Toniolo and Linder, Equation (13)

Average Accuracy: 46.5% → 61.3%

Time: 1.3min

Precision: binary64

Cost: 64396

Math

\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

↓

\[\begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;{\left(\sqrt[3]{t + t} \cdot \sqrt[3]{n \cdot U}\right)}^{1.5}\\ \mathbf{elif}\;t_1 \leq 10^{+151}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\left|\frac{\left(\ell \cdot \sqrt{U \cdot U*}\right) \cdot \left(n \cdot \sqrt{2}\right)}{Om}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\ell, -2, \frac{n \cdot \ell}{\frac{Om}{U* - U}}\right), \left(n \cdot \ell\right) \cdot \frac{U}{Om}, t \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (sqrt
  (*
   (* (* 2.0 n) U)
   (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))

↓

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* (* 2.0 n) U)
           (+
            (- t (* 2.0 (/ (* l l) Om)))
            (* (* n (pow (/ l Om) 2.0)) (- U* U)))))))
   (if (<= t_1 0.0)
     (pow (* (cbrt (+ t t)) (cbrt (* n U))) 1.5)
     (if (<= t_1 1e+151)
       t_1
       (if (<= t_1 INFINITY)
         (fabs (/ (* (* l (sqrt (* U U*))) (* n (sqrt 2.0))) Om))
         (sqrt
          (*
           2.0
           (fma
            (fma l -2.0 (/ (* n l) (/ Om (- U* U))))
            (* (* n l) (/ U Om))
            (* t (* n U))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
}

↓

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + ((n * pow((l / Om), 2.0)) * (U_42_ - U)))));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = pow((cbrt((t + t)) * cbrt((n * U))), 1.5);
	} else if (t_1 <= 1e+151) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fabs((((l * sqrt((U * U_42_))) * (n * sqrt(2.0))) / Om));
	} else {
		tmp = sqrt((2.0 * fma(fma(l, -2.0, ((n * l) / (Om / (U_42_ - U)))), ((n * l) * (U / Om)), (t * (n * U)))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_)))))
end

↓

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U_42_ - U)))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(cbrt(Float64(t + t)) * cbrt(Float64(n * U))) ^ 1.5;
	elseif (t_1 <= 1e+151)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = abs(Float64(Float64(Float64(l * sqrt(Float64(U * U_42_))) * Float64(n * sqrt(2.0))) / Om));
	else
		tmp = sqrt(Float64(2.0 * fma(fma(l, -2.0, Float64(Float64(n * l) / Float64(Om / Float64(U_42_ - U)))), Float64(Float64(n * l) * Float64(U / Om)), Float64(t * Float64(n * U)))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

↓

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[Power[N[(N[Power[N[(t + t), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(n * U), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], If[LessEqual[t$95$1, 1e+151], t$95$1, If[LessEqual[t$95$1, Infinity], N[Abs[N[(N[(N[(l * N[Sqrt[N[(U * U$42$), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(n * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(N[(l * -2.0 + N[(N[(n * l), $MachinePrecision] / N[(Om / N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(n * l), $MachinePrecision] * N[(U / Om), $MachinePrecision]), $MachinePrecision] + N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 0.0

   1. Initial program 12.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Simplified 39.5%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)}} \]
      Proof

      [Start] 12.3	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 42.0	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      sub-neg [=>] 42.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(-\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      associate-+l- [=>] 42.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [=>] 42.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(-\left(-\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)\right)} \]
      associate-/l* [=>] 42.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(-\left(-\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)} \]
      remove-double-neg [=>] 42.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)\right)\right)} \]
      associate-*l* [=>] 39.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in t around inf 35.0%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)}} \]

   4. Simplified 35.0%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
      Proof

      [Start] 35.0	\[ \sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)} \]
      *-commutative [=>] 35.0	\[ \sqrt{2 \cdot \left(n \cdot \color{blue}{\left(U \cdot t\right)}\right)} \]
      associate-*r* [=>] 12.3	\[ \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot U\right) \cdot t\right)}} \]
      *-commutative [=>] 12.3	\[ \sqrt{2 \cdot \left(\color{blue}{\left(U \cdot n\right)} \cdot t\right)} \]
      associate-*l* [=>] 35.0	\[ \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

   5. Applied egg-rr 32.9%

      \[\leadsto \color{blue}{e^{\left(1.5 \cdot \log \left(2 \cdot \left(n \cdot \left(t \cdot U\right)\right)\right)\right) \cdot 0.3333333333333333}} \]
      Proof

      [Start] 35.0	\[ \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \]
      add-cbrt-cube [=>] 24.4	\[ \color{blue}{\sqrt[3]{\left(\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\right) \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}}} \]
      pow1/3 [=>] 23.4	\[ \color{blue}{{\left(\left(\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\right) \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\right)}^{0.3333333333333333}} \]
      pow-to-exp [=>] 23.5	\[ \color{blue}{e^{\log \left(\left(\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\right) \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\right) \cdot 0.3333333333333333}} \]
      add-sqr-sqrt [<=] 23.5	\[ e^{\log \left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\right) \cdot 0.3333333333333333} \]
      sum-log [<=] 32.9	\[ e^{\color{blue}{\left(\log \left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right) + \log \left(\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\right)\right)} \cdot 0.3333333333333333} \]

   6. Applied egg-rr 12.3%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{2 \cdot \left(t \cdot \left(U \cdot n\right)\right)}\right)}^{1.5}} \]
      Proof

      [Start] 32.9	\[ e^{\left(1.5 \cdot \log \left(2 \cdot \left(n \cdot \left(t \cdot U\right)\right)\right)\right) \cdot 0.3333333333333333} \]
      *-commutative [=>] 32.9	\[ e^{\color{blue}{\left(\log \left(2 \cdot \left(n \cdot \left(t \cdot U\right)\right)\right) \cdot 1.5\right)} \cdot 0.3333333333333333} \]
      associate-*l* [=>] 33.1	\[ e^{\color{blue}{\log \left(2 \cdot \left(n \cdot \left(t \cdot U\right)\right)\right) \cdot \left(1.5 \cdot 0.3333333333333333\right)}} \]
      metadata-eval [=>] 33.1	\[ e^{\log \left(2 \cdot \left(n \cdot \left(t \cdot U\right)\right)\right) \cdot \color{blue}{0.5}} \]
      pow-to-exp [<=] 35.0	\[ \color{blue}{{\left(2 \cdot \left(n \cdot \left(t \cdot U\right)\right)\right)}^{0.5}} \]
      add-cube-cbrt [=>] 34.7	\[ {\color{blue}{\left(\left(\sqrt[3]{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)} \cdot \sqrt[3]{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)}\right) \cdot \sqrt[3]{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)}\right)}}^{0.5} \]
      pow3 [=>] 34.7	\[ {\color{blue}{\left({\left(\sqrt[3]{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)}\right)}^{3}\right)}}^{0.5} \]
      metadata-eval [<=] 34.7	\[ {\left({\left(\sqrt[3]{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)}\right)}^{\color{blue}{\left(1 + 2\right)}}\right)}^{0.5} \]
      pow-pow [=>] 34.7	\[ \color{blue}{{\left(\sqrt[3]{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)}\right)}^{\left(\left(1 + 2\right) \cdot 0.5\right)}} \]
      *-commutative [=>] 34.7	\[ {\left(\sqrt[3]{2 \cdot \color{blue}{\left(\left(t \cdot U\right) \cdot n\right)}}\right)}^{\left(\left(1 + 2\right) \cdot 0.5\right)} \]
      associate-*l* [=>] 12.3	\[ {\left(\sqrt[3]{2 \cdot \color{blue}{\left(t \cdot \left(U \cdot n\right)\right)}}\right)}^{\left(\left(1 + 2\right) \cdot 0.5\right)} \]
      metadata-eval [=>] 12.3	\[ {\left(\sqrt[3]{2 \cdot \left(t \cdot \left(U \cdot n\right)\right)}\right)}^{\left(\color{blue}{3} \cdot 0.5\right)} \]
      metadata-eval [=>] 12.3	\[ {\left(\sqrt[3]{2 \cdot \left(t \cdot \left(U \cdot n\right)\right)}\right)}^{\color{blue}{1.5}} \]

   7. Applied egg-rr 31.8%

      \[\leadsto {\color{blue}{\left(\sqrt[3]{2 \cdot t} \cdot \sqrt[3]{U \cdot n}\right)}}^{1.5} \]
      Proof

      [Start] 12.3	\[ {\left(\sqrt[3]{2 \cdot \left(t \cdot \left(U \cdot n\right)\right)}\right)}^{1.5} \]
      associate-*r* [=>] 12.3	\[ {\left(\sqrt[3]{\color{blue}{\left(2 \cdot t\right) \cdot \left(U \cdot n\right)}}\right)}^{1.5} \]
      cbrt-prod [=>] 31.8	\[ {\color{blue}{\left(\sqrt[3]{2 \cdot t} \cdot \sqrt[3]{U \cdot n}\right)}}^{1.5} \]

   8. Simplified 31.8%

      \[\leadsto {\color{blue}{\left(\sqrt[3]{t + t} \cdot \sqrt[3]{n \cdot U}\right)}}^{1.5} \]
      Proof

      [Start] 31.8	\[ {\left(\sqrt[3]{2 \cdot t} \cdot \sqrt[3]{U \cdot n}\right)}^{1.5} \]
      count-2 [<=] 31.8	\[ {\left(\sqrt[3]{\color{blue}{t + t}} \cdot \sqrt[3]{U \cdot n}\right)}^{1.5} \]
      *-commutative [=>] 31.8	\[ {\left(\sqrt[3]{t + t} \cdot \sqrt[3]{\color{blue}{n \cdot U}}\right)}^{1.5} \]

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 1.00000000000000002e151

   1. Initial program 97.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   if 1.00000000000000002e151 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < +inf.0

   1. Initial program 1.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in U* around inf 2.0%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \frac{{n}^{2} \cdot \left({\ell}^{2} \cdot \left(U \cdot U*\right)\right)}{{Om}^{2}}}} \]

   3. Simplified 2.2%

      \[\leadsto \sqrt{\color{blue}{\frac{\left(2 \cdot \left(n \cdot n\right)\right) \cdot \left(\ell \cdot \left(\ell \cdot \left(U \cdot U*\right)\right)\right)}{Om \cdot Om}}} \]
      Proof

      [Start] 2.0	\[ \sqrt{2 \cdot \frac{{n}^{2} \cdot \left({\ell}^{2} \cdot \left(U \cdot U*\right)\right)}{{Om}^{2}}} \]
      associate-*r/ [=>] 2.0	\[ \sqrt{\color{blue}{\frac{2 \cdot \left({n}^{2} \cdot \left({\ell}^{2} \cdot \left(U \cdot U*\right)\right)\right)}{{Om}^{2}}}} \]
      associate-*r* [=>] 1.9	\[ \sqrt{\frac{\color{blue}{\left(2 \cdot {n}^{2}\right) \cdot \left({\ell}^{2} \cdot \left(U \cdot U*\right)\right)}}{{Om}^{2}}} \]
      unpow2 [=>] 1.9	\[ \sqrt{\frac{\left(2 \cdot \color{blue}{\left(n \cdot n\right)}\right) \cdot \left({\ell}^{2} \cdot \left(U \cdot U*\right)\right)}{{Om}^{2}}} \]
      unpow2 [=>] 1.9	\[ \sqrt{\frac{\left(2 \cdot \left(n \cdot n\right)\right) \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(U \cdot U*\right)\right)}{{Om}^{2}}} \]
      associate-*l* [=>] 2.2	\[ \sqrt{\frac{\left(2 \cdot \left(n \cdot n\right)\right) \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \left(U \cdot U*\right)\right)\right)}}{{Om}^{2}}} \]
      unpow2 [=>] 2.2	\[ \sqrt{\frac{\left(2 \cdot \left(n \cdot n\right)\right) \cdot \left(\ell \cdot \left(\ell \cdot \left(U \cdot U*\right)\right)\right)}{\color{blue}{Om \cdot Om}}} \]

   4. Applied egg-rr 25.2%

      \[\leadsto \color{blue}{\left|\frac{\left(\ell \cdot \sqrt{U \cdot U*}\right) \cdot \left(n \cdot \sqrt{2}\right)}{Om}\right|} \]
      Proof

      [Start] 2.2	\[ \sqrt{\frac{\left(2 \cdot \left(n \cdot n\right)\right) \cdot \left(\ell \cdot \left(\ell \cdot \left(U \cdot U*\right)\right)\right)}{Om \cdot Om}} \]
      add-sqr-sqrt [=>] 2.2	\[ \sqrt{\color{blue}{\sqrt{\frac{\left(2 \cdot \left(n \cdot n\right)\right) \cdot \left(\ell \cdot \left(\ell \cdot \left(U \cdot U*\right)\right)\right)}{Om \cdot Om}} \cdot \sqrt{\frac{\left(2 \cdot \left(n \cdot n\right)\right) \cdot \left(\ell \cdot \left(\ell \cdot \left(U \cdot U*\right)\right)\right)}{Om \cdot Om}}}} \]
      rem-sqrt-square [=>] 2.2	\[ \color{blue}{\left|\sqrt{\frac{\left(2 \cdot \left(n \cdot n\right)\right) \cdot \left(\ell \cdot \left(\ell \cdot \left(U \cdot U*\right)\right)\right)}{Om \cdot Om}}\right|} \]
      sqrt-div [=>] 4.2	\[ \left|\color{blue}{\frac{\sqrt{\left(2 \cdot \left(n \cdot n\right)\right) \cdot \left(\ell \cdot \left(\ell \cdot \left(U \cdot U*\right)\right)\right)}}{\sqrt{Om \cdot Om}}}\right| \]
      *-commutative [=>] 4.2	\[ \left|\frac{\sqrt{\color{blue}{\left(\ell \cdot \left(\ell \cdot \left(U \cdot U*\right)\right)\right) \cdot \left(2 \cdot \left(n \cdot n\right)\right)}}}{\sqrt{Om \cdot Om}}\right| \]
      sqrt-prod [=>] 5.5	\[ \left|\frac{\color{blue}{\sqrt{\ell \cdot \left(\ell \cdot \left(U \cdot U*\right)\right)} \cdot \sqrt{2 \cdot \left(n \cdot n\right)}}}{\sqrt{Om \cdot Om}}\right| \]
      associate-*r* [=>] 4.9	\[ \left|\frac{\sqrt{\color{blue}{\left(\ell \cdot \ell\right) \cdot \left(U \cdot U*\right)}} \cdot \sqrt{2 \cdot \left(n \cdot n\right)}}{\sqrt{Om \cdot Om}}\right| \]
      sqrt-prod [=>] 6.1	\[ \left|\frac{\color{blue}{\left(\sqrt{\ell \cdot \ell} \cdot \sqrt{U \cdot U*}\right)} \cdot \sqrt{2 \cdot \left(n \cdot n\right)}}{\sqrt{Om \cdot Om}}\right| \]
      sqrt-unprod [<=] 3.7	\[ \left|\frac{\left(\color{blue}{\left(\sqrt{\ell} \cdot \sqrt{\ell}\right)} \cdot \sqrt{U \cdot U*}\right) \cdot \sqrt{2 \cdot \left(n \cdot n\right)}}{\sqrt{Om \cdot Om}}\right| \]
      add-sqr-sqrt [<=] 7.6	\[ \left|\frac{\left(\color{blue}{\ell} \cdot \sqrt{U \cdot U*}\right) \cdot \sqrt{2 \cdot \left(n \cdot n\right)}}{\sqrt{Om \cdot Om}}\right| \]
      *-commutative [=>] 7.6	\[ \left|\frac{\left(\ell \cdot \sqrt{U \cdot U*}\right) \cdot \sqrt{\color{blue}{\left(n \cdot n\right) \cdot 2}}}{\sqrt{Om \cdot Om}}\right| \]
      sqrt-prod [=>] 7.7	\[ \left|\frac{\left(\ell \cdot \sqrt{U \cdot U*}\right) \cdot \color{blue}{\left(\sqrt{n \cdot n} \cdot \sqrt{2}\right)}}{\sqrt{Om \cdot Om}}\right| \]
      sqrt-prod [=>] 6.7	\[ \left|\frac{\left(\ell \cdot \sqrt{U \cdot U*}\right) \cdot \left(\color{blue}{\left(\sqrt{n} \cdot \sqrt{n}\right)} \cdot \sqrt{2}\right)}{\sqrt{Om \cdot Om}}\right| \]
      add-sqr-sqrt [<=] 13.0	\[ \left|\frac{\left(\ell \cdot \sqrt{U \cdot U*}\right) \cdot \left(\color{blue}{n} \cdot \sqrt{2}\right)}{\sqrt{Om \cdot Om}}\right| \]
      sqrt-prod [=>] 12.7	\[ \left|\frac{\left(\ell \cdot \sqrt{U \cdot U*}\right) \cdot \left(n \cdot \sqrt{2}\right)}{\color{blue}{\sqrt{Om} \cdot \sqrt{Om}}}\right| \]
      add-sqr-sqrt [<=] 25.2	\[ \left|\frac{\left(\ell \cdot \sqrt{U \cdot U*}\right) \cdot \left(n \cdot \sqrt{2}\right)}{\color{blue}{Om}}\right| \]

   if +inf.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))))

   1. Initial program 0.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Simplified 16.5%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]
      Proof

      [Start] 0.0	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 0.0	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      sub-neg [=>] 0.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]
      associate--l+ [=>] 0.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      *-commutative [=>] 0.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      distribute-rgt-neg-in [=>] 0.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l/ [<=] 7.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 7.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [<=] 7.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [=>] 7.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 8.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]
      unpow2 [=>] 8.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]
      associate-*l* [=>] 16.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in t around inf 33.6%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(t \cdot U\right)\right) + 2 \cdot \frac{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(n \cdot \left(\ell \cdot U\right)\right)}{Om}}} \]

   4. Simplified 36.3%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}{Om}}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}\right)}} \]
      Proof

      [Start] 33.6	\[ \sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right)\right) + 2 \cdot \frac{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(n \cdot \left(\ell \cdot U\right)\right)}{Om}} \]
      distribute-lft-out [=>] 33.6	\[ \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(n \cdot \left(\ell \cdot U\right)\right)}{Om}\right)}} \]
      *-commutative [<=] 33.6	\[ \sqrt{2 \cdot \left(n \cdot \color{blue}{\left(U \cdot t\right)} + \frac{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(n \cdot \left(\ell \cdot U\right)\right)}{Om}\right)} \]
      associate-/l* [=>] 32.8	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \color{blue}{\frac{\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}}\right)} \]
      +-commutative [=>] 32.8	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\color{blue}{-2 \cdot \ell + \frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om}}}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}\right)} \]
      *-commutative [=>] 32.8	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\color{blue}{\ell \cdot -2} + \frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om}}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}\right)} \]
      associate-*r* [=>] 36.3	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{\color{blue}{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}}{Om}}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}\right)} \]

   5. Applied egg-rr 43.3%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}{Om}}{\color{blue}{\frac{1}{n \cdot \ell} \cdot \frac{Om}{U}}}\right)} \]
      Proof

      [Start] 36.3	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}{Om}}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}\right)} \]
      *-un-lft-identity [=>] 36.3	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}{Om}}{\frac{\color{blue}{1 \cdot Om}}{n \cdot \left(\ell \cdot U\right)}}\right)} \]
      associate-*r* [=>] 37.5	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}{Om}}{\frac{1 \cdot Om}{\color{blue}{\left(n \cdot \ell\right) \cdot U}}}\right)} \]
      times-frac [=>] 43.3	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}{Om}}{\color{blue}{\frac{1}{n \cdot \ell} \cdot \frac{Om}{U}}}\right)} \]

   6. Taylor expanded in n around 0 36.3%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}{Om}}{\color{blue}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}}\right)} \]

   7. Simplified 43.7%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}{Om}}{\color{blue}{\frac{\frac{Om}{U}}{n \cdot \ell}}}\right)} \]
      Proof

      [Start] 36.3	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}{Om}}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}\right)} \]
      associate-*r* [=>] 37.5	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}{Om}}{\frac{Om}{\color{blue}{\left(n \cdot \ell\right) \cdot U}}}\right)} \]
      associate-/l/ [<=] 43.7	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}{Om}}{\color{blue}{\frac{\frac{Om}{U}}{n \cdot \ell}}}\right)} \]

   8. Applied egg-rr 28.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\ell, -2, \frac{n \cdot \ell}{\frac{Om}{U* - U}}\right), \left(n \cdot \ell\right) \cdot \frac{U}{Om}, n \cdot \left(U \cdot t\right)\right)}\right)} - 1} \]
      Proof

      [Start] 43.7	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}{Om}}{\frac{\frac{Om}{U}}{n \cdot \ell}}\right)} \]
      expm1-log1p-u [=>] 41.5	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}{Om}}{\frac{\frac{Om}{U}}{n \cdot \ell}}\right)}\right)\right)} \]
      expm1-udef [=>] 30.0	\[ \color{blue}{e^{\mathsf{log1p}\left(\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}{Om}}{\frac{\frac{Om}{U}}{n \cdot \ell}}\right)}\right)} - 1} \]

   9. Simplified 40.9%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\ell, -2, \frac{n \cdot \ell}{\frac{Om}{U* - U}}\right), \left(n \cdot \ell\right) \cdot \frac{U}{Om}, \left(n \cdot U\right) \cdot t\right)}} \]
      Proof

      [Start] 28.8	\[ e^{\mathsf{log1p}\left(\sqrt{2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\ell, -2, \frac{n \cdot \ell}{\frac{Om}{U* - U}}\right), \left(n \cdot \ell\right) \cdot \frac{U}{Om}, n \cdot \left(U \cdot t\right)\right)}\right)} - 1 \]
      expm1-def [=>] 39.6	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\ell, -2, \frac{n \cdot \ell}{\frac{Om}{U* - U}}\right), \left(n \cdot \ell\right) \cdot \frac{U}{Om}, n \cdot \left(U \cdot t\right)\right)}\right)\right)} \]
      expm1-log1p [=>] 41.7	\[ \color{blue}{\sqrt{2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\ell, -2, \frac{n \cdot \ell}{\frac{Om}{U* - U}}\right), \left(n \cdot \ell\right) \cdot \frac{U}{Om}, n \cdot \left(U \cdot t\right)\right)}} \]
      associate-*r* [=>] 40.9	\[ \sqrt{2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\ell, -2, \frac{n \cdot \ell}{\frac{Om}{U* - U}}\right), \left(n \cdot \ell\right) \cdot \frac{U}{Om}, \color{blue}{\left(n \cdot U\right) \cdot t}\right)} \]

3. Recombined 4 regimes into one program.

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;{\left(\sqrt[3]{t + t} \cdot \sqrt[3]{n \cdot U}\right)}^{1.5}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 10^{+151}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;\left|\frac{\left(\ell \cdot \sqrt{U \cdot U*}\right) \cdot \left(n \cdot \sqrt{2}\right)}{Om}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\ell, -2, \frac{n \cdot \ell}{\frac{Om}{U* - U}}\right), \left(n \cdot \ell\right) \cdot \frac{U}{Om}, t \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	61.4%
Cost	44300

\[\begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;{\left(\sqrt[3]{t + t} \cdot \sqrt[3]{n \cdot U}\right)}^{1.5}\\ \mathbf{elif}\;t_1 \leq 10^{+302}:\\ \;\;\;\;\sqrt{t_1}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\left|\frac{\left(\ell \cdot \sqrt{U \cdot U*}\right) \cdot \left(n \cdot \sqrt{2}\right)}{Om}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}{Om}}{\frac{\frac{Om}{U}}{n \cdot \ell}}\right)}\\ \end{array} \]

Alternative 2
Accuracy	61.1%
Cost	30728

\[\begin{array}{l} t_1 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot t_1\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_2 \leq 4 \cdot 10^{-316}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(n \cdot \left(t_1 \cdot \left(U - U*\right)\right) + 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;t_2 \leq 4 \cdot 10^{+281}:\\ \;\;\;\;\sqrt{t_2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}{Om}}{\frac{\frac{Om}{U}}{n \cdot \ell}}\right)}\\ \end{array} \]

Alternative 3
Accuracy	55.4%
Cost	20372

\[\begin{array}{l} t_1 := {\left(\sqrt[3]{t + t} \cdot \sqrt[3]{n \cdot U}\right)}^{1.5}\\ t_2 := \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}{Om}}{\frac{\frac{Om}{U}}{n \cdot \ell}}\right)}\\ \mathbf{if}\;\ell \leq -4.4 \cdot 10^{+142}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -5500:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \left(U \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\left(U* - U\right) \cdot \frac{n}{Om} - 2}{Om}\right)}\\ \mathbf{elif}\;\ell \leq -5.9 \cdot 10^{-95}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.4 \cdot 10^{-302}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{-183}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Accuracy	53.8%
Cost	14808

\[\begin{array}{l} t_1 := \ell \cdot -2 + \frac{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}{Om}\\ t_2 := n \cdot \left(U \cdot t\right)\\ t_3 := \sqrt{2 \cdot \left(t_2 + \frac{t_1}{\frac{\frac{Om}{U}}{n \cdot \ell}}\right)}\\ \mathbf{if}\;\ell \leq -7.2 \cdot 10^{+142}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\ell \leq -2.8 \cdot 10^{-64}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \left(U \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\left(U* - U\right) \cdot \frac{n}{Om} - 2}{Om}\right)}\\ \mathbf{elif}\;\ell \leq -7.2 \cdot 10^{-95}:\\ \;\;\;\;\sqrt{t + t} \cdot \sqrt{n \cdot U}\\ \mathbf{elif}\;\ell \leq 10^{-147}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{+79}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\ell \leq 4.8 \cdot 10^{+230}:\\ \;\;\;\;\left(\ell \cdot \sqrt{\frac{n}{\frac{\frac{Om}{U}}{-2 + \frac{n}{\frac{Om}{U* - U}}}}}\right) \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_2 + \frac{t_1}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)}\\ \end{array} \]

Alternative 5
Accuracy	54.5%
Cost	13964

\[\begin{array}{l} t_1 := \ell \cdot -2 + \frac{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}{Om}\\ t_2 := \frac{\ell}{\frac{Om}{\ell}}\\ t_3 := n \cdot \left(U \cdot t\right)\\ \mathbf{if}\;Om \leq -5 \cdot 10^{+106}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)}\\ \mathbf{elif}\;Om \leq -1.05 \cdot 10^{-186}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_3 + \frac{t_1}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)}\\ \mathbf{elif}\;Om \leq 1.2 \cdot 10^{-231}:\\ \;\;\;\;\frac{\ell \cdot \left(\sqrt{U \cdot U*} \cdot \left(n \cdot \left(-\sqrt{2}\right)\right)\right)}{Om}\\ \mathbf{elif}\;Om \leq 5 \cdot 10^{+163}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_3 + \frac{t_1}{\frac{\frac{Om}{U}}{n \cdot \ell}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(n \cdot \left(t_2 \cdot \frac{U* - U}{Om}\right) - 2 \cdot t_2\right)\right)\right)}\\ \end{array} \]

Alternative 6
Accuracy	54.8%
Cost	8784

\[\begin{array}{l} t_1 := \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)}\\ t_2 := \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}{Om}}{\frac{\frac{Om}{U}}{n \cdot \ell}}\right)}\\ \mathbf{if}\;Om \leq -1.7 \cdot 10^{+107}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;Om \leq -1.9 \cdot 10^{-67}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;Om \leq 6.8 \cdot 10^{-230}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;Om \leq 1.5 \cdot 10^{+165}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;Om \leq 5 \cdot 10^{+287}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \end{array} \]

Alternative 7
Accuracy	54.9%
Cost	8784

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}{Om}}{\frac{\frac{Om}{U}}{n \cdot \ell}}\right)}\\ \mathbf{if}\;Om \leq -2.8 \cdot 10^{+107}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)}\\ \mathbf{elif}\;Om \leq -1.9 \cdot 10^{-67}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;Om \leq 5.1 \cdot 10^{-228}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;Om \leq 2.75 \cdot 10^{+163}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(n \cdot \left(t_1 \cdot \frac{U* - U}{Om}\right) - 2 \cdot t_1\right)\right)\right)}\\ \end{array} \]

Alternative 8
Accuracy	54.9%
Cost	8784

\[\begin{array}{l} t_1 := \ell \cdot -2 + \frac{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}{Om}\\ t_2 := \frac{\ell}{\frac{Om}{\ell}}\\ t_3 := n \cdot \left(U \cdot t\right)\\ \mathbf{if}\;Om \leq -1.75 \cdot 10^{+107}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)}\\ \mathbf{elif}\;Om \leq -1.9 \cdot 10^{-67}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_3 + \frac{t_1}{\frac{Om}{U} \cdot \frac{1}{n \cdot \ell}}\right)}\\ \mathbf{elif}\;Om \leq 1.08 \cdot 10^{-221}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;Om \leq 3.1 \cdot 10^{+163}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_3 + \frac{t_1}{\frac{\frac{Om}{U}}{n \cdot \ell}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(n \cdot \left(t_2 \cdot \frac{U* - U}{Om}\right) - 2 \cdot t_2\right)\right)\right)}\\ \end{array} \]

Alternative 9
Accuracy	52.1%
Cost	8656

\[\begin{array}{l} t_1 := \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)}\\ \mathbf{if}\;U \leq -1.6 \cdot 10^{+69}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;U \leq -2.3 \cdot 10^{-227}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + U* \cdot \frac{n \cdot \ell}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;U \leq 3.3 \cdot 10^{-298}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\frac{n}{Om} \cdot \left(\left(2 - \frac{U*}{\frac{Om}{n}}\right) \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \mathbf{elif}\;U \leq 5.2 \cdot 10^{+83}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}}{\frac{\frac{Om}{U}}{n \cdot \ell}}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Accuracy	51.4%
Cost	8525

\[\begin{array}{l} t_1 := n \cdot \left(U \cdot \ell\right)\\ \mathbf{if}\;Om \leq -3.4 \cdot 10^{+103}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)}\\ \mathbf{elif}\;Om \leq -5 \cdot 10^{-11} \lor \neg \left(Om \leq 1.8 \cdot 10^{-89}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 - \frac{t_1}{Om}}{\frac{Om}{t_1}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \end{array} \]

Alternative 11
Accuracy	51.1%
Cost	8524

\[\begin{array}{l} t_1 := n \cdot \left(U \cdot \ell\right)\\ t_2 := \ell \cdot -2 - \frac{t_1}{Om}\\ t_3 := n \cdot \left(U \cdot t\right)\\ \mathbf{if}\;Om \leq -8.8 \cdot 10^{+103}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)}\\ \mathbf{elif}\;Om \leq -1 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_3 + \frac{t_2}{\frac{\frac{Om}{U}}{n \cdot \ell}}\right)}\\ \mathbf{elif}\;Om \leq 2.1 \cdot 10^{-91}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_3 + \frac{t_2}{\frac{Om}{t_1}}\right)}\\ \end{array} \]

Alternative 12
Accuracy	52.2%
Cost	8524

\[\begin{array}{l} t_1 := n \cdot \left(U \cdot \ell\right)\\ t_2 := \frac{Om}{t_1}\\ t_3 := n \cdot \left(U \cdot t\right)\\ \mathbf{if}\;Om \leq -3.9 \cdot 10^{+102}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)}\\ \mathbf{elif}\;Om \leq -1.3 \cdot 10^{-87}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_3 + \frac{\ell \cdot -2 + \frac{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}{Om}}{t_2}\right)}\\ \mathbf{elif}\;Om \leq 3.2 \cdot 10^{-94}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_3 + \frac{\ell \cdot -2 - \frac{t_1}{Om}}{t_2}\right)}\\ \end{array} \]

Alternative 13
Accuracy	51.4%
Cost	8392

\[\begin{array}{l} t_1 := \ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\\ \mathbf{if}\;\ell \leq -3.1 \cdot 10^{+130}:\\ \;\;\;\;\sqrt{\left(\frac{n}{Om} \cdot \frac{U - U*}{Om} + \frac{2}{Om}\right) \cdot \left(-2 \cdot \left(n \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 10^{-8}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot t_1}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{t_1}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)}\\ \end{array} \]

Alternative 14
Accuracy	51.1%
Cost	8136

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.85 \cdot 10^{+128}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\frac{n}{Om} \cdot \left(\left(2 - \frac{U*}{\frac{Om}{n}}\right) \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 11200000000:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + U* \cdot \frac{n \cdot \ell}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)}\\ \end{array} \]

Alternative 15
Accuracy	50.3%
Cost	8136

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.06 \cdot 10^{+130}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\frac{n}{Om} \cdot \left(\left(2 - \frac{U*}{\frac{Om}{n}}\right) \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 3600000000:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)}\\ \end{array} \]

Alternative 16
Accuracy	50.2%
Cost	8136

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.5 \cdot 10^{+130}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\frac{n}{Om} \cdot \left(\left(2 - \frac{U*}{\frac{Om}{n}}\right) \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.65 \cdot 10^{+128}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \frac{\ell \cdot -2 + \frac{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}{Om}}{\frac{Om}{U \cdot \ell}}}\\ \end{array} \]

Alternative 17
Accuracy	50.4%
Cost	8136

\[\begin{array}{l} \mathbf{if}\;\ell \leq -4.15 \cdot 10^{+130}:\\ \;\;\;\;\sqrt{\left(\frac{n}{Om} \cdot \frac{U - U*}{Om} + \frac{2}{Om}\right) \cdot \left(-2 \cdot \left(n \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.2 \cdot 10^{+129}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \frac{\ell \cdot -2 + \frac{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}{Om}}{\frac{Om}{U \cdot \ell}}}\\ \end{array} \]

Alternative 18
Accuracy	46.8%
Cost	8013

\[\begin{array}{l} \mathbf{if}\;\ell \leq -3.1 \cdot 10^{+141}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\frac{n}{Om} \cdot \left(\left(2 - \frac{U*}{\frac{Om}{n}}\right) \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{-140} \lor \neg \left(\ell \leq 3.8 \cdot 10^{-36}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{n \cdot \left(\ell \cdot \ell\right)}{Om} \cdot \frac{U*}{Om}\right)\right)}\\ \end{array} \]

Alternative 19
Accuracy	50.0%
Cost	7625

\[\begin{array}{l} \mathbf{if}\;U \leq -1.2 \cdot 10^{-157} \lor \neg \left(U \leq 10^{-55}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)}\\ \end{array} \]

Alternative 20
Accuracy	39.6%
Cost	7500

\[\begin{array}{l} t_1 := \sqrt{-4 \cdot \frac{\ell \cdot \left(\ell \cdot \left(n \cdot U\right)\right)}{Om}}\\ \mathbf{if}\;\ell \leq -3 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.06 \cdot 10^{-61}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.4 \cdot 10^{+129}:\\ \;\;\;\;\sqrt{t \cdot \left(n \cdot \left(U + U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 21
Accuracy	46.6%
Cost	7492

\[\begin{array}{l} \mathbf{if}\;U \leq -1.95 \cdot 10^{+83}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \end{array} \]

Alternative 22
Accuracy	46.6%
Cost	7492

\[\begin{array}{l} \mathbf{if}\;U \leq -7.1 \cdot 10^{+84}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)}\\ \end{array} \]

Alternative 23
Accuracy	41.0%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;\ell \leq -6.4 \cdot 10^{+53}:\\ \;\;\;\;\sqrt{-2 \cdot \left(2 \cdot \frac{n}{\frac{Om}{\ell \cdot \left(U \cdot \ell\right)}}\right)}\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{+127}:\\ \;\;\;\;\sqrt{t \cdot \left(n \cdot \left(U + U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{\ell \cdot \left(\ell \cdot \left(n \cdot U\right)\right)}{Om}}\\ \end{array} \]

Alternative 24
Accuracy	40.1%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;U \leq -3.4 \cdot 10^{+69}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;U \leq 4.8 \cdot 10^{-49}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(n \cdot \left(U + U\right)\right)}\\ \end{array} \]

Alternative 25
Accuracy	38.8%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;n \leq 6.2 \cdot 10^{-131}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(n \cdot \left(U + U\right)\right)}\\ \end{array} \]

Alternative 26
Accuracy	37.9%
Cost	6848

\[\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \]

Alternative 27
Accuracy	5.7%
Cost	704

\[\begin{array}{l} t_1 := t \cdot \left(n \cdot U\right)\\ t_1 + t_1 \end{array} \]

Alternative 28
Accuracy	5.5%
Cost	448

\[\left(n \cdot t\right) \cdot \left(U + U\right) \]

Alternative 29
Accuracy	5.0%
Cost	320

\[\left(U \cdot U\right) \cdot 4 \]

Alternative 30
Accuracy	2.4%
Cost	192

\[U + -2 \]

Alternative 31
Accuracy	3.8%
Cost	192

\[U + U \]

Reproduce

herbie shell --seed 2023151 
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  :precision binary64
  (sqrt (* (* (* 2.0 n) U) (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))