Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.2% → 90.0%

Time: 10.6s

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: 73.2% 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: 90.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := {\left(-A\right)}^{0.25}\\ \mathbf{if}\;A \leq -5 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \left(\frac{t\_0}{\sqrt{-V}} \cdot \frac{t\_0}{\sqrt{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;{A}^{0.25} \cdot \left(\frac{\sqrt{\sqrt{A}}}{\sqrt{\ell \cdot V}} \cdot c0\right)\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (pow (- A) 0.25)))
   (if (<= A -5e-310)
     (* c0 (* (/ t_0 (sqrt (- V))) (/ t_0 (sqrt l))))
     (* (pow A 0.25) (* (/ (sqrt (sqrt A)) (sqrt (* l V))) c0)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = pow(-A, 0.25);
	double tmp;
	if (A <= -5e-310) {
		tmp = c0 * ((t_0 / sqrt(-V)) * (t_0 / sqrt(l)));
	} else {
		tmp = pow(A, 0.25) * ((sqrt(sqrt(A)) / sqrt((l * V))) * c0);
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -a ** 0.25d0
    if (a <= (-5d-310)) then
        tmp = c0 * ((t_0 / sqrt(-v)) * (t_0 / sqrt(l)))
    else
        tmp = (a ** 0.25d0) * ((sqrt(sqrt(a)) / sqrt((l * v))) * c0)
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = Math.pow(-A, 0.25);
	double tmp;
	if (A <= -5e-310) {
		tmp = c0 * ((t_0 / Math.sqrt(-V)) * (t_0 / Math.sqrt(l)));
	} else {
		tmp = Math.pow(A, 0.25) * ((Math.sqrt(Math.sqrt(A)) / Math.sqrt((l * V))) * c0);
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = math.pow(-A, 0.25)
	tmp = 0
	if A <= -5e-310:
		tmp = c0 * ((t_0 / math.sqrt(-V)) * (t_0 / math.sqrt(l)))
	else:
		tmp = math.pow(A, 0.25) * ((math.sqrt(math.sqrt(A)) / math.sqrt((l * V))) * c0)
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(-A) ^ 0.25
	tmp = 0.0
	if (A <= -5e-310)
		tmp = Float64(c0 * Float64(Float64(t_0 / sqrt(Float64(-V))) * Float64(t_0 / sqrt(l))));
	else
		tmp = Float64((A ^ 0.25) * Float64(Float64(sqrt(sqrt(A)) / sqrt(Float64(l * V))) * c0));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = -A ^ 0.25;
	tmp = 0.0;
	if (A <= -5e-310)
		tmp = c0 * ((t_0 / sqrt(-V)) * (t_0 / sqrt(l)));
	else
		tmp = (A ^ 0.25) * ((sqrt(sqrt(A)) / sqrt((l * V))) * c0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[Power[(-A), 0.25], $MachinePrecision]}, If[LessEqual[A, -5e-310], N[(c0 * N[(N[(t$95$0 / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[A, 0.25], $MachinePrecision] * N[(N[(N[Sqrt[N[Sqrt[A], $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if A < -4.999999999999985e-310

   1. Initial program 73.1%

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

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

      4. sqrt-div N/A

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

      5. pow1/2 N/A

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

      6. sqr-pow N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. sqrt-prod N/A

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

      10. pow1/2 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. metadata-eval N/A

          \[\leadsto c0 \cdot \left(\frac{{\left(-A\right)}^{\color{blue}{\frac{1}{4}}}}{{\left(\mathsf{neg}\left(V\right)\right)}^{\frac{1}{2}}} \cdot \frac{{\left(\mathsf{neg}\left(A\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}{\sqrt{\ell}}\right) \]

      17. pow1/2 N/A

          \[\leadsto c0 \cdot \left(\frac{{\left(-A\right)}^{\frac{1}{4}}}{\color{blue}{\sqrt{\mathsf{neg}\left(V\right)}}} \cdot \frac{{\left(\mathsf{neg}\left(A\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}{\sqrt{\ell}}\right) \]

      18. lower-sqrt.f64 N/A

          \[\leadsto c0 \cdot \left(\frac{{\left(-A\right)}^{\frac{1}{4}}}{\color{blue}{\sqrt{\mathsf{neg}\left(V\right)}}} \cdot \frac{{\left(\mathsf{neg}\left(A\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}{\sqrt{\ell}}\right) \]

      19. lower-neg.f64 N/A

          \[\leadsto c0 \cdot \left(\frac{{\left(-A\right)}^{\frac{1}{4}}}{\sqrt{\color{blue}{-V}}} \cdot \frac{{\left(\mathsf{neg}\left(A\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}{\sqrt{\ell}}\right) \]

      20. lower-/.f64 N/A

          \[\leadsto c0 \cdot \left(\frac{{\left(-A\right)}^{\frac{1}{4}}}{\sqrt{-V}} \cdot \color{blue}{\frac{{\left(\mathsf{neg}\left(A\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}{\sqrt{\ell}}}\right) \]

   4. Applied rewrites 46.5%

      \[\leadsto c0 \cdot \color{blue}{\left(\frac{{\left(-A\right)}^{0.25}}{\sqrt{-V}} \cdot \frac{{\left(-A\right)}^{0.25}}{\sqrt{\ell}}\right)} \]

   if -4.999999999999985e-310 < A

   1. Initial program 79.0%

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

      1. lift-*.f64 N/A

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

      2. *-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. pow1/2 N/A

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

      7. sqr-pow N/A

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

      8. associate-/l* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval N/A

          \[\leadsto {A}^{\color{blue}{\frac{1}{4}}} \cdot \left(\frac{{A}^{\left(\frac{\frac{1}{2}}{2}\right)}}{\sqrt{V \cdot \ell}} \cdot c0\right) \]

      13. lower-*.f64 N/A

          \[\leadsto {A}^{\frac{1}{4}} \cdot \color{blue}{\left(\frac{{A}^{\left(\frac{\frac{1}{2}}{2}\right)}}{\sqrt{V \cdot \ell}} \cdot c0\right)} \]

      14. lower-/.f64 N/A

          \[\leadsto {A}^{\frac{1}{4}} \cdot \left(\color{blue}{\frac{{A}^{\left(\frac{\frac{1}{2}}{2}\right)}}{\sqrt{V \cdot \ell}}} \cdot c0\right) \]

      15. lower-pow.f64 N/A

          \[\leadsto {A}^{\frac{1}{4}} \cdot \left(\frac{\color{blue}{{A}^{\left(\frac{\frac{1}{2}}{2}\right)}}}{\sqrt{V \cdot \ell}} \cdot c0\right) \]

      16. metadata-eval N/A

          \[\leadsto {A}^{\frac{1}{4}} \cdot \left(\frac{{A}^{\color{blue}{\frac{1}{4}}}}{\sqrt{V \cdot \ell}} \cdot c0\right) \]

      17. lower-sqrt.f64 89.0

          \[\leadsto {A}^{0.25} \cdot \left(\frac{{A}^{0.25}}{\color{blue}{\sqrt{V \cdot \ell}}} \cdot c0\right) \]

      18. lift-*.f64 N/A

          \[\leadsto {A}^{\frac{1}{4}} \cdot \left(\frac{{A}^{\frac{1}{4}}}{\sqrt{\color{blue}{V \cdot \ell}}} \cdot c0\right) \]

      19. *-commutative N/A

          \[\leadsto {A}^{\frac{1}{4}} \cdot \left(\frac{{A}^{\frac{1}{4}}}{\sqrt{\color{blue}{\ell \cdot V}}} \cdot c0\right) \]

      20. lower-*.f64 89.0

          \[\leadsto {A}^{0.25} \cdot \left(\frac{{A}^{0.25}}{\sqrt{\color{blue}{\ell \cdot V}}} \cdot c0\right) \]

   4. Applied rewrites 89.0%

      \[\leadsto \color{blue}{{A}^{0.25} \cdot \left(\frac{{A}^{0.25}}{\sqrt{\ell \cdot V}} \cdot c0\right)} \]
   5. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto {A}^{\frac{1}{4}} \cdot \left(\frac{\color{blue}{\sqrt{{A}^{\frac{1}{4}}} \cdot \sqrt{{A}^{\frac{1}{4}}}}}{\sqrt{\ell \cdot V}} \cdot c0\right) \]

      2. sqrt-unprod N/A

         \[\leadsto {A}^{\frac{1}{4}} \cdot \left(\frac{\color{blue}{\sqrt{{A}^{\frac{1}{4}} \cdot {A}^{\frac{1}{4}}}}}{\sqrt{\ell \cdot V}} \cdot c0\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto {A}^{\frac{1}{4}} \cdot \left(\frac{\sqrt{\color{blue}{{A}^{\frac{1}{4}}} \cdot {A}^{\frac{1}{4}}}}{\sqrt{\ell \cdot V}} \cdot c0\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto {A}^{\frac{1}{4}} \cdot \left(\frac{\sqrt{{A}^{\frac{1}{4}} \cdot \color{blue}{{A}^{\frac{1}{4}}}}}{\sqrt{\ell \cdot V}} \cdot c0\right) \]

      5. pow-sqr N/A

         \[\leadsto {A}^{\frac{1}{4}} \cdot \left(\frac{\sqrt{\color{blue}{{A}^{\left(2 \cdot \frac{1}{4}\right)}}}}{\sqrt{\ell \cdot V}} \cdot c0\right) \]

      6. metadata-eval N/A

         \[\leadsto {A}^{\frac{1}{4}} \cdot \left(\frac{\sqrt{{A}^{\color{blue}{\frac{1}{2}}}}}{\sqrt{\ell \cdot V}} \cdot c0\right) \]

      7. pow1/2 N/A

         \[\leadsto {A}^{\frac{1}{4}} \cdot \left(\frac{\sqrt{\color{blue}{\sqrt{A}}}}{\sqrt{\ell \cdot V}} \cdot c0\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto {A}^{\frac{1}{4}} \cdot \left(\frac{\sqrt{\color{blue}{\sqrt{A}}}}{\sqrt{\ell \cdot V}} \cdot c0\right) \]

      9. lower-sqrt.f64 89.1

         \[\leadsto {A}^{0.25} \cdot \left(\frac{\color{blue}{\sqrt{\sqrt{A}}}}{\sqrt{\ell \cdot V}} \cdot c0\right) \]

   6. Applied rewrites 89.1%

      \[\leadsto {A}^{0.25} \cdot \left(\frac{\color{blue}{\sqrt{\sqrt{A}}}}{\sqrt{\ell \cdot V}} \cdot c0\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 88.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\frac{\sqrt{-A}}{\sqrt{-V}} \cdot c0}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{A}^{0.25} \cdot \left(\frac{\sqrt{\sqrt{A}}}{\sqrt{\ell \cdot V}} \cdot c0\right)\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= A -5e-310)
   (/ (* (/ (sqrt (- A)) (sqrt (- V))) c0) (sqrt l))
   (* (pow A 0.25) (* (/ (sqrt (sqrt A)) (sqrt (* l V))) c0))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -5e-310) {
		tmp = ((sqrt(-A) / sqrt(-V)) * c0) / sqrt(l);
	} else {
		tmp = pow(A, 0.25) * ((sqrt(sqrt(A)) / sqrt((l * V))) * c0);
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (a <= (-5d-310)) then
        tmp = ((sqrt(-a) / sqrt(-v)) * c0) / sqrt(l)
    else
        tmp = (a ** 0.25d0) * ((sqrt(sqrt(a)) / sqrt((l * v))) * c0)
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -5e-310) {
		tmp = ((Math.sqrt(-A) / Math.sqrt(-V)) * c0) / Math.sqrt(l);
	} else {
		tmp = Math.pow(A, 0.25) * ((Math.sqrt(Math.sqrt(A)) / Math.sqrt((l * V))) * c0);
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if A <= -5e-310:
		tmp = ((math.sqrt(-A) / math.sqrt(-V)) * c0) / math.sqrt(l)
	else:
		tmp = math.pow(A, 0.25) * ((math.sqrt(math.sqrt(A)) / math.sqrt((l * V))) * c0)
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (A <= -5e-310)
		tmp = Float64(Float64(Float64(sqrt(Float64(-A)) / sqrt(Float64(-V))) * c0) / sqrt(l));
	else
		tmp = Float64((A ^ 0.25) * Float64(Float64(sqrt(sqrt(A)) / sqrt(Float64(l * V))) * c0));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (A <= -5e-310)
		tmp = ((sqrt(-A) / sqrt(-V)) * c0) / sqrt(l);
	else
		tmp = (A ^ 0.25) * ((sqrt(sqrt(A)) / sqrt((l * V))) * c0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[A, -5e-310], N[(N[(N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], N[(N[Power[A, 0.25], $MachinePrecision] * N[(N[(N[Sqrt[N[Sqrt[A], $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -4.999999999999985e-310

   1. Initial program 73.1%

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

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

      8. sqrt-prod N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. sqrt-div N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-sqrt.f64 40.2

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

   4. Applied rewrites 40.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. sqrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-neg.f64 45.9

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

   6. Applied rewrites 45.9%

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

   if -4.999999999999985e-310 < A

   1. Initial program 79.0%

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

      1. lift-*.f64 N/A

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

      2. *-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. pow1/2 N/A

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

      7. sqr-pow N/A

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

      8. associate-/l* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval N/A

          \[\leadsto {A}^{\color{blue}{\frac{1}{4}}} \cdot \left(\frac{{A}^{\left(\frac{\frac{1}{2}}{2}\right)}}{\sqrt{V \cdot \ell}} \cdot c0\right) \]

      13. lower-*.f64 N/A

          \[\leadsto {A}^{\frac{1}{4}} \cdot \color{blue}{\left(\frac{{A}^{\left(\frac{\frac{1}{2}}{2}\right)}}{\sqrt{V \cdot \ell}} \cdot c0\right)} \]

      14. lower-/.f64 N/A

          \[\leadsto {A}^{\frac{1}{4}} \cdot \left(\color{blue}{\frac{{A}^{\left(\frac{\frac{1}{2}}{2}\right)}}{\sqrt{V \cdot \ell}}} \cdot c0\right) \]

      15. lower-pow.f64 N/A

          \[\leadsto {A}^{\frac{1}{4}} \cdot \left(\frac{\color{blue}{{A}^{\left(\frac{\frac{1}{2}}{2}\right)}}}{\sqrt{V \cdot \ell}} \cdot c0\right) \]

      16. metadata-eval N/A

          \[\leadsto {A}^{\frac{1}{4}} \cdot \left(\frac{{A}^{\color{blue}{\frac{1}{4}}}}{\sqrt{V \cdot \ell}} \cdot c0\right) \]

      17. lower-sqrt.f64 89.0

          \[\leadsto {A}^{0.25} \cdot \left(\frac{{A}^{0.25}}{\color{blue}{\sqrt{V \cdot \ell}}} \cdot c0\right) \]

      18. lift-*.f64 N/A

          \[\leadsto {A}^{\frac{1}{4}} \cdot \left(\frac{{A}^{\frac{1}{4}}}{\sqrt{\color{blue}{V \cdot \ell}}} \cdot c0\right) \]

      19. *-commutative N/A

          \[\leadsto {A}^{\frac{1}{4}} \cdot \left(\frac{{A}^{\frac{1}{4}}}{\sqrt{\color{blue}{\ell \cdot V}}} \cdot c0\right) \]

      20. lower-*.f64 89.0

          \[\leadsto {A}^{0.25} \cdot \left(\frac{{A}^{0.25}}{\sqrt{\color{blue}{\ell \cdot V}}} \cdot c0\right) \]

   4. Applied rewrites 89.0%

      \[\leadsto \color{blue}{{A}^{0.25} \cdot \left(\frac{{A}^{0.25}}{\sqrt{\ell \cdot V}} \cdot c0\right)} \]
   5. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto {A}^{\frac{1}{4}} \cdot \left(\frac{\color{blue}{\sqrt{{A}^{\frac{1}{4}}} \cdot \sqrt{{A}^{\frac{1}{4}}}}}{\sqrt{\ell \cdot V}} \cdot c0\right) \]

      2. sqrt-unprod N/A

         \[\leadsto {A}^{\frac{1}{4}} \cdot \left(\frac{\color{blue}{\sqrt{{A}^{\frac{1}{4}} \cdot {A}^{\frac{1}{4}}}}}{\sqrt{\ell \cdot V}} \cdot c0\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto {A}^{\frac{1}{4}} \cdot \left(\frac{\sqrt{\color{blue}{{A}^{\frac{1}{4}}} \cdot {A}^{\frac{1}{4}}}}{\sqrt{\ell \cdot V}} \cdot c0\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto {A}^{\frac{1}{4}} \cdot \left(\frac{\sqrt{{A}^{\frac{1}{4}} \cdot \color{blue}{{A}^{\frac{1}{4}}}}}{\sqrt{\ell \cdot V}} \cdot c0\right) \]

      5. pow-sqr N/A

         \[\leadsto {A}^{\frac{1}{4}} \cdot \left(\frac{\sqrt{\color{blue}{{A}^{\left(2 \cdot \frac{1}{4}\right)}}}}{\sqrt{\ell \cdot V}} \cdot c0\right) \]

      6. metadata-eval N/A

         \[\leadsto {A}^{\frac{1}{4}} \cdot \left(\frac{\sqrt{{A}^{\color{blue}{\frac{1}{2}}}}}{\sqrt{\ell \cdot V}} \cdot c0\right) \]

      7. pow1/2 N/A

         \[\leadsto {A}^{\frac{1}{4}} \cdot \left(\frac{\sqrt{\color{blue}{\sqrt{A}}}}{\sqrt{\ell \cdot V}} \cdot c0\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto {A}^{\frac{1}{4}} \cdot \left(\frac{\sqrt{\color{blue}{\sqrt{A}}}}{\sqrt{\ell \cdot V}} \cdot c0\right) \]

      9. lower-sqrt.f64 89.1

         \[\leadsto {A}^{0.25} \cdot \left(\frac{\color{blue}{\sqrt{\sqrt{A}}}}{\sqrt{\ell \cdot V}} \cdot c0\right) \]

   6. Applied rewrites 89.1%

      \[\leadsto {A}^{0.25} \cdot \left(\frac{\color{blue}{\sqrt{\sqrt{A}}}}{\sqrt{\ell \cdot V}} \cdot c0\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 76.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 5 \cdot 10^{+258}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (or (<= t_0 0.0) (not (<= t_0 5e+258)))
     (* c0 (sqrt (/ (/ A V) l)))
     t_0)))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 5e+258)) {
		tmp = c0 * sqrt(((A / V) / l));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    if ((t_0 <= 0.0d0) .or. (.not. (t_0 <= 5d+258))) then
        tmp = c0 * sqrt(((a / v) / l))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 5e+258)) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 5e+258):
		tmp = c0 * math.sqrt(((A / V) / l))
	else:
		tmp = t_0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 5e+258))
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 5e+258)))
		tmp = c0 * sqrt(((A / V) / l));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 5e+258]], $MachinePrecision]], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 0.0 or 5e258 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 69.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 70.1

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

   4. Applied rewrites 70.1%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 5e258

   1. Initial program 98.1%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 0 \lor \neg \left(c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 5 \cdot 10^{+258}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-291}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+258}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A} \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (<= t_0 5e-291)
     (* c0 (sqrt (/ (/ A l) V)))
     (if (<= t_0 5e+258) t_0 (/ c0 (sqrt (* (/ V A) l)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 5e-291) {
		tmp = c0 * sqrt(((A / l) / V));
	} else if (t_0 <= 5e+258) {
		tmp = t_0;
	} else {
		tmp = c0 / sqrt(((V / A) * l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    if (t_0 <= 5d-291) then
        tmp = c0 * sqrt(((a / l) / v))
    else if (t_0 <= 5d+258) then
        tmp = t_0
    else
        tmp = c0 / sqrt(((v / a) * l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 5e-291) {
		tmp = c0 * Math.sqrt(((A / l) / V));
	} else if (t_0 <= 5e+258) {
		tmp = t_0;
	} else {
		tmp = c0 / Math.sqrt(((V / A) * l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 5e-291:
		tmp = c0 * math.sqrt(((A / l) / V))
	elif t_0 <= 5e+258:
		tmp = t_0
	else:
		tmp = c0 / math.sqrt(((V / A) * l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if (t_0 <= 5e-291)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	elseif (t_0 <= 5e+258)
		tmp = t_0;
	else
		tmp = Float64(c0 / sqrt(Float64(Float64(V / A) * l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= 5e-291)
		tmp = c0 * sqrt(((A / l) / V));
	elseif (t_0 <= 5e+258)
		tmp = t_0;
	else
		tmp = c0 / sqrt(((V / A) * l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-291], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+258], t$95$0, N[(c0 / N[Sqrt[N[(N[(V / A), $MachinePrecision] * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 5.0000000000000003e-291

   1. Initial program 71.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 70.2

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

   4. Applied rewrites 70.2%

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

   if 5.0000000000000003e-291 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 5e258

   1. Initial program 98.1%

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

   if 5e258 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 65.0%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. pow-to-exp N/A

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

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

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

      6. flip-+ N/A

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

      7. sinh-cosh N/A

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

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

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. exp-neg N/A

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

      13. pow-to-exp N/A

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

      14. pow1/2 N/A

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

      15. lift-sqrt.f64 N/A

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

   4. Applied rewrites 64.9%

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

   5. Taylor expanded in A around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 66.7

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

   7. Applied rewrites 66.7%

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

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 66.7

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

   9. Applied rewrites 73.8%

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

Alternative 5: 76.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-291}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+258}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (<= t_0 5e-291)
     (* c0 (sqrt (/ (/ A l) V)))
     (if (<= t_0 5e+258) t_0 (* c0 (sqrt (/ (/ A V) l)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 5e-291) {
		tmp = c0 * sqrt(((A / l) / V));
	} else if (t_0 <= 5e+258) {
		tmp = t_0;
	} else {
		tmp = c0 * sqrt(((A / V) / l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    if (t_0 <= 5d-291) then
        tmp = c0 * sqrt(((a / l) / v))
    else if (t_0 <= 5d+258) then
        tmp = t_0
    else
        tmp = c0 * sqrt(((a / v) / l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 5e-291) {
		tmp = c0 * Math.sqrt(((A / l) / V));
	} else if (t_0 <= 5e+258) {
		tmp = t_0;
	} else {
		tmp = c0 * Math.sqrt(((A / V) / l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 5e-291:
		tmp = c0 * math.sqrt(((A / l) / V))
	elif t_0 <= 5e+258:
		tmp = t_0
	else:
		tmp = c0 * math.sqrt(((A / V) / l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if (t_0 <= 5e-291)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	elseif (t_0 <= 5e+258)
		tmp = t_0;
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= 5e-291)
		tmp = c0 * sqrt(((A / l) / V));
	elseif (t_0 <= 5e+258)
		tmp = t_0;
	else
		tmp = c0 * sqrt(((A / V) / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-291], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+258], t$95$0, N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 5.0000000000000003e-291

   1. Initial program 71.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 70.2

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

   4. Applied rewrites 70.2%

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

   if 5.0000000000000003e-291 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 5e258

   1. Initial program 98.1%

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

   if 5e258 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 65.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 72.1

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

   4. Applied rewrites 72.1%

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

Alternative 6: 88.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if A < -4.999999999999985e-310

   1. Initial program 73.1%

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

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

      8. sqrt-prod N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. sqrt-div N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-sqrt.f64 40.2

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

   4. Applied rewrites 40.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. sqrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-neg.f64 45.9

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

   6. Applied rewrites 45.9%

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

   if -4.999999999999985e-310 < A

   1. Initial program 79.0%

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

      1. lift-*.f64 N/A

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

      2. *-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. pow1/2 N/A

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

      7. sqr-pow N/A

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

      8. associate-/l* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval N/A

          \[\leadsto {A}^{\color{blue}{\frac{1}{4}}} \cdot \left(\frac{{A}^{\left(\frac{\frac{1}{2}}{2}\right)}}{\sqrt{V \cdot \ell}} \cdot c0\right) \]

      13. lower-*.f64 N/A

          \[\leadsto {A}^{\frac{1}{4}} \cdot \color{blue}{\left(\frac{{A}^{\left(\frac{\frac{1}{2}}{2}\right)}}{\sqrt{V \cdot \ell}} \cdot c0\right)} \]

      14. lower-/.f64 N/A

          \[\leadsto {A}^{\frac{1}{4}} \cdot \left(\color{blue}{\frac{{A}^{\left(\frac{\frac{1}{2}}{2}\right)}}{\sqrt{V \cdot \ell}}} \cdot c0\right) \]

      15. lower-pow.f64 N/A

          \[\leadsto {A}^{\frac{1}{4}} \cdot \left(\frac{\color{blue}{{A}^{\left(\frac{\frac{1}{2}}{2}\right)}}}{\sqrt{V \cdot \ell}} \cdot c0\right) \]

      16. metadata-eval N/A

          \[\leadsto {A}^{\frac{1}{4}} \cdot \left(\frac{{A}^{\color{blue}{\frac{1}{4}}}}{\sqrt{V \cdot \ell}} \cdot c0\right) \]

      17. lower-sqrt.f64 89.0

          \[\leadsto {A}^{0.25} \cdot \left(\frac{{A}^{0.25}}{\color{blue}{\sqrt{V \cdot \ell}}} \cdot c0\right) \]

      18. lift-*.f64 N/A

          \[\leadsto {A}^{\frac{1}{4}} \cdot \left(\frac{{A}^{\frac{1}{4}}}{\sqrt{\color{blue}{V \cdot \ell}}} \cdot c0\right) \]

      19. *-commutative N/A

          \[\leadsto {A}^{\frac{1}{4}} \cdot \left(\frac{{A}^{\frac{1}{4}}}{\sqrt{\color{blue}{\ell \cdot V}}} \cdot c0\right) \]

      20. lower-*.f64 89.0

          \[\leadsto {A}^{0.25} \cdot \left(\frac{{A}^{0.25}}{\sqrt{\color{blue}{\ell \cdot V}}} \cdot c0\right) \]

   4. Applied rewrites 89.0%

      \[\leadsto \color{blue}{{A}^{0.25} \cdot \left(\frac{{A}^{0.25}}{\sqrt{\ell \cdot V}} \cdot c0\right)} \]
   5. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto {A}^{\frac{1}{4}} \cdot \left(\frac{\color{blue}{\sqrt{{A}^{\frac{1}{4}}} \cdot \sqrt{{A}^{\frac{1}{4}}}}}{\sqrt{\ell \cdot V}} \cdot c0\right) \]

      2. sqrt-unprod N/A

         \[\leadsto {A}^{\frac{1}{4}} \cdot \left(\frac{\color{blue}{\sqrt{{A}^{\frac{1}{4}} \cdot {A}^{\frac{1}{4}}}}}{\sqrt{\ell \cdot V}} \cdot c0\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto {A}^{\frac{1}{4}} \cdot \left(\frac{\sqrt{\color{blue}{{A}^{\frac{1}{4}}} \cdot {A}^{\frac{1}{4}}}}{\sqrt{\ell \cdot V}} \cdot c0\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto {A}^{\frac{1}{4}} \cdot \left(\frac{\sqrt{{A}^{\frac{1}{4}} \cdot \color{blue}{{A}^{\frac{1}{4}}}}}{\sqrt{\ell \cdot V}} \cdot c0\right) \]

      5. pow-sqr N/A

         \[\leadsto {A}^{\frac{1}{4}} \cdot \left(\frac{\sqrt{\color{blue}{{A}^{\left(2 \cdot \frac{1}{4}\right)}}}}{\sqrt{\ell \cdot V}} \cdot c0\right) \]

      6. metadata-eval N/A

         \[\leadsto {A}^{\frac{1}{4}} \cdot \left(\frac{\sqrt{{A}^{\color{blue}{\frac{1}{2}}}}}{\sqrt{\ell \cdot V}} \cdot c0\right) \]

      7. pow1/2 N/A

         \[\leadsto {A}^{\frac{1}{4}} \cdot \left(\frac{\sqrt{\color{blue}{\sqrt{A}}}}{\sqrt{\ell \cdot V}} \cdot c0\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto {A}^{\frac{1}{4}} \cdot \left(\frac{\sqrt{\color{blue}{\sqrt{A}}}}{\sqrt{\ell \cdot V}} \cdot c0\right) \]

      9. lower-sqrt.f64 89.1

         \[\leadsto {A}^{0.25} \cdot \left(\frac{\color{blue}{\sqrt{\sqrt{A}}}}{\sqrt{\ell \cdot V}} \cdot c0\right) \]

   6. Applied rewrites 89.1%

      \[\leadsto {A}^{0.25} \cdot \left(\frac{\color{blue}{\sqrt{\sqrt{A}}}}{\sqrt{\ell \cdot V}} \cdot c0\right) \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \color{blue}{{A}^{\frac{1}{4}}} \cdot \left(\frac{\sqrt{\sqrt{A}}}{\sqrt{\ell \cdot V}} \cdot c0\right) \]

      2. metadata-eval N/A

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

      3. sqrt-pow1 N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 89.0

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

   8. Applied rewrites 89.0%

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

Alternative 7: 84.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+295}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-171}:\\ \;\;\;\;\frac{\sqrt{-A} \cdot c0}{\sqrt{\left(-V\right) \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{-314}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A} \cdot c0}{\sqrt{\ell \cdot V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -1e+295)
   (* c0 (sqrt (/ (/ A l) V)))
   (if (<= (* V l) -1e-171)
     (/ (* (sqrt (- A)) c0) (sqrt (* (- V) l)))
     (if (<= (* V l) 4e-314)
       (* c0 (sqrt (/ (/ A V) l)))
       (/ (* (sqrt A) c0) (sqrt (* l V)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+295) {
		tmp = c0 * sqrt(((A / l) / V));
	} else if ((V * l) <= -1e-171) {
		tmp = (sqrt(-A) * c0) / sqrt((-V * l));
	} else if ((V * l) <= 4e-314) {
		tmp = c0 * sqrt(((A / V) / l));
	} else {
		tmp = (sqrt(A) * c0) / sqrt((l * V));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-1d+295)) then
        tmp = c0 * sqrt(((a / l) / v))
    else if ((v * l) <= (-1d-171)) then
        tmp = (sqrt(-a) * c0) / sqrt((-v * l))
    else if ((v * l) <= 4d-314) then
        tmp = c0 * sqrt(((a / v) / l))
    else
        tmp = (sqrt(a) * c0) / sqrt((l * v))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+295) {
		tmp = c0 * Math.sqrt(((A / l) / V));
	} else if ((V * l) <= -1e-171) {
		tmp = (Math.sqrt(-A) * c0) / Math.sqrt((-V * l));
	} else if ((V * l) <= 4e-314) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else {
		tmp = (Math.sqrt(A) * c0) / Math.sqrt((l * V));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -1e+295:
		tmp = c0 * math.sqrt(((A / l) / V))
	elif (V * l) <= -1e-171:
		tmp = (math.sqrt(-A) * c0) / math.sqrt((-V * l))
	elif (V * l) <= 4e-314:
		tmp = c0 * math.sqrt(((A / V) / l))
	else:
		tmp = (math.sqrt(A) * c0) / math.sqrt((l * V))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -1e+295)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	elseif (Float64(V * l) <= -1e-171)
		tmp = Float64(Float64(sqrt(Float64(-A)) * c0) / sqrt(Float64(Float64(-V) * l)));
	elseif (Float64(V * l) <= 4e-314)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	else
		tmp = Float64(Float64(sqrt(A) * c0) / sqrt(Float64(l * V)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -1e+295)
		tmp = c0 * sqrt(((A / l) / V));
	elseif ((V * l) <= -1e-171)
		tmp = (sqrt(-A) * c0) / sqrt((-V * l));
	elseif ((V * l) <= 4e-314)
		tmp = c0 * sqrt(((A / V) / l));
	else
		tmp = (sqrt(A) * c0) / sqrt((l * V));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -1e+295], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-171], N[(N[(N[Sqrt[(-A)], $MachinePrecision] * c0), $MachinePrecision] / N[Sqrt[N[((-V) * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 4e-314], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[A], $MachinePrecision] * c0), $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -9.9999999999999998e294

   1. Initial program 32.9%

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

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

   if -9.9999999999999998e294 < (*.f64 V l) < -9.9999999999999998e-172

   1. Initial program 82.4%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. pow-to-exp N/A

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

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

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

      6. flip-+ N/A

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

      7. sinh-cosh N/A

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

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

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. exp-neg N/A

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

      13. pow-to-exp N/A

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

      14. pow1/2 N/A

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

      15. lift-sqrt.f64 N/A

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

   4. Applied rewrites 82.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow-1 N/A

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

      7. remove-double-div N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-2neg N/A

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

      11. sqrt-div N/A

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

      12. associate-*l/ N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

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

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

      21. lower-*.f64 N/A

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

      22. lower-neg.f64 94.5

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

   6. Applied rewrites 94.5%

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

   if -9.9999999999999998e-172 < (*.f64 V l) < 3.9999999999e-314

   1. Initial program 70.4%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 78.7

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

   4. Applied rewrites 78.7%

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

   if 3.9999999999e-314 < (*.f64 V l)

   1. Initial program 82.0%

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

      1. lift-*.f64 N/A

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

      2. *-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. frac-2neg N/A

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

      8. frac-2neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-/.f64 N/A

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

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

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

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

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

      13. remove-double-neg N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-sqrt.f64 92.4

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 92.4

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

   4. Applied rewrites 92.4%

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

Alternative 8: 76.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\sqrt{\frac{A}{V \cdot \ell}} \leq 4 \cdot 10^{-155}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \end{array} \end{array} \]

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 (<= (sqrt (/ A (* V l))) 4e-155)
   (* c0 (sqrt (/ (/ A l) V)))
   (/ c0 (sqrt (/ (* V l) A)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (sqrt((A / (V * l))) <= 4e-155) {
		tmp = c0 * sqrt(((A / l) / V));
	} else {
		tmp = c0 / sqrt(((V * l) / 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 (sqrt((a / (v * l))) <= 4d-155) then
        tmp = c0 * sqrt(((a / l) / v))
    else
        tmp = c0 / sqrt(((v * l) / 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 (Math.sqrt((A / (V * l))) <= 4e-155) {
		tmp = c0 * Math.sqrt(((A / l) / V));
	} else {
		tmp = c0 / Math.sqrt(((V * l) / A));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if math.sqrt((A / (V * l))) <= 4e-155:
		tmp = c0 * math.sqrt(((A / l) / V))
	else:
		tmp = c0 / math.sqrt(((V * l) / A))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (sqrt(Float64(A / Float64(V * l))) <= 4e-155)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	else
		tmp = Float64(c0 / sqrt(Float64(Float64(V * l) / 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 (sqrt((A / (V * l))) <= 4e-155)
		tmp = c0 * sqrt(((A / l) / V));
	else
		tmp = c0 / sqrt(((V * l) / 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[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 4e-155], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (/.f64 A (*.f64 V l))) < 4.00000000000000006e-155

   1. Initial program 35.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 43.2

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

   4. Applied rewrites 43.2%

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

   if 4.00000000000000006e-155 < (sqrt.f64 (/.f64 A (*.f64 V l)))

   1. Initial program 87.5%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. pow-to-exp N/A

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

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

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

      6. flip-+ N/A

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

      7. sinh-cosh N/A

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

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

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. exp-neg N/A

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

      13. pow-to-exp N/A

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

      14. pow1/2 N/A

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

      15. lift-sqrt.f64 N/A

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

   4. Applied rewrites 87.4%

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

   5. Taylor expanded in A around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 88.0

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

   7. Applied rewrites 88.0%

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

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 88.0

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

   9. Applied rewrites 88.0%

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

Alternative 9: 88.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if A < -4.999999999999985e-310

   1. Initial program 73.1%

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

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

      8. sqrt-prod N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. sqrt-div N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-sqrt.f64 40.2

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

   4. Applied rewrites 40.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. sqrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-neg.f64 45.9

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

   6. Applied rewrites 45.9%

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

   if -4.999999999999985e-310 < A

   1. Initial program 79.0%

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

      1. lift-*.f64 N/A

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

      2. *-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. frac-2neg N/A

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

      8. frac-2neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-/.f64 N/A

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

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

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

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

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

      13. remove-double-neg N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-sqrt.f64 88.4

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 88.4

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

   4. Applied rewrites 88.4%

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

Alternative 10: 88.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if A < -4.999999999999985e-310

   1. Initial program 73.1%

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

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

      4. pow-to-exp N/A

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

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

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

      6. flip-+ N/A

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

      7. sinh-cosh N/A

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

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

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. exp-neg N/A

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

      13. pow-to-exp N/A

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

      14. pow1/2 N/A

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

      15. lift-sqrt.f64 N/A

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

   4. Applied rewrites 73.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow-1 N/A

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

      6. associate-/r/ N/A

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

      7. /-rgt-identity N/A

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

      8. *-commutative N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. sqrt-div N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. associate-*l/ N/A

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

      15. lift-sqrt.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. sqrt-prod N/A

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

      19. lift-sqrt.f64 N/A

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

      20. frac-times N/A

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

      21. frac-2neg N/A

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

   6. Applied rewrites 44.7%

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

   if -4.999999999999985e-310 < A

   1. Initial program 79.0%

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

      1. lift-*.f64 N/A

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

      2. *-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. frac-2neg N/A

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

      8. frac-2neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-/.f64 N/A

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

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

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

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

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

      13. remove-double-neg N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-sqrt.f64 88.4

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 88.4

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

   4. Applied rewrites 88.4%

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

4. Final simplification 66.6%

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

Alternative 11: 82.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if A < -4.999999999999985e-310

   1. Initial program 73.1%

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

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

      4. associate-/r* N/A

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

      5. sqrt-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 41.6

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

   4. Applied rewrites 41.6%

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

   if -4.999999999999985e-310 < A

   1. Initial program 79.0%

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

      1. lift-*.f64 N/A

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

      2. *-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. frac-2neg N/A

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

      8. frac-2neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-/.f64 N/A

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

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

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

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

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

      13. remove-double-neg N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-sqrt.f64 88.4

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 88.4

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

   4. Applied rewrites 88.4%

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

Alternative 12: 78.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 4 \cdot 10^{-314}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A} \cdot c0}{\sqrt{\ell \cdot V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) 4e-314)
   (* c0 (sqrt (/ (/ A l) V)))
   (/ (* (sqrt A) c0) (sqrt (* l V)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= 4e-314) {
		tmp = c0 * sqrt(((A / l) / V));
	} else {
		tmp = (sqrt(A) * c0) / sqrt((l * V));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= 4d-314) then
        tmp = c0 * sqrt(((a / l) / v))
    else
        tmp = (sqrt(a) * c0) / sqrt((l * v))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= 4e-314) {
		tmp = c0 * Math.sqrt(((A / l) / V));
	} else {
		tmp = (Math.sqrt(A) * c0) / Math.sqrt((l * V));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= 4e-314:
		tmp = c0 * math.sqrt(((A / l) / V))
	else:
		tmp = (math.sqrt(A) * c0) / math.sqrt((l * V))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= 4e-314)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	else
		tmp = Float64(Float64(sqrt(A) * c0) / sqrt(Float64(l * V)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= 4e-314)
		tmp = c0 * sqrt(((A / l) / V));
	else
		tmp = (sqrt(A) * c0) / sqrt((l * V));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], 4e-314], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[A], $MachinePrecision] * c0), $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 V l) < 3.9999999999e-314

   1. Initial program 71.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 74.2

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

   4. Applied rewrites 74.2%

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

   if 3.9999999999e-314 < (*.f64 V l)

   1. Initial program 82.0%

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

      1. lift-*.f64 N/A

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

      2. *-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. frac-2neg N/A

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

      8. frac-2neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-/.f64 N/A

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

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

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

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

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

      13. remove-double-neg N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-sqrt.f64 92.4

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 92.4

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

   4. Applied rewrites 92.4%

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

Alternative 13: 78.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 4 \cdot 10^{-314}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot V}} \cdot \sqrt{A}\\ \end{array} \end{array} \]

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) 4e-314)
   (* c0 (sqrt (/ (/ A l) 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) <= 4e-314) {
		tmp = c0 * sqrt(((A / l) / 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) <= 4d-314) then
        tmp = c0 * sqrt(((a / l) / 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) <= 4e-314) {
		tmp = c0 * Math.sqrt(((A / l) / 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) <= 4e-314:
		tmp = c0 * math.sqrt(((A / l) / 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) <= 4e-314)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / 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) <= 4e-314)
		tmp = c0 * sqrt(((A / l) / 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], 4e-314], N[(c0 * N[Sqrt[N[(N[(A / l), $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 2 regimes

2. if (*.f64 V l) < 3.9999999999e-314

   1. Initial program 71.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 74.2

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

   4. Applied rewrites 74.2%

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

   if 3.9999999999e-314 < (*.f64 V l)

   1. Initial program 82.0%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sqrt.f64 92.4

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

   4. Applied rewrites 92.4%

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

Alternative 14: 73.2% 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 76.1%

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

Reproduce

herbie shell --seed 2024332 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))