Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.5% → 89.7%

Time: 14.1s

Alternatives: 16

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: 73.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{-V}} \cdot \left(c0 \cdot {\ell}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= A -1e-310)
   (* (/ (sqrt (- A)) (sqrt (- V))) (* c0 (pow l -0.5)))
   (* 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 tmp;
	if (A <= -1e-310) {
		tmp = (sqrt(-A) / sqrt(-V)) * (c0 * pow(l, -0.5));
	} 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) :: tmp
    if (a <= (-1d-310)) then
        tmp = (sqrt(-a) / sqrt(-v)) * (c0 * (l ** (-0.5d0)))
    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 tmp;
	if (A <= -1e-310) {
		tmp = (Math.sqrt(-A) / Math.sqrt(-V)) * (c0 * Math.pow(l, -0.5));
	} 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):
	tmp = 0
	if A <= -1e-310:
		tmp = (math.sqrt(-A) / math.sqrt(-V)) * (c0 * math.pow(l, -0.5))
	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)
	tmp = 0.0
	if (A <= -1e-310)
		tmp = Float64(Float64(sqrt(Float64(-A)) / sqrt(Float64(-V))) * Float64(c0 * (l ^ -0.5)));
	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)
	tmp = 0.0;
	if (A <= -1e-310)
		tmp = (sqrt(-A) / sqrt(-V)) * (c0 * (l ^ -0.5));
	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_] := If[LessEqual[A, -1e-310], N[(N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision] * N[(c0 * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -9.999999999999969e-311

   1. Initial program 79.8%

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

      1. *-commutative 79.8%

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

      2. associate-/r* 76.5%

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

      3. sqrt-div 39.9%

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

      4. associate-*l/ 40.0%

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

   4. Applied egg-rr 40.0%

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

      1. div-inv 39.9%

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

      2. *-commutative 39.9%

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

      3. pow1/2 39.9%

         \[\leadsto \left(c0 \cdot \sqrt{\frac{A}{V}}\right) \cdot \frac{1}{\color{blue}{{\ell}^{0.5}}} \]

      4. pow-flip 40.0%

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

      5. metadata-eval 40.0%

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

   6. Applied egg-rr 40.0%

      \[\leadsto \color{blue}{\left(c0 \cdot \sqrt{\frac{A}{V}}\right) \cdot {\ell}^{-0.5}} \]
   7. Step-by-step derivation

      1. *-commutative 40.0%

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

      2. associate-*l* 38.6%

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

   8. Simplified 38.6%

      \[\leadsto \color{blue}{\sqrt{\frac{A}{V}} \cdot \left(c0 \cdot {\ell}^{-0.5}\right)} \]
   9. Step-by-step derivation

      1. frac-2neg 38.6%

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

      2. sqrt-div 45.2%

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

   10. Applied egg-rr 45.2%

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

   if -9.999999999999969e-311 < A

   1. Initial program 72.7%

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

      1. sqrt-div 83.9%

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

      2. div-inv 83.9%

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

   4. Applied egg-rr 83.9%

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

      1. associate-*r/ 83.9%

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

      2. *-rgt-identity 83.9%

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

   6. Simplified 83.9%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{-V}} \cdot \left(c0 \cdot {\ell}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 10^{-305}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+241}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (<= t_0 1e-305)
     (/ c0 (sqrt (* V (/ l A))))
     (if (<= t_0 5e+241) t_0 (/ c0 (sqrt (* l (/ V A))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 1e-305) {
		tmp = c0 / sqrt((V * (l / A)));
	} else if (t_0 <= 5e+241) {
		tmp = t_0;
	} else {
		tmp = c0 / sqrt((l * (V / A)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    if (t_0 <= 1d-305) then
        tmp = c0 / sqrt((v * (l / a)))
    else if (t_0 <= 5d+241) then
        tmp = t_0
    else
        tmp = c0 / sqrt((l * (v / a)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 1e-305) {
		tmp = c0 / Math.sqrt((V * (l / A)));
	} else if (t_0 <= 5e+241) {
		tmp = t_0;
	} else {
		tmp = c0 / Math.sqrt((l * (V / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 1e-305:
		tmp = c0 / math.sqrt((V * (l / A)))
	elif t_0 <= 5e+241:
		tmp = t_0
	else:
		tmp = c0 / math.sqrt((l * (V / A)))
	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 <= 1e-305)
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	elseif (t_0 <= 5e+241)
		tmp = t_0;
	else
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / A))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= 1e-305)
		tmp = c0 / sqrt((V * (l / A)));
	elseif (t_0 <= 5e+241)
		tmp = t_0;
	else
		tmp = c0 / sqrt((l * (V / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-305], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+241], t$95$0, N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 9.99999999999999996e-306

   1. Initial program 71.0%

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

      1. associate-/r* 69.7%

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

      2. clear-num 69.7%

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

      3. sqrt-div 70.0%

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

      4. metadata-eval 70.0%

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

      5. div-inv 70.0%

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

      6. clear-num 70.3%

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

   4. Applied egg-rr 70.3%

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

      1. *-commutative 70.3%

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

      2. associate-*l/ 71.2%

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

      3. associate-/l* 73.2%

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

   6. Simplified 73.2%

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

      1. un-div-inv 73.3%

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

      2. associate-*r/ 71.3%

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

   8. Applied egg-rr 71.3%

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

      1. associate-/l* 73.3%

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

   10. Simplified 73.3%

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

   if 9.99999999999999996e-306 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 5.00000000000000025e241

   1. Initial program 98.7%

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

   if 5.00000000000000025e241 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 51.4%

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

      1. associate-/r* 50.8%

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

      2. clear-num 50.7%

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

      3. sqrt-div 52.0%

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

      4. metadata-eval 52.0%

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

      5. div-inv 52.1%

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

      6. clear-num 52.0%

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

   4. Applied egg-rr 52.0%

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

      1. *-commutative 52.0%

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

      2. associate-*l/ 57.7%

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

      3. associate-/l* 59.9%

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

   6. Simplified 59.9%

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

      1. un-div-inv 59.9%

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

      2. associate-*r/ 57.7%

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

   8. Applied egg-rr 57.7%

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

      1. *-commutative 57.7%

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

      2. associate-/l* 52.1%

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

   10. Applied egg-rr 52.1%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 10^{-305}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 5 \cdot 10^{+241}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.6% accurate, 0.3× speedup

Math

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

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= A -1e-310)
   (/ (* (/ (sqrt (- A)) (sqrt (- V))) c0) (sqrt l))
   (* 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 tmp;
	if (A <= -1e-310) {
		tmp = ((sqrt(-A) / sqrt(-V)) * c0) / sqrt(l);
	} 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) :: tmp
    if (a <= (-1d-310)) then
        tmp = ((sqrt(-a) / sqrt(-v)) * c0) / sqrt(l)
    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 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 * l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if A <= -1e-310:
		tmp = ((math.sqrt(-A) / math.sqrt(-V)) * c0) / math.sqrt(l)
	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)
	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(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)
	tmp = 0.0;
	if (A <= -1e-310)
		tmp = ((sqrt(-A) / sqrt(-V)) * c0) / sqrt(l);
	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_] := 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[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -9.999999999999969e-311

   1. Initial program 79.8%

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

      1. *-commutative 79.8%

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

      2. associate-/r* 76.5%

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

      3. sqrt-div 39.9%

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

      4. associate-*l/ 40.0%

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

   4. Applied egg-rr 40.0%

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

      1. frac-2neg 38.6%

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

      2. sqrt-div 45.2%

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

   6. Applied egg-rr 45.9%

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

   if -9.999999999999969e-311 < A

   1. Initial program 72.7%

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

      1. sqrt-div 83.9%

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

      2. div-inv 83.9%

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

   4. Applied egg-rr 83.9%

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

      1. associate-*r/ 83.9%

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

      2. *-rgt-identity 83.9%

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

   6. Simplified 83.9%

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

4. Final simplification 64.9%

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

Alternative 4: 84.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{V}}\\ \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+144}:\\ \;\;\;\;\left(c0 \cdot {\ell}^{-0.5}\right) \cdot t\_0\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-139}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\frac{V \cdot \ell}{A}}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-276}:\\ \;\;\;\;t\_0 \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+226}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{1}{\ell}}{\frac{V}{A}}}\\ \end{array} \end{array} \]

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 (sqrt (/ A V))))
   (if (<= (* V l) -5e+144)
     (* (* c0 (pow l -0.5)) t_0)
     (if (<= (* V l) -1e-139)
       (/ 1.0 (/ (sqrt (/ (* V l) A)) c0))
       (if (<= (* V l) 1e-276)
         (* t_0 (/ c0 (sqrt l)))
         (if (<= (* V l) 1e+226)
           (* c0 (* (sqrt A) (sqrt (/ (/ 1.0 V) l))))
           (* c0 (sqrt (/ (/ 1.0 l) (/ V A))))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt((A / V));
	double tmp;
	if ((V * l) <= -5e+144) {
		tmp = (c0 * pow(l, -0.5)) * t_0;
	} else if ((V * l) <= -1e-139) {
		tmp = 1.0 / (sqrt(((V * l) / A)) / c0);
	} else if ((V * l) <= 1e-276) {
		tmp = t_0 * (c0 / sqrt(l));
	} else if ((V * l) <= 1e+226) {
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	} else {
		tmp = c0 * sqrt(((1.0 / l) / (V / 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 = sqrt((a / v))
    if ((v * l) <= (-5d+144)) then
        tmp = (c0 * (l ** (-0.5d0))) * t_0
    else if ((v * l) <= (-1d-139)) then
        tmp = 1.0d0 / (sqrt(((v * l) / a)) / c0)
    else if ((v * l) <= 1d-276) then
        tmp = t_0 * (c0 / sqrt(l))
    else if ((v * l) <= 1d+226) then
        tmp = c0 * (sqrt(a) * sqrt(((1.0d0 / v) / l)))
    else
        tmp = c0 * sqrt(((1.0d0 / l) / (v / 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 = Math.sqrt((A / V));
	double tmp;
	if ((V * l) <= -5e+144) {
		tmp = (c0 * Math.pow(l, -0.5)) * t_0;
	} else if ((V * l) <= -1e-139) {
		tmp = 1.0 / (Math.sqrt(((V * l) / A)) / c0);
	} else if ((V * l) <= 1e-276) {
		tmp = t_0 * (c0 / Math.sqrt(l));
	} else if ((V * l) <= 1e+226) {
		tmp = c0 * (Math.sqrt(A) * Math.sqrt(((1.0 / V) / l)));
	} else {
		tmp = c0 * Math.sqrt(((1.0 / l) / (V / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt((A / V))
	tmp = 0
	if (V * l) <= -5e+144:
		tmp = (c0 * math.pow(l, -0.5)) * t_0
	elif (V * l) <= -1e-139:
		tmp = 1.0 / (math.sqrt(((V * l) / A)) / c0)
	elif (V * l) <= 1e-276:
		tmp = t_0 * (c0 / math.sqrt(l))
	elif (V * l) <= 1e+226:
		tmp = c0 * (math.sqrt(A) * math.sqrt(((1.0 / V) / l)))
	else:
		tmp = c0 * math.sqrt(((1.0 / l) / (V / A)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = sqrt(Float64(A / V))
	tmp = 0.0
	if (Float64(V * l) <= -5e+144)
		tmp = Float64(Float64(c0 * (l ^ -0.5)) * t_0);
	elseif (Float64(V * l) <= -1e-139)
		tmp = Float64(1.0 / Float64(sqrt(Float64(Float64(V * l) / A)) / c0));
	elseif (Float64(V * l) <= 1e-276)
		tmp = Float64(t_0 * Float64(c0 / sqrt(l)));
	elseif (Float64(V * l) <= 1e+226)
		tmp = Float64(c0 * Float64(sqrt(A) * sqrt(Float64(Float64(1.0 / V) / l))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(1.0 / l) / Float64(V / 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 = sqrt((A / V));
	tmp = 0.0;
	if ((V * l) <= -5e+144)
		tmp = (c0 * (l ^ -0.5)) * t_0;
	elseif ((V * l) <= -1e-139)
		tmp = 1.0 / (sqrt(((V * l) / A)) / c0);
	elseif ((V * l) <= 1e-276)
		tmp = t_0 * (c0 / sqrt(l));
	elseif ((V * l) <= 1e+226)
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	else
		tmp = c0 * sqrt(((1.0 / l) / (V / 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[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -5e+144], N[(N[(c0 * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-139], N[(1.0 / N[(N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision] / c0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e-276], N[(t$95$0 * N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+226], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Sqrt[N[(N[(1.0 / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -4.9999999999999999e144

   1. Initial program 69.6%

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

      1. *-commutative 69.6%

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

      2. associate-/r* 67.3%

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

      3. sqrt-div 45.1%

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

      4. associate-*l/ 45.3%

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

   4. Applied egg-rr 45.3%

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

      1. div-inv 45.2%

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

      2. *-commutative 45.2%

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

      3. pow1/2 45.2%

         \[\leadsto \left(c0 \cdot \sqrt{\frac{A}{V}}\right) \cdot \frac{1}{\color{blue}{{\ell}^{0.5}}} \]

      4. pow-flip 45.3%

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

      5. metadata-eval 45.3%

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

   6. Applied egg-rr 45.3%

      \[\leadsto \color{blue}{\left(c0 \cdot \sqrt{\frac{A}{V}}\right) \cdot {\ell}^{-0.5}} \]
   7. Step-by-step derivation

      1. *-commutative 45.3%

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

      2. associate-*l* 45.3%

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

   8. Simplified 45.3%

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

   if -4.9999999999999999e144 < (*.f64 V l) < -1.00000000000000003e-139

   1. Initial program 95.7%

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

      1. associate-/r* 85.4%

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

      2. clear-num 85.4%

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

      3. sqrt-div 85.4%

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

      4. metadata-eval 85.4%

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

      5. div-inv 84.3%

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

      6. clear-num 85.3%

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

   4. Applied egg-rr 85.3%

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

      1. *-commutative 85.3%

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

      2. associate-*l/ 97.2%

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

      3. associate-/l* 84.1%

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

   6. Simplified 84.1%

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

      1. un-div-inv 84.1%

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

      2. clear-num 84.1%

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

      3. associate-*r/ 97.4%

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

   8. Applied egg-rr 97.4%

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

   if -1.00000000000000003e-139 < (*.f64 V l) < 1e-276

   1. Initial program 66.2%

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

      1. associate-/r* 71.8%

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

      2. sqrt-div 39.6%

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

      3. associate-*r/ 39.6%

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

   4. Applied egg-rr 39.6%

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

      1. *-commutative 39.6%

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

      2. associate-/l* 39.7%

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

   6. Simplified 39.7%

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

   if 1e-276 < (*.f64 V l) < 9.99999999999999961e225

   1. Initial program 83.0%

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

      1. div-inv 82.9%

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

      2. sqrt-prod 99.5%

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

      3. associate-/r* 99.6%

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

   4. Applied egg-rr 99.6%

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

   if 9.99999999999999961e225 < (*.f64 V l)

   1. Initial program 52.3%

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

      1. *-un-lft-identity 52.3%

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

      2. times-frac 79.7%

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

   4. Applied egg-rr 79.7%

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

      1. associate-*r/ 79.7%

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

      2. div-inv 79.7%

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

      3. associate-*l/ 79.6%

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

      4. *-un-lft-identity 79.6%

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

      5. clear-num 79.7%

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

      6. associate-*l/ 79.8%

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

      7. *-un-lft-identity 79.8%

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

   6. Applied egg-rr 79.8%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+144}:\\ \;\;\;\;\left(c0 \cdot {\ell}^{-0.5}\right) \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-139}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\frac{V \cdot \ell}{A}}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-276}:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+226}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{1}{\ell}}{\frac{V}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+143}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-139}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\frac{V \cdot \ell}{A}}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-318}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+226}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{1}{\ell}}{\frac{V}{A}}}\\ \end{array} \end{array} \]

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 (* (sqrt (/ A V)) (/ c0 (sqrt l)))))
   (if (<= (* V l) -5e+143)
     t_0
     (if (<= (* V l) -1e-139)
       (/ 1.0 (/ (sqrt (/ (* V l) A)) c0))
       (if (<= (* V l) 2e-318)
         t_0
         (if (<= (* V l) 1e+226)
           (* c0 (/ (sqrt A) (sqrt (* V l))))
           (* c0 (sqrt (/ (/ 1.0 l) (/ V A))))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt((A / V)) * (c0 / sqrt(l));
	double tmp;
	if ((V * l) <= -5e+143) {
		tmp = t_0;
	} else if ((V * l) <= -1e-139) {
		tmp = 1.0 / (sqrt(((V * l) / A)) / c0);
	} else if ((V * l) <= 2e-318) {
		tmp = t_0;
	} else if ((V * l) <= 1e+226) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0 * sqrt(((1.0 / l) / (V / 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 = sqrt((a / v)) * (c0 / sqrt(l))
    if ((v * l) <= (-5d+143)) then
        tmp = t_0
    else if ((v * l) <= (-1d-139)) then
        tmp = 1.0d0 / (sqrt(((v * l) / a)) / c0)
    else if ((v * l) <= 2d-318) then
        tmp = t_0
    else if ((v * l) <= 1d+226) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = c0 * sqrt(((1.0d0 / l) / (v / 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 = Math.sqrt((A / V)) * (c0 / Math.sqrt(l));
	double tmp;
	if ((V * l) <= -5e+143) {
		tmp = t_0;
	} else if ((V * l) <= -1e-139) {
		tmp = 1.0 / (Math.sqrt(((V * l) / A)) / c0);
	} else if ((V * l) <= 2e-318) {
		tmp = t_0;
	} else if ((V * l) <= 1e+226) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0 * Math.sqrt(((1.0 / l) / (V / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt((A / V)) * (c0 / math.sqrt(l))
	tmp = 0
	if (V * l) <= -5e+143:
		tmp = t_0
	elif (V * l) <= -1e-139:
		tmp = 1.0 / (math.sqrt(((V * l) / A)) / c0)
	elif (V * l) <= 2e-318:
		tmp = t_0
	elif (V * l) <= 1e+226:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0 * math.sqrt(((1.0 / l) / (V / A)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(sqrt(Float64(A / V)) * Float64(c0 / sqrt(l)))
	tmp = 0.0
	if (Float64(V * l) <= -5e+143)
		tmp = t_0;
	elseif (Float64(V * l) <= -1e-139)
		tmp = Float64(1.0 / Float64(sqrt(Float64(Float64(V * l) / A)) / c0));
	elseif (Float64(V * l) <= 2e-318)
		tmp = t_0;
	elseif (Float64(V * l) <= 1e+226)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(1.0 / l) / Float64(V / 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 = sqrt((A / V)) * (c0 / sqrt(l));
	tmp = 0.0;
	if ((V * l) <= -5e+143)
		tmp = t_0;
	elseif ((V * l) <= -1e-139)
		tmp = 1.0 / (sqrt(((V * l) / A)) / c0);
	elseif ((V * l) <= 2e-318)
		tmp = t_0;
	elseif ((V * l) <= 1e+226)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0 * sqrt(((1.0 / l) / (V / 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[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] * N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -5e+143], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -1e-139], N[(1.0 / N[(N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision] / c0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 2e-318], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], 1e+226], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -5.00000000000000012e143 or -1.00000000000000003e-139 < (*.f64 V l) < 2.0000024e-318

   1. Initial program 68.2%

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

      1. associate-/r* 69.6%

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

      2. sqrt-div 40.2%

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

      3. associate-*r/ 40.3%

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

   4. Applied egg-rr 40.3%

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

      1. *-commutative 40.3%

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

      2. associate-/l* 40.3%

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

   6. Simplified 40.3%

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

   if -5.00000000000000012e143 < (*.f64 V l) < -1.00000000000000003e-139

   1. Initial program 95.6%

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

      1. associate-/r* 87.1%

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

      2. clear-num 87.1%

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

      3. sqrt-div 87.1%

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

      4. metadata-eval 87.1%

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

      5. div-inv 86.0%

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

      6. clear-num 86.0%

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

   4. Applied egg-rr 86.0%

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

      1. *-commutative 86.0%

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

      2. associate-*l/ 97.2%

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

      3. associate-/l* 83.7%

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

   6. Simplified 83.7%

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

      1. un-div-inv 83.7%

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

      2. clear-num 83.8%

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

      3. associate-*r/ 97.4%

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

   8. Applied egg-rr 97.4%

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

   if 2.0000024e-318 < (*.f64 V l) < 9.99999999999999961e225

   1. Initial program 82.3%

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

      1. sqrt-div 99.4%

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

      2. div-inv 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. associate-*r/ 99.4%

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

      2. *-rgt-identity 99.4%

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

   6. Simplified 99.4%

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

   if 9.99999999999999961e225 < (*.f64 V l)

   1. Initial program 52.3%

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

      1. *-un-lft-identity 52.3%

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

      2. times-frac 79.7%

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

   4. Applied egg-rr 79.7%

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

      1. associate-*r/ 79.7%

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

      2. div-inv 79.7%

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

      3. associate-*l/ 79.6%

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

      4. *-un-lft-identity 79.6%

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

      5. clear-num 79.7%

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

      6. associate-*l/ 79.8%

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

      7. *-un-lft-identity 79.8%

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

   6. Applied egg-rr 79.8%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+143}:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-139}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\frac{V \cdot \ell}{A}}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-318}:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+226}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{1}{\ell}}{\frac{V}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{V}}\\ \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+143}:\\ \;\;\;\;\frac{c0 \cdot t\_0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-139}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\frac{V \cdot \ell}{A}}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-318}:\\ \;\;\;\;t\_0 \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+226}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{1}{\ell}}{\frac{V}{A}}}\\ \end{array} \end{array} \]

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 (sqrt (/ A V))))
   (if (<= (* V l) -5e+143)
     (/ (* c0 t_0) (sqrt l))
     (if (<= (* V l) -1e-139)
       (/ 1.0 (/ (sqrt (/ (* V l) A)) c0))
       (if (<= (* V l) 2e-318)
         (* t_0 (/ c0 (sqrt l)))
         (if (<= (* V l) 1e+226)
           (* c0 (/ (sqrt A) (sqrt (* V l))))
           (* c0 (sqrt (/ (/ 1.0 l) (/ V A))))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt((A / V));
	double tmp;
	if ((V * l) <= -5e+143) {
		tmp = (c0 * t_0) / sqrt(l);
	} else if ((V * l) <= -1e-139) {
		tmp = 1.0 / (sqrt(((V * l) / A)) / c0);
	} else if ((V * l) <= 2e-318) {
		tmp = t_0 * (c0 / sqrt(l));
	} else if ((V * l) <= 1e+226) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0 * sqrt(((1.0 / l) / (V / 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 = sqrt((a / v))
    if ((v * l) <= (-5d+143)) then
        tmp = (c0 * t_0) / sqrt(l)
    else if ((v * l) <= (-1d-139)) then
        tmp = 1.0d0 / (sqrt(((v * l) / a)) / c0)
    else if ((v * l) <= 2d-318) then
        tmp = t_0 * (c0 / sqrt(l))
    else if ((v * l) <= 1d+226) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = c0 * sqrt(((1.0d0 / l) / (v / 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 = Math.sqrt((A / V));
	double tmp;
	if ((V * l) <= -5e+143) {
		tmp = (c0 * t_0) / Math.sqrt(l);
	} else if ((V * l) <= -1e-139) {
		tmp = 1.0 / (Math.sqrt(((V * l) / A)) / c0);
	} else if ((V * l) <= 2e-318) {
		tmp = t_0 * (c0 / Math.sqrt(l));
	} else if ((V * l) <= 1e+226) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0 * Math.sqrt(((1.0 / l) / (V / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt((A / V))
	tmp = 0
	if (V * l) <= -5e+143:
		tmp = (c0 * t_0) / math.sqrt(l)
	elif (V * l) <= -1e-139:
		tmp = 1.0 / (math.sqrt(((V * l) / A)) / c0)
	elif (V * l) <= 2e-318:
		tmp = t_0 * (c0 / math.sqrt(l))
	elif (V * l) <= 1e+226:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0 * math.sqrt(((1.0 / l) / (V / A)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = sqrt(Float64(A / V))
	tmp = 0.0
	if (Float64(V * l) <= -5e+143)
		tmp = Float64(Float64(c0 * t_0) / sqrt(l));
	elseif (Float64(V * l) <= -1e-139)
		tmp = Float64(1.0 / Float64(sqrt(Float64(Float64(V * l) / A)) / c0));
	elseif (Float64(V * l) <= 2e-318)
		tmp = Float64(t_0 * Float64(c0 / sqrt(l)));
	elseif (Float64(V * l) <= 1e+226)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(1.0 / l) / Float64(V / 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 = sqrt((A / V));
	tmp = 0.0;
	if ((V * l) <= -5e+143)
		tmp = (c0 * t_0) / sqrt(l);
	elseif ((V * l) <= -1e-139)
		tmp = 1.0 / (sqrt(((V * l) / A)) / c0);
	elseif ((V * l) <= 2e-318)
		tmp = t_0 * (c0 / sqrt(l));
	elseif ((V * l) <= 1e+226)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0 * sqrt(((1.0 / l) / (V / 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[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -5e+143], N[(N[(c0 * t$95$0), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-139], N[(1.0 / N[(N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision] / c0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 2e-318], N[(t$95$0 * N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+226], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -5.00000000000000012e143

   1. Initial program 70.4%

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

      1. *-commutative 70.4%

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

      2. associate-/r* 65.7%

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

      3. sqrt-div 44.0%

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

      4. associate-*l/ 44.2%

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

   4. Applied egg-rr 44.2%

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

   if -5.00000000000000012e143 < (*.f64 V l) < -1.00000000000000003e-139

   1. Initial program 95.6%

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

      1. associate-/r* 87.1%

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

      2. clear-num 87.1%

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

      3. sqrt-div 87.1%

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

      4. metadata-eval 87.1%

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

      5. div-inv 86.0%

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

      6. clear-num 86.0%

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

   4. Applied egg-rr 86.0%

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

      1. *-commutative 86.0%

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

      2. associate-*l/ 97.2%

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

      3. associate-/l* 83.7%

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

   6. Simplified 83.7%

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

      1. un-div-inv 83.7%

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

      2. clear-num 83.8%

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

      3. associate-*r/ 97.4%

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

   8. Applied egg-rr 97.4%

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

   if -1.00000000000000003e-139 < (*.f64 V l) < 2.0000024e-318

   1. Initial program 66.6%

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

      1. associate-/r* 72.5%

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

      2. sqrt-div 37.5%

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

      3. associate-*r/ 37.5%

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

   4. Applied egg-rr 37.5%

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

      1. *-commutative 37.5%

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

      2. associate-/l* 37.5%

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

   6. Simplified 37.5%

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

   if 2.0000024e-318 < (*.f64 V l) < 9.99999999999999961e225

   1. Initial program 82.3%

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

      1. sqrt-div 99.4%

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

      2. div-inv 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. associate-*r/ 99.4%

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

      2. *-rgt-identity 99.4%

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

   6. Simplified 99.4%

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

   if 9.99999999999999961e225 < (*.f64 V l)

   1. Initial program 52.3%

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

      1. *-un-lft-identity 52.3%

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

      2. times-frac 79.7%

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

   4. Applied egg-rr 79.7%

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

      1. associate-*r/ 79.7%

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

      2. div-inv 79.7%

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

      3. associate-*l/ 79.6%

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

      4. *-un-lft-identity 79.6%

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

      5. clear-num 79.7%

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

      6. associate-*l/ 79.8%

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

      7. *-un-lft-identity 79.8%

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

   6. Applied egg-rr 79.8%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+143}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-139}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\frac{V \cdot \ell}{A}}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-318}:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+226}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{1}{\ell}}{\frac{V}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{V}}\\ \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+144}:\\ \;\;\;\;\left(c0 \cdot {\ell}^{-0.5}\right) \cdot t\_0\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-139}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\frac{V \cdot \ell}{A}}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-318}:\\ \;\;\;\;t\_0 \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+226}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{1}{\ell}}{\frac{V}{A}}}\\ \end{array} \end{array} \]

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 (sqrt (/ A V))))
   (if (<= (* V l) -5e+144)
     (* (* c0 (pow l -0.5)) t_0)
     (if (<= (* V l) -1e-139)
       (/ 1.0 (/ (sqrt (/ (* V l) A)) c0))
       (if (<= (* V l) 2e-318)
         (* t_0 (/ c0 (sqrt l)))
         (if (<= (* V l) 1e+226)
           (* c0 (/ (sqrt A) (sqrt (* V l))))
           (* c0 (sqrt (/ (/ 1.0 l) (/ V A))))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt((A / V));
	double tmp;
	if ((V * l) <= -5e+144) {
		tmp = (c0 * pow(l, -0.5)) * t_0;
	} else if ((V * l) <= -1e-139) {
		tmp = 1.0 / (sqrt(((V * l) / A)) / c0);
	} else if ((V * l) <= 2e-318) {
		tmp = t_0 * (c0 / sqrt(l));
	} else if ((V * l) <= 1e+226) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0 * sqrt(((1.0 / l) / (V / 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 = sqrt((a / v))
    if ((v * l) <= (-5d+144)) then
        tmp = (c0 * (l ** (-0.5d0))) * t_0
    else if ((v * l) <= (-1d-139)) then
        tmp = 1.0d0 / (sqrt(((v * l) / a)) / c0)
    else if ((v * l) <= 2d-318) then
        tmp = t_0 * (c0 / sqrt(l))
    else if ((v * l) <= 1d+226) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = c0 * sqrt(((1.0d0 / l) / (v / 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 = Math.sqrt((A / V));
	double tmp;
	if ((V * l) <= -5e+144) {
		tmp = (c0 * Math.pow(l, -0.5)) * t_0;
	} else if ((V * l) <= -1e-139) {
		tmp = 1.0 / (Math.sqrt(((V * l) / A)) / c0);
	} else if ((V * l) <= 2e-318) {
		tmp = t_0 * (c0 / Math.sqrt(l));
	} else if ((V * l) <= 1e+226) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0 * Math.sqrt(((1.0 / l) / (V / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt((A / V))
	tmp = 0
	if (V * l) <= -5e+144:
		tmp = (c0 * math.pow(l, -0.5)) * t_0
	elif (V * l) <= -1e-139:
		tmp = 1.0 / (math.sqrt(((V * l) / A)) / c0)
	elif (V * l) <= 2e-318:
		tmp = t_0 * (c0 / math.sqrt(l))
	elif (V * l) <= 1e+226:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0 * math.sqrt(((1.0 / l) / (V / A)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = sqrt(Float64(A / V))
	tmp = 0.0
	if (Float64(V * l) <= -5e+144)
		tmp = Float64(Float64(c0 * (l ^ -0.5)) * t_0);
	elseif (Float64(V * l) <= -1e-139)
		tmp = Float64(1.0 / Float64(sqrt(Float64(Float64(V * l) / A)) / c0));
	elseif (Float64(V * l) <= 2e-318)
		tmp = Float64(t_0 * Float64(c0 / sqrt(l)));
	elseif (Float64(V * l) <= 1e+226)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(1.0 / l) / Float64(V / 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 = sqrt((A / V));
	tmp = 0.0;
	if ((V * l) <= -5e+144)
		tmp = (c0 * (l ^ -0.5)) * t_0;
	elseif ((V * l) <= -1e-139)
		tmp = 1.0 / (sqrt(((V * l) / A)) / c0);
	elseif ((V * l) <= 2e-318)
		tmp = t_0 * (c0 / sqrt(l));
	elseif ((V * l) <= 1e+226)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0 * sqrt(((1.0 / l) / (V / 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[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -5e+144], N[(N[(c0 * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-139], N[(1.0 / N[(N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision] / c0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 2e-318], N[(t$95$0 * N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+226], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -4.9999999999999999e144

   1. Initial program 69.6%

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

      1. *-commutative 69.6%

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

      2. associate-/r* 67.3%

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

      3. sqrt-div 45.1%

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

      4. associate-*l/ 45.3%

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

   4. Applied egg-rr 45.3%

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

      1. div-inv 45.2%

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

      2. *-commutative 45.2%

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

      3. pow1/2 45.2%

         \[\leadsto \left(c0 \cdot \sqrt{\frac{A}{V}}\right) \cdot \frac{1}{\color{blue}{{\ell}^{0.5}}} \]

      4. pow-flip 45.3%

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

      5. metadata-eval 45.3%

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

   6. Applied egg-rr 45.3%

      \[\leadsto \color{blue}{\left(c0 \cdot \sqrt{\frac{A}{V}}\right) \cdot {\ell}^{-0.5}} \]
   7. Step-by-step derivation

      1. *-commutative 45.3%

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

      2. associate-*l* 45.3%

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

   8. Simplified 45.3%

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

   if -4.9999999999999999e144 < (*.f64 V l) < -1.00000000000000003e-139

   1. Initial program 95.7%

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

      1. associate-/r* 85.4%

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

      2. clear-num 85.4%

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

      3. sqrt-div 85.4%

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

      4. metadata-eval 85.4%

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

      5. div-inv 84.3%

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

      6. clear-num 85.3%

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

   4. Applied egg-rr 85.3%

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

      1. *-commutative 85.3%

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

      2. associate-*l/ 97.2%

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

      3. associate-/l* 84.1%

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

   6. Simplified 84.1%

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

      1. un-div-inv 84.1%

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

      2. clear-num 84.1%

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

      3. associate-*r/ 97.4%

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

   8. Applied egg-rr 97.4%

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

   if -1.00000000000000003e-139 < (*.f64 V l) < 2.0000024e-318

   1. Initial program 66.6%

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

      1. associate-/r* 72.5%

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

      2. sqrt-div 37.5%

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

      3. associate-*r/ 37.5%

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

   4. Applied egg-rr 37.5%

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

      1. *-commutative 37.5%

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

      2. associate-/l* 37.5%

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

   6. Simplified 37.5%

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

   if 2.0000024e-318 < (*.f64 V l) < 9.99999999999999961e225

   1. Initial program 82.3%

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

      1. sqrt-div 99.4%

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

      2. div-inv 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. associate-*r/ 99.4%

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

      2. *-rgt-identity 99.4%

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

   6. Simplified 99.4%

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

   if 9.99999999999999961e225 < (*.f64 V l)

   1. Initial program 52.3%

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

      1. *-un-lft-identity 52.3%

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

      2. times-frac 79.7%

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

   4. Applied egg-rr 79.7%

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

      1. associate-*r/ 79.7%

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

      2. div-inv 79.7%

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

      3. associate-*l/ 79.6%

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

      4. *-un-lft-identity 79.6%

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

      5. clear-num 79.7%

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

      6. associate-*l/ 79.8%

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

      7. *-un-lft-identity 79.8%

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

   6. Applied egg-rr 79.8%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+144}:\\ \;\;\;\;\left(c0 \cdot {\ell}^{-0.5}\right) \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-139}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\frac{V \cdot \ell}{A}}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-318}:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+226}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{1}{\ell}}{\frac{V}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 87.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{-313}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-322}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+226}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{1}{\ell}}{\frac{V}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -2e-313)
   (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
   (if (<= (* V l) 5e-322)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= (* V l) 1e+226)
       (* c0 (/ (sqrt A) (sqrt (* V l))))
       (* c0 (sqrt (/ (/ 1.0 l) (/ V A))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e-313) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 5e-322) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if ((V * l) <= 1e+226) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0 * sqrt(((1.0 / l) / (V / A)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-2d-313)) then
        tmp = c0 * (sqrt(-a) / sqrt((v * -l)))
    else if ((v * l) <= 5d-322) then
        tmp = c0 * sqrt(((a / v) / l))
    else if ((v * l) <= 1d+226) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = c0 * sqrt(((1.0d0 / l) / (v / a)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e-313) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 5e-322) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if ((V * l) <= 1e+226) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0 * Math.sqrt(((1.0 / l) / (V / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -2e-313:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 5e-322:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif (V * l) <= 1e+226:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0 * math.sqrt(((1.0 / l) / (V / A)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -2e-313)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 5e-322)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (Float64(V * l) <= 1e+226)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(1.0 / l) / Float64(V / A))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -2e-313)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 5e-322)
		tmp = c0 * sqrt(((A / V) / l));
	elseif ((V * l) <= 1e+226)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0 * sqrt(((1.0 / l) / (V / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -2e-313], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e-322], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+226], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -1.99999999998e-313

   1. Initial program 85.1%

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

      1. frac-2neg 85.1%

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

      2. sqrt-div 94.2%

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

      3. distribute-rgt-neg-in 94.2%

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

   4. Applied egg-rr 94.2%

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

      1. distribute-rgt-neg-out 94.2%

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

      2. *-commutative 94.2%

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

      3. distribute-rgt-neg-in 94.2%

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

   6. Simplified 94.2%

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

   if -1.99999999998e-313 < (*.f64 V l) < 4.99006e-322

   1. Initial program 48.9%

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

      1. associate-/r* 60.1%

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

   3. Simplified 60.1%

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

   if 4.99006e-322 < (*.f64 V l) < 9.99999999999999961e225

   1. Initial program 81.4%

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

      1. sqrt-div 98.9%

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

      2. div-inv 98.7%

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

   4. Applied egg-rr 98.7%

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

      1. associate-*r/ 98.9%

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

      2. *-rgt-identity 98.9%

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

   6. Simplified 98.9%

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

   if 9.99999999999999961e225 < (*.f64 V l)

   1. Initial program 52.3%

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

      1. *-un-lft-identity 52.3%

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

      2. times-frac 79.7%

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

   4. Applied egg-rr 79.7%

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

      1. associate-*r/ 79.7%

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

      2. div-inv 79.7%

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

      3. associate-*l/ 79.6%

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

      4. *-un-lft-identity 79.6%

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

      5. clear-num 79.7%

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

      6. associate-*l/ 79.8%

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

      7. *-un-lft-identity 79.8%

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

   6. Applied egg-rr 79.8%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{-313}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-322}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+226}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{1}{\ell}}{\frac{V}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 5 \cdot 10^{-322}:\\ \;\;\;\;c0 \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+226}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{1}{\ell}}{\frac{V}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) 5e-322)
   (* c0 (pow (/ (* V l) A) -0.5))
   (if (<= (* V l) 1e+226)
     (* c0 (/ (sqrt A) (sqrt (* V l))))
     (* c0 (sqrt (/ (/ 1.0 l) (/ V A)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= 5e-322) {
		tmp = c0 * pow(((V * l) / A), -0.5);
	} else if ((V * l) <= 1e+226) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0 * sqrt(((1.0 / l) / (V / A)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= 5d-322) then
        tmp = c0 * (((v * l) / a) ** (-0.5d0))
    else if ((v * l) <= 1d+226) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = c0 * sqrt(((1.0d0 / l) / (v / a)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= 5e-322) {
		tmp = c0 * Math.pow(((V * l) / A), -0.5);
	} else if ((V * l) <= 1e+226) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0 * Math.sqrt(((1.0 / l) / (V / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= 5e-322:
		tmp = c0 * math.pow(((V * l) / A), -0.5)
	elif (V * l) <= 1e+226:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0 * math.sqrt(((1.0 / l) / (V / A)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= 5e-322)
		tmp = Float64(c0 * (Float64(Float64(V * l) / A) ^ -0.5));
	elseif (Float64(V * l) <= 1e+226)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(1.0 / l) / Float64(V / A))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= 5e-322)
		tmp = c0 * (((V * l) / A) ^ -0.5);
	elseif ((V * l) <= 1e+226)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0 * sqrt(((1.0 / l) / (V / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], 5e-322], N[(c0 * N[Power[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+226], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < 4.99006e-322

   1. Initial program 77.9%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. clear-num 75.7%

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

      3. sqrt-div 75.6%

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

      4. metadata-eval 75.6%

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

      5. div-inv 75.2%

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

      6. clear-num 75.6%

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

   4. Applied egg-rr 75.6%

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

      1. *-commutative 75.6%

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

      2. associate-*l/ 78.1%

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

      3. associate-/l* 75.8%

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

   6. Simplified 75.8%

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

      1. pow1/2 75.8%

         \[\leadsto c0 \cdot \frac{1}{\color{blue}{{\left(V \cdot \frac{\ell}{A}\right)}^{0.5}}} \]

      2. pow-flip 75.8%

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

      3. associate-*r/ 78.2%

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

      4. metadata-eval 78.2%

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

   8. Applied egg-rr 78.2%

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

   if 4.99006e-322 < (*.f64 V l) < 9.99999999999999961e225

   1. Initial program 81.4%

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

      1. sqrt-div 98.9%

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

      2. div-inv 98.7%

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

   4. Applied egg-rr 98.7%

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

      1. associate-*r/ 98.9%

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

      2. *-rgt-identity 98.9%

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

   6. Simplified 98.9%

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

   if 9.99999999999999961e225 < (*.f64 V l)

   1. Initial program 52.3%

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

      1. *-un-lft-identity 52.3%

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

      2. times-frac 79.7%

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

   4. Applied egg-rr 79.7%

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

      1. associate-*r/ 79.7%

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

      2. div-inv 79.7%

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

      3. associate-*l/ 79.6%

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

      4. *-un-lft-identity 79.6%

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

      5. clear-num 79.7%

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

      6. associate-*l/ 79.8%

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

      7. *-un-lft-identity 79.8%

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

   6. Applied egg-rr 79.8%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 5 \cdot 10^{-322}:\\ \;\;\;\;c0 \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+226}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{1}{\ell}}{\frac{V}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.2% accurate, 0.5× speedup

Math

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

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 (<= t_0 0.0) (* 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) {
		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) 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) {
		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:
		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)
		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)
		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[LessEqual[t$95$0, 0.0], 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

   1. Initial program 70.9%

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

      1. associate-/r* 69.5%

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

   3. Simplified 69.5%

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

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

   1. Initial program 84.3%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 10^{-305}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 (<= t_0 1e-305) (/ c0 (sqrt (* V (/ l A)))) 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 <= 1e-305) {
		tmp = c0 / sqrt((V * (l / A)));
	} 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 <= 1d-305) then
        tmp = c0 / sqrt((v * (l / a)))
    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 <= 1e-305) {
		tmp = c0 / Math.sqrt((V * (l / A)));
	} 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 <= 1e-305:
		tmp = c0 / math.sqrt((V * (l / A)))
	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 <= 1e-305)
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	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 <= 1e-305)
		tmp = c0 / sqrt((V * (l / A)));
	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[LessEqual[t$95$0, 1e-305], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 9.99999999999999996e-306

   1. Initial program 71.0%

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

      1. associate-/r* 69.7%

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

      2. clear-num 69.7%

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

      3. sqrt-div 70.0%

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

      4. metadata-eval 70.0%

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

      5. div-inv 70.0%

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

      6. clear-num 70.3%

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

   4. Applied egg-rr 70.3%

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

      1. *-commutative 70.3%

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

      2. associate-*l/ 71.2%

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

      3. associate-/l* 73.2%

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

   6. Simplified 73.2%

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

      1. un-div-inv 73.3%

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

      2. associate-*r/ 71.3%

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

   8. Applied egg-rr 71.3%

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

      1. associate-/l* 73.3%

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

   10. Simplified 73.3%

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

   if 9.99999999999999996e-306 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 84.2%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 10^{-305}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 83.8% accurate, 0.5× speedup

Math

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

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= A -1e-310)
   (* c0 (* (sqrt (/ A V)) (sqrt (/ 1.0 l))))
   (* 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 tmp;
	if (A <= -1e-310) {
		tmp = c0 * (sqrt((A / V)) * sqrt((1.0 / l)));
	} 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) :: tmp
    if (a <= (-1d-310)) then
        tmp = c0 * (sqrt((a / v)) * sqrt((1.0d0 / l)))
    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 tmp;
	if (A <= -1e-310) {
		tmp = c0 * (Math.sqrt((A / V)) * Math.sqrt((1.0 / l)));
	} 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):
	tmp = 0
	if A <= -1e-310:
		tmp = c0 * (math.sqrt((A / V)) * math.sqrt((1.0 / l)))
	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)
	tmp = 0.0
	if (A <= -1e-310)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) * sqrt(Float64(1.0 / l))));
	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)
	tmp = 0.0;
	if (A <= -1e-310)
		tmp = c0 * (sqrt((A / V)) * sqrt((1.0 / l)));
	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_] := If[LessEqual[A, -1e-310], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -9.999999999999969e-311

   1. Initial program 79.8%

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

      1. pow1/2 79.8%

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

      2. associate-/r* 76.5%

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

      3. div-inv 76.5%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{A}{V} \cdot \frac{1}{\ell}\right)}}^{0.5} \]

      4. unpow-prod-down 39.9%

         \[\leadsto c0 \cdot \color{blue}{\left({\left(\frac{A}{V}\right)}^{0.5} \cdot {\left(\frac{1}{\ell}\right)}^{0.5}\right)} \]

      5. pow1/2 39.9%

         \[\leadsto c0 \cdot \left(\color{blue}{\sqrt{\frac{A}{V}}} \cdot {\left(\frac{1}{\ell}\right)}^{0.5}\right) \]

   4. Applied egg-rr 39.9%

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{\frac{A}{V}} \cdot {\left(\frac{1}{\ell}\right)}^{0.5}\right)} \]
   5. Step-by-step derivation

      1. unpow1/2 39.9%

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

   6. Simplified 39.9%

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

   if -9.999999999999969e-311 < A

   1. Initial program 72.7%

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

      1. sqrt-div 83.9%

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

      2. div-inv 83.9%

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

   4. Applied egg-rr 83.9%

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

      1. associate-*r/ 83.9%

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

      2. *-rgt-identity 83.9%

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

   6. Simplified 83.9%

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

4. Final simplification 61.9%

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

Alternative 13: 76.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{1}{\ell}}{\frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (/ A (* V l)) 2e-310)
   (* c0 (sqrt (/ (/ 1.0 l) (/ V A))))
   (* c0 (pow (/ (* V l) A) -0.5))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((A / (V * l)) <= 2e-310) {
		tmp = c0 * sqrt(((1.0 / l) / (V / A)));
	} else {
		tmp = c0 * pow(((V * l) / A), -0.5);
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((a / (v * l)) <= 2d-310) then
        tmp = c0 * sqrt(((1.0d0 / l) / (v / a)))
    else
        tmp = c0 * (((v * l) / a) ** (-0.5d0))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((A / (V * l)) <= 2e-310) {
		tmp = c0 * Math.sqrt(((1.0 / l) / (V / A)));
	} else {
		tmp = c0 * Math.pow(((V * l) / A), -0.5);
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (A / (V * l)) <= 2e-310:
		tmp = c0 * math.sqrt(((1.0 / l) / (V / A)))
	else:
		tmp = c0 * math.pow(((V * l) / A), -0.5)
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(A / Float64(V * l)) <= 2e-310)
		tmp = Float64(c0 * sqrt(Float64(Float64(1.0 / l) / Float64(V / A))));
	else
		tmp = Float64(c0 * (Float64(Float64(V * l) / A) ^ -0.5));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((A / (V * l)) <= 2e-310)
		tmp = c0 * sqrt(((1.0 / l) / (V / A)));
	else
		tmp = c0 * (((V * l) / A) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision], 2e-310], N[(c0 * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[Power[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 1.999999999999994e-310

   1. Initial program 38.9%

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

      1. *-un-lft-identity 38.9%

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

      2. times-frac 55.8%

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

   4. Applied egg-rr 55.8%

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

      1. associate-*r/ 54.1%

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

      2. div-inv 54.1%

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

      3. associate-*l/ 54.1%

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

      4. *-un-lft-identity 54.1%

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

      5. clear-num 54.1%

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

      6. associate-*l/ 54.1%

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

      7. *-un-lft-identity 54.1%

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

   6. Applied egg-rr 54.1%

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

   if 1.999999999999994e-310 < (/.f64 A (*.f64 V l))

   1. Initial program 85.5%

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

      1. associate-/r* 77.4%

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

      2. clear-num 77.3%

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

      3. sqrt-div 77.7%

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

      4. metadata-eval 77.7%

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

      5. div-inv 76.8%

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

      6. clear-num 77.0%

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

   4. Applied egg-rr 77.0%

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

      1. *-commutative 77.0%

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

      2. associate-*l/ 87.0%

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

      3. associate-/l* 82.6%

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

   6. Simplified 82.6%

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

      1. pow1/2 82.6%

         \[\leadsto c0 \cdot \frac{1}{\color{blue}{{\left(V \cdot \frac{\ell}{A}\right)}^{0.5}}} \]

      2. pow-flip 82.7%

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

      3. associate-*r/ 87.1%

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

      4. metadata-eval 87.1%

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

   8. Applied egg-rr 87.1%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{1}{\ell}}{\frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 77.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (/ A (* V l)) 2e-310)
   (* c0 (sqrt (/ (/ A V) l)))
   (* c0 (pow (/ (* V l) A) -0.5))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((A / (V * l)) <= 2e-310) {
		tmp = c0 * sqrt(((A / V) / l));
	} else {
		tmp = c0 * pow(((V * l) / A), -0.5);
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((a / (v * l)) <= 2d-310) then
        tmp = c0 * sqrt(((a / v) / l))
    else
        tmp = c0 * (((v * l) / a) ** (-0.5d0))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((A / (V * l)) <= 2e-310) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else {
		tmp = c0 * Math.pow(((V * l) / A), -0.5);
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (A / (V * l)) <= 2e-310:
		tmp = c0 * math.sqrt(((A / V) / l))
	else:
		tmp = c0 * math.pow(((V * l) / A), -0.5)
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(A / Float64(V * l)) <= 2e-310)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	else
		tmp = Float64(c0 * (Float64(Float64(V * l) / A) ^ -0.5));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((A / (V * l)) <= 2e-310)
		tmp = c0 * sqrt(((A / V) / l));
	else
		tmp = c0 * (((V * l) / A) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision], 2e-310], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[Power[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 1.999999999999994e-310

   1. Initial program 38.9%

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

      1. associate-/r* 54.1%

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

   3. Simplified 54.1%

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

   if 1.999999999999994e-310 < (/.f64 A (*.f64 V l))

   1. Initial program 85.5%

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

      1. associate-/r* 77.4%

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

      2. clear-num 77.3%

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

      3. sqrt-div 77.7%

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

      4. metadata-eval 77.7%

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

      5. div-inv 76.8%

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

      6. clear-num 77.0%

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

   4. Applied egg-rr 77.0%

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

      1. *-commutative 77.0%

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

      2. associate-*l/ 87.0%

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

      3. associate-/l* 82.6%

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

   6. Simplified 82.6%

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

      1. pow1/2 82.6%

         \[\leadsto c0 \cdot \frac{1}{\color{blue}{{\left(V \cdot \frac{\ell}{A}\right)}^{0.5}}} \]

      2. pow-flip 82.7%

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

      3. associate-*r/ 87.1%

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

      4. metadata-eval 87.1%

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

   8. Applied egg-rr 87.1%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 77.0% accurate, 0.9× speedup

Math

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

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 (<= (/ A (* V l)) 2e-310)
   (* c0 (sqrt (/ (/ A V) l)))
   (/ 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 ((A / (V * l)) <= 2e-310) {
		tmp = c0 * sqrt(((A / V) / l));
	} 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 ((a / (v * l)) <= 2d-310) then
        tmp = c0 * sqrt(((a / v) / l))
    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 ((A / (V * l)) <= 2e-310) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} 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 (A / (V * l)) <= 2e-310:
		tmp = c0 * math.sqrt(((A / V) / l))
	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 (Float64(A / Float64(V * l)) <= 2e-310)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	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 ((A / (V * l)) <= 2e-310)
		tmp = c0 * sqrt(((A / V) / l));
	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[(A / N[(V * l), $MachinePrecision]), $MachinePrecision], 2e-310], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $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 (/.f64 A (*.f64 V l)) < 1.999999999999994e-310

   1. Initial program 38.9%

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

      1. associate-/r* 54.1%

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

   3. Simplified 54.1%

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

   if 1.999999999999994e-310 < (/.f64 A (*.f64 V l))

   1. Initial program 85.5%

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

      1. associate-/r* 77.4%

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

      2. clear-num 77.3%

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

      3. sqrt-div 77.7%

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

      4. metadata-eval 77.7%

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

      5. div-inv 76.8%

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

      6. clear-num 77.0%

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

   4. Applied egg-rr 77.0%

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

      1. *-commutative 77.0%

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

      2. associate-*l/ 87.0%

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

      3. associate-/l* 82.6%

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

   6. Simplified 82.6%

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

      1. un-div-inv 82.7%

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

      2. associate-*r/ 87.0%

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

   8. Applied egg-rr 87.0%

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

4. Final simplification 80.5%

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

Alternative 16: 73.5% 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 76.2%

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

3. Final simplification 76.2%

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

Reproduce

herbie shell --seed 2024041 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))