f64 50.6

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{V}}}{\ell}} \]

   4. Applied rewrites 50.6%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

   if 0.0 < (/.f64 A (*.f64 V l)) < 4.9999999999999999e266

   1. Initial program 98.8%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{\ell \cdot V} \leq 0:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \mathbf{elif}\;\frac{A}{\ell \cdot V} \leq 5 \cdot 10^{+266}:\\ \;\;\;\;\sqrt{\frac{A}{\ell \cdot V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 81.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq 5 \cdot 10^{-292}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{+291}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot V}} \cdot \sqrt{A}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{-1}{V} \cdot A}{-\ell}} \cdot c0\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* l V) 5e-292)
   (/ c0 (sqrt (* (/ l A) V)))
   (if (<= (* l V) 5e+291)
     (* (/ c0 (sqrt (* l V))) (sqrt A))
     (* (sqrt (/ (* (/ -1.0 V) A) (- l))) c0))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= 5e-292) {
		tmp = c0 / sqrt(((l / A) * V));
	} else if ((l * V) <= 5e+291) {
		tmp = (c0 / sqrt((l * V))) * sqrt(A);
	} else {
		tmp = sqrt((((-1.0 / V) * A) / -l)) * c0;
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((l * v) <= 5d-292) then
        tmp = c0 / sqrt(((l / a) * v))
    else if ((l * v) <= 5d+291) then
        tmp = (c0 / sqrt((l * v))) * sqrt(a)
    else
        tmp = sqrt(((((-1.0d0) / v) * a) / -l)) * c0
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= 5e-292) {
		tmp = c0 / Math.sqrt(((l / A) * V));
	} else if ((l * V) <= 5e+291) {
		tmp = (c0 / Math.sqrt((l * V))) * Math.sqrt(A);
	} else {
		tmp = Math.sqrt((((-1.0 / V) * A) / -l)) * c0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (l * V) <= 5e-292:
		tmp = c0 / math.sqrt(((l / A) * V))
	elif (l * V) <= 5e+291:
		tmp = (c0 / math.sqrt((l * V))) * math.sqrt(A)
	else:
		tmp = math.sqrt((((-1.0 / V) * A) / -l)) * c0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(l * V) <= 5e-292)
		tmp = Float64(c0 / sqrt(Float64(Float64(l / A) * V)));
	elseif (Float64(l * V) <= 5e+291)
		tmp = Float64(Float64(c0 / sqrt(Float64(l * V))) * sqrt(A));
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(-1.0 / V) * A) / Float64(-l))) * c0);
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((l * V) <= 5e-292)
		tmp = c0 / sqrt(((l / A) * V));
	elseif ((l * V) <= 5e+291)
		tmp = (c0 / sqrt((l * V))) * sqrt(A);
	else
		tmp = sqrt((((-1.0 / V) * A) / -l)) * c0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(l * V), $MachinePrecision], 5e-292], N[(c0 / N[Sqrt[N[(N[(l / A), $MachinePrecision] * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 5e+291], N[(N[(c0 / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[A], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(-1.0 / V), $MachinePrecision] * A), $MachinePrecision] / (-l)), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < 4.99999999999999981e-292

   1. Initial program 67.3%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      3. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      4. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      5. lower-/.f64 71.4

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{V}}}{\ell}} \]

   4. Applied rewrites 71.4%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\sqrt{\frac{\frac{A}{V}}{\ell}}} \]

      3. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      4. sqrt-div N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{\frac{A}{V}}}}{\sqrt{\ell}} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\color{blue}{\sqrt{\ell}}} \]

      7. clear-num N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{1}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}} \]

      8. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}} \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \frac{c0}{\frac{\color{blue}{\sqrt{\ell}}}{\sqrt{\frac{A}{V}}}} \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \frac{c0}{\frac{\sqrt{\ell}}{\color{blue}{\sqrt{\frac{A}{V}}}}} \]

      12. sqrt-undiv N/A

          \[\leadsto \frac{c0}{\color{blue}{\sqrt{\frac{\ell}{\frac{A}{V}}}}} \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \frac{c0}{\color{blue}{\sqrt{\frac{\ell}{\frac{A}{V}}}}} \]

      14. lift-/.f64 N/A

          \[\leadsto \frac{c0}{\sqrt{\frac{\ell}{\color{blue}{\frac{A}{V}}}}} \]

      15. associate-/r/ N/A

          \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{\ell}{A} \cdot V}}} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{\ell}{A} \cdot V}}} \]

      17. lower-/.f64 71.3

          \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{\ell}{A}} \cdot V}} \]

   6. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}} \]

   if 4.99999999999999981e-292 < (*.f64 V l) < 5.0000000000000001e291

   1. Initial program 83.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{A}{V \cdot \ell}} \cdot c0} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{A}{V \cdot \ell}}} \cdot c0 \]

      4. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \cdot c0 \]

      5. sqrt-div N/A

         \[\leadsto \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}} \cdot c0 \]

      6. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sqrt{A} \cdot c0}{\sqrt{V \cdot \ell}}} \]

      7. associate-/l* N/A

         \[\leadsto \color{blue}{\sqrt{A} \cdot \frac{c0}{\sqrt{V \cdot \ell}}} \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\frac{c0}{\sqrt{V \cdot \ell}} \cdot \sqrt{A}} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{c0}{\sqrt{V \cdot \ell}} \cdot \sqrt{A}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{c0}{\sqrt{V \cdot \ell}}} \cdot \sqrt{A} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \frac{c0}{\color{blue}{\sqrt{V \cdot \ell}}} \cdot \sqrt{A} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{c0}{\sqrt{\color{blue}{V \cdot \ell}}} \cdot \sqrt{A} \]

      13. *-commutative N/A

          \[\leadsto \frac{c0}{\sqrt{\color{blue}{\ell \cdot V}}} \cdot \sqrt{A} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{c0}{\sqrt{\color{blue}{\ell \cdot V}}} \cdot \sqrt{A} \]

      15. lower-sqrt.f64 98.4

          \[\leadsto \frac{c0}{\sqrt{\ell \cdot V}} \cdot \color{blue}{\sqrt{A}} \]

   4. Applied rewrites 98.4%

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{\ell \cdot V}} \cdot \sqrt{A}} \]

   if 5.0000000000000001e291 < (*.f64 V l)

   1. Initial program 44.4%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      3. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      4. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      5. lower-/.f64 69.0

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{V}}}{\ell}} \]

   4. Applied rewrites 69.0%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{V}}}{\ell}} \]

      2. clear-num N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{1}{\frac{V}{A}}}}{\ell}} \]

      3. frac-2neg N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{1}{\color{blue}{\frac{\mathsf{neg}\left(V\right)}{\mathsf{neg}\left(A\right)}}}}{\ell}} \]

      4. associate-/r/ N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{1}{\mathsf{neg}\left(V\right)} \cdot \left(\mathsf{neg}\left(A\right)\right)}}{\ell}} \]

      5. lower-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{1}{\mathsf{neg}\left(V\right)} \cdot \left(\mathsf{neg}\left(A\right)\right)}}{\ell}} \]

      6. metadata-eval N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\mathsf{neg}\left(V\right)} \cdot \left(\mathsf{neg}\left(A\right)\right)}{\ell}} \]

      7. frac-2neg N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{-1}{V}} \cdot \left(\mathsf{neg}\left(A\right)\right)}{\ell}} \]

      8. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{-1}{V}} \cdot \left(\mathsf{neg}\left(A\right)\right)}{\ell}} \]

      9. lower-neg.f64 69.1

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{-1}{V} \cdot \color{blue}{\left(-A\right)}}{\ell}} \]

   6. Applied rewrites 69.1%

      \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{-1}{V} \cdot \left(-A\right)}}{\ell}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq 5 \cdot 10^{-292}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{+291}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot V}} \cdot \sqrt{A}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{-1}{V} \cdot A}{-\ell}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 81.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq 5 \cdot 10^{-292}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{+291}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot V}} \cdot \sqrt{A}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* l V) 5e-292)
   (/ c0 (sqrt (* (/ l A) V)))
   (if (<= (* l V) 5e+291)
     (* (/ c0 (sqrt (* l V))) (sqrt A))
     (* (sqrt (/ (/ A V) l)) c0))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= 5e-292) {
		tmp = c0 / sqrt(((l / A) * V));
	} else if ((l * V) <= 5e+291) {
		tmp = (c0 / sqrt((l * V))) * sqrt(A);
	} else {
		tmp = sqrt(((A / V) / l)) * c0;
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((l * v) <= 5d-292) then
        tmp = c0 / sqrt(((l / a) * v))
    else if ((l * v) <= 5d+291) then
        tmp = (c0 / sqrt((l * v))) * sqrt(a)
    else
        tmp = sqrt(((a / v) / l)) * c0
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= 5e-292) {
		tmp = c0 / Math.sqrt(((l / A) * V));
	} else if ((l * V) <= 5e+291) {
		tmp = (c0 / Math.sqrt((l * V))) * Math.sqrt(A);
	} else {
		tmp = Math.sqrt(((A / V) / l)) * c0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (l * V) <= 5e-292:
		tmp = c0 / math.sqrt(((l / A) * V))
	elif (l * V) <= 5e+291:
		tmp = (c0 / math.sqrt((l * V))) * math.sqrt(A)
	else:
		tmp = math.sqrt(((A / V) / l)) * c0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(l * V) <= 5e-292)
		tmp = Float64(c0 / sqrt(Float64(Float64(l / A) * V)));
	elseif (Float64(l * V) <= 5e+291)
		tmp = Float64(Float64(c0 / sqrt(Float64(l * V))) * sqrt(A));
	else
		tmp = Float64(sqrt(Float64(Float64(A / V) / l)) * c0);
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((l * V) <= 5e-292)
		tmp = c0 / sqrt(((l / A) * V));
	elseif ((l * V) <= 5e+291)
		tmp = (c0 / sqrt((l * V))) * sqrt(A);
	else
		tmp = sqrt(((A / V) / l)) * c0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(l * V), $MachinePrecision], 5e-292], N[(c0 / N[Sqrt[N[(N[(l / A), $MachinePrecision] * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 5e+291], N[(N[(c0 / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[A], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < 4.99999999999999981e-292

   1. Initial program 67.3%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      3. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      4. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      5. lower-/.f64 71.4

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{V}}}{\ell}} \]

   4. Applied rewrites 71.4%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\sqrt{\frac{\frac{A}{V}}{\ell}}} \]

      3. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      4. sqrt-div N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{\frac{A}{V}}}}{\sqrt{\ell}} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\color{blue}{\sqrt{\ell}}} \]

      7. clear-num N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{1}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}} \]

      8. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}} \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \frac{c0}{\frac{\color{blue}{\sqrt{\ell}}}{\sqrt{\frac{A}{V}}}} \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \frac{c0}{\frac{\sqrt{\ell}}{\color{blue}{\sqrt{\frac{A}{V}}}}} \]

      12. sqrt-undiv N/A

          \[\leadsto \frac{c0}{\color{blue}{\sqrt{\frac{\ell}{\frac{A}{V}}}}} \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \frac{c0}{\color{blue}{\sqrt{\frac{\ell}{\frac{A}{V}}}}} \]

      14. lift-/.f64 N/A

          \[\leadsto \frac{c0}{\sqrt{\frac{\ell}{\color{blue}{\frac{A}{V}}}}} \]

      15. associate-/r/ N/A

          \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{\ell}{A} \cdot V}}} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{\ell}{A} \cdot V}}} \]

      17. lower-/.f64 71.3

          \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{\ell}{A}} \cdot V}} \]

   6. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}} \]

   if 4.99999999999999981e-292 < (*.f64 V l) < 5.0000000000000001e291

   1. Initial program 83.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{A}{V \cdot \ell}} \cdot c0} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{A}{V \cdot \ell}}} \cdot c0 \]

      4. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \cdot c0 \]

      5. sqrt-div N/A

         \[\leadsto \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}} \cdot c0 \]

      6. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sqrt{A} \cdot c0}{\sqrt{V \cdot \ell}}} \]

      7. associate-/l* N/A

         \[\leadsto \color{blue}{\sqrt{A} \cdot \frac{c0}{\sqrt{V \cdot \ell}}} \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\frac{c0}{\sqrt{V \cdot \ell}} \cdot \sqrt{A}} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{c0}{\sqrt{V \cdot \ell}} \cdot \sqrt{A}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{c0}{\sqrt{V \cdot \ell}}} \cdot \sqrt{A} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \frac{c0}{\color{blue}{\sqrt{V \cdot \ell}}} \cdot \sqrt{A} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{c0}{\sqrt{\color{blue}{V \cdot \ell}}} \cdot \sqrt{A} \]

      13. *-commutative N/A

          \[\leadsto \frac{c0}{\sqrt{\color{blue}{\ell \cdot V}}} \cdot \sqrt{A} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{c0}{\sqrt{\color{blue}{\ell \cdot V}}} \cdot \sqrt{A} \]

      15. lower-sqrt.f64 98.4

          \[\leadsto \frac{c0}{\sqrt{\ell \cdot V}} \cdot \color{blue}{\sqrt{A}} \]

   4. Applied rewrites 98.4%

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{\ell \cdot V}} \cdot \sqrt{A}} \]

   if 5.0000000000000001e291 < (*.f64 V l)

   1. Initial program 44.4%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      3. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      4. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      5. lower-/.f64 69.0

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{V}}}{\ell}} \]

   4. Applied rewrites 69.0%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq 5 \cdot 10^{-292}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{+291}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot V}} \cdot \sqrt{A}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 74.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \sqrt{\frac{A}{\ell \cdot V}} \cdot c0 \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l) :precision binary64 (* (sqrt (/ A (* l V))) c0))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	return sqrt((A / (l * V))) * c0;
}

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 = sqrt((a / (l * v))) * c0
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	return Math.sqrt((A / (l * V))) * c0;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	return math.sqrt((A / (l * V))) * c0

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	return Float64(sqrt(Float64(A / Float64(l * V))) * c0)
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp = code(c0, A, V, l)
	tmp = sqrt((A / (l * V))) * c0;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := N[(N[Sqrt[N[(A / N[(l * V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]

Derivation

1. Initial program 71.9%

   \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
2. Add Preprocessing

3. Final simplification 71.9%

   \[\leadsto \sqrt{\frac{A}{\ell \cdot V}} \cdot c0 \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024268 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))