Henrywood and Agarwal, Equation (3)

Average Error: 29.87% → 12.48%

Time: 13.5s

Precision: binary64

Cost: 20688

\[ \begin{array}{c}[V, l] = \mathsf{sort}([V, l])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+283}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-75}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{\frac{c0}{\sqrt{\ell}}}{\sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+295}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c0 \cdot \sqrt{A}}{\sqrt{\ell}}}{\sqrt{V}}\\ \end{array} \]

FPCore

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

↓

(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -5e+283)
   (* c0 (/ (sqrt (/ A (- l))) (sqrt (- V))))
   (if (<= (* V l) -2e-75)
     (/ (* c0 (sqrt (- A))) (sqrt (* V (- l))))
     (if (<= (* V l) 0.0)
       (/ (/ c0 (sqrt l)) (sqrt (/ V A)))
       (if (<= (* V l) 5e+295)
         (/ c0 (/ (sqrt (* V l)) (sqrt A)))
         (/ (/ (* c0 (sqrt A)) (sqrt l)) (sqrt V)))))))

C

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

↓

double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -5e+283) {
		tmp = c0 * (sqrt((A / -l)) / sqrt(-V));
	} else if ((V * l) <= -2e-75) {
		tmp = (c0 * sqrt(-A)) / sqrt((V * -l));
	} else if ((V * l) <= 0.0) {
		tmp = (c0 / sqrt(l)) / sqrt((V / A));
	} else if ((V * l) <= 5e+295) {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	} else {
		tmp = ((c0 * sqrt(A)) / sqrt(l)) / sqrt(V);
	}
	return tmp;
}

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

↓

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+283)) then
        tmp = c0 * (sqrt((a / -l)) / sqrt(-v))
    else if ((v * l) <= (-2d-75)) then
        tmp = (c0 * sqrt(-a)) / sqrt((v * -l))
    else if ((v * l) <= 0.0d0) then
        tmp = (c0 / sqrt(l)) / sqrt((v / a))
    else if ((v * l) <= 5d+295) then
        tmp = c0 / (sqrt((v * l)) / sqrt(a))
    else
        tmp = ((c0 * sqrt(a)) / sqrt(l)) / sqrt(v)
    end if
    code = tmp
end function

Java

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

↓

public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -5e+283) {
		tmp = c0 * (Math.sqrt((A / -l)) / Math.sqrt(-V));
	} else if ((V * l) <= -2e-75) {
		tmp = (c0 * Math.sqrt(-A)) / Math.sqrt((V * -l));
	} else if ((V * l) <= 0.0) {
		tmp = (c0 / Math.sqrt(l)) / Math.sqrt((V / A));
	} else if ((V * l) <= 5e+295) {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	} else {
		tmp = ((c0 * Math.sqrt(A)) / Math.sqrt(l)) / Math.sqrt(V);
	}
	return tmp;
}

Python

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

↓

def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -5e+283:
		tmp = c0 * (math.sqrt((A / -l)) / math.sqrt(-V))
	elif (V * l) <= -2e-75:
		tmp = (c0 * math.sqrt(-A)) / math.sqrt((V * -l))
	elif (V * l) <= 0.0:
		tmp = (c0 / math.sqrt(l)) / math.sqrt((V / A))
	elif (V * l) <= 5e+295:
		tmp = c0 / (math.sqrt((V * l)) / math.sqrt(A))
	else:
		tmp = ((c0 * math.sqrt(A)) / math.sqrt(l)) / math.sqrt(V)
	return tmp

Julia

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

↓

function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -5e+283)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / Float64(-l))) / sqrt(Float64(-V))));
	elseif (Float64(V * l) <= -2e-75)
		tmp = Float64(Float64(c0 * sqrt(Float64(-A))) / sqrt(Float64(V * Float64(-l))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(Float64(c0 / sqrt(l)) / sqrt(Float64(V / A)));
	elseif (Float64(V * l) <= 5e+295)
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	else
		tmp = Float64(Float64(Float64(c0 * sqrt(A)) / sqrt(l)) / sqrt(V));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -5e+283)
		tmp = c0 * (sqrt((A / -l)) / sqrt(-V));
	elseif ((V * l) <= -2e-75)
		tmp = (c0 * sqrt(-A)) / sqrt((V * -l));
	elseif ((V * l) <= 0.0)
		tmp = (c0 / sqrt(l)) / sqrt((V / A));
	elseif ((V * l) <= 5e+295)
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	else
		tmp = ((c0 * sqrt(A)) / sqrt(l)) / sqrt(V);
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -5e+283], N[(c0 * N[(N[Sqrt[N[(A / (-l)), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-75], N[(N[(c0 * N[Sqrt[(-A)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e+295], N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c0 * N[Sqrt[A], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[V], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -5.0000000000000004e283

   1. Initial program 56.36

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

   2. Applied egg-rr 56.36

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

   3. Applied egg-rr 31.77

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

   4. Applied egg-rr 12.94

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

   if -5.0000000000000004e283 < (*.f64 V l) < -1.9999999999999999e-75

   1. Initial program 10.68

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

   2. Applied egg-rr 3.53

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

   if -1.9999999999999999e-75 < (*.f64 V l) < -0.0

   1. Initial program 56.28

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

   2. Applied egg-rr 43.12

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

   3. Simplified 42.38

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

      [Start] 43.12	\[ \frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}} \]
      associate-/r/ [=>] 42.38	\[ \frac{c0}{\sqrt{\color{blue}{\frac{V}{A} \cdot \ell}}} \]

   4. Applied egg-rr 63.95

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

   5. Simplified 30.62

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

      [Start] 63.95	\[ \frac{\sqrt{c0}}{\sqrt{\frac{V}{A}}} \cdot \frac{\sqrt{c0}}{\sqrt{\ell}} \]
      associate-*l/ [=>] 64.9	\[ \color{blue}{\frac{\sqrt{c0} \cdot \frac{\sqrt{c0}}{\sqrt{\ell}}}{\sqrt{\frac{V}{A}}}} \]
      associate-*r/ [=>] 64.9	\[ \frac{\color{blue}{\frac{\sqrt{c0} \cdot \sqrt{c0}}{\sqrt{\ell}}}}{\sqrt{\frac{V}{A}}} \]
      rem-square-sqrt [=>] 30.62	\[ \frac{\frac{\color{blue}{c0}}{\sqrt{\ell}}}{\sqrt{\frac{V}{A}}} \]

   if -0.0 < (*.f64 V l) < 4.99999999999999991e295

   1. Initial program 15.65

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

   2. Applied egg-rr 1.08

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

   if 4.99999999999999991e295 < (*.f64 V l)

   1. Initial program 62.61

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

   2. Applied egg-rr 48.25

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

4. Final simplification 12.48

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+283}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-75}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{\frac{c0}{\sqrt{\ell}}}{\sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+295}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c0 \cdot \sqrt{A}}{\sqrt{\ell}}}{\sqrt{V}}\\ \end{array} \]

Alternatives

Alternative 1
Error	24.6%
Cost	27596

\[\begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-323}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+198}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \end{array} \]

Alternative 2
Error	25.89%
Cost	20681

\[\begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-323} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]

Alternative 3
Error	25.89%
Cost	20680

\[\begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-323}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \end{array} \]

Alternative 4
Error	10.64%
Cost	20168

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{\frac{c0}{\sqrt{\ell}}}{\frac{\sqrt{-V}}{\sqrt{-A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+295}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c0 \cdot \sqrt{A}}{\sqrt{\ell}}}{\sqrt{V}}\\ \end{array} \]

Alternative 5
Error	12.01%
Cost	20168

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{-V} \cdot \frac{\sqrt{\ell}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+295}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c0 \cdot \sqrt{A}}{\sqrt{\ell}}}{\sqrt{V}}\\ \end{array} \]

Alternative 6
Error	14.5%
Cost	14288

\[\begin{array}{l} t_0 := c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+223}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-31}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+295}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	14.58%
Cost	14288

\[\begin{array}{l} t_0 := \sqrt{\frac{A}{V}}\\ \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+223}:\\ \;\;\;\;c0 \cdot \frac{t_0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-31}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+295}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot t_0}{\sqrt{\ell}}\\ \end{array} \]

Alternative 8
Error	15.03%
Cost	14288

\[\begin{array}{l} t_0 := \sqrt{\frac{A}{V}}\\ \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+223}:\\ \;\;\;\;c0 \cdot \frac{t_0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-41}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{\frac{c0}{\sqrt{\ell}}}{\sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+295}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot t_0}{\sqrt{\ell}}\\ \end{array} \]

Alternative 9
Error	15.01%
Cost	14288

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+200}:\\ \;\;\;\;c0 \cdot \frac{{\left(\frac{V}{A}\right)}^{-0.5}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-41}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{\frac{c0}{\sqrt{\ell}}}{\sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+295}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 10
Error	14.74%
Cost	14288

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+118}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-41}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{\frac{c0}{\sqrt{\ell}}}{\sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+295}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 11
Error	12.84%
Cost	14288

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+283}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-75}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{\frac{c0}{\sqrt{\ell}}}{\sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+295}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 12
Error	18.25%
Cost	14025

\[\begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 4 \cdot 10^{-320} \lor \neg \left(t_0 \leq 5 \cdot 10^{+289}\right):\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\frac{V}{\frac{1}{\ell}}}{A}}}\\ \end{array} \]

Alternative 13
Error	18%
Cost	14024

\[\begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 4 \cdot 10^{-320}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+289}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\frac{V}{\frac{1}{\ell}}}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \end{array} \]

Alternative 14
Error	25.52%
Cost	7753

\[\begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0 \lor \neg \left(t_0 \leq 5 \cdot 10^{+289}\right):\\ \;\;\;\;\sqrt{c0 \cdot \frac{c0 \cdot \frac{A}{\ell}}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \end{array} \]

Alternative 15
Error	25.27%
Cost	7752

\[\begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;\sqrt{c0 \cdot \frac{c0 \cdot \frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+289}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{c0 \cdot \frac{A}{V \cdot \frac{\ell}{c0}}}\\ \end{array} \]

Alternative 16
Error	24.62%
Cost	7752

\[\begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;\sqrt{\frac{c0 \cdot A}{\ell} \cdot \frac{c0}{V}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+289}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{c0 \cdot \frac{A}{V \cdot \frac{\ell}{c0}}}\\ \end{array} \]

Alternative 17
Error	25.07%
Cost	7752

\[\begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;\sqrt{\frac{c0 \cdot A}{\ell} \cdot \frac{c0}{V}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+289}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \frac{1}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{c0 \cdot \frac{A}{V \cdot \frac{\ell}{c0}}}\\ \end{array} \]

Alternative 18
Error	29.87%
Cost	6848

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

Reproduce

herbie shell --seed 2023102 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))