Henrywood and Agarwal, Equation (3)

Average Accuracy: 63.2% → 81.7%

Time: 18.2s

Precision: binary64

Cost: 14348

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

Math

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

↓

\[\begin{array}{l} t_0 := c0 \cdot \left({\left(\frac{-1}{V}\right)}^{0.5} \cdot {\left(\frac{-\ell}{A}\right)}^{-0.5}\right)\\ \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-295}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+296}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \end{array} \]

FPCore

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

↓

(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (* (pow (/ -1.0 V) 0.5) (pow (/ (- l) A) -0.5)))))
   (if (<= (* V l) (- INFINITY))
     t_0
     (if (<= (* V l) -2e-295)
       (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
       (if (<= (* V l) 0.0)
         t_0
         (if (<= (* V l) 4e+296)
           (/ c0 (/ (sqrt (* V l)) (sqrt A)))
           (* c0 (/ (sqrt (/ A V)) (sqrt l)))))))))

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 = c0 * (pow((-1.0 / V), 0.5) * pow((-l / A), -0.5));
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = t_0;
	} else if ((V * l) <= -2e-295) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else if ((V * l) <= 4e+296) {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	} else {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	}
	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 = c0 * (Math.pow((-1.0 / V), 0.5) * Math.pow((-l / A), -0.5));
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else if ((V * l) <= -2e-295) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else if ((V * l) <= 4e+296) {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	} else {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	}
	return tmp;
}

Python

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

↓

def code(c0, A, V, l):
	t_0 = c0 * (math.pow((-1.0 / V), 0.5) * math.pow((-l / A), -0.5))
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = t_0
	elif (V * l) <= -2e-295:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 0.0:
		tmp = t_0
	elif (V * l) <= 4e+296:
		tmp = c0 / (math.sqrt((V * l)) / math.sqrt(A))
	else:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	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 = Float64(c0 * Float64((Float64(-1.0 / V) ^ 0.5) * (Float64(Float64(-l) / A) ^ -0.5)))
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = t_0;
	elseif (Float64(V * l) <= -2e-295)
		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) <= 4e+296)
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	else
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	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 = c0 * (((-1.0 / V) ^ 0.5) * ((-l / A) ^ -0.5));
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = t_0;
	elseif ((V * l) <= -2e-295)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 0.0)
		tmp = t_0;
	elseif ((V * l) <= 4e+296)
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	else
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	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[(c0 * N[(N[Power[N[(-1.0 / V), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[N[((-l) / A), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -2e-295], 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], 4e+296], N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -inf.0 or -2.00000000000000012e-295 < (*.f64 V l) < 0.0

   1. Initial program 14.9%

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

   2. Applied egg-rr 43.0%

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

      [Start] 14.9	\[ c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
      pow1/2 [=>] 14.9	\[ c0 \cdot \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{0.5}} \]
      clear-num [=>] 14.9	\[ c0 \cdot {\color{blue}{\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)}}^{0.5} \]
      inv-pow [=>] 14.9	\[ c0 \cdot {\color{blue}{\left({\left(\frac{V \cdot \ell}{A}\right)}^{-1}\right)}}^{0.5} \]
      pow-pow [=>] 14.9	\[ c0 \cdot \color{blue}{{\left(\frac{V \cdot \ell}{A}\right)}^{\left(-1 \cdot 0.5\right)}} \]
      associate-/l* [=>] 43.0	\[ c0 \cdot {\color{blue}{\left(\frac{V}{\frac{A}{\ell}}\right)}}^{\left(-1 \cdot 0.5\right)} \]
      metadata-eval [=>] 43.0	\[ c0 \cdot {\left(\frac{V}{\frac{A}{\ell}}\right)}^{\color{blue}{-0.5}} \]

   3. Simplified 43.0%

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

      [Start] 43.0	\[ c0 \cdot {\left(\frac{V}{\frac{A}{\ell}}\right)}^{-0.5} \]
      associate-/l* [<=] 14.9	\[ c0 \cdot {\color{blue}{\left(\frac{V \cdot \ell}{A}\right)}}^{-0.5} \]
      *-lft-identity [<=] 14.9	\[ c0 \cdot {\left(\frac{V \cdot \ell}{\color{blue}{1 \cdot A}}\right)}^{-0.5} \]
      times-frac [=>] 43.0	\[ c0 \cdot {\color{blue}{\left(\frac{V}{1} \cdot \frac{\ell}{A}\right)}}^{-0.5} \]
      /-rgt-identity [=>] 43.0	\[ c0 \cdot {\left(\color{blue}{V} \cdot \frac{\ell}{A}\right)}^{-0.5} \]

   4. Taylor expanded in V around -inf 52.1%

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

   5. Simplified 56.1%

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

      [Start] 52.1	\[ c0 \cdot e^{-0.5 \cdot \left(\log \left(-1 \cdot \frac{\ell}{A}\right) + -1 \cdot \log \left(\frac{-1}{V}\right)\right)} \]
      +-commutative [=>] 52.1	\[ c0 \cdot e^{-0.5 \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{-1}{V}\right) + \log \left(-1 \cdot \frac{\ell}{A}\right)\right)}} \]
      distribute-lft-in [=>] 52.1	\[ c0 \cdot e^{\color{blue}{-0.5 \cdot \left(-1 \cdot \log \left(\frac{-1}{V}\right)\right) + -0.5 \cdot \log \left(-1 \cdot \frac{\ell}{A}\right)}} \]
      *-commutative [<=] 52.1	\[ c0 \cdot e^{-0.5 \cdot \left(-1 \cdot \log \left(\frac{-1}{V}\right)\right) + \color{blue}{\log \left(-1 \cdot \frac{\ell}{A}\right) \cdot -0.5}} \]
      exp-sum [=>] 52.5	\[ c0 \cdot \color{blue}{\left(e^{-0.5 \cdot \left(-1 \cdot \log \left(\frac{-1}{V}\right)\right)} \cdot e^{\log \left(-1 \cdot \frac{\ell}{A}\right) \cdot -0.5}\right)} \]
      associate-*r* [=>] 52.5	\[ c0 \cdot \left(e^{\color{blue}{\left(-0.5 \cdot -1\right) \cdot \log \left(\frac{-1}{V}\right)}} \cdot e^{\log \left(-1 \cdot \frac{\ell}{A}\right) \cdot -0.5}\right) \]
      metadata-eval [=>] 52.5	\[ c0 \cdot \left(e^{\color{blue}{0.5} \cdot \log \left(\frac{-1}{V}\right)} \cdot e^{\log \left(-1 \cdot \frac{\ell}{A}\right) \cdot -0.5}\right) \]
      *-commutative [<=] 52.5	\[ c0 \cdot \left(e^{\color{blue}{\log \left(\frac{-1}{V}\right) \cdot 0.5}} \cdot e^{\log \left(-1 \cdot \frac{\ell}{A}\right) \cdot -0.5}\right) \]
      exp-to-pow [=>] 53.4	\[ c0 \cdot \left(\color{blue}{{\left(\frac{-1}{V}\right)}^{0.5}} \cdot e^{\log \left(-1 \cdot \frac{\ell}{A}\right) \cdot -0.5}\right) \]
      exp-to-pow [=>] 56.1	\[ c0 \cdot \left({\left(\frac{-1}{V}\right)}^{0.5} \cdot \color{blue}{{\left(-1 \cdot \frac{\ell}{A}\right)}^{-0.5}}\right) \]
      associate-*r/ [=>] 56.1	\[ c0 \cdot \left({\left(\frac{-1}{V}\right)}^{0.5} \cdot {\color{blue}{\left(\frac{-1 \cdot \ell}{A}\right)}}^{-0.5}\right) \]
      neg-mul-1 [<=] 56.1	\[ c0 \cdot \left({\left(\frac{-1}{V}\right)}^{0.5} \cdot {\left(\frac{\color{blue}{-\ell}}{A}\right)}^{-0.5}\right) \]

   if -inf.0 < (*.f64 V l) < -2.00000000000000012e-295

   1. Initial program 77.5%

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

   2. Applied egg-rr 92.1%

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

      [Start] 77.5	\[ c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
      frac-2neg [=>] 77.5	\[ c0 \cdot \sqrt{\color{blue}{\frac{-A}{-V \cdot \ell}}} \]
      sqrt-div [=>] 92.1	\[ c0 \cdot \color{blue}{\frac{\sqrt{-A}}{\sqrt{-V \cdot \ell}}} \]
      *-commutative [=>] 92.1	\[ c0 \cdot \frac{\sqrt{-A}}{\sqrt{-\color{blue}{\ell \cdot V}}} \]
      distribute-rgt-neg-in [=>] 92.1	\[ c0 \cdot \frac{\sqrt{-A}}{\sqrt{\color{blue}{\ell \cdot \left(-V\right)}}} \]

   if 0.0 < (*.f64 V l) < 3.99999999999999993e296

   1. Initial program 76.3%

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

   2. Applied egg-rr 87.1%

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

      [Start] 76.3	\[ c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
      sqrt-div [=>] 90.5	\[ c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}} \]
      associate-*r/ [=>] 87.1	\[ \color{blue}{\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}} \]

   3. Simplified 90.6%

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

      [Start] 87.1	\[ \frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}} \]
      associate-/l* [=>] 90.6	\[ \color{blue}{\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}} \]

   if 3.99999999999999993e296 < (*.f64 V l)

   1. Initial program 40.4%

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

   2. Applied egg-rr 45.1%

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

      [Start] 40.4	\[ c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
      associate-/r* [=>] 66.5	\[ c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]
      sqrt-div [=>] 45.1	\[ c0 \cdot \color{blue}{\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \left({\left(\frac{-1}{V}\right)}^{0.5} \cdot {\left(\frac{-\ell}{A}\right)}^{-0.5}\right)\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-295}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \left({\left(\frac{-1}{V}\right)}^{0.5} \cdot {\left(\frac{-\ell}{A}\right)}^{-0.5}\right)\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+296}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	81.5%
Cost	20036

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\frac{\sqrt{-A}}{\sqrt{-V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+296}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 2
Accuracy	75.5%
Cost	14808

\[\begin{array}{l} t_0 := c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ t_1 := \sqrt{A} \cdot \frac{c0}{\sqrt{V \cdot \ell}}\\ \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+221}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-137}:\\ \;\;\;\;\frac{c0}{{\left(\frac{A}{V \cdot \ell}\right)}^{-0.5}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-316}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-259}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-181}:\\ \;\;\;\;c0 \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+296}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Accuracy	75.5%
Cost	14808

\[\begin{array}{l} t_0 := \sqrt{\frac{A}{V}}\\ t_1 := \frac{c0}{\frac{\sqrt{\ell}}{t_0}}\\ t_2 := \sqrt{A} \cdot \frac{c0}{\sqrt{V \cdot \ell}}\\ \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+221}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-137}:\\ \;\;\;\;\frac{c0}{{\left(\frac{A}{V \cdot \ell}\right)}^{-0.5}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-316}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-259}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-181}:\\ \;\;\;\;c0 \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+296}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{t_0}{\sqrt{\ell}}\\ \end{array} \]

Alternative 4
Accuracy	77.6%
Cost	14288

\[\begin{array}{l} t_0 := \sqrt{\frac{A}{V}}\\ t_1 := \frac{c0}{\frac{\sqrt{\ell}}{t_0}}\\ \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+221}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-137}:\\ \;\;\;\;\frac{c0}{{\left(\frac{A}{V \cdot \ell}\right)}^{-0.5}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-316}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+296}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{t_0}{\sqrt{\ell}}\\ \end{array} \]

Alternative 5
Accuracy	77.6%
Cost	14288

\[\begin{array}{l} t_0 := \sqrt{\frac{A}{V}}\\ \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+221}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{t_0}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-162}:\\ \;\;\;\;\frac{c0}{{\left(\frac{A}{V \cdot \ell}\right)}^{-0.5}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-316}:\\ \;\;\;\;c0 \cdot \left(t_0 \cdot {\ell}^{-0.5}\right)\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+296}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{t_0}{\sqrt{\ell}}\\ \end{array} \]

Alternative 6
Accuracy	81.8%
Cost	14288

\[\begin{array}{l} t_0 := \sqrt{\frac{A}{V}}\\ t_1 := \frac{c0}{\frac{\sqrt{\ell}}{t_0}}\\ \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-318}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-316}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+296}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{t_0}{\sqrt{\ell}}\\ \end{array} \]

Alternative 7
Accuracy	81.4%
Cost	14288

\[\begin{array}{l} t_0 := \sqrt{\frac{A}{V}}\\ \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{t_0}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-268}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+296}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{t_0}{\sqrt{\ell}}\\ \end{array} \]

Alternative 8
Accuracy	71.3%
Cost	13645

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.2 \cdot 10^{-300}:\\ \;\;\;\;\frac{c0}{{\left(\frac{\frac{A}{V}}{\ell}\right)}^{-0.5}}\\ \mathbf{elif}\;\ell \leq 1.05 \cdot 10^{+123} \lor \neg \left(\ell \leq 1.9 \cdot 10^{+170}\right):\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{{\left(\frac{A}{V \cdot \ell}\right)}^{-0.5}}\\ \end{array} \]

Alternative 9
Accuracy	70.4%
Cost	7752

\[\begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-324}:\\ \;\;\;\;\frac{c0}{{\left(\frac{\frac{A}{V}}{\ell}\right)}^{-0.5}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\frac{c0}{{t_0}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(A \cdot \frac{c0}{V}\right) \cdot \frac{c0}{\ell}}\\ \end{array} \]

Alternative 10
Accuracy	70.4%
Cost	7752

\[\begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-324}:\\ \;\;\;\;\frac{c0}{\frac{1}{\sqrt{\frac{A}{V} \cdot \frac{1}{\ell}}}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\frac{c0}{{t_0}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(A \cdot \frac{c0}{V}\right) \cdot \frac{c0}{\ell}}\\ \end{array} \]

Alternative 11
Accuracy	70.4%
Cost	7688

\[\begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+302}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(\frac{V}{\frac{A}{\ell}}\right)}^{-0.5}\\ \end{array} \]

Alternative 12
Accuracy	70.4%
Cost	7688

\[\begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\frac{c0}{{t_0}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(\frac{V}{\frac{A}{\ell}}\right)}^{-0.5}\\ \end{array} \]

Alternative 13
Accuracy	70.4%
Cost	7688

\[\begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-324}:\\ \;\;\;\;\frac{c0}{{\left(\frac{\frac{A}{V}}{\ell}\right)}^{-0.5}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\frac{c0}{{t_0}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(\frac{V}{\frac{A}{\ell}}\right)}^{-0.5}\\ \end{array} \]

Alternative 14
Accuracy	69.9%
Cost	7625

\[\begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0 \lor \neg \left(t_0 \leq 5 \cdot 10^{+302}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \end{array} \]

Alternative 15
Accuracy	70.1%
Cost	7624

\[\begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+291}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \]

Alternative 16
Accuracy	70.4%
Cost	7624

\[\begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+302}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]

Alternative 17
Accuracy	70.4%
Cost	7624

\[\begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+302}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \end{array} \]

Alternative 18
Accuracy	63.2%
Cost	6848

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

Reproduce

herbie shell --seed 2023153 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))