Henrywood and Agarwal, Equation (3)

Average Error: 19.7 → 5.5

Time: 11.8s

Precision: binary64

Cost: 20036

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

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{-A}\\ \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+298}:\\ \;\;\;\;\frac{\frac{c0}{\sqrt{\ell}}}{\frac{\sqrt{-V}}{t_0}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-319}:\\ \;\;\;\;c0 \cdot \frac{t_0}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+296}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot \left(\frac{c0}{\ell} \cdot \frac{c0}{V}\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (sqrt (- A))))
   (if (<= (* V l) -1e+298)
     (/ (/ c0 (sqrt l)) (/ (sqrt (- V)) t_0))
     (if (<= (* V l) -5e-319)
       (* c0 (/ t_0 (sqrt (* l (- V)))))
       (if (<= (* V l) 0.0)
         (/ c0 (* (sqrt l) (sqrt (/ V A))))
         (if (<= (* V l) 1e+296)
           (/ c0 (/ (sqrt (* V l)) (sqrt A)))
           (sqrt (* A (* (/ c0 l) (/ c0 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(-A);
	double tmp;
	if ((V * l) <= -1e+298) {
		tmp = (c0 / sqrt(l)) / (sqrt(-V) / t_0);
	} else if ((V * l) <= -5e-319) {
		tmp = c0 * (t_0 / sqrt((l * -V)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	} else if ((V * l) <= 1e+296) {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	} else {
		tmp = sqrt((A * ((c0 / l) * (c0 / 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) :: tmp
    t_0 = sqrt(-a)
    if ((v * l) <= (-1d+298)) then
        tmp = (c0 / sqrt(l)) / (sqrt(-v) / t_0)
    else if ((v * l) <= (-5d-319)) then
        tmp = c0 * (t_0 / sqrt((l * -v)))
    else if ((v * l) <= 0.0d0) then
        tmp = c0 / (sqrt(l) * sqrt((v / a)))
    else if ((v * l) <= 1d+296) then
        tmp = c0 / (sqrt((v * l)) / sqrt(a))
    else
        tmp = sqrt((a * ((c0 / l) * (c0 / 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(-A);
	double tmp;
	if ((V * l) <= -1e+298) {
		tmp = (c0 / Math.sqrt(l)) / (Math.sqrt(-V) / t_0);
	} else if ((V * l) <= -5e-319) {
		tmp = c0 * (t_0 / Math.sqrt((l * -V)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 / (Math.sqrt(l) * Math.sqrt((V / A)));
	} else if ((V * l) <= 1e+296) {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	} else {
		tmp = Math.sqrt((A * ((c0 / l) * (c0 / 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(-A)
	tmp = 0
	if (V * l) <= -1e+298:
		tmp = (c0 / math.sqrt(l)) / (math.sqrt(-V) / t_0)
	elif (V * l) <= -5e-319:
		tmp = c0 * (t_0 / math.sqrt((l * -V)))
	elif (V * l) <= 0.0:
		tmp = c0 / (math.sqrt(l) * math.sqrt((V / A)))
	elif (V * l) <= 1e+296:
		tmp = c0 / (math.sqrt((V * l)) / math.sqrt(A))
	else:
		tmp = math.sqrt((A * ((c0 / l) * (c0 / 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(-A))
	tmp = 0.0
	if (Float64(V * l) <= -1e+298)
		tmp = Float64(Float64(c0 / sqrt(l)) / Float64(sqrt(Float64(-V)) / t_0));
	elseif (Float64(V * l) <= -5e-319)
		tmp = Float64(c0 * Float64(t_0 / sqrt(Float64(l * Float64(-V)))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 / Float64(sqrt(l) * sqrt(Float64(V / A))));
	elseif (Float64(V * l) <= 1e+296)
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	else
		tmp = sqrt(Float64(A * Float64(Float64(c0 / l) * Float64(c0 / 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(-A);
	tmp = 0.0;
	if ((V * l) <= -1e+298)
		tmp = (c0 / sqrt(l)) / (sqrt(-V) / t_0);
	elseif ((V * l) <= -5e-319)
		tmp = c0 * (t_0 / sqrt((l * -V)));
	elseif ((V * l) <= 0.0)
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	elseif ((V * l) <= 1e+296)
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	else
		tmp = sqrt((A * ((c0 / l) * (c0 / 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[(-A)], $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -1e+298], N[(N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[(-V)], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -5e-319], N[(c0 * N[(t$95$0 / N[Sqrt[N[(l * (-V)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+296], N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(A * N[(N[(c0 / l), $MachinePrecision] * N[(c0 / V), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -9.9999999999999996e297

   1. Initial program 40.7

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

   2. Applied egg-rr 23.6

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

   3. Simplified 23.6

      \[\leadsto c0 \cdot \color{blue}{\frac{1}{\sqrt{\ell \cdot \frac{V}{A}}}} \]
      Proof
      (/.f64 1 (sqrt.f64 (*.f64 l (/.f64 V A)))): 0 points increase in error, 0 points decrease in error
      (/.f64 1 (sqrt.f64 (*.f64 (Rewrite<= /-rgt-identity_binary64 (/.f64 l 1)) (/.f64 V A)))): 0 points increase in error, 0 points decrease in error
      (/.f64 1 (sqrt.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 l V) (*.f64 1 A))))): 32 points increase in error, 36 points decrease in error
      (/.f64 1 (sqrt.f64 (/.f64 (*.f64 l V) (Rewrite=> *-lft-identity_binary64 A)))): 0 points increase in error, 0 points decrease in error
      (/.f64 1 (sqrt.f64 (Rewrite=> associate-/l*_binary64 (/.f64 l (/.f64 A V))))): 33 points increase in error, 34 points decrease in error

   4. Applied egg-rr 12.8

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

   5. Applied egg-rr 1.1

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

   if -9.9999999999999996e297 < (*.f64 V l) < -4.9999937e-319

   1. Initial program 10.6

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

   2. Applied egg-rr 0.5

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

   if -4.9999937e-319 < (*.f64 V l) < 0.0

   1. Initial program 63.6

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

   2. Applied egg-rr 37.9

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

   3. Simplified 37.9

      \[\leadsto c0 \cdot \color{blue}{\frac{1}{\sqrt{\ell \cdot \frac{V}{A}}}} \]
      Proof
      (/.f64 1 (sqrt.f64 (*.f64 l (/.f64 V A)))): 0 points increase in error, 0 points decrease in error
      (/.f64 1 (sqrt.f64 (*.f64 (Rewrite<= /-rgt-identity_binary64 (/.f64 l 1)) (/.f64 V A)))): 0 points increase in error, 0 points decrease in error
      (/.f64 1 (sqrt.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 l V) (*.f64 1 A))))): 32 points increase in error, 36 points decrease in error
      (/.f64 1 (sqrt.f64 (/.f64 (*.f64 l V) (Rewrite=> *-lft-identity_binary64 A)))): 0 points increase in error, 0 points decrease in error
      (/.f64 1 (sqrt.f64 (Rewrite=> associate-/l*_binary64 (/.f64 l (/.f64 A V))))): 33 points increase in error, 34 points decrease in error

   4. Applied egg-rr 30.2

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

   5. Simplified 29.7

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}} \]
      Proof
      (/.f64 c0 (*.f64 (sqrt.f64 (/.f64 V A)) (sqrt.f64 l))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 c0 (sqrt.f64 l)) (sqrt.f64 (/.f64 V A)))): 32 points increase in error, 26 points decrease in error

   if 0.0 < (*.f64 V l) < 9.99999999999999981e295

   1. Initial program 10.6

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

   2. Applied egg-rr 3.3

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

   3. Simplified 0.7

      \[\leadsto \color{blue}{\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}} \]
      Proof
      (/.f64 c0 (/.f64 (sqrt.f64 (*.f64 V l)) (sqrt.f64 A))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 c0 (sqrt.f64 A)) (sqrt.f64 (*.f64 V l)))): 18 points increase in error, 15 points decrease in error
      (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 (sqrt.f64 A) c0)) (sqrt.f64 (*.f64 V l))): 0 points increase in error, 0 points decrease in error

   if 9.99999999999999981e295 < (*.f64 V l)

   1. Initial program 40.4

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

   2. Applied egg-rr 26.4

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

   3. Simplified 26.4

      \[\leadsto c0 \cdot \color{blue}{\frac{1}{\sqrt{\ell \cdot \frac{V}{A}}}} \]
      Proof
      (/.f64 1 (sqrt.f64 (*.f64 l (/.f64 V A)))): 0 points increase in error, 0 points decrease in error
      (/.f64 1 (sqrt.f64 (*.f64 (Rewrite<= /-rgt-identity_binary64 (/.f64 l 1)) (/.f64 V A)))): 0 points increase in error, 0 points decrease in error
      (/.f64 1 (sqrt.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 l V) (*.f64 1 A))))): 32 points increase in error, 36 points decrease in error
      (/.f64 1 (sqrt.f64 (/.f64 (*.f64 l V) (Rewrite=> *-lft-identity_binary64 A)))): 0 points increase in error, 0 points decrease in error
      (/.f64 1 (sqrt.f64 (Rewrite=> associate-/l*_binary64 (/.f64 l (/.f64 A V))))): 33 points increase in error, 34 points decrease in error

   4. Applied egg-rr 26.3

      \[\leadsto \color{blue}{0 + \frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}} \]

   5. Simplified 26.3

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}} \]
      Proof
      (/.f64 c0 (sqrt.f64 (*.f64 l (/.f64 V A)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-lft-identity_binary64 (+.f64 0 (/.f64 c0 (sqrt.f64 (*.f64 l (/.f64 V A)))))): 0 points increase in error, 0 points decrease in error

   6. Applied egg-rr 39.1

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

   7. Simplified 32.0

      \[\leadsto \color{blue}{\sqrt{A \cdot \left(\frac{c0}{\ell} \cdot \frac{c0}{V}\right)}} \]
      Proof
      (sqrt.f64 (*.f64 A (*.f64 (/.f64 c0 l) (/.f64 c0 V)))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 A (Rewrite<= times-frac_binary64 (/.f64 (*.f64 c0 c0) (*.f64 l V))))): 45 points increase in error, 19 points decrease in error
      (sqrt.f64 (*.f64 A (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 (*.f64 c0 c0) l) V)))): 16 points increase in error, 23 points decrease in error
      (sqrt.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (/.f64 (*.f64 c0 c0) l) V) A))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (Rewrite<= associate-/r/_binary64 (/.f64 (/.f64 (*.f64 c0 c0) l) (/.f64 V A)))): 27 points increase in error, 11 points decrease in error
      (sqrt.f64 (Rewrite<= associate-/r*_binary64 (/.f64 (*.f64 c0 c0) (*.f64 l (/.f64 V A))))): 18 points increase in error, 13 points decrease in error
      (Rewrite<= unpow1/2_binary64 (pow.f64 (/.f64 (*.f64 c0 c0) (*.f64 l (/.f64 V A))) 1/2)): 0 points increase in error, 0 points decrease in error
3. Recombined 5 regimes into one program.

4. Final simplification 5.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+298}:\\ \;\;\;\;\frac{\frac{c0}{\sqrt{\ell}}}{\frac{\sqrt{-V}}{\sqrt{-A}}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-319}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+296}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot \left(\frac{c0}{\ell} \cdot \frac{c0}{V}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	13.9
Cost	34640

\[\begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ t_1 := \sqrt{A \cdot \left(\frac{c0}{\ell} \cdot \frac{c0}{V}\right)}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;c0 \cdot {\left(\ell \cdot \frac{V}{A}\right)}^{-0.5}\\ \mathbf{elif}\;t_0 \leq -4 \cdot 10^{-312}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+300}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	11.7
Cost	34640

\[\begin{array}{l} t_0 := c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ t_1 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_1 \leq -5 \cdot 10^{-305}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+300}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot \left(\frac{c0}{\ell} \cdot \frac{c0}{V}\right)}\\ \end{array} \]

Alternative 3
Error	11.8
Cost	34640

\[\begin{array}{l} t_0 := \frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ t_1 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_1 \leq -5 \cdot 10^{-305}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+300}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot \left(\frac{c0}{\ell} \cdot \frac{c0}{V}\right)}\\ \end{array} \]

Alternative 4
Error	9.2
Cost	14288

\[\begin{array}{l} t_0 := \frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+188}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-103}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+296}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot \left(\frac{c0}{\ell} \cdot \frac{c0}{V}\right)}\\ \end{array} \]

Alternative 5
Error	6.3
Cost	14288

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-319}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+296}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot \left(\frac{c0}{\ell} \cdot \frac{c0}{V}\right)}\\ \end{array} \]

Alternative 6
Error	15.3
Cost	7624

\[\begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ t_1 := c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+297}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	15.0
Cost	7624

\[\begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+297}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \]

Alternative 8
Error	14.9
Cost	7624

\[\begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+305}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \end{array} \]

Alternative 9
Error	19.7
Cost	6848

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

Reproduce

herbie shell --seed 2022332 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))