Henrywood and Agarwal, Equation (3)

Percentage Accurate: 74.4% → 91.4%

Time: 6.5s

Alternatives: 14

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\left(-c0\right) \cdot \frac{\sqrt{-A} \cdot -1}{\sqrt{-V} \cdot \sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{{\left(\ell \cdot V\right)}^{-0.5}}{{A}^{-0.5}}\\ \end{array} \end{array} \]

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-311)
   (* (- c0) (/ (* (sqrt (- A)) -1.0) (* (sqrt (- V)) (sqrt l))))
   (* c0 (/ (pow (* l V) -0.5) (pow A -0.5)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -2e-311) {
		tmp = -c0 * ((sqrt(-A) * -1.0) / (sqrt(-V) * sqrt(l)));
	} else {
		tmp = c0 * (pow((l * V), -0.5) / pow(A, -0.5));
	}
	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-311)) then
        tmp = -c0 * ((sqrt(-a) * (-1.0d0)) / (sqrt(-v) * sqrt(l)))
    else
        tmp = c0 * (((l * v) ** (-0.5d0)) / (a ** (-0.5d0)))
    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-311) {
		tmp = -c0 * ((Math.sqrt(-A) * -1.0) / (Math.sqrt(-V) * Math.sqrt(l)));
	} else {
		tmp = c0 * (Math.pow((l * V), -0.5) / Math.pow(A, -0.5));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if A <= -2e-311:
		tmp = -c0 * ((math.sqrt(-A) * -1.0) / (math.sqrt(-V) * math.sqrt(l)))
	else:
		tmp = c0 * (math.pow((l * V), -0.5) / math.pow(A, -0.5))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (A <= -2e-311)
		tmp = Float64(Float64(-c0) * Float64(Float64(sqrt(Float64(-A)) * -1.0) / Float64(sqrt(Float64(-V)) * sqrt(l))));
	else
		tmp = Float64(c0 * Float64((Float64(l * V) ^ -0.5) / (A ^ -0.5)));
	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-311)
		tmp = -c0 * ((sqrt(-A) * -1.0) / (sqrt(-V) * sqrt(l)));
	else
		tmp = c0 * (((l * V) ^ -0.5) / (A ^ -0.5));
	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-311], N[((-c0) * N[(N[(N[Sqrt[(-A)], $MachinePrecision] * -1.0), $MachinePrecision] / N[(N[Sqrt[(-V)], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Power[N[(l * V), $MachinePrecision], -0.5], $MachinePrecision] / N[Power[A, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -1.9999999999999e-311

   1. Initial program 72.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 72.7

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

   4. Applied rewrites 72.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. div-inv N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-2neg N/A

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

      10. sqrt-div N/A

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

      11. frac-2neg N/A

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

      12. metadata-eval N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-sqrt.f64 N/A

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

      20. lower-neg.f64 N/A

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

      21. lower-neg.f64 57.9

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

   6. Applied rewrites 57.9%

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

   if -1.9999999999999e-311 < A

   1. Initial program 74.1%

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

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

      4. div-inv N/A

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

      5. associate-/r* N/A

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

      6. sqrt-div N/A

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

      7. pow1/2 N/A

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

      8. lower-/.f64 N/A

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

      9. inv-pow N/A

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

      10. pow-pow N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. pow1/2 N/A

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

      17. inv-pow N/A

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

      18. pow-pow N/A

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

      19. lower-pow.f64 N/A

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

      20. metadata-eval 85.4

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

   4. Applied rewrites 85.4%

      \[\leadsto c0 \cdot \color{blue}{\frac{{\left(\ell \cdot V\right)}^{-0.5}}{{A}^{-0.5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\left(-c0\right) \cdot \frac{\sqrt{-A} \cdot -1}{\sqrt{-V} \cdot \sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{{\left(\ell \cdot V\right)}^{-0.5}}{{A}^{-0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-252} \lor \neg \left(t\_0 \leq 10^{+288}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (or (<= t_0 4e-252) (not (<= t_0 1e+288)))
     (* c0 (sqrt (/ (/ A V) l)))
     t_0)))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if ((t_0 <= 4e-252) || !(t_0 <= 1e+288)) {
		tmp = c0 * sqrt(((A / V) / l));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    if ((t_0 <= 4d-252) .or. (.not. (t_0 <= 1d+288))) then
        tmp = c0 * sqrt(((a / v) / l))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if ((t_0 <= 4e-252) || !(t_0 <= 1e+288)) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if (t_0 <= 4e-252) or not (t_0 <= 1e+288):
		tmp = c0 * math.sqrt(((A / V) / l))
	else:
		tmp = t_0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if ((t_0 <= 4e-252) || !(t_0 <= 1e+288))
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if ((t_0 <= 4e-252) || ~((t_0 <= 1e+288)))
		tmp = c0 * sqrt(((A / V) / l));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 4e-252], N[Not[LessEqual[t$95$0, 1e+288]], $MachinePrecision]], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 3.99999999999999977e-252 or 1e288 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 64.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 70.4

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

   4. Applied rewrites 70.4%

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

   if 3.99999999999999977e-252 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 1e288

   1. Initial program 98.4%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 4 \cdot 10^{-252} \lor \neg \left(c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 10^{+288}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-252}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+291}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (<= t_0 4e-252)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 4e+291) t_0 (/ c0 (sqrt (* (/ l A) V)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 4e-252) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 4e+291) {
		tmp = t_0;
	} else {
		tmp = c0 / sqrt(((l / A) * V));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    if (t_0 <= 4d-252) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 4d+291) then
        tmp = t_0
    else
        tmp = c0 / sqrt(((l / a) * v))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 4e-252) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 4e+291) {
		tmp = t_0;
	} else {
		tmp = c0 / Math.sqrt(((l / A) * V));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 4e-252:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 4e+291:
		tmp = t_0
	else:
		tmp = c0 / math.sqrt(((l / A) * V))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if (t_0 <= 4e-252)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 4e+291)
		tmp = t_0;
	else
		tmp = Float64(c0 / sqrt(Float64(Float64(l / A) * V)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= 4e-252)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 4e+291)
		tmp = t_0;
	else
		tmp = c0 / sqrt(((l / A) * V));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 4e-252], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+291], t$95$0, N[(c0 / N[Sqrt[N[(N[(l / A), $MachinePrecision] * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 3.99999999999999977e-252

   1. Initial program 65.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 71.2

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

   4. Applied rewrites 71.2%

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

   if 3.99999999999999977e-252 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 3.9999999999999998e291

   1. Initial program 98.4%

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

   if 3.9999999999999998e291 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 55.3%

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

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 55.3

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 55.3

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

   4. Applied rewrites 55.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 65.0

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

   6. Applied rewrites 65.0%

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

Alternative 4: 77.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-252}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+291}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (<= t_0 4e-252)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 4e+291) t_0 (* c0 (sqrt (/ (/ A l) V)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 4e-252) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 4e+291) {
		tmp = t_0;
	} else {
		tmp = c0 * sqrt(((A / l) / V));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    if (t_0 <= 4d-252) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 4d+291) then
        tmp = t_0
    else
        tmp = c0 * sqrt(((a / l) / v))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 4e-252) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 4e+291) {
		tmp = t_0;
	} else {
		tmp = c0 * Math.sqrt(((A / l) / V));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 4e-252:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 4e+291:
		tmp = t_0
	else:
		tmp = c0 * math.sqrt(((A / l) / V))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if (t_0 <= 4e-252)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 4e+291)
		tmp = t_0;
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= 4e-252)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 4e+291)
		tmp = t_0;
	else
		tmp = c0 * sqrt(((A / l) / V));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 4e-252], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+291], t$95$0, N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 3.99999999999999977e-252

   1. Initial program 65.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 71.2

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

   4. Applied rewrites 71.2%

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

   if 3.99999999999999977e-252 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 3.9999999999999998e291

   1. Initial program 98.4%

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

   if 3.9999999999999998e291 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 55.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 64.9

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

   4. Applied rewrites 64.9%

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

Alternative 5: 90.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-314}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\left(-V\right) \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-316} \lor \neg \left(V \cdot \ell \leq 10^{+262}\right):\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{--1}{\sqrt{V \cdot \ell}} \cdot c0\right) \cdot \sqrt{A}\\ \end{array} \end{array} \]

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) -5e-314)
     (* c0 (/ (sqrt (- A)) (sqrt (* (- V) l))))
     (if (or (<= (* V l) 2e-316) (not (<= (* V l) 1e+262)))
       (/ c0 (sqrt (* (/ l A) V)))
       (* (* (/ (- -1.0) (sqrt (* V l))) c0) (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) <= -5e-314) {
		tmp = c0 * (sqrt(-A) / sqrt((-V * l)));
	} else if (((V * l) <= 2e-316) || !((V * l) <= 1e+262)) {
		tmp = c0 / sqrt(((l / A) * V));
	} else {
		tmp = ((-(-1.0) / sqrt((V * l))) * c0) * 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) <= -5e-314) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((-V * l)));
	} else if (((V * l) <= 2e-316) || !((V * l) <= 1e+262)) {
		tmp = c0 / Math.sqrt(((l / A) * V));
	} else {
		tmp = ((-(-1.0) / Math.sqrt((V * l))) * c0) * 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) <= -5e-314:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((-V * l)))
	elif ((V * l) <= 2e-316) or not ((V * l) <= 1e+262):
		tmp = c0 / math.sqrt(((l / A) * V))
	else:
		tmp = ((-(-1.0) / math.sqrt((V * l))) * c0) * 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) <= -5e-314)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(Float64(-V) * l))));
	elseif ((Float64(V * l) <= 2e-316) || !(Float64(V * l) <= 1e+262))
		tmp = Float64(c0 / sqrt(Float64(Float64(l / A) * V)));
	else
		tmp = Float64(Float64(Float64(Float64(-(-1.0)) / sqrt(Float64(V * l))) * c0) * 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) <= -5e-314)
		tmp = c0 * (sqrt(-A) / sqrt((-V * l)));
	elseif (((V * l) <= 2e-316) || ~(((V * l) <= 1e+262)))
		tmp = c0 / sqrt(((l / A) * V));
	else
		tmp = ((-(-1.0) / sqrt((V * l))) * c0) * 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], -5e-314], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[((-V) * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(V * l), $MachinePrecision], 2e-316], N[Not[LessEqual[N[(V * l), $MachinePrecision], 1e+262]], $MachinePrecision]], N[(c0 / N[Sqrt[N[(N[(l / A), $MachinePrecision] * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[((--1.0) / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision] * N[Sqrt[A], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 37.7%

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

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 37.7

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 37.7

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

   4. Applied rewrites 37.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. sqrt-prod N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 58.5

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

   6. Applied rewrites 58.5%

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

   if -inf.0 < (*.f64 V l) < -4.99999999982e-314

   1. Initial program 85.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 74.8

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

   4. Applied rewrites 74.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

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

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 99.2

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

   6. Applied rewrites 99.2%

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

   if -4.99999999982e-314 < (*.f64 V l) < 2.000000017e-316 or 1e262 < (*.f64 V l)

   1. Initial program 44.9%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 44.9

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 44.9

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

   4. Applied rewrites 44.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 70.3

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

   6. Applied rewrites 70.3%

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

   if 2.000000017e-316 < (*.f64 V l) < 1e262

   1. Initial program 83.5%

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

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 95.1

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

   4. Applied rewrites 95.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 95.1

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

   6. Applied rewrites 95.1%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-314}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\left(-V\right) \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-316} \lor \neg \left(V \cdot \ell \leq 10^{+262}\right):\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{--1}{\sqrt{V \cdot \ell}} \cdot c0\right) \cdot \sqrt{A}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}}\\ \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-277}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\left(-V\right) \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+262}:\\ \;\;\;\;\left(\frac{--1}{\sqrt{V \cdot \ell}} \cdot c0\right) \cdot \sqrt{A}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (/ (sqrt (/ (- A) l)) (sqrt (- V))))))
   (if (<= (* V l) (- INFINITY))
     t_0
     (if (<= (* V l) -5e-277)
       (* c0 (/ (sqrt (- A)) (sqrt (* (- V) l))))
       (if (<= (* V l) 0.0)
         t_0
         (if (<= (* V l) 1e+262)
           (* (* (/ (- -1.0) (sqrt (* V l))) c0) (sqrt A))
           (/ c0 (sqrt (* (/ l A) V)))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * (sqrt((-A / l)) / sqrt(-V));
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = t_0;
	} else if ((V * l) <= -5e-277) {
		tmp = c0 * (sqrt(-A) / sqrt((-V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else if ((V * l) <= 1e+262) {
		tmp = ((-(-1.0) / sqrt((V * l))) * c0) * sqrt(A);
	} else {
		tmp = c0 / sqrt(((l / A) * V));
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * (Math.sqrt((-A / l)) / Math.sqrt(-V));
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else if ((V * l) <= -5e-277) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((-V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else if ((V * l) <= 1e+262) {
		tmp = ((-(-1.0) / Math.sqrt((V * l))) * c0) * Math.sqrt(A);
	} else {
		tmp = c0 / Math.sqrt(((l / A) * V));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * (math.sqrt((-A / l)) / math.sqrt(-V))
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = t_0
	elif (V * l) <= -5e-277:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((-V * l)))
	elif (V * l) <= 0.0:
		tmp = t_0
	elif (V * l) <= 1e+262:
		tmp = ((-(-1.0) / math.sqrt((V * l))) * c0) * math.sqrt(A)
	else:
		tmp = c0 / math.sqrt(((l / A) * V))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * Float64(sqrt(Float64(Float64(-A) / l)) / sqrt(Float64(-V))))
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = t_0;
	elseif (Float64(V * l) <= -5e-277)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(Float64(-V) * l))));
	elseif (Float64(V * l) <= 0.0)
		tmp = t_0;
	elseif (Float64(V * l) <= 1e+262)
		tmp = Float64(Float64(Float64(Float64(-(-1.0)) / sqrt(Float64(V * l))) * c0) * sqrt(A));
	else
		tmp = Float64(c0 / sqrt(Float64(Float64(l / A) * V)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * (sqrt((-A / l)) / sqrt(-V));
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = t_0;
	elseif ((V * l) <= -5e-277)
		tmp = c0 * (sqrt(-A) / sqrt((-V * l)));
	elseif ((V * l) <= 0.0)
		tmp = t_0;
	elseif ((V * l) <= 1e+262)
		tmp = ((-(-1.0) / sqrt((V * l))) * c0) * sqrt(A);
	else
		tmp = c0 / sqrt(((l / A) * V));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[(N[Sqrt[N[((-A) / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -5e-277], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[((-V) * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], 1e+262], N[(N[(N[((--1.0) / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision] * N[Sqrt[A], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(N[(l / A), $MachinePrecision] * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -inf.0 or -5e-277 < (*.f64 V l) < -0.0

   1. Initial program 42.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 66.2

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

   4. Applied rewrites 66.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-/l/ N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. sqrt-div N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. distribute-neg-frac N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-neg.f64 51.9

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

   6. Applied rewrites 51.9%

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

   if -inf.0 < (*.f64 V l) < -5e-277

   1. Initial program 85.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 75.0

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

   4. Applied rewrites 75.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

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

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 99.4

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

   6. Applied rewrites 99.4%

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

   if -0.0 < (*.f64 V l) < 1e262

   1. Initial program 82.9%

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

      1. lift-*.f64 N/A

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

      2. *-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. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 94.3

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

   4. Applied rewrites 94.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 94.3

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

   6. Applied rewrites 94.3%

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

   if 1e262 < (*.f64 V l)

   1. Initial program 46.1%

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

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 46.1

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 46.1

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

   4. Applied rewrites 46.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 75.8

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

   6. Applied rewrites 75.8%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-277}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\left(-V\right) \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+262}:\\ \;\;\;\;\left(\frac{--1}{\sqrt{V \cdot \ell}} \cdot c0\right) \cdot \sqrt{A}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-314}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\left(-V\right) \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-316} \lor \neg \left(V \cdot \ell \leq 10^{+262}\right):\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot V}} \cdot \sqrt{A}\\ \end{array} \end{array} \]

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) -5e-314)
     (* c0 (/ (sqrt (- A)) (sqrt (* (- V) l))))
     (if (or (<= (* V l) 2e-316) (not (<= (* V l) 1e+262)))
       (/ c0 (sqrt (* (/ l A) V)))
       (* (/ 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) <= -5e-314) {
		tmp = c0 * (sqrt(-A) / sqrt((-V * l)));
	} else if (((V * l) <= 2e-316) || !((V * l) <= 1e+262)) {
		tmp = c0 / sqrt(((l / A) * V));
	} 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) <= -5e-314) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((-V * l)));
	} else if (((V * l) <= 2e-316) || !((V * l) <= 1e+262)) {
		tmp = c0 / Math.sqrt(((l / A) * V));
	} 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) <= -5e-314:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((-V * l)))
	elif ((V * l) <= 2e-316) or not ((V * l) <= 1e+262):
		tmp = c0 / math.sqrt(((l / A) * V))
	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) <= -5e-314)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(Float64(-V) * l))));
	elseif ((Float64(V * l) <= 2e-316) || !(Float64(V * l) <= 1e+262))
		tmp = Float64(c0 / sqrt(Float64(Float64(l / A) * V)));
	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) <= -5e-314)
		tmp = c0 * (sqrt(-A) / sqrt((-V * l)));
	elseif (((V * l) <= 2e-316) || ~(((V * l) <= 1e+262)))
		tmp = c0 / sqrt(((l / A) * V));
	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], -5e-314], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[((-V) * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(V * l), $MachinePrecision], 2e-316], N[Not[LessEqual[N[(V * l), $MachinePrecision], 1e+262]], $MachinePrecision]], N[(c0 / N[Sqrt[N[(N[(l / A), $MachinePrecision] * V), $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 37.7%

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

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 37.7

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 37.7

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

   4. Applied rewrites 37.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. sqrt-prod N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 58.5

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

   6. Applied rewrites 58.5%

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

   if -inf.0 < (*.f64 V l) < -4.99999999982e-314

   1. Initial program 85.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 74.8

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

   4. Applied rewrites 74.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

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

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 99.2

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

   6. Applied rewrites 99.2%

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

   if -4.99999999982e-314 < (*.f64 V l) < 2.000000017e-316 or 1e262 < (*.f64 V l)

   1. Initial program 44.9%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 44.9

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 44.9

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

   4. Applied rewrites 44.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 70.3

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

   6. Applied rewrites 70.3%

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

   if 2.000000017e-316 < (*.f64 V l) < 1e262

   1. Initial program 83.5%

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

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 95.1

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

   4. Applied rewrites 95.1%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-314}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\left(-V\right) \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-316} \lor \neg \left(V \cdot \ell \leq 10^{+262}\right):\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot V}} \cdot \sqrt{A}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-314}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\left(-V\right) \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-316} \lor \neg \left(V \cdot \ell \leq 10^{+262}\right):\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot V}} \cdot \sqrt{A}\\ \end{array} \end{array} \]

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) -5e-314)
     (* c0 (/ (sqrt (- A)) (sqrt (* (- V) l))))
     (if (or (<= (* V l) 2e-316) (not (<= (* V l) 1e+262)))
       (/ c0 (sqrt (* (/ l A) V)))
       (* (/ 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) <= -5e-314) {
		tmp = c0 * (sqrt(-A) / sqrt((-V * l)));
	} else if (((V * l) <= 2e-316) || !((V * l) <= 1e+262)) {
		tmp = c0 / sqrt(((l / A) * V));
	} 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) <= -5e-314) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((-V * l)));
	} else if (((V * l) <= 2e-316) || !((V * l) <= 1e+262)) {
		tmp = c0 / Math.sqrt(((l / A) * V));
	} 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) <= -5e-314:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((-V * l)))
	elif ((V * l) <= 2e-316) or not ((V * l) <= 1e+262):
		tmp = c0 / math.sqrt(((l / A) * V))
	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) <= -5e-314)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(Float64(-V) * l))));
	elseif ((Float64(V * l) <= 2e-316) || !(Float64(V * l) <= 1e+262))
		tmp = Float64(c0 / sqrt(Float64(Float64(l / A) * V)));
	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) <= -5e-314)
		tmp = c0 * (sqrt(-A) / sqrt((-V * l)));
	elseif (((V * l) <= 2e-316) || ~(((V * l) <= 1e+262)))
		tmp = c0 / sqrt(((l / A) * V));
	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], -5e-314], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[((-V) * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(V * l), $MachinePrecision], 2e-316], N[Not[LessEqual[N[(V * l), $MachinePrecision], 1e+262]], $MachinePrecision]], N[(c0 / N[Sqrt[N[(N[(l / A), $MachinePrecision] * V), $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 37.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 58.4

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

   4. Applied rewrites 58.4%

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

   if -inf.0 < (*.f64 V l) < -4.99999999982e-314

   1. Initial program 85.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 74.8

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

   4. Applied rewrites 74.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

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

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 99.2

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

   6. Applied rewrites 99.2%

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

   if -4.99999999982e-314 < (*.f64 V l) < 2.000000017e-316 or 1e262 < (*.f64 V l)

   1. Initial program 44.9%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 44.9

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 44.9

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

   4. Applied rewrites 44.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 70.3

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

   6. Applied rewrites 70.3%

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

   if 2.000000017e-316 < (*.f64 V l) < 1e262

   1. Initial program 83.5%

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

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 95.1

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

   4. Applied rewrites 95.1%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-314}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\left(-V\right) \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-316} \lor \neg \left(V \cdot \ell \leq 10^{+262}\right):\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot V}} \cdot \sqrt{A}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 88.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-314}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\left(-V\right) \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-316} \lor \neg \left(V \cdot \ell \leq 10^{+262}\right):\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot V}} \cdot \sqrt{A}\\ \end{array} \end{array} \]

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 l))))
   (if (<= (* V l) -5e-314)
     (* c0 (/ (sqrt (- A)) (sqrt (* (- V) l))))
     (if (or (<= (* V l) 2e-316) (not (<= (* V l) 1e+262)))
       (/ c0 (sqrt (* (/ l A) V)))
       (* (/ 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 / l)));
	} else if ((V * l) <= -5e-314) {
		tmp = c0 * (sqrt(-A) / sqrt((-V * l)));
	} else if (((V * l) <= 2e-316) || !((V * l) <= 1e+262)) {
		tmp = c0 / sqrt(((l / A) * V));
	} 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 / l)));
	} else if ((V * l) <= -5e-314) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((-V * l)));
	} else if (((V * l) <= 2e-316) || !((V * l) <= 1e+262)) {
		tmp = c0 / Math.sqrt(((l / A) * V));
	} 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 / l)))
	elif (V * l) <= -5e-314:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((-V * l)))
	elif ((V * l) <= 2e-316) or not ((V * l) <= 1e+262):
		tmp = c0 / math.sqrt(((l / A) * V))
	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(V / Float64(A / l))));
	elseif (Float64(V * l) <= -5e-314)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(Float64(-V) * l))));
	elseif ((Float64(V * l) <= 2e-316) || !(Float64(V * l) <= 1e+262))
		tmp = Float64(c0 / sqrt(Float64(Float64(l / A) * V)));
	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 / l)));
	elseif ((V * l) <= -5e-314)
		tmp = c0 * (sqrt(-A) / sqrt((-V * l)));
	elseif (((V * l) <= 2e-316) || ~(((V * l) <= 1e+262)))
		tmp = c0 / sqrt(((l / A) * V));
	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[(V / N[(A / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -5e-314], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[((-V) * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(V * l), $MachinePrecision], 2e-316], N[Not[LessEqual[N[(V * l), $MachinePrecision], 1e+262]], $MachinePrecision]], N[(c0 / N[Sqrt[N[(N[(l / A), $MachinePrecision] * V), $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 37.7%

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

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 37.7

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 37.7

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

   4. Applied rewrites 37.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. lift-/.f64 N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 71.6

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

   6. Applied rewrites 71.6%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. *-commutative N/A

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

      6. associate-/r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 71.6

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

   8. Applied rewrites 71.6%

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

   if -inf.0 < (*.f64 V l) < -4.99999999982e-314

   1. Initial program 85.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 74.8

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

   4. Applied rewrites 74.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

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

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 99.2

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

   6. Applied rewrites 99.2%

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

   if -4.99999999982e-314 < (*.f64 V l) < 2.000000017e-316 or 1e262 < (*.f64 V l)

   1. Initial program 44.9%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 44.9

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 44.9

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

   4. Applied rewrites 44.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 70.3

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

   6. Applied rewrites 70.3%

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

   if 2.000000017e-316 < (*.f64 V l) < 1e262

   1. Initial program 83.5%

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

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 95.1

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

   4. Applied rewrites 95.1%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-314}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\left(-V\right) \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-316} \lor \neg \left(V \cdot \ell \leq 10^{+262}\right):\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot V}} \cdot \sqrt{A}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 82.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+75}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-114}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell \cdot V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-316} \lor \neg \left(V \cdot \ell \leq 10^{+262}\right):\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot V}} \cdot \sqrt{A}\\ \end{array} \end{array} \]

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) -2e+75)
   (* c0 (sqrt (/ (/ A V) l)))
   (if (<= (* V l) -1e-114)
     (/ c0 (sqrt (/ (* l V) A)))
     (if (or (<= (* V l) 2e-316) (not (<= (* V l) 1e+262)))
       (/ c0 (sqrt (* (/ l A) V)))
       (* (/ 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) <= -2e+75) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if ((V * l) <= -1e-114) {
		tmp = c0 / sqrt(((l * V) / A));
	} else if (((V * l) <= 2e-316) || !((V * l) <= 1e+262)) {
		tmp = c0 / sqrt(((l / A) * V));
	} 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) <= (-2d+75)) then
        tmp = c0 * sqrt(((a / v) / l))
    else if ((v * l) <= (-1d-114)) then
        tmp = c0 / sqrt(((l * v) / a))
    else if (((v * l) <= 2d-316) .or. (.not. ((v * l) <= 1d+262))) then
        tmp = c0 / sqrt(((l / a) * v))
    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) <= -2e+75) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if ((V * l) <= -1e-114) {
		tmp = c0 / Math.sqrt(((l * V) / A));
	} else if (((V * l) <= 2e-316) || !((V * l) <= 1e+262)) {
		tmp = c0 / Math.sqrt(((l / A) * V));
	} 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) <= -2e+75:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif (V * l) <= -1e-114:
		tmp = c0 / math.sqrt(((l * V) / A))
	elif ((V * l) <= 2e-316) or not ((V * l) <= 1e+262):
		tmp = c0 / math.sqrt(((l / A) * V))
	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) <= -2e+75)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (Float64(V * l) <= -1e-114)
		tmp = Float64(c0 / sqrt(Float64(Float64(l * V) / A)));
	elseif ((Float64(V * l) <= 2e-316) || !(Float64(V * l) <= 1e+262))
		tmp = Float64(c0 / sqrt(Float64(Float64(l / A) * V)));
	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) <= -2e+75)
		tmp = c0 * sqrt(((A / V) / l));
	elseif ((V * l) <= -1e-114)
		tmp = c0 / sqrt(((l * V) / A));
	elseif (((V * l) <= 2e-316) || ~(((V * l) <= 1e+262)))
		tmp = c0 / sqrt(((l / A) * V));
	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], -2e+75], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-114], N[(c0 / N[Sqrt[N[(N[(l * V), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(V * l), $MachinePrecision], 2e-316], N[Not[LessEqual[N[(V * l), $MachinePrecision], 1e+262]], $MachinePrecision]], N[(c0 / N[Sqrt[N[(N[(l / A), $MachinePrecision] * V), $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) < -1.99999999999999985e75

   1. Initial program 58.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 70.6

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

   4. Applied rewrites 70.6%

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

   if -1.99999999999999985e75 < (*.f64 V l) < -1.0000000000000001e-114

   1. Initial program 95.1%

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

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 96.7

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 96.7

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

   4. Applied rewrites 96.7%

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

   if -1.0000000000000001e-114 < (*.f64 V l) < 2.000000017e-316 or 1e262 < (*.f64 V l)

   1. Initial program 60.1%

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

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 60.2

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 60.2

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

   4. Applied rewrites 60.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 75.1

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

   6. Applied rewrites 75.1%

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

   if 2.000000017e-316 < (*.f64 V l) < 1e262

   1. Initial program 83.5%

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

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 95.1

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

   4. Applied rewrites 95.1%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+75}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-114}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell \cdot V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-316} \lor \neg \left(V \cdot \ell \leq 10^{+262}\right):\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot V}} \cdot \sqrt{A}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 82.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+75}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-114}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell \cdot V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-316}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+262}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot V}} \cdot \sqrt{A}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -2e+75)
   (* c0 (sqrt (/ (/ A V) l)))
   (if (<= (* V l) -1e-114)
     (/ c0 (sqrt (/ (* l V) A)))
     (if (<= (* V l) 2e-316)
       (/ c0 (sqrt (/ V (/ A l))))
       (if (<= (* V l) 1e+262)
         (* (/ c0 (sqrt (* l V))) (sqrt A))
         (/ c0 (sqrt (* (/ l A) V))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e+75) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if ((V * l) <= -1e-114) {
		tmp = c0 / sqrt(((l * V) / A));
	} else if ((V * l) <= 2e-316) {
		tmp = c0 / sqrt((V / (A / l)));
	} else if ((V * l) <= 1e+262) {
		tmp = (c0 / sqrt((l * V))) * sqrt(A);
	} else {
		tmp = c0 / sqrt(((l / A) * V));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-2d+75)) then
        tmp = c0 * sqrt(((a / v) / l))
    else if ((v * l) <= (-1d-114)) then
        tmp = c0 / sqrt(((l * v) / a))
    else if ((v * l) <= 2d-316) then
        tmp = c0 / sqrt((v / (a / l)))
    else if ((v * l) <= 1d+262) then
        tmp = (c0 / sqrt((l * v))) * sqrt(a)
    else
        tmp = c0 / sqrt(((l / a) * v))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e+75) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if ((V * l) <= -1e-114) {
		tmp = c0 / Math.sqrt(((l * V) / A));
	} else if ((V * l) <= 2e-316) {
		tmp = c0 / Math.sqrt((V / (A / l)));
	} else if ((V * l) <= 1e+262) {
		tmp = (c0 / Math.sqrt((l * V))) * Math.sqrt(A);
	} else {
		tmp = c0 / Math.sqrt(((l / A) * V));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -2e+75:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif (V * l) <= -1e-114:
		tmp = c0 / math.sqrt(((l * V) / A))
	elif (V * l) <= 2e-316:
		tmp = c0 / math.sqrt((V / (A / l)))
	elif (V * l) <= 1e+262:
		tmp = (c0 / math.sqrt((l * V))) * math.sqrt(A)
	else:
		tmp = c0 / math.sqrt(((l / A) * V))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -2e+75)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (Float64(V * l) <= -1e-114)
		tmp = Float64(c0 / sqrt(Float64(Float64(l * V) / A)));
	elseif (Float64(V * l) <= 2e-316)
		tmp = Float64(c0 / sqrt(Float64(V / Float64(A / l))));
	elseif (Float64(V * l) <= 1e+262)
		tmp = Float64(Float64(c0 / sqrt(Float64(l * V))) * sqrt(A));
	else
		tmp = Float64(c0 / sqrt(Float64(Float64(l / A) * V)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -2e+75)
		tmp = c0 * sqrt(((A / V) / l));
	elseif ((V * l) <= -1e-114)
		tmp = c0 / sqrt(((l * V) / A));
	elseif ((V * l) <= 2e-316)
		tmp = c0 / sqrt((V / (A / l)));
	elseif ((V * l) <= 1e+262)
		tmp = (c0 / sqrt((l * V))) * sqrt(A);
	else
		tmp = c0 / sqrt(((l / A) * V));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -2e+75], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-114], N[(c0 / N[Sqrt[N[(N[(l * V), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 2e-316], N[(c0 / N[Sqrt[N[(V / N[(A / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+262], N[(N[(c0 / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[A], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(N[(l / A), $MachinePrecision] * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -1.99999999999999985e75

   1. Initial program 58.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 70.6

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

   4. Applied rewrites 70.6%

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

   if -1.99999999999999985e75 < (*.f64 V l) < -1.0000000000000001e-114

   1. Initial program 95.1%

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

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 96.7

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 96.7

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

   4. Applied rewrites 96.7%

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

   if -1.0000000000000001e-114 < (*.f64 V l) < 2.000000017e-316

   1. Initial program 63.5%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 63.6

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 63.6

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

   4. Applied rewrites 63.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. lift-/.f64 N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 76.5

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

   6. Applied rewrites 76.5%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. *-commutative N/A

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

      6. associate-/r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 76.0

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

   8. Applied rewrites 76.0%

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

   if 2.000000017e-316 < (*.f64 V l) < 1e262

   1. Initial program 83.5%

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

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 95.1

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

   4. Applied rewrites 95.1%

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

   if 1e262 < (*.f64 V l)

   1. Initial program 46.1%

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

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 46.1

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 46.1

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

   4. Applied rewrites 46.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 75.8

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

   6. Applied rewrites 75.8%

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

Alternative 12: 90.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\left(-c0\right) \cdot \frac{\sqrt{-A} \cdot -1}{\sqrt{-V} \cdot \sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{--1}{\sqrt{V \cdot \ell}} \cdot c0\right) \cdot \sqrt{A}\\ \end{array} \end{array} \]

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-311)
   (* (- c0) (/ (* (sqrt (- A)) -1.0) (* (sqrt (- V)) (sqrt l))))
   (* (* (/ (- -1.0) (sqrt (* V l))) c0) (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-311) {
		tmp = -c0 * ((sqrt(-A) * -1.0) / (sqrt(-V) * sqrt(l)));
	} else {
		tmp = ((-(-1.0) / sqrt((V * l))) * c0) * 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-311)) then
        tmp = -c0 * ((sqrt(-a) * (-1.0d0)) / (sqrt(-v) * sqrt(l)))
    else
        tmp = ((-(-1.0d0) / sqrt((v * l))) * c0) * 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-311) {
		tmp = -c0 * ((Math.sqrt(-A) * -1.0) / (Math.sqrt(-V) * Math.sqrt(l)));
	} else {
		tmp = ((-(-1.0) / Math.sqrt((V * l))) * c0) * 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-311:
		tmp = -c0 * ((math.sqrt(-A) * -1.0) / (math.sqrt(-V) * math.sqrt(l)))
	else:
		tmp = ((-(-1.0) / math.sqrt((V * l))) * c0) * 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-311)
		tmp = Float64(Float64(-c0) * Float64(Float64(sqrt(Float64(-A)) * -1.0) / Float64(sqrt(Float64(-V)) * sqrt(l))));
	else
		tmp = Float64(Float64(Float64(Float64(-(-1.0)) / sqrt(Float64(V * l))) * c0) * 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-311)
		tmp = -c0 * ((sqrt(-A) * -1.0) / (sqrt(-V) * sqrt(l)));
	else
		tmp = ((-(-1.0) / sqrt((V * l))) * c0) * 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-311], N[((-c0) * N[(N[(N[Sqrt[(-A)], $MachinePrecision] * -1.0), $MachinePrecision] / N[(N[Sqrt[(-V)], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[((--1.0) / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision] * N[Sqrt[A], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -1.9999999999999e-311

   1. Initial program 72.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 72.7

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

   4. Applied rewrites 72.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. div-inv N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-2neg N/A

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

      10. sqrt-div N/A

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

      11. frac-2neg N/A

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

      12. metadata-eval N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-sqrt.f64 N/A

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

      20. lower-neg.f64 N/A

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

      21. lower-neg.f64 57.9

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

   6. Applied rewrites 57.9%

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

   if -1.9999999999999e-311 < A

   1. Initial program 74.1%

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

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 82.7

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

   4. Applied rewrites 82.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 82.7

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

   6. Applied rewrites 82.7%

      \[\leadsto \color{blue}{\left(\frac{-1}{\sqrt{V \cdot \ell}} \cdot \left(-c0\right)\right)} \cdot \sqrt{A} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\left(-c0\right) \cdot \frac{\sqrt{-A} \cdot -1}{\sqrt{-V} \cdot \sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{--1}{\sqrt{V \cdot \ell}} \cdot c0\right) \cdot \sqrt{A}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 87.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\frac{\frac{c0}{\sqrt{\ell}} \cdot \sqrt{-A}}{\sqrt{-V}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{--1}{\sqrt{V \cdot \ell}} \cdot c0\right) \cdot \sqrt{A}\\ \end{array} \end{array} \]

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-311)
   (/ (* (/ c0 (sqrt l)) (sqrt (- A))) (sqrt (- V)))
   (* (* (/ (- -1.0) (sqrt (* V l))) c0) (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-311) {
		tmp = ((c0 / sqrt(l)) * sqrt(-A)) / sqrt(-V);
	} else {
		tmp = ((-(-1.0) / sqrt((V * l))) * c0) * 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-311)) then
        tmp = ((c0 / sqrt(l)) * sqrt(-a)) / sqrt(-v)
    else
        tmp = ((-(-1.0d0) / sqrt((v * l))) * c0) * 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-311) {
		tmp = ((c0 / Math.sqrt(l)) * Math.sqrt(-A)) / Math.sqrt(-V);
	} else {
		tmp = ((-(-1.0) / Math.sqrt((V * l))) * c0) * 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-311:
		tmp = ((c0 / math.sqrt(l)) * math.sqrt(-A)) / math.sqrt(-V)
	else:
		tmp = ((-(-1.0) / math.sqrt((V * l))) * c0) * 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-311)
		tmp = Float64(Float64(Float64(c0 / sqrt(l)) * sqrt(Float64(-A))) / sqrt(Float64(-V)));
	else
		tmp = Float64(Float64(Float64(Float64(-(-1.0)) / sqrt(Float64(V * l))) * c0) * 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-311)
		tmp = ((c0 / sqrt(l)) * sqrt(-A)) / sqrt(-V);
	else
		tmp = ((-(-1.0) / sqrt((V * l))) * c0) * 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-311], N[(N[(N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[Sqrt[(-A)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision], N[(N[(N[((--1.0) / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision] * N[Sqrt[A], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -1.9999999999999e-311

   1. Initial program 72.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 72.7

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

   4. Applied rewrites 72.7%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. frac-2neg N/A

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

      8. lift-neg.f64 N/A

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

      9. associate-*r/ N/A

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

      10. associate-*l/ N/A

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

      11. un-div-inv N/A

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

      12. metadata-eval N/A

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

      13. frac-2neg N/A

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

      14. lift-/.f64 N/A

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

      15. lift-neg.f64 N/A

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

      16. neg-mul-1 N/A

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

      17. associate-*r* N/A

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

      18. lift-sqrt.f64 N/A

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

      19. lift-/.f64 N/A

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

      20. frac-2neg N/A

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

      21. sqrt-div N/A

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

   6. Applied rewrites 55.2%

      \[\leadsto \color{blue}{\frac{\left(c0 \cdot {\ell}^{-0.5}\right) \cdot \sqrt{-A}}{\sqrt{-V}}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-flip N/A

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

      5. pow1/2 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. div-inv N/A

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

      8. lower-/.f64 55.2

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

   8. Applied rewrites 55.2%

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

   if -1.9999999999999e-311 < A

   1. Initial program 74.1%

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

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 82.7

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

   4. Applied rewrites 82.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 82.7

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

   6. Applied rewrites 82.7%

      \[\leadsto \color{blue}{\left(\frac{-1}{\sqrt{V \cdot \ell}} \cdot \left(-c0\right)\right)} \cdot \sqrt{A} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\frac{\frac{c0}{\sqrt{\ell}} \cdot \sqrt{-A}}{\sqrt{-V}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{--1}{\sqrt{V \cdot \ell}} \cdot c0\right) \cdot \sqrt{A}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 74.4% 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 73.3%

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

Reproduce

herbie shell --seed 2024298 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))