Henrywood and Agarwal, Equation (3)

Percentage Accurate: 74.3% → 91.6%

Time: 14.1s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}}\\ \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-302}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+285}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \end{array} \]

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) l)) (sqrt (- V))))))
   (if (<= (* V l) (- INFINITY))
     t_0
     (if (<= (* V l) -1e-302)
       (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
       (if (<= (* V l) 0.0)
         t_0
         (if (<= (* V l) 1e+285)
           (* c0 (/ (sqrt A) (sqrt (* V l))))
           (/ 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 / l)) / sqrt(-V));
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = t_0;
	} else if ((V * l) <= -1e-302) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else if ((V * l) <= 1e+285) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0 / sqrt((l * (V / A)));
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * (Math.sqrt((-A / l)) / Math.sqrt(-V));
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else if ((V * l) <= -1e-302) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else if ((V * l) <= 1e+285) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} 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 / l)) / math.sqrt(-V))
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = t_0
	elif (V * l) <= -1e-302:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 0.0:
		tmp = t_0
	elif (V * l) <= 1e+285:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	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 * Float64(sqrt(Float64(Float64(-A) / l)) / sqrt(Float64(-V))))
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = t_0;
	elseif (Float64(V * l) <= -1e-302)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 0.0)
		tmp = t_0;
	elseif (Float64(V * l) <= 1e+285)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	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 / l)) / sqrt(-V));
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = t_0;
	elseif ((V * l) <= -1e-302)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 0.0)
		tmp = t_0;
	elseif ((V * l) <= 1e+285)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	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[(N[Sqrt[N[((-A) / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -1e-302], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], 1e+285], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -inf.0 or -9.9999999999999996e-303 < (*.f64 V l) < -0.0

   1. Initial program 47.9%

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

      1. add-sqr-sqrt 47.9%

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

      2. pow2 47.9%

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

      3. pow1/2 47.9%

         \[\leadsto c0 \cdot {\left(\sqrt{\color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{0.5}}}\right)}^{2} \]

      4. sqrt-pow1 47.9%

         \[\leadsto c0 \cdot {\color{blue}{\left({\left(\frac{A}{V \cdot \ell}\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} \]

      5. metadata-eval 47.9%

         \[\leadsto c0 \cdot {\left({\left(\frac{A}{V \cdot \ell}\right)}^{\color{blue}{0.25}}\right)}^{2} \]

   4. Applied egg-rr 47.9%

      \[\leadsto c0 \cdot \color{blue}{{\left({\left(\frac{A}{V \cdot \ell}\right)}^{0.25}\right)}^{2}} \]

   5. Taylor expanded in c0 around 0 47.9%

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

      1. *-commutative 47.9%

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

      2. associate-/r* 68.6%

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

   7. Simplified 68.6%

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

      1. frac-2neg 68.6%

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

      2. sqrt-div 33.8%

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

      3. distribute-neg-frac 33.8%

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

   9. Applied egg-rr 33.8%

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

   if -inf.0 < (*.f64 V l) < -9.9999999999999996e-303

   1. Initial program 85.7%

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

      1. frac-2neg 85.7%

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

      2. sqrt-div 99.5%

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

      3. distribute-rgt-neg-in 99.5%

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

   4. Applied egg-rr 99.5%

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

      1. distribute-rgt-neg-out 99.5%

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

      2. *-commutative 99.5%

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

      3. distribute-rgt-neg-in 99.5%

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

   6. Simplified 99.5%

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

   if -0.0 < (*.f64 V l) < 9.9999999999999998e284

   1. Initial program 87.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. associate-*r/ 97.6%

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

   4. Applied egg-rr 97.6%

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

      1. *-commutative 97.6%

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

      2. associate-/l* 95.2%

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

      3. associate-/r/ 99.4%

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

   6. Simplified 99.4%

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

   if 9.9999999999999998e284 < (*.f64 V l)

   1. Initial program 44.3%

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

      1. add-sqr-sqrt 44.3%

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

      2. pow2 44.3%

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

      3. pow1/2 44.3%

         \[\leadsto c0 \cdot {\left(\sqrt{\color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{0.5}}}\right)}^{2} \]

      4. sqrt-pow1 44.3%

         \[\leadsto c0 \cdot {\color{blue}{\left({\left(\frac{A}{V \cdot \ell}\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} \]

      5. metadata-eval 44.3%

         \[\leadsto c0 \cdot {\left({\left(\frac{A}{V \cdot \ell}\right)}^{\color{blue}{0.25}}\right)}^{2} \]

   4. Applied egg-rr 44.3%

      \[\leadsto c0 \cdot \color{blue}{{\left({\left(\frac{A}{V \cdot \ell}\right)}^{0.25}\right)}^{2}} \]
   5. Step-by-step derivation

      1. pow-pow 44.3%

         \[\leadsto c0 \cdot \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{\left(0.25 \cdot 2\right)}} \]

      2. metadata-eval 44.3%

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

      3. pow1/2 44.3%

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

      4. clear-num 44.3%

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

      5. associate-*r/ 83.5%

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

      6. sqrt-div 83.4%

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

      7. metadata-eval 83.4%

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

      8. un-div-inv 83.8%

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

      9. clear-num 83.7%

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

      10. un-div-inv 83.5%

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

   6. Applied egg-rr 83.5%

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

      1. associate-/r/ 83.7%

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

   8. Simplified 83.7%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-302}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+285}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.9% accurate, 0.3× speedup

Math

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

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 -5e-310)
   (/ (/ (sqrt (- A)) (sqrt (- V))) (/ (sqrt l) c0))
   (* 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 <= -5e-310) {
		tmp = (sqrt(-A) / sqrt(-V)) / (sqrt(l) / c0);
	} 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 <= (-5d-310)) then
        tmp = (sqrt(-a) / sqrt(-v)) / (sqrt(l) / c0)
    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 <= -5e-310) {
		tmp = (Math.sqrt(-A) / Math.sqrt(-V)) / (Math.sqrt(l) / c0);
	} 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 <= -5e-310:
		tmp = (math.sqrt(-A) / math.sqrt(-V)) / (math.sqrt(l) / c0)
	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 <= -5e-310)
		tmp = Float64(Float64(sqrt(Float64(-A)) / sqrt(Float64(-V))) / Float64(sqrt(l) / c0));
	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 <= -5e-310)
		tmp = (sqrt(-A) / sqrt(-V)) / (sqrt(l) / c0);
	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, -5e-310], N[(N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] / c0), $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 < -4.999999999999985e-310

   1. Initial program 78.1%

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

      1. add-sqr-sqrt 77.7%

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

      2. pow2 77.7%

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

      3. pow1/2 77.7%

         \[\leadsto c0 \cdot {\left(\sqrt{\color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{0.5}}}\right)}^{2} \]

      4. sqrt-pow1 77.8%

         \[\leadsto c0 \cdot {\color{blue}{\left({\left(\frac{A}{V \cdot \ell}\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} \]

      5. metadata-eval 77.8%

         \[\leadsto c0 \cdot {\left({\left(\frac{A}{V \cdot \ell}\right)}^{\color{blue}{0.25}}\right)}^{2} \]

   4. Applied egg-rr 77.8%

      \[\leadsto c0 \cdot \color{blue}{{\left({\left(\frac{A}{V \cdot \ell}\right)}^{0.25}\right)}^{2}} \]
   5. Step-by-step derivation

      1. *-commutative 77.8%

         \[\leadsto \color{blue}{{\left({\left(\frac{A}{V \cdot \ell}\right)}^{0.25}\right)}^{2} \cdot c0} \]

      2. pow-pow 78.1%

         \[\leadsto \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{\left(0.25 \cdot 2\right)}} \cdot c0 \]

      3. metadata-eval 78.1%

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

      4. pow1/2 78.1%

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

      5. associate-/r* 76.0%

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

      6. sqrt-div 41.9%

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

      7. associate-/r/ 41.1%

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

   6. Applied egg-rr 41.1%

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

      1. frac-2neg 41.1%

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

      2. sqrt-div 47.5%

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

   8. Applied egg-rr 47.5%

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

   if -4.999999999999985e-310 < A

   1. Initial program 78.6%

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

      1. sqrt-div 88.1%

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

      2. associate-*r/ 86.6%

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

   4. Applied egg-rr 86.6%

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

      1. *-commutative 86.6%

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

      2. associate-/l* 84.7%

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

      3. associate-/r/ 88.1%

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

   6. Simplified 88.1%

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

4. Final simplification 67.5%

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

Alternative 3: 90.0% accurate, 0.3× speedup

Math

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

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 -5e-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 <= -5e-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 <= (-5d-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 <= -5e-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 <= -5e-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 <= -5e-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 <= -5e-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, -5e-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 < -4.999999999999985e-310

   1. Initial program 78.1%

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

      1. *-commutative 78.1%

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

      2. associate-/r* 76.0%

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

      3. sqrt-div 41.9%

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

      4. associate-*l/ 39.7%

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

   4. Applied egg-rr 39.7%

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

      1. frac-2neg 41.1%

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

      2. sqrt-div 47.5%

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

   6. Applied egg-rr 46.0%

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

   if -4.999999999999985e-310 < A

   1. Initial program 78.6%

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

      1. sqrt-div 88.1%

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

      2. associate-*r/ 86.6%

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

   4. Applied egg-rr 86.6%

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

      1. *-commutative 86.6%

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

      2. associate-/l* 84.7%

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

      3. associate-/r/ 88.1%

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

   6. Simplified 88.1%

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

4. Final simplification 66.7%

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

Alternative 4: 88.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+176}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-160}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+285}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \end{array} \]

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 l) (sqrt (/ A V))))))
   (if (<= (* V l) -2e+176)
     t_0
     (if (<= (* V l) -2e-160)
       (* c0 (sqrt (/ A (* V l))))
       (if (<= (* V l) 0.0)
         t_0
         (if (<= (* V l) 1e+285)
           (* c0 (/ (sqrt A) (sqrt (* V l))))
           (/ 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(l) / sqrt((A / V)));
	double tmp;
	if ((V * l) <= -2e+176) {
		tmp = t_0;
	} else if ((V * l) <= -2e-160) {
		tmp = c0 * sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else if ((V * l) <= 1e+285) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} 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(l) / sqrt((a / v)))
    if ((v * l) <= (-2d+176)) then
        tmp = t_0
    else if ((v * l) <= (-2d-160)) then
        tmp = c0 * sqrt((a / (v * l)))
    else if ((v * l) <= 0.0d0) then
        tmp = t_0
    else if ((v * l) <= 1d+285) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    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(l) / Math.sqrt((A / V)));
	double tmp;
	if ((V * l) <= -2e+176) {
		tmp = t_0;
	} else if ((V * l) <= -2e-160) {
		tmp = c0 * Math.sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else if ((V * l) <= 1e+285) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} 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(l) / math.sqrt((A / V)))
	tmp = 0
	if (V * l) <= -2e+176:
		tmp = t_0
	elif (V * l) <= -2e-160:
		tmp = c0 * math.sqrt((A / (V * l)))
	elif (V * l) <= 0.0:
		tmp = t_0
	elif (V * l) <= 1e+285:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	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 / Float64(sqrt(l) / sqrt(Float64(A / V))))
	tmp = 0.0
	if (Float64(V * l) <= -2e+176)
		tmp = t_0;
	elseif (Float64(V * l) <= -2e-160)
		tmp = Float64(c0 * sqrt(Float64(A / Float64(V * l))));
	elseif (Float64(V * l) <= 0.0)
		tmp = t_0;
	elseif (Float64(V * l) <= 1e+285)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	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(l) / sqrt((A / V)));
	tmp = 0.0;
	if ((V * l) <= -2e+176)
		tmp = t_0;
	elseif ((V * l) <= -2e-160)
		tmp = c0 * sqrt((A / (V * l)));
	elseif ((V * l) <= 0.0)
		tmp = t_0;
	elseif ((V * l) <= 1e+285)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	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[(N[Sqrt[l], $MachinePrecision] / N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -2e+176], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -2e-160], N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], 1e+285], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -2e176 or -2e-160 < (*.f64 V l) < -0.0

   1. Initial program 59.2%

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

      1. associate-/r* 69.8%

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

      2. sqrt-div 38.4%

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

      3. associate-*r/ 38.4%

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

   4. Applied egg-rr 38.4%

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

      1. associate-/l* 38.4%

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

   6. Simplified 38.4%

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

   if -2e176 < (*.f64 V l) < -2e-160

   1. Initial program 95.6%

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

   if -0.0 < (*.f64 V l) < 9.9999999999999998e284

   1. Initial program 87.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. associate-*r/ 97.6%

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

   4. Applied egg-rr 97.6%

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

      1. *-commutative 97.6%

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

      2. associate-/l* 95.2%

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

      3. associate-/r/ 99.4%

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

   6. Simplified 99.4%

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

   if 9.9999999999999998e284 < (*.f64 V l)

   1. Initial program 44.3%

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

      1. add-sqr-sqrt 44.3%

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

      2. pow2 44.3%

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

      3. pow1/2 44.3%

         \[\leadsto c0 \cdot {\left(\sqrt{\color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{0.5}}}\right)}^{2} \]

      4. sqrt-pow1 44.3%

         \[\leadsto c0 \cdot {\color{blue}{\left({\left(\frac{A}{V \cdot \ell}\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} \]

      5. metadata-eval 44.3%

         \[\leadsto c0 \cdot {\left({\left(\frac{A}{V \cdot \ell}\right)}^{\color{blue}{0.25}}\right)}^{2} \]

   4. Applied egg-rr 44.3%

      \[\leadsto c0 \cdot \color{blue}{{\left({\left(\frac{A}{V \cdot \ell}\right)}^{0.25}\right)}^{2}} \]
   5. Step-by-step derivation

      1. pow-pow 44.3%

         \[\leadsto c0 \cdot \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{\left(0.25 \cdot 2\right)}} \]

      2. metadata-eval 44.3%

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

      3. pow1/2 44.3%

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

      4. clear-num 44.3%

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

      5. associate-*r/ 83.5%

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

      6. sqrt-div 83.4%

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

      7. metadata-eval 83.4%

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

      8. un-div-inv 83.8%

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

      9. clear-num 83.7%

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

      10. un-div-inv 83.5%

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

   6. Applied egg-rr 83.5%

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

      1. associate-/r/ 83.7%

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

   8. Simplified 83.7%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+176}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-160}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+285}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{\sqrt{\frac{A}{V}}}{\frac{\sqrt{\ell}}{c0}}\\ \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+176}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-155}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+285}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \end{array} \]

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 (/ (sqrt (/ A V)) (/ (sqrt l) c0))))
   (if (<= (* V l) -2e+176)
     t_0
     (if (<= (* V l) -2e-155)
       (* c0 (sqrt (/ A (* V l))))
       (if (<= (* V l) 0.0)
         t_0
         (if (<= (* V l) 1e+285)
           (* c0 (/ (sqrt A) (sqrt (* V l))))
           (/ 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 = sqrt((A / V)) / (sqrt(l) / c0);
	double tmp;
	if ((V * l) <= -2e+176) {
		tmp = t_0;
	} else if ((V * l) <= -2e-155) {
		tmp = c0 * sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else if ((V * l) <= 1e+285) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} 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 = sqrt((a / v)) / (sqrt(l) / c0)
    if ((v * l) <= (-2d+176)) then
        tmp = t_0
    else if ((v * l) <= (-2d-155)) then
        tmp = c0 * sqrt((a / (v * l)))
    else if ((v * l) <= 0.0d0) then
        tmp = t_0
    else if ((v * l) <= 1d+285) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    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 = Math.sqrt((A / V)) / (Math.sqrt(l) / c0);
	double tmp;
	if ((V * l) <= -2e+176) {
		tmp = t_0;
	} else if ((V * l) <= -2e-155) {
		tmp = c0 * Math.sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else if ((V * l) <= 1e+285) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} 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 = math.sqrt((A / V)) / (math.sqrt(l) / c0)
	tmp = 0
	if (V * l) <= -2e+176:
		tmp = t_0
	elif (V * l) <= -2e-155:
		tmp = c0 * math.sqrt((A / (V * l)))
	elif (V * l) <= 0.0:
		tmp = t_0
	elif (V * l) <= 1e+285:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	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(sqrt(Float64(A / V)) / Float64(sqrt(l) / c0))
	tmp = 0.0
	if (Float64(V * l) <= -2e+176)
		tmp = t_0;
	elseif (Float64(V * l) <= -2e-155)
		tmp = Float64(c0 * sqrt(Float64(A / Float64(V * l))));
	elseif (Float64(V * l) <= 0.0)
		tmp = t_0;
	elseif (Float64(V * l) <= 1e+285)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	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 = sqrt((A / V)) / (sqrt(l) / c0);
	tmp = 0.0;
	if ((V * l) <= -2e+176)
		tmp = t_0;
	elseif ((V * l) <= -2e-155)
		tmp = c0 * sqrt((A / (V * l)));
	elseif ((V * l) <= 0.0)
		tmp = t_0;
	elseif ((V * l) <= 1e+285)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	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[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] / c0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -2e+176], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -2e-155], N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], 1e+285], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -2e176 or -2.00000000000000003e-155 < (*.f64 V l) < -0.0

   1. Initial program 60.2%

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

      1. add-sqr-sqrt 60.0%

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

      2. pow2 60.0%

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

      3. pow1/2 60.0%

         \[\leadsto c0 \cdot {\left(\sqrt{\color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{0.5}}}\right)}^{2} \]

      4. sqrt-pow1 60.0%

         \[\leadsto c0 \cdot {\color{blue}{\left({\left(\frac{A}{V \cdot \ell}\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} \]

      5. metadata-eval 60.0%

         \[\leadsto c0 \cdot {\left({\left(\frac{A}{V \cdot \ell}\right)}^{\color{blue}{0.25}}\right)}^{2} \]

   4. Applied egg-rr 60.0%

      \[\leadsto c0 \cdot \color{blue}{{\left({\left(\frac{A}{V \cdot \ell}\right)}^{0.25}\right)}^{2}} \]
   5. Step-by-step derivation

      1. *-commutative 60.0%

         \[\leadsto \color{blue}{{\left({\left(\frac{A}{V \cdot \ell}\right)}^{0.25}\right)}^{2} \cdot c0} \]

      2. pow-pow 60.2%

         \[\leadsto \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{\left(0.25 \cdot 2\right)}} \cdot c0 \]

      3. metadata-eval 60.2%

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

      4. pow1/2 60.2%

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

      5. associate-/r* 70.5%

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

      6. sqrt-div 38.6%

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

      7. associate-/r/ 38.6%

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

   6. Applied egg-rr 38.6%

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

   if -2e176 < (*.f64 V l) < -2.00000000000000003e-155

   1. Initial program 95.5%

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

   if -0.0 < (*.f64 V l) < 9.9999999999999998e284

   1. Initial program 87.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. associate-*r/ 97.6%

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

   4. Applied egg-rr 97.6%

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

      1. *-commutative 97.6%

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

      2. associate-/l* 95.2%

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

      3. associate-/r/ 99.4%

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

   6. Simplified 99.4%

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

   if 9.9999999999999998e284 < (*.f64 V l)

   1. Initial program 44.3%

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

      1. add-sqr-sqrt 44.3%

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

      2. pow2 44.3%

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

      3. pow1/2 44.3%

         \[\leadsto c0 \cdot {\left(\sqrt{\color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{0.5}}}\right)}^{2} \]

      4. sqrt-pow1 44.3%

         \[\leadsto c0 \cdot {\color{blue}{\left({\left(\frac{A}{V \cdot \ell}\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} \]

      5. metadata-eval 44.3%

         \[\leadsto c0 \cdot {\left({\left(\frac{A}{V \cdot \ell}\right)}^{\color{blue}{0.25}}\right)}^{2} \]

   4. Applied egg-rr 44.3%

      \[\leadsto c0 \cdot \color{blue}{{\left({\left(\frac{A}{V \cdot \ell}\right)}^{0.25}\right)}^{2}} \]
   5. Step-by-step derivation

      1. pow-pow 44.3%

         \[\leadsto c0 \cdot \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{\left(0.25 \cdot 2\right)}} \]

      2. metadata-eval 44.3%

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

      3. pow1/2 44.3%

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

      4. clear-num 44.3%

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

      5. associate-*r/ 83.5%

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

      6. sqrt-div 83.4%

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

      7. metadata-eval 83.4%

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

      8. un-div-inv 83.8%

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

      9. clear-num 83.7%

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

      10. un-div-inv 83.5%

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

   6. Applied egg-rr 83.5%

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

      1. associate-/r/ 83.7%

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

   8. Simplified 83.7%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+176}:\\ \;\;\;\;\frac{\sqrt{\frac{A}{V}}}{\frac{\sqrt{\ell}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-155}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{\sqrt{\frac{A}{V}}}{\frac{\sqrt{\ell}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+285}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{V}}\\ \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+176}:\\ \;\;\;\;c0 \cdot \left(t_0 \cdot \sqrt{\frac{1}{\ell}}\right)\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-155}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{t_0}{\frac{\sqrt{\ell}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+285}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \end{array} \]

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 (sqrt (/ A V))))
   (if (<= (* V l) -2e+176)
     (* c0 (* t_0 (sqrt (/ 1.0 l))))
     (if (<= (* V l) -2e-155)
       (* c0 (sqrt (/ A (* V l))))
       (if (<= (* V l) 0.0)
         (/ t_0 (/ (sqrt l) c0))
         (if (<= (* V l) 1e+285)
           (* c0 (/ (sqrt A) (sqrt (* V l))))
           (/ 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 = sqrt((A / V));
	double tmp;
	if ((V * l) <= -2e+176) {
		tmp = c0 * (t_0 * sqrt((1.0 / l)));
	} else if ((V * l) <= -2e-155) {
		tmp = c0 * sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0 / (sqrt(l) / c0);
	} else if ((V * l) <= 1e+285) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} 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 = sqrt((a / v))
    if ((v * l) <= (-2d+176)) then
        tmp = c0 * (t_0 * sqrt((1.0d0 / l)))
    else if ((v * l) <= (-2d-155)) then
        tmp = c0 * sqrt((a / (v * l)))
    else if ((v * l) <= 0.0d0) then
        tmp = t_0 / (sqrt(l) / c0)
    else if ((v * l) <= 1d+285) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    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 = Math.sqrt((A / V));
	double tmp;
	if ((V * l) <= -2e+176) {
		tmp = c0 * (t_0 * Math.sqrt((1.0 / l)));
	} else if ((V * l) <= -2e-155) {
		tmp = c0 * Math.sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0 / (Math.sqrt(l) / c0);
	} else if ((V * l) <= 1e+285) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} 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 = math.sqrt((A / V))
	tmp = 0
	if (V * l) <= -2e+176:
		tmp = c0 * (t_0 * math.sqrt((1.0 / l)))
	elif (V * l) <= -2e-155:
		tmp = c0 * math.sqrt((A / (V * l)))
	elif (V * l) <= 0.0:
		tmp = t_0 / (math.sqrt(l) / c0)
	elif (V * l) <= 1e+285:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	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 = sqrt(Float64(A / V))
	tmp = 0.0
	if (Float64(V * l) <= -2e+176)
		tmp = Float64(c0 * Float64(t_0 * sqrt(Float64(1.0 / l))));
	elseif (Float64(V * l) <= -2e-155)
		tmp = Float64(c0 * sqrt(Float64(A / Float64(V * l))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(t_0 / Float64(sqrt(l) / c0));
	elseif (Float64(V * l) <= 1e+285)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	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 = sqrt((A / V));
	tmp = 0.0;
	if ((V * l) <= -2e+176)
		tmp = c0 * (t_0 * sqrt((1.0 / l)));
	elseif ((V * l) <= -2e-155)
		tmp = c0 * sqrt((A / (V * l)));
	elseif ((V * l) <= 0.0)
		tmp = t_0 / (sqrt(l) / c0);
	elseif ((V * l) <= 1e+285)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	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[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -2e+176], N[(c0 * N[(t$95$0 * N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-155], N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(t$95$0 / N[(N[Sqrt[l], $MachinePrecision] / c0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+285], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -2e176

   1. Initial program 53.4%

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

      1. pow1/2 53.4%

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

      2. associate-/r* 69.0%

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

      3. div-inv 69.1%

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

      4. unpow-prod-down 33.8%

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

      5. pow1/2 33.8%

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

   4. Applied egg-rr 33.8%

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

      1. unpow1/2 33.8%

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

   6. Simplified 33.8%

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

   if -2e176 < (*.f64 V l) < -2.00000000000000003e-155

   1. Initial program 95.5%

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

   if -2.00000000000000003e-155 < (*.f64 V l) < -0.0

   1. Initial program 65.5%

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

      1. add-sqr-sqrt 65.2%

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

      2. pow2 65.2%

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

      3. pow1/2 65.2%

         \[\leadsto c0 \cdot {\left(\sqrt{\color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{0.5}}}\right)}^{2} \]

      4. sqrt-pow1 65.2%

         \[\leadsto c0 \cdot {\color{blue}{\left({\left(\frac{A}{V \cdot \ell}\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} \]

      5. metadata-eval 65.2%

         \[\leadsto c0 \cdot {\left({\left(\frac{A}{V \cdot \ell}\right)}^{\color{blue}{0.25}}\right)}^{2} \]

   4. Applied egg-rr 65.2%

      \[\leadsto c0 \cdot \color{blue}{{\left({\left(\frac{A}{V \cdot \ell}\right)}^{0.25}\right)}^{2}} \]
   5. Step-by-step derivation

      1. *-commutative 65.2%

         \[\leadsto \color{blue}{{\left({\left(\frac{A}{V \cdot \ell}\right)}^{0.25}\right)}^{2} \cdot c0} \]

      2. pow-pow 65.5%

         \[\leadsto \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{\left(0.25 \cdot 2\right)}} \cdot c0 \]

      3. metadata-eval 65.5%

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

      4. pow1/2 65.5%

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

      5. associate-/r* 71.7%

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

      6. sqrt-div 42.5%

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

      7. associate-/r/ 42.4%

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

   6. Applied egg-rr 42.4%

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

   if -0.0 < (*.f64 V l) < 9.9999999999999998e284

   1. Initial program 87.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. associate-*r/ 97.6%

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

   4. Applied egg-rr 97.6%

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

      1. *-commutative 97.6%

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

      2. associate-/l* 95.2%

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

      3. associate-/r/ 99.4%

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

   6. Simplified 99.4%

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

   if 9.9999999999999998e284 < (*.f64 V l)

   1. Initial program 44.3%

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

      1. add-sqr-sqrt 44.3%

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

      2. pow2 44.3%

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

      3. pow1/2 44.3%

         \[\leadsto c0 \cdot {\left(\sqrt{\color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{0.5}}}\right)}^{2} \]

      4. sqrt-pow1 44.3%

         \[\leadsto c0 \cdot {\color{blue}{\left({\left(\frac{A}{V \cdot \ell}\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} \]

      5. metadata-eval 44.3%

         \[\leadsto c0 \cdot {\left({\left(\frac{A}{V \cdot \ell}\right)}^{\color{blue}{0.25}}\right)}^{2} \]

   4. Applied egg-rr 44.3%

      \[\leadsto c0 \cdot \color{blue}{{\left({\left(\frac{A}{V \cdot \ell}\right)}^{0.25}\right)}^{2}} \]
   5. Step-by-step derivation

      1. pow-pow 44.3%

         \[\leadsto c0 \cdot \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{\left(0.25 \cdot 2\right)}} \]

      2. metadata-eval 44.3%

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

      3. pow1/2 44.3%

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

      4. clear-num 44.3%

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

      5. associate-*r/ 83.5%

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

      6. sqrt-div 83.4%

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

      7. metadata-eval 83.4%

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

      8. un-div-inv 83.8%

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

      9. clear-num 83.7%

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

      10. un-div-inv 83.5%

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

   6. Applied egg-rr 83.5%

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

      1. associate-/r/ 83.7%

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

   8. Simplified 83.7%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+176}:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{A}{V}} \cdot \sqrt{\frac{1}{\ell}}\right)\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-155}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{\sqrt{\frac{A}{V}}}{\frac{\sqrt{\ell}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+285}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{V}}\\ \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+210}:\\ \;\;\;\;c0 \cdot \left(t_0 \cdot \sqrt{\frac{1}{\ell}}\right)\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-265}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{t_0}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+285}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \end{array} \]

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 (sqrt (/ A V))))
   (if (<= (* V l) -5e+210)
     (* c0 (* t_0 (sqrt (/ 1.0 l))))
     (if (<= (* V l) -2e-265)
       (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
       (if (<= (* V l) 0.0)
         (/ c0 (/ (sqrt l) t_0))
         (if (<= (* V l) 1e+285)
           (* c0 (/ (sqrt A) (sqrt (* V l))))
           (/ 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 = sqrt((A / V));
	double tmp;
	if ((V * l) <= -5e+210) {
		tmp = c0 * (t_0 * sqrt((1.0 / l)));
	} else if ((V * l) <= -2e-265) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 / (sqrt(l) / t_0);
	} else if ((V * l) <= 1e+285) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} 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 = sqrt((a / v))
    if ((v * l) <= (-5d+210)) then
        tmp = c0 * (t_0 * sqrt((1.0d0 / l)))
    else if ((v * l) <= (-2d-265)) then
        tmp = c0 * (sqrt(-a) / sqrt((v * -l)))
    else if ((v * l) <= 0.0d0) then
        tmp = c0 / (sqrt(l) / t_0)
    else if ((v * l) <= 1d+285) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    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 = Math.sqrt((A / V));
	double tmp;
	if ((V * l) <= -5e+210) {
		tmp = c0 * (t_0 * Math.sqrt((1.0 / l)));
	} else if ((V * l) <= -2e-265) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 / (Math.sqrt(l) / t_0);
	} else if ((V * l) <= 1e+285) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} 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 = math.sqrt((A / V))
	tmp = 0
	if (V * l) <= -5e+210:
		tmp = c0 * (t_0 * math.sqrt((1.0 / l)))
	elif (V * l) <= -2e-265:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 0.0:
		tmp = c0 / (math.sqrt(l) / t_0)
	elif (V * l) <= 1e+285:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	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 = sqrt(Float64(A / V))
	tmp = 0.0
	if (Float64(V * l) <= -5e+210)
		tmp = Float64(c0 * Float64(t_0 * sqrt(Float64(1.0 / l))));
	elseif (Float64(V * l) <= -2e-265)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 / Float64(sqrt(l) / t_0));
	elseif (Float64(V * l) <= 1e+285)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	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 = sqrt((A / V));
	tmp = 0.0;
	if ((V * l) <= -5e+210)
		tmp = c0 * (t_0 * sqrt((1.0 / l)));
	elseif ((V * l) <= -2e-265)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 0.0)
		tmp = c0 / (sqrt(l) / t_0);
	elseif ((V * l) <= 1e+285)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	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[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -5e+210], N[(c0 * N[(t$95$0 * N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-265], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 / N[(N[Sqrt[l], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+285], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -4.9999999999999998e210

   1. Initial program 56.8%

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

      1. pow1/2 56.8%

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

      2. associate-/r* 75.5%

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

      3. div-inv 75.6%

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

      4. unpow-prod-down 39.4%

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

      5. pow1/2 39.4%

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

   4. Applied egg-rr 39.4%

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

      1. unpow1/2 39.4%

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

   6. Simplified 39.4%

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

   if -4.9999999999999998e210 < (*.f64 V l) < -1.99999999999999997e-265

   1. Initial program 87.6%

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

      1. frac-2neg 87.6%

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

      2. sqrt-div 99.4%

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

      3. distribute-rgt-neg-in 99.4%

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

   4. Applied egg-rr 99.4%

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

      1. distribute-rgt-neg-out 99.4%

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

      2. *-commutative 99.4%

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

      3. distribute-rgt-neg-in 99.4%

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

   6. Simplified 99.4%

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

   if -1.99999999999999997e-265 < (*.f64 V l) < -0.0

   1. Initial program 54.2%

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

      1. associate-/r* 64.9%

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

      2. sqrt-div 33.3%

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

      3. associate-*r/ 33.4%

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

   4. Applied egg-rr 33.4%

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

      1. associate-/l* 33.4%

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

   6. Simplified 33.4%

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

   if -0.0 < (*.f64 V l) < 9.9999999999999998e284

   1. Initial program 87.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. associate-*r/ 97.6%

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

   4. Applied egg-rr 97.6%

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

      1. *-commutative 97.6%

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

      2. associate-/l* 95.2%

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

      3. associate-/r/ 99.4%

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

   6. Simplified 99.4%

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

   if 9.9999999999999998e284 < (*.f64 V l)

   1. Initial program 44.3%

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

      1. add-sqr-sqrt 44.3%

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

      2. pow2 44.3%

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

      3. pow1/2 44.3%

         \[\leadsto c0 \cdot {\left(\sqrt{\color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{0.5}}}\right)}^{2} \]

      4. sqrt-pow1 44.3%

         \[\leadsto c0 \cdot {\color{blue}{\left({\left(\frac{A}{V \cdot \ell}\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} \]

      5. metadata-eval 44.3%

         \[\leadsto c0 \cdot {\left({\left(\frac{A}{V \cdot \ell}\right)}^{\color{blue}{0.25}}\right)}^{2} \]

   4. Applied egg-rr 44.3%

      \[\leadsto c0 \cdot \color{blue}{{\left({\left(\frac{A}{V \cdot \ell}\right)}^{0.25}\right)}^{2}} \]
   5. Step-by-step derivation

      1. pow-pow 44.3%

         \[\leadsto c0 \cdot \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{\left(0.25 \cdot 2\right)}} \]

      2. metadata-eval 44.3%

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

      3. pow1/2 44.3%

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

      4. clear-num 44.3%

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

      5. associate-*r/ 83.5%

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

      6. sqrt-div 83.4%

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

      7. metadata-eval 83.4%

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

      8. un-div-inv 83.8%

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

      9. clear-num 83.7%

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

      10. un-div-inv 83.5%

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

   6. Applied egg-rr 83.5%

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

      1. associate-/r/ 83.7%

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

   8. Simplified 83.7%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+210}:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{A}{V}} \cdot \sqrt{\frac{1}{\ell}}\right)\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-265}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+285}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+210}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+285}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -5e+210)
   (* c0 (sqrt (/ (/ A l) V)))
   (if (<= (* V l) 0.0)
     (* c0 (sqrt (/ A (* V l))))
     (if (<= (* V l) 1e+285)
       (* c0 (/ (sqrt A) (sqrt (* V l))))
       (/ c0 (sqrt (* 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+210) {
		tmp = c0 * sqrt(((A / l) / V));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * sqrt((A / (V * l)));
	} else if ((V * l) <= 1e+285) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} 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) :: tmp
    if ((v * l) <= (-5d+210)) then
        tmp = c0 * sqrt(((a / l) / v))
    else if ((v * l) <= 0.0d0) then
        tmp = c0 * sqrt((a / (v * l)))
    else if ((v * l) <= 1d+285) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    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 tmp;
	if ((V * l) <= -5e+210) {
		tmp = c0 * Math.sqrt(((A / l) / V));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * Math.sqrt((A / (V * l)));
	} else if ((V * l) <= 1e+285) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} 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):
	tmp = 0
	if (V * l) <= -5e+210:
		tmp = c0 * math.sqrt(((A / l) / V))
	elif (V * l) <= 0.0:
		tmp = c0 * math.sqrt((A / (V * l)))
	elif (V * l) <= 1e+285:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	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)
	tmp = 0.0
	if (Float64(V * l) <= -5e+210)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(A / Float64(V * l))));
	elseif (Float64(V * l) <= 1e+285)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	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)
	tmp = 0.0;
	if ((V * l) <= -5e+210)
		tmp = c0 * sqrt(((A / l) / V));
	elseif ((V * l) <= 0.0)
		tmp = c0 * sqrt((A / (V * l)));
	elseif ((V * l) <= 1e+285)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	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_] := If[LessEqual[N[(V * l), $MachinePrecision], -5e+210], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+285], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -4.9999999999999998e210

   1. Initial program 56.8%

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

      1. add-sqr-sqrt 56.8%

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

      2. pow2 56.8%

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

      3. pow1/2 56.8%

         \[\leadsto c0 \cdot {\left(\sqrt{\color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{0.5}}}\right)}^{2} \]

      4. sqrt-pow1 56.8%

         \[\leadsto c0 \cdot {\color{blue}{\left({\left(\frac{A}{V \cdot \ell}\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} \]

      5. metadata-eval 56.8%

         \[\leadsto c0 \cdot {\left({\left(\frac{A}{V \cdot \ell}\right)}^{\color{blue}{0.25}}\right)}^{2} \]

   4. Applied egg-rr 56.8%

      \[\leadsto c0 \cdot \color{blue}{{\left({\left(\frac{A}{V \cdot \ell}\right)}^{0.25}\right)}^{2}} \]

   5. Taylor expanded in c0 around 0 56.8%

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

      1. *-commutative 56.8%

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

      2. associate-/r* 75.6%

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

   7. Simplified 75.6%

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

   if -4.9999999999999998e210 < (*.f64 V l) < -0.0

   1. Initial program 79.4%

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

   if -0.0 < (*.f64 V l) < 9.9999999999999998e284

   1. Initial program 87.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. associate-*r/ 97.6%

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

   4. Applied egg-rr 97.6%

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

      1. *-commutative 97.6%

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

      2. associate-/l* 95.2%

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

      3. associate-/r/ 99.4%

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

   6. Simplified 99.4%

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

   if 9.9999999999999998e284 < (*.f64 V l)

   1. Initial program 44.3%

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

      1. add-sqr-sqrt 44.3%

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

      2. pow2 44.3%

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

      3. pow1/2 44.3%

         \[\leadsto c0 \cdot {\left(\sqrt{\color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{0.5}}}\right)}^{2} \]

      4. sqrt-pow1 44.3%

         \[\leadsto c0 \cdot {\color{blue}{\left({\left(\frac{A}{V \cdot \ell}\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} \]

      5. metadata-eval 44.3%

         \[\leadsto c0 \cdot {\left({\left(\frac{A}{V \cdot \ell}\right)}^{\color{blue}{0.25}}\right)}^{2} \]

   4. Applied egg-rr 44.3%

      \[\leadsto c0 \cdot \color{blue}{{\left({\left(\frac{A}{V \cdot \ell}\right)}^{0.25}\right)}^{2}} \]
   5. Step-by-step derivation

      1. pow-pow 44.3%

         \[\leadsto c0 \cdot \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{\left(0.25 \cdot 2\right)}} \]

      2. metadata-eval 44.3%

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

      3. pow1/2 44.3%

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

      4. clear-num 44.3%

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

      5. associate-*r/ 83.5%

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

      6. sqrt-div 83.4%

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

      7. metadata-eval 83.4%

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

      8. un-div-inv 83.8%

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

      9. clear-num 83.7%

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

      10. un-div-inv 83.5%

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

   6. Applied egg-rr 83.5%

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

      1. associate-/r/ 83.7%

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

   8. Simplified 83.7%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+210}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+285}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \frac{\frac{1}{V}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (/ A (* V l)) 0.0)
   (/ c0 (sqrt (* l (/ V A))))
   (* c0 (sqrt (* A (/ (/ 1.0 V) l))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((A / (V * l)) <= 0.0) {
		tmp = c0 / sqrt((l * (V / A)));
	} else {
		tmp = c0 * sqrt((A * ((1.0 / 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 / (v * l)) <= 0.0d0) then
        tmp = c0 / sqrt((l * (v / a)))
    else
        tmp = c0 * sqrt((a * ((1.0d0 / 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 / (V * l)) <= 0.0) {
		tmp = c0 / Math.sqrt((l * (V / A)));
	} else {
		tmp = c0 * Math.sqrt((A * ((1.0 / V) / l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (A / (V * l)) <= 0.0:
		tmp = c0 / math.sqrt((l * (V / A)))
	else:
		tmp = c0 * math.sqrt((A * ((1.0 / V) / l)))
	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)) <= 0.0)
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / A))));
	else
		tmp = Float64(c0 * sqrt(Float64(A * Float64(Float64(1.0 / 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 / (V * l)) <= 0.0)
		tmp = c0 / sqrt((l * (V / A)));
	else
		tmp = c0 * sqrt((A * ((1.0 / 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[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision], 0.0], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(A * N[(N[(1.0 / V), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 41.6%

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

      1. add-sqr-sqrt 41.6%

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

      2. pow2 41.6%

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

      3. pow1/2 41.6%

         \[\leadsto c0 \cdot {\left(\sqrt{\color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{0.5}}}\right)}^{2} \]

      4. sqrt-pow1 41.6%

         \[\leadsto c0 \cdot {\color{blue}{\left({\left(\frac{A}{V \cdot \ell}\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} \]

      5. metadata-eval 41.6%

         \[\leadsto c0 \cdot {\left({\left(\frac{A}{V \cdot \ell}\right)}^{\color{blue}{0.25}}\right)}^{2} \]

   4. Applied egg-rr 41.6%

      \[\leadsto c0 \cdot \color{blue}{{\left({\left(\frac{A}{V \cdot \ell}\right)}^{0.25}\right)}^{2}} \]
   5. Step-by-step derivation

      1. pow-pow 41.6%

         \[\leadsto c0 \cdot \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{\left(0.25 \cdot 2\right)}} \]

      2. metadata-eval 41.6%

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

      3. pow1/2 41.6%

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

      4. clear-num 41.6%

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

      5. associate-*r/ 61.0%

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

      6. sqrt-div 61.0%

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

      7. metadata-eval 61.0%

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

      8. un-div-inv 61.1%

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

      9. clear-num 61.0%

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

      10. un-div-inv 60.9%

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

   6. Applied egg-rr 60.9%

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

      1. associate-/r/ 61.0%

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

   8. Simplified 61.0%

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

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

   1. Initial program 88.2%

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

      1. clear-num 86.9%

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

      2. associate-/r/ 88.2%

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

      3. associate-/r* 88.2%

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

   4. Applied egg-rr 88.2%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{1}{V}}{\ell} \cdot A}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \frac{\frac{1}{V}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 10^{-309}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (<= t_0 1e-309) (* c0 (sqrt (/ (/ A V) l))) (* c0 (sqrt t_0)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 1e-309) {
		tmp = c0 * sqrt(((A / V) / l));
	} else {
		tmp = c0 * sqrt(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 = a / (v * l)
    if (t_0 <= 1d-309) then
        tmp = c0 * sqrt(((a / v) / l))
    else
        tmp = c0 * sqrt(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 = A / (V * l);
	double tmp;
	if (t_0 <= 1e-309) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else {
		tmp = c0 * Math.sqrt(t_0);
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 1e-309:
		tmp = c0 * math.sqrt(((A / V) / l))
	else:
		tmp = c0 * math.sqrt(t_0)
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 1e-309)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	else
		tmp = Float64(c0 * sqrt(t_0));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 1e-309)
		tmp = c0 * sqrt(((A / V) / l));
	else
		tmp = c0 * sqrt(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-309], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 1.000000000000002e-309

   1. Initial program 43.3%

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

      1. *-commutative 43.3%

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

      2. associate-/l/ 60.3%

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

   3. Simplified 60.3%

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

   if 1.000000000000002e-309 < (/.f64 A (*.f64 V l))

   1. Initial program 88.6%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 10^{-309}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 77.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 / sqrt((l * (V / A)));
	} else {
		tmp = c0 * sqrt(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 = a / (v * l)
    if (t_0 <= 0.0d0) then
        tmp = c0 / sqrt((l * (v / a)))
    else
        tmp = c0 * sqrt(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 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 / Math.sqrt((l * (V / A)));
	} else {
		tmp = c0 * Math.sqrt(t_0);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 41.6%

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

      1. add-sqr-sqrt 41.6%

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

      2. pow2 41.6%

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

      3. pow1/2 41.6%

         \[\leadsto c0 \cdot {\left(\sqrt{\color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{0.5}}}\right)}^{2} \]

      4. sqrt-pow1 41.6%

         \[\leadsto c0 \cdot {\color{blue}{\left({\left(\frac{A}{V \cdot \ell}\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} \]

      5. metadata-eval 41.6%

         \[\leadsto c0 \cdot {\left({\left(\frac{A}{V \cdot \ell}\right)}^{\color{blue}{0.25}}\right)}^{2} \]

   4. Applied egg-rr 41.6%

      \[\leadsto c0 \cdot \color{blue}{{\left({\left(\frac{A}{V \cdot \ell}\right)}^{0.25}\right)}^{2}} \]
   5. Step-by-step derivation

      1. pow-pow 41.6%

         \[\leadsto c0 \cdot \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{\left(0.25 \cdot 2\right)}} \]

      2. metadata-eval 41.6%

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

      3. pow1/2 41.6%

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

      4. clear-num 41.6%

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

      5. associate-*r/ 61.0%

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

      6. sqrt-div 61.0%

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

      7. metadata-eval 61.0%

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

      8. un-div-inv 61.1%

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

      9. clear-num 61.0%

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

      10. un-div-inv 60.9%

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

   6. Applied egg-rr 60.9%

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

      1. associate-/r/ 61.0%

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

   8. Simplified 61.0%

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

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

   1. Initial program 88.2%

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

4. Final simplification 82.5%

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

Alternative 12: 74.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.3%

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

3. Final simplification 78.3%

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

Reproduce

herbie shell --seed 2024018 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))