Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.3% → 87.9%

Time: 6.3s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 87.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}}\\ \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+261}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-248}:\\ \;\;\;\;\frac{\sqrt{-A} \cdot c0}{\sqrt{\left(-\ell\right) \cdot V}}\\ \mathbf{elif}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq 2 \cdot 10^{+305}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot V}} \cdot \sqrt{A}\\ \end{array} \end{array} \]

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) l)) (sqrt (- V))))))
   (if (<= (* V l) -1e+261)
     t_0
     (if (<= (* V l) -1e-248)
       (/ (* (sqrt (- A)) c0) (sqrt (* (- l) V)))
       (if (or (<= (* V l) 0.0) (not (<= (* V l) 2e+305)))
         t_0
         (* (/ c0 (sqrt (* l V))) (sqrt A)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * (sqrt((-A / l)) / sqrt(-V));
	double tmp;
	if ((V * l) <= -1e+261) {
		tmp = t_0;
	} else if ((V * l) <= -1e-248) {
		tmp = (sqrt(-A) * c0) / sqrt((-l * V));
	} else if (((V * l) <= 0.0) || !((V * l) <= 2e+305)) {
		tmp = t_0;
	} else {
		tmp = (c0 / sqrt((l * V))) * sqrt(A);
	}
	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 / l)) / sqrt(-v))
    if ((v * l) <= (-1d+261)) then
        tmp = t_0
    else if ((v * l) <= (-1d-248)) then
        tmp = (sqrt(-a) * c0) / sqrt((-l * v))
    else if (((v * l) <= 0.0d0) .or. (.not. ((v * l) <= 2d+305))) then
        tmp = t_0
    else
        tmp = (c0 / sqrt((l * v))) * sqrt(a)
    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 / l)) / Math.sqrt(-V));
	double tmp;
	if ((V * l) <= -1e+261) {
		tmp = t_0;
	} else if ((V * l) <= -1e-248) {
		tmp = (Math.sqrt(-A) * c0) / Math.sqrt((-l * V));
	} else if (((V * l) <= 0.0) || !((V * l) <= 2e+305)) {
		tmp = t_0;
	} else {
		tmp = (c0 / Math.sqrt((l * V))) * Math.sqrt(A);
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * (math.sqrt((-A / l)) / math.sqrt(-V))
	tmp = 0
	if (V * l) <= -1e+261:
		tmp = t_0
	elif (V * l) <= -1e-248:
		tmp = (math.sqrt(-A) * c0) / math.sqrt((-l * V))
	elif ((V * l) <= 0.0) or not ((V * l) <= 2e+305):
		tmp = t_0
	else:
		tmp = (c0 / math.sqrt((l * V))) * math.sqrt(A)
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * Float64(sqrt(Float64(Float64(-A) / l)) / sqrt(Float64(-V))))
	tmp = 0.0
	if (Float64(V * l) <= -1e+261)
		tmp = t_0;
	elseif (Float64(V * l) <= -1e-248)
		tmp = Float64(Float64(sqrt(Float64(-A)) * c0) / sqrt(Float64(Float64(-l) * V)));
	elseif ((Float64(V * l) <= 0.0) || !(Float64(V * l) <= 2e+305))
		tmp = t_0;
	else
		tmp = Float64(Float64(c0 / sqrt(Float64(l * V))) * sqrt(A));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * (sqrt((-A / l)) / sqrt(-V));
	tmp = 0.0;
	if ((V * l) <= -1e+261)
		tmp = t_0;
	elseif ((V * l) <= -1e-248)
		tmp = (sqrt(-A) * c0) / sqrt((-l * V));
	elseif (((V * l) <= 0.0) || ~(((V * l) <= 2e+305)))
		tmp = t_0;
	else
		tmp = (c0 / sqrt((l * V))) * sqrt(A);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[(N[Sqrt[N[((-A) / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -1e+261], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -1e-248], N[(N[(N[Sqrt[(-A)], $MachinePrecision] * c0), $MachinePrecision] / N[Sqrt[N[((-l) * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[Not[LessEqual[N[(V * l), $MachinePrecision], 2e+305]], $MachinePrecision]], t$95$0, N[(N[(c0 / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[A], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -9.9999999999999993e260 or -9.9999999999999998e-249 < (*.f64 V l) < 0.0 or 1.9999999999999999e305 < (*.f64 V l)

   1. Initial program 45.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 18.1

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

      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 18.1

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

   4. Applied rewrites 18.1%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-undiv N/A

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

      5. frac-2neg-rev N/A

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

      6. distribute-frac-neg2 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. distribute-frac-neg2 N/A

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

      10. sqrt-div N/A

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

      11. pow1/2 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. pow1/2 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lower-neg.f64 44.4

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

   6. Applied rewrites 44.4%

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

   if -9.9999999999999993e260 < (*.f64 V l) < -9.9999999999999998e-249

   1. Initial program 82.5%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. pow1/2 N/A

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

      4. pow-to-exp N/A

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

      5. sinh-+-cosh-rev N/A

         \[\leadsto c0 \cdot \color{blue}{\left(\cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) + \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right)\right)} \]

      6. flip-+ N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) \cdot \cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) - \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) \cdot \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right)}{\cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) - \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right)}} \]

      7. sinh-cosh N/A

         \[\leadsto c0 \cdot \frac{\color{blue}{1}}{\cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) - \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right)} \]

      8. sinh---cosh-rev N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. exp-neg N/A

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

      13. pow-to-exp N/A

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

      14. pow1/2 N/A

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

      15. lift-sqrt.f64 N/A

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

   4. Applied rewrites 82.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow-1 N/A

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

      7. remove-double-div N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-2neg N/A

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

      11. sqrt-div N/A

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

      12. associate-*l/ N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lift-*.f64 N/A

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

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

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

      20. lower-*.f64 N/A

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

      21. lower-neg.f64 99.5

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

   6. Applied rewrites 99.5%

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

   if 0.0 < (*.f64 V l) < 1.9999999999999999e305

   1. Initial program 89.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sqrt.f64 99.0

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

   4. Applied rewrites 99.0%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+261}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-248}:\\ \;\;\;\;\frac{\sqrt{-A} \cdot c0}{\sqrt{\left(-\ell\right) \cdot V}}\\ \mathbf{elif}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq 2 \cdot 10^{+305}\right):\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot V}} \cdot \sqrt{A}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 4 \cdot 10^{+189}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (or (<= t_0 0.0) (not (<= t_0 4e+189)))
     (* c0 (sqrt (/ (/ A V) l)))
     t_0)))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 4e+189)) {
		tmp = c0 * sqrt(((A / V) / l));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    if ((t_0 <= 0.0d0) .or. (.not. (t_0 <= 4d+189))) then
        tmp = c0 * sqrt(((a / v) / l))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 4e+189)) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 4e+189):
		tmp = c0 * math.sqrt(((A / V) / l))
	else:
		tmp = t_0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 4e+189))
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 4e+189)))
		tmp = c0 * sqrt(((A / V) / l));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 4e+189]], $MachinePrecision]], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 0.0 or 4.0000000000000001e189 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 68.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 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)))) < 4.0000000000000001e189

   1. Initial program 99.5%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 0 \lor \neg \left(c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 4 \cdot 10^{+189}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+189}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A} \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (<= t_0 0.0)
     (* c0 (sqrt (/ (/ A l) V)))
     (if (<= t_0 4e+189) t_0 (/ c0 (sqrt (* (/ V A) l)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * sqrt(((A / l) / V));
	} else if (t_0 <= 4e+189) {
		tmp = t_0;
	} else {
		tmp = c0 / sqrt(((V / A) * l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    if (t_0 <= 0.0d0) then
        tmp = c0 * sqrt(((a / l) / v))
    else if (t_0 <= 4d+189) then
        tmp = t_0
    else
        tmp = c0 / sqrt(((v / a) * l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * Math.sqrt(((A / l) / V));
	} else if (t_0 <= 4e+189) {
		tmp = t_0;
	} else {
		tmp = c0 / Math.sqrt(((V / A) * l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0 * math.sqrt(((A / l) / V))
	elif t_0 <= 4e+189:
		tmp = t_0
	else:
		tmp = c0 / math.sqrt(((V / A) * l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	elseif (t_0 <= 4e+189)
		tmp = t_0;
	else
		tmp = Float64(c0 / sqrt(Float64(Float64(V / A) * l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0 * sqrt(((A / l) / V));
	elseif (t_0 <= 4e+189)
		tmp = t_0;
	else
		tmp = c0 / sqrt(((V / A) * l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+189], t$95$0, N[(c0 / N[Sqrt[N[(N[(V / A), $MachinePrecision] * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 68.7%

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

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 71.6

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

   4. Applied rewrites 71.6%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 4.0000000000000001e189

   1. Initial program 99.5%

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

   if 4.0000000000000001e189 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 66.3%

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

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

      3. pow1/2 N/A

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

      4. pow-to-exp N/A

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

      5. sinh-+-cosh-rev N/A

         \[\leadsto c0 \cdot \color{blue}{\left(\cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) + \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right)\right)} \]

      6. flip-+ N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) \cdot \cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) - \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) \cdot \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right)}{\cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) - \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right)}} \]

      7. sinh-cosh N/A

         \[\leadsto c0 \cdot \frac{\color{blue}{1}}{\cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) - \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right)} \]

      8. sinh---cosh-rev N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. exp-neg N/A

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

      13. pow-to-exp N/A

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

      14. pow1/2 N/A

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

      15. lift-sqrt.f64 N/A

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

   4. Applied rewrites 66.3%

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

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 66.3

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

      3. lift-pow.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 66.4

          \[\leadsto \frac{c0}{{\left(\frac{A}{V \cdot \ell}\right)}^{\color{blue}{-0.5}}} \]

   6. Applied rewrites 66.4%

      \[\leadsto \color{blue}{\frac{c0}{{\left(\frac{A}{V \cdot \ell}\right)}^{-0.5}}} \]

   7. Taylor expanded in A around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 67.5

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

   9. Applied rewrites 67.5%

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

   11. Applied rewrites 74.6%

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

Alternative 4: 76.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+189}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (<= t_0 0.0)
     (* c0 (sqrt (/ (/ A l) V)))
     (if (<= t_0 4e+189) t_0 (* c0 (sqrt (/ (/ A V) l)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * sqrt(((A / l) / V));
	} else if (t_0 <= 4e+189) {
		tmp = t_0;
	} else {
		tmp = c0 * sqrt(((A / V) / l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    if (t_0 <= 0.0d0) then
        tmp = c0 * sqrt(((a / l) / v))
    else if (t_0 <= 4d+189) then
        tmp = t_0
    else
        tmp = c0 * sqrt(((a / v) / l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * Math.sqrt(((A / l) / V));
	} else if (t_0 <= 4e+189) {
		tmp = t_0;
	} else {
		tmp = c0 * Math.sqrt(((A / V) / l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0 * math.sqrt(((A / l) / V))
	elif t_0 <= 4e+189:
		tmp = t_0
	else:
		tmp = c0 * math.sqrt(((A / V) / l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	elseif (t_0 <= 4e+189)
		tmp = t_0;
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0 * sqrt(((A / l) / V));
	elseif (t_0 <= 4e+189)
		tmp = t_0;
	else
		tmp = c0 * sqrt(((A / V) / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+189], t$95$0, N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 68.7%

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

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 71.6

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

   4. Applied rewrites 71.6%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 4.0000000000000001e189

   1. Initial program 99.5%

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

   if 4.0000000000000001e189 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 66.3%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 73.5

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

   4. Applied rewrites 73.5%

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

Alternative 5: 87.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\frac{A}{V}} \cdot c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-248}:\\ \;\;\;\;\frac{\sqrt{-A} \cdot c0}{\sqrt{\left(-\ell\right) \cdot V}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-276}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A} \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot V}} \cdot \sqrt{A}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) (- INFINITY))
   (/ (* (sqrt (/ A V)) c0) (sqrt l))
   (if (<= (* V l) -1e-248)
     (/ (* (sqrt (- A)) c0) (sqrt (* (- l) V)))
     (if (<= (* V l) 5e-276)
       (/ c0 (sqrt (* (/ V A) l)))
       (* (/ c0 (sqrt (* l V))) (sqrt A))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = (sqrt((A / V)) * c0) / sqrt(l);
	} else if ((V * l) <= -1e-248) {
		tmp = (sqrt(-A) * c0) / sqrt((-l * V));
	} else if ((V * l) <= 5e-276) {
		tmp = c0 / sqrt(((V / A) * l));
	} else {
		tmp = (c0 / sqrt((l * V))) * sqrt(A);
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = (Math.sqrt((A / V)) * c0) / Math.sqrt(l);
	} else if ((V * l) <= -1e-248) {
		tmp = (Math.sqrt(-A) * c0) / Math.sqrt((-l * V));
	} else if ((V * l) <= 5e-276) {
		tmp = c0 / Math.sqrt(((V / A) * l));
	} else {
		tmp = (c0 / Math.sqrt((l * V))) * Math.sqrt(A);
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = (math.sqrt((A / V)) * c0) / math.sqrt(l)
	elif (V * l) <= -1e-248:
		tmp = (math.sqrt(-A) * c0) / math.sqrt((-l * V))
	elif (V * l) <= 5e-276:
		tmp = c0 / math.sqrt(((V / A) * l))
	else:
		tmp = (c0 / math.sqrt((l * V))) * math.sqrt(A)
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(Float64(A / V)) * c0) / sqrt(l));
	elseif (Float64(V * l) <= -1e-248)
		tmp = Float64(Float64(sqrt(Float64(-A)) * c0) / sqrt(Float64(Float64(-l) * V)));
	elseif (Float64(V * l) <= 5e-276)
		tmp = Float64(c0 / sqrt(Float64(Float64(V / A) * l)));
	else
		tmp = Float64(Float64(c0 / sqrt(Float64(l * V))) * sqrt(A));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = (sqrt((A / V)) * c0) / sqrt(l);
	elseif ((V * l) <= -1e-248)
		tmp = (sqrt(-A) * c0) / sqrt((-l * V));
	elseif ((V * l) <= 5e-276)
		tmp = c0 / sqrt(((V / A) * l));
	else
		tmp = (c0 / sqrt((l * V))) * sqrt(A);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-248], N[(N[(N[Sqrt[(-A)], $MachinePrecision] * c0), $MachinePrecision] / N[Sqrt[N[((-l) * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e-276], N[(c0 / N[Sqrt[N[(N[(V / A), $MachinePrecision] * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(c0 / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[A], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 35.8%

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

      1. lift-*.f64 N/A

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

      2. *-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. lift-*.f64 N/A

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

      8. sqrt-prod N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. sqrt-div N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-sqrt.f64 40.8

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

   4. Applied rewrites 40.8%

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

   if -inf.0 < (*.f64 V l) < -9.9999999999999998e-249

   1. Initial program 81.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. pow1/2 N/A

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

      4. pow-to-exp N/A

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

      5. sinh-+-cosh-rev N/A

         \[\leadsto c0 \cdot \color{blue}{\left(\cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) + \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right)\right)} \]

      6. flip-+ N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) \cdot \cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) - \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) \cdot \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right)}{\cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) - \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right)}} \]

      7. sinh-cosh N/A

         \[\leadsto c0 \cdot \frac{\color{blue}{1}}{\cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) - \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right)} \]

      8. sinh---cosh-rev N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. exp-neg N/A

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

      13. pow-to-exp N/A

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

      14. pow1/2 N/A

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

      15. lift-sqrt.f64 N/A

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

   4. Applied rewrites 81.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow-1 N/A

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

      7. remove-double-div N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-2neg N/A

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

      11. sqrt-div N/A

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

      12. associate-*l/ N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lift-*.f64 N/A

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

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

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

      20. lower-*.f64 N/A

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

      21. lower-neg.f64 98.4

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

   6. Applied rewrites 98.4%

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

   if -9.9999999999999998e-249 < (*.f64 V l) < 4.99999999999999967e-276

   1. Initial program 59.4%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. pow1/2 N/A

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

      4. pow-to-exp N/A

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

      5. sinh-+-cosh-rev N/A

         \[\leadsto c0 \cdot \color{blue}{\left(\cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) + \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right)\right)} \]

      6. flip-+ N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) \cdot \cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) - \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) \cdot \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right)}{\cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) - \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right)}} \]

      7. sinh-cosh N/A

         \[\leadsto c0 \cdot \frac{\color{blue}{1}}{\cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) - \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right)} \]

      8. sinh---cosh-rev N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. exp-neg N/A

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

      13. pow-to-exp N/A

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

      14. pow1/2 N/A

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

      15. lift-sqrt.f64 N/A

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

   4. Applied rewrites 59.3%

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

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 59.3

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

      3. lift-pow.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 59.4

          \[\leadsto \frac{c0}{{\left(\frac{A}{V \cdot \ell}\right)}^{\color{blue}{-0.5}}} \]

   6. Applied rewrites 59.4%

      \[\leadsto \color{blue}{\frac{c0}{{\left(\frac{A}{V \cdot \ell}\right)}^{-0.5}}} \]

   7. Taylor expanded in A around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 59.4

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

   9. Applied rewrites 59.4%

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

   11. Applied rewrites 74.5%

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

   if 4.99999999999999967e-276 < (*.f64 V l)

   1. Initial program 81.8%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sqrt.f64 92.1

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

   4. Applied rewrites 92.1%

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

Alternative 6: 86.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-248}:\\ \;\;\;\;\frac{\sqrt{-A} \cdot c0}{\sqrt{\left(-\ell\right) \cdot V}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-276}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A} \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot V}} \cdot \sqrt{A}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) (- INFINITY))
   (/ c0 (* (sqrt (/ V A)) (sqrt l)))
   (if (<= (* V l) -1e-248)
     (/ (* (sqrt (- A)) c0) (sqrt (* (- l) V)))
     (if (<= (* V l) 5e-276)
       (/ c0 (sqrt (* (/ V A) l)))
       (* (/ c0 (sqrt (* l V))) (sqrt A))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = c0 / (sqrt((V / A)) * sqrt(l));
	} else if ((V * l) <= -1e-248) {
		tmp = (sqrt(-A) * c0) / sqrt((-l * V));
	} else if ((V * l) <= 5e-276) {
		tmp = c0 / sqrt(((V / A) * l));
	} else {
		tmp = (c0 / sqrt((l * V))) * sqrt(A);
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = c0 / (Math.sqrt((V / A)) * Math.sqrt(l));
	} else if ((V * l) <= -1e-248) {
		tmp = (Math.sqrt(-A) * c0) / Math.sqrt((-l * V));
	} else if ((V * l) <= 5e-276) {
		tmp = c0 / Math.sqrt(((V / A) * l));
	} else {
		tmp = (c0 / Math.sqrt((l * V))) * Math.sqrt(A);
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = c0 / (math.sqrt((V / A)) * math.sqrt(l))
	elif (V * l) <= -1e-248:
		tmp = (math.sqrt(-A) * c0) / math.sqrt((-l * V))
	elif (V * l) <= 5e-276:
		tmp = c0 / math.sqrt(((V / A) * l))
	else:
		tmp = (c0 / math.sqrt((l * V))) * math.sqrt(A)
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(c0 / Float64(sqrt(Float64(V / A)) * sqrt(l)));
	elseif (Float64(V * l) <= -1e-248)
		tmp = Float64(Float64(sqrt(Float64(-A)) * c0) / sqrt(Float64(Float64(-l) * V)));
	elseif (Float64(V * l) <= 5e-276)
		tmp = Float64(c0 / sqrt(Float64(Float64(V / A) * l)));
	else
		tmp = Float64(Float64(c0 / sqrt(Float64(l * V))) * sqrt(A));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = c0 / (sqrt((V / A)) * sqrt(l));
	elseif ((V * l) <= -1e-248)
		tmp = (sqrt(-A) * c0) / sqrt((-l * V));
	elseif ((V * l) <= 5e-276)
		tmp = c0 / sqrt(((V / A) * l));
	else
		tmp = (c0 / sqrt((l * V))) * sqrt(A);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(c0 / N[(N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-248], N[(N[(N[Sqrt[(-A)], $MachinePrecision] * c0), $MachinePrecision] / N[Sqrt[N[((-l) * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e-276], N[(c0 / N[Sqrt[N[(N[(V / A), $MachinePrecision] * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(c0 / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[A], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 35.8%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. pow-to-exp N/A

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

      5. sinh-+-cosh-rev N/A

         \[\leadsto c0 \cdot \color{blue}{\left(\cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) + \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right)\right)} \]

      6. flip-+ N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) \cdot \cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) - \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) \cdot \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right)}{\cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) - \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right)}} \]

      7. sinh-cosh N/A

         \[\leadsto c0 \cdot \frac{\color{blue}{1}}{\cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) - \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right)} \]

      8. sinh---cosh-rev N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. exp-neg N/A

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

      13. pow-to-exp N/A

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

      14. pow1/2 N/A

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

      15. lift-sqrt.f64 N/A

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

   4. Applied rewrites 35.8%

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

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 35.8

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

      3. lift-pow.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 35.8

          \[\leadsto \frac{c0}{{\left(\frac{A}{V \cdot \ell}\right)}^{\color{blue}{-0.5}}} \]

   6. Applied rewrites 35.8%

      \[\leadsto \color{blue}{\frac{c0}{{\left(\frac{A}{V \cdot \ell}\right)}^{-0.5}}} \]

   7. Taylor expanded in A around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 35.8

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

   9. Applied rewrites 35.8%

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

   11. Applied rewrites 40.9%

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

   if -inf.0 < (*.f64 V l) < -9.9999999999999998e-249

   1. Initial program 81.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. pow1/2 N/A

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

      4. pow-to-exp N/A

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

      5. sinh-+-cosh-rev N/A

         \[\leadsto c0 \cdot \color{blue}{\left(\cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) + \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right)\right)} \]

      6. flip-+ N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) \cdot \cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) - \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) \cdot \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right)}{\cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) - \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right)}} \]

      7. sinh-cosh N/A

         \[\leadsto c0 \cdot \frac{\color{blue}{1}}{\cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) - \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right)} \]

      8. sinh---cosh-rev N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. exp-neg N/A

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

      13. pow-to-exp N/A

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

      14. pow1/2 N/A

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

      15. lift-sqrt.f64 N/A

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

   4. Applied rewrites 81.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow-1 N/A

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

      7. remove-double-div N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-2neg N/A

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

      11. sqrt-div N/A

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

      12. associate-*l/ N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lift-*.f64 N/A

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

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

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

      20. lower-*.f64 N/A

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

      21. lower-neg.f64 98.4

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

   6. Applied rewrites 98.4%

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

   if -9.9999999999999998e-249 < (*.f64 V l) < 4.99999999999999967e-276

   1. Initial program 59.4%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. pow1/2 N/A

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

      4. pow-to-exp N/A

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

      5. sinh-+-cosh-rev N/A

         \[\leadsto c0 \cdot \color{blue}{\left(\cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) + \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right)\right)} \]

      6. flip-+ N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) \cdot \cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) - \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) \cdot \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right)}{\cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) - \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right)}} \]

      7. sinh-cosh N/A

         \[\leadsto c0 \cdot \frac{\color{blue}{1}}{\cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) - \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right)} \]

      8. sinh---cosh-rev N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. exp-neg N/A

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

      13. pow-to-exp N/A

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

      14. pow1/2 N/A

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

      15. lift-sqrt.f64 N/A

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

   4. Applied rewrites 59.3%

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

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 59.3

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

      3. lift-pow.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 59.4

          \[\leadsto \frac{c0}{{\left(\frac{A}{V \cdot \ell}\right)}^{\color{blue}{-0.5}}} \]

   6. Applied rewrites 59.4%

      \[\leadsto \color{blue}{\frac{c0}{{\left(\frac{A}{V \cdot \ell}\right)}^{-0.5}}} \]

   7. Taylor expanded in A around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 59.4

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

   9. Applied rewrites 59.4%

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

   11. Applied rewrites 74.5%

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

   if 4.99999999999999967e-276 < (*.f64 V l)

   1. Initial program 81.8%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sqrt.f64 92.1

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

   4. Applied rewrites 92.1%

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

Alternative 7: 87.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-248}:\\ \;\;\;\;\frac{\sqrt{-A} \cdot c0}{\sqrt{\left(-\ell\right) \cdot V}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-276}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A} \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot V}} \cdot \sqrt{A}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) (- INFINITY))
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))
   (if (<= (* V l) -1e-248)
     (/ (* (sqrt (- A)) c0) (sqrt (* (- l) V)))
     (if (<= (* V l) 5e-276)
       (/ c0 (sqrt (* (/ V A) l)))
       (* (/ c0 (sqrt (* l V))) (sqrt A))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else if ((V * l) <= -1e-248) {
		tmp = (sqrt(-A) * c0) / sqrt((-l * V));
	} else if ((V * l) <= 5e-276) {
		tmp = c0 / sqrt(((V / A) * l));
	} else {
		tmp = (c0 / sqrt((l * V))) * sqrt(A);
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if ((V * l) <= -1e-248) {
		tmp = (Math.sqrt(-A) * c0) / Math.sqrt((-l * V));
	} else if ((V * l) <= 5e-276) {
		tmp = c0 / Math.sqrt(((V / A) * l));
	} else {
		tmp = (c0 / Math.sqrt((l * V))) * Math.sqrt(A);
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	elif (V * l) <= -1e-248:
		tmp = (math.sqrt(-A) * c0) / math.sqrt((-l * V))
	elif (V * l) <= 5e-276:
		tmp = c0 / math.sqrt(((V / A) * l))
	else:
		tmp = (c0 / math.sqrt((l * V))) * math.sqrt(A)
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (Float64(V * l) <= -1e-248)
		tmp = Float64(Float64(sqrt(Float64(-A)) * c0) / sqrt(Float64(Float64(-l) * V)));
	elseif (Float64(V * l) <= 5e-276)
		tmp = Float64(c0 / sqrt(Float64(Float64(V / A) * l)));
	else
		tmp = Float64(Float64(c0 / sqrt(Float64(l * V))) * sqrt(A));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	elseif ((V * l) <= -1e-248)
		tmp = (sqrt(-A) * c0) / sqrt((-l * V));
	elseif ((V * l) <= 5e-276)
		tmp = c0 / sqrt(((V / A) * l));
	else
		tmp = (c0 / sqrt((l * V))) * sqrt(A);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-248], N[(N[(N[Sqrt[(-A)], $MachinePrecision] * c0), $MachinePrecision] / N[Sqrt[N[((-l) * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e-276], N[(c0 / N[Sqrt[N[(N[(V / A), $MachinePrecision] * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(c0 / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[A], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 35.8%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 40.8

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

   4. Applied rewrites 40.8%

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

   if -inf.0 < (*.f64 V l) < -9.9999999999999998e-249

   1. Initial program 81.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. pow1/2 N/A

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

      4. pow-to-exp N/A

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

      5. sinh-+-cosh-rev N/A

         \[\leadsto c0 \cdot \color{blue}{\left(\cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) + \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right)\right)} \]

      6. flip-+ N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) \cdot \cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) - \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) \cdot \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right)}{\cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) - \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right)}} \]

      7. sinh-cosh N/A

         \[\leadsto c0 \cdot \frac{\color{blue}{1}}{\cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) - \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right)} \]

      8. sinh---cosh-rev N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. exp-neg N/A

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

      13. pow-to-exp N/A

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

      14. pow1/2 N/A

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

      15. lift-sqrt.f64 N/A

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

   4. Applied rewrites 81.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow-1 N/A

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

      7. remove-double-div N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-2neg N/A

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

      11. sqrt-div N/A

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

      12. associate-*l/ N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lift-*.f64 N/A

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

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

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

      20. lower-*.f64 N/A

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

      21. lower-neg.f64 98.4

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

   6. Applied rewrites 98.4%

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

   if -9.9999999999999998e-249 < (*.f64 V l) < 4.99999999999999967e-276

   1. Initial program 59.4%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. pow1/2 N/A

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

      4. pow-to-exp N/A

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

      5. sinh-+-cosh-rev N/A

         \[\leadsto c0 \cdot \color{blue}{\left(\cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) + \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right)\right)} \]

      6. flip-+ N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) \cdot \cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) - \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) \cdot \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right)}{\cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) - \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right)}} \]

      7. sinh-cosh N/A

         \[\leadsto c0 \cdot \frac{\color{blue}{1}}{\cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) - \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right)} \]

      8. sinh---cosh-rev N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. exp-neg N/A

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

      13. pow-to-exp N/A

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

      14. pow1/2 N/A

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

      15. lift-sqrt.f64 N/A

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

   4. Applied rewrites 59.3%

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

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 59.3

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

      3. lift-pow.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 59.4

          \[\leadsto \frac{c0}{{\left(\frac{A}{V \cdot \ell}\right)}^{\color{blue}{-0.5}}} \]

   6. Applied rewrites 59.4%

      \[\leadsto \color{blue}{\frac{c0}{{\left(\frac{A}{V \cdot \ell}\right)}^{-0.5}}} \]

   7. Taylor expanded in A around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 59.4

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

   9. Applied rewrites 59.4%

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

   11. Applied rewrites 74.5%

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

   if 4.99999999999999967e-276 < (*.f64 V l)

   1. Initial program 81.8%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sqrt.f64 92.1

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

   4. Applied rewrites 92.1%

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

Alternative 8: 85.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-248}:\\ \;\;\;\;\frac{\sqrt{-A} \cdot c0}{\sqrt{\left(-\ell\right) \cdot V}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-276}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A} \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot V}} \cdot \sqrt{A}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) (- INFINITY))
   (* c0 (sqrt (/ (/ A l) V)))
   (if (<= (* V l) -1e-248)
     (/ (* (sqrt (- A)) c0) (sqrt (* (- l) V)))
     (if (<= (* V l) 5e-276)
       (/ c0 (sqrt (* (/ V A) l)))
       (* (/ c0 (sqrt (* l V))) (sqrt A))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = c0 * sqrt(((A / l) / V));
	} else if ((V * l) <= -1e-248) {
		tmp = (sqrt(-A) * c0) / sqrt((-l * V));
	} else if ((V * l) <= 5e-276) {
		tmp = c0 / sqrt(((V / A) * l));
	} else {
		tmp = (c0 / sqrt((l * V))) * sqrt(A);
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = c0 * Math.sqrt(((A / l) / V));
	} else if ((V * l) <= -1e-248) {
		tmp = (Math.sqrt(-A) * c0) / Math.sqrt((-l * V));
	} else if ((V * l) <= 5e-276) {
		tmp = c0 / Math.sqrt(((V / A) * l));
	} else {
		tmp = (c0 / Math.sqrt((l * V))) * Math.sqrt(A);
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = c0 * math.sqrt(((A / l) / V))
	elif (V * l) <= -1e-248:
		tmp = (math.sqrt(-A) * c0) / math.sqrt((-l * V))
	elif (V * l) <= 5e-276:
		tmp = c0 / math.sqrt(((V / A) * l))
	else:
		tmp = (c0 / math.sqrt((l * V))) * math.sqrt(A)
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	elseif (Float64(V * l) <= -1e-248)
		tmp = Float64(Float64(sqrt(Float64(-A)) * c0) / sqrt(Float64(Float64(-l) * V)));
	elseif (Float64(V * l) <= 5e-276)
		tmp = Float64(c0 / sqrt(Float64(Float64(V / A) * l)));
	else
		tmp = Float64(Float64(c0 / sqrt(Float64(l * V))) * sqrt(A));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = c0 * sqrt(((A / l) / V));
	elseif ((V * l) <= -1e-248)
		tmp = (sqrt(-A) * c0) / sqrt((-l * V));
	elseif ((V * l) <= 5e-276)
		tmp = c0 / sqrt(((V / A) * l));
	else
		tmp = (c0 / sqrt((l * V))) * sqrt(A);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-248], N[(N[(N[Sqrt[(-A)], $MachinePrecision] * c0), $MachinePrecision] / N[Sqrt[N[((-l) * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e-276], N[(c0 / N[Sqrt[N[(N[(V / A), $MachinePrecision] * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(c0 / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[A], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 35.8%

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

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 60.4

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

   4. Applied rewrites 60.4%

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

   if -inf.0 < (*.f64 V l) < -9.9999999999999998e-249

   1. Initial program 81.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. pow1/2 N/A

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

      4. pow-to-exp N/A

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

      5. sinh-+-cosh-rev N/A

         \[\leadsto c0 \cdot \color{blue}{\left(\cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) + \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right)\right)} \]

      6. flip-+ N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) \cdot \cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) - \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) \cdot \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right)}{\cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) - \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right)}} \]

      7. sinh-cosh N/A

         \[\leadsto c0 \cdot \frac{\color{blue}{1}}{\cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) - \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right)} \]

      8. sinh---cosh-rev N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. exp-neg N/A

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

      13. pow-to-exp N/A

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

      14. pow1/2 N/A

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

      15. lift-sqrt.f64 N/A

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

   4. Applied rewrites 81.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow-1 N/A

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

      7. remove-double-div N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-2neg N/A

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

      11. sqrt-div N/A

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

      12. associate-*l/ N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lift-*.f64 N/A

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

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

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

      20. lower-*.f64 N/A

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

      21. lower-neg.f64 98.4

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

   6. Applied rewrites 98.4%

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

   if -9.9999999999999998e-249 < (*.f64 V l) < 4.99999999999999967e-276

   1. Initial program 59.4%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. pow1/2 N/A

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

      4. pow-to-exp N/A

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

      5. sinh-+-cosh-rev N/A

         \[\leadsto c0 \cdot \color{blue}{\left(\cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) + \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right)\right)} \]

      6. flip-+ N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) \cdot \cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) - \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) \cdot \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right)}{\cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) - \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right)}} \]

      7. sinh-cosh N/A

         \[\leadsto c0 \cdot \frac{\color{blue}{1}}{\cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) - \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right)} \]

      8. sinh---cosh-rev N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. exp-neg N/A

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

      13. pow-to-exp N/A

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

      14. pow1/2 N/A

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

      15. lift-sqrt.f64 N/A

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

   4. Applied rewrites 59.3%

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

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 59.3

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

      3. lift-pow.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 59.4

          \[\leadsto \frac{c0}{{\left(\frac{A}{V \cdot \ell}\right)}^{\color{blue}{-0.5}}} \]

   6. Applied rewrites 59.4%

      \[\leadsto \color{blue}{\frac{c0}{{\left(\frac{A}{V \cdot \ell}\right)}^{-0.5}}} \]

   7. Taylor expanded in A around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 59.4

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

   9. Applied rewrites 59.4%

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

   11. Applied rewrites 74.5%

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

   if 4.99999999999999967e-276 < (*.f64 V l)

   1. Initial program 81.8%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sqrt.f64 92.1

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

   4. Applied rewrites 92.1%

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

Alternative 9: 90.1% accurate, 0.5× speedup

Math

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

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 -2e-310)
   (* c0 (/ (/ (sqrt (- A)) (sqrt (- V))) (sqrt l)))
   (* (/ c0 (sqrt (* l V))) (sqrt A))))

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if A < -1.999999999999994e-310

   1. Initial program 71.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. sqrt-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 41.3

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

   4. Applied rewrites 41.3%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}} \]
   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. frac-2neg N/A

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

      4. sqrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-neg.f64 50.1

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

   6. Applied rewrites 50.1%

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

   if -1.999999999999994e-310 < A

   1. Initial program 78.0%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sqrt.f64 86.3

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

   4. Applied rewrites 86.3%

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

Alternative 10: 89.0% accurate, 0.5× speedup

Math

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

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 -2e-310)
   (/ (* (sqrt (- A)) c0) (* (sqrt (- V)) (sqrt l)))
   (* (/ c0 (sqrt (* l V))) (sqrt A))))

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if A < -1.999999999999994e-310

   1. Initial program 71.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. pow1/2 N/A

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

      4. pow-to-exp N/A

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

      5. sinh-+-cosh-rev N/A

         \[\leadsto c0 \cdot \color{blue}{\left(\cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) + \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right)\right)} \]

      6. flip-+ N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) \cdot \cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) - \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) \cdot \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right)}{\cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) - \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right)}} \]

      7. sinh-cosh N/A

         \[\leadsto c0 \cdot \frac{\color{blue}{1}}{\cosh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right) - \sinh \left(\log \left(\frac{A}{V \cdot \ell}\right) \cdot \frac{1}{2}\right)} \]

      8. sinh---cosh-rev N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. exp-neg N/A

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

      13. pow-to-exp N/A

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

      14. pow1/2 N/A

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

      15. lift-sqrt.f64 N/A

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

   4. Applied rewrites 71.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow-1 N/A

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

      6. associate-/r/ N/A

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

      7. /-rgt-identity N/A

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

      8. *-commutative N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. sqrt-div N/A

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

      12. lift-sqrt.f64 N/A

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

      13. associate-*l/ N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. sqrt-prod N/A

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

      17. lift-sqrt.f64 N/A

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

      18. lift-sqrt.f64 N/A

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

      19. frac-times N/A

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

   6. Applied rewrites 49.2%

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

   if -1.999999999999994e-310 < A

   1. Initial program 78.0%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sqrt.f64 86.3

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

   4. Applied rewrites 86.3%

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

4. Final simplification 67.6%

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

Alternative 11: 78.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 5 \cdot 10^{-276}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot V}} \cdot \sqrt{A}\\ \end{array} \end{array} \]

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-276)
   (* c0 (sqrt (/ (/ A V) l)))
   (* (/ c0 (sqrt (* l V))) (sqrt A))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= 5e-276) {
		tmp = c0 * sqrt(((A / V) / l));
	} else {
		tmp = (c0 / sqrt((l * V))) * sqrt(A);
	}
	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-276) then
        tmp = c0 * sqrt(((a / v) / l))
    else
        tmp = (c0 / sqrt((l * v))) * sqrt(a)
    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-276) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else {
		tmp = (c0 / Math.sqrt((l * V))) * Math.sqrt(A);
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= 5e-276:
		tmp = c0 * math.sqrt(((A / V) / l))
	else:
		tmp = (c0 / math.sqrt((l * V))) * math.sqrt(A)
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= 5e-276)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	else
		tmp = Float64(Float64(c0 / sqrt(Float64(l * V))) * sqrt(A));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= 5e-276)
		tmp = c0 * sqrt(((A / V) / l));
	else
		tmp = (c0 / sqrt((l * V))) * sqrt(A);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], 5e-276], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(c0 / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[A], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 V l) < 4.99999999999999967e-276

   1. Initial program 70.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 71.0

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

   4. Applied rewrites 71.0%

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

   if 4.99999999999999967e-276 < (*.f64 V l)

   1. Initial program 81.8%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sqrt.f64 92.1

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

   4. Applied rewrites 92.1%

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

Alternative 12: 80.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 2 \cdot 10^{-281}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) 2e-281)
   (* 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) <= 2e-281) {
		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) <= 2d-281) 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) <= 2e-281) {
		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) <= 2e-281:
		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) <= 2e-281)
		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) <= 2e-281)
		tmp = c0 * sqrt(((A / V) / l));
	else
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], 2e-281], 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) < 2e-281

   1. Initial program 70.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 70.8

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

   4. Applied rewrites 70.8%

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

   if 2e-281 < (*.f64 V l)

   1. Initial program 81.9%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 92.1

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

      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 92.1

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

   4. Applied rewrites 92.1%

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

Alternative 13: 73.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.8%

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

Reproduce

herbie shell --seed 2024329 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))