Henrywood and Agarwal, Equation (3)

Average Error: 19.4 → 4.9

Time: 15.1s

Precision: binary64

Cost: 20036

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

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{-A}\\ \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{t_0 \cdot \frac{c0}{\sqrt{-V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \frac{t_0}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-321}:\\ \;\;\;\;c0 \cdot \left({\left(\frac{-\ell}{A}\right)}^{-0.5} \cdot {\left(\frac{-1}{V}\right)}^{0.5}\right)\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+301}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{c0}{V} \cdot \frac{c0}{\ell}}}{{A}^{-0.5}}\\ \end{array} \]

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))))
   (if (<= (* V l) (- INFINITY))
     (/ (* t_0 (/ c0 (sqrt (- V)))) (sqrt l))
     (if (<= (* V l) -2e-310)
       (* c0 (/ t_0 (sqrt (* V (- l)))))
       (if (<= (* V l) 2e-321)
         (* c0 (* (pow (/ (- l) A) -0.5) (pow (/ -1.0 V) 0.5)))
         (if (<= (* V l) 1e+301)
           (* c0 (/ (sqrt A) (sqrt (* V l))))
           (/ (sqrt (* (/ c0 V) (/ c0 l))) (pow A -0.5))))))))

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 tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = (t_0 * (c0 / sqrt(-V))) / sqrt(l);
	} else if ((V * l) <= -2e-310) {
		tmp = c0 * (t_0 / sqrt((V * -l)));
	} else if ((V * l) <= 2e-321) {
		tmp = c0 * (pow((-l / A), -0.5) * pow((-1.0 / V), 0.5));
	} else if ((V * l) <= 1e+301) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = sqrt(((c0 / V) * (c0 / l))) / pow(A, -0.5);
	}
	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 tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = (t_0 * (c0 / Math.sqrt(-V))) / Math.sqrt(l);
	} else if ((V * l) <= -2e-310) {
		tmp = c0 * (t_0 / Math.sqrt((V * -l)));
	} else if ((V * l) <= 2e-321) {
		tmp = c0 * (Math.pow((-l / A), -0.5) * Math.pow((-1.0 / V), 0.5));
	} else if ((V * l) <= 1e+301) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = Math.sqrt(((c0 / V) * (c0 / l))) / Math.pow(A, -0.5);
	}
	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)
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = (t_0 * (c0 / math.sqrt(-V))) / math.sqrt(l)
	elif (V * l) <= -2e-310:
		tmp = c0 * (t_0 / math.sqrt((V * -l)))
	elif (V * l) <= 2e-321:
		tmp = c0 * (math.pow((-l / A), -0.5) * math.pow((-1.0 / V), 0.5))
	elif (V * l) <= 1e+301:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = math.sqrt(((c0 / V) * (c0 / l))) / math.pow(A, -0.5)
	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))
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(Float64(t_0 * Float64(c0 / sqrt(Float64(-V)))) / sqrt(l));
	elseif (Float64(V * l) <= -2e-310)
		tmp = Float64(c0 * Float64(t_0 / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 2e-321)
		tmp = Float64(c0 * Float64((Float64(Float64(-l) / A) ^ -0.5) * (Float64(-1.0 / V) ^ 0.5)));
	elseif (Float64(V * l) <= 1e+301)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(sqrt(Float64(Float64(c0 / V) * Float64(c0 / l))) / (A ^ -0.5));
	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);
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = (t_0 * (c0 / sqrt(-V))) / sqrt(l);
	elseif ((V * l) <= -2e-310)
		tmp = c0 * (t_0 / sqrt((V * -l)));
	elseif ((V * l) <= 2e-321)
		tmp = c0 * (((-l / A) ^ -0.5) * ((-1.0 / V) ^ 0.5));
	elseif ((V * l) <= 1e+301)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = sqrt(((c0 / V) * (c0 / l))) / (A ^ -0.5);
	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]}, If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(N[(t$95$0 * N[(c0 / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-310], N[(c0 * N[(t$95$0 / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 2e-321], N[(c0 * N[(N[Power[N[((-l) / A), $MachinePrecision], -0.5], $MachinePrecision] * N[Power[N[(-1.0 / V), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+301], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(c0 / V), $MachinePrecision] * N[(c0 / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Power[A, -0.5], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -inf.0

   1. Initial program 42.0

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

   2. Applied egg-rr 10.5

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

   3. Applied egg-rr 4.3

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

   4. Simplified 1.5

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

      [Start] 4.3	\[ \frac{\frac{\sqrt{-A} \cdot c0}{\sqrt{-V}}}{\sqrt{\ell}} \]
      *-lft-identity [<=] 4.3	\[ \frac{\frac{\sqrt{-A} \cdot c0}{\color{blue}{1 \cdot \sqrt{-V}}}}{\sqrt{\ell}} \]
      times-frac [=>] 1.5	\[ \frac{\color{blue}{\frac{\sqrt{-A}}{1} \cdot \frac{c0}{\sqrt{-V}}}}{\sqrt{\ell}} \]
      /-rgt-identity [=>] 1.5	\[ \frac{\color{blue}{\sqrt{-A}} \cdot \frac{c0}{\sqrt{-V}}}{\sqrt{\ell}} \]

   if -inf.0 < (*.f64 V l) < -1.999999999999994e-310

   1. Initial program 10.4

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

   2. Applied egg-rr 2.8

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

   3. Simplified 0.4

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

      [Start] 2.8	\[ \frac{c0 \cdot \sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}} \]
      *-commutative [<=] 2.8	\[ \frac{\color{blue}{\sqrt{-A} \cdot c0}}{\sqrt{V \cdot \left(-\ell\right)}} \]
      associate-/l* [=>] 2.6	\[ \color{blue}{\frac{\sqrt{-A}}{\frac{\sqrt{V \cdot \left(-\ell\right)}}{c0}}} \]
      associate-/r/ [=>] 0.4	\[ \color{blue}{\frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}} \cdot c0} \]

   if -1.999999999999994e-310 < (*.f64 V l) < 2.00097e-321

   1. Initial program 63.4

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

   2. Applied egg-rr 37.6

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

   3. Simplified 37.6

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

      [Start] 37.6	\[ \frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}} \]
      associate-/r/ [=>] 37.6	\[ \frac{c0}{\sqrt{\color{blue}{\frac{V}{A} \cdot \ell}}} \]

   4. Applied egg-rr 37.6

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

   5. Taylor expanded in V around -inf 30.3

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

   6. Simplified 27.0

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

      [Start] 30.3	\[ e^{-0.5 \cdot \left(\log \left(-1 \cdot \frac{\ell}{A}\right) + -1 \cdot \log \left(\frac{-1}{V}\right)\right)} \cdot c0 \]
      distribute-rgt-in [=>] 30.3	\[ e^{\color{blue}{\log \left(-1 \cdot \frac{\ell}{A}\right) \cdot -0.5 + \left(-1 \cdot \log \left(\frac{-1}{V}\right)\right) \cdot -0.5}} \cdot c0 \]
      exp-sum [=>] 30.0	\[ \color{blue}{\left(e^{\log \left(-1 \cdot \frac{\ell}{A}\right) \cdot -0.5} \cdot e^{\left(-1 \cdot \log \left(\frac{-1}{V}\right)\right) \cdot -0.5}\right)} \cdot c0 \]
      exp-to-pow [=>] 29.9	\[ \left(\color{blue}{{\left(-1 \cdot \frac{\ell}{A}\right)}^{-0.5}} \cdot e^{\left(-1 \cdot \log \left(\frac{-1}{V}\right)\right) \cdot -0.5}\right) \cdot c0 \]
      mul-1-neg [=>] 29.9	\[ \left({\color{blue}{\left(-\frac{\ell}{A}\right)}}^{-0.5} \cdot e^{\left(-1 \cdot \log \left(\frac{-1}{V}\right)\right) \cdot -0.5}\right) \cdot c0 \]
      distribute-neg-frac [=>] 29.9	\[ \left({\color{blue}{\left(\frac{-\ell}{A}\right)}}^{-0.5} \cdot e^{\left(-1 \cdot \log \left(\frac{-1}{V}\right)\right) \cdot -0.5}\right) \cdot c0 \]
      *-commutative [=>] 29.9	\[ \left({\left(\frac{-\ell}{A}\right)}^{-0.5} \cdot e^{\color{blue}{\left(\log \left(\frac{-1}{V}\right) \cdot -1\right)} \cdot -0.5}\right) \cdot c0 \]
      associate-*l* [=>] 29.9	\[ \left({\left(\frac{-\ell}{A}\right)}^{-0.5} \cdot e^{\color{blue}{\log \left(\frac{-1}{V}\right) \cdot \left(-1 \cdot -0.5\right)}}\right) \cdot c0 \]
      metadata-eval [=>] 29.9	\[ \left({\left(\frac{-\ell}{A}\right)}^{-0.5} \cdot e^{\log \left(\frac{-1}{V}\right) \cdot \color{blue}{0.5}}\right) \cdot c0 \]
      metadata-eval [<=] 29.9	\[ \left({\left(\frac{-\ell}{A}\right)}^{-0.5} \cdot e^{\log \left(\frac{-1}{V}\right) \cdot \color{blue}{\frac{1}{2}}}\right) \cdot c0 \]
      exp-to-pow [=>] 27.0	\[ \left({\left(\frac{-\ell}{A}\right)}^{-0.5} \cdot \color{blue}{{\left(\frac{-1}{V}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot c0 \]
      metadata-eval [=>] 27.0	\[ \left({\left(\frac{-\ell}{A}\right)}^{-0.5} \cdot {\left(\frac{-1}{V}\right)}^{\color{blue}{0.5}}\right) \cdot c0 \]

   if 2.00097e-321 < (*.f64 V l) < 1.00000000000000005e301

   1. Initial program 10.6

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

   2. Applied egg-rr 0.6

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

   3. Simplified 0.6

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

      [Start] 0.6	\[ c0 \cdot \left(\sqrt{A} \cdot \frac{1}{\sqrt{V \cdot \ell}}\right) \]
      *-commutative [<=] 0.6	\[ c0 \cdot \color{blue}{\left(\frac{1}{\sqrt{V \cdot \ell}} \cdot \sqrt{A}\right)} \]
      associate-*l/ [=>] 0.6	\[ c0 \cdot \color{blue}{\frac{1 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}} \]
      *-lft-identity [=>] 0.6	\[ c0 \cdot \frac{\color{blue}{\sqrt{A}}}{\sqrt{V \cdot \ell}} \]

   if 1.00000000000000005e301 < (*.f64 V l)

   1. Initial program 39.8

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

   2. Applied egg-rr 64.0

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

   3. Simplified 64.0

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

      [Start] 64.0	\[ \frac{c0 \cdot \sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}} \]
      associate-/l* [=>] 64.0	\[ \color{blue}{\frac{c0}{\frac{\sqrt{V \cdot \left(-\ell\right)}}{\sqrt{-A}}}} \]

   4. Applied egg-rr 39.2

      \[\leadsto \color{blue}{\frac{1}{\sqrt{V \cdot \ell}} \cdot \frac{c0}{{A}^{-0.5}}} \]

   5. Simplified 39.0

      \[\leadsto \color{blue}{\frac{\frac{c0}{\sqrt{V \cdot \ell}}}{{A}^{-0.5}}} \]
      Proof

      [Start] 39.2	\[ \frac{1}{\sqrt{V \cdot \ell}} \cdot \frac{c0}{{A}^{-0.5}} \]
      associate-*r/ [=>] 39.0	\[ \color{blue}{\frac{\frac{1}{\sqrt{V \cdot \ell}} \cdot c0}{{A}^{-0.5}}} \]
      associate-*l/ [=>] 39.0	\[ \frac{\color{blue}{\frac{1 \cdot c0}{\sqrt{V \cdot \ell}}}}{{A}^{-0.5}} \]
      *-lft-identity [=>] 39.0	\[ \frac{\frac{\color{blue}{c0}}{\sqrt{V \cdot \ell}}}{{A}^{-0.5}} \]

   6. Applied egg-rr 41.2

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

   7. Simplified 28.8

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{c0}{V} \cdot \frac{c0}{\ell}}}}{{A}^{-0.5}} \]
      Proof

      [Start] 41.2	\[ \frac{\sqrt{\frac{c0 \cdot c0}{V \cdot \ell}}}{{A}^{-0.5}} \]
      times-frac [=>] 28.8	\[ \frac{\sqrt{\color{blue}{\frac{c0}{V} \cdot \frac{c0}{\ell}}}}{{A}^{-0.5}} \]

3. Recombined 5 regimes into one program.

4. Final simplification 4.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{\sqrt{-A} \cdot \frac{c0}{\sqrt{-V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-321}:\\ \;\;\;\;c0 \cdot \left({\left(\frac{-\ell}{A}\right)}^{-0.5} \cdot {\left(\frac{-1}{V}\right)}^{0.5}\right)\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+301}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{c0}{V} \cdot \frac{c0}{\ell}}}{{A}^{-0.5}}\\ \end{array} \]

Alternatives

Alternative 1
Error	14.0
Cost	34641

\[\begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+268}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{elif}\;t_0 \leq -5 \cdot 10^{-293} \lor \neg \left(t_0 \leq 0\right) \land t_0 \leq 10^{+307}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot \left(\frac{c0}{V} \cdot \frac{c0}{\ell}\right)}\\ \end{array} \]

Alternative 2
Error	14.1
Cost	34641

\[\begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+268}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{elif}\;t_0 \leq -5 \cdot 10^{-293} \lor \neg \left(t_0 \leq 0\right) \land t_0 \leq 10^{+307}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{c0}{\ell} \cdot \frac{A \cdot c0}{V}}\\ \end{array} \]

Alternative 3
Error	5.7
Cost	14480

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -4 \cdot 10^{+276}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-321}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{-V}}{\sqrt{\frac{-A}{\ell}}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+301}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{c0}{V} \cdot \frac{c0}{\ell}}}{{A}^{-0.5}}\\ \end{array} \]

Alternative 4
Error	5.6
Cost	14480

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -4 \cdot 10^{+276}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-321}:\\ \;\;\;\;c0 \cdot \left({\left(\frac{-\ell}{A}\right)}^{-0.5} \cdot {\left(\frac{-1}{V}\right)}^{0.5}\right)\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+301}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{c0}{V} \cdot \frac{c0}{\ell}}}{{A}^{-0.5}}\\ \end{array} \]

Alternative 5
Error	9.5
Cost	14289

\[\begin{array}{l} t_0 := \sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+76}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-141}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq 5 \cdot 10^{+303}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \]

Alternative 6
Error	9.0
Cost	14288

\[\begin{array}{l} t_0 := \sqrt{\frac{A}{V}}\\ t_1 := \frac{c0}{\frac{\sqrt{\ell}}{t_0}}\\ \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+76}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-108}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t_1\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+303}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{c0}{\sqrt{\ell}}\\ \end{array} \]

Alternative 7
Error	7.2
Cost	14288

\[\begin{array}{l} t_0 := \sqrt{\frac{A}{V}}\\ \mathbf{if}\;V \cdot \ell \leq -6 \cdot 10^{+212}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{t_0}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{-A} \cdot \frac{c0}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0 \cdot t_0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+303}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{c0}{\sqrt{\ell}}\\ \end{array} \]

Alternative 8
Error	6.2
Cost	14288

\[\begin{array}{l} t_0 := \sqrt{\frac{A}{V}}\\ \mathbf{if}\;V \cdot \ell \leq -4 \cdot 10^{+276}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{t_0}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0 \cdot t_0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+303}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{c0}{\sqrt{\ell}}\\ \end{array} \]

Alternative 9
Error	6.1
Cost	14288

\[\begin{array}{l} t_0 := \sqrt{\frac{A}{V}}\\ \mathbf{if}\;V \cdot \ell \leq -4 \cdot 10^{+276}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{t_0}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-321}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{-V}}{\sqrt{\frac{-A}{\ell}}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+303}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{c0}{\sqrt{\ell}}\\ \end{array} \]

Alternative 10
Error	13.0
Cost	14028

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -10000000:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \frac{\frac{1}{\ell}}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-321}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+303}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]

Alternative 11
Error	15.0
Cost	7625

\[\begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0 \lor \neg \left(t_0 \leq 10^{+277}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \end{array} \]

Alternative 12
Error	15.0
Cost	7624

\[\begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t_0 \leq 4 \cdot 10^{+246}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \]

Alternative 13
Error	19.4
Cost	6848

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

Reproduce

herbie shell --seed 2023017 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))