Henrywood and Agarwal, Equation (3)

Average Error: 18.8 → 5.4

Time: 15.1s

Precision: binary64

Cost: 20556

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

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{-A}\\ t_1 := \frac{t_0 \cdot \frac{c0}{\sqrt{-V}}}{\sqrt{\ell}}\\ \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;V \cdot \ell \leq -4 \cdot 10^{-182}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \left(-\ell\right)}}{t_0}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t_1\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+294}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \ell} \cdot \sqrt{\frac{1}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]

FPCore

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

↓

(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (sqrt (- A))) (t_1 (/ (* t_0 (/ c0 (sqrt (- V)))) (sqrt l))))
   (if (<= (* V l) (- INFINITY))
     t_1
     (if (<= (* V l) -4e-182)
       (/ c0 (/ (sqrt (* V (- l))) t_0))
       (if (<= (* V l) 0.0)
         t_1
         (if (<= (* V l) 5e+294)
           (/ c0 (* (sqrt (* V l)) (sqrt (/ 1.0 A))))
           (* c0 (sqrt (/ (/ A l) 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 t_0 = sqrt(-A);
	double t_1 = (t_0 * (c0 / sqrt(-V))) / sqrt(l);
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = t_1;
	} else if ((V * l) <= -4e-182) {
		tmp = c0 / (sqrt((V * -l)) / t_0);
	} else if ((V * l) <= 0.0) {
		tmp = t_1;
	} else if ((V * l) <= 5e+294) {
		tmp = c0 / (sqrt((V * l)) * sqrt((1.0 / A)));
	} else {
		tmp = c0 * sqrt(((A / l) / V));
	}
	return tmp;
}

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 t_0 = Math.sqrt(-A);
	double t_1 = (t_0 * (c0 / Math.sqrt(-V))) / Math.sqrt(l);
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if ((V * l) <= -4e-182) {
		tmp = c0 / (Math.sqrt((V * -l)) / t_0);
	} else if ((V * l) <= 0.0) {
		tmp = t_1;
	} else if ((V * l) <= 5e+294) {
		tmp = c0 / (Math.sqrt((V * l)) * Math.sqrt((1.0 / A)));
	} else {
		tmp = c0 * Math.sqrt(((A / l) / V));
	}
	return tmp;
}

Python

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

↓

def code(c0, A, V, l):
	t_0 = math.sqrt(-A)
	t_1 = (t_0 * (c0 / math.sqrt(-V))) / math.sqrt(l)
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = t_1
	elif (V * l) <= -4e-182:
		tmp = c0 / (math.sqrt((V * -l)) / t_0)
	elif (V * l) <= 0.0:
		tmp = t_1
	elif (V * l) <= 5e+294:
		tmp = c0 / (math.sqrt((V * l)) * math.sqrt((1.0 / A)))
	else:
		tmp = c0 * math.sqrt(((A / l) / 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)
	t_0 = sqrt(Float64(-A))
	t_1 = Float64(Float64(t_0 * Float64(c0 / sqrt(Float64(-V)))) / sqrt(l))
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = t_1;
	elseif (Float64(V * l) <= -4e-182)
		tmp = Float64(c0 / Float64(sqrt(Float64(V * Float64(-l))) / t_0));
	elseif (Float64(V * l) <= 0.0)
		tmp = t_1;
	elseif (Float64(V * l) <= 5e+294)
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) * sqrt(Float64(1.0 / A))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / 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)
	t_0 = sqrt(-A);
	t_1 = (t_0 * (c0 / sqrt(-V))) / sqrt(l);
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = t_1;
	elseif ((V * l) <= -4e-182)
		tmp = c0 / (sqrt((V * -l)) / t_0);
	elseif ((V * l) <= 0.0)
		tmp = t_1;
	elseif ((V * l) <= 5e+294)
		tmp = c0 / (sqrt((V * l)) * sqrt((1.0 / A)));
	else
		tmp = c0 * sqrt(((A / l) / 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_] := Block[{t$95$0 = N[Sqrt[(-A)], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[(c0 / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], t$95$1, If[LessEqual[N[(V * l), $MachinePrecision], -4e-182], N[(c0 / N[(N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], t$95$1, If[LessEqual[N[(V * l), $MachinePrecision], 5e+294], N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(1.0 / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -inf.0 or -4.0000000000000002e-182 < (*.f64 V l) < -0.0

   1. Initial program 44.6

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

   2. Applied egg-rr 18.5

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

   3. Applied egg-rr 17.0

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

   4. Simplified 15.8

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

      [Start] 17.0	\[ \frac{\frac{\sqrt{-A} \cdot c0}{\sqrt{-V}}}{\sqrt{\ell}} \]
      *-lft-identity [<=] 17.0	\[ \frac{\frac{\sqrt{-A} \cdot c0}{\color{blue}{1 \cdot \sqrt{-V}}}}{\sqrt{\ell}} \]
      times-frac [=>] 15.8	\[ \frac{\color{blue}{\frac{\sqrt{-A}}{1} \cdot \frac{c0}{\sqrt{-V}}}}{\sqrt{\ell}} \]
      /-rgt-identity [=>] 15.8	\[ \frac{\color{blue}{\sqrt{-A}} \cdot \frac{c0}{\sqrt{-V}}}{\sqrt{\ell}} \]

   if -inf.0 < (*.f64 V l) < -4.0000000000000002e-182

   1. Initial program 8.0

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

   2. Applied egg-rr 2.5

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

   3. Simplified 0.4

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

      [Start] 2.5	\[ \frac{c0 \cdot \sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}} \]
      associate-/l* [=>] 0.4	\[ \color{blue}{\frac{c0}{\frac{\sqrt{V \cdot \left(-\ell\right)}}{\sqrt{-A}}}} \]

   if -0.0 < (*.f64 V l) < 4.9999999999999999e294

   1. Initial program 10.2

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

   2. Applied egg-rr 16.3

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

   3. Simplified 15.8

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

      [Start] 16.3	\[ \frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}} \]
      associate-/r/ [=>] 15.8	\[ \frac{c0}{\sqrt{\color{blue}{\frac{V}{A} \cdot \ell}}} \]

   4. Applied egg-rr 10.1

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

   5. Applied egg-rr 0.7

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

   if 4.9999999999999999e294 < (*.f64 V l)

   1. Initial program 39.3

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

   2. Applied egg-rr 39.3

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

   3. Applied egg-rr 22.8

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

4. Final simplification 5.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{\sqrt{-A} \cdot \frac{c0}{\sqrt{-V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -4 \cdot 10^{-182}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \left(-\ell\right)}}{\sqrt{-A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{\sqrt{-A} \cdot \frac{c0}{\sqrt{-V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+294}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \ell} \cdot \sqrt{\frac{1}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]

Alternatives

Alternative 1
Error	12.5
Cost	34640

\[\begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{elif}\;t_0 \leq -5 \cdot 10^{-276}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{-244}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V} \cdot \frac{1}{\ell}}\\ \mathbf{elif}\;t_0 \leq 10^{+307}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \frac{1}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{A}{\frac{\ell}{c0} \cdot \frac{V}{c0}}}\\ \end{array} \]

Alternative 2
Error	15.6
Cost	27724

\[\begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ t_1 := \frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-307}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t_0 \leq 10^{+307}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{c0 \cdot \frac{A \cdot c0}{V \cdot \ell}}\\ \end{array} \]

Alternative 3
Error	14.5
Cost	27724

\[\begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ t_1 := \frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-307}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t_0 \leq 10^{+307}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{A}{\frac{\ell}{c0} \cdot \frac{V}{c0}}}\\ \end{array} \]

Alternative 4
Error	14.4
Cost	27724

\[\begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-307}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t_0 \leq 10^{+307}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \frac{1}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{A}{\frac{\ell}{c0} \cdot \frac{V}{c0}}}\\ \end{array} \]

Alternative 5
Error	14.6
Cost	27724

\[\begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-276}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{-244}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V} \cdot \frac{1}{\ell}}\\ \mathbf{elif}\;t_0 \leq 10^{+307}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \frac{1}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{A}{\frac{\ell}{c0} \cdot \frac{V}{c0}}}\\ \end{array} \]

Alternative 6
Error	14.7
Cost	27724

\[\begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-276}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\frac{V}{\frac{1}{\ell}}}{A}}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{-244}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V} \cdot \frac{1}{\ell}}\\ \mathbf{elif}\;t_0 \leq 10^{+307}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \frac{1}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{A}{\frac{\ell}{c0} \cdot \frac{V}{c0}}}\\ \end{array} \]

Alternative 7
Error	6.5
Cost	14416

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+253}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-299}:\\ \;\;\;\;\sqrt{-A} \cdot \frac{c0}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+294}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \ell} \cdot \sqrt{\frac{1}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]

Alternative 8
Error	5.4
Cost	14416

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{{\ell}^{-0.5}}{\frac{\sqrt{\frac{V}{A}}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq -8 \cdot 10^{-309}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \left(-\ell\right)}}{\sqrt{-A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+294}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \ell} \cdot \sqrt{\frac{1}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]

Alternative 9
Error	8.8
Cost	14288

\[\begin{array}{l} t_0 := \frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+92}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-79}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+294}:\\ \;\;\;\;\frac{\sqrt{A}}{\frac{\sqrt{V \cdot \ell}}{c0}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]

Alternative 10
Error	9.2
Cost	14288

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+92}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-253}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+294}:\\ \;\;\;\;\frac{\sqrt{A}}{\frac{\sqrt{V \cdot \ell}}{c0}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]

Alternative 11
Error	7.5
Cost	14288

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+253}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-299}:\\ \;\;\;\;\sqrt{-A} \cdot \frac{c0}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+294}:\\ \;\;\;\;\frac{\sqrt{A}}{\frac{\sqrt{V \cdot \ell}}{c0}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]

Alternative 12
Error	14.3
Cost	7890

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+164} \lor \neg \left(V \cdot \ell \leq -5 \cdot 10^{-253}\right) \land \left(V \cdot \ell \leq 5 \cdot 10^{-303} \lor \neg \left(V \cdot \ell \leq 2 \cdot 10^{+252}\right)\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]

Alternative 13
Error	14.3
Cost	7888

\[\begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ t_1 := c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+164}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-285}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+252}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Error	14.2
Cost	7888

\[\begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ t_1 := c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+164}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-285}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-223}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+252}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 15
Error	14.1
Cost	7888

\[\begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+250}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-253}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-223}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+252}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 16
Error	18.8
Cost	6848

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

Reproduce

herbie shell --seed 2023027 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))