Toniolo and Linder, Equation (13)

Average Accuracy: 46.2% → 61.2%

Time: 51.9s

Precision: binary64

Cost: 38796

Math

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

↓

\[\begin{array}{l} t_1 := \left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right) + \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right)\\ \mathbf{if}\;t_1 \leq 2 \cdot 10^{-312}:\\ \;\;\;\;{\left(\sqrt[3]{n + n} \cdot \sqrt[3]{U \cdot t}\right)}^{1.5}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{t_1}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + \ell \cdot \left(\ell \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{2}{Om} + \frac{U - U*}{\frac{Om}{\frac{n}{Om}}}\right) \cdot \left(\left(U \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)\right) \cdot -2\right)}\\ \end{array} \]

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
         (*
          (* U (* n -2.0))
          (+
           (* (* n (pow (/ l Om) 2.0)) (- U U*))
           (- (* 2.0 (/ (* l l) Om)) t)))))
   (if (<= t_1 2e-312)
     (pow (* (cbrt (+ n n)) (cbrt (* U t))) 1.5)
     (if (<= t_1 5e+306)
       (sqrt t_1)
       (if (<= t_1 INFINITY)
         (*
          (sqrt (* 2.0 n))
          (sqrt
           (*
            U
            (+ t (* l (* l (+ (* (/ n Om) (/ (- U* U) Om)) (/ -2.0 Om))))))))
         (sqrt
          (*
           (+ (/ 2.0 Om) (/ (- U U*) (/ Om (/ n Om))))
           (* (* U (* l (* n l))) -2.0))))))))

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 = (U * (n * -2.0)) * (((n * pow((l / Om), 2.0)) * (U - U_42_)) + ((2.0 * ((l * l) / Om)) - t));
	double tmp;
	if (t_1 <= 2e-312) {
		tmp = pow((cbrt((n + n)) * cbrt((U * t))), 1.5);
	} else if (t_1 <= 5e+306) {
		tmp = sqrt(t_1);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = sqrt((2.0 * n)) * sqrt((U * (t + (l * (l * (((n / Om) * ((U_42_ - U) / Om)) + (-2.0 / Om)))))));
	} else {
		tmp = sqrt((((2.0 / Om) + ((U - U_42_) / (Om / (n / Om)))) * ((U * (l * (n * l))) * -2.0)));
	}
	return tmp;
}

Java

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

↓

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (U * (n * -2.0)) * (((n * Math.pow((l / Om), 2.0)) * (U - U_42_)) + ((2.0 * ((l * l) / Om)) - t));
	double tmp;
	if (t_1 <= 2e-312) {
		tmp = Math.pow((Math.cbrt((n + n)) * Math.cbrt((U * t))), 1.5);
	} else if (t_1 <= 5e+306) {
		tmp = Math.sqrt(t_1);
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((2.0 * n)) * Math.sqrt((U * (t + (l * (l * (((n / Om) * ((U_42_ - U) / Om)) + (-2.0 / Om)))))));
	} else {
		tmp = Math.sqrt((((2.0 / Om) + ((U - U_42_) / (Om / (n / Om)))) * ((U * (l * (n * l))) * -2.0)));
	}
	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 = Float64(Float64(U * Float64(n * -2.0)) * Float64(Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_)) + Float64(Float64(2.0 * Float64(Float64(l * l) / Om)) - t)))
	tmp = 0.0
	if (t_1 <= 2e-312)
		tmp = Float64(cbrt(Float64(n + n)) * cbrt(Float64(U * t))) ^ 1.5;
	elseif (t_1 <= 5e+306)
		tmp = sqrt(t_1);
	elseif (t_1 <= Inf)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * Float64(t + Float64(l * Float64(l * Float64(Float64(Float64(n / Om) * Float64(Float64(U_42_ - U) / Om)) + Float64(-2.0 / Om))))))));
	else
		tmp = sqrt(Float64(Float64(Float64(2.0 / Om) + Float64(Float64(U - U_42_) / Float64(Om / Float64(n / Om)))) * Float64(Float64(U * Float64(l * Float64(n * l))) * -2.0)));
	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[(N[(U * N[(n * -2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-312], N[Power[N[(N[Power[N[(n + n), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(U * t), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], If[LessEqual[t$95$1, 5e+306], N[Sqrt[t$95$1], $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t + N[(l * N[(l * N[(N[(N[(n / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(N[(2.0 / Om), $MachinePrecision] + N[(N[(U - U$42$), $MachinePrecision] / N[(Om / N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(U * N[(l * N[(n * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.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*)))) < 2.0000000000019e-312

   1. Initial program 11.2%

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

   2. Simplified 13.9%

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

      [Start] 11.2	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 11.2	\[ \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 11.1	\[ \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      *-commutative [=>] 11.1	\[ \sqrt{2 \cdot \color{blue}{\left(\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot \left(n \cdot U\right)\right)}} \]

   3. Taylor expanded in l around 0 30.0%

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

   4. Applied egg-rr 29.8%

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

      [Start] 30.0	\[ \sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)} \]
      rem-cube-cbrt [<=] 29.8	\[ \sqrt{\color{blue}{{\left(\sqrt[3]{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)}\right)}^{3}}} \]
      sqrt-pow1 [=>] 29.8	\[ \color{blue}{{\left(\sqrt[3]{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)}\right)}^{\left(\frac{3}{2}\right)}} \]
      associate-*r* [=>] 29.8	\[ {\left(\sqrt[3]{\color{blue}{\left(2 \cdot n\right) \cdot \left(t \cdot U\right)}}\right)}^{\left(\frac{3}{2}\right)} \]
      *-commutative [=>] 29.8	\[ {\left(\sqrt[3]{\color{blue}{\left(t \cdot U\right) \cdot \left(2 \cdot n\right)}}\right)}^{\left(\frac{3}{2}\right)} \]
      metadata-eval [=>] 29.8	\[ {\left(\sqrt[3]{\left(t \cdot U\right) \cdot \left(2 \cdot n\right)}\right)}^{\color{blue}{1.5}} \]

   5. Applied egg-rr 56.1%

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

      [Start] 29.8	\[ {\left(\sqrt[3]{\left(t \cdot U\right) \cdot \left(2 \cdot n\right)}\right)}^{1.5} \]
      cbrt-prod [=>] 56.1	\[ {\color{blue}{\left(\sqrt[3]{t \cdot U} \cdot \sqrt[3]{2 \cdot n}\right)}}^{1.5} \]
      *-commutative [=>] 56.1	\[ {\color{blue}{\left(\sqrt[3]{2 \cdot n} \cdot \sqrt[3]{t \cdot U}\right)}}^{1.5} \]

   6. Simplified 56.1%

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

      [Start] 56.1	\[ {\left(\sqrt[3]{2 \cdot n} \cdot \sqrt[3]{t \cdot U}\right)}^{1.5} \]
      count-2 [<=] 56.1	\[ {\left(\sqrt[3]{\color{blue}{n + n}} \cdot \sqrt[3]{t \cdot U}\right)}^{1.5} \]

   if 2.0000000000019e-312 < (*.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*)))) < 4.99999999999999993e306

   1. Initial program 97.7%

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

   if 4.99999999999999993e306 < (*.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 0.1%

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

   2. Applied egg-rr 10.6%

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

      [Start] 0.1	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 3.1	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      sqrt-prod [=>] 10.3	\[ \color{blue}{\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]
      associate--l- [=>] 10.3	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      fma-def [=>] 10.3	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \color{blue}{\mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)} \]
      div-inv [=>] 10.3	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{Om}}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]
      associate-*l* [=>] 16.6	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \color{blue}{\ell \cdot \left(\ell \cdot \frac{1}{Om}\right)}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]
      div-inv [<=] 16.6	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \color{blue}{\frac{\ell}{Om}}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]
      *-commutative [=>] 16.6	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)} \]
      associate-*l* [=>] 10.6	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)} \]

   3. Simplified 17.1%

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

      [Start] 10.6	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*r/ [=>] 4.3	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \color{blue}{\frac{\ell \cdot \ell}{Om}}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-/l* [=>] 10.6	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*r* [=>] 16.6	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U - U*\right)}\right)\right)} \]
      *-commutative [<=] 16.6	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)} \cdot \left(U - U*\right)\right)\right)} \]
      associate-*l* [=>] 17.1	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)} \]

   4. Taylor expanded in t around 0 8.3%

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

   5. Simplified 14.6%

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

      [Start] 8.3	\[ \sqrt{2 \cdot n} \cdot \sqrt{t \cdot U + -1 \cdot \left(\left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{n \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right) \cdot U\right)} \]
      associate-*r* [=>] 8.3	\[ \sqrt{2 \cdot n} \cdot \sqrt{t \cdot U + \color{blue}{\left(-1 \cdot \left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{n \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right) \cdot U}} \]
      distribute-rgt-out [=>] 8.3	\[ \sqrt{2 \cdot n} \cdot \sqrt{\color{blue}{U \cdot \left(t + -1 \cdot \left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{n \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)}} \]
      mul-1-neg [=>] 8.3	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + \color{blue}{\left(-\left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{n \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)}\right)} \]
      associate-*r/ [=>] 8.3	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + \left(-\left(\color{blue}{\frac{2 \cdot {\ell}^{2}}{Om}} + \frac{n \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)} \]
      associate-/l* [=>] 8.3	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + \left(-\left(\color{blue}{\frac{2}{\frac{Om}{{\ell}^{2}}}} + \frac{n \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)} \]
      associate-/r/ [=>] 8.3	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + \left(-\left(\color{blue}{\frac{2}{Om} \cdot {\ell}^{2}} + \frac{n \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)} \]
      metadata-eval [<=] 8.3	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + \left(-\left(\frac{\color{blue}{2 \cdot 1}}{Om} \cdot {\ell}^{2} + \frac{n \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)} \]
      associate-*r/ [<=] 8.3	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + \left(-\left(\color{blue}{\left(2 \cdot \frac{1}{Om}\right)} \cdot {\ell}^{2} + \frac{n \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)} \]
      associate-/l* [=>] 9.3	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + \left(-\left(\left(2 \cdot \frac{1}{Om}\right) \cdot {\ell}^{2} + \color{blue}{\frac{n}{\frac{{Om}^{2}}{{\ell}^{2} \cdot \left(U - U*\right)}}}\right)\right)\right)} \]
      associate-/l/ [<=] 9.5	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + \left(-\left(\left(2 \cdot \frac{1}{Om}\right) \cdot {\ell}^{2} + \frac{n}{\color{blue}{\frac{\frac{{Om}^{2}}{U - U*}}{{\ell}^{2}}}}\right)\right)\right)} \]
      associate-/r/ [=>] 8.0	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + \left(-\left(\left(2 \cdot \frac{1}{Om}\right) \cdot {\ell}^{2} + \color{blue}{\frac{n}{\frac{{Om}^{2}}{U - U*}} \cdot {\ell}^{2}}\right)\right)\right)} \]
      associate-/l* [<=] 2.9	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + \left(-\left(\left(2 \cdot \frac{1}{Om}\right) \cdot {\ell}^{2} + \color{blue}{\frac{n \cdot \left(U - U*\right)}{{Om}^{2}}} \cdot {\ell}^{2}\right)\right)\right)} \]

   if +inf.0 < (*.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. Taylor expanded in l around inf 4.8%

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

   3. Simplified 22.9%

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

      [Start] 4.8	\[ \sqrt{-2 \cdot \left(\left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(n \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      *-commutative [=>] 4.8	\[ \sqrt{\color{blue}{\left(\left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(n \cdot \left({\ell}^{2} \cdot U\right)\right)\right) \cdot -2}} \]
      associate-*l* [=>] 4.8	\[ \sqrt{\color{blue}{\left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(\left(n \cdot \left({\ell}^{2} \cdot U\right)\right) \cdot -2\right)}} \]
      associate-/l* [=>] 4.1	\[ \sqrt{\left(\color{blue}{\frac{n}{\frac{{Om}^{2}}{U - U*}}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(\left(n \cdot \left({\ell}^{2} \cdot U\right)\right) \cdot -2\right)} \]
      associate-/r/ [=>] 4.8	\[ \sqrt{\left(\color{blue}{\frac{n}{{Om}^{2}} \cdot \left(U - U*\right)} + 2 \cdot \frac{1}{Om}\right) \cdot \left(\left(n \cdot \left({\ell}^{2} \cdot U\right)\right) \cdot -2\right)} \]
      unpow2 [=>] 4.8	\[ \sqrt{\left(\frac{n}{\color{blue}{Om \cdot Om}} \cdot \left(U - U*\right) + 2 \cdot \frac{1}{Om}\right) \cdot \left(\left(n \cdot \left({\ell}^{2} \cdot U\right)\right) \cdot -2\right)} \]
      associate-*r/ [=>] 4.8	\[ \sqrt{\left(\frac{n}{Om \cdot Om} \cdot \left(U - U*\right) + \color{blue}{\frac{2 \cdot 1}{Om}}\right) \cdot \left(\left(n \cdot \left({\ell}^{2} \cdot U\right)\right) \cdot -2\right)} \]
      metadata-eval [=>] 4.8	\[ \sqrt{\left(\frac{n}{Om \cdot Om} \cdot \left(U - U*\right) + \frac{\color{blue}{2}}{Om}\right) \cdot \left(\left(n \cdot \left({\ell}^{2} \cdot U\right)\right) \cdot -2\right)} \]
      associate-*r* [=>] 5.2	\[ \sqrt{\left(\frac{n}{Om \cdot Om} \cdot \left(U - U*\right) + \frac{2}{Om}\right) \cdot \left(\color{blue}{\left(\left(n \cdot {\ell}^{2}\right) \cdot U\right)} \cdot -2\right)} \]
      *-commutative [=>] 5.2	\[ \sqrt{\left(\frac{n}{Om \cdot Om} \cdot \left(U - U*\right) + \frac{2}{Om}\right) \cdot \left(\color{blue}{\left(U \cdot \left(n \cdot {\ell}^{2}\right)\right)} \cdot -2\right)} \]
      *-commutative [=>] 5.2	\[ \sqrt{\left(\frac{n}{Om \cdot Om} \cdot \left(U - U*\right) + \frac{2}{Om}\right) \cdot \left(\left(U \cdot \color{blue}{\left({\ell}^{2} \cdot n\right)}\right) \cdot -2\right)} \]
      unpow2 [=>] 5.2	\[ \sqrt{\left(\frac{n}{Om \cdot Om} \cdot \left(U - U*\right) + \frac{2}{Om}\right) \cdot \left(\left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)\right) \cdot -2\right)} \]
      associate-*l* [=>] 22.9	\[ \sqrt{\left(\frac{n}{Om \cdot Om} \cdot \left(U - U*\right) + \frac{2}{Om}\right) \cdot \left(\left(U \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot n\right)\right)}\right) \cdot -2\right)} \]
      *-commutative [<=] 22.9	\[ \sqrt{\left(\frac{n}{Om \cdot Om} \cdot \left(U - U*\right) + \frac{2}{Om}\right) \cdot \left(\left(U \cdot \left(\ell \cdot \color{blue}{\left(n \cdot \ell\right)}\right)\right) \cdot -2\right)} \]

   4. Applied egg-rr 29.2%

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

      [Start] 22.9	\[ \sqrt{\left(\frac{n}{Om \cdot Om} \cdot \left(U - U*\right) + \frac{2}{Om}\right) \cdot \left(\left(U \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)\right) \cdot -2\right)} \]
      *-commutative [=>] 22.9	\[ \sqrt{\left(\color{blue}{\left(U - U*\right) \cdot \frac{n}{Om \cdot Om}} + \frac{2}{Om}\right) \cdot \left(\left(U \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)\right) \cdot -2\right)} \]
      associate-/r* [=>] 28.9	\[ \sqrt{\left(\left(U - U*\right) \cdot \color{blue}{\frac{\frac{n}{Om}}{Om}} + \frac{2}{Om}\right) \cdot \left(\left(U \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)\right) \cdot -2\right)} \]
      associate-*r/ [=>] 30.6	\[ \sqrt{\left(\color{blue}{\frac{\left(U - U*\right) \cdot \frac{n}{Om}}{Om}} + \frac{2}{Om}\right) \cdot \left(\left(U \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)\right) \cdot -2\right)} \]
      associate-/l* [=>] 29.2	\[ \sqrt{\left(\color{blue}{\frac{U - U*}{\frac{Om}{\frac{n}{Om}}}} + \frac{2}{Om}\right) \cdot \left(\left(U \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)\right) \cdot -2\right)} \]

3. Recombined 4 regimes into one program.

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right) + \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right) \leq 2 \cdot 10^{-312}:\\ \;\;\;\;{\left(\sqrt[3]{n + n} \cdot \sqrt[3]{U \cdot t}\right)}^{1.5}\\ \mathbf{elif}\;\left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right) + \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right) \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right) + \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right)}\\ \mathbf{elif}\;\left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right) + \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + \ell \cdot \left(\ell \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{2}{Om} + \frac{U - U*}{\frac{Om}{\frac{n}{Om}}}\right) \cdot \left(\left(U \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)\right) \cdot -2\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	57.8%
Cost	38796

\[\begin{array}{l} t_1 := \left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right) + \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right)\\ t_2 := \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + \ell \cdot \left(\ell \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)\right)\right)}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{t_1}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{2}{Om} + \frac{U - U*}{\frac{Om}{\frac{n}{Om}}}\right) \cdot \left(\left(U \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)\right) \cdot -2\right)}\\ \end{array} \]

Alternative 2
Accuracy	61.2%
Cost	38796

\[\begin{array}{l} t_1 := \left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right) + \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right)\\ \mathbf{if}\;t_1 \leq 2 \cdot 10^{-312}:\\ \;\;\;\;{\left(\sqrt[3]{t \cdot \left(2 \cdot U\right)} \cdot \sqrt[3]{n}\right)}^{1.5}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{t_1}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + \ell \cdot \left(\ell \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{2}{Om} + \frac{U - U*}{\frac{Om}{\frac{n}{Om}}}\right) \cdot \left(\left(U \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)\right) \cdot -2\right)}\\ \end{array} \]

Alternative 3
Accuracy	59.1%
Cost	20108

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := \sqrt{n \cdot \left(U \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right)\right)}\\ \mathbf{if}\;\ell \leq -1.9 \cdot 10^{+90}:\\ \;\;\;\;t_2 \cdot \left(\ell \cdot \left(-\sqrt{2}\right)\right)\\ \mathbf{elif}\;\ell \leq -2.6 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot t_1 + \frac{n \cdot \left(\ell \cdot \frac{U* - U}{Om}\right)}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.2 \cdot 10^{-142}:\\ \;\;\;\;{\left(\sqrt[3]{t} \cdot \sqrt[3]{n \cdot \left(2 \cdot U\right)}\right)}^{1.5}\\ \mathbf{elif}\;\ell \leq 1.85 \cdot 10^{+180}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot t_1 - \frac{U* \cdot \left(\ell \cdot \frac{n}{\frac{Om}{\ell}}\right)}{Om}\right) - t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]

Alternative 4
Accuracy	52.3%
Cost	14420

\[\begin{array}{l} t_1 := \frac{n}{\frac{Om}{\ell}}\\ t_2 := \frac{\ell}{\frac{Om}{\ell}}\\ t_3 := \sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot t_2 - \frac{U* \cdot \left(\ell \cdot t_1\right)}{Om}\right) - t\right)\right)}\\ \mathbf{if}\;n \leq -9 \cdot 10^{+159}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot t_2 - n \cdot \frac{\ell}{\frac{\frac{Om}{\ell}}{\frac{U - U*}{Om}}}\right)\right)\right)}\\ \mathbf{elif}\;n \leq -6.6 \cdot 10^{+96}:\\ \;\;\;\;\sqrt{2 \cdot t} \cdot \sqrt{n \cdot U}\\ \mathbf{elif}\;n \leq -3 \cdot 10^{-74}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;n \leq 7.2 \cdot 10^{-299}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), -4 \cdot \left(t_1 \cdot \left(U \cdot \ell\right)\right)\right)}\\ \mathbf{elif}\;n \leq 10^{-165}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + \ell \cdot \left(\frac{\ell}{Om} \cdot -2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 5
Accuracy	54.5%
Cost	14412

\[\begin{array}{l} t_1 := \sqrt{n \cdot \left(U \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right)\right)}\\ \mathbf{if}\;\ell \leq -2.85 \cdot 10^{+37}:\\ \;\;\;\;t_1 \cdot \left(\ell \cdot \left(-\sqrt{2}\right)\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-243}:\\ \;\;\;\;\sqrt{\left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \left(-2 + \frac{n}{Om} \cdot \left(U* - U\right)\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{+170}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \frac{U* \cdot \left(\ell \cdot \frac{n}{\frac{Om}{\ell}}\right)}{Om}\right) - t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]

Alternative 6
Accuracy	52.4%
Cost	14164

\[\begin{array}{l} t_1 := 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := \sqrt{\left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right) \cdot \left(2 \cdot \left(\ell \cdot \left(U \cdot \left(n \cdot \ell\right)\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -2.2 \cdot 10^{+141}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -8.2 \cdot 10^{-134}:\\ \;\;\;\;\sqrt{\left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \left(-2 + \frac{n}{Om} \cdot \left(U* - U\right)\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;\ell \leq -1.02 \cdot 10^{-277}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(t_1 - t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{+170}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(t_1 - \frac{U* \cdot \left(\ell \cdot \frac{n}{\frac{Om}{\ell}}\right)}{Om}\right) - t\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.8 \cdot 10^{+238}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{-2 \cdot \frac{n \cdot U}{Om}}\\ \end{array} \]

Alternative 7
Accuracy	52.6%
Cost	14164

\[\begin{array}{l} t_1 := \sqrt{-2 \cdot \frac{n \cdot U}{Om}}\\ t_2 := 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;\ell \leq -1.35 \cdot 10^{+123}:\\ \;\;\;\;\left(\ell \cdot \left(-\sqrt{2}\right)\right) \cdot t_1\\ \mathbf{elif}\;\ell \leq -9 \cdot 10^{-134}:\\ \;\;\;\;\sqrt{\left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \left(-2 + \frac{n}{Om} \cdot \left(U* - U\right)\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;\ell \leq -2.25 \cdot 10^{-278}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(t_2 - t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.4 \cdot 10^{+176}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(t_2 - \frac{U* \cdot \left(\ell \cdot \frac{n}{\frac{Om}{\ell}}\right)}{Om}\right) - t\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{+238}:\\ \;\;\;\;\sqrt{\left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right) \cdot \left(2 \cdot \left(\ell \cdot \left(U \cdot \left(n \cdot \ell\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot t_1\\ \end{array} \]

Alternative 8
Accuracy	51.6%
Cost	8656

\[\begin{array}{l} t_1 := 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := \sqrt{\left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right) \cdot \left(2 \cdot \left(\ell \cdot \left(U \cdot \left(n \cdot \ell\right)\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -3 \cdot 10^{+142}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -8.5 \cdot 10^{-134}:\\ \;\;\;\;\sqrt{\left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \left(-2 + \frac{n}{Om} \cdot \left(U* - U\right)\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;\ell \leq -2.7 \cdot 10^{-278}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(t_1 - t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.8 \cdot 10^{+169}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(t_1 - \frac{U* \cdot \left(\ell \cdot \frac{n}{\frac{Om}{\ell}}\right)}{Om}\right) - t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 9
Accuracy	50.1%
Cost	8524

\[\begin{array}{l} t_1 := \sqrt{\left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right) \cdot \left(2 \cdot \left(\ell \cdot \left(U \cdot \left(n \cdot \ell\right)\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -2.8 \cdot 10^{+140}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -2.15 \cdot 10^{-244}:\\ \;\;\;\;\sqrt{\left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \left(-2 + \frac{n}{Om} \cdot \left(U* - U\right)\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.1 \cdot 10^{+81}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\frac{n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om} + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Accuracy	50.1%
Cost	8524

\[\begin{array}{l} t_1 := \sqrt{\left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right) \cdot \left(2 \cdot \left(\ell \cdot \left(U \cdot \left(n \cdot \ell\right)\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -2.8 \cdot 10^{+140}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -2.8 \cdot 10^{-245}:\\ \;\;\;\;\sqrt{\left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \left(-2 + \frac{n}{Om} \cdot \left(U* - U\right)\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{+83}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\frac{n}{\frac{Om}{\left(\ell \cdot \ell\right) \cdot U*}}}{Om} + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Accuracy	50.1%
Cost	8404

\[\begin{array}{l} t_1 := \sqrt{\left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right) \cdot \left(2 \cdot \left(\ell \cdot \left(U \cdot \left(n \cdot \ell\right)\right)\right)\right)}\\ t_2 := \sqrt{\left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \left(-2 + \frac{n}{Om} \cdot \left(U* - U\right)\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{if}\;\ell \leq -2.9 \cdot 10^{+138}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -6.4 \cdot 10^{-244}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{-165}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.8 \cdot 10^{+35}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{+89}:\\ \;\;\;\;\sqrt{n \cdot \left(-2 \cdot \left(U \cdot \left(\frac{\ell}{Om} \cdot \left(2 \cdot \ell\right) - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Accuracy	47.2%
Cost	8272

\[\begin{array}{l} t_1 := \sqrt{\left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right) \cdot \left(2 \cdot \left(\ell \cdot \left(U \cdot \left(n \cdot \ell\right)\right)\right)\right)}\\ t_2 := \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - t\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -1.4 \cdot 10^{+61}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -1.28 \cdot 10^{-185}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{-164}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.2 \cdot 10^{+89}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Accuracy	46.1%
Cost	7876

\[\begin{array}{l} \mathbf{if}\;\ell \leq -3.2 \cdot 10^{+62}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(\ell \cdot \left(\left(U \cdot \ell\right) \cdot \left(n \cdot \frac{U*}{Om \cdot Om} + \frac{-2}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -1.28 \cdot 10^{-185} \lor \neg \left(\ell \leq 5 \cdot 10^{-164}\right):\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right)\right)\right)}\\ \end{array} \]

Alternative 14
Accuracy	45.7%
Cost	7876

\[\begin{array}{l} \mathbf{if}\;\ell \leq -3.5 \cdot 10^{+62}:\\ \;\;\;\;\sqrt{\left(\left(U \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)\right) \cdot -2\right) \cdot \left(\frac{2}{Om} - \frac{n}{Om} \cdot \frac{U*}{Om}\right)}\\ \mathbf{elif}\;\ell \leq -1.28 \cdot 10^{-185} \lor \neg \left(\ell \leq 2.95 \cdot 10^{-164}\right):\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right)\right)\right)}\\ \end{array} \]

Alternative 15
Accuracy	47.0%
Cost	7756

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{if}\;U \leq -1.3 \cdot 10^{-211}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;U \leq 7.1 \cdot 10^{-230}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - t\right)\right)\right)}\\ \mathbf{elif}\;U \leq 4.1 \cdot 10^{+156}:\\ \;\;\;\;\sqrt{n \cdot \left(-2 \cdot \left(U \cdot \left(\frac{\ell}{Om} \cdot \left(2 \cdot \ell\right) - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 16
Accuracy	47.0%
Cost	7756

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{if}\;U \leq -1.3 \cdot 10^{-211}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;U \leq 1.9 \cdot 10^{-230}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot t_1 - t\right)\right)\right)}\\ \mathbf{elif}\;U \leq 4.1 \cdot 10^{+156}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 17
Accuracy	38.7%
Cost	7632

\[\begin{array}{l} t_1 := \sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ t_2 := \sqrt{-4 \cdot \left(\frac{n}{Om} \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -1.1 \cdot 10^{+34}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -1.06 \cdot 10^{-268}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{-74}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 3 \cdot 10^{+44}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 18
Accuracy	41.9%
Cost	7632

\[\begin{array}{l} t_1 := \sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ t_2 := \sqrt{-4 \cdot \frac{n}{\frac{\frac{Om}{\ell}}{U \cdot \ell}}}\\ \mathbf{if}\;\ell \leq -7.4 \cdot 10^{+33}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -4 \cdot 10^{-266}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.01 \cdot 10^{-72}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.7 \cdot 10^{+44}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 19
Accuracy	47.0%
Cost	7492

\[\begin{array}{l} \mathbf{if}\;n \leq 2.1 \cdot 10^{+80}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 20
Accuracy	37.8%
Cost	7244

\[\begin{array}{l} t_1 := \sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{if}\;U \leq -1.15 \cdot 10^{-211}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;U \leq 1.15 \cdot 10^{-250}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;U \leq 2 \cdot 10^{+157}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 21
Accuracy	37.7%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;n \leq 4.3 \cdot 10^{-156}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]

Alternative 22
Accuracy	37.0%
Cost	6848

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

Reproduce

herbie shell --seed 2023137 
(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*))))))