Henrywood and Agarwal, Equation (3)

Average Error: 19.2 → 6.2

Time: 13.2s

Precision: binary64

Cost: 14352

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+254}:\\ \;\;\;\;\frac{\frac{c0}{\sqrt{\ell}}}{{\left(\frac{A}{V}\right)}^{-0.5}}\\ \mathbf{elif}\;V \cdot \ell \leq -4 \cdot 10^{-304}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell \cdot \left(-V\right)}}{\sqrt{-A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+253}:\\ \;\;\;\;c0 \cdot \left({\left(V \cdot \ell\right)}^{-0.5} \cdot \sqrt{A}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(\ell \cdot \frac{V}{A}\right)}^{-0.5}\\ \end{array} \]

FPCore

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

↓

(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -5e+254)
   (/ (/ c0 (sqrt l)) (pow (/ A V) -0.5))
   (if (<= (* V l) -4e-304)
     (/ c0 (/ (sqrt (* l (- V))) (sqrt (- A))))
     (if (<= (* V l) 0.0)
       (/ c0 (/ (sqrt l) (sqrt (/ A V))))
       (if (<= (* V l) 1e+253)
         (* c0 (* (pow (* V l) -0.5) (sqrt A)))
         (* c0 (pow (* l (/ V 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 tmp;
	if ((V * l) <= -5e+254) {
		tmp = (c0 / sqrt(l)) / pow((A / V), -0.5);
	} else if ((V * l) <= -4e-304) {
		tmp = c0 / (sqrt((l * -V)) / sqrt(-A));
	} else if ((V * l) <= 0.0) {
		tmp = c0 / (sqrt(l) / sqrt((A / V)));
	} else if ((V * l) <= 1e+253) {
		tmp = c0 * (pow((V * l), -0.5) * sqrt(A));
	} else {
		tmp = c0 * pow((l * (V / A)), -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    code = c0 * sqrt((a / (v * l)))
end function

↓

real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-5d+254)) then
        tmp = (c0 / sqrt(l)) / ((a / v) ** (-0.5d0))
    else if ((v * l) <= (-4d-304)) then
        tmp = c0 / (sqrt((l * -v)) / sqrt(-a))
    else if ((v * l) <= 0.0d0) then
        tmp = c0 / (sqrt(l) / sqrt((a / v)))
    else if ((v * l) <= 1d+253) then
        tmp = c0 * (((v * l) ** (-0.5d0)) * sqrt(a))
    else
        tmp = c0 * ((l * (v / a)) ** (-0.5d0))
    end if
    code = tmp
end function

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) <= -5e+254) {
		tmp = (c0 / Math.sqrt(l)) / Math.pow((A / V), -0.5);
	} else if ((V * l) <= -4e-304) {
		tmp = c0 / (Math.sqrt((l * -V)) / Math.sqrt(-A));
	} else if ((V * l) <= 0.0) {
		tmp = c0 / (Math.sqrt(l) / Math.sqrt((A / V)));
	} else if ((V * l) <= 1e+253) {
		tmp = c0 * (Math.pow((V * l), -0.5) * Math.sqrt(A));
	} else {
		tmp = c0 * Math.pow((l * (V / 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):
	tmp = 0
	if (V * l) <= -5e+254:
		tmp = (c0 / math.sqrt(l)) / math.pow((A / V), -0.5)
	elif (V * l) <= -4e-304:
		tmp = c0 / (math.sqrt((l * -V)) / math.sqrt(-A))
	elif (V * l) <= 0.0:
		tmp = c0 / (math.sqrt(l) / math.sqrt((A / V)))
	elif (V * l) <= 1e+253:
		tmp = c0 * (math.pow((V * l), -0.5) * math.sqrt(A))
	else:
		tmp = c0 * math.pow((l * (V / 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)
	tmp = 0.0
	if (Float64(V * l) <= -5e+254)
		tmp = Float64(Float64(c0 / sqrt(l)) / (Float64(A / V) ^ -0.5));
	elseif (Float64(V * l) <= -4e-304)
		tmp = Float64(c0 / Float64(sqrt(Float64(l * Float64(-V))) / sqrt(Float64(-A))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 / Float64(sqrt(l) / sqrt(Float64(A / V))));
	elseif (Float64(V * l) <= 1e+253)
		tmp = Float64(c0 * Float64((Float64(V * l) ^ -0.5) * sqrt(A)));
	else
		tmp = Float64(c0 * (Float64(l * Float64(V / 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)
	tmp = 0.0;
	if ((V * l) <= -5e+254)
		tmp = (c0 / sqrt(l)) / ((A / V) ^ -0.5);
	elseif ((V * l) <= -4e-304)
		tmp = c0 / (sqrt((l * -V)) / sqrt(-A));
	elseif ((V * l) <= 0.0)
		tmp = c0 / (sqrt(l) / sqrt((A / V)));
	elseif ((V * l) <= 1e+253)
		tmp = c0 * (((V * l) ^ -0.5) * sqrt(A));
	else
		tmp = c0 * ((l * (V / 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_] := If[LessEqual[N[(V * l), $MachinePrecision], -5e+254], N[(N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Power[N[(A / V), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -4e-304], N[(c0 / N[(N[Sqrt[N[(l * (-V)), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-A)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 / N[(N[Sqrt[l], $MachinePrecision] / N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+253], N[(c0 * N[(N[Power[N[(V * l), $MachinePrecision], -0.5], $MachinePrecision] * N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Power[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -4.99999999999999994e254

   1. Initial program 32.9

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

   2. Applied egg-rr 9.4

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

   3. Simplified 8.1

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

      [Start] 9.4	\[ \frac{c0 \cdot \sqrt{\frac{A}{V}}}{\sqrt{\ell}} \]
      associate-/l* [=>] 8.1	\[ \color{blue}{\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}} \]

   4. Applied egg-rr 9.4

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

   5. Simplified 8.9

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

      [Start] 9.4	\[ \frac{1}{\sqrt{\ell}} \cdot \frac{c0}{{\left(\frac{A}{V}\right)}^{-0.5}} \]
      associate-*r/ [=>] 8.9	\[ \color{blue}{\frac{\frac{1}{\sqrt{\ell}} \cdot c0}{{\left(\frac{A}{V}\right)}^{-0.5}}} \]
      associate-*l/ [=>] 8.9	\[ \frac{\color{blue}{\frac{1 \cdot c0}{\sqrt{\ell}}}}{{\left(\frac{A}{V}\right)}^{-0.5}} \]
      associate-*r/ [<=] 8.9	\[ \frac{\color{blue}{1 \cdot \frac{c0}{\sqrt{\ell}}}}{{\left(\frac{A}{V}\right)}^{-0.5}} \]
      *-lft-identity [=>] 8.9	\[ \frac{\color{blue}{\frac{c0}{\sqrt{\ell}}}}{{\left(\frac{A}{V}\right)}^{-0.5}} \]

   if -4.99999999999999994e254 < (*.f64 V l) < -3.99999999999999988e-304

   1. Initial program 9.8

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

   2. Applied egg-rr 2.8

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

   3. Simplified 0.4

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

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

   if -3.99999999999999988e-304 < (*.f64 V l) < 0.0

   1. Initial program 61.6

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

   2. Applied egg-rr 29.4

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

   3. Simplified 28.5

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

      [Start] 29.4	\[ \frac{c0 \cdot \sqrt{\frac{A}{V}}}{\sqrt{\ell}} \]
      associate-/l* [=>] 28.5	\[ \color{blue}{\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}} \]

   if 0.0 < (*.f64 V l) < 9.9999999999999994e252

   1. Initial program 9.6

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

   2. Applied egg-rr 1.2

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

   3. Applied egg-rr 26.0

      \[\leadsto c0 \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left({\left(V \cdot \ell\right)}^{-0.5}\right)} - 1\right)} \cdot \sqrt{A}\right) \]

   4. Simplified 0.7

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

      [Start] 26.0	\[ c0 \cdot \left(\left(e^{\mathsf{log1p}\left({\left(V \cdot \ell\right)}^{-0.5}\right)} - 1\right) \cdot \sqrt{A}\right) \]
      expm1-def [=>] 2.7	\[ c0 \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(V \cdot \ell\right)}^{-0.5}\right)\right)} \cdot \sqrt{A}\right) \]
      expm1-log1p [=>] 0.7	\[ c0 \cdot \left(\color{blue}{{\left(V \cdot \ell\right)}^{-0.5}} \cdot \sqrt{A}\right) \]

   if 9.9999999999999994e252 < (*.f64 V l)

   1. Initial program 35.6

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

   2. Applied egg-rr 23.4

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

   3. Simplified 23.5

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

      [Start] 23.4	\[ \frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}} \]
      associate-/r/ [=>] 23.5	\[ \frac{c0}{\sqrt{\color{blue}{\frac{V}{A} \cdot \ell}}} \]

   4. Applied egg-rr 23.5

      \[\leadsto \color{blue}{{\left(\frac{V}{A} \cdot \ell\right)}^{-0.5} \cdot c0} \]
3. Recombined 5 regimes into one program.

4. Final simplification 6.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+254}:\\ \;\;\;\;\frac{\frac{c0}{\sqrt{\ell}}}{{\left(\frac{A}{V}\right)}^{-0.5}}\\ \mathbf{elif}\;V \cdot \ell \leq -4 \cdot 10^{-304}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell \cdot \left(-V\right)}}{\sqrt{-A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+253}:\\ \;\;\;\;c0 \cdot \left({\left(V \cdot \ell\right)}^{-0.5} \cdot \sqrt{A}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(\ell \cdot \frac{V}{A}\right)}^{-0.5}\\ \end{array} \]

Alternatives

Alternative 1
Error	14.8
Cost	34640

\[\begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+159}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \mathbf{elif}\;t_0 \leq -2 \cdot 10^{-192}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{-222}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+291}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot \frac{c0}{\frac{V}{\frac{c0}{\ell}}}}\\ \end{array} \]

Alternative 2
Error	15.1
Cost	34640

\[\begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+159}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \mathbf{elif}\;t_0 \leq -2 \cdot 10^{-192}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{-222}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+291}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{A \cdot \left(c0 \cdot \frac{c0}{V}\right)}{\ell}}\\ \end{array} \]

Alternative 3
Error	15.1
Cost	34640

\[\begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+159}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \mathbf{elif}\;t_0 \leq -2 \cdot 10^{-192}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{-222}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+291}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{c0 \cdot \frac{A}{\ell}}{\frac{V}{c0}}}\\ \end{array} \]

Alternative 4
Error	15.1
Cost	34640

\[\begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+159}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \mathbf{elif}\;t_0 \leq -2 \cdot 10^{-192}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{-222}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\ell} \cdot \frac{1}{V}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+291}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{c0 \cdot \frac{A}{\ell}}{\frac{V}{c0}}}\\ \end{array} \]

Alternative 5
Error	15.0
Cost	34512

\[\begin{array}{l} t_0 := \frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ t_1 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{-192}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{-222}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+288}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	15.5
Cost	34512

\[\begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ t_1 := \frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+159}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq -2 \cdot 10^{-192}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{-222}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+289}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	15.5
Cost	34512

\[\begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ t_1 := \frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+159}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq -2 \cdot 10^{-192}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{-222}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+289}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	5.9
Cost	20036

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{-V} \cdot \sqrt{\ell}}{\sqrt{-A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+253}:\\ \;\;\;\;c0 \cdot \left({\left(V \cdot \ell\right)}^{-0.5} \cdot \sqrt{A}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(\ell \cdot \frac{V}{A}\right)}^{-0.5}\\ \end{array} \]

Alternative 9
Error	8.9
Cost	14352

\[\begin{array}{l} t_0 := \frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+223}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-210}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+253}:\\ \;\;\;\;c0 \cdot \left({\left(V \cdot \ell\right)}^{-0.5} \cdot \sqrt{A}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(\ell \cdot \frac{V}{A}\right)}^{-0.5}\\ \end{array} \]

Alternative 10
Error	8.9
Cost	14352

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -4 \cdot 10^{+204}:\\ \;\;\;\;\frac{\frac{c0}{\sqrt{\ell}}}{{\left(\frac{A}{V}\right)}^{-0.5}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-210}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+253}:\\ \;\;\;\;c0 \cdot \left({\left(V \cdot \ell\right)}^{-0.5} \cdot \sqrt{A}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(\ell \cdot \frac{V}{A}\right)}^{-0.5}\\ \end{array} \]

Alternative 11
Error	9.3
Cost	14352

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+180}:\\ \;\;\;\;\frac{\frac{c0}{\sqrt{\ell}}}{{\left(\frac{A}{V}\right)}^{-0.5}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-7}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{-V}}{\sqrt{\frac{-A}{\ell}}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+253}:\\ \;\;\;\;c0 \cdot \left({\left(V \cdot \ell\right)}^{-0.5} \cdot \sqrt{A}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(\ell \cdot \frac{V}{A}\right)}^{-0.5}\\ \end{array} \]

Alternative 12
Error	11.5
Cost	14288

\[\begin{array}{l} t_0 := \ell \cdot \frac{V}{A}\\ \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+177}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-210}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{t_0}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+253}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {t_0}^{-0.5}\\ \end{array} \]

Alternative 13
Error	9.0
Cost	14288

\[\begin{array}{l} t_0 := c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+239}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-219}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+253}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(\ell \cdot \frac{V}{A}\right)}^{-0.5}\\ \end{array} \]

Alternative 14
Error	8.9
Cost	14288

\[\begin{array}{l} t_0 := \frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+223}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-210}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+253}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(\ell \cdot \frac{V}{A}\right)}^{-0.5}\\ \end{array} \]

Alternative 15
Error	15.3
Cost	7890

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+306} \lor \neg \left(V \cdot \ell \leq -2 \cdot 10^{-210} \lor \neg \left(V \cdot \ell \leq 0\right) \land V \cdot \ell \leq 2 \cdot 10^{+63}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]

Alternative 16
Error	15.6
Cost	7889

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+134}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-210} \lor \neg \left(V \cdot \ell \leq 0\right) \land V \cdot \ell \leq 2 \cdot 10^{+63}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]

Alternative 17
Error	19.2
Cost	6848

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

Reproduce

herbie shell --seed 2023012 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))