Henrywood and Agarwal, Equation (3)

Average Accuracy: 70.1% → 91.3%

Time: 18.5s

Precision: binary64

Cost: 14416

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-258}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-319}:\\ \;\;\;\;c0 \cdot \left({\left(\frac{-1}{V}\right)}^{0.5} \cdot {\left(\frac{-\ell}{A}\right)}^{-0.5}\right)\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+307}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A} \cdot \sqrt{\frac{c0}{V} \cdot \frac{c0}{\ell}}\\ \end{array} \]

FPCore

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

↓

(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) (- INFINITY))
   (* (sqrt (/ A V)) (/ c0 (sqrt l)))
   (if (<= (* V l) -1e-258)
     (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
     (if (<= (* V l) 2e-319)
       (* c0 (* (pow (/ -1.0 V) 0.5) (pow (/ (- l) A) -0.5)))
       (if (<= (* V l) 1e+307)
         (/ c0 (/ (sqrt (* V l)) (sqrt A)))
         (* (sqrt A) (sqrt (* (/ c0 V) (/ c0 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 tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = sqrt((A / V)) * (c0 / sqrt(l));
	} else if ((V * l) <= -1e-258) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 2e-319) {
		tmp = c0 * (pow((-1.0 / V), 0.5) * pow((-l / A), -0.5));
	} else if ((V * l) <= 1e+307) {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	} else {
		tmp = sqrt(A) * sqrt(((c0 / V) * (c0 / 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 tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((A / V)) * (c0 / Math.sqrt(l));
	} else if ((V * l) <= -1e-258) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 2e-319) {
		tmp = c0 * (Math.pow((-1.0 / V), 0.5) * Math.pow((-l / A), -0.5));
	} else if ((V * l) <= 1e+307) {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	} else {
		tmp = Math.sqrt(A) * Math.sqrt(((c0 / V) * (c0 / l)));
	}
	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) <= -math.inf:
		tmp = math.sqrt((A / V)) * (c0 / math.sqrt(l))
	elif (V * l) <= -1e-258:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 2e-319:
		tmp = c0 * (math.pow((-1.0 / V), 0.5) * math.pow((-l / A), -0.5))
	elif (V * l) <= 1e+307:
		tmp = c0 / (math.sqrt((V * l)) / math.sqrt(A))
	else:
		tmp = math.sqrt(A) * math.sqrt(((c0 / V) * (c0 / 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)
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(sqrt(Float64(A / V)) * Float64(c0 / sqrt(l)));
	elseif (Float64(V * l) <= -1e-258)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 2e-319)
		tmp = Float64(c0 * Float64((Float64(-1.0 / V) ^ 0.5) * (Float64(Float64(-l) / A) ^ -0.5)));
	elseif (Float64(V * l) <= 1e+307)
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	else
		tmp = Float64(sqrt(A) * sqrt(Float64(Float64(c0 / V) * Float64(c0 / 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)
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = sqrt((A / V)) * (c0 / sqrt(l));
	elseif ((V * l) <= -1e-258)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 2e-319)
		tmp = c0 * (((-1.0 / V) ^ 0.5) * ((-l / A) ^ -0.5));
	elseif ((V * l) <= 1e+307)
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	else
		tmp = sqrt(A) * sqrt(((c0 / V) * (c0 / 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_] := If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] * N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-258], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 2e-319], 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], 1e+307], N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[A], $MachinePrecision] * N[Sqrt[N[(N[(c0 / V), $MachinePrecision] * N[(c0 / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 35.4%

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

   2. Applied egg-rr 85.2%

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

      [Start] 35.4	\[ c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
      associate-/r* [=>] 62.6	\[ c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]
      sqrt-div [=>] 85.2	\[ c0 \cdot \color{blue}{\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}} \]

   3. Applied egg-rr 84.4%

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

      [Start] 85.2	\[ c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}} \]
      associate-*r/ [=>] 85.2	\[ \color{blue}{\frac{c0 \cdot \sqrt{\frac{A}{V}}}{\sqrt{\ell}}} \]
      clear-num [=>] 84.4	\[ \color{blue}{\frac{1}{\frac{\sqrt{\ell}}{c0 \cdot \sqrt{\frac{A}{V}}}}} \]
      *-commutative [=>] 84.4	\[ \frac{1}{\frac{\sqrt{\ell}}{\color{blue}{\sqrt{\frac{A}{V}} \cdot c0}}} \]

   4. Simplified 84.6%

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

      [Start] 84.4	\[ \frac{1}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}} \cdot c0}} \]
      associate-/r* [=>] 84.3	\[ \frac{1}{\color{blue}{\frac{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}{c0}}} \]
      associate-/r/ [=>] 85.0	\[ \color{blue}{\frac{1}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}} \cdot c0} \]
      associate-*l/ [=>] 85.1	\[ \color{blue}{\frac{1 \cdot c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}} \]
      *-lft-identity [=>] 85.1	\[ \frac{\color{blue}{c0}}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}} \]
      associate-/r/ [=>] 84.6	\[ \color{blue}{\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}} \]
      *-commutative [=>] 84.6	\[ \color{blue}{\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}} \]

   if -inf.0 < (*.f64 V l) < -9.99999999999999954e-259

   1. Initial program 85.1%

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

   2. Applied egg-rr 99.4%

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

      [Start] 85.1	\[ c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
      frac-2neg [=>] 85.1	\[ c0 \cdot \sqrt{\color{blue}{\frac{-A}{-V \cdot \ell}}} \]
      sqrt-div [=>] 99.4	\[ c0 \cdot \color{blue}{\frac{\sqrt{-A}}{\sqrt{-V \cdot \ell}}} \]
      *-commutative [=>] 99.4	\[ c0 \cdot \frac{\sqrt{-A}}{\sqrt{-\color{blue}{\ell \cdot V}}} \]
      distribute-rgt-neg-in [=>] 99.4	\[ c0 \cdot \frac{\sqrt{-A}}{\sqrt{\color{blue}{\ell \cdot \left(-V\right)}}} \]

   if -9.99999999999999954e-259 < (*.f64 V l) < 1.99998e-319

   1. Initial program 11.9%

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

   2. Applied egg-rr 46.4%

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

      [Start] 11.9	\[ c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
      pow1/2 [=>] 11.9	\[ c0 \cdot \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{0.5}} \]
      clear-num [=>] 11.9	\[ c0 \cdot {\color{blue}{\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)}}^{0.5} \]
      inv-pow [=>] 11.9	\[ c0 \cdot {\color{blue}{\left({\left(\frac{V \cdot \ell}{A}\right)}^{-1}\right)}}^{0.5} \]
      pow-pow [=>] 12.0	\[ c0 \cdot \color{blue}{{\left(\frac{V \cdot \ell}{A}\right)}^{\left(-1 \cdot 0.5\right)}} \]
      associate-/l* [=>] 46.4	\[ c0 \cdot {\color{blue}{\left(\frac{V}{\frac{A}{\ell}}\right)}}^{\left(-1 \cdot 0.5\right)} \]
      metadata-eval [=>] 46.4	\[ c0 \cdot {\left(\frac{V}{\frac{A}{\ell}}\right)}^{\color{blue}{-0.5}} \]

   3. Simplified 46.5%

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

      [Start] 46.4	\[ c0 \cdot {\left(\frac{V}{\frac{A}{\ell}}\right)}^{-0.5} \]
      associate-/l* [<=] 12.0	\[ c0 \cdot {\color{blue}{\left(\frac{V \cdot \ell}{A}\right)}}^{-0.5} \]
      *-lft-identity [<=] 12.0	\[ c0 \cdot {\left(\frac{V \cdot \ell}{\color{blue}{1 \cdot A}}\right)}^{-0.5} \]
      times-frac [=>] 46.5	\[ c0 \cdot {\color{blue}{\left(\frac{V}{1} \cdot \frac{\ell}{A}\right)}}^{-0.5} \]
      /-rgt-identity [=>] 46.5	\[ c0 \cdot {\left(\color{blue}{V} \cdot \frac{\ell}{A}\right)}^{-0.5} \]

   4. Taylor expanded in V around -inf 53.7%

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

   5. Simplified 58.9%

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

      [Start] 53.7	\[ c0 \cdot e^{-0.5 \cdot \left(\log \left(-1 \cdot \frac{\ell}{A}\right) + -1 \cdot \log \left(\frac{-1}{V}\right)\right)} \]
      +-commutative [=>] 53.7	\[ c0 \cdot e^{-0.5 \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{-1}{V}\right) + \log \left(-1 \cdot \frac{\ell}{A}\right)\right)}} \]
      distribute-lft-in [=>] 53.7	\[ c0 \cdot e^{\color{blue}{-0.5 \cdot \left(-1 \cdot \log \left(\frac{-1}{V}\right)\right) + -0.5 \cdot \log \left(-1 \cdot \frac{\ell}{A}\right)}} \]
      *-commutative [<=] 53.7	\[ c0 \cdot e^{-0.5 \cdot \left(-1 \cdot \log \left(\frac{-1}{V}\right)\right) + \color{blue}{\log \left(-1 \cdot \frac{\ell}{A}\right) \cdot -0.5}} \]
      exp-sum [=>] 54.3	\[ c0 \cdot \color{blue}{\left(e^{-0.5 \cdot \left(-1 \cdot \log \left(\frac{-1}{V}\right)\right)} \cdot e^{\log \left(-1 \cdot \frac{\ell}{A}\right) \cdot -0.5}\right)} \]
      associate-*r* [=>] 54.3	\[ c0 \cdot \left(e^{\color{blue}{\left(-0.5 \cdot -1\right) \cdot \log \left(\frac{-1}{V}\right)}} \cdot e^{\log \left(-1 \cdot \frac{\ell}{A}\right) \cdot -0.5}\right) \]
      metadata-eval [=>] 54.3	\[ c0 \cdot \left(e^{\color{blue}{0.5} \cdot \log \left(\frac{-1}{V}\right)} \cdot e^{\log \left(-1 \cdot \frac{\ell}{A}\right) \cdot -0.5}\right) \]
      *-commutative [<=] 54.3	\[ c0 \cdot \left(e^{\color{blue}{\log \left(\frac{-1}{V}\right) \cdot 0.5}} \cdot e^{\log \left(-1 \cdot \frac{\ell}{A}\right) \cdot -0.5}\right) \]
      exp-to-pow [=>] 55.3	\[ c0 \cdot \left(\color{blue}{{\left(\frac{-1}{V}\right)}^{0.5}} \cdot e^{\log \left(-1 \cdot \frac{\ell}{A}\right) \cdot -0.5}\right) \]
      exp-to-pow [=>] 58.9	\[ c0 \cdot \left({\left(\frac{-1}{V}\right)}^{0.5} \cdot \color{blue}{{\left(-1 \cdot \frac{\ell}{A}\right)}^{-0.5}}\right) \]
      associate-*r/ [=>] 58.9	\[ c0 \cdot \left({\left(\frac{-1}{V}\right)}^{0.5} \cdot {\color{blue}{\left(\frac{-1 \cdot \ell}{A}\right)}}^{-0.5}\right) \]
      neg-mul-1 [<=] 58.9	\[ c0 \cdot \left({\left(\frac{-1}{V}\right)}^{0.5} \cdot {\left(\frac{\color{blue}{-\ell}}{A}\right)}^{-0.5}\right) \]

   if 1.99998e-319 < (*.f64 V l) < 9.99999999999999986e306

   1. Initial program 83.5%

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

   2. Applied egg-rr 95.8%

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

      [Start] 83.5	\[ c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
      sqrt-div [=>] 99.2	\[ c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}} \]
      associate-*r/ [=>] 95.8	\[ \color{blue}{\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}} \]

   3. Simplified 99.2%

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

      [Start] 95.8	\[ \frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}} \]
      associate-/l* [=>] 99.2	\[ \color{blue}{\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}} \]

   if 9.99999999999999986e306 < (*.f64 V l)

   1. Initial program 39.4%

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

   2. Applied egg-rr 39.3%

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

      [Start] 39.4	\[ c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
      sqrt-div [=>] 39.4	\[ c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}} \]
      associate-*r/ [=>] 39.3	\[ \color{blue}{\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}} \]

   3. Simplified 39.4%

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

      [Start] 39.3	\[ \frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}} \]
      associate-*l/ [<=] 39.4	\[ \color{blue}{\frac{c0}{\sqrt{V \cdot \ell}} \cdot \sqrt{A}} \]

   4. Applied egg-rr 38.6%

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

      [Start] 39.4	\[ \frac{c0}{\sqrt{V \cdot \ell}} \cdot \sqrt{A} \]
      add-sqr-sqrt [=>] 39.4	\[ \color{blue}{\left(\sqrt{\frac{c0}{\sqrt{V \cdot \ell}}} \cdot \sqrt{\frac{c0}{\sqrt{V \cdot \ell}}}\right)} \cdot \sqrt{A} \]
      sqrt-unprod [=>] 39.4	\[ \color{blue}{\sqrt{\frac{c0}{\sqrt{V \cdot \ell}} \cdot \frac{c0}{\sqrt{V \cdot \ell}}}} \cdot \sqrt{A} \]
      pow1/2 [=>] 39.4	\[ \color{blue}{{\left(\frac{c0}{\sqrt{V \cdot \ell}} \cdot \frac{c0}{\sqrt{V \cdot \ell}}\right)}^{0.5}} \cdot \sqrt{A} \]
      frac-times [=>] 38.6	\[ {\color{blue}{\left(\frac{c0 \cdot c0}{\sqrt{V \cdot \ell} \cdot \sqrt{V \cdot \ell}}\right)}}^{0.5} \cdot \sqrt{A} \]
      add-sqr-sqrt [<=] 38.6	\[ {\left(\frac{c0 \cdot c0}{\color{blue}{V \cdot \ell}}\right)}^{0.5} \cdot \sqrt{A} \]

   5. Simplified 60.1%

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

      [Start] 38.6	\[ {\left(\frac{c0 \cdot c0}{V \cdot \ell}\right)}^{0.5} \cdot \sqrt{A} \]
      unpow1/2 [=>] 38.6	\[ \color{blue}{\sqrt{\frac{c0 \cdot c0}{V \cdot \ell}}} \cdot \sqrt{A} \]
      times-frac [=>] 60.1	\[ \sqrt{\color{blue}{\frac{c0}{V} \cdot \frac{c0}{\ell}}} \cdot \sqrt{A} \]

3. Recombined 5 regimes into one program.

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-258}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-319}:\\ \;\;\;\;c0 \cdot \left({\left(\frac{-1}{V}\right)}^{0.5} \cdot {\left(\frac{-\ell}{A}\right)}^{-0.5}\right)\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+307}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A} \cdot \sqrt{\frac{c0}{V} \cdot \frac{c0}{\ell}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	74.7%
Cost	27724

\[\begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t_0 \leq -0.001:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t_0 \leq 10^{+291}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot \left(\frac{c0}{V} \cdot \frac{c0}{\ell}\right)}\\ \end{array} \]

Alternative 2
Accuracy	74.9%
Cost	27724

\[\begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t_0 \leq -0.001:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t_0 \leq 10^{+291}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{c0}{V} \cdot \left(A \cdot \frac{c0}{\ell}\right)}\\ \end{array} \]

Alternative 3
Accuracy	74.9%
Cost	27724

\[\begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t_0 \leq -0.001:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t_0 \leq 10^{+291}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{c0}{V} \cdot \frac{A}{\frac{\ell}{c0}}}\\ \end{array} \]

Alternative 4
Accuracy	74.9%
Cost	27724

\[\begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t_0 \leq -0.005:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\ell} \cdot \frac{1}{V}}\\ \mathbf{elif}\;t_0 \leq 10^{+291}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{c0}{V} \cdot \frac{A}{\frac{\ell}{c0}}}\\ \end{array} \]

Alternative 5
Accuracy	91.3%
Cost	20036

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

Alternative 6
Accuracy	91.2%
Cost	14416

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-258}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-319}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+307}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A} \cdot \sqrt{\frac{c0}{V} \cdot \frac{c0}{\ell}}\\ \end{array} \]

Alternative 7
Accuracy	86.1%
Cost	14288

\[\begin{array}{l} t_0 := \sqrt{\frac{A}{V}}\\ \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+178}:\\ \;\;\;\;t_0 \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-82}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{t_0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+307}:\\ \;\;\;\;\sqrt{A} \cdot \frac{c0}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]

Alternative 8
Accuracy	87.4%
Cost	14288

\[\begin{array}{l} t_0 := \sqrt{\frac{A}{V}}\\ \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+178}:\\ \;\;\;\;t_0 \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-82}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{t_0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+307}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]

Alternative 9
Accuracy	92.0%
Cost	14288

\[\begin{array}{l} t_0 := \sqrt{\frac{A}{V}}\\ \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;t_0 \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-297}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{t_0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+307}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]

Alternative 10
Accuracy	91.6%
Cost	14288

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-258}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-319}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+307}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]

Alternative 11
Accuracy	81.2%
Cost	14028

\[\begin{array}{l} t_0 := c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+178}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-82}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+136}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]

Alternative 12
Accuracy	81.0%
Cost	14028

\[\begin{array}{l} t_0 := \sqrt{\frac{A}{V}}\\ \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+178}:\\ \;\;\;\;t_0 \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-82}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{t_0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+136}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]

Alternative 13
Accuracy	76.7%
Cost	7890

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+178} \lor \neg \left(V \cdot \ell \leq -5 \cdot 10^{-189}\right) \land \left(V \cdot \ell \leq 2 \cdot 10^{-235} \lor \neg \left(V \cdot \ell \leq 5 \cdot 10^{+136}\right)\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]

Alternative 14
Accuracy	76.2%
Cost	7888

\[\begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ t_1 := c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+178}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-80}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-201}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+136}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 15
Accuracy	76.4%
Cost	7888

\[\begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+178}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-80}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-319}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+136}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 16
Accuracy	70.1%
Cost	6848

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

Reproduce

herbie shell --seed 2023146 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))