Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.4% → 89.3%

Time: 7.5s

Alternatives: 12

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: 73.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\frac{-A}{\ell}} \cdot c0}{\sqrt{-V}}\\ \mathbf{elif}\;\ell \cdot V \leq -2 \cdot 10^{-257}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\left(-\ell\right) \cdot V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 10^{-317}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{\ell \cdot V}} \cdot c0\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* l V) (- INFINITY))
   (/ (* (sqrt (/ (- A) l)) c0) (sqrt (- V)))
   (if (<= (* l V) -2e-257)
     (* (/ (sqrt (- A)) (sqrt (* (- l) V))) c0)
     (if (<= (* l V) 1e-317)
       (/ c0 (* (sqrt (/ V A)) (sqrt l)))
       (* (/ (sqrt A) (sqrt (* l V))) c0)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -((double) INFINITY)) {
		tmp = (sqrt((-A / l)) * c0) / sqrt(-V);
	} else if ((l * V) <= -2e-257) {
		tmp = (sqrt(-A) / sqrt((-l * V))) * c0;
	} else if ((l * V) <= 1e-317) {
		tmp = c0 / (sqrt((V / A)) * sqrt(l));
	} else {
		tmp = (sqrt(A) / sqrt((l * V))) * c0;
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -Double.POSITIVE_INFINITY) {
		tmp = (Math.sqrt((-A / l)) * c0) / Math.sqrt(-V);
	} else if ((l * V) <= -2e-257) {
		tmp = (Math.sqrt(-A) / Math.sqrt((-l * V))) * c0;
	} else if ((l * V) <= 1e-317) {
		tmp = c0 / (Math.sqrt((V / A)) * Math.sqrt(l));
	} else {
		tmp = (Math.sqrt(A) / Math.sqrt((l * V))) * c0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (l * V) <= -math.inf:
		tmp = (math.sqrt((-A / l)) * c0) / math.sqrt(-V)
	elif (l * V) <= -2e-257:
		tmp = (math.sqrt(-A) / math.sqrt((-l * V))) * c0
	elif (l * V) <= 1e-317:
		tmp = c0 / (math.sqrt((V / A)) * math.sqrt(l))
	else:
		tmp = (math.sqrt(A) / math.sqrt((l * V))) * c0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(l * V) <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(Float64(Float64(-A) / l)) * c0) / sqrt(Float64(-V)));
	elseif (Float64(l * V) <= -2e-257)
		tmp = Float64(Float64(sqrt(Float64(-A)) / sqrt(Float64(Float64(-l) * V))) * c0);
	elseif (Float64(l * V) <= 1e-317)
		tmp = Float64(c0 / Float64(sqrt(Float64(V / A)) * sqrt(l)));
	else
		tmp = Float64(Float64(sqrt(A) / sqrt(Float64(l * V))) * c0);
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((l * V) <= -Inf)
		tmp = (sqrt((-A / l)) * c0) / sqrt(-V);
	elseif ((l * V) <= -2e-257)
		tmp = (sqrt(-A) / sqrt((-l * V))) * c0;
	elseif ((l * V) <= 1e-317)
		tmp = c0 / (sqrt((V / A)) * sqrt(l));
	else
		tmp = (sqrt(A) / sqrt((l * V))) * c0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(l * V), $MachinePrecision], (-Infinity)], N[(N[(N[Sqrt[N[((-A) / l), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], -2e-257], N[(N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[((-l) * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 1e-317], N[(c0 / N[(N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 39.6%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 75.8

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

   4. Applied rewrites 75.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. sqrt-div N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. distribute-neg-frac N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-neg.f64 43.8

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

   6. Applied rewrites 43.8%

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

   if -inf.0 < (*.f64 V l) < -2e-257

   1. Initial program 83.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/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 70.6

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

   4. Applied rewrites 70.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

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

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 99.5

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

   6. Applied rewrites 99.5%

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

   if -2e-257 < (*.f64 V l) < 1.00000023e-317

   1. Initial program 26.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/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 57.7

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

   4. Applied rewrites 57.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r* N/A

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

      7. lift-*.f64 N/A

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

      8. sqrt-div N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. associate-/r/ N/A

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

      12. div-inv N/A

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

      13. associate-/r* N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. sqrt-div N/A

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

      17. lift-*.f64 N/A

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

      18. associate-/l/ N/A

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

      19. lift-/.f64 N/A

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

      20. lift-/.f64 N/A

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

      21. lift-sqrt.f64 N/A

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

      22. lower-/.f64 N/A

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

   6. Applied rewrites 26.3%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-neg-out N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. sqrt-undiv N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. clear-num N/A

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

      15. associate-/r* N/A

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

      16. associate-/l/ N/A

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

      17. lift-/.f64 N/A

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

      18. clear-num N/A

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

      19. /-rgt-identity N/A

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

      20. lower-/.f64 N/A

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

   8. Applied rewrites 57.7%

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

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

      6. *-commutative N/A

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

      7. sqrt-prod N/A

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

      8. pow1/2 N/A

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

      9. lower-*.f64 N/A

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

      10. pow1/2 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sqrt.f64 41.9

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

   10. Applied rewrites 41.9%

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

   if 1.00000023e-317 < (*.f64 V l)

   1. Initial program 84.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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-/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 80.0

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

   4. Applied rewrites 80.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lower-/.f64 91.4

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 91.4

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

   6. Applied rewrites 91.4%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\frac{-A}{\ell}} \cdot c0}{\sqrt{-V}}\\ \mathbf{elif}\;\ell \cdot V \leq -2 \cdot 10^{-257}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\left(-\ell\right) \cdot V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 10^{-317}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{\ell \cdot V}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{\ell \cdot V}} \cdot c0\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \mathbf{elif}\;t\_0 \leq 10^{+299}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* (sqrt (/ A (* l V))) c0)))
   (if (<= t_0 0.0)
     (* (sqrt (/ (/ A V) l)) c0)
     (if (<= t_0 1e+299) t_0 (/ c0 (sqrt (* (/ l A) V)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt((A / (l * V))) * c0;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = sqrt(((A / V) / l)) * c0;
	} else if (t_0 <= 1e+299) {
		tmp = t_0;
	} else {
		tmp = c0 / sqrt(((l / A) * V));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((a / (l * v))) * c0
    if (t_0 <= 0.0d0) then
        tmp = sqrt(((a / v) / l)) * c0
    else if (t_0 <= 1d+299) then
        tmp = t_0
    else
        tmp = c0 / sqrt(((l / a) * v))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = Math.sqrt((A / (l * V))) * c0;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = Math.sqrt(((A / V) / l)) * c0;
	} else if (t_0 <= 1e+299) {
		tmp = t_0;
	} else {
		tmp = c0 / Math.sqrt(((l / A) * V));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt((A / (l * V))) * c0
	tmp = 0
	if t_0 <= 0.0:
		tmp = math.sqrt(((A / V) / l)) * c0
	elif t_0 <= 1e+299:
		tmp = t_0
	else:
		tmp = c0 / math.sqrt(((l / A) * V))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(sqrt(Float64(A / Float64(l * V))) * c0)
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(sqrt(Float64(Float64(A / V) / l)) * c0);
	elseif (t_0 <= 1e+299)
		tmp = t_0;
	else
		tmp = Float64(c0 / sqrt(Float64(Float64(l / A) * V)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = sqrt((A / (l * V))) * c0;
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = sqrt(((A / V) / l)) * c0;
	elseif (t_0 <= 1e+299)
		tmp = t_0;
	else
		tmp = c0 / sqrt(((l / A) * V));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(N[Sqrt[N[(A / N[(l * V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[t$95$0, 1e+299], t$95$0, N[(c0 / N[Sqrt[N[(N[(l / A), $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 67.6%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 71.3

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

   4. Applied rewrites 71.3%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 1.0000000000000001e299

   1. Initial program 97.0%

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

   if 1.0000000000000001e299 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 43.6%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 64.2

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

   4. Applied rewrites 64.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r* N/A

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

      7. lift-*.f64 N/A

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

      8. sqrt-div N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. associate-/r/ N/A

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

      12. div-inv N/A

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

      13. associate-/r* N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. sqrt-div N/A

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

      17. lift-*.f64 N/A

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

      18. associate-/l/ N/A

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

      19. lift-/.f64 N/A

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

      20. lift-/.f64 N/A

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

      21. lift-sqrt.f64 N/A

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

      22. lower-/.f64 N/A

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

   6. Applied rewrites 43.6%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-neg-out N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. sqrt-undiv N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. clear-num N/A

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

      15. associate-/r* N/A

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

      16. associate-/l/ N/A

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

      17. lift-/.f64 N/A

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

      18. clear-num N/A

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

      19. /-rgt-identity N/A

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

      20. lower-/.f64 N/A

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

   8. Applied rewrites 66.1%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\frac{A}{\ell \cdot V}} \cdot c0 \leq 0:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \mathbf{elif}\;\sqrt{\frac{A}{\ell \cdot V}} \cdot c0 \leq 10^{+299}:\\ \;\;\;\;\sqrt{\frac{A}{\ell \cdot V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{\ell \cdot V}} \cdot c0\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+294}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{\ell}}{V}} \cdot c0\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* (sqrt (/ A (* l V))) c0)))
   (if (<= t_0 0.0)
     (* (sqrt (/ (/ A V) l)) c0)
     (if (<= t_0 5e+294) t_0 (* (sqrt (/ (/ A l) V)) c0)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt((A / (l * V))) * c0;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = sqrt(((A / V) / l)) * c0;
	} else if (t_0 <= 5e+294) {
		tmp = t_0;
	} else {
		tmp = sqrt(((A / l) / V)) * c0;
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((a / (l * v))) * c0
    if (t_0 <= 0.0d0) then
        tmp = sqrt(((a / v) / l)) * c0
    else if (t_0 <= 5d+294) then
        tmp = t_0
    else
        tmp = sqrt(((a / l) / v)) * c0
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = Math.sqrt((A / (l * V))) * c0;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = Math.sqrt(((A / V) / l)) * c0;
	} else if (t_0 <= 5e+294) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt(((A / l) / V)) * c0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt((A / (l * V))) * c0
	tmp = 0
	if t_0 <= 0.0:
		tmp = math.sqrt(((A / V) / l)) * c0
	elif t_0 <= 5e+294:
		tmp = t_0
	else:
		tmp = math.sqrt(((A / l) / V)) * c0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(sqrt(Float64(A / Float64(l * V))) * c0)
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(sqrt(Float64(Float64(A / V) / l)) * c0);
	elseif (t_0 <= 5e+294)
		tmp = t_0;
	else
		tmp = Float64(sqrt(Float64(Float64(A / l) / V)) * c0);
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = sqrt((A / (l * V))) * c0;
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = sqrt(((A / V) / l)) * c0;
	elseif (t_0 <= 5e+294)
		tmp = t_0;
	else
		tmp = sqrt(((A / l) / V)) * c0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(N[Sqrt[N[(A / N[(l * V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[t$95$0, 5e+294], t$95$0, N[(N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 67.6%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 71.3

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

   4. Applied rewrites 71.3%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 4.9999999999999999e294

   1. Initial program 97.0%

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

   if 4.9999999999999999e294 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 43.6%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 64.2

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

   4. Applied rewrites 64.2%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\frac{A}{\ell \cdot V}} \cdot c0 \leq 0:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \mathbf{elif}\;\sqrt{\frac{A}{\ell \cdot V}} \cdot c0 \leq 5 \cdot 10^{+294}:\\ \;\;\;\;\sqrt{\frac{A}{\ell \cdot V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{\ell}}{V}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{\ell \cdot V}} \cdot c0\\ t_1 := \sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+304}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* (sqrt (/ A (* l V))) c0)) (t_1 (* (sqrt (/ (/ A V) l)) c0)))
   (if (<= t_0 0.0) t_1 (if (<= t_0 5e+304) t_0 t_1))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt((A / (l * V))) * c0;
	double t_1 = sqrt(((A / V) / l)) * c0;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 5e+304) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((a / (l * v))) * c0
    t_1 = sqrt(((a / v) / l)) * c0
    if (t_0 <= 0.0d0) then
        tmp = t_1
    else if (t_0 <= 5d+304) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = Math.sqrt((A / (l * V))) * c0;
	double t_1 = Math.sqrt(((A / V) / l)) * c0;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 5e+304) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt((A / (l * V))) * c0
	t_1 = math.sqrt(((A / V) / l)) * c0
	tmp = 0
	if t_0 <= 0.0:
		tmp = t_1
	elif t_0 <= 5e+304:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(sqrt(Float64(A / Float64(l * V))) * c0)
	t_1 = Float64(sqrt(Float64(Float64(A / V) / l)) * c0)
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 5e+304)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = sqrt((A / (l * V))) * c0;
	t_1 = sqrt(((A / V) / l)) * c0;
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 5e+304)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(N[Sqrt[N[(A / N[(l * V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 5e+304], t$95$0, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 0.0 or 4.9999999999999997e304 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 64.4%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 70.4

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

   4. Applied rewrites 70.4%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 4.9999999999999997e304

   1. Initial program 97.0%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\frac{A}{\ell \cdot V}} \cdot c0 \leq 0:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \mathbf{elif}\;\sqrt{\frac{A}{\ell \cdot V}} \cdot c0 \leq 5 \cdot 10^{+304}:\\ \;\;\;\;\sqrt{\frac{A}{\ell \cdot V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq -2 \cdot 10^{-257}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\left(-\ell\right) \cdot V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 10^{-317}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{\ell \cdot V}} \cdot c0\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* l V) (- INFINITY))
   (* (/ (sqrt (/ (- A) l)) (sqrt (- V))) c0)
   (if (<= (* l V) -2e-257)
     (* (/ (sqrt (- A)) (sqrt (* (- l) V))) c0)
     (if (<= (* l V) 1e-317)
       (/ c0 (* (sqrt (/ V A)) (sqrt l)))
       (* (/ (sqrt A) (sqrt (* l V))) c0)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -((double) INFINITY)) {
		tmp = (sqrt((-A / l)) / sqrt(-V)) * c0;
	} else if ((l * V) <= -2e-257) {
		tmp = (sqrt(-A) / sqrt((-l * V))) * c0;
	} else if ((l * V) <= 1e-317) {
		tmp = c0 / (sqrt((V / A)) * sqrt(l));
	} else {
		tmp = (sqrt(A) / sqrt((l * V))) * c0;
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -Double.POSITIVE_INFINITY) {
		tmp = (Math.sqrt((-A / l)) / Math.sqrt(-V)) * c0;
	} else if ((l * V) <= -2e-257) {
		tmp = (Math.sqrt(-A) / Math.sqrt((-l * V))) * c0;
	} else if ((l * V) <= 1e-317) {
		tmp = c0 / (Math.sqrt((V / A)) * Math.sqrt(l));
	} else {
		tmp = (Math.sqrt(A) / Math.sqrt((l * V))) * c0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (l * V) <= -math.inf:
		tmp = (math.sqrt((-A / l)) / math.sqrt(-V)) * c0
	elif (l * V) <= -2e-257:
		tmp = (math.sqrt(-A) / math.sqrt((-l * V))) * c0
	elif (l * V) <= 1e-317:
		tmp = c0 / (math.sqrt((V / A)) * math.sqrt(l))
	else:
		tmp = (math.sqrt(A) / math.sqrt((l * V))) * c0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(l * V) <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(Float64(Float64(-A) / l)) / sqrt(Float64(-V))) * c0);
	elseif (Float64(l * V) <= -2e-257)
		tmp = Float64(Float64(sqrt(Float64(-A)) / sqrt(Float64(Float64(-l) * V))) * c0);
	elseif (Float64(l * V) <= 1e-317)
		tmp = Float64(c0 / Float64(sqrt(Float64(V / A)) * sqrt(l)));
	else
		tmp = Float64(Float64(sqrt(A) / sqrt(Float64(l * V))) * c0);
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((l * V) <= -Inf)
		tmp = (sqrt((-A / l)) / sqrt(-V)) * c0;
	elseif ((l * V) <= -2e-257)
		tmp = (sqrt(-A) / sqrt((-l * V))) * c0;
	elseif ((l * V) <= 1e-317)
		tmp = c0 / (sqrt((V / A)) * sqrt(l));
	else
		tmp = (sqrt(A) / sqrt((l * V))) * c0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(l * V), $MachinePrecision], (-Infinity)], N[(N[(N[Sqrt[N[((-A) / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], -2e-257], N[(N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[((-l) * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 1e-317], N[(c0 / N[(N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 39.6%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 75.8

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

   4. Applied rewrites 75.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. sqrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 43.9

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

   6. Applied rewrites 43.9%

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

   if -inf.0 < (*.f64 V l) < -2e-257

   1. Initial program 83.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/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 70.6

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

   4. Applied rewrites 70.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

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

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 99.5

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

   6. Applied rewrites 99.5%

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

   if -2e-257 < (*.f64 V l) < 1.00000023e-317

   1. Initial program 26.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/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 57.7

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

   4. Applied rewrites 57.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r* N/A

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

      7. lift-*.f64 N/A

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

      8. sqrt-div N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. associate-/r/ N/A

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

      12. div-inv N/A

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

      13. associate-/r* N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. sqrt-div N/A

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

      17. lift-*.f64 N/A

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

      18. associate-/l/ N/A

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

      19. lift-/.f64 N/A

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

      20. lift-/.f64 N/A

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

      21. lift-sqrt.f64 N/A

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

      22. lower-/.f64 N/A

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

   6. Applied rewrites 26.3%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-neg-out N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. sqrt-undiv N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. clear-num N/A

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

      15. associate-/r* N/A

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

      16. associate-/l/ N/A

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

      17. lift-/.f64 N/A

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

      18. clear-num N/A

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

      19. /-rgt-identity N/A

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

      20. lower-/.f64 N/A

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

   8. Applied rewrites 57.7%

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

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

      6. *-commutative N/A

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

      7. sqrt-prod N/A

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

      8. pow1/2 N/A

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

      9. lower-*.f64 N/A

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

      10. pow1/2 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sqrt.f64 41.9

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

   10. Applied rewrites 41.9%

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

   if 1.00000023e-317 < (*.f64 V l)

   1. Initial program 84.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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-/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 80.0

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

   4. Applied rewrites 80.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lower-/.f64 91.4

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 91.4

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

   6. Applied rewrites 91.4%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq -2 \cdot 10^{-257}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\left(-\ell\right) \cdot V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 10^{-317}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{\ell \cdot V}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq -2 \cdot 10^{-257}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\left(-\ell\right) \cdot V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 10^{-317}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{\ell \cdot V}} \cdot c0\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* l V) (- INFINITY))
   (* (/ (sqrt (/ A V)) (sqrt l)) c0)
   (if (<= (* l V) -2e-257)
     (* (/ (sqrt (- A)) (sqrt (* (- l) V))) c0)
     (if (<= (* l V) 1e-317)
       (/ c0 (* (sqrt (/ V A)) (sqrt l)))
       (* (/ (sqrt A) (sqrt (* l V))) c0)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -((double) INFINITY)) {
		tmp = (sqrt((A / V)) / sqrt(l)) * c0;
	} else if ((l * V) <= -2e-257) {
		tmp = (sqrt(-A) / sqrt((-l * V))) * c0;
	} else if ((l * V) <= 1e-317) {
		tmp = c0 / (sqrt((V / A)) * sqrt(l));
	} else {
		tmp = (sqrt(A) / sqrt((l * V))) * c0;
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -Double.POSITIVE_INFINITY) {
		tmp = (Math.sqrt((A / V)) / Math.sqrt(l)) * c0;
	} else if ((l * V) <= -2e-257) {
		tmp = (Math.sqrt(-A) / Math.sqrt((-l * V))) * c0;
	} else if ((l * V) <= 1e-317) {
		tmp = c0 / (Math.sqrt((V / A)) * Math.sqrt(l));
	} else {
		tmp = (Math.sqrt(A) / Math.sqrt((l * V))) * c0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (l * V) <= -math.inf:
		tmp = (math.sqrt((A / V)) / math.sqrt(l)) * c0
	elif (l * V) <= -2e-257:
		tmp = (math.sqrt(-A) / math.sqrt((-l * V))) * c0
	elif (l * V) <= 1e-317:
		tmp = c0 / (math.sqrt((V / A)) * math.sqrt(l))
	else:
		tmp = (math.sqrt(A) / math.sqrt((l * V))) * c0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(l * V) <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(Float64(A / V)) / sqrt(l)) * c0);
	elseif (Float64(l * V) <= -2e-257)
		tmp = Float64(Float64(sqrt(Float64(-A)) / sqrt(Float64(Float64(-l) * V))) * c0);
	elseif (Float64(l * V) <= 1e-317)
		tmp = Float64(c0 / Float64(sqrt(Float64(V / A)) * sqrt(l)));
	else
		tmp = Float64(Float64(sqrt(A) / sqrt(Float64(l * V))) * c0);
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((l * V) <= -Inf)
		tmp = (sqrt((A / V)) / sqrt(l)) * c0;
	elseif ((l * V) <= -2e-257)
		tmp = (sqrt(-A) / sqrt((-l * V))) * c0;
	elseif ((l * V) <= 1e-317)
		tmp = c0 / (sqrt((V / A)) * sqrt(l));
	else
		tmp = (sqrt(A) / sqrt((l * V))) * c0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(l * V), $MachinePrecision], (-Infinity)], N[(N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], -2e-257], N[(N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[((-l) * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 1e-317], N[(c0 / N[(N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 39.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. sqrt-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 43.9

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

   4. Applied rewrites 43.9%

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

   if -inf.0 < (*.f64 V l) < -2e-257

   1. Initial program 83.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/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 70.6

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

   4. Applied rewrites 70.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

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

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 99.5

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

   6. Applied rewrites 99.5%

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

   if -2e-257 < (*.f64 V l) < 1.00000023e-317

   1. Initial program 26.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/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 57.7

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

   4. Applied rewrites 57.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r* N/A

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

      7. lift-*.f64 N/A

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

      8. sqrt-div N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. associate-/r/ N/A

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

      12. div-inv N/A

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

      13. associate-/r* N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. sqrt-div N/A

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

      17. lift-*.f64 N/A

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

      18. associate-/l/ N/A

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

      19. lift-/.f64 N/A

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

      20. lift-/.f64 N/A

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

      21. lift-sqrt.f64 N/A

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

      22. lower-/.f64 N/A

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

   6. Applied rewrites 26.3%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-neg-out N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. sqrt-undiv N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. clear-num N/A

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

      15. associate-/r* N/A

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

      16. associate-/l/ N/A

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

      17. lift-/.f64 N/A

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

      18. clear-num N/A

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

      19. /-rgt-identity N/A

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

      20. lower-/.f64 N/A

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

   8. Applied rewrites 57.7%

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

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

      6. *-commutative N/A

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

      7. sqrt-prod N/A

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

      8. pow1/2 N/A

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

      9. lower-*.f64 N/A

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

      10. pow1/2 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sqrt.f64 41.9

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

   10. Applied rewrites 41.9%

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

   if 1.00000023e-317 < (*.f64 V l)

   1. Initial program 84.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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-/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 80.0

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

   4. Applied rewrites 80.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lower-/.f64 91.4

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 91.4

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

   6. Applied rewrites 91.4%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq -2 \cdot 10^{-257}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\left(-\ell\right) \cdot V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 10^{-317}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{\ell \cdot V}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}} \cdot c0\\ \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \cdot V \leq -2 \cdot 10^{-286}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\left(-\ell\right) \cdot V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 10^{-317}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{\ell \cdot V}} \cdot c0\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* (/ (sqrt (/ A V)) (sqrt l)) c0)))
   (if (<= (* l V) (- INFINITY))
     t_0
     (if (<= (* l V) -2e-286)
       (* (/ (sqrt (- A)) (sqrt (* (- l) V))) c0)
       (if (<= (* l V) 1e-317) t_0 (* (/ (sqrt A) (sqrt (* l V))) c0))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = (sqrt((A / V)) / sqrt(l)) * c0;
	double tmp;
	if ((l * V) <= -((double) INFINITY)) {
		tmp = t_0;
	} else if ((l * V) <= -2e-286) {
		tmp = (sqrt(-A) / sqrt((-l * V))) * c0;
	} else if ((l * V) <= 1e-317) {
		tmp = t_0;
	} else {
		tmp = (sqrt(A) / sqrt((l * V))) * c0;
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = (Math.sqrt((A / V)) / Math.sqrt(l)) * c0;
	double tmp;
	if ((l * V) <= -Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else if ((l * V) <= -2e-286) {
		tmp = (Math.sqrt(-A) / Math.sqrt((-l * V))) * c0;
	} else if ((l * V) <= 1e-317) {
		tmp = t_0;
	} else {
		tmp = (Math.sqrt(A) / Math.sqrt((l * V))) * c0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = (math.sqrt((A / V)) / math.sqrt(l)) * c0
	tmp = 0
	if (l * V) <= -math.inf:
		tmp = t_0
	elif (l * V) <= -2e-286:
		tmp = (math.sqrt(-A) / math.sqrt((-l * V))) * c0
	elif (l * V) <= 1e-317:
		tmp = t_0
	else:
		tmp = (math.sqrt(A) / math.sqrt((l * V))) * c0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(Float64(sqrt(Float64(A / V)) / sqrt(l)) * c0)
	tmp = 0.0
	if (Float64(l * V) <= Float64(-Inf))
		tmp = t_0;
	elseif (Float64(l * V) <= -2e-286)
		tmp = Float64(Float64(sqrt(Float64(-A)) / sqrt(Float64(Float64(-l) * V))) * c0);
	elseif (Float64(l * V) <= 1e-317)
		tmp = t_0;
	else
		tmp = Float64(Float64(sqrt(A) / sqrt(Float64(l * V))) * c0);
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = (sqrt((A / V)) / sqrt(l)) * c0;
	tmp = 0.0;
	if ((l * V) <= -Inf)
		tmp = t_0;
	elseif ((l * V) <= -2e-286)
		tmp = (sqrt(-A) / sqrt((-l * V))) * c0;
	elseif ((l * V) <= 1e-317)
		tmp = t_0;
	else
		tmp = (sqrt(A) / sqrt((l * V))) * c0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision]}, If[LessEqual[N[(l * V), $MachinePrecision], (-Infinity)], t$95$0, If[LessEqual[N[(l * V), $MachinePrecision], -2e-286], N[(N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[((-l) * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 1e-317], t$95$0, N[(N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -inf.0 or -2.0000000000000001e-286 < (*.f64 V l) < 1.00000023e-317

   1. Initial program 27.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. sqrt-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 42.5

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

   4. Applied rewrites 42.5%

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

   if -inf.0 < (*.f64 V l) < -2.0000000000000001e-286

   1. Initial program 83.6%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 71.2

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

   4. Applied rewrites 71.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

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

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 99.5

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

   6. Applied rewrites 99.5%

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

   if 1.00000023e-317 < (*.f64 V l)

   1. Initial program 84.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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-/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 80.0

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

   4. Applied rewrites 80.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lower-/.f64 91.4

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 91.4

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

   6. Applied rewrites 91.4%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq -2 \cdot 10^{-286}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\left(-\ell\right) \cdot V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 10^{-317}:\\ \;\;\;\;\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{\ell \cdot V}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 87.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq -2 \cdot 10^{-257}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\left(-\ell\right) \cdot V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A} \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{\ell \cdot V}} \cdot c0\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* l V) (- INFINITY))
   (* (sqrt (/ (/ A V) l)) c0)
   (if (<= (* l V) -2e-257)
     (* (/ (sqrt (- A)) (sqrt (* (- l) V))) c0)
     (if (<= (* l V) 0.0)
       (/ c0 (sqrt (* (/ V A) l)))
       (* (/ (sqrt A) (sqrt (* l V))) c0)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -((double) INFINITY)) {
		tmp = sqrt(((A / V) / l)) * c0;
	} else if ((l * V) <= -2e-257) {
		tmp = (sqrt(-A) / sqrt((-l * V))) * c0;
	} else if ((l * V) <= 0.0) {
		tmp = c0 / sqrt(((V / A) * l));
	} else {
		tmp = (sqrt(A) / sqrt((l * V))) * c0;
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt(((A / V) / l)) * c0;
	} else if ((l * V) <= -2e-257) {
		tmp = (Math.sqrt(-A) / Math.sqrt((-l * V))) * c0;
	} else if ((l * V) <= 0.0) {
		tmp = c0 / Math.sqrt(((V / A) * l));
	} else {
		tmp = (Math.sqrt(A) / Math.sqrt((l * V))) * c0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (l * V) <= -math.inf:
		tmp = math.sqrt(((A / V) / l)) * c0
	elif (l * V) <= -2e-257:
		tmp = (math.sqrt(-A) / math.sqrt((-l * V))) * c0
	elif (l * V) <= 0.0:
		tmp = c0 / math.sqrt(((V / A) * l))
	else:
		tmp = (math.sqrt(A) / math.sqrt((l * V))) * c0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(l * V) <= Float64(-Inf))
		tmp = Float64(sqrt(Float64(Float64(A / V) / l)) * c0);
	elseif (Float64(l * V) <= -2e-257)
		tmp = Float64(Float64(sqrt(Float64(-A)) / sqrt(Float64(Float64(-l) * V))) * c0);
	elseif (Float64(l * V) <= 0.0)
		tmp = Float64(c0 / sqrt(Float64(Float64(V / A) * l)));
	else
		tmp = Float64(Float64(sqrt(A) / sqrt(Float64(l * V))) * c0);
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((l * V) <= -Inf)
		tmp = sqrt(((A / V) / l)) * c0;
	elseif ((l * V) <= -2e-257)
		tmp = (sqrt(-A) / sqrt((-l * V))) * c0;
	elseif ((l * V) <= 0.0)
		tmp = c0 / sqrt(((V / A) * l));
	else
		tmp = (sqrt(A) / sqrt((l * V))) * c0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(l * V), $MachinePrecision], (-Infinity)], N[(N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], -2e-257], N[(N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[((-l) * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 0.0], N[(c0 / N[Sqrt[N[(N[(V / A), $MachinePrecision] * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 39.6%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 75.6

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

   4. Applied rewrites 75.6%

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

   if -inf.0 < (*.f64 V l) < -2e-257

   1. Initial program 83.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/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 70.6

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

   4. Applied rewrites 70.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

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

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 99.5

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

   6. Applied rewrites 99.5%

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

   if -2e-257 < (*.f64 V l) < 0.0

   1. Initial program 24.4%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 56.9

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

   4. Applied rewrites 56.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r* N/A

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

      7. lift-*.f64 N/A

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

      8. sqrt-div N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. associate-/r/ N/A

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

      12. div-inv N/A

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

      13. associate-/r* N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. sqrt-div N/A

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

      17. lift-*.f64 N/A

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

      18. associate-/l/ N/A

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

      19. lift-/.f64 N/A

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

      20. lift-/.f64 N/A

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

      21. lift-sqrt.f64 N/A

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

      22. lower-/.f64 N/A

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

   6. Applied rewrites 24.4%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-neg-out N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. sqrt-undiv N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. clear-num N/A

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

      15. associate-/r* N/A

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

      16. associate-/l/ N/A

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

      17. lift-/.f64 N/A

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

      18. clear-num N/A

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

      19. /-rgt-identity N/A

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

      20. lower-/.f64 N/A

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

   8. Applied rewrites 57.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 57.0

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

   10. Applied rewrites 57.0%

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

   if 0.0 < (*.f64 V l)

   1. Initial program 83.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/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 79.7

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

   4. Applied rewrites 79.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lower-/.f64 90.7

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 90.7

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

   6. Applied rewrites 90.7%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq -2 \cdot 10^{-257}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\left(-\ell\right) \cdot V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A} \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{\ell \cdot V}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if A < -4.999999999999985e-310

   1. Initial program 66.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 67.0

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

   4. Applied rewrites 67.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

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

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 77.0

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

   6. Applied rewrites 77.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-neg.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. associate-*r* N/A

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

      7. sqrt-prod N/A

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

      8. *-commutative N/A

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

      9. neg-mul-1 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-sqrt.f64 50.2

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

   8. Applied rewrites 50.2%

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

   if -4.999999999999985e-310 < A

   1. Initial program 77.6%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 79.1

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

   4. Applied rewrites 79.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lower-/.f64 84.3

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 84.3

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

   6. Applied rewrites 84.3%

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

4. Final simplification 65.8%

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

Alternative 10: 87.1% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if V < -1.999999999999994e-310

   1. Initial program 70.8%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 73.3

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

   4. Applied rewrites 73.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. sqrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 85.7

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

   6. Applied rewrites 85.7%

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

   if -1.999999999999994e-310 < V

   1. Initial program 71.9%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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-/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 71.8

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

   4. Applied rewrites 71.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. associate-/r* N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-div N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. frac-2neg N/A

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

      11. div-inv N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. sqrt-div N/A

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

      15. lift-sqrt.f64 N/A

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

      16. distribute-neg-frac2 N/A

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

      17. frac-times N/A

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

      18. *-rgt-identity N/A

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

      19. lower-/.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lower-neg.f64 N/A

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

      22. lower-sqrt.f64 N/A

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

      23. lower-neg.f64 49.7

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

   6. Applied rewrites 49.7%

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

4. Final simplification 67.1%

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

Alternative 11: 80.2% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 62.9%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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-/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 67.5

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

   4. Applied rewrites 67.5%

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

   if 0.0 < (*.f64 V l)

   1. Initial program 83.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/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 79.7

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

   4. Applied rewrites 79.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lower-/.f64 90.7

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 90.7

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

   6. Applied rewrites 90.7%

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

4. Final simplification 77.1%

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

Alternative 12: 73.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.4%

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

3. Final simplification 71.4%

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

Reproduce

herbie shell --seed 2024240 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))