Henrywood and Agarwal, Equation (3)

Average Error: 18.9 → 5.9

Time: 23.9s

Precision: binary64

Cost: 20944

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

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{-A}\\ \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+161}:\\ \;\;\;\;\frac{\frac{t_0}{\sqrt{-V}}}{\frac{\sqrt{\ell}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-308}:\\ \;\;\;\;c0 \cdot \frac{t_0}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+279}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \left(\sqrt{\frac{-1}{V}} \cdot {\left(-\ell\right)}^{-0.5}\right)\right)\\ \end{array} \]

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) -2e+161)
     (/ (/ t_0 (sqrt (- V))) (/ (sqrt l) c0))
     (if (<= (* V l) -5e-308)
       (* c0 (/ t_0 (sqrt (* l (- V)))))
       (if (<= (* V l) 0.0)
         (/ c0 (* (sqrt l) (sqrt (/ V A))))
         (if (<= (* V l) 4e+279)
           (* c0 (* (sqrt A) (pow (* V l) -0.5)))
           (* c0 (* (sqrt A) (* (sqrt (/ -1.0 V)) (pow (- l) -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 t_0 = sqrt(-A);
	double tmp;
	if ((V * l) <= -2e+161) {
		tmp = (t_0 / sqrt(-V)) / (sqrt(l) / c0);
	} else if ((V * l) <= -5e-308) {
		tmp = c0 * (t_0 / sqrt((l * -V)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	} else if ((V * l) <= 4e+279) {
		tmp = c0 * (sqrt(A) * pow((V * l), -0.5));
	} else {
		tmp = c0 * (sqrt(A) * (sqrt((-1.0 / V)) * pow(-l, -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) :: t_0
    real(8) :: tmp
    t_0 = sqrt(-a)
    if ((v * l) <= (-2d+161)) then
        tmp = (t_0 / sqrt(-v)) / (sqrt(l) / c0)
    else if ((v * l) <= (-5d-308)) 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) <= 4d+279) then
        tmp = c0 * (sqrt(a) * ((v * l) ** (-0.5d0)))
    else
        tmp = c0 * (sqrt(a) * (sqrt(((-1.0d0) / v)) * (-l ** (-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 t_0 = Math.sqrt(-A);
	double tmp;
	if ((V * l) <= -2e+161) {
		tmp = (t_0 / Math.sqrt(-V)) / (Math.sqrt(l) / c0);
	} else if ((V * l) <= -5e-308) {
		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) <= 4e+279) {
		tmp = c0 * (Math.sqrt(A) * Math.pow((V * l), -0.5));
	} else {
		tmp = c0 * (Math.sqrt(A) * (Math.sqrt((-1.0 / V)) * Math.pow(-l, -0.5)));
	}
	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) <= -2e+161:
		tmp = (t_0 / math.sqrt(-V)) / (math.sqrt(l) / c0)
	elif (V * l) <= -5e-308:
		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) <= 4e+279:
		tmp = c0 * (math.sqrt(A) * math.pow((V * l), -0.5))
	else:
		tmp = c0 * (math.sqrt(A) * (math.sqrt((-1.0 / V)) * math.pow(-l, -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)
	t_0 = sqrt(Float64(-A))
	tmp = 0.0
	if (Float64(V * l) <= -2e+161)
		tmp = Float64(Float64(t_0 / sqrt(Float64(-V))) / Float64(sqrt(l) / c0));
	elseif (Float64(V * l) <= -5e-308)
		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) <= 4e+279)
		tmp = Float64(c0 * Float64(sqrt(A) * (Float64(V * l) ^ -0.5)));
	else
		tmp = Float64(c0 * Float64(sqrt(A) * Float64(sqrt(Float64(-1.0 / V)) * (Float64(-l) ^ -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)
	t_0 = sqrt(-A);
	tmp = 0.0;
	if ((V * l) <= -2e+161)
		tmp = (t_0 / sqrt(-V)) / (sqrt(l) / c0);
	elseif ((V * l) <= -5e-308)
		tmp = c0 * (t_0 / sqrt((l * -V)));
	elseif ((V * l) <= 0.0)
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	elseif ((V * l) <= 4e+279)
		tmp = c0 * (sqrt(A) * ((V * l) ^ -0.5));
	else
		tmp = c0 * (sqrt(A) * (sqrt((-1.0 / V)) * (-l ^ -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_] := Block[{t$95$0 = N[Sqrt[(-A)], $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -2e+161], N[(N[(t$95$0 / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] / c0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -5e-308], 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], 4e+279], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Power[N[(V * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[(N[Sqrt[N[(-1.0 / V), $MachinePrecision]], $MachinePrecision] * N[Power[(-l), -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -2.0000000000000001e161

   1. Initial program 26.0

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

   2. Applied egg-rr 19.3

      \[\leadsto c0 \cdot \color{blue}{{\left(\sqrt[3]{\sqrt{\frac{\frac{A}{V}}{\ell}}}\right)}^{3}} \]

   3. Applied egg-rr 9.8

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

   4. Simplified 9.5

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{A}{V}}}{\frac{\sqrt{\ell}}{c0}}} \]
      Proof
      (/.f64 (sqrt.f64 (/.f64 A V)) (/.f64 (sqrt.f64 l) c0)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (sqrt.f64 (/.f64 A V)) c0) (sqrt.f64 l))): 33 points increase in error, 24 points decrease in error

   5. Applied egg-rr 1.7

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

   if -2.0000000000000001e161 < (*.f64 V l) < -4.99999999999999955e-308

   1. Initial program 8.6

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

   2. Applied egg-rr 0.4

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

   if -4.99999999999999955e-308 < (*.f64 V l) < 0.0

   1. Initial program 62.3

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

   2. Applied egg-rr 64.0

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

   3. Simplified 64.0

      \[\leadsto \color{blue}{\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}} \]
      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)))): 17 points increase in error, 11 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

   4. Applied egg-rr 62.3

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

   5. Simplified 36.8

      \[\leadsto \frac{c0}{\color{blue}{\sqrt{\frac{V}{A} \cdot \ell}}} \]
      Proof
      (sqrt.f64 (*.f64 (/.f64 V A) l)): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 V l) A))): 36 points increase in error, 31 points decrease in error
      (Rewrite<= +-lft-identity_binary64 (+.f64 0 (sqrt.f64 (/.f64 (*.f64 V l) A)))): 0 points increase in error, 0 points decrease in error

   6. Applied egg-rr 27.6

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

   if 0.0 < (*.f64 V l) < 4.00000000000000023e279

   1. Initial program 10.4

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

   2. Applied egg-rr 1.2

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

   3. Applied egg-rr 0.7

      \[\leadsto c0 \cdot \left(\sqrt{A} \cdot \color{blue}{\left(0 + {\left(V \cdot \ell\right)}^{-0.5}\right)}\right) \]

   4. Simplified 0.7

      \[\leadsto c0 \cdot \left(\sqrt{A} \cdot \color{blue}{{\left(V \cdot \ell\right)}^{-0.5}}\right) \]
      Proof
      (pow.f64 (*.f64 V l) -1/2): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-lft-identity_binary64 (+.f64 0 (pow.f64 (*.f64 V l) -1/2))): 0 points increase in error, 0 points decrease in error

   if 4.00000000000000023e279 < (*.f64 V l)

   1. Initial program 37.4

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

   2. Applied egg-rr 34.4

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

   3. Applied egg-rr 34.4

      \[\leadsto c0 \cdot \left(\sqrt{A} \cdot \color{blue}{\left(0 + {\left(V \cdot \ell\right)}^{-0.5}\right)}\right) \]

   4. Simplified 34.4

      \[\leadsto c0 \cdot \left(\sqrt{A} \cdot \color{blue}{{\left(V \cdot \ell\right)}^{-0.5}}\right) \]
      Proof
      (pow.f64 (*.f64 V l) -1/2): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-lft-identity_binary64 (+.f64 0 (pow.f64 (*.f64 V l) -1/2))): 0 points increase in error, 0 points decrease in error

   5. Taylor expanded in V around -inf 36.5

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

   6. Simplified 34.7

      \[\leadsto c0 \cdot \left(\sqrt{A} \cdot \color{blue}{\left(\sqrt{\frac{-1}{V}} \cdot {\left(-\ell\right)}^{-0.5}\right)}\right) \]
      Proof
      (*.f64 (sqrt.f64 (/.f64 -1 V)) (pow.f64 (neg.f64 l) -1/2)): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= unpow1/2_binary64 (pow.f64 (/.f64 -1 V) 1/2)) (pow.f64 (neg.f64 l) -1/2)): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= exp-to-pow_binary64 (exp.f64 (*.f64 (log.f64 (/.f64 -1 V)) 1/2))) (pow.f64 (neg.f64 l) -1/2)): 29 points increase in error, 30 points decrease in error
      (*.f64 (exp.f64 (*.f64 (log.f64 (/.f64 -1 V)) (Rewrite<= metadata-eval (*.f64 -1 -1/2)))) (pow.f64 (neg.f64 l) -1/2)): 0 points increase in error, 0 points decrease in error
      (*.f64 (exp.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (log.f64 (/.f64 -1 V)) -1) -1/2))) (pow.f64 (neg.f64 l) -1/2)): 0 points increase in error, 0 points decrease in error
      (*.f64 (exp.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1 (log.f64 (/.f64 -1 V)))) -1/2)) (pow.f64 (neg.f64 l) -1/2)): 0 points increase in error, 0 points decrease in error
      (*.f64 (exp.f64 (*.f64 (*.f64 -1 (log.f64 (/.f64 -1 V))) -1/2)) (pow.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 l)) -1/2)): 0 points increase in error, 0 points decrease in error
      (*.f64 (exp.f64 (*.f64 (*.f64 -1 (log.f64 (/.f64 -1 V))) -1/2)) (Rewrite<= exp-to-pow_binary64 (exp.f64 (*.f64 (log.f64 (*.f64 -1 l)) -1/2)))): 29 points increase in error, 30 points decrease in error
      (Rewrite<= exp-sum_binary64 (exp.f64 (+.f64 (*.f64 (*.f64 -1 (log.f64 (/.f64 -1 V))) -1/2) (*.f64 (log.f64 (*.f64 -1 l)) -1/2)))): 21 points increase in error, 17 points decrease in error
      (exp.f64 (Rewrite<= distribute-rgt-in_binary64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 V))) (log.f64 (*.f64 -1 l)))))): 0 points increase in error, 0 points decrease in error
3. Recombined 5 regimes into one program.

4. Final simplification 5.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+161}:\\ \;\;\;\;\frac{\frac{\sqrt{-A}}{\sqrt{-V}}}{\frac{\sqrt{\ell}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-308}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+279}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \left(\sqrt{\frac{-1}{V}} \cdot {\left(-\ell\right)}^{-0.5}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	5.0
Cost	20036

\[\begin{array}{l} t_0 := \sqrt{-A}\\ \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+161}:\\ \;\;\;\;\frac{\frac{t_0}{\sqrt{-V}}}{\frac{\sqrt{\ell}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-308}:\\ \;\;\;\;c0 \cdot \frac{t_0}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+271}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]

Alternative 2
Error	8.0
Cost	14352

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+146}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-162}:\\ \;\;\;\;\frac{c0}{{\left(\frac{A}{V \cdot \ell}\right)}^{-0.5}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+271}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]

Alternative 3
Error	5.5
Cost	14352

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+290}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-308}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+271}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]

Alternative 4
Error	13.0
Cost	14288

\[\begin{array}{l} t_0 := \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+219}:\\ \;\;\;\;\frac{c0}{\frac{1}{t_0}}\\ \mathbf{elif}\;V \cdot \ell \leq -4 \cdot 10^{-227}:\\ \;\;\;\;\frac{c0}{{\left(\frac{A}{V \cdot \ell}\right)}^{-0.5}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\sqrt{\left(A \cdot \frac{c0}{V}\right) \cdot \frac{c0}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+271}:\\ \;\;\;\;\sqrt{A} \cdot \frac{c0}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot t_0\\ \end{array} \]

Alternative 5
Error	8.8
Cost	14288

\[\begin{array}{l} t_0 := \frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+146}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-162}:\\ \;\;\;\;\frac{c0}{{\left(\frac{A}{V \cdot \ell}\right)}^{-0.5}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+271}:\\ \;\;\;\;\sqrt{A} \cdot \frac{c0}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]

Alternative 6
Error	8.9
Cost	14288

\[\begin{array}{l} t_0 := \frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+146}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-162}:\\ \;\;\;\;\frac{c0}{{\left(\frac{A}{V \cdot \ell}\right)}^{-0.5}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+271}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]

Alternative 7
Error	8.9
Cost	14288

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+146}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-162}:\\ \;\;\;\;\frac{c0}{{\left(\frac{A}{V \cdot \ell}\right)}^{-0.5}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+271}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]

Alternative 8
Error	8.0
Cost	14288

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+146}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-162}:\\ \;\;\;\;\frac{c0}{{\left(\frac{A}{V \cdot \ell}\right)}^{-0.5}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+271}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]

Alternative 9
Error	7.9
Cost	14288

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+146}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-162}:\\ \;\;\;\;\frac{c0}{{\left(\frac{A}{V \cdot \ell}\right)}^{-0.5}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+271}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]

Alternative 10
Error	14.1
Cost	7752

\[\begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-314}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t_0 \leq 10^{+308}:\\ \;\;\;\;\frac{c0}{{t_0}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{A}{\frac{\ell}{c0} \cdot \frac{V}{c0}}}\\ \end{array} \]

Alternative 11
Error	14.0
Cost	7752

\[\begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-314}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t_0 \leq 10^{+308}:\\ \;\;\;\;\frac{c0}{{t_0}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(A \cdot \frac{c0}{V}\right) \cdot \frac{c0}{\ell}}\\ \end{array} \]

Alternative 12
Error	14.0
Cost	7752

\[\begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-314}:\\ \;\;\;\;\frac{c0}{\frac{1}{\sqrt{\frac{\frac{A}{V}}{\ell}}}}\\ \mathbf{elif}\;t_0 \leq 10^{+308}:\\ \;\;\;\;\frac{c0}{{t_0}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(A \cdot \frac{c0}{V}\right) \cdot \frac{c0}{\ell}}\\ \end{array} \]

Alternative 13
Error	14.1
Cost	7688

\[\begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-314}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t_0 \leq 10^{+308}:\\ \;\;\;\;\frac{c0}{{t_0}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \]

Alternative 14
Error	14.1
Cost	7688

\[\begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-314}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t_0 \leq 10^{+308}:\\ \;\;\;\;\frac{c0}{{t_0}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(\ell \cdot \frac{V}{A}\right)}^{-0.5}\\ \end{array} \]

Alternative 15
Error	14.1
Cost	7624

\[\begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-314}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t_0 \leq 10^{+308}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \]

Alternative 16
Error	19.0
Cost	7112

\[\begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{if}\;c0 \leq 2.25 \cdot 10^{-287}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c0 \leq 4 \cdot 10^{+111}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 17
Error	19.0
Cost	7112

\[\begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{if}\;c0 \leq 3.25 \cdot 10^{-288}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c0 \leq 6 \cdot 10^{+111}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 18
Error	19.0
Cost	6980

\[\begin{array}{l} \mathbf{if}\;V \leq -1 \cdot 10^{-93}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \end{array} \]

Alternative 19
Error	19.1
Cost	6848

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

Alternative 20
Error	19.2
Cost	6848

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

Reproduce

herbie shell --seed 2022318 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))