Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.3% → 90.7%

Time: 5.9s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.3% 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: 90.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{-308}:\\ \;\;\;\;\frac{\sqrt{-A} \cdot c0}{\sqrt{-V} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-309}:\\ \;\;\;\;\frac{c0}{\sqrt{\left(\frac{-1}{A} \cdot V\right) \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+277}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A} \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -2e-308)
   (/ (* (sqrt (- A)) c0) (* (sqrt (- V)) (sqrt l)))
   (if (<= (* V l) 1e-309)
     (/ c0 (sqrt (* (* (/ -1.0 A) V) (- l))))
     (if (<= (* V l) 1e+277)
       (* c0 (/ (sqrt A) (sqrt (* l V))))
       (/ c0 (sqrt (* (/ V A) l)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e-308) {
		tmp = (sqrt(-A) * c0) / (sqrt(-V) * sqrt(l));
	} else if ((V * l) <= 1e-309) {
		tmp = c0 / sqrt((((-1.0 / A) * V) * -l));
	} else if ((V * l) <= 1e+277) {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	} else {
		tmp = c0 / sqrt(((V / A) * l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-2d-308)) then
        tmp = (sqrt(-a) * c0) / (sqrt(-v) * sqrt(l))
    else if ((v * l) <= 1d-309) then
        tmp = c0 / sqrt(((((-1.0d0) / a) * v) * -l))
    else if ((v * l) <= 1d+277) then
        tmp = c0 * (sqrt(a) / sqrt((l * v)))
    else
        tmp = c0 / sqrt(((v / a) * l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e-308) {
		tmp = (Math.sqrt(-A) * c0) / (Math.sqrt(-V) * Math.sqrt(l));
	} else if ((V * l) <= 1e-309) {
		tmp = c0 / Math.sqrt((((-1.0 / A) * V) * -l));
	} else if ((V * l) <= 1e+277) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((l * V)));
	} else {
		tmp = c0 / Math.sqrt(((V / A) * l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -2e-308:
		tmp = (math.sqrt(-A) * c0) / (math.sqrt(-V) * math.sqrt(l))
	elif (V * l) <= 1e-309:
		tmp = c0 / math.sqrt((((-1.0 / A) * V) * -l))
	elif (V * l) <= 1e+277:
		tmp = c0 * (math.sqrt(A) / math.sqrt((l * V)))
	else:
		tmp = c0 / math.sqrt(((V / A) * l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -2e-308)
		tmp = Float64(Float64(sqrt(Float64(-A)) * c0) / Float64(sqrt(Float64(-V)) * sqrt(l)));
	elseif (Float64(V * l) <= 1e-309)
		tmp = Float64(c0 / sqrt(Float64(Float64(Float64(-1.0 / A) * V) * Float64(-l))));
	elseif (Float64(V * l) <= 1e+277)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(l * V))));
	else
		tmp = Float64(c0 / sqrt(Float64(Float64(V / A) * l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -2e-308)
		tmp = (sqrt(-A) * c0) / (sqrt(-V) * sqrt(l));
	elseif ((V * l) <= 1e-309)
		tmp = c0 / sqrt((((-1.0 / A) * V) * -l));
	elseif ((V * l) <= 1e+277)
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	else
		tmp = c0 / sqrt(((V / A) * l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -2e-308], N[(N[(N[Sqrt[(-A)], $MachinePrecision] * c0), $MachinePrecision] / N[(N[Sqrt[(-V)], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e-309], N[(c0 / N[Sqrt[N[(N[(N[(-1.0 / A), $MachinePrecision] * V), $MachinePrecision] * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+277], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(N[(V / A), $MachinePrecision] * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -1.9999999999999998e-308

   1. Initial program 79.5%

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

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. frac-2neg N/A

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

      13. sqrt-div N/A

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

      14. frac-times N/A

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

      15. lift-sqrt.f64 N/A

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

      16. sqrt-prod N/A

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

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

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

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

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

      22. sqrt-prod N/A

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

      23. lift-sqrt.f64 N/A

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

   6. Applied rewrites 88.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 47.7

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

   8. Applied rewrites 47.7%

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

   if -1.9999999999999998e-308 < (*.f64 V l) < 1.000000000000002e-309

   1. Initial program 41.8%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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-/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 67.2

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

   4. Applied rewrites 67.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. distribute-frac-neg2 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 67.2

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

   6. Applied rewrites 67.2%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. clear-num N/A

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

      6. sqrt-div N/A

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

      7. div-inv N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. neg-mul-1 N/A

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

      12. lift-neg.f64 N/A

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

      13. remove-double-neg N/A

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

      14. clear-num N/A

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

      15. lift-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-sqrt.f64 N/A

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

   8. Applied rewrites 67.4%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-neg.f64 N/A

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

      4. div-inv N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lift-neg.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. associate-/r* N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 67.4

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

   10. Applied rewrites 67.4%

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

   if 1.000000000000002e-309 < (*.f64 V l) < 1e277

   1. Initial program 82.2%

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

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 99.4

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.4

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

   4. Applied rewrites 99.4%

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

   if 1e277 < (*.f64 V l)

   1. Initial program 36.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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-/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 64.3

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

   4. Applied rewrites 64.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. distribute-frac-neg2 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 64.3

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

   6. Applied rewrites 64.3%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. clear-num N/A

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

      6. sqrt-div N/A

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

      7. div-inv N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. neg-mul-1 N/A

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

      12. lift-neg.f64 N/A

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

      13. remove-double-neg N/A

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

      14. clear-num N/A

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

      15. lift-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-sqrt.f64 N/A

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

   8. Applied rewrites 64.5%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{-308}:\\ \;\;\;\;\frac{\sqrt{-A} \cdot c0}{\sqrt{-V} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-309}:\\ \;\;\;\;\frac{c0}{\sqrt{\left(\frac{-1}{A} \cdot V\right) \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+277}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A} \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 10^{+252}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A} \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (<= t_0 0.0)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 1e+252) t_0 (/ c0 (sqrt (* (/ V A) l)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 1e+252) {
		tmp = t_0;
	} else {
		tmp = c0 / sqrt(((V / A) * l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    if (t_0 <= 0.0d0) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 1d+252) then
        tmp = t_0
    else
        tmp = c0 / sqrt(((v / a) * l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 1e+252) {
		tmp = t_0;
	} else {
		tmp = c0 / Math.sqrt(((V / A) * l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 1e+252:
		tmp = t_0
	else:
		tmp = c0 / math.sqrt(((V / A) * l))
	return tmp

Julia

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

MATLAB

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

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+252], t$95$0, N[(c0 / N[Sqrt[N[(N[(V / A), $MachinePrecision] * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 67.8%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 71.4

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

   4. Applied rewrites 71.4%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 1.0000000000000001e252

   1. Initial program 97.4%

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

   if 1.0000000000000001e252 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 37.4%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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-/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 54.2

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

   4. Applied rewrites 54.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. distribute-frac-neg2 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 54.1

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

   6. Applied rewrites 54.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. clear-num N/A

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

      6. sqrt-div N/A

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

      7. div-inv N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. neg-mul-1 N/A

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

      12. lift-neg.f64 N/A

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

      13. remove-double-neg N/A

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

      14. clear-num N/A

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

      15. lift-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-sqrt.f64 N/A

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

   8. Applied rewrites 54.2%

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

Alternative 3: 76.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 10^{+252}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (<= t_0 0.0)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 1e+252) t_0 (* c0 (sqrt (/ (/ A l) V)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 1e+252) {
		tmp = t_0;
	} else {
		tmp = c0 * sqrt(((A / l) / V));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    if (t_0 <= 0.0d0) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 1d+252) then
        tmp = t_0
    else
        tmp = c0 * sqrt(((a / l) / v))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 1e+252) {
		tmp = t_0;
	} else {
		tmp = c0 * Math.sqrt(((A / l) / V));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 1e+252:
		tmp = t_0
	else:
		tmp = c0 * math.sqrt(((A / l) / V))
	return tmp

Julia

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

MATLAB

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

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+252], t$95$0, N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 67.8%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 71.4

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

   4. Applied rewrites 71.4%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 1.0000000000000001e252

   1. Initial program 97.4%

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

   if 1.0000000000000001e252 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 37.4%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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-/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 54.2

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

   4. Applied rewrites 54.2%

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

Alternative 4: 91.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-308}:\\ \;\;\;\;c0 \cdot \left(\sqrt{-A} \cdot \sqrt{\frac{-1}{V \cdot \ell}}\right)\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-309}:\\ \;\;\;\;\frac{c0}{\sqrt{\left(\frac{-1}{A} \cdot V\right) \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+277}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A} \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) (- INFINITY))
   (* c0 (/ (sqrt (/ (- A) l)) (sqrt (- V))))
   (if (<= (* V l) -2e-308)
     (* c0 (* (sqrt (- A)) (sqrt (/ -1.0 (* V l)))))
     (if (<= (* V l) 1e-309)
       (/ c0 (sqrt (* (* (/ -1.0 A) V) (- l))))
       (if (<= (* V l) 1e+277)
         (* c0 (/ (sqrt A) (sqrt (* l V))))
         (/ c0 (sqrt (* (/ V A) l))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = c0 * (sqrt((-A / l)) / sqrt(-V));
	} else if ((V * l) <= -2e-308) {
		tmp = c0 * (sqrt(-A) * sqrt((-1.0 / (V * l))));
	} else if ((V * l) <= 1e-309) {
		tmp = c0 / sqrt((((-1.0 / A) * V) * -l));
	} else if ((V * l) <= 1e+277) {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	} else {
		tmp = c0 / sqrt(((V / A) * l));
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = c0 * (Math.sqrt((-A / l)) / Math.sqrt(-V));
	} else if ((V * l) <= -2e-308) {
		tmp = c0 * (Math.sqrt(-A) * Math.sqrt((-1.0 / (V * l))));
	} else if ((V * l) <= 1e-309) {
		tmp = c0 / Math.sqrt((((-1.0 / A) * V) * -l));
	} else if ((V * l) <= 1e+277) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((l * V)));
	} else {
		tmp = c0 / Math.sqrt(((V / A) * l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = c0 * (math.sqrt((-A / l)) / math.sqrt(-V))
	elif (V * l) <= -2e-308:
		tmp = c0 * (math.sqrt(-A) * math.sqrt((-1.0 / (V * l))))
	elif (V * l) <= 1e-309:
		tmp = c0 / math.sqrt((((-1.0 / A) * V) * -l))
	elif (V * l) <= 1e+277:
		tmp = c0 * (math.sqrt(A) / math.sqrt((l * V)))
	else:
		tmp = c0 / math.sqrt(((V / A) * l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(c0 * Float64(sqrt(Float64(Float64(-A) / l)) / sqrt(Float64(-V))));
	elseif (Float64(V * l) <= -2e-308)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) * sqrt(Float64(-1.0 / Float64(V * l)))));
	elseif (Float64(V * l) <= 1e-309)
		tmp = Float64(c0 / sqrt(Float64(Float64(Float64(-1.0 / A) * V) * Float64(-l))));
	elseif (Float64(V * l) <= 1e+277)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(l * V))));
	else
		tmp = Float64(c0 / sqrt(Float64(Float64(V / A) * l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = c0 * (sqrt((-A / l)) / sqrt(-V));
	elseif ((V * l) <= -2e-308)
		tmp = c0 * (sqrt(-A) * sqrt((-1.0 / (V * l))));
	elseif ((V * l) <= 1e-309)
		tmp = c0 / sqrt((((-1.0 / A) * V) * -l));
	elseif ((V * l) <= 1e+277)
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	else
		tmp = c0 / sqrt(((V / A) * l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(c0 * N[(N[Sqrt[N[((-A) / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-308], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] * N[Sqrt[N[(-1.0 / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e-309], N[(c0 / N[Sqrt[N[(N[(N[(-1.0 / A), $MachinePrecision] * V), $MachinePrecision] * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+277], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(N[(V / A), $MachinePrecision] * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 41.4%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 66.9

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

   4. Applied rewrites 66.9%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]
   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. 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 31.7

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

   6. Applied rewrites 31.7%

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

   if -inf.0 < (*.f64 V l) < -1.9999999999999998e-308

   1. Initial program 86.5%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 75.9

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

   4. Applied rewrites 75.9%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]
   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. 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. lift-/.f64 N/A

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

      8. frac-2neg N/A

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

      9. div-inv N/A

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

      10. associate-/l* N/A

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

      11. sqrt-prod N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. distribute-frac-neg2 N/A

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

      18. distribute-neg-frac N/A

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

      19. metadata-eval N/A

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

      20. lower-/.f64 99.5

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

   6. Applied rewrites 99.5%

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

   8. Applied rewrites 99.5%

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

   if -1.9999999999999998e-308 < (*.f64 V l) < 1.000000000000002e-309

   1. Initial program 41.8%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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-/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 67.2

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

   4. Applied rewrites 67.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. distribute-frac-neg2 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 67.2

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

   6. Applied rewrites 67.2%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. clear-num N/A

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

      6. sqrt-div N/A

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

      7. div-inv N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. neg-mul-1 N/A

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

      12. lift-neg.f64 N/A

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

      13. remove-double-neg N/A

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

      14. clear-num N/A

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

      15. lift-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-sqrt.f64 N/A

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

   8. Applied rewrites 67.4%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-neg.f64 N/A

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

      4. div-inv N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lift-neg.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. associate-/r* N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 67.4

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

   10. Applied rewrites 67.4%

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

   if 1.000000000000002e-309 < (*.f64 V l) < 1e277

   1. Initial program 82.2%

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

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 99.4

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.4

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

   4. Applied rewrites 99.4%

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

   if 1e277 < (*.f64 V l)

   1. Initial program 36.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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-/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 64.3

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

   4. Applied rewrites 64.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. distribute-frac-neg2 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 64.3

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

   6. Applied rewrites 64.3%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. clear-num N/A

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

      6. sqrt-div N/A

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

      7. div-inv N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. neg-mul-1 N/A

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

      12. lift-neg.f64 N/A

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

      13. remove-double-neg N/A

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

      14. clear-num N/A

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

      15. lift-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-sqrt.f64 N/A

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

   8. Applied rewrites 64.5%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-308}:\\ \;\;\;\;c0 \cdot \left(\sqrt{-A} \cdot \sqrt{\frac{-1}{V \cdot \ell}}\right)\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-309}:\\ \;\;\;\;\frac{c0}{\sqrt{\left(\frac{-1}{A} \cdot V\right) \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+277}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A} \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-308}:\\ \;\;\;\;c0 \cdot \left(\sqrt{-A} \cdot \sqrt{\frac{-1}{V \cdot \ell}}\right)\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-309}:\\ \;\;\;\;\frac{c0}{\sqrt{\left(\frac{-1}{A} \cdot V\right) \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+277}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A} \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) (- INFINITY))
   (/ c0 (* (sqrt (/ V A)) (sqrt l)))
   (if (<= (* V l) -2e-308)
     (* c0 (* (sqrt (- A)) (sqrt (/ -1.0 (* V l)))))
     (if (<= (* V l) 1e-309)
       (/ c0 (sqrt (* (* (/ -1.0 A) V) (- l))))
       (if (<= (* V l) 1e+277)
         (* c0 (/ (sqrt A) (sqrt (* l V))))
         (/ c0 (sqrt (* (/ V A) l))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = c0 / (sqrt((V / A)) * sqrt(l));
	} else if ((V * l) <= -2e-308) {
		tmp = c0 * (sqrt(-A) * sqrt((-1.0 / (V * l))));
	} else if ((V * l) <= 1e-309) {
		tmp = c0 / sqrt((((-1.0 / A) * V) * -l));
	} else if ((V * l) <= 1e+277) {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	} else {
		tmp = c0 / sqrt(((V / A) * l));
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = c0 / (Math.sqrt((V / A)) * Math.sqrt(l));
	} else if ((V * l) <= -2e-308) {
		tmp = c0 * (Math.sqrt(-A) * Math.sqrt((-1.0 / (V * l))));
	} else if ((V * l) <= 1e-309) {
		tmp = c0 / Math.sqrt((((-1.0 / A) * V) * -l));
	} else if ((V * l) <= 1e+277) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((l * V)));
	} else {
		tmp = c0 / Math.sqrt(((V / A) * l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = c0 / (math.sqrt((V / A)) * math.sqrt(l))
	elif (V * l) <= -2e-308:
		tmp = c0 * (math.sqrt(-A) * math.sqrt((-1.0 / (V * l))))
	elif (V * l) <= 1e-309:
		tmp = c0 / math.sqrt((((-1.0 / A) * V) * -l))
	elif (V * l) <= 1e+277:
		tmp = c0 * (math.sqrt(A) / math.sqrt((l * V)))
	else:
		tmp = c0 / math.sqrt(((V / A) * l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(c0 / Float64(sqrt(Float64(V / A)) * sqrt(l)));
	elseif (Float64(V * l) <= -2e-308)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) * sqrt(Float64(-1.0 / Float64(V * l)))));
	elseif (Float64(V * l) <= 1e-309)
		tmp = Float64(c0 / sqrt(Float64(Float64(Float64(-1.0 / A) * V) * Float64(-l))));
	elseif (Float64(V * l) <= 1e+277)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(l * V))));
	else
		tmp = Float64(c0 / sqrt(Float64(Float64(V / A) * l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = c0 / (sqrt((V / A)) * sqrt(l));
	elseif ((V * l) <= -2e-308)
		tmp = c0 * (sqrt(-A) * sqrt((-1.0 / (V * l))));
	elseif ((V * l) <= 1e-309)
		tmp = c0 / sqrt((((-1.0 / A) * V) * -l));
	elseif ((V * l) <= 1e+277)
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	else
		tmp = c0 / sqrt(((V / A) * l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(c0 / N[(N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-308], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] * N[Sqrt[N[(-1.0 / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e-309], N[(c0 / N[Sqrt[N[(N[(N[(-1.0 / A), $MachinePrecision] * V), $MachinePrecision] * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+277], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(N[(V / A), $MachinePrecision] * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 41.4%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 66.9

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

   4. Applied rewrites 66.9%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]
   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. associate-*r/ N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. clear-num N/A

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

      11. lift-/.f64 N/A

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

      12. sqrt-div N/A

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

      13. metadata-eval N/A

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

      14. lift-sqrt.f64 N/A

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

      15. div-inv N/A

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

      16. associate-/r* N/A

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

      17. lower-/.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-sqrt.f64 N/A

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

      20. lift-sqrt.f64 N/A

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

      21. sqrt-prod N/A

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

      22. lift-/.f64 N/A

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

      23. clear-num N/A

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

      24. lift-/.f64 N/A

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

   6. Applied rewrites 64.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. frac-2neg N/A

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

      6. distribute-rgt-neg-out N/A

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

      7. lift-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. sqrt-div N/A

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

      12. lift-*.f64 N/A

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

      13. sqrt-prod N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. associate-*l/ N/A

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

      17. lower-*.f64 N/A

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

   8. Applied rewrites 31.7%

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

   if -inf.0 < (*.f64 V l) < -1.9999999999999998e-308

   1. Initial program 86.5%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 75.9

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

   4. Applied rewrites 75.9%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]
   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. 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. lift-/.f64 N/A

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

      8. frac-2neg N/A

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

      9. div-inv N/A

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

      10. associate-/l* N/A

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

      11. sqrt-prod N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. distribute-frac-neg2 N/A

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

      18. distribute-neg-frac N/A

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

      19. metadata-eval N/A

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

      20. lower-/.f64 99.5

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

   6. Applied rewrites 99.5%

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

   8. Applied rewrites 99.5%

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

   if -1.9999999999999998e-308 < (*.f64 V l) < 1.000000000000002e-309

   1. Initial program 41.8%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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-/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 67.2

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

   4. Applied rewrites 67.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. distribute-frac-neg2 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 67.2

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

   6. Applied rewrites 67.2%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. clear-num N/A

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

      6. sqrt-div N/A

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

      7. div-inv N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. neg-mul-1 N/A

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

      12. lift-neg.f64 N/A

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

      13. remove-double-neg N/A

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

      14. clear-num N/A

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

      15. lift-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-sqrt.f64 N/A

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

   8. Applied rewrites 67.4%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-neg.f64 N/A

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

      4. div-inv N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lift-neg.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. associate-/r* N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 67.4

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

   10. Applied rewrites 67.4%

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

   if 1.000000000000002e-309 < (*.f64 V l) < 1e277

   1. Initial program 82.2%

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

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 99.4

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.4

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

   4. Applied rewrites 99.4%

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

   if 1e277 < (*.f64 V l)

   1. Initial program 36.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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-/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 64.3

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

   4. Applied rewrites 64.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. distribute-frac-neg2 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 64.3

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

   6. Applied rewrites 64.3%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. clear-num N/A

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

      6. sqrt-div N/A

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

      7. div-inv N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. neg-mul-1 N/A

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

      12. lift-neg.f64 N/A

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

      13. remove-double-neg N/A

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

      14. clear-num N/A

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

      15. lift-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-sqrt.f64 N/A

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

   8. Applied rewrites 64.5%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-308}:\\ \;\;\;\;c0 \cdot \left(\sqrt{-A} \cdot \sqrt{\frac{-1}{V \cdot \ell}}\right)\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-309}:\\ \;\;\;\;\frac{c0}{\sqrt{\left(\frac{-1}{A} \cdot V\right) \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+277}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A} \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-308}:\\ \;\;\;\;\frac{\sqrt{-A} \cdot c0}{\sqrt{\left(-V\right) \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-309}:\\ \;\;\;\;\frac{c0}{\sqrt{\left(\frac{-1}{A} \cdot V\right) \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+277}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A} \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) (- INFINITY))
   (/ c0 (* (sqrt (/ V A)) (sqrt l)))
   (if (<= (* V l) -2e-308)
     (/ (* (sqrt (- A)) c0) (sqrt (* (- V) l)))
     (if (<= (* V l) 1e-309)
       (/ c0 (sqrt (* (* (/ -1.0 A) V) (- l))))
       (if (<= (* V l) 1e+277)
         (* c0 (/ (sqrt A) (sqrt (* l V))))
         (/ c0 (sqrt (* (/ V A) l))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = c0 / (sqrt((V / A)) * sqrt(l));
	} else if ((V * l) <= -2e-308) {
		tmp = (sqrt(-A) * c0) / sqrt((-V * l));
	} else if ((V * l) <= 1e-309) {
		tmp = c0 / sqrt((((-1.0 / A) * V) * -l));
	} else if ((V * l) <= 1e+277) {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	} else {
		tmp = c0 / sqrt(((V / A) * l));
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = c0 / (Math.sqrt((V / A)) * Math.sqrt(l));
	} else if ((V * l) <= -2e-308) {
		tmp = (Math.sqrt(-A) * c0) / Math.sqrt((-V * l));
	} else if ((V * l) <= 1e-309) {
		tmp = c0 / Math.sqrt((((-1.0 / A) * V) * -l));
	} else if ((V * l) <= 1e+277) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((l * V)));
	} else {
		tmp = c0 / Math.sqrt(((V / A) * l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = c0 / (math.sqrt((V / A)) * math.sqrt(l))
	elif (V * l) <= -2e-308:
		tmp = (math.sqrt(-A) * c0) / math.sqrt((-V * l))
	elif (V * l) <= 1e-309:
		tmp = c0 / math.sqrt((((-1.0 / A) * V) * -l))
	elif (V * l) <= 1e+277:
		tmp = c0 * (math.sqrt(A) / math.sqrt((l * V)))
	else:
		tmp = c0 / math.sqrt(((V / A) * l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(c0 / Float64(sqrt(Float64(V / A)) * sqrt(l)));
	elseif (Float64(V * l) <= -2e-308)
		tmp = Float64(Float64(sqrt(Float64(-A)) * c0) / sqrt(Float64(Float64(-V) * l)));
	elseif (Float64(V * l) <= 1e-309)
		tmp = Float64(c0 / sqrt(Float64(Float64(Float64(-1.0 / A) * V) * Float64(-l))));
	elseif (Float64(V * l) <= 1e+277)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(l * V))));
	else
		tmp = Float64(c0 / sqrt(Float64(Float64(V / A) * l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = c0 / (sqrt((V / A)) * sqrt(l));
	elseif ((V * l) <= -2e-308)
		tmp = (sqrt(-A) * c0) / sqrt((-V * l));
	elseif ((V * l) <= 1e-309)
		tmp = c0 / sqrt((((-1.0 / A) * V) * -l));
	elseif ((V * l) <= 1e+277)
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	else
		tmp = c0 / sqrt(((V / A) * l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(c0 / N[(N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-308], N[(N[(N[Sqrt[(-A)], $MachinePrecision] * c0), $MachinePrecision] / N[Sqrt[N[((-V) * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e-309], N[(c0 / N[Sqrt[N[(N[(N[(-1.0 / A), $MachinePrecision] * V), $MachinePrecision] * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+277], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(N[(V / A), $MachinePrecision] * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 41.4%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 66.9

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

   4. Applied rewrites 66.9%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]
   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. associate-*r/ N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. clear-num N/A

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

      11. lift-/.f64 N/A

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

      12. sqrt-div N/A

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

      13. metadata-eval N/A

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

      14. lift-sqrt.f64 N/A

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

      15. div-inv N/A

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

      16. associate-/r* N/A

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

      17. lower-/.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-sqrt.f64 N/A

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

      20. lift-sqrt.f64 N/A

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

      21. sqrt-prod N/A

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

      22. lift-/.f64 N/A

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

      23. clear-num N/A

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

      24. lift-/.f64 N/A

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

   6. Applied rewrites 64.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. frac-2neg N/A

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

      6. distribute-rgt-neg-out N/A

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

      7. lift-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. sqrt-div N/A

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

      12. lift-*.f64 N/A

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

      13. sqrt-prod N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. associate-*l/ N/A

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

      17. lower-*.f64 N/A

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

   8. Applied rewrites 31.7%

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

   if -inf.0 < (*.f64 V l) < -1.9999999999999998e-308

   1. Initial program 86.5%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 75.9

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

   4. Applied rewrites 75.9%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]
   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. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. frac-2neg N/A

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

      13. sqrt-div N/A

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

      14. frac-times N/A

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

      15. lift-sqrt.f64 N/A

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

      16. sqrt-prod N/A

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

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

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

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

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

      22. sqrt-prod N/A

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

      23. lift-sqrt.f64 N/A

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

   6. Applied rewrites 97.3%

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

   if -1.9999999999999998e-308 < (*.f64 V l) < 1.000000000000002e-309

   1. Initial program 41.8%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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-/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 67.2

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

   4. Applied rewrites 67.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. distribute-frac-neg2 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 67.2

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

   6. Applied rewrites 67.2%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. clear-num N/A

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

      6. sqrt-div N/A

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

      7. div-inv N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. neg-mul-1 N/A

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

      12. lift-neg.f64 N/A

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

      13. remove-double-neg N/A

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

      14. clear-num N/A

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

      15. lift-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-sqrt.f64 N/A

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

   8. Applied rewrites 67.4%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-neg.f64 N/A

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

      4. div-inv N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lift-neg.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. associate-/r* N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 67.4

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

   10. Applied rewrites 67.4%

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

   if 1.000000000000002e-309 < (*.f64 V l) < 1e277

   1. Initial program 82.2%

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

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 99.4

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.4

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

   4. Applied rewrites 99.4%

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

   if 1e277 < (*.f64 V l)

   1. Initial program 36.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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-/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 64.3

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

   4. Applied rewrites 64.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. distribute-frac-neg2 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 64.3

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

   6. Applied rewrites 64.3%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. clear-num N/A

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

      6. sqrt-div N/A

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

      7. div-inv N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. neg-mul-1 N/A

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

      12. lift-neg.f64 N/A

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

      13. remove-double-neg N/A

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

      14. clear-num N/A

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

      15. lift-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-sqrt.f64 N/A

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

   8. Applied rewrites 64.5%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-308}:\\ \;\;\;\;\frac{\sqrt{-A} \cdot c0}{\sqrt{\left(-V\right) \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-309}:\\ \;\;\;\;\frac{c0}{\sqrt{\left(\frac{-1}{A} \cdot V\right) \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+277}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A} \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{-1}{\ell} \cdot A}{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-308}:\\ \;\;\;\;\frac{\sqrt{-A} \cdot c0}{\sqrt{\left(-V\right) \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-309}:\\ \;\;\;\;\frac{c0}{\sqrt{\left(\frac{-1}{A} \cdot V\right) \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+277}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A} \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) (- INFINITY))
   (* c0 (sqrt (/ (* (/ -1.0 l) A) (- V))))
   (if (<= (* V l) -2e-308)
     (/ (* (sqrt (- A)) c0) (sqrt (* (- V) l)))
     (if (<= (* V l) 1e-309)
       (/ c0 (sqrt (* (* (/ -1.0 A) V) (- l))))
       (if (<= (* V l) 1e+277)
         (* c0 (/ (sqrt A) (sqrt (* l V))))
         (/ c0 (sqrt (* (/ V A) l))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = c0 * sqrt((((-1.0 / l) * A) / -V));
	} else if ((V * l) <= -2e-308) {
		tmp = (sqrt(-A) * c0) / sqrt((-V * l));
	} else if ((V * l) <= 1e-309) {
		tmp = c0 / sqrt((((-1.0 / A) * V) * -l));
	} else if ((V * l) <= 1e+277) {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	} else {
		tmp = c0 / sqrt(((V / A) * l));
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = c0 * Math.sqrt((((-1.0 / l) * A) / -V));
	} else if ((V * l) <= -2e-308) {
		tmp = (Math.sqrt(-A) * c0) / Math.sqrt((-V * l));
	} else if ((V * l) <= 1e-309) {
		tmp = c0 / Math.sqrt((((-1.0 / A) * V) * -l));
	} else if ((V * l) <= 1e+277) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((l * V)));
	} else {
		tmp = c0 / Math.sqrt(((V / A) * l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = c0 * math.sqrt((((-1.0 / l) * A) / -V))
	elif (V * l) <= -2e-308:
		tmp = (math.sqrt(-A) * c0) / math.sqrt((-V * l))
	elif (V * l) <= 1e-309:
		tmp = c0 / math.sqrt((((-1.0 / A) * V) * -l))
	elif (V * l) <= 1e+277:
		tmp = c0 * (math.sqrt(A) / math.sqrt((l * V)))
	else:
		tmp = c0 / math.sqrt(((V / A) * l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(c0 * sqrt(Float64(Float64(Float64(-1.0 / l) * A) / Float64(-V))));
	elseif (Float64(V * l) <= -2e-308)
		tmp = Float64(Float64(sqrt(Float64(-A)) * c0) / sqrt(Float64(Float64(-V) * l)));
	elseif (Float64(V * l) <= 1e-309)
		tmp = Float64(c0 / sqrt(Float64(Float64(Float64(-1.0 / A) * V) * Float64(-l))));
	elseif (Float64(V * l) <= 1e+277)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(l * V))));
	else
		tmp = Float64(c0 / sqrt(Float64(Float64(V / A) * l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = c0 * sqrt((((-1.0 / l) * A) / -V));
	elseif ((V * l) <= -2e-308)
		tmp = (sqrt(-A) * c0) / sqrt((-V * l));
	elseif ((V * l) <= 1e-309)
		tmp = c0 / sqrt((((-1.0 / A) * V) * -l));
	elseif ((V * l) <= 1e+277)
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	else
		tmp = c0 / sqrt(((V / A) * l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(c0 * N[Sqrt[N[(N[(N[(-1.0 / l), $MachinePrecision] * A), $MachinePrecision] / (-V)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-308], N[(N[(N[Sqrt[(-A)], $MachinePrecision] * c0), $MachinePrecision] / N[Sqrt[N[((-V) * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e-309], N[(c0 / N[Sqrt[N[(N[(N[(-1.0 / A), $MachinePrecision] * V), $MachinePrecision] * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+277], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(N[(V / A), $MachinePrecision] * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 41.4%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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-/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 66.9

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

   4. Applied rewrites 66.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. distribute-frac-neg2 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 66.9

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

   6. Applied rewrites 66.9%

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

   if -inf.0 < (*.f64 V l) < -1.9999999999999998e-308

   1. Initial program 86.5%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 75.9

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

   4. Applied rewrites 75.9%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]
   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. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. frac-2neg N/A

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

      13. sqrt-div N/A

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

      14. frac-times N/A

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

      15. lift-sqrt.f64 N/A

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

      16. sqrt-prod N/A

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

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

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

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

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

      22. sqrt-prod N/A

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

      23. lift-sqrt.f64 N/A

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

   6. Applied rewrites 97.3%

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

   if -1.9999999999999998e-308 < (*.f64 V l) < 1.000000000000002e-309

   1. Initial program 41.8%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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-/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 67.2

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

   4. Applied rewrites 67.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. distribute-frac-neg2 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 67.2

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

   6. Applied rewrites 67.2%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. clear-num N/A

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

      6. sqrt-div N/A

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

      7. div-inv N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. neg-mul-1 N/A

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

      12. lift-neg.f64 N/A

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

      13. remove-double-neg N/A

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

      14. clear-num N/A

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

      15. lift-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-sqrt.f64 N/A

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

   8. Applied rewrites 67.4%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-neg.f64 N/A

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

      4. div-inv N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lift-neg.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. associate-/r* N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 67.4

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

   10. Applied rewrites 67.4%

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

   if 1.000000000000002e-309 < (*.f64 V l) < 1e277

   1. Initial program 82.2%

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

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 99.4

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.4

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

   4. Applied rewrites 99.4%

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

   if 1e277 < (*.f64 V l)

   1. Initial program 36.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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-/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 64.3

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

   4. Applied rewrites 64.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. distribute-frac-neg2 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 64.3

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

   6. Applied rewrites 64.3%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. clear-num N/A

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

      6. sqrt-div N/A

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

      7. div-inv N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. neg-mul-1 N/A

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

      12. lift-neg.f64 N/A

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

      13. remove-double-neg N/A

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

      14. clear-num N/A

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

      15. lift-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-sqrt.f64 N/A

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

   8. Applied rewrites 64.5%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{-1}{\ell} \cdot A}{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-308}:\\ \;\;\;\;\frac{\sqrt{-A} \cdot c0}{\sqrt{\left(-V\right) \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-309}:\\ \;\;\;\;\frac{c0}{\sqrt{\left(\frac{-1}{A} \cdot V\right) \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+277}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A} \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{-1}{\ell} \cdot A}{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-308}:\\ \;\;\;\;\frac{c0}{\sqrt{\left(-\ell\right) \cdot V}} \cdot \sqrt{-A}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-309}:\\ \;\;\;\;\frac{c0}{\sqrt{\left(\frac{-1}{A} \cdot V\right) \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+277}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A} \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) (- INFINITY))
   (* c0 (sqrt (/ (* (/ -1.0 l) A) (- V))))
   (if (<= (* V l) -2e-308)
     (* (/ c0 (sqrt (* (- l) V))) (sqrt (- A)))
     (if (<= (* V l) 1e-309)
       (/ c0 (sqrt (* (* (/ -1.0 A) V) (- l))))
       (if (<= (* V l) 1e+277)
         (* c0 (/ (sqrt A) (sqrt (* l V))))
         (/ c0 (sqrt (* (/ V A) l))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = c0 * sqrt((((-1.0 / l) * A) / -V));
	} else if ((V * l) <= -2e-308) {
		tmp = (c0 / sqrt((-l * V))) * sqrt(-A);
	} else if ((V * l) <= 1e-309) {
		tmp = c0 / sqrt((((-1.0 / A) * V) * -l));
	} else if ((V * l) <= 1e+277) {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	} else {
		tmp = c0 / sqrt(((V / A) * l));
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = c0 * Math.sqrt((((-1.0 / l) * A) / -V));
	} else if ((V * l) <= -2e-308) {
		tmp = (c0 / Math.sqrt((-l * V))) * Math.sqrt(-A);
	} else if ((V * l) <= 1e-309) {
		tmp = c0 / Math.sqrt((((-1.0 / A) * V) * -l));
	} else if ((V * l) <= 1e+277) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((l * V)));
	} else {
		tmp = c0 / Math.sqrt(((V / A) * l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = c0 * math.sqrt((((-1.0 / l) * A) / -V))
	elif (V * l) <= -2e-308:
		tmp = (c0 / math.sqrt((-l * V))) * math.sqrt(-A)
	elif (V * l) <= 1e-309:
		tmp = c0 / math.sqrt((((-1.0 / A) * V) * -l))
	elif (V * l) <= 1e+277:
		tmp = c0 * (math.sqrt(A) / math.sqrt((l * V)))
	else:
		tmp = c0 / math.sqrt(((V / A) * l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(c0 * sqrt(Float64(Float64(Float64(-1.0 / l) * A) / Float64(-V))));
	elseif (Float64(V * l) <= -2e-308)
		tmp = Float64(Float64(c0 / sqrt(Float64(Float64(-l) * V))) * sqrt(Float64(-A)));
	elseif (Float64(V * l) <= 1e-309)
		tmp = Float64(c0 / sqrt(Float64(Float64(Float64(-1.0 / A) * V) * Float64(-l))));
	elseif (Float64(V * l) <= 1e+277)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(l * V))));
	else
		tmp = Float64(c0 / sqrt(Float64(Float64(V / A) * l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = c0 * sqrt((((-1.0 / l) * A) / -V));
	elseif ((V * l) <= -2e-308)
		tmp = (c0 / sqrt((-l * V))) * sqrt(-A);
	elseif ((V * l) <= 1e-309)
		tmp = c0 / sqrt((((-1.0 / A) * V) * -l));
	elseif ((V * l) <= 1e+277)
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	else
		tmp = c0 / sqrt(((V / A) * l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(c0 * N[Sqrt[N[(N[(N[(-1.0 / l), $MachinePrecision] * A), $MachinePrecision] / (-V)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-308], N[(N[(c0 / N[Sqrt[N[((-l) * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[(-A)], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e-309], N[(c0 / N[Sqrt[N[(N[(N[(-1.0 / A), $MachinePrecision] * V), $MachinePrecision] * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+277], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(N[(V / A), $MachinePrecision] * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 41.4%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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-/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 66.9

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

   4. Applied rewrites 66.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. distribute-frac-neg2 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 66.9

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

   6. Applied rewrites 66.9%

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

   if -inf.0 < (*.f64 V l) < -1.9999999999999998e-308

   1. Initial program 86.5%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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-/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 74.3

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

   4. Applied rewrites 74.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. distribute-frac-neg2 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 74.3

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

   6. Applied rewrites 74.3%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. clear-num N/A

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

      6. sqrt-div N/A

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

      7. div-inv N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. neg-mul-1 N/A

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

      12. lift-neg.f64 N/A

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

      13. remove-double-neg N/A

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

      14. clear-num N/A

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

      15. lift-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-sqrt.f64 N/A

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

   8. Applied rewrites 74.2%

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

   10. Applied rewrites 94.9%

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

   if -1.9999999999999998e-308 < (*.f64 V l) < 1.000000000000002e-309

   1. Initial program 41.8%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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-/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 67.2

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

   4. Applied rewrites 67.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. distribute-frac-neg2 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 67.2

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

   6. Applied rewrites 67.2%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. clear-num N/A

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

      6. sqrt-div N/A

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

      7. div-inv N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. neg-mul-1 N/A

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

      12. lift-neg.f64 N/A

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

      13. remove-double-neg N/A

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

      14. clear-num N/A

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

      15. lift-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-sqrt.f64 N/A

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

   8. Applied rewrites 67.4%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-neg.f64 N/A

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

      4. div-inv N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lift-neg.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. associate-/r* N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 67.4

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

   10. Applied rewrites 67.4%

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

   if 1.000000000000002e-309 < (*.f64 V l) < 1e277

   1. Initial program 82.2%

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

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 99.4

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.4

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

   4. Applied rewrites 99.4%

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

   if 1e277 < (*.f64 V l)

   1. Initial program 36.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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-/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 64.3

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

   4. Applied rewrites 64.3%

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

      1. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{\ell}}}{V}} \]

      2. clear-num N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{1}{\frac{\ell}{A}}}}{V}} \]

      3. frac-2neg N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{1}{\color{blue}{\frac{\mathsf{neg}\left(\ell\right)}{\mathsf{neg}\left(A\right)}}}}{V}} \]

      4. associate-/r/ N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot \left(\mathsf{neg}\left(A\right)\right)}}{V}} \]

      5. lower-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot \left(\mathsf{neg}\left(A\right)\right)}}{V}} \]

      6. distribute-frac-neg2 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{\ell}\right)\right)} \cdot \left(\mathsf{neg}\left(A\right)\right)}{V}} \]

      7. distribute-neg-frac N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{\mathsf{neg}\left(1\right)}{\ell}} \cdot \left(\mathsf{neg}\left(A\right)\right)}{V}} \]

      8. metadata-eval N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{\color{blue}{-1}}{\ell} \cdot \left(\mathsf{neg}\left(A\right)\right)}{V}} \]

      9. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{-1}{\ell}} \cdot \left(\mathsf{neg}\left(A\right)\right)}{V}} \]

      10. lower-neg.f64 64.3

          \[\leadsto c0 \cdot \sqrt{\frac{\frac{-1}{\ell} \cdot \color{blue}{\left(-A\right)}}{V}} \]

   6. Applied rewrites 64.3%

      \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{-1}{\ell} \cdot \left(-A\right)}}{V}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{c0 \cdot \sqrt{\frac{\frac{-1}{\ell} \cdot \left(-A\right)}{V}}} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\sqrt{\frac{\frac{-1}{\ell} \cdot \left(-A\right)}{V}}} \]

      3. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{-1}{\ell} \cdot \left(-A\right)}{V}}} \]

      4. sqrt-div N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{-1}{\ell} \cdot \left(-A\right)}}{\sqrt{V}}} \]

      5. clear-num N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{1}{\frac{\sqrt{V}}{\sqrt{\frac{-1}{\ell} \cdot \left(-A\right)}}}} \]

      6. sqrt-div N/A

         \[\leadsto c0 \cdot \frac{1}{\color{blue}{\sqrt{\frac{V}{\frac{-1}{\ell} \cdot \left(-A\right)}}}} \]

      7. div-inv N/A

         \[\leadsto c0 \cdot \frac{1}{\sqrt{\color{blue}{V \cdot \frac{1}{\frac{-1}{\ell} \cdot \left(-A\right)}}}} \]

      8. lift-*.f64 N/A

         \[\leadsto c0 \cdot \frac{1}{\sqrt{V \cdot \frac{1}{\color{blue}{\frac{-1}{\ell} \cdot \left(-A\right)}}}} \]

      9. lift-/.f64 N/A

         \[\leadsto c0 \cdot \frac{1}{\sqrt{V \cdot \frac{1}{\color{blue}{\frac{-1}{\ell}} \cdot \left(-A\right)}}} \]

      10. associate-*l/ N/A

          \[\leadsto c0 \cdot \frac{1}{\sqrt{V \cdot \frac{1}{\color{blue}{\frac{-1 \cdot \left(-A\right)}{\ell}}}}} \]

      11. neg-mul-1 N/A

          \[\leadsto c0 \cdot \frac{1}{\sqrt{V \cdot \frac{1}{\frac{\color{blue}{\mathsf{neg}\left(\left(-A\right)\right)}}{\ell}}}} \]

      12. lift-neg.f64 N/A

          \[\leadsto c0 \cdot \frac{1}{\sqrt{V \cdot \frac{1}{\frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(A\right)\right)}\right)}{\ell}}}} \]

      13. remove-double-neg N/A

          \[\leadsto c0 \cdot \frac{1}{\sqrt{V \cdot \frac{1}{\frac{\color{blue}{A}}{\ell}}}} \]

      14. clear-num N/A

          \[\leadsto c0 \cdot \frac{1}{\sqrt{V \cdot \color{blue}{\frac{\ell}{A}}}} \]

      15. lift-/.f64 N/A

          \[\leadsto c0 \cdot \frac{1}{\sqrt{V \cdot \color{blue}{\frac{\ell}{A}}}} \]

      16. *-commutative N/A

          \[\leadsto c0 \cdot \frac{1}{\sqrt{\color{blue}{\frac{\ell}{A} \cdot V}}} \]

      17. lift-*.f64 N/A

          \[\leadsto c0 \cdot \frac{1}{\sqrt{\color{blue}{\frac{\ell}{A} \cdot V}}} \]

      18. lift-sqrt.f64 N/A

          \[\leadsto c0 \cdot \frac{1}{\color{blue}{\sqrt{\frac{\ell}{A} \cdot V}}} \]

   8. Applied rewrites 64.5%

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{\frac{V}{A} \cdot \ell}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{-1}{\ell} \cdot A}{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-308}:\\ \;\;\;\;\frac{c0}{\sqrt{\left(-\ell\right) \cdot V}} \cdot \sqrt{-A}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-309}:\\ \;\;\;\;\frac{c0}{\sqrt{\left(\frac{-1}{A} \cdot V\right) \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+277}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A} \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 2 \cdot 10^{+307}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (or (<= t_0 0.0) (not (<= t_0 2e+307)))
     (* c0 (sqrt (/ (/ A V) l)))
     (* c0 (sqrt t_0)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 2e+307)) {
		tmp = c0 * sqrt(((A / V) / l));
	} else {
		tmp = c0 * sqrt(t_0);
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if ((t_0 <= 0.0d0) .or. (.not. (t_0 <= 2d+307))) then
        tmp = c0 * sqrt(((a / v) / l))
    else
        tmp = c0 * sqrt(t_0)
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 2e+307)) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else {
		tmp = c0 * Math.sqrt(t_0);
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 2e+307):
		tmp = c0 * math.sqrt(((A / V) / l))
	else:
		tmp = c0 * math.sqrt(t_0)
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 2e+307))
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	else
		tmp = Float64(c0 * sqrt(t_0));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 2e+307)))
		tmp = c0 * sqrt(((A / V) / l));
	else
		tmp = c0 * sqrt(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 2e+307]], $MachinePrecision]], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 0.0 or 1.99999999999999997e307 < (/.f64 A (*.f64 V l))

   1. Initial program 35.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 52.6

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{V}}}{\ell}} \]

   4. Applied rewrites 52.6%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

   if 0.0 < (/.f64 A (*.f64 V l)) < 1.99999999999999997e307

   1. Initial program 97.5%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0 \lor \neg \left(\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{+307}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 10^{-309}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+277}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A} \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) 1e-309)
   (* c0 (sqrt (/ (/ A V) l)))
   (if (<= (* V l) 1e+277)
     (* c0 (/ (sqrt A) (sqrt (* l V))))
     (/ c0 (sqrt (* (/ V A) l))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= 1e-309) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if ((V * l) <= 1e+277) {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	} else {
		tmp = c0 / sqrt(((V / A) * l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= 1d-309) then
        tmp = c0 * sqrt(((a / v) / l))
    else if ((v * l) <= 1d+277) then
        tmp = c0 * (sqrt(a) / sqrt((l * v)))
    else
        tmp = c0 / sqrt(((v / a) * l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= 1e-309) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if ((V * l) <= 1e+277) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((l * V)));
	} else {
		tmp = c0 / Math.sqrt(((V / A) * l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= 1e-309:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif (V * l) <= 1e+277:
		tmp = c0 * (math.sqrt(A) / math.sqrt((l * V)))
	else:
		tmp = c0 / math.sqrt(((V / A) * l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= 1e-309)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (Float64(V * l) <= 1e+277)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(l * V))));
	else
		tmp = Float64(c0 / sqrt(Float64(Float64(V / A) * l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= 1e-309)
		tmp = c0 * sqrt(((A / V) / l));
	elseif ((V * l) <= 1e+277)
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	else
		tmp = c0 / sqrt(((V / A) * l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], 1e-309], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+277], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(N[(V / A), $MachinePrecision] * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < 1.000000000000002e-309

   1. Initial program 69.9%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 72.6

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{V}}}{\ell}} \]

   4. Applied rewrites 72.6%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

   if 1.000000000000002e-309 < (*.f64 V l) < 1e277

   1. Initial program 82.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. sqrt-div N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}} \]

      4. lower-/.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{A}}}{\sqrt{V \cdot \ell}} \]

      6. lower-sqrt.f64 99.4

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\color{blue}{\sqrt{V \cdot \ell}}} \]

      7. lift-*.f64 N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{V \cdot \ell}}} \]

      8. *-commutative N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{\ell \cdot V}}} \]

      9. lower-*.f64 99.4

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{\ell \cdot V}}} \]

   4. Applied rewrites 99.4%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{\ell \cdot V}}} \]

   if 1e277 < (*.f64 V l)

   1. Initial program 36.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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-/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 64.3

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{\ell}}}{V}} \]

   4. Applied rewrites 64.3%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{\ell}}{V}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{\ell}}}{V}} \]

      2. clear-num N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{1}{\frac{\ell}{A}}}}{V}} \]

      3. frac-2neg N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{1}{\color{blue}{\frac{\mathsf{neg}\left(\ell\right)}{\mathsf{neg}\left(A\right)}}}}{V}} \]

      4. associate-/r/ N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot \left(\mathsf{neg}\left(A\right)\right)}}{V}} \]

      5. lower-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot \left(\mathsf{neg}\left(A\right)\right)}}{V}} \]

      6. distribute-frac-neg2 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{\ell}\right)\right)} \cdot \left(\mathsf{neg}\left(A\right)\right)}{V}} \]

      7. distribute-neg-frac N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{\mathsf{neg}\left(1\right)}{\ell}} \cdot \left(\mathsf{neg}\left(A\right)\right)}{V}} \]

      8. metadata-eval N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{\color{blue}{-1}}{\ell} \cdot \left(\mathsf{neg}\left(A\right)\right)}{V}} \]

      9. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{-1}{\ell}} \cdot \left(\mathsf{neg}\left(A\right)\right)}{V}} \]

      10. lower-neg.f64 64.3

          \[\leadsto c0 \cdot \sqrt{\frac{\frac{-1}{\ell} \cdot \color{blue}{\left(-A\right)}}{V}} \]

   6. Applied rewrites 64.3%

      \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{-1}{\ell} \cdot \left(-A\right)}}{V}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{c0 \cdot \sqrt{\frac{\frac{-1}{\ell} \cdot \left(-A\right)}{V}}} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\sqrt{\frac{\frac{-1}{\ell} \cdot \left(-A\right)}{V}}} \]

      3. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{-1}{\ell} \cdot \left(-A\right)}{V}}} \]

      4. sqrt-div N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{-1}{\ell} \cdot \left(-A\right)}}{\sqrt{V}}} \]

      5. clear-num N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{1}{\frac{\sqrt{V}}{\sqrt{\frac{-1}{\ell} \cdot \left(-A\right)}}}} \]

      6. sqrt-div N/A

         \[\leadsto c0 \cdot \frac{1}{\color{blue}{\sqrt{\frac{V}{\frac{-1}{\ell} \cdot \left(-A\right)}}}} \]

      7. div-inv N/A

         \[\leadsto c0 \cdot \frac{1}{\sqrt{\color{blue}{V \cdot \frac{1}{\frac{-1}{\ell} \cdot \left(-A\right)}}}} \]

      8. lift-*.f64 N/A

         \[\leadsto c0 \cdot \frac{1}{\sqrt{V \cdot \frac{1}{\color{blue}{\frac{-1}{\ell} \cdot \left(-A\right)}}}} \]

      9. lift-/.f64 N/A

         \[\leadsto c0 \cdot \frac{1}{\sqrt{V \cdot \frac{1}{\color{blue}{\frac{-1}{\ell}} \cdot \left(-A\right)}}} \]

      10. associate-*l/ N/A

          \[\leadsto c0 \cdot \frac{1}{\sqrt{V \cdot \frac{1}{\color{blue}{\frac{-1 \cdot \left(-A\right)}{\ell}}}}} \]

      11. neg-mul-1 N/A

          \[\leadsto c0 \cdot \frac{1}{\sqrt{V \cdot \frac{1}{\frac{\color{blue}{\mathsf{neg}\left(\left(-A\right)\right)}}{\ell}}}} \]

      12. lift-neg.f64 N/A

          \[\leadsto c0 \cdot \frac{1}{\sqrt{V \cdot \frac{1}{\frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(A\right)\right)}\right)}{\ell}}}} \]

      13. remove-double-neg N/A

          \[\leadsto c0 \cdot \frac{1}{\sqrt{V \cdot \frac{1}{\frac{\color{blue}{A}}{\ell}}}} \]

      14. clear-num N/A

          \[\leadsto c0 \cdot \frac{1}{\sqrt{V \cdot \color{blue}{\frac{\ell}{A}}}} \]

      15. lift-/.f64 N/A

          \[\leadsto c0 \cdot \frac{1}{\sqrt{V \cdot \color{blue}{\frac{\ell}{A}}}} \]

      16. *-commutative N/A

          \[\leadsto c0 \cdot \frac{1}{\sqrt{\color{blue}{\frac{\ell}{A} \cdot V}}} \]

      17. lift-*.f64 N/A

          \[\leadsto c0 \cdot \frac{1}{\sqrt{\color{blue}{\frac{\ell}{A} \cdot V}}} \]

      18. lift-sqrt.f64 N/A

          \[\leadsto c0 \cdot \frac{1}{\color{blue}{\sqrt{\frac{\ell}{A} \cdot V}}} \]

   8. Applied rewrites 64.5%

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{\frac{V}{A} \cdot \ell}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 73.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l) :precision binary64 (* c0 (sqrt (/ A (* V l)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	return c0 * sqrt((A / (V * l)));
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    code = c0 * sqrt((a / (v * l)))
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	return c0 * Math.sqrt((A / (V * l)));
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	return c0 * math.sqrt((A / (V * l)))

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	return Float64(c0 * sqrt(Float64(A / Float64(V * l))))
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp = code(c0, A, V, l)
	tmp = c0 * sqrt((A / (V * l)));
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.0%

   \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
2. Add Preprocessing
3. Add Preprocessing

Reproduce

herbie shell --seed 2024321 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))