Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.7% → 89.8%

Time: 6.3s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.7% 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: 89.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-322}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\left(-V\right) \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-291}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left({\left(\sqrt{\ell \cdot V}\right)}^{-1} \cdot \sqrt{A}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) (- INFINITY))
   (* (/ c0 (sqrt l)) (sqrt (/ A V)))
   (if (<= (* V l) -2e-322)
     (* c0 (/ (sqrt (- A)) (sqrt (* (- V) l))))
     (if (<= (* V l) 5e-291)
       (* c0 (sqrt (/ (/ A l) V)))
       (* c0 (* (pow (sqrt (* l V)) -1.0) (sqrt A)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = (c0 / sqrt(l)) * sqrt((A / V));
	} else if ((V * l) <= -2e-322) {
		tmp = c0 * (sqrt(-A) / sqrt((-V * l)));
	} else if ((V * l) <= 5e-291) {
		tmp = c0 * sqrt(((A / l) / V));
	} else {
		tmp = c0 * (pow(sqrt((l * V)), -1.0) * sqrt(A));
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = (c0 / Math.sqrt(l)) * Math.sqrt((A / V));
	} else if ((V * l) <= -2e-322) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((-V * l)));
	} else if ((V * l) <= 5e-291) {
		tmp = c0 * Math.sqrt(((A / l) / V));
	} else {
		tmp = c0 * (Math.pow(Math.sqrt((l * V)), -1.0) * Math.sqrt(A));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = (c0 / math.sqrt(l)) * math.sqrt((A / V))
	elif (V * l) <= -2e-322:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((-V * l)))
	elif (V * l) <= 5e-291:
		tmp = c0 * math.sqrt(((A / l) / V))
	else:
		tmp = c0 * (math.pow(math.sqrt((l * V)), -1.0) * math.sqrt(A))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(Float64(c0 / sqrt(l)) * sqrt(Float64(A / V)));
	elseif (Float64(V * l) <= -2e-322)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(Float64(-V) * l))));
	elseif (Float64(V * l) <= 5e-291)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	else
		tmp = Float64(c0 * Float64((sqrt(Float64(l * V)) ^ -1.0) * sqrt(A)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = (c0 / sqrt(l)) * sqrt((A / V));
	elseif ((V * l) <= -2e-322)
		tmp = c0 * (sqrt(-A) / sqrt((-V * l)));
	elseif ((V * l) <= 5e-291)
		tmp = c0 * sqrt(((A / l) / V));
	else
		tmp = c0 * ((sqrt((l * V)) ^ -1.0) * sqrt(A));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-322], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[((-V) * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e-291], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Power[N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision] * N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 31.5%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. sqrt-div N/A

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

      7. associate-*r/ N/A

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

      8. associate-*l/ N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-/.f64 50.3

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

   4. Applied rewrites 50.3%

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

   if -inf.0 < (*.f64 V l) < -1.97626e-322

   1. Initial program 89.6%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 80.1

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

   4. Applied rewrites 80.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. div-inv N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-2neg N/A

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

      10. sqrt-div N/A

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

      11. frac-times N/A

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

      12. metadata-eval N/A

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

      13. sqrt-prod N/A

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

      14. *-rgt-identity N/A

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

      15. lower-/.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. lift-sqrt.f64 N/A

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

      19. sqrt-prod N/A

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

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

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

      21. *-commutative N/A

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

      22. lift-*.f64 N/A

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

      23. lower-sqrt.f64 N/A

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

      24. lift-*.f64 N/A

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

      25. *-commutative N/A

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

   6. Applied rewrites 98.4%

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

   if -1.97626e-322 < (*.f64 V l) < 5.0000000000000003e-291

   1. Initial program 40.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/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 5.0000000000000003e-291 < (*.f64 V l)

   1. Initial program 78.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. clear-num N/A

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

      5. frac-2neg N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. associate-/r* N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-sqrt.f64 90.6

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

   4. Applied rewrites 90.6%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-322}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\left(-V\right) \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-291}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left({\left(\sqrt{\ell \cdot V}\right)}^{-1} \cdot \sqrt{A}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-302} \lor \neg \left(t\_0 \leq 5 \cdot 10^{+226}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (or (<= t_0 4e-302) (not (<= t_0 5e+226)))
     (* c0 (sqrt (/ (/ A V) l)))
     t_0)))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if ((t_0 <= 4e-302) || !(t_0 <= 5e+226)) {
		tmp = c0 * sqrt(((A / V) / l));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    if ((t_0 <= 4d-302) .or. (.not. (t_0 <= 5d+226))) then
        tmp = c0 * sqrt(((a / v) / l))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if ((t_0 <= 4e-302) || !(t_0 <= 5e+226)) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if (t_0 <= 4e-302) or not (t_0 <= 5e+226):
		tmp = c0 * math.sqrt(((A / V) / l))
	else:
		tmp = t_0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if ((t_0 <= 4e-302) || !(t_0 <= 5e+226))
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if ((t_0 <= 4e-302) || ~((t_0 <= 5e+226)))
		tmp = c0 * sqrt(((A / V) / l));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 4e-302], N[Not[LessEqual[t$95$0, 5e+226]], $MachinePrecision]], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 3.9999999999999999e-302 or 5.0000000000000005e226 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 67.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 69.3

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

   4. Applied rewrites 69.3%

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

   if 3.9999999999999999e-302 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 5.0000000000000005e226

   1. Initial program 99.4%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 4 \cdot 10^{-302} \lor \neg \left(c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 5 \cdot 10^{+226}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-302}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+235}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A} \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (<= t_0 4e-302)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 5e+235) t_0 (/ c0 (sqrt (* (/ V A) l)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 4e-302) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 5e+235) {
		tmp = t_0;
	} else {
		tmp = c0 / sqrt(((V / A) * l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    if (t_0 <= 4d-302) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 5d+235) then
        tmp = t_0
    else
        tmp = c0 / sqrt(((v / a) * l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 4e-302) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 5e+235) {
		tmp = t_0;
	} else {
		tmp = c0 / Math.sqrt(((V / A) * l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 4e-302:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 5e+235:
		tmp = t_0
	else:
		tmp = c0 / math.sqrt(((V / A) * l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if (t_0 <= 4e-302)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 5e+235)
		tmp = t_0;
	else
		tmp = Float64(c0 / sqrt(Float64(Float64(V / A) * l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= 4e-302)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 5e+235)
		tmp = t_0;
	else
		tmp = c0 / sqrt(((V / A) * l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 4e-302], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+235], t$95$0, N[(c0 / N[Sqrt[N[(N[(V / A), $MachinePrecision] * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 3.9999999999999999e-302

   1. Initial program 71.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 70.6

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

   4. Applied rewrites 70.6%

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

   if 3.9999999999999999e-302 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 5.00000000000000027e235

   1. Initial program 99.4%

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

   if 5.00000000000000027e235 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 46.4%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 61.0

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

   4. Applied rewrites 61.0%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. clear-num N/A

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

      11. lift-/.f64 N/A

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

      12. sqrt-div N/A

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

      13. metadata-eval N/A

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

      14. lift-sqrt.f64 N/A

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

      15. div-inv N/A

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

      16. associate-/r* N/A

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

      17. lower-/.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-sqrt.f64 N/A

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

      20. lift-sqrt.f64 N/A

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

      21. sqrt-prod N/A

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

      22. lift-/.f64 N/A

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

      23. clear-num N/A

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

      24. lift-/.f64 N/A

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

   6. Applied rewrites 62.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. associate-*r* N/A

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

      7. div-inv N/A

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

      8. remove-double-neg N/A

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

      9. lift-neg.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lift-neg.f64 N/A

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

      14. remove-double-neg N/A

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

      15. lower-/.f64 62.3

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

   8. Applied rewrites 62.3%

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

Alternative 4: 76.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-302}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+297}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (<= t_0 4e-302)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 5e+297) t_0 (* c0 (sqrt (/ (/ A l) V)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 4e-302) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 5e+297) {
		tmp = t_0;
	} else {
		tmp = c0 * sqrt(((A / l) / V));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    if (t_0 <= 4d-302) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 5d+297) then
        tmp = t_0
    else
        tmp = c0 * sqrt(((a / l) / v))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 4e-302) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 5e+297) {
		tmp = t_0;
	} else {
		tmp = c0 * Math.sqrt(((A / l) / V));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 4e-302:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 5e+297:
		tmp = t_0
	else:
		tmp = c0 * math.sqrt(((A / l) / V))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if (t_0 <= 4e-302)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 5e+297)
		tmp = t_0;
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= 4e-302)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 5e+297)
		tmp = t_0;
	else
		tmp = c0 * sqrt(((A / l) / V));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 4e-302], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+297], t$95$0, N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 3.9999999999999999e-302

   1. Initial program 71.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 70.6

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

   4. Applied rewrites 70.6%

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

   if 3.9999999999999999e-302 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 4.9999999999999998e297

   1. Initial program 99.4%

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

   if 4.9999999999999998e297 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 40.4%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 56.6

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

   4. Applied rewrites 56.6%

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

Alternative 5: 89.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-322}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\left(-V\right) \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-296}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) (- INFINITY))
   (* (/ c0 (sqrt l)) (sqrt (/ A V)))
   (if (<= (* V l) -2e-322)
     (* c0 (/ (sqrt (- A)) (sqrt (* (- V) l))))
     (if (<= (* V l) 5e-296)
       (* c0 (sqrt (/ (/ A l) V)))
       (* c0 (/ (sqrt A) (sqrt (* l V))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = (c0 / sqrt(l)) * sqrt((A / V));
	} else if ((V * l) <= -2e-322) {
		tmp = c0 * (sqrt(-A) / sqrt((-V * l)));
	} else if ((V * l) <= 5e-296) {
		tmp = c0 * sqrt(((A / l) / V));
	} else {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = (c0 / Math.sqrt(l)) * Math.sqrt((A / V));
	} else if ((V * l) <= -2e-322) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((-V * l)));
	} else if ((V * l) <= 5e-296) {
		tmp = c0 * Math.sqrt(((A / l) / V));
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((l * V)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = (c0 / math.sqrt(l)) * math.sqrt((A / V))
	elif (V * l) <= -2e-322:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((-V * l)))
	elif (V * l) <= 5e-296:
		tmp = c0 * math.sqrt(((A / l) / V))
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((l * V)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(Float64(c0 / sqrt(l)) * sqrt(Float64(A / V)));
	elseif (Float64(V * l) <= -2e-322)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(Float64(-V) * l))));
	elseif (Float64(V * l) <= 5e-296)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(l * V))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = (c0 / sqrt(l)) * sqrt((A / V));
	elseif ((V * l) <= -2e-322)
		tmp = c0 * (sqrt(-A) / sqrt((-V * l)));
	elseif ((V * l) <= 5e-296)
		tmp = c0 * sqrt(((A / l) / V));
	else
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-322], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[((-V) * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e-296], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 31.5%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. sqrt-div N/A

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

      7. associate-*r/ N/A

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

      8. associate-*l/ N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-/.f64 50.3

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

   4. Applied rewrites 50.3%

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

   if -inf.0 < (*.f64 V l) < -1.97626e-322

   1. Initial program 89.6%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 80.1

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

   4. Applied rewrites 80.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. div-inv N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-2neg N/A

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

      10. sqrt-div N/A

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

      11. frac-times N/A

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

      12. metadata-eval N/A

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

      13. sqrt-prod N/A

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

      14. *-rgt-identity N/A

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

      15. lower-/.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. lift-sqrt.f64 N/A

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

      19. sqrt-prod N/A

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

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

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

      21. *-commutative N/A

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

      22. lift-*.f64 N/A

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

      23. lower-sqrt.f64 N/A

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

      24. lift-*.f64 N/A

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

      25. *-commutative N/A

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

   6. Applied rewrites 98.4%

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

   if -1.97626e-322 < (*.f64 V l) < 5.0000000000000003e-296

   1. Initial program 38.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 66.6

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

   4. Applied rewrites 66.6%

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

   if 5.0000000000000003e-296 < (*.f64 V l)

   1. Initial program 78.6%

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

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 90.7

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 90.7

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

   4. Applied rewrites 90.7%

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

Alternative 6: 88.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-322}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\left(-V\right) \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-296}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) (- INFINITY))
   (* c0 (sqrt (/ (/ A V) l)))
   (if (<= (* V l) -2e-322)
     (* c0 (/ (sqrt (- A)) (sqrt (* (- V) l))))
     (if (<= (* V l) 5e-296)
       (* c0 (sqrt (/ (/ A l) V)))
       (* c0 (/ (sqrt A) (sqrt (* l V))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if ((V * l) <= -2e-322) {
		tmp = c0 * (sqrt(-A) / sqrt((-V * l)));
	} else if ((V * l) <= 5e-296) {
		tmp = c0 * sqrt(((A / l) / V));
	} else {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if ((V * l) <= -2e-322) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((-V * l)));
	} else if ((V * l) <= 5e-296) {
		tmp = c0 * Math.sqrt(((A / l) / V));
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((l * V)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif (V * l) <= -2e-322:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((-V * l)))
	elif (V * l) <= 5e-296:
		tmp = c0 * math.sqrt(((A / l) / V))
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((l * V)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (Float64(V * l) <= -2e-322)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(Float64(-V) * l))));
	elseif (Float64(V * l) <= 5e-296)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(l * V))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = c0 * sqrt(((A / V) / l));
	elseif ((V * l) <= -2e-322)
		tmp = c0 * (sqrt(-A) / sqrt((-V * l)));
	elseif ((V * l) <= 5e-296)
		tmp = c0 * sqrt(((A / l) / V));
	else
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-322], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[((-V) * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e-296], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 31.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 61.4

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

   4. Applied rewrites 61.4%

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

   if -inf.0 < (*.f64 V l) < -1.97626e-322

   1. Initial program 89.6%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 80.1

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

   4. Applied rewrites 80.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. div-inv N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-2neg N/A

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

      10. sqrt-div N/A

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

      11. frac-times N/A

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

      12. metadata-eval N/A

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

      13. sqrt-prod N/A

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

      14. *-rgt-identity N/A

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

      15. lower-/.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. lift-sqrt.f64 N/A

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

      19. sqrt-prod N/A

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

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

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

      21. *-commutative N/A

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

      22. lift-*.f64 N/A

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

      23. lower-sqrt.f64 N/A

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

      24. lift-*.f64 N/A

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

      25. *-commutative N/A

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

   6. Applied rewrites 98.4%

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

   if -1.97626e-322 < (*.f64 V l) < 5.0000000000000003e-296

   1. Initial program 38.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 66.6

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

   4. Applied rewrites 66.6%

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

   if 5.0000000000000003e-296 < (*.f64 V l)

   1. Initial program 78.6%

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

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 90.7

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 90.7

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

   4. Applied rewrites 90.7%

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

Alternative 7: 87.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{-V}\\ \mathbf{if}\;V \leq -9.4 \cdot 10^{-200}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{-A}{\ell}}}{t\_0}\\ \mathbf{elif}\;V \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-A} \cdot c0}{t\_0 \cdot \sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V}} \cdot \frac{\sqrt{A}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (sqrt (- V))))
   (if (<= V -9.4e-200)
     (* c0 (/ (sqrt (/ (- A) l)) t_0))
     (if (<= V -5e-310)
       (/ (* (sqrt (- A)) c0) (* t_0 (sqrt l)))
       (* (/ c0 (sqrt V)) (/ (sqrt A) (sqrt l)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt(-V);
	double tmp;
	if (V <= -9.4e-200) {
		tmp = c0 * (sqrt((-A / l)) / t_0);
	} else if (V <= -5e-310) {
		tmp = (sqrt(-A) * c0) / (t_0 * sqrt(l));
	} else {
		tmp = (c0 / sqrt(V)) * (sqrt(A) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(-v)
    if (v <= (-9.4d-200)) then
        tmp = c0 * (sqrt((-a / l)) / t_0)
    else if (v <= (-5d-310)) then
        tmp = (sqrt(-a) * c0) / (t_0 * sqrt(l))
    else
        tmp = (c0 / sqrt(v)) * (sqrt(a) / sqrt(l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = Math.sqrt(-V);
	double tmp;
	if (V <= -9.4e-200) {
		tmp = c0 * (Math.sqrt((-A / l)) / t_0);
	} else if (V <= -5e-310) {
		tmp = (Math.sqrt(-A) * c0) / (t_0 * Math.sqrt(l));
	} else {
		tmp = (c0 / Math.sqrt(V)) * (Math.sqrt(A) / Math.sqrt(l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt(-V)
	tmp = 0
	if V <= -9.4e-200:
		tmp = c0 * (math.sqrt((-A / l)) / t_0)
	elif V <= -5e-310:
		tmp = (math.sqrt(-A) * c0) / (t_0 * math.sqrt(l))
	else:
		tmp = (c0 / math.sqrt(V)) * (math.sqrt(A) / math.sqrt(l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = sqrt(Float64(-V))
	tmp = 0.0
	if (V <= -9.4e-200)
		tmp = Float64(c0 * Float64(sqrt(Float64(Float64(-A) / l)) / t_0));
	elseif (V <= -5e-310)
		tmp = Float64(Float64(sqrt(Float64(-A)) * c0) / Float64(t_0 * sqrt(l)));
	else
		tmp = Float64(Float64(c0 / sqrt(V)) * Float64(sqrt(A) / sqrt(l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = sqrt(-V);
	tmp = 0.0;
	if (V <= -9.4e-200)
		tmp = c0 * (sqrt((-A / l)) / t_0);
	elseif (V <= -5e-310)
		tmp = (sqrt(-A) * c0) / (t_0 * sqrt(l));
	else
		tmp = (c0 / sqrt(V)) * (sqrt(A) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[Sqrt[(-V)], $MachinePrecision]}, If[LessEqual[V, -9.4e-200], N[(c0 * N[(N[Sqrt[N[((-A) / l), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[V, -5e-310], N[(N[(N[Sqrt[(-A)], $MachinePrecision] * c0), $MachinePrecision] / N[(t$95$0 * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c0 / N[Sqrt[V], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if V < -9.4000000000000002e-200

   1. Initial program 79.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 77.5

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

   4. Applied rewrites 77.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-/l/ N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. sqrt-div N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. distribute-neg-frac N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-neg.f64 85.8

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

   6. Applied rewrites 85.8%

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

   if -9.4000000000000002e-200 < V < -4.999999999999985e-310

   1. Initial program 65.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 72.3

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

   4. Applied rewrites 72.3%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. frac-2neg N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. frac-2neg N/A

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

      14. sqrt-div N/A

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

      15. frac-times N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-sqrt.f64 N/A

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

      19. lower-neg.f64 N/A

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

      20. lower-neg.f64 N/A

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

      21. lower-*.f64 N/A

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

   6. Applied rewrites 63.8%

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

   if -4.999999999999985e-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

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. associate-*r/ N/A

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

      6. lift-*.f64 N/A

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

      7. sqrt-prod N/A

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

      8. times-frac N/A

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

      9. sqrt-div N/A

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

      10. pow1/2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. pow1/2 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-/.f64 83.9

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

   4. Applied rewrites 83.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 52.4

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

   6. Applied rewrites 52.4%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \leq -9.4 \cdot 10^{-200}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-A} \cdot c0}{\sqrt{-V} \cdot \sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V}} \cdot \frac{\sqrt{A}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 87.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if V < -4.999999999999985e-310

   1. Initial program 76.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 76.4

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

   4. Applied rewrites 76.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-/l/ N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. sqrt-div N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. distribute-neg-frac N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-neg.f64 82.9

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

   6. Applied rewrites 82.9%

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

   if -4.999999999999985e-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

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. associate-*r/ N/A

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

      6. lift-*.f64 N/A

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

      7. sqrt-prod N/A

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

      8. times-frac N/A

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

      9. sqrt-div N/A

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

      10. pow1/2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. pow1/2 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-/.f64 83.9

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

   4. Applied rewrites 83.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 52.4

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

   6. Applied rewrites 52.4%

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

Alternative 9: 80.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 5 \cdot 10^{-296}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) 5e-296)
   (* c0 (sqrt (/ (/ A V) l)))
   (* c0 (/ (sqrt A) (sqrt (* l V))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= 5e-296) {
		tmp = c0 * sqrt(((A / V) / l));
	} else {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= 5d-296) then
        tmp = c0 * sqrt(((a / v) / l))
    else
        tmp = c0 * (sqrt(a) / sqrt((l * v)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= 5e-296) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((l * V)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= 5e-296:
		tmp = c0 * math.sqrt(((A / V) / l))
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((l * V)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= 5e-296)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(l * V))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= 5e-296)
		tmp = c0 * sqrt(((A / V) / l));
	else
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], 5e-296], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 V l) < 5.0000000000000003e-296

   1. Initial program 71.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 74.9

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

   4. Applied rewrites 74.9%

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

   if 5.0000000000000003e-296 < (*.f64 V l)

   1. Initial program 78.6%

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

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 90.7

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 90.7

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

   4. Applied rewrites 90.7%

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

Alternative 10: 73.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.3%

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

Reproduce

herbie shell --seed 2024307 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))