Henrywood and Agarwal, Equation (3)

Average Error: 18.6 → 6.2

Time: 15.9s

Precision: binary64

Cost: 20556

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

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{-V}\\ t_1 := \sqrt{-A}\\ \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+209}:\\ \;\;\;\;\frac{\frac{t_1 \cdot c0}{t_0}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-294}:\\ \;\;\;\;t_1 \cdot \frac{c0}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{t_1}{t_0 \cdot \frac{\sqrt{\ell}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+290}:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{1}{V \cdot \ell}} \cdot \sqrt{A}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]

FPCore

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

↓

(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (sqrt (- V))) (t_1 (sqrt (- A))))
   (if (<= (* V l) -2e+209)
     (/ (/ (* t_1 c0) t_0) (sqrt l))
     (if (<= (* V l) -2e-294)
       (* t_1 (/ c0 (sqrt (* V (- l)))))
       (if (<= (* V l) 0.0)
         (/ t_1 (* t_0 (/ (sqrt l) c0)))
         (if (<= (* V l) 5e+290)
           (* c0 (* (sqrt (/ 1.0 (* V l))) (sqrt A)))
           (* c0 (sqrt (/ (/ A l) V)))))))))

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(-V);
	double t_1 = sqrt(-A);
	double tmp;
	if ((V * l) <= -2e+209) {
		tmp = ((t_1 * c0) / t_0) / sqrt(l);
	} else if ((V * l) <= -2e-294) {
		tmp = t_1 * (c0 / sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_1 / (t_0 * (sqrt(l) / c0));
	} else if ((V * l) <= 5e+290) {
		tmp = c0 * (sqrt((1.0 / (V * l))) * sqrt(A));
	} else {
		tmp = c0 * sqrt(((A / l) / V));
	}
	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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt(-v)
    t_1 = sqrt(-a)
    if ((v * l) <= (-2d+209)) then
        tmp = ((t_1 * c0) / t_0) / sqrt(l)
    else if ((v * l) <= (-2d-294)) then
        tmp = t_1 * (c0 / sqrt((v * -l)))
    else if ((v * l) <= 0.0d0) then
        tmp = t_1 / (t_0 * (sqrt(l) / c0))
    else if ((v * l) <= 5d+290) then
        tmp = c0 * (sqrt((1.0d0 / (v * l))) * sqrt(a))
    else
        tmp = c0 * sqrt(((a / l) / v))
    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 t_0 = Math.sqrt(-V);
	double t_1 = Math.sqrt(-A);
	double tmp;
	if ((V * l) <= -2e+209) {
		tmp = ((t_1 * c0) / t_0) / Math.sqrt(l);
	} else if ((V * l) <= -2e-294) {
		tmp = t_1 * (c0 / Math.sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_1 / (t_0 * (Math.sqrt(l) / c0));
	} else if ((V * l) <= 5e+290) {
		tmp = c0 * (Math.sqrt((1.0 / (V * l))) * Math.sqrt(A));
	} else {
		tmp = c0 * Math.sqrt(((A / l) / V));
	}
	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(-V)
	t_1 = math.sqrt(-A)
	tmp = 0
	if (V * l) <= -2e+209:
		tmp = ((t_1 * c0) / t_0) / math.sqrt(l)
	elif (V * l) <= -2e-294:
		tmp = t_1 * (c0 / math.sqrt((V * -l)))
	elif (V * l) <= 0.0:
		tmp = t_1 / (t_0 * (math.sqrt(l) / c0))
	elif (V * l) <= 5e+290:
		tmp = c0 * (math.sqrt((1.0 / (V * l))) * math.sqrt(A))
	else:
		tmp = c0 * math.sqrt(((A / l) / V))
	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(-V))
	t_1 = sqrt(Float64(-A))
	tmp = 0.0
	if (Float64(V * l) <= -2e+209)
		tmp = Float64(Float64(Float64(t_1 * c0) / t_0) / sqrt(l));
	elseif (Float64(V * l) <= -2e-294)
		tmp = Float64(t_1 * Float64(c0 / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(t_1 / Float64(t_0 * Float64(sqrt(l) / c0)));
	elseif (Float64(V * l) <= 5e+290)
		tmp = Float64(c0 * Float64(sqrt(Float64(1.0 / Float64(V * l))) * sqrt(A)));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	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(-V);
	t_1 = sqrt(-A);
	tmp = 0.0;
	if ((V * l) <= -2e+209)
		tmp = ((t_1 * c0) / t_0) / sqrt(l);
	elseif ((V * l) <= -2e-294)
		tmp = t_1 * (c0 / sqrt((V * -l)));
	elseif ((V * l) <= 0.0)
		tmp = t_1 / (t_0 * (sqrt(l) / c0));
	elseif ((V * l) <= 5e+290)
		tmp = c0 * (sqrt((1.0 / (V * l))) * sqrt(A));
	else
		tmp = c0 * sqrt(((A / l) / V));
	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[(-V)], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[(-A)], $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -2e+209], N[(N[(N[(t$95$1 * c0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-294], N[(t$95$1 * N[(c0 / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(t$95$1 / N[(t$95$0 * N[(N[Sqrt[l], $MachinePrecision] / c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e+290], N[(c0 * N[(N[Sqrt[N[(1.0 / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -2.0000000000000001e209

   1. Initial program 30.4

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

   2. Applied egg-rr 9.6

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

   3. Applied egg-rr 4.7

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

   if -2.0000000000000001e209 < (*.f64 V l) < -2.00000000000000003e-294

   1. Initial program 7.9

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

   2. Applied egg-rr 2.5

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

   3. Simplified 2.6

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

      [Start] 2.5	\[ \frac{c0 \cdot \sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}} \]
      associate-/l* [=>] 0.4	\[ \color{blue}{\frac{c0}{\frac{\sqrt{V \cdot \left(-\ell\right)}}{\sqrt{-A}}}} \]
      associate-/r/ [=>] 2.6	\[ \color{blue}{\frac{c0}{\sqrt{V \cdot \left(-\ell\right)}} \cdot \sqrt{-A}} \]

   if -2.00000000000000003e-294 < (*.f64 V l) < 0.0

   1. Initial program 60.9

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

   2. Applied egg-rr 26.7

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

   3. Simplified 26.4

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

      [Start] 26.7	\[ \frac{c0 \cdot \sqrt{\frac{A}{V}}}{\sqrt{\ell}} \]
      *-commutative [=>] 26.7	\[ \frac{\color{blue}{\sqrt{\frac{A}{V}} \cdot c0}}{\sqrt{\ell}} \]
      associate-/l* [=>] 26.5	\[ \color{blue}{\frac{\sqrt{\frac{A}{V}}}{\frac{\sqrt{\ell}}{c0}}} \]
      associate-/r/ [=>] 26.4	\[ \color{blue}{\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}} \cdot c0} \]

   4. Applied egg-rr 29.4

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

   if 0.0 < (*.f64 V l) < 4.9999999999999998e290

   1. Initial program 10.1

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

   2. Applied egg-rr 1.1

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

   if 4.9999999999999998e290 < (*.f64 V l)

   1. Initial program 38.1

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

   2. Applied egg-rr 21.1

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

   3. Applied egg-rr 21.1

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{\ell}}{V}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 6.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+209}:\\ \;\;\;\;\frac{\frac{\sqrt{-A} \cdot c0}{\sqrt{-V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-294}:\\ \;\;\;\;\sqrt{-A} \cdot \frac{c0}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{-V} \cdot \frac{\sqrt{\ell}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+290}:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{1}{V \cdot \ell}} \cdot \sqrt{A}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]

Alternatives

Alternative 1
Error	12.7
Cost	34640

\[\begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{+299}:\\ \;\;\;\;c0 \cdot {\left(\ell \cdot \frac{V}{A}\right)}^{-0.5}\\ \mathbf{elif}\;t_0 \leq -1.2 \cdot 10^{-318}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;t_0 \leq 10^{-276}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+300}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot \left(\frac{c0}{V} \cdot \frac{c0}{\ell}\right)}\\ \end{array} \]

Alternative 2
Error	12.7
Cost	34640

\[\begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{+299}:\\ \;\;\;\;c0 \cdot {\left(\ell \cdot \frac{V}{A}\right)}^{-0.5}\\ \mathbf{elif}\;t_0 \leq -1.2 \cdot 10^{-318}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;t_0 \leq 10^{-276}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+300}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{A \cdot c0}{\ell} \cdot \frac{c0}{V}}\\ \end{array} \]

Alternative 3
Error	12.7
Cost	34640

\[\begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{+299}:\\ \;\;\;\;c0 \cdot {\left(\ell \cdot \frac{V}{A}\right)}^{-0.5}\\ \mathbf{elif}\;t_0 \leq -1.2 \cdot 10^{-318}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\frac{c0}{\frac{1}{\sqrt{\frac{\frac{A}{V}}{\ell}}}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+300}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{A \cdot c0}{\ell} \cdot \frac{c0}{V}}\\ \end{array} \]

Alternative 4
Error	6.5
Cost	20556

\[\begin{array}{l} t_0 := \sqrt{-A}\\ \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+297}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-294}:\\ \;\;\;\;t_0 \cdot \frac{c0}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{t_0}{\sqrt{-V} \cdot \frac{\sqrt{\ell}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+290}:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{1}{V \cdot \ell}} \cdot \sqrt{A}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]

Alternative 5
Error	7.9
Cost	14416

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -4 \cdot 10^{+134}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1.8 \cdot 10^{-114}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;{\ell}^{-0.5} \cdot \frac{c0}{\sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+290}:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{1}{V \cdot \ell}} \cdot \sqrt{A}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]

Alternative 6
Error	6.2
Cost	14416

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+297}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-321}:\\ \;\;\;\;\sqrt{-A} \cdot \frac{c0}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;{\ell}^{-0.5} \cdot \frac{c0}{\sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+290}:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{1}{V \cdot \ell}} \cdot \sqrt{A}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]

Alternative 7
Error	10.0
Cost	14288

\[\begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+211}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-225}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+290}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	7.9
Cost	14288

\[\begin{array}{l} t_0 := \sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{if}\;V \cdot \ell \leq -4 \cdot 10^{+134}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-196}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+290}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]

Alternative 9
Error	7.6
Cost	14288

\[\begin{array}{l} t_0 := \sqrt{\frac{A}{V}}\\ \mathbf{if}\;V \cdot \ell \leq -4 \cdot 10^{+134}:\\ \;\;\;\;c0 \cdot \frac{t_0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-196}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t_0 \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+290}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]

Alternative 10
Error	7.7
Cost	14288

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -4 \cdot 10^{+134}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1.8 \cdot 10^{-114}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{\frac{c0}{\sqrt{\frac{V}{A}}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+290}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]

Alternative 11
Error	7.7
Cost	14288

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -4 \cdot 10^{+134}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1.8 \cdot 10^{-114}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;{\ell}^{-0.5} \cdot \frac{c0}{\sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+290}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]

Alternative 12
Error	14.1
Cost	7890

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+223} \lor \neg \left(V \cdot \ell \leq -2 \cdot 10^{-167}\right) \land \left(V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq 5 \cdot 10^{+118}\right)\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]

Alternative 13
Error	13.9
Cost	7888

\[\begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ t_1 := c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{if}\;V \cdot \ell \leq -4 \cdot 10^{+300}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-167}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+160}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Error	13.8
Cost	7888

\[\begin{array}{l} t_0 := \frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ t_1 := c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+211}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-225}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+118}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 15
Error	13.6
Cost	7624

\[\begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+290}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \]

Alternative 16
Error	13.4
Cost	7624

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

Alternative 17
Error	13.4
Cost	7624

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

Alternative 18
Error	18.6
Cost	6848

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

Reproduce

herbie shell --seed 2023187 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))