Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.2% → 86.2%

Time: 8.2s

Precision: binary64

Cost: 14028

• Unsound rule application detected in e-graph. Results may not be sound.

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\left(\sqrt{\frac{A}{V}} \cdot c0\right) \cdot {\ell}^{-0.5}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-323}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{-320}:\\ \;\;\;\;\sqrt{\frac{A \cdot \frac{c0}{\frac{\ell}{c0}}}{V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \end{array} \]

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) (pow l -0.5))
   (if (<= (* V l) -2e-323)
     (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
     (if (<= (* V l) 4e-320)
       (sqrt (/ (* A (/ c0 (/ l c0))) V))
       (/ c0 (/ (sqrt (* V l)) (sqrt A)))))))

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) * pow(l, -0.5);
	} else if ((V * l) <= -2e-323) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 4e-320) {
		tmp = sqrt(((A * (c0 / (l / c0))) / V));
	} else {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	}
	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.pow(l, -0.5);
	} else if ((V * l) <= -2e-323) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 4e-320) {
		tmp = Math.sqrt(((A * (c0 / (l / c0))) / V));
	} else {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	}
	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.pow(l, -0.5)
	elif (V * l) <= -2e-323:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 4e-320:
		tmp = math.sqrt(((A * (c0 / (l / c0))) / V))
	else:
		tmp = c0 / (math.sqrt((V * l)) / math.sqrt(A))
	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(Float64(sqrt(Float64(A / V)) * c0) * (l ^ -0.5));
	elseif (Float64(V * l) <= -2e-323)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 4e-320)
		tmp = sqrt(Float64(Float64(A * Float64(c0 / Float64(l / c0))) / V));
	else
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	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) * (l ^ -0.5);
	elseif ((V * l) <= -2e-323)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 4e-320)
		tmp = sqrt(((A * (c0 / (l / c0))) / V));
	else
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	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[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-323], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 4e-320], N[Sqrt[N[(N[(A * N[(c0 / N[(l / c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision], N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 40.8%

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

   2. Applied egg-rr 68.0%

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

      [Start] 40.8	\[ c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
      pow1/2 [=>] 40.8	\[ c0 \cdot \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{0.5}} \]
      clear-num [=>] 40.8	\[ c0 \cdot {\color{blue}{\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)}}^{0.5} \]
      inv-pow [=>] 40.8	\[ c0 \cdot {\color{blue}{\left({\left(\frac{V \cdot \ell}{A}\right)}^{-1}\right)}}^{0.5} \]
      pow-pow [=>] 40.8	\[ c0 \cdot \color{blue}{{\left(\frac{V \cdot \ell}{A}\right)}^{\left(-1 \cdot 0.5\right)}} \]
      associate-/l* [=>] 68.0	\[ c0 \cdot {\color{blue}{\left(\frac{V}{\frac{A}{\ell}}\right)}}^{\left(-1 \cdot 0.5\right)} \]
      metadata-eval [=>] 68.0	\[ c0 \cdot {\left(\frac{V}{\frac{A}{\ell}}\right)}^{\color{blue}{-0.5}} \]

   3. Simplified 40.8%

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

      [Start] 68.0	\[ c0 \cdot {\left(\frac{V}{\frac{A}{\ell}}\right)}^{-0.5} \]
      associate-/l* [<=] 40.8	\[ c0 \cdot {\color{blue}{\left(\frac{V \cdot \ell}{A}\right)}}^{-0.5} \]

   4. Applied egg-rr 68.2%

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

      [Start] 40.8	\[ c0 \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5} \]
      associate-*r/ [<=] 68.2	\[ c0 \cdot {\color{blue}{\left(V \cdot \frac{\ell}{A}\right)}}^{-0.5} \]
      metadata-eval [<=] 68.2	\[ c0 \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{\color{blue}{\left(-0.5\right)}} \]
      pow-flip [<=] 68.2	\[ c0 \cdot \color{blue}{\frac{1}{{\left(V \cdot \frac{\ell}{A}\right)}^{0.5}}} \]
      pow1/2 [<=] 68.2	\[ c0 \cdot \frac{1}{\color{blue}{\sqrt{V \cdot \frac{\ell}{A}}}} \]
      un-div-inv [=>] 68.2	\[ \color{blue}{\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}} \]

   5. Simplified 68.2%

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

      [Start] 68.2	\[ \frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}} \]
      associate-*r/ [=>] 40.8	\[ \frac{c0}{\sqrt{\color{blue}{\frac{V \cdot \ell}{A}}}} \]
      associate-*l/ [<=] 68.2	\[ \frac{c0}{\sqrt{\color{blue}{\frac{V}{A} \cdot \ell}}} \]
      *-commutative [<=] 68.2	\[ \frac{c0}{\sqrt{\color{blue}{\ell \cdot \frac{V}{A}}}} \]

   6. Applied egg-rr 40.0%

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

      [Start] 68.2	\[ \frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}} \]
      *-un-lft-identity [=>] 68.2	\[ \frac{\color{blue}{1 \cdot c0}}{\sqrt{\ell \cdot \frac{V}{A}}} \]
      add-sqr-sqrt [=>] 68.0	\[ \frac{1 \cdot c0}{\color{blue}{\sqrt{\sqrt{\ell \cdot \frac{V}{A}}} \cdot \sqrt{\sqrt{\ell \cdot \frac{V}{A}}}}} \]
      times-frac [=>] 68.0	\[ \color{blue}{\frac{1}{\sqrt{\sqrt{\ell \cdot \frac{V}{A}}}} \cdot \frac{c0}{\sqrt{\sqrt{\ell \cdot \frac{V}{A}}}}} \]

   7. Simplified 43.1%

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

      [Start] 40.0	\[ \sqrt{\frac{A}{\ell \cdot V} \cdot \left(c0 \cdot c0\right)} \]
      associate-*l/ [=>] 40.0	\[ \sqrt{\color{blue}{\frac{A \cdot \left(c0 \cdot c0\right)}{\ell \cdot V}}} \]
      *-commutative [<=] 40.0	\[ \sqrt{\frac{A \cdot \left(c0 \cdot c0\right)}{\color{blue}{V \cdot \ell}}} \]
      times-frac [=>] 43.1	\[ \sqrt{\color{blue}{\frac{A}{V} \cdot \frac{c0 \cdot c0}{\ell}}} \]

   8. Applied egg-rr 49.9%

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

      [Start] 43.1	\[ \sqrt{\frac{A}{V} \cdot \frac{c0 \cdot c0}{\ell}} \]
      associate-*r/ [=>] 48.0	\[ \sqrt{\color{blue}{\frac{\frac{A}{V} \cdot \left(c0 \cdot c0\right)}{\ell}}} \]
      add-sqr-sqrt [=>] 47.3	\[ \sqrt{\frac{\color{blue}{\left(\sqrt{\frac{A}{V}} \cdot \sqrt{\frac{A}{V}}\right)} \cdot \left(c0 \cdot c0\right)}{\ell}} \]
      unswap-sqr [=>] 47.8	\[ \sqrt{\frac{\color{blue}{\left(\sqrt{\frac{A}{V}} \cdot c0\right) \cdot \left(\sqrt{\frac{A}{V}} \cdot c0\right)}}{\ell}} \]
      add-sqr-sqrt [=>] 21.9	\[ \sqrt{\frac{\left(\sqrt{\frac{A}{V}} \cdot c0\right) \cdot \left(\sqrt{\frac{A}{V}} \cdot c0\right)}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}} \]
      frac-times [<=] 21.9	\[ \sqrt{\color{blue}{\frac{\sqrt{\frac{A}{V}} \cdot c0}{\sqrt{\ell}} \cdot \frac{\sqrt{\frac{A}{V}} \cdot c0}{\sqrt{\ell}}}} \]
      associate-*l/ [<=] 21.7	\[ \sqrt{\color{blue}{\left(\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}} \cdot c0\right)} \cdot \frac{\sqrt{\frac{A}{V}} \cdot c0}{\sqrt{\ell}}} \]
      associate-*l/ [<=] 21.7	\[ \sqrt{\left(\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}} \cdot c0\right) \cdot \color{blue}{\left(\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}} \cdot c0\right)}} \]
      sqrt-unprod [<=] 33.1	\[ \color{blue}{\sqrt{\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}} \cdot c0} \cdot \sqrt{\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}} \cdot c0}} \]
      add-sqr-sqrt [<=] 49.6	\[ \color{blue}{\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}} \cdot c0} \]
      associate-*l/ [=>] 49.7	\[ \color{blue}{\frac{\sqrt{\frac{A}{V}} \cdot c0}{\sqrt{\ell}}} \]
      div-inv [=>] 49.7	\[ \color{blue}{\left(\sqrt{\frac{A}{V}} \cdot c0\right) \cdot \frac{1}{\sqrt{\ell}}} \]
      pow1/2 [=>] 49.7	\[ \left(\sqrt{\frac{A}{V}} \cdot c0\right) \cdot \frac{1}{\color{blue}{{\ell}^{0.5}}} \]
      pow-flip [=>] 49.9	\[ \left(\sqrt{\frac{A}{V}} \cdot c0\right) \cdot \color{blue}{{\ell}^{\left(-0.5\right)}} \]
      metadata-eval [=>] 49.9	\[ \left(\sqrt{\frac{A}{V}} \cdot c0\right) \cdot {\ell}^{\color{blue}{-0.5}} \]

   if -inf.0 < (*.f64 V l) < -1.97626e-323

   1. Initial program 83.8%

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

   2. Applied egg-rr 98.8%

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

      [Start] 83.8	\[ c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
      frac-2neg [=>] 83.8	\[ c0 \cdot \sqrt{\color{blue}{\frac{-A}{-V \cdot \ell}}} \]
      sqrt-div [=>] 98.8	\[ c0 \cdot \color{blue}{\frac{\sqrt{-A}}{\sqrt{-V \cdot \ell}}} \]
      *-commutative [=>] 98.8	\[ c0 \cdot \frac{\sqrt{-A}}{\sqrt{-\color{blue}{\ell \cdot V}}} \]
      distribute-rgt-neg-in [=>] 98.8	\[ c0 \cdot \frac{\sqrt{-A}}{\sqrt{\color{blue}{\ell \cdot \left(-V\right)}}} \]

   if -1.97626e-323 < (*.f64 V l) < 3.99996e-320

   1. Initial program 35.9%

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

   2. Applied egg-rr 50.5%

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

      [Start] 35.9	\[ c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
      pow1/2 [=>] 35.9	\[ c0 \cdot \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{0.5}} \]
      clear-num [=>] 35.9	\[ c0 \cdot {\color{blue}{\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)}}^{0.5} \]
      inv-pow [=>] 35.9	\[ c0 \cdot {\color{blue}{\left({\left(\frac{V \cdot \ell}{A}\right)}^{-1}\right)}}^{0.5} \]
      pow-pow [=>] 35.9	\[ c0 \cdot \color{blue}{{\left(\frac{V \cdot \ell}{A}\right)}^{\left(-1 \cdot 0.5\right)}} \]
      associate-/l* [=>] 50.5	\[ c0 \cdot {\color{blue}{\left(\frac{V}{\frac{A}{\ell}}\right)}}^{\left(-1 \cdot 0.5\right)} \]
      metadata-eval [=>] 50.5	\[ c0 \cdot {\left(\frac{V}{\frac{A}{\ell}}\right)}^{\color{blue}{-0.5}} \]

   3. Simplified 35.9%

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

      [Start] 50.5	\[ c0 \cdot {\left(\frac{V}{\frac{A}{\ell}}\right)}^{-0.5} \]
      associate-/l* [<=] 35.9	\[ c0 \cdot {\color{blue}{\left(\frac{V \cdot \ell}{A}\right)}}^{-0.5} \]

   4. Applied egg-rr 50.5%

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

      [Start] 35.9	\[ c0 \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5} \]
      associate-*r/ [<=] 50.4	\[ c0 \cdot {\color{blue}{\left(V \cdot \frac{\ell}{A}\right)}}^{-0.5} \]
      metadata-eval [<=] 50.4	\[ c0 \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{\color{blue}{\left(-0.5\right)}} \]
      pow-flip [<=] 50.5	\[ c0 \cdot \color{blue}{\frac{1}{{\left(V \cdot \frac{\ell}{A}\right)}^{0.5}}} \]
      pow1/2 [<=] 50.5	\[ c0 \cdot \frac{1}{\color{blue}{\sqrt{V \cdot \frac{\ell}{A}}}} \]
      un-div-inv [=>] 50.5	\[ \color{blue}{\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}} \]

   5. Simplified 50.6%

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

      [Start] 50.5	\[ \frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}} \]
      associate-*r/ [=>] 35.9	\[ \frac{c0}{\sqrt{\color{blue}{\frac{V \cdot \ell}{A}}}} \]
      associate-*l/ [<=] 50.6	\[ \frac{c0}{\sqrt{\color{blue}{\frac{V}{A} \cdot \ell}}} \]
      *-commutative [<=] 50.6	\[ \frac{c0}{\sqrt{\color{blue}{\ell \cdot \frac{V}{A}}}} \]

   6. Applied egg-rr 24.7%

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

      [Start] 50.6	\[ \frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}} \]
      *-un-lft-identity [=>] 50.6	\[ \frac{\color{blue}{1 \cdot c0}}{\sqrt{\ell \cdot \frac{V}{A}}} \]
      add-sqr-sqrt [=>] 50.4	\[ \frac{1 \cdot c0}{\color{blue}{\sqrt{\sqrt{\ell \cdot \frac{V}{A}}} \cdot \sqrt{\sqrt{\ell \cdot \frac{V}{A}}}}} \]
      times-frac [=>] 50.4	\[ \color{blue}{\frac{1}{\sqrt{\sqrt{\ell \cdot \frac{V}{A}}}} \cdot \frac{c0}{\sqrt{\sqrt{\ell \cdot \frac{V}{A}}}}} \]

   7. Simplified 41.4%

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

      [Start] 24.7	\[ \sqrt{\frac{A}{\ell \cdot V} \cdot \left(c0 \cdot c0\right)} \]
      associate-*l/ [=>] 23.2	\[ \sqrt{\color{blue}{\frac{A \cdot \left(c0 \cdot c0\right)}{\ell \cdot V}}} \]
      *-commutative [<=] 23.2	\[ \sqrt{\frac{A \cdot \left(c0 \cdot c0\right)}{\color{blue}{V \cdot \ell}}} \]
      times-frac [=>] 41.4	\[ \sqrt{\color{blue}{\frac{A}{V} \cdot \frac{c0 \cdot c0}{\ell}}} \]

   8. Applied egg-rr 57.0%

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

      [Start] 41.4	\[ \sqrt{\frac{A}{V} \cdot \frac{c0 \cdot c0}{\ell}} \]
      associate-*l/ [=>] 40.9	\[ \sqrt{\color{blue}{\frac{A \cdot \frac{c0 \cdot c0}{\ell}}{V}}} \]
      associate-/l* [=>] 57.0	\[ \sqrt{\frac{A \cdot \color{blue}{\frac{c0}{\frac{\ell}{c0}}}}{V}} \]

   if 3.99996e-320 < (*.f64 V l)

   1. Initial program 82.3%

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

   2. Applied egg-rr 91.3%

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

      [Start] 82.3	\[ c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
      sqrt-div [=>] 94.9	\[ c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}} \]
      associate-*r/ [=>] 91.3	\[ \color{blue}{\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}} \]

   3. Simplified 95.0%

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

      [Start] 91.3	\[ \frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}} \]
      associate-/l* [=>] 95.0	\[ \color{blue}{\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\left(\sqrt{\frac{A}{V}} \cdot c0\right) \cdot {\ell}^{-0.5}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-323}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{-320}:\\ \;\;\;\;\sqrt{\frac{A \cdot \frac{c0}{\frac{\ell}{c0}}}{V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	78.4%
Cost	27724

\[\begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ t_1 := c0 \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-215}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{-302}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+305}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{A \cdot \frac{c0}{\frac{\ell}{c0}}}{V}}\\ \end{array} \]

Alternative 2
Accuracy	78.1%
Cost	27724

\[\begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-124}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\frac{V \cdot \ell}{A}}}{c0}}\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \frac{1}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+305}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{A \cdot \frac{c0}{\frac{\ell}{c0}}}{V}}\\ \end{array} \]

Alternative 3
Accuracy	81.4%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 4
Accuracy	83.7%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{A} \cdot \frac{c0}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 5
Accuracy	79.2%
Cost	7625

\[\begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0 \lor \neg \left(t_0 \leq 10^{+241}\right):\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \end{array} \]

Alternative 6
Accuracy	78.1%
Cost	7625

\[\begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-234} \lor \neg \left(t_0 \leq 10^{+276}\right):\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \end{array} \]

Alternative 7
Accuracy	75.3%
Cost	7300

\[\begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 4 \cdot 10^{-230}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}\\ \end{array} \]

Alternative 8
Accuracy	75.8%
Cost	7236

\[\begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-246}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \end{array} \]

Alternative 9
Accuracy	73.2%
Cost	6848

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

Reproduce

herbie shell --seed 2023162 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))