Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.7% → 94.4%

Time: 11.3s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{\frac{\sqrt{-A}}{\sqrt{-V}} \cdot c0}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\frac{\sqrt{A}}{\sqrt{V \cdot \sqrt[3]{\ell}}} \cdot \sqrt{{\left(\sqrt[3]{\ell}\right)}^{-2}}\right)\\ \end{array} \end{array} \]

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 -1e-310)
   (/ (* (/ (sqrt (- A)) (sqrt (- V))) c0) (sqrt l))
   (* c0 (* (/ (sqrt A) (sqrt (* V (cbrt l)))) (sqrt (pow (cbrt l) -2.0))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -1e-310) {
		tmp = ((sqrt(-A) / sqrt(-V)) * c0) / sqrt(l);
	} else {
		tmp = c0 * ((sqrt(A) / sqrt((V * cbrt(l)))) * sqrt(pow(cbrt(l), -2.0)));
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -1e-310) {
		tmp = ((Math.sqrt(-A) / Math.sqrt(-V)) * c0) / Math.sqrt(l);
	} else {
		tmp = c0 * ((Math.sqrt(A) / Math.sqrt((V * Math.cbrt(l)))) * Math.sqrt(Math.pow(Math.cbrt(l), -2.0)));
	}
	return tmp;
}

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (A <= -1e-310)
		tmp = Float64(Float64(Float64(sqrt(Float64(-A)) / sqrt(Float64(-V))) * c0) / sqrt(l));
	else
		tmp = Float64(c0 * Float64(Float64(sqrt(A) / sqrt(Float64(V * cbrt(l)))) * sqrt((cbrt(l) ^ -2.0))));
	end
	return 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, -1e-310], N[(N[(N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -9.999999999999969e-311

   1. Initial program 78.3%

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

      1. *-commutative 78.3%

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

      2. associate-/r* 77.5%

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

      3. sqrt-div 44.4%

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

      4. associate-*l/ 44.5%

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

   4. Applied egg-rr 44.5%

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

      1. frac-2neg 44.5%

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

      2. sqrt-div 54.5%

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

   6. Applied egg-rr 54.5%

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

   if -9.999999999999969e-311 < A

   1. Initial program 71.2%

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

      1. associate-/r* 69.1%

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

      2. div-inv 69.1%

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

      3. add-cube-cbrt 68.7%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \frac{1}{V}}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}} \]

      4. times-frac 73.8%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]

      5. pow2 73.8%

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}} \]

   4. Applied egg-rr 73.8%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]
   5. Step-by-step derivation

      1. associate-*l/ 72.2%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

      2. associate-/l/ 71.5%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \color{blue}{\frac{1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      3. associate-*r/ 71.6%

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A \cdot 1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      4. *-rgt-identity 71.6%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{\color{blue}{A}}{\sqrt[3]{\ell} \cdot V}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      5. *-commutative 71.6%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{A}{\color{blue}{V \cdot \sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

   6. Simplified 71.6%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V \cdot \sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]
   7. Step-by-step derivation

      1. div-inv 71.6%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \sqrt[3]{\ell}} \cdot \frac{1}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

      2. sqrt-prod 76.8%

         \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{\frac{A}{V \cdot \sqrt[3]{\ell}}} \cdot \sqrt{\frac{1}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\right)} \]

      3. associate-/r* 74.5%

         \[\leadsto c0 \cdot \left(\sqrt{\color{blue}{\frac{\frac{A}{V}}{\sqrt[3]{\ell}}}} \cdot \sqrt{\frac{1}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\right) \]

      4. pow-flip 74.5%

         \[\leadsto c0 \cdot \left(\sqrt{\frac{\frac{A}{V}}{\sqrt[3]{\ell}}} \cdot \sqrt{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{\left(-2\right)}}}\right) \]

      5. metadata-eval 74.5%

         \[\leadsto c0 \cdot \left(\sqrt{\frac{\frac{A}{V}}{\sqrt[3]{\ell}}} \cdot \sqrt{{\left(\sqrt[3]{\ell}\right)}^{\color{blue}{-2}}}\right) \]

   8. Applied egg-rr 74.5%

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{\frac{\frac{A}{V}}{\sqrt[3]{\ell}}} \cdot \sqrt{{\left(\sqrt[3]{\ell}\right)}^{-2}}\right)} \]
   9. Step-by-step derivation

      1. associate-/l/ 76.9%

         \[\leadsto c0 \cdot \left(\sqrt{\color{blue}{\frac{A}{\sqrt[3]{\ell} \cdot V}}} \cdot \sqrt{{\left(\sqrt[3]{\ell}\right)}^{-2}}\right) \]

      2. sqrt-div 93.2%

         \[\leadsto c0 \cdot \left(\color{blue}{\frac{\sqrt{A}}{\sqrt{\sqrt[3]{\ell} \cdot V}}} \cdot \sqrt{{\left(\sqrt[3]{\ell}\right)}^{-2}}\right) \]

   10. Applied egg-rr 93.2%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{\frac{\sqrt{-A}}{\sqrt{-V}} \cdot c0}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\frac{\sqrt{A}}{\sqrt{V \cdot \sqrt[3]{\ell}}} \cdot \sqrt{{\left(\sqrt[3]{\ell}\right)}^{-2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 5 \cdot 10^{+303}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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+303)))
     (* 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+303)) {
		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+303))) 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+303)) {
		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+303):
		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+303))
		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+303)))
		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+303]], $MachinePrecision]], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 0.0 or 4.9999999999999997e303 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 65.8%

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

      1. associate-/r* 68.3%

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

   3. Simplified 68.3%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 4.9999999999999997e303

   1. Initial program 99.1%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 0 \lor \neg \left(c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 5 \cdot 10^{+303}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 10^{+242}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (<= t_0 0.0)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 1e+242) t_0 (* c0 (sqrt (/ (/ A l) V)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 1e+242) {
		tmp = t_0;
	} else {
		tmp = c0 * sqrt(((A / l) / V));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    if (t_0 <= 0.0d0) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 1d+242) then
        tmp = t_0
    else
        tmp = c0 * sqrt(((a / l) / v))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 1e+242) {
		tmp = t_0;
	} else {
		tmp = c0 * Math.sqrt(((A / l) / V));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 1e+242:
		tmp = t_0
	else:
		tmp = c0 * math.sqrt(((A / l) / V))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 1e+242)
		tmp = t_0;
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 1e+242)
		tmp = t_0;
	else
		tmp = c0 * sqrt(((A / l) / V));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+242], t$95$0, N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 69.3%

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

      1. associate-/r* 71.5%

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

   3. Simplified 71.5%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 1.00000000000000005e242

   1. Initial program 99.1%

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

   if 1.00000000000000005e242 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 46.4%

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

      1. *-un-lft-identity 46.4%

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

      2. times-frac 50.2%

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

   4. Applied egg-rr 50.2%

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

      1. associate-*l/ 50.2%

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

      2. *-un-lft-identity 50.2%

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

   6. Applied egg-rr 50.2%

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

Alternative 4: 87.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \leq -4 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell} \cdot \sqrt{V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= V -4e-310)
   (* c0 (/ (sqrt (/ A (- l))) (sqrt (- V))))
   (* c0 (/ (sqrt A) (* (sqrt l) (sqrt V))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (V <= -4e-310) {
		tmp = c0 * (sqrt((A / -l)) / sqrt(-V));
	} else {
		tmp = c0 * (sqrt(A) / (sqrt(l) * sqrt(V)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (v <= (-4d-310)) then
        tmp = c0 * (sqrt((a / -l)) / sqrt(-v))
    else
        tmp = c0 * (sqrt(a) / (sqrt(l) * sqrt(v)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (V <= -4e-310) {
		tmp = c0 * (Math.sqrt((A / -l)) / Math.sqrt(-V));
	} else {
		tmp = c0 * (Math.sqrt(A) / (Math.sqrt(l) * Math.sqrt(V)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if V <= -4e-310:
		tmp = c0 * (math.sqrt((A / -l)) / math.sqrt(-V))
	else:
		tmp = c0 * (math.sqrt(A) / (math.sqrt(l) * math.sqrt(V)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (V <= -4e-310)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / Float64(-l))) / sqrt(Float64(-V))));
	else
		tmp = Float64(c0 * Float64(sqrt(A) / Float64(sqrt(l) * sqrt(V))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (V <= -4e-310)
		tmp = c0 * (sqrt((A / -l)) / sqrt(-V));
	else
		tmp = c0 * (sqrt(A) / (sqrt(l) * sqrt(V)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[V, -4e-310], N[(c0 * N[(N[Sqrt[N[(A / (-l)), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[V], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if V < -3.999999999999988e-310

   1. Initial program 76.0%

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

      1. *-un-lft-identity 76.0%

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

      2. times-frac 76.9%

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

   4. Applied egg-rr 76.9%

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

      1. associate-*l/ 76.9%

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

      2. *-un-lft-identity 76.9%

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

   6. Applied egg-rr 76.9%

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

      1. frac-2neg 76.9%

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

      2. sqrt-div 91.7%

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

   8. Applied egg-rr 91.7%

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

      1. distribute-frac-neg2 91.7%

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

   10. Simplified 91.7%

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

   if -3.999999999999988e-310 < V

   1. Initial program 72.6%

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

      1. associate-/r* 73.6%

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

      2. div-inv 73.5%

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

      3. add-cube-cbrt 73.1%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \frac{1}{V}}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}} \]

      4. times-frac 75.0%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]

      5. pow2 75.0%

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}} \]

   4. Applied egg-rr 75.0%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]
   5. Step-by-step derivation

      1. associate-*l/ 76.2%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

      2. associate-/l/ 75.6%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \color{blue}{\frac{1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      3. associate-*r/ 75.6%

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A \cdot 1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      4. *-rgt-identity 75.6%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{\color{blue}{A}}{\sqrt[3]{\ell} \cdot V}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      5. *-commutative 75.6%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{A}{\color{blue}{V \cdot \sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

   6. Simplified 75.6%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V \cdot \sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]
   7. Step-by-step derivation

      1. associate-/r* 73.1%

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{\frac{A}{V}}{\sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      2. associate-/l/ 73.1%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{\ell}}}} \]

      3. unpow2 73.1%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{A}{V}}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)} \cdot \sqrt[3]{\ell}}} \]

      4. add-cube-cbrt 73.6%

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

      5. un-div-inv 73.6%

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

      6. sqrt-unprod 45.8%

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

      7. *-commutative 45.8%

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

      8. sqrt-div 57.3%

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

      9. sqrt-div 57.3%

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

      10. metadata-eval 57.3%

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

      11. frac-times 57.4%

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

      12. *-un-lft-identity 57.4%

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

   8. Applied egg-rr 57.4%

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

Alternative 5: 83.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{A \cdot \left(\frac{c0}{\ell} \cdot \frac{c0}{V}\right)}\\ \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+292}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-300}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq 2 \cdot 10^{+256}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

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 (sqrt (* A (* (/ c0 l) (/ c0 V))))))
   (if (<= (* V l) -5e+292)
     t_0
     (if (<= (* V l) -5e-300)
       (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
       (if (or (<= (* V l) 0.0) (not (<= (* V l) 2e+256)))
         t_0
         (* c0 (/ (sqrt A) (sqrt (* V l)))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt((A * ((c0 / l) * (c0 / V))));
	double tmp;
	if ((V * l) <= -5e+292) {
		tmp = t_0;
	} else if ((V * l) <= -5e-300) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if (((V * l) <= 0.0) || !((V * l) <= 2e+256)) {
		tmp = t_0;
	} else {
		tmp = c0 * (sqrt(A) / sqrt((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 = sqrt((a * ((c0 / l) * (c0 / v))))
    if ((v * l) <= (-5d+292)) then
        tmp = t_0
    else if ((v * l) <= (-5d-300)) then
        tmp = c0 * (sqrt(-a) / sqrt((v * -l)))
    else if (((v * l) <= 0.0d0) .or. (.not. ((v * l) <= 2d+256))) then
        tmp = t_0
    else
        tmp = c0 * (sqrt(a) / sqrt((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 = Math.sqrt((A * ((c0 / l) * (c0 / V))));
	double tmp;
	if ((V * l) <= -5e+292) {
		tmp = t_0;
	} else if ((V * l) <= -5e-300) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if (((V * l) <= 0.0) || !((V * l) <= 2e+256)) {
		tmp = t_0;
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt((A * ((c0 / l) * (c0 / V))))
	tmp = 0
	if (V * l) <= -5e+292:
		tmp = t_0
	elif (V * l) <= -5e-300:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif ((V * l) <= 0.0) or not ((V * l) <= 2e+256):
		tmp = t_0
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = sqrt(Float64(A * Float64(Float64(c0 / l) * Float64(c0 / V))))
	tmp = 0.0
	if (Float64(V * l) <= -5e+292)
		tmp = t_0;
	elseif (Float64(V * l) <= -5e-300)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif ((Float64(V * l) <= 0.0) || !(Float64(V * l) <= 2e+256))
		tmp = t_0;
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(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 = sqrt((A * ((c0 / l) * (c0 / V))));
	tmp = 0.0;
	if ((V * l) <= -5e+292)
		tmp = t_0;
	elseif ((V * l) <= -5e-300)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif (((V * l) <= 0.0) || ~(((V * l) <= 2e+256)))
		tmp = t_0;
	else
		tmp = c0 * (sqrt(A) / sqrt((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[Sqrt[N[(A * N[(N[(c0 / l), $MachinePrecision] * N[(c0 / V), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -5e+292], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -5e-300], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[Not[LessEqual[N[(V * l), $MachinePrecision], 2e+256]], $MachinePrecision]], t$95$0, N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -4.9999999999999996e292 or -4.99999999999999996e-300 < (*.f64 V l) < 0.0 or 2.0000000000000001e256 < (*.f64 V l)

   1. Initial program 39.7%

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

      1. associate-/r* 59.2%

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

      2. div-inv 59.2%

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

      3. add-cube-cbrt 59.1%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \frac{1}{V}}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}} \]

      4. times-frac 54.5%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]

      5. pow2 54.5%

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}} \]

   4. Applied egg-rr 54.5%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]
   5. Step-by-step derivation

      1. associate-*l/ 54.5%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

      2. associate-/l/ 53.1%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \color{blue}{\frac{1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      3. associate-*r/ 53.1%

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A \cdot 1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      4. *-rgt-identity 53.1%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{\color{blue}{A}}{\sqrt[3]{\ell} \cdot V}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      5. *-commutative 53.1%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{A}{\color{blue}{V \cdot \sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

   6. Simplified 53.1%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V \cdot \sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

   8. Applied egg-rr 21.9%

      \[\leadsto \color{blue}{\sqrt{\frac{A}{V \cdot \ell} \cdot {c0}^{2}}} \]
   9. Step-by-step derivation

      1. associate-*l/ 20.9%

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

      2. associate-/l* 22.0%

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

   10. Simplified 22.0%

       \[\leadsto \color{blue}{\sqrt{A \cdot \frac{{c0}^{2}}{V \cdot \ell}}} \]
   11. Step-by-step derivation

       1. unpow2 22.0%

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

       2. *-commutative 22.0%

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

       3. times-frac 39.2%

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

   12. Applied egg-rr 39.2%

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

   if -4.9999999999999996e292 < (*.f64 V l) < -4.99999999999999996e-300

   1. Initial program 92.8%

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

      1. frac-2neg 92.8%

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

      2. sqrt-div 99.4%

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

      3. distribute-rgt-neg-in 99.4%

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

   4. Applied egg-rr 99.4%

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

      1. distribute-rgt-neg-out 99.4%

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

      2. *-commutative 99.4%

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

      3. distribute-rgt-neg-in 99.4%

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

   6. Simplified 99.4%

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

   if 0.0 < (*.f64 V l) < 2.0000000000000001e256

   1. Initial program 81.2%

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

      1. sqrt-div 99.0%

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

      2. div-inv 98.9%

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

   4. Applied egg-rr 98.9%

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

      1. associate-*r/ 99.0%

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

      2. *-rgt-identity 99.0%

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

   6. Simplified 99.0%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+292}:\\ \;\;\;\;\sqrt{A \cdot \left(\frac{c0}{\ell} \cdot \frac{c0}{V}\right)}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-300}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq 2 \cdot 10^{+256}\right):\\ \;\;\;\;\sqrt{A \cdot \left(\frac{c0}{\ell} \cdot \frac{c0}{V}\right)}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{A \cdot \left(\frac{c0}{\ell} \cdot \frac{c0}{V}\right)}\\ \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+292}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-300}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq 2 \cdot 10^{+256}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

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 (sqrt (* A (* (/ c0 l) (/ c0 V))))))
   (if (<= (* V l) -5e+292)
     t_0
     (if (<= (* V l) -5e-300)
       (/ c0 (sqrt (/ (* V l) A)))
       (if (or (<= (* V l) 0.0) (not (<= (* V l) 2e+256)))
         t_0
         (* c0 (/ (sqrt A) (sqrt (* V l)))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt((A * ((c0 / l) * (c0 / V))));
	double tmp;
	if ((V * l) <= -5e+292) {
		tmp = t_0;
	} else if ((V * l) <= -5e-300) {
		tmp = c0 / sqrt(((V * l) / A));
	} else if (((V * l) <= 0.0) || !((V * l) <= 2e+256)) {
		tmp = t_0;
	} else {
		tmp = c0 * (sqrt(A) / sqrt((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 = sqrt((a * ((c0 / l) * (c0 / v))))
    if ((v * l) <= (-5d+292)) then
        tmp = t_0
    else if ((v * l) <= (-5d-300)) then
        tmp = c0 / sqrt(((v * l) / a))
    else if (((v * l) <= 0.0d0) .or. (.not. ((v * l) <= 2d+256))) then
        tmp = t_0
    else
        tmp = c0 * (sqrt(a) / sqrt((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 = Math.sqrt((A * ((c0 / l) * (c0 / V))));
	double tmp;
	if ((V * l) <= -5e+292) {
		tmp = t_0;
	} else if ((V * l) <= -5e-300) {
		tmp = c0 / Math.sqrt(((V * l) / A));
	} else if (((V * l) <= 0.0) || !((V * l) <= 2e+256)) {
		tmp = t_0;
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt((A * ((c0 / l) * (c0 / V))))
	tmp = 0
	if (V * l) <= -5e+292:
		tmp = t_0
	elif (V * l) <= -5e-300:
		tmp = c0 / math.sqrt(((V * l) / A))
	elif ((V * l) <= 0.0) or not ((V * l) <= 2e+256):
		tmp = t_0
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = sqrt(Float64(A * Float64(Float64(c0 / l) * Float64(c0 / V))))
	tmp = 0.0
	if (Float64(V * l) <= -5e+292)
		tmp = t_0;
	elseif (Float64(V * l) <= -5e-300)
		tmp = Float64(c0 / sqrt(Float64(Float64(V * l) / A)));
	elseif ((Float64(V * l) <= 0.0) || !(Float64(V * l) <= 2e+256))
		tmp = t_0;
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(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 = sqrt((A * ((c0 / l) * (c0 / V))));
	tmp = 0.0;
	if ((V * l) <= -5e+292)
		tmp = t_0;
	elseif ((V * l) <= -5e-300)
		tmp = c0 / sqrt(((V * l) / A));
	elseif (((V * l) <= 0.0) || ~(((V * l) <= 2e+256)))
		tmp = t_0;
	else
		tmp = c0 * (sqrt(A) / sqrt((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[Sqrt[N[(A * N[(N[(c0 / l), $MachinePrecision] * N[(c0 / V), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -5e+292], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -5e-300], N[(c0 / N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[Not[LessEqual[N[(V * l), $MachinePrecision], 2e+256]], $MachinePrecision]], t$95$0, N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -4.9999999999999996e292 or -4.99999999999999996e-300 < (*.f64 V l) < 0.0 or 2.0000000000000001e256 < (*.f64 V l)

   1. Initial program 39.7%

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

      1. associate-/r* 59.2%

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

      2. div-inv 59.2%

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

      3. add-cube-cbrt 59.1%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \frac{1}{V}}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}} \]

      4. times-frac 54.5%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]

      5. pow2 54.5%

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}} \]

   4. Applied egg-rr 54.5%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]
   5. Step-by-step derivation

      1. associate-*l/ 54.5%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

      2. associate-/l/ 53.1%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \color{blue}{\frac{1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      3. associate-*r/ 53.1%

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A \cdot 1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      4. *-rgt-identity 53.1%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{\color{blue}{A}}{\sqrt[3]{\ell} \cdot V}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      5. *-commutative 53.1%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{A}{\color{blue}{V \cdot \sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

   6. Simplified 53.1%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V \cdot \sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

   8. Applied egg-rr 21.9%

      \[\leadsto \color{blue}{\sqrt{\frac{A}{V \cdot \ell} \cdot {c0}^{2}}} \]
   9. Step-by-step derivation

      1. associate-*l/ 20.9%

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

      2. associate-/l* 22.0%

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

   10. Simplified 22.0%

       \[\leadsto \color{blue}{\sqrt{A \cdot \frac{{c0}^{2}}{V \cdot \ell}}} \]
   11. Step-by-step derivation

       1. unpow2 22.0%

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

       2. *-commutative 22.0%

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

       3. times-frac 39.2%

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

   12. Applied egg-rr 39.2%

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

   if -4.9999999999999996e292 < (*.f64 V l) < -4.99999999999999996e-300

   1. Initial program 92.8%

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

      1. *-un-lft-identity 92.8%

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

      2. times-frac 85.3%

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

   4. Applied egg-rr 85.3%

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

      1. associate-*l/ 85.3%

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

      2. *-un-lft-identity 85.3%

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

   6. Applied egg-rr 85.3%

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

      1. div-inv 85.3%

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

      2. clear-num 85.0%

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

      3. frac-times 85.0%

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

      4. *-commutative 85.0%

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

      5. add-sqr-sqrt 84.8%

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

      6. frac-times 84.8%

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

      7. sqrt-unprod 85.4%

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

      8. add-sqr-sqrt 85.8%

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

      9. un-div-inv 85.9%

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

   8. Applied egg-rr 85.9%

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

      1. associate-*r/ 93.7%

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

      2. *-commutative 93.7%

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

      3. associate-/l* 86.6%

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

   10. Simplified 86.6%

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

   11. Taylor expanded in l around 0 93.7%

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

   if 0.0 < (*.f64 V l) < 2.0000000000000001e256

   1. Initial program 81.2%

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

      1. sqrt-div 99.0%

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

      2. div-inv 98.9%

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

   4. Applied egg-rr 98.9%

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

      1. associate-*r/ 99.0%

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

      2. *-rgt-identity 99.0%

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

   6. Simplified 99.0%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+292}:\\ \;\;\;\;\sqrt{A \cdot \left(\frac{c0}{\ell} \cdot \frac{c0}{V}\right)}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-300}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq 2 \cdot 10^{+256}\right):\\ \;\;\;\;\sqrt{A \cdot \left(\frac{c0}{\ell} \cdot \frac{c0}{V}\right)}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \leq -4 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \leq 1.05 \cdot 10^{-167}:\\ \;\;\;\;\frac{\frac{c0}{\sqrt{V}}}{\sqrt{\frac{\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= V -4e-310)
   (* c0 (/ (sqrt (/ A (- l))) (sqrt (- V))))
   (if (<= V 1.05e-167)
     (/ (/ c0 (sqrt V)) (sqrt (/ l A)))
     (* c0 (/ (sqrt (/ A V)) (sqrt l))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (V <= -4e-310) {
		tmp = c0 * (sqrt((A / -l)) / sqrt(-V));
	} else if (V <= 1.05e-167) {
		tmp = (c0 / sqrt(V)) / sqrt((l / A));
	} else {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (v <= (-4d-310)) then
        tmp = c0 * (sqrt((a / -l)) / sqrt(-v))
    else if (v <= 1.05d-167) then
        tmp = (c0 / sqrt(v)) / sqrt((l / a))
    else
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (V <= -4e-310) {
		tmp = c0 * (Math.sqrt((A / -l)) / Math.sqrt(-V));
	} else if (V <= 1.05e-167) {
		tmp = (c0 / Math.sqrt(V)) / Math.sqrt((l / A));
	} else {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if V <= -4e-310:
		tmp = c0 * (math.sqrt((A / -l)) / math.sqrt(-V))
	elif V <= 1.05e-167:
		tmp = (c0 / math.sqrt(V)) / math.sqrt((l / A))
	else:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (V <= -4e-310)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / Float64(-l))) / sqrt(Float64(-V))));
	elseif (V <= 1.05e-167)
		tmp = Float64(Float64(c0 / sqrt(V)) / sqrt(Float64(l / A)));
	else
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (V <= -4e-310)
		tmp = c0 * (sqrt((A / -l)) / sqrt(-V));
	elseif (V <= 1.05e-167)
		tmp = (c0 / sqrt(V)) / sqrt((l / A));
	else
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[V, -4e-310], N[(c0 * N[(N[Sqrt[N[(A / (-l)), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[V, 1.05e-167], N[(N[(c0 / N[Sqrt[V], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if V < -3.999999999999988e-310

   1. Initial program 76.0%

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

      1. *-un-lft-identity 76.0%

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

      2. times-frac 76.9%

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

   4. Applied egg-rr 76.9%

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

      1. associate-*l/ 76.9%

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

      2. *-un-lft-identity 76.9%

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

   6. Applied egg-rr 76.9%

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

      1. frac-2neg 76.9%

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

      2. sqrt-div 91.7%

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

   8. Applied egg-rr 91.7%

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

      1. distribute-frac-neg2 91.7%

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

   10. Simplified 91.7%

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

   if -3.999999999999988e-310 < V < 1.05000000000000009e-167

   1. Initial program 69.2%

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

      1. *-un-lft-identity 69.2%

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

      2. times-frac 75.3%

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

   4. Applied egg-rr 75.3%

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

      1. associate-*l/ 75.3%

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

      2. *-un-lft-identity 75.3%

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

   6. Applied egg-rr 75.3%

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

      1. div-inv 75.3%

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

      2. clear-num 75.3%

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

      3. frac-times 75.2%

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

      4. *-commutative 75.2%

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

      5. add-sqr-sqrt 75.1%

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

      6. frac-times 75.1%

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

      7. sqrt-unprod 74.9%

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

      8. add-sqr-sqrt 75.1%

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

      9. associate-*r/ 75.2%

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

      10. sqrt-prod 95.4%

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

      11. times-frac 92.1%

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

      12. metadata-eval 92.1%

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

      13. sqrt-div 92.1%

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

      14. clear-num 92.1%

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

      15. sqrt-div 70.6%

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

      16. times-frac 67.3%

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

      17. *-commutative 67.3%

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

      18. sqrt-prod 45.5%

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

      19. clear-num 45.6%

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

   8. Applied egg-rr 92.0%

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

      1. associate-/r* 92.1%

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

      2. associate-/r/ 91.9%

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

      3. associate-*l/ 92.2%

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

      4. *-lft-identity 92.2%

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

   10. Simplified 92.2%

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

   if 1.05000000000000009e-167 < V

   1. Initial program 73.6%

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

      1. associate-/r* 75.9%

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

      2. sqrt-div 43.3%

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

      3. div-inv 43.3%

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

   4. Applied egg-rr 43.3%

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

      1. associate-*r/ 43.3%

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

      2. *-rgt-identity 43.3%

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

   6. Simplified 43.3%

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

Alternative 8: 76.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-315} \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;\sqrt{A \cdot \left(\frac{c0}{\ell} \cdot \frac{c0}{V}\right)}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (or (<= t_0 5e-315) (not (<= t_0 INFINITY)))
     (sqrt (* A (* (/ c0 l) (/ c0 V))))
     (* c0 (sqrt t_0)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 5e-315) || !(t_0 <= ((double) INFINITY))) {
		tmp = sqrt((A * ((c0 / l) * (c0 / V))));
	} else {
		tmp = c0 * sqrt(t_0);
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 5e-315) || !(t_0 <= Double.POSITIVE_INFINITY)) {
		tmp = Math.sqrt((A * ((c0 / l) * (c0 / V))));
	} else {
		tmp = c0 * Math.sqrt(t_0);
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if (t_0 <= 5e-315) or not (t_0 <= math.inf):
		tmp = math.sqrt((A * ((c0 / l) * (c0 / V))))
	else:
		tmp = c0 * math.sqrt(t_0)
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if ((t_0 <= 5e-315) || !(t_0 <= Inf))
		tmp = sqrt(Float64(A * Float64(Float64(c0 / l) * Float64(c0 / V))));
	else
		tmp = Float64(c0 * sqrt(t_0));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if ((t_0 <= 5e-315) || ~((t_0 <= Inf)))
		tmp = sqrt((A * ((c0 / l) * (c0 / V))));
	else
		tmp = c0 * sqrt(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 5e-315], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[Sqrt[N[(A * N[(N[(c0 / l), $MachinePrecision] * N[(c0 / V), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 5.0000000023e-315 or +inf.0 < (/.f64 A (*.f64 V l))

   1. Initial program 38.2%

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

      1. associate-/r* 51.4%

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

      2. div-inv 51.5%

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

      3. add-cube-cbrt 51.4%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \frac{1}{V}}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}} \]

      4. times-frac 47.4%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]

      5. pow2 47.4%

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}} \]

   4. Applied egg-rr 47.4%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]
   5. Step-by-step derivation

      1. associate-*l/ 47.4%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

      2. associate-/l/ 45.6%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \color{blue}{\frac{1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      3. associate-*r/ 45.6%

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A \cdot 1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      4. *-rgt-identity 45.6%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{\color{blue}{A}}{\sqrt[3]{\ell} \cdot V}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      5. *-commutative 45.6%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{A}{\color{blue}{V \cdot \sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

   6. Simplified 45.6%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V \cdot \sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

   8. Applied egg-rr 35.8%

      \[\leadsto \color{blue}{\sqrt{\frac{A}{V \cdot \ell} \cdot {c0}^{2}}} \]
   9. Step-by-step derivation

      1. associate-*l/ 39.6%

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

      2. associate-/l* 39.7%

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

   10. Simplified 39.7%

       \[\leadsto \color{blue}{\sqrt{A \cdot \frac{{c0}^{2}}{V \cdot \ell}}} \]
   11. Step-by-step derivation

       1. unpow2 39.7%

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

       2. *-commutative 39.7%

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

       3. times-frac 55.0%

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

   12. Applied egg-rr 55.0%

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

   if 5.0000000023e-315 < (/.f64 A (*.f64 V l)) < +inf.0

   1. Initial program 83.6%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{-315} \lor \neg \left(\frac{A}{V \cdot \ell} \leq \infty\right):\\ \;\;\;\;\sqrt{A \cdot \left(\frac{c0}{\ell} \cdot \frac{c0}{V}\right)}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-315}:\\ \;\;\;\;\sqrt{A \cdot \left(c0 \cdot \frac{c0}{V \cdot \ell}\right)}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+285}:\\ \;\;\;\;c0 \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (<= t_0 5e-315)
     (sqrt (* A (* c0 (/ c0 (* V l)))))
     (if (<= t_0 5e+285) (* c0 (sqrt t_0)) (/ c0 (sqrt (* V (/ l A))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 5e-315) {
		tmp = sqrt((A * (c0 * (c0 / (V * l)))));
	} else if (t_0 <= 5e+285) {
		tmp = c0 * sqrt(t_0);
	} 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) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if (t_0 <= 5d-315) then
        tmp = sqrt((a * (c0 * (c0 / (v * l)))))
    else if (t_0 <= 5d+285) then
        tmp = c0 * sqrt(t_0)
    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 t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 5e-315) {
		tmp = Math.sqrt((A * (c0 * (c0 / (V * l)))));
	} else if (t_0 <= 5e+285) {
		tmp = c0 * Math.sqrt(t_0);
	} 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):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 5e-315:
		tmp = math.sqrt((A * (c0 * (c0 / (V * l)))))
	elif t_0 <= 5e+285:
		tmp = c0 * math.sqrt(t_0)
	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)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 5e-315)
		tmp = sqrt(Float64(A * Float64(c0 * Float64(c0 / Float64(V * l)))));
	elseif (t_0 <= 5e+285)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 / sqrt(Float64(V * Float64(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)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 5e-315)
		tmp = sqrt((A * (c0 * (c0 / (V * l)))));
	elseif (t_0 <= 5e+285)
		tmp = c0 * sqrt(t_0);
	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_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-315], N[Sqrt[N[(A * N[(c0 * N[(c0 / N[(V * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 5e+285], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 A (*.f64 V l)) < 5.0000000023e-315

   1. Initial program 38.2%

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

      1. associate-/r* 51.4%

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

      2. div-inv 51.5%

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

      3. add-cube-cbrt 51.4%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \frac{1}{V}}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}} \]

      4. times-frac 47.4%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]

      5. pow2 47.4%

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}} \]

   4. Applied egg-rr 47.4%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]
   5. Step-by-step derivation

      1. associate-*l/ 47.4%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

      2. associate-/l/ 45.6%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \color{blue}{\frac{1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      3. associate-*r/ 45.6%

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A \cdot 1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      4. *-rgt-identity 45.6%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{\color{blue}{A}}{\sqrt[3]{\ell} \cdot V}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      5. *-commutative 45.6%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{A}{\color{blue}{V \cdot \sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

   6. Simplified 45.6%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V \cdot \sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

   8. Applied egg-rr 35.8%

      \[\leadsto \color{blue}{\sqrt{\frac{A}{V \cdot \ell} \cdot {c0}^{2}}} \]
   9. Step-by-step derivation

      1. associate-*l/ 39.6%

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

      2. associate-/l* 39.7%

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

   10. Simplified 39.7%

       \[\leadsto \color{blue}{\sqrt{A \cdot \frac{{c0}^{2}}{V \cdot \ell}}} \]
   11. Step-by-step derivation

       1. unpow2 39.7%

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

       2. associate-/l* 47.9%

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

   12. Applied egg-rr 47.9%

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

   if 5.0000000023e-315 < (/.f64 A (*.f64 V l)) < 5.00000000000000016e285

   1. Initial program 99.4%

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

   if 5.00000000000000016e285 < (/.f64 A (*.f64 V l))

   1. Initial program 37.6%

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

      1. *-un-lft-identity 37.6%

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

      2. times-frac 48.6%

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

   4. Applied egg-rr 48.6%

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

      1. associate-*l/ 48.6%

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

      2. *-un-lft-identity 48.6%

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

   6. Applied egg-rr 48.6%

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

      1. div-inv 48.6%

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

      2. clear-num 48.5%

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

      3. frac-times 48.5%

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

      4. *-commutative 48.5%

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

      5. add-sqr-sqrt 48.5%

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

      6. frac-times 48.6%

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

      7. sqrt-unprod 50.7%

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

      8. add-sqr-sqrt 50.7%

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

      9. un-div-inv 50.8%

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

   8. Applied egg-rr 50.8%

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

Alternative 10: 79.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 10^{-272}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+285}:\\ \;\;\;\;c0 \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (<= t_0 1e-272)
     (* c0 (sqrt (/ (/ A l) V)))
     (if (<= t_0 5e+285) (* c0 (sqrt t_0)) (/ c0 (sqrt (* V (/ l A))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 1e-272) {
		tmp = c0 * sqrt(((A / l) / V));
	} else if (t_0 <= 5e+285) {
		tmp = c0 * sqrt(t_0);
	} 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) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if (t_0 <= 1d-272) then
        tmp = c0 * sqrt(((a / l) / v))
    else if (t_0 <= 5d+285) then
        tmp = c0 * sqrt(t_0)
    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 t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 1e-272) {
		tmp = c0 * Math.sqrt(((A / l) / V));
	} else if (t_0 <= 5e+285) {
		tmp = c0 * Math.sqrt(t_0);
	} 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):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 1e-272:
		tmp = c0 * math.sqrt(((A / l) / V))
	elif t_0 <= 5e+285:
		tmp = c0 * math.sqrt(t_0)
	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)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 1e-272)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	elseif (t_0 <= 5e+285)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 / sqrt(Float64(V * Float64(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)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 1e-272)
		tmp = c0 * sqrt(((A / l) / V));
	elseif (t_0 <= 5e+285)
		tmp = c0 * sqrt(t_0);
	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_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-272], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+285], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 A (*.f64 V l)) < 9.9999999999999993e-273

   1. Initial program 42.5%

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

      1. *-un-lft-identity 42.5%

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

      2. times-frac 53.7%

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

   4. Applied egg-rr 53.7%

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

      1. associate-*l/ 53.7%

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

      2. *-un-lft-identity 53.7%

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

   6. Applied egg-rr 53.7%

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

   if 9.9999999999999993e-273 < (/.f64 A (*.f64 V l)) < 5.00000000000000016e285

   1. Initial program 99.4%

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

   if 5.00000000000000016e285 < (/.f64 A (*.f64 V l))

   1. Initial program 37.6%

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

      1. *-un-lft-identity 37.6%

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

      2. times-frac 48.6%

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

   4. Applied egg-rr 48.6%

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

      1. associate-*l/ 48.6%

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

      2. *-un-lft-identity 48.6%

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

   6. Applied egg-rr 48.6%

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

      1. div-inv 48.6%

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

      2. clear-num 48.5%

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

      3. frac-times 48.5%

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

      4. *-commutative 48.5%

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

      5. add-sqr-sqrt 48.5%

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

      6. frac-times 48.6%

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

      7. sqrt-unprod 50.7%

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

      8. add-sqr-sqrt 50.7%

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

      9. un-div-inv 50.8%

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

   8. Applied egg-rr 50.8%

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

Alternative 11: 73.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.4%

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

Reproduce

herbie shell --seed 2024107 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))