Henrywood and Agarwal, Equation (3)

Percentage Accurate: 74.1% → 89.5%

Time: 6.2s

Alternatives: 14

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.1% 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: 89.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+263}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-295}:\\ \;\;\;\;\frac{\sqrt{-A} \cdot c0}{\sqrt{\left(-V\right) \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+304}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(\sqrt{V \cdot \frac{\ell}{A}}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -5e+263)
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))
   (if (<= (* V l) -2e-295)
     (/ (* (sqrt (- A)) c0) (sqrt (* (- V) l)))
     (if (<= (* V l) 0.0)
       (* c0 (/ (sqrt (/ (- A) l)) (sqrt (- V))))
       (if (<= (* V l) 5e+304)
         (* c0 (/ (sqrt A) (sqrt (* l V))))
         (* c0 (pow (sqrt (* V (/ l A))) -1.0)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -5e+263) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else if ((V * l) <= -2e-295) {
		tmp = (sqrt(-A) * c0) / sqrt((-V * l));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * (sqrt((-A / l)) / sqrt(-V));
	} else if ((V * l) <= 5e+304) {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	} else {
		tmp = c0 * pow(sqrt((V * (l / A))), -1.0);
	}
	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) :: tmp
    if ((v * l) <= (-5d+263)) then
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    else if ((v * l) <= (-2d-295)) then
        tmp = (sqrt(-a) * c0) / sqrt((-v * l))
    else if ((v * l) <= 0.0d0) then
        tmp = c0 * (sqrt((-a / l)) / sqrt(-v))
    else if ((v * l) <= 5d+304) then
        tmp = c0 * (sqrt(a) / sqrt((l * v)))
    else
        tmp = c0 * (sqrt((v * (l / a))) ** (-1.0d0))
    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+263) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if ((V * l) <= -2e-295) {
		tmp = (Math.sqrt(-A) * c0) / Math.sqrt((-V * l));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * (Math.sqrt((-A / l)) / Math.sqrt(-V));
	} else if ((V * l) <= 5e+304) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((l * V)));
	} else {
		tmp = c0 * Math.pow(Math.sqrt((V * (l / A))), -1.0);
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -5e+263:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	elif (V * l) <= -2e-295:
		tmp = (math.sqrt(-A) * c0) / math.sqrt((-V * l))
	elif (V * l) <= 0.0:
		tmp = c0 * (math.sqrt((-A / l)) / math.sqrt(-V))
	elif (V * l) <= 5e+304:
		tmp = c0 * (math.sqrt(A) / math.sqrt((l * V)))
	else:
		tmp = c0 * math.pow(math.sqrt((V * (l / A))), -1.0)
	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+263)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (Float64(V * l) <= -2e-295)
		tmp = Float64(Float64(sqrt(Float64(-A)) * c0) / sqrt(Float64(Float64(-V) * l)));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 * Float64(sqrt(Float64(Float64(-A) / l)) / sqrt(Float64(-V))));
	elseif (Float64(V * l) <= 5e+304)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(l * V))));
	else
		tmp = Float64(c0 * (sqrt(Float64(V * Float64(l / A))) ^ -1.0));
	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+263)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	elseif ((V * l) <= -2e-295)
		tmp = (sqrt(-A) * c0) / sqrt((-V * l));
	elseif ((V * l) <= 0.0)
		tmp = c0 * (sqrt((-A / l)) / sqrt(-V));
	elseif ((V * l) <= 5e+304)
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	else
		tmp = c0 * (sqrt((V * (l / A))) ^ -1.0);
	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_] := If[LessEqual[N[(V * l), $MachinePrecision], -5e+263], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-295], N[(N[(N[Sqrt[(-A)], $MachinePrecision] * c0), $MachinePrecision] / N[Sqrt[N[((-V) * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 * N[(N[Sqrt[N[((-A) / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e+304], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Power[N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -5.00000000000000022e263

   1. Initial program 37.8%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 35.5

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

   4. Applied rewrites 35.5%

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

   if -5.00000000000000022e263 < (*.f64 V l) < -2.00000000000000012e-295

   1. Initial program 83.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/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 75.6

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

   4. Applied rewrites 75.6%

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

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

      6. lift-sqrt.f64 N/A

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

      7. associate-*l/ N/A

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

      8. associate-*r/ N/A

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

      9. lift-/.f64 N/A

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

      10. frac-2neg N/A

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

      11. sqrt-div N/A

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

      12. frac-times N/A

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

      13. lift-sqrt.f64 N/A

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

      14. sqrt-prod N/A

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

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

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

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

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

      20. sqrt-prod N/A

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

      21. lift-sqrt.f64 N/A

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

   6. Applied rewrites 99.6%

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

   if -2.00000000000000012e-295 < (*.f64 V l) < -0.0

   1. Initial program 27.3%

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

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

   4. Applied rewrites 71.4%

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

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 47.1

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

   6. Applied rewrites 47.1%

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

   if -0.0 < (*.f64 V l) < 4.9999999999999997e304

   1. Initial program 83.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. 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 98.6

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

      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 98.6

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

   4. Applied rewrites 98.6%

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

   if 4.9999999999999997e304 < (*.f64 V l)

   1. Initial program 34.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 34.2

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

      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 34.2

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

   4. Applied rewrites 34.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-div N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lower-/.f64 61.6

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 61.6

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

   6. Applied rewrites 61.6%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+263}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-295}:\\ \;\;\;\;\frac{\sqrt{-A} \cdot c0}{\sqrt{\left(-V\right) \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+304}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(\sqrt{V \cdot \frac{\ell}{A}}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+263}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-295}:\\ \;\;\;\;\frac{\sqrt{-A} \cdot c0}{\sqrt{\left(-V\right) \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+304}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(\sqrt{V \cdot \frac{\ell}{A}}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -5e+263)
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))
   (if (<= (* V l) -2e-295)
     (/ (* (sqrt (- A)) c0) (sqrt (* (- V) l)))
     (if (<= (* V l) 0.0)
       (* c0 (sqrt (/ (/ A V) l)))
       (if (<= (* V l) 5e+304)
         (* c0 (/ (sqrt A) (sqrt (* l V))))
         (* c0 (pow (sqrt (* V (/ l A))) -1.0)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -5e+263) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else if ((V * l) <= -2e-295) {
		tmp = (sqrt(-A) * c0) / sqrt((-V * l));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if ((V * l) <= 5e+304) {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	} else {
		tmp = c0 * pow(sqrt((V * (l / A))), -1.0);
	}
	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) :: tmp
    if ((v * l) <= (-5d+263)) then
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    else if ((v * l) <= (-2d-295)) then
        tmp = (sqrt(-a) * c0) / sqrt((-v * l))
    else if ((v * l) <= 0.0d0) then
        tmp = c0 * sqrt(((a / v) / l))
    else if ((v * l) <= 5d+304) then
        tmp = c0 * (sqrt(a) / sqrt((l * v)))
    else
        tmp = c0 * (sqrt((v * (l / a))) ** (-1.0d0))
    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+263) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if ((V * l) <= -2e-295) {
		tmp = (Math.sqrt(-A) * c0) / Math.sqrt((-V * l));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if ((V * l) <= 5e+304) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((l * V)));
	} else {
		tmp = c0 * Math.pow(Math.sqrt((V * (l / A))), -1.0);
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -5e+263:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	elif (V * l) <= -2e-295:
		tmp = (math.sqrt(-A) * c0) / math.sqrt((-V * l))
	elif (V * l) <= 0.0:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif (V * l) <= 5e+304:
		tmp = c0 * (math.sqrt(A) / math.sqrt((l * V)))
	else:
		tmp = c0 * math.pow(math.sqrt((V * (l / A))), -1.0)
	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+263)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (Float64(V * l) <= -2e-295)
		tmp = Float64(Float64(sqrt(Float64(-A)) * c0) / sqrt(Float64(Float64(-V) * l)));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (Float64(V * l) <= 5e+304)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(l * V))));
	else
		tmp = Float64(c0 * (sqrt(Float64(V * Float64(l / A))) ^ -1.0));
	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+263)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	elseif ((V * l) <= -2e-295)
		tmp = (sqrt(-A) * c0) / sqrt((-V * l));
	elseif ((V * l) <= 0.0)
		tmp = c0 * sqrt(((A / V) / l));
	elseif ((V * l) <= 5e+304)
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	else
		tmp = c0 * (sqrt((V * (l / A))) ^ -1.0);
	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_] := If[LessEqual[N[(V * l), $MachinePrecision], -5e+263], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-295], N[(N[(N[Sqrt[(-A)], $MachinePrecision] * c0), $MachinePrecision] / N[Sqrt[N[((-V) * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e+304], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Power[N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -5.00000000000000022e263

   1. Initial program 37.8%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 35.5

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

   4. Applied rewrites 35.5%

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

   if -5.00000000000000022e263 < (*.f64 V l) < -2.00000000000000012e-295

   1. Initial program 83.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/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 75.6

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

   4. Applied rewrites 75.6%

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

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

      6. lift-sqrt.f64 N/A

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

      7. associate-*l/ N/A

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

      8. associate-*r/ N/A

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

      9. lift-/.f64 N/A

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

      10. frac-2neg N/A

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

      11. sqrt-div N/A

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

      12. frac-times N/A

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

      13. lift-sqrt.f64 N/A

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

      14. sqrt-prod N/A

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

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

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

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

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

      20. sqrt-prod N/A

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

      21. lift-sqrt.f64 N/A

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

   6. Applied rewrites 99.6%

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

   if -2.00000000000000012e-295 < (*.f64 V l) < -0.0

   1. Initial program 27.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 71.3

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

   4. Applied rewrites 71.3%

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

   if -0.0 < (*.f64 V l) < 4.9999999999999997e304

   1. Initial program 83.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. 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 98.6

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

      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 98.6

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

   4. Applied rewrites 98.6%

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

   if 4.9999999999999997e304 < (*.f64 V l)

   1. Initial program 34.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 34.2

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

      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 34.2

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

   4. Applied rewrites 34.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-div N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lower-/.f64 61.6

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 61.6

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

   6. Applied rewrites 61.6%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+263}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-295}:\\ \;\;\;\;\frac{\sqrt{-A} \cdot c0}{\sqrt{\left(-V\right) \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+304}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(\sqrt{V \cdot \frac{\ell}{A}}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+263}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-295}:\\ \;\;\;\;\frac{\sqrt{-A} \cdot c0}{\sqrt{\left(-V\right) \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+304}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(\sqrt{V \cdot \frac{\ell}{A}}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ (/ A V) l)))))
   (if (<= (* V l) -5e+263)
     t_0
     (if (<= (* V l) -2e-295)
       (/ (* (sqrt (- A)) c0) (sqrt (* (- V) l)))
       (if (<= (* V l) 0.0)
         t_0
         (if (<= (* V l) 5e+304)
           (* c0 (/ (sqrt A) (sqrt (* l V))))
           (* c0 (pow (sqrt (* V (/ l A))) -1.0))))))))

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 ((V * l) <= -5e+263) {
		tmp = t_0;
	} else if ((V * l) <= -2e-295) {
		tmp = (sqrt(-A) * c0) / sqrt((-V * l));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else if ((V * l) <= 5e+304) {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	} else {
		tmp = c0 * pow(sqrt((V * (l / A))), -1.0);
	}
	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 ((v * l) <= (-5d+263)) then
        tmp = t_0
    else if ((v * l) <= (-2d-295)) then
        tmp = (sqrt(-a) * c0) / sqrt((-v * l))
    else if ((v * l) <= 0.0d0) then
        tmp = t_0
    else if ((v * l) <= 5d+304) then
        tmp = c0 * (sqrt(a) / sqrt((l * v)))
    else
        tmp = c0 * (sqrt((v * (l / a))) ** (-1.0d0))
    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 ((V * l) <= -5e+263) {
		tmp = t_0;
	} else if ((V * l) <= -2e-295) {
		tmp = (Math.sqrt(-A) * c0) / Math.sqrt((-V * l));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else if ((V * l) <= 5e+304) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((l * V)));
	} else {
		tmp = c0 * Math.pow(Math.sqrt((V * (l / A))), -1.0);
	}
	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 (V * l) <= -5e+263:
		tmp = t_0
	elif (V * l) <= -2e-295:
		tmp = (math.sqrt(-A) * c0) / math.sqrt((-V * l))
	elif (V * l) <= 0.0:
		tmp = t_0
	elif (V * l) <= 5e+304:
		tmp = c0 * (math.sqrt(A) / math.sqrt((l * V)))
	else:
		tmp = c0 * math.pow(math.sqrt((V * (l / A))), -1.0)
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(Float64(A / V) / l)))
	tmp = 0.0
	if (Float64(V * l) <= -5e+263)
		tmp = t_0;
	elseif (Float64(V * l) <= -2e-295)
		tmp = Float64(Float64(sqrt(Float64(-A)) * c0) / sqrt(Float64(Float64(-V) * l)));
	elseif (Float64(V * l) <= 0.0)
		tmp = t_0;
	elseif (Float64(V * l) <= 5e+304)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(l * V))));
	else
		tmp = Float64(c0 * (sqrt(Float64(V * Float64(l / A))) ^ -1.0));
	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 ((V * l) <= -5e+263)
		tmp = t_0;
	elseif ((V * l) <= -2e-295)
		tmp = (sqrt(-A) * c0) / sqrt((-V * l));
	elseif ((V * l) <= 0.0)
		tmp = t_0;
	elseif ((V * l) <= 5e+304)
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	else
		tmp = c0 * (sqrt((V * (l / A))) ^ -1.0);
	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[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -5e+263], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -2e-295], N[(N[(N[Sqrt[(-A)], $MachinePrecision] * c0), $MachinePrecision] / N[Sqrt[N[((-V) * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], 5e+304], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Power[N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -5.00000000000000022e263 or -2.00000000000000012e-295 < (*.f64 V l) < -0.0

   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 69.5

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

   4. Applied rewrites 69.5%

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

   if -5.00000000000000022e263 < (*.f64 V l) < -2.00000000000000012e-295

   1. Initial program 83.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/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 75.6

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

   4. Applied rewrites 75.6%

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

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

      6. lift-sqrt.f64 N/A

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

      7. associate-*l/ N/A

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

      8. associate-*r/ N/A

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

      9. lift-/.f64 N/A

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

      10. frac-2neg N/A

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

      11. sqrt-div N/A

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

      12. frac-times N/A

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

      13. lift-sqrt.f64 N/A

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

      14. sqrt-prod N/A

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

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

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

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

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

      20. sqrt-prod N/A

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

      21. lift-sqrt.f64 N/A

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

   6. Applied rewrites 99.6%

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

   if -0.0 < (*.f64 V l) < 4.9999999999999997e304

   1. Initial program 83.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. 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 98.6

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

      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 98.6

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

   4. Applied rewrites 98.6%

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

   if 4.9999999999999997e304 < (*.f64 V l)

   1. Initial program 34.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 34.2

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

      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 34.2

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

   4. Applied rewrites 34.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-div N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lower-/.f64 61.6

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 61.6

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

   6. Applied rewrites 61.6%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+263}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-295}:\\ \;\;\;\;\frac{\sqrt{-A} \cdot c0}{\sqrt{\left(-V\right) \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+304}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(\sqrt{V \cdot \frac{\ell}{A}}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+263}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-295}:\\ \;\;\;\;\frac{c0}{\sqrt{\left(-\ell\right) \cdot V}} \cdot \sqrt{-A}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+304}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(\sqrt{V \cdot \frac{\ell}{A}}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ (/ A V) l)))))
   (if (<= (* V l) -5e+263)
     t_0
     (if (<= (* V l) -2e-295)
       (* (/ c0 (sqrt (* (- l) V))) (sqrt (- A)))
       (if (<= (* V l) 0.0)
         t_0
         (if (<= (* V l) 5e+304)
           (* c0 (/ (sqrt A) (sqrt (* l V))))
           (* c0 (pow (sqrt (* V (/ l A))) -1.0))))))))

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 ((V * l) <= -5e+263) {
		tmp = t_0;
	} else if ((V * l) <= -2e-295) {
		tmp = (c0 / sqrt((-l * V))) * sqrt(-A);
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else if ((V * l) <= 5e+304) {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	} else {
		tmp = c0 * pow(sqrt((V * (l / A))), -1.0);
	}
	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 ((v * l) <= (-5d+263)) then
        tmp = t_0
    else if ((v * l) <= (-2d-295)) then
        tmp = (c0 / sqrt((-l * v))) * sqrt(-a)
    else if ((v * l) <= 0.0d0) then
        tmp = t_0
    else if ((v * l) <= 5d+304) then
        tmp = c0 * (sqrt(a) / sqrt((l * v)))
    else
        tmp = c0 * (sqrt((v * (l / a))) ** (-1.0d0))
    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 ((V * l) <= -5e+263) {
		tmp = t_0;
	} else if ((V * l) <= -2e-295) {
		tmp = (c0 / Math.sqrt((-l * V))) * Math.sqrt(-A);
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else if ((V * l) <= 5e+304) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((l * V)));
	} else {
		tmp = c0 * Math.pow(Math.sqrt((V * (l / A))), -1.0);
	}
	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 (V * l) <= -5e+263:
		tmp = t_0
	elif (V * l) <= -2e-295:
		tmp = (c0 / math.sqrt((-l * V))) * math.sqrt(-A)
	elif (V * l) <= 0.0:
		tmp = t_0
	elif (V * l) <= 5e+304:
		tmp = c0 * (math.sqrt(A) / math.sqrt((l * V)))
	else:
		tmp = c0 * math.pow(math.sqrt((V * (l / A))), -1.0)
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(Float64(A / V) / l)))
	tmp = 0.0
	if (Float64(V * l) <= -5e+263)
		tmp = t_0;
	elseif (Float64(V * l) <= -2e-295)
		tmp = Float64(Float64(c0 / sqrt(Float64(Float64(-l) * V))) * sqrt(Float64(-A)));
	elseif (Float64(V * l) <= 0.0)
		tmp = t_0;
	elseif (Float64(V * l) <= 5e+304)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(l * V))));
	else
		tmp = Float64(c0 * (sqrt(Float64(V * Float64(l / A))) ^ -1.0));
	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 ((V * l) <= -5e+263)
		tmp = t_0;
	elseif ((V * l) <= -2e-295)
		tmp = (c0 / sqrt((-l * V))) * sqrt(-A);
	elseif ((V * l) <= 0.0)
		tmp = t_0;
	elseif ((V * l) <= 5e+304)
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	else
		tmp = c0 * (sqrt((V * (l / A))) ^ -1.0);
	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[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -5e+263], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -2e-295], N[(N[(c0 / N[Sqrt[N[((-l) * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[(-A)], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], 5e+304], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Power[N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -5.00000000000000022e263 or -2.00000000000000012e-295 < (*.f64 V l) < -0.0

   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 69.5

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

   4. Applied rewrites 69.5%

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

   if -5.00000000000000022e263 < (*.f64 V l) < -2.00000000000000012e-295

   1. Initial program 83.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/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 75.6

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

   4. Applied rewrites 75.6%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. lift-sqrt.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. lift-/.f64 N/A

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

      10. sqrt-div N/A

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

      11. metadata-eval N/A

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

      12. lift-sqrt.f64 N/A

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

      13. div-inv N/A

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

      14. associate-/r* N/A

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

      15. lower-/.f64 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lift-sqrt.f64 N/A

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

      18. sqrt-unprod N/A

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

      19. lift-/.f64 N/A

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

      20. associate-/r/ N/A

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

      21. lift-/.f64 N/A

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

      22. lower-sqrt.f64 N/A

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

      23. div-inv N/A

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

   6. Applied rewrites 79.7%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-*.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift-neg.f64 N/A

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

      10. sqrt-div N/A

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

      11. associate-/r/ N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lift-*.f64 N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. lower-neg.f64 N/A

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

      19. lower-sqrt.f64 99.5

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

   8. Applied rewrites 99.5%

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

   if -0.0 < (*.f64 V l) < 4.9999999999999997e304

   1. Initial program 83.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. 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 98.6

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

      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 98.6

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

   4. Applied rewrites 98.6%

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

   if 4.9999999999999997e304 < (*.f64 V l)

   1. Initial program 34.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 34.2

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

      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 34.2

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

   4. Applied rewrites 34.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-div N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lower-/.f64 61.6

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 61.6

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

   6. Applied rewrites 61.6%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+263}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-295}:\\ \;\;\;\;\frac{c0}{\sqrt{\left(-\ell\right) \cdot V}} \cdot \sqrt{-A}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+304}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(\sqrt{V \cdot \frac{\ell}{A}}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+274}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-295}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\left(-V\right) \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+304}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(\sqrt{V \cdot \frac{\ell}{A}}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ (/ A V) l)))))
   (if (<= (* V l) -5e+274)
     t_0
     (if (<= (* V l) -2e-295)
       (* c0 (/ (sqrt (- A)) (sqrt (* (- V) l))))
       (if (<= (* V l) 0.0)
         t_0
         (if (<= (* V l) 5e+304)
           (* c0 (/ (sqrt A) (sqrt (* l V))))
           (* c0 (pow (sqrt (* V (/ l A))) -1.0))))))))

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 ((V * l) <= -5e+274) {
		tmp = t_0;
	} else if ((V * l) <= -2e-295) {
		tmp = c0 * (sqrt(-A) / sqrt((-V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else if ((V * l) <= 5e+304) {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	} else {
		tmp = c0 * pow(sqrt((V * (l / A))), -1.0);
	}
	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 ((v * l) <= (-5d+274)) then
        tmp = t_0
    else if ((v * l) <= (-2d-295)) then
        tmp = c0 * (sqrt(-a) / sqrt((-v * l)))
    else if ((v * l) <= 0.0d0) then
        tmp = t_0
    else if ((v * l) <= 5d+304) then
        tmp = c0 * (sqrt(a) / sqrt((l * v)))
    else
        tmp = c0 * (sqrt((v * (l / a))) ** (-1.0d0))
    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 ((V * l) <= -5e+274) {
		tmp = t_0;
	} else if ((V * l) <= -2e-295) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((-V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else if ((V * l) <= 5e+304) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((l * V)));
	} else {
		tmp = c0 * Math.pow(Math.sqrt((V * (l / A))), -1.0);
	}
	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 (V * l) <= -5e+274:
		tmp = t_0
	elif (V * l) <= -2e-295:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((-V * l)))
	elif (V * l) <= 0.0:
		tmp = t_0
	elif (V * l) <= 5e+304:
		tmp = c0 * (math.sqrt(A) / math.sqrt((l * V)))
	else:
		tmp = c0 * math.pow(math.sqrt((V * (l / A))), -1.0)
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(Float64(A / V) / l)))
	tmp = 0.0
	if (Float64(V * l) <= -5e+274)
		tmp = t_0;
	elseif (Float64(V * l) <= -2e-295)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(Float64(-V) * l))));
	elseif (Float64(V * l) <= 0.0)
		tmp = t_0;
	elseif (Float64(V * l) <= 5e+304)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(l * V))));
	else
		tmp = Float64(c0 * (sqrt(Float64(V * Float64(l / A))) ^ -1.0));
	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 ((V * l) <= -5e+274)
		tmp = t_0;
	elseif ((V * l) <= -2e-295)
		tmp = c0 * (sqrt(-A) / sqrt((-V * l)));
	elseif ((V * l) <= 0.0)
		tmp = t_0;
	elseif ((V * l) <= 5e+304)
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	else
		tmp = c0 * (sqrt((V * (l / A))) ^ -1.0);
	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[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -5e+274], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -2e-295], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[((-V) * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], 5e+304], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Power[N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -4.9999999999999998e274 or -2.00000000000000012e-295 < (*.f64 V l) < -0.0

   1. Initial program 30.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 68.9

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

   4. Applied rewrites 68.9%

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

   if -4.9999999999999998e274 < (*.f64 V l) < -2.00000000000000012e-295

   1. Initial program 83.8%

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

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. sqrt-div N/A

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

      7. lift-sqrt.f64 N/A

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

      8. associate-/l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. sqrt-prod N/A

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

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

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

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

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

      21. lower-*.f64 N/A

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

      22. lower-neg.f64 99.4

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

   6. Applied rewrites 99.4%

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

   if -0.0 < (*.f64 V l) < 4.9999999999999997e304

   1. Initial program 83.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. 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 98.6

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

      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 98.6

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

   4. Applied rewrites 98.6%

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

   if 4.9999999999999997e304 < (*.f64 V l)

   1. Initial program 34.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 34.2

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

      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 34.2

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

   4. Applied rewrites 34.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-div N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lower-/.f64 61.6

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 61.6

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

   6. Applied rewrites 61.6%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+274}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-295}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\left(-V\right) \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+304}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(\sqrt{V \cdot \frac{\ell}{A}}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+304}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(\sqrt{V \cdot \frac{\ell}{A}}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) 0.0)
   (* c0 (sqrt (/ (/ A V) l)))
   (if (<= (* V l) 5e+304)
     (* c0 (/ (sqrt A) (sqrt (* l V))))
     (* c0 (pow (sqrt (* V (/ l A))) -1.0)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= 0.0) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if ((V * l) <= 5e+304) {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	} else {
		tmp = c0 * pow(sqrt((V * (l / A))), -1.0);
	}
	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) :: tmp
    if ((v * l) <= 0.0d0) then
        tmp = c0 * sqrt(((a / v) / l))
    else if ((v * l) <= 5d+304) then
        tmp = c0 * (sqrt(a) / sqrt((l * v)))
    else
        tmp = c0 * (sqrt((v * (l / a))) ** (-1.0d0))
    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) <= 0.0) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if ((V * l) <= 5e+304) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((l * V)));
	} else {
		tmp = c0 * Math.pow(Math.sqrt((V * (l / A))), -1.0);
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= 0.0:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif (V * l) <= 5e+304:
		tmp = c0 * (math.sqrt(A) / math.sqrt((l * V)))
	else:
		tmp = c0 * math.pow(math.sqrt((V * (l / A))), -1.0)
	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) <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (Float64(V * l) <= 5e+304)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(l * V))));
	else
		tmp = Float64(c0 * (sqrt(Float64(V * Float64(l / A))) ^ -1.0));
	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) <= 0.0)
		tmp = c0 * sqrt(((A / V) / l));
	elseif ((V * l) <= 5e+304)
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	else
		tmp = c0 * (sqrt((V * (l / A))) ^ -1.0);
	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_] := If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e+304], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Power[N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 65.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 76.2

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

   4. Applied rewrites 76.2%

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

   if -0.0 < (*.f64 V l) < 4.9999999999999997e304

   1. Initial program 83.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. 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 98.6

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

      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 98.6

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

   4. Applied rewrites 98.6%

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

   if 4.9999999999999997e304 < (*.f64 V l)

   1. Initial program 34.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 34.2

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

      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 34.2

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

   4. Applied rewrites 34.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-div N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lower-/.f64 61.6

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 61.6

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

   6. Applied rewrites 61.6%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+304}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(\sqrt{V \cdot \frac{\ell}{A}}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.5% accurate, 0.3× speedup

Math

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

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 (* c0 (sqrt (/ A (* V l))))))
   (if (<= t_0 0.0)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 2e+279)
       (/ c0 (sqrt (/ (* l V) A)))
       (/ 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 = c0 * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 2e+279) {
		tmp = c0 / sqrt(((l * V) / A));
	} 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 = c0 * sqrt((a / (v * l)))
    if (t_0 <= 0.0d0) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 2d+279) then
        tmp = c0 / sqrt(((l * v) / a))
    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 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 2e+279) {
		tmp = c0 / Math.sqrt(((l * V) / A));
	} 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 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 2e+279:
		tmp = c0 / math.sqrt(((l * V) / A))
	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(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 2e+279)
		tmp = Float64(c0 / sqrt(Float64(Float64(l * V) / A)));
	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 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 2e+279)
		tmp = c0 / sqrt(((l * V) / A));
	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[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+279], N[(c0 / N[Sqrt[N[(N[(l * V), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 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 65.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 69.9

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

   4. Applied rewrites 69.9%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 2.00000000000000012e279

   1. Initial program 98.5%

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

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 98.7

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 98.7

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

   4. Applied rewrites 98.7%

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

   if 2.00000000000000012e279 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 35.6%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 53.7

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

   4. Applied rewrites 53.7%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. lift-sqrt.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. lift-/.f64 N/A

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

      10. sqrt-div N/A

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

      11. metadata-eval N/A

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

      12. lift-sqrt.f64 N/A

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

      13. div-inv N/A

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

      14. associate-/r* N/A

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

      15. lower-/.f64 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lift-sqrt.f64 N/A

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

      18. sqrt-unprod N/A

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

      19. lift-/.f64 N/A

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

      20. associate-/r/ N/A

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

      21. lift-/.f64 N/A

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

      22. lower-sqrt.f64 N/A

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

      23. div-inv N/A

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

   6. Applied rewrites 53.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 55.9

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

   8. Applied rewrites 55.9%

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

Alternative 8: 80.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 2 \cdot 10^{+287}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (or (<= t_0 0.0) (not (<= t_0 2e+287)))
     (* c0 (sqrt (/ (/ A V) l)))
     (* c0 (sqrt t_0)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 2e+287)) {
		tmp = c0 * sqrt(((A / V) / l));
	} else {
		tmp = c0 * sqrt(t_0);
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if ((t_0 <= 0.0d0) .or. (.not. (t_0 <= 2d+287))) then
        tmp = c0 * sqrt(((a / v) / l))
    else
        tmp = c0 * sqrt(t_0)
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 2e+287)) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else {
		tmp = c0 * Math.sqrt(t_0);
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 2e+287):
		tmp = c0 * math.sqrt(((A / V) / l))
	else:
		tmp = c0 * math.sqrt(t_0)
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 2e+287))
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	else
		tmp = Float64(c0 * sqrt(t_0));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 2e+287)))
		tmp = c0 * sqrt(((A / V) / l));
	else
		tmp = c0 * sqrt(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 2e+287]], $MachinePrecision]], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 0.0 or 2.0000000000000002e287 < (/.f64 A (*.f64 V l))

   1. Initial program 31.4%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 52.0

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

   4. Applied rewrites 52.0%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 2.0000000000000002e287

   1. Initial program 99.2%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0 \lor \neg \left(\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{+287}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+297}:\\ \;\;\;\;c0 \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \end{array} \end{array} \]

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 (/ A (* V l))))
   (if (<= t_0 0.0)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 2e+297) (* c0 (sqrt 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 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 2e+297) {
		tmp = c0 * sqrt(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 = a / (v * l)
    if (t_0 <= 0.0d0) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 2d+297) then
        tmp = c0 * sqrt(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 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 2e+297) {
		tmp = c0 * Math.sqrt(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 = A / (V * l)
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 2e+297:
		tmp = c0 * math.sqrt(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(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 2e+297)
		tmp = Float64(c0 * sqrt(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 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 2e+297)
		tmp = c0 * sqrt(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[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+297], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(N[(l / A), $MachinePrecision] * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 24.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 44.8

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

   4. Applied rewrites 44.8%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 2e297

   1. Initial program 99.2%

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

   if 2e297 < (/.f64 A (*.f64 V l))

   1. Initial program 36.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 59.4

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

   4. Applied rewrites 59.4%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. lift-sqrt.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. lift-/.f64 N/A

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

      10. sqrt-div N/A

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

      11. metadata-eval N/A

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

      12. lift-sqrt.f64 N/A

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

      13. div-inv N/A

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

      14. associate-/r* N/A

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

      15. lower-/.f64 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lift-sqrt.f64 N/A

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

      18. sqrt-unprod N/A

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

      19. lift-/.f64 N/A

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

      20. associate-/r/ N/A

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

      21. lift-/.f64 N/A

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

      22. lower-sqrt.f64 N/A

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

      23. div-inv N/A

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

   6. Applied rewrites 59.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 60.8

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

   8. Applied rewrites 60.8%

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

Alternative 10: 82.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq 5 \cdot 10^{+304}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \end{array} \end{array} \]

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 (or (<= (* V l) 0.0) (not (<= (* V l) 5e+304)))
   (* 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) <= 0.0) || !((V * l) <= 5e+304)) {
		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) <= 0.0d0) .or. (.not. ((v * l) <= 5d+304))) 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) <= 0.0) || !((V * l) <= 5e+304)) {
		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) <= 0.0) or not ((V * l) <= 5e+304):
		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) <= 0.0) || !(Float64(V * l) <= 5e+304))
		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) <= 0.0) || ~(((V * l) <= 5e+304)))
		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[Or[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[Not[LessEqual[N[(V * l), $MachinePrecision], 5e+304]], $MachinePrecision]], 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) < -0.0 or 4.9999999999999997e304 < (*.f64 V l)

   1. Initial program 61.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 74.3

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

   4. Applied rewrites 74.3%

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

   if -0.0 < (*.f64 V l) < 4.9999999999999997e304

   1. Initial program 83.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. 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 98.6

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

      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 98.6

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

   4. Applied rewrites 98.6%

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

4. Final simplification 82.9%

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

Alternative 11: 87.5% accurate, 0.5× speedup

Math

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

FPCore

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if V < -4.999999999999985e-310

   1. Initial program 67.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 74.6

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

   4. Applied rewrites 74.6%

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

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 83.7

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

   6. Applied rewrites 83.7%

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

   if -4.999999999999985e-310 < V

   1. Initial program 71.3%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 44.7

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

   4. Applied rewrites 44.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 52.5

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

   6. Applied rewrites 52.5%

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

Alternative 12: 89.8% accurate, 0.5× speedup

Math

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

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 (<= A -5e-310)
   (/ (* (sqrt (- A)) c0) (* (sqrt (- V)) (sqrt 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 (A <= -5e-310) {
		tmp = (sqrt(-A) * c0) / (sqrt(-V) * sqrt(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 (a <= (-5d-310)) then
        tmp = (sqrt(-a) * c0) / (sqrt(-v) * sqrt(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 (A <= -5e-310) {
		tmp = (Math.sqrt(-A) * c0) / (Math.sqrt(-V) * Math.sqrt(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 A <= -5e-310:
		tmp = (math.sqrt(-A) * c0) / (math.sqrt(-V) * math.sqrt(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 (A <= -5e-310)
		tmp = Float64(Float64(sqrt(Float64(-A)) * c0) / Float64(sqrt(Float64(-V)) * sqrt(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 (A <= -5e-310)
		tmp = (sqrt(-A) * c0) / (sqrt(-V) * sqrt(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[A, -5e-310], N[(N[(N[Sqrt[(-A)], $MachinePrecision] * c0), $MachinePrecision] / N[(N[Sqrt[(-V)], $MachinePrecision] * N[Sqrt[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 A < -4.999999999999985e-310

   1. Initial program 68.1%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 72.9

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

   4. Applied rewrites 72.9%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. lift-sqrt.f64 N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. sqrt-div N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. frac-times N/A

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

      13. *-commutative N/A

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

      14. frac-times N/A

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

      15. frac-2neg N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lift-sqrt.f64 N/A

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

      18. sqrt-div N/A

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

      19. frac-2neg N/A

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

      20. sqrt-div N/A

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

      21. frac-times N/A

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

   6. Applied rewrites 47.9%

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

   if -4.999999999999985e-310 < A

   1. Initial program 71.0%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 82.0

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

      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 82.0

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

   4. Applied rewrites 82.0%

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

4. Final simplification 64.0%

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

Alternative 13: 87.1% accurate, 0.6× speedup

Math

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

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 -5e-310)
   (* c0 (/ (sqrt (/ (- A) l)) (sqrt (- V))))
   (/ (* (sqrt A) c0) (* (sqrt l) (sqrt V)))))

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 = (sqrt(A) * c0) / (sqrt(l) * sqrt(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 <= (-5d-310)) then
        tmp = c0 * (sqrt((-a / l)) / sqrt(-v))
    else
        tmp = (sqrt(a) * c0) / (sqrt(l) * sqrt(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 <= -5e-310) {
		tmp = c0 * (Math.sqrt((-A / l)) / Math.sqrt(-V));
	} else {
		tmp = (Math.sqrt(A) * c0) / (Math.sqrt(l) * Math.sqrt(V));
	}
	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 = (math.sqrt(A) * c0) / (math.sqrt(l) * math.sqrt(V))
	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(sqrt(A) * c0) / Float64(sqrt(l) * sqrt(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 <= -5e-310)
		tmp = c0 * (sqrt((-A / l)) / sqrt(-V));
	else
		tmp = (sqrt(A) * c0) / (sqrt(l) * sqrt(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[V, -5e-310], N[(c0 * N[(N[Sqrt[N[((-A) / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[A], $MachinePrecision] * c0), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[V], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if V < -4.999999999999985e-310

   1. Initial program 67.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 74.6

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

   4. Applied rewrites 74.6%

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

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 83.7

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

   6. Applied rewrites 83.7%

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

   if -4.999999999999985e-310 < V

   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

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

      2. *-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. sqrt-div N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 43.8

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 43.8

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

   4. Applied rewrites 43.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lower-*.f64 51.3

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

   6. Applied rewrites 51.3%

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

Alternative 14: 74.1% 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 69.5%

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

Reproduce

herbie shell --seed 2024340 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))