Toniolo and Linder, Equation (13)

Average Accuracy: 45.6% → 59.9%

Time: 57.9s

Precision: binary64

Cost: 51532

Math

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

↓

\[\begin{array}{l} t_1 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot t_1\right) \cdot \left(U* - U\right)\right)\\ t_3 := \left(U \cdot \ell\right) \cdot \left(n \cdot \ell\right)\\ \mathbf{if}\;t_2 \leq 0:\\ \;\;\;\;\sqrt{n \cdot \mathsf{fma}\left(\ell, \frac{\ell}{Om} \cdot -2, t\right)} \cdot \sqrt{2 \cdot U}\\ \mathbf{elif}\;t_2 \leq 4 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{t_2}\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;\sqrt{t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, n \cdot \left(t_1 \cdot \left(U - U*\right)\right)\right)} \cdot \sqrt{2 \cdot \left(n \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_3 \cdot \frac{n \cdot \left(U* - U\right)}{Om \cdot Om} + t_3 \cdot \frac{-2}{Om}\right)}\\ \end{array} \]

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 (pow (/ l Om) 2.0))
        (t_2
         (*
          (* (* 2.0 n) U)
          (+ (+ t (* (/ (* l l) Om) -2.0)) (* (* n t_1) (- U* U)))))
        (t_3 (* (* U l) (* n l))))
   (if (<= t_2 0.0)
     (* (sqrt (* n (fma l (* (/ l Om) -2.0) t))) (sqrt (* 2.0 U)))
     (if (<= t_2 4e+306)
       (sqrt t_2)
       (if (<= t_2 INFINITY)
         (*
          (sqrt (- t (fma 2.0 (* l (/ l Om)) (* n (* t_1 (- U U*))))))
          (sqrt (* 2.0 (* n U))))
         (sqrt
          (*
           2.0
           (+ (* t_3 (/ (* n (- U* U)) (* Om Om))) (* t_3 (/ -2.0 Om))))))))))

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 = pow((l / Om), 2.0);
	double t_2 = ((2.0 * n) * U) * ((t + (((l * l) / Om) * -2.0)) + ((n * t_1) * (U_42_ - U)));
	double t_3 = (U * l) * (n * l);
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt((n * fma(l, ((l / Om) * -2.0), t))) * sqrt((2.0 * U));
	} else if (t_2 <= 4e+306) {
		tmp = sqrt(t_2);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt((t - fma(2.0, (l * (l / Om)), (n * (t_1 * (U - U_42_)))))) * sqrt((2.0 * (n * U)));
	} else {
		tmp = sqrt((2.0 * ((t_3 * ((n * (U_42_ - U)) / (Om * Om))) + (t_3 * (-2.0 / Om)))));
	}
	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(l / Om) ^ 2.0
	t_2 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t + Float64(Float64(Float64(l * l) / Om) * -2.0)) + Float64(Float64(n * t_1) * Float64(U_42_ - U))))
	t_3 = Float64(Float64(U * l) * Float64(n * l))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(sqrt(Float64(n * fma(l, Float64(Float64(l / Om) * -2.0), t))) * sqrt(Float64(2.0 * U)));
	elseif (t_2 <= 4e+306)
		tmp = sqrt(t_2);
	elseif (t_2 <= Inf)
		tmp = Float64(sqrt(Float64(t - fma(2.0, Float64(l * Float64(l / Om)), Float64(n * Float64(t_1 * Float64(U - U_42_)))))) * sqrt(Float64(2.0 * Float64(n * U))));
	else
		tmp = sqrt(Float64(2.0 * Float64(Float64(t_3 * Float64(Float64(n * Float64(U_42_ - U)) / Float64(Om * Om))) + Float64(t_3 * Float64(-2.0 / Om)))));
	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[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t + N[(N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] + N[(N[(n * t$95$1), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(U * l), $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(N[Sqrt[N[(n * N[(l * N[(N[(l / Om), $MachinePrecision] * -2.0), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e+306], N[Sqrt[t$95$2], $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[Sqrt[N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + N[(n * N[(t$95$1 * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(2.0 * N[(N[(t$95$3 * N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 * N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $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*)))) < 0.0

   1. Initial program 9.7%

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

   2. Taylor expanded in n around 0 33.8%

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

   3. Simplified 35.8%

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

      [Start] 33.8	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot U\right)\right)} \]
      associate-*r* [=>] 34.3	\[ \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right)}} \]
      *-commutative [=>] 34.3	\[ \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
      cancel-sign-sub-inv [=>] 34.3	\[ \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{\left(t + \left(-2\right) \cdot \frac{{\ell}^{2}}{Om}\right)}\right)\right)} \]
      metadata-eval [=>] 34.3	\[ \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \color{blue}{-2} \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)} \]
      unpow2 [=>] 34.3	\[ \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + -2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right)\right)\right)} \]
      associate-*r/ [<=] 35.8	\[ \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + -2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)}\right)\right)\right)} \]
      *-commutative [<=] 35.8	\[ \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2}\right)\right)\right)} \]
      associate-*l* [=>] 35.8	\[ \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \color{blue}{\ell \cdot \left(\frac{\ell}{Om} \cdot -2\right)}\right)\right)\right)} \]
      associate-*l/ [=>] 35.8	\[ \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \ell \cdot \color{blue}{\frac{\ell \cdot -2}{Om}}\right)\right)\right)} \]

   4. Applied egg-rr 35.5%

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

      [Start] 35.8	\[ \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \ell \cdot \frac{\ell \cdot -2}{Om}\right)\right)\right)} \]
      associate-*r* [=>] 35.9	\[ \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + \ell \cdot \frac{\ell \cdot -2}{Om}\right)\right)}} \]
      sqrt-prod [=>] 35.5	\[ \color{blue}{\sqrt{2 \cdot U} \cdot \sqrt{n \cdot \left(t + \ell \cdot \frac{\ell \cdot -2}{Om}\right)}} \]
      +-commutative [=>] 35.5	\[ \sqrt{2 \cdot U} \cdot \sqrt{n \cdot \color{blue}{\left(\ell \cdot \frac{\ell \cdot -2}{Om} + t\right)}} \]
      fma-def [=>] 35.5	\[ \sqrt{2 \cdot U} \cdot \sqrt{n \cdot \color{blue}{\mathsf{fma}\left(\ell, \frac{\ell \cdot -2}{Om}, t\right)}} \]
      *-commutative [=>] 35.5	\[ \sqrt{2 \cdot U} \cdot \sqrt{n \cdot \mathsf{fma}\left(\ell, \frac{\color{blue}{-2 \cdot \ell}}{Om}, t\right)} \]
      *-un-lft-identity [=>] 35.5	\[ \sqrt{2 \cdot U} \cdot \sqrt{n \cdot \mathsf{fma}\left(\ell, \frac{-2 \cdot \ell}{\color{blue}{1 \cdot Om}}, t\right)} \]
      times-frac [=>] 35.5	\[ \sqrt{2 \cdot U} \cdot \sqrt{n \cdot \mathsf{fma}\left(\ell, \color{blue}{\frac{-2}{1} \cdot \frac{\ell}{Om}}, t\right)} \]
      metadata-eval [=>] 35.5	\[ \sqrt{2 \cdot U} \cdot \sqrt{n \cdot \mathsf{fma}\left(\ell, \color{blue}{-2} \cdot \frac{\ell}{Om}, t\right)} \]

   5. Simplified 35.5%

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

      [Start] 35.5	\[ \sqrt{2 \cdot U} \cdot \sqrt{n \cdot \mathsf{fma}\left(\ell, -2 \cdot \frac{\ell}{Om}, t\right)} \]
      *-commutative [=>] 35.5	\[ \color{blue}{\sqrt{n \cdot \mathsf{fma}\left(\ell, -2 \cdot \frac{\ell}{Om}, t\right)} \cdot \sqrt{2 \cdot U}} \]

   if 0.0 < (*.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*)))) < 4.00000000000000007e306

   1. Initial program 97.2%

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

   if 4.00000000000000007e306 < (*.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 0.3%

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

   2. Simplified 15.6%

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

      [Start] 0.3	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 3.5	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 3.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      sub-neg [=>] 3.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(-\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [<=] 3.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      cancel-sign-sub [<=] 3.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      cancel-sign-sub [=>] 3.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-/l* [=>] 14.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 15.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Applied egg-rr 11.3%

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

      [Start] 15.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)} \]
      associate-*r* [=>] 15.5	\[ \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)}} \]
      sqrt-prod [=>] 21.8	\[ \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)}} \]
      associate-*l* [=>] 21.8	\[ \sqrt{\color{blue}{2 \cdot \left(n \cdot U\right)}} \cdot \sqrt{t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)} \]
      fma-def [=>] 21.8	\[ \sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{t - \color{blue}{\mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)}} \]
      associate-/r/ [=>] 21.8	\[ \sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{t - \mathsf{fma}\left(2, \color{blue}{\frac{\ell}{Om} \cdot \ell}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)} \]
      associate-*l/ [=>] 13.5	\[ \sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{t - \mathsf{fma}\left(2, \color{blue}{\frac{\ell \cdot \ell}{Om}}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)} \]
      *-commutative [=>] 13.5	\[ \sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right) \cdot n}\right)} \]
      associate-*l* [=>] 11.3	\[ \sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U - U*\right) \cdot n\right)}\right)} \]

   4. Simplified 21.8%

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

      [Start] 11.3	\[ \sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U - U*\right) \cdot n\right)\right)} \]
      *-commutative [=>] 11.3	\[ \color{blue}{\sqrt{t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U - U*\right) \cdot n\right)\right)} \cdot \sqrt{2 \cdot \left(n \cdot U\right)}} \]
      associate-/l* [=>] 20.3	\[ \sqrt{t - \mathsf{fma}\left(2, \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U - U*\right) \cdot n\right)\right)} \cdot \sqrt{2 \cdot \left(n \cdot U\right)} \]
      associate-/r/ [=>] 20.3	\[ \sqrt{t - \mathsf{fma}\left(2, \color{blue}{\frac{\ell}{Om} \cdot \ell}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U - U*\right) \cdot n\right)\right)} \cdot \sqrt{2 \cdot \left(n \cdot U\right)} \]
      associate-*r* [=>] 21.8	\[ \sqrt{t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right) \cdot n}\right)} \cdot \sqrt{2 \cdot \left(n \cdot U\right)} \]
      *-commutative [=>] 21.8	\[ \sqrt{t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)} \cdot \sqrt{2 \cdot \left(n \cdot U\right)} \]
      *-commutative [=>] 21.8	\[ \sqrt{t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, n \cdot \color{blue}{\left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right)} \cdot \sqrt{2 \cdot \left(n \cdot U\right)} \]

   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. Simplified 6.1%

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

      [Start] 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	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 0.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      sub-neg [=>] 0.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(-\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [<=] 0.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      cancel-sign-sub [<=] 0.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      cancel-sign-sub [=>] 0.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-/l* [=>] 6.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 6.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in l around inf 7.0%

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

   4. Simplified 23.0%

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

      [Start] 7.0	\[ \sqrt{-2 \cdot \left(\left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(n \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      associate-/l* [=>] 5.8	\[ \sqrt{-2 \cdot \left(\left(\color{blue}{\frac{n}{\frac{{Om}^{2}}{U - U*}}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(n \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      unpow2 [=>] 5.8	\[ \sqrt{-2 \cdot \left(\left(\frac{n}{\frac{\color{blue}{Om \cdot Om}}{U - U*}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(n \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      associate-*r/ [=>] 5.8	\[ \sqrt{-2 \cdot \left(\left(\frac{n}{\frac{Om \cdot Om}{U - U*}} + \color{blue}{\frac{2 \cdot 1}{Om}}\right) \cdot \left(n \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      metadata-eval [=>] 5.8	\[ \sqrt{-2 \cdot \left(\left(\frac{n}{\frac{Om \cdot Om}{U - U*}} + \frac{\color{blue}{2}}{Om}\right) \cdot \left(n \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      unpow2 [=>] 5.8	\[ \sqrt{-2 \cdot \left(\left(\frac{n}{\frac{Om \cdot Om}{U - U*}} + \frac{2}{Om}\right) \cdot \left(n \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot U\right)\right)\right)} \]
      associate-*l* [=>] 23.0	\[ \sqrt{-2 \cdot \left(\left(\frac{n}{\frac{Om \cdot Om}{U - U*}} + \frac{2}{Om}\right) \cdot \left(n \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot U\right)\right)}\right)\right)} \]

   5. Applied egg-rr 35.8%

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

      [Start] 23.0	\[ \sqrt{-2 \cdot \left(\left(\frac{n}{\frac{Om \cdot Om}{U - U*}} + \frac{2}{Om}\right) \cdot \left(n \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right)\right)} \]
      *-commutative [=>] 23.0	\[ \sqrt{-2 \cdot \color{blue}{\left(\left(n \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right) \cdot \left(\frac{n}{\frac{Om \cdot Om}{U - U*}} + \frac{2}{Om}\right)\right)}} \]
      distribute-rgt-in [=>] 23.0	\[ \sqrt{-2 \cdot \color{blue}{\left(\frac{n}{\frac{Om \cdot Om}{U - U*}} \cdot \left(n \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right) + \frac{2}{Om} \cdot \left(n \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right)\right)}} \]
      associate-/r/ [=>] 24.9	\[ \sqrt{-2 \cdot \left(\color{blue}{\left(\frac{n}{Om \cdot Om} \cdot \left(U - U*\right)\right)} \cdot \left(n \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right) + \frac{2}{Om} \cdot \left(n \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right)\right)} \]
      associate-*l/ [=>] 25.1	\[ \sqrt{-2 \cdot \left(\color{blue}{\frac{n \cdot \left(U - U*\right)}{Om \cdot Om}} \cdot \left(n \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right) + \frac{2}{Om} \cdot \left(n \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right)\right)} \]
      associate-*r* [=>] 24.6	\[ \sqrt{-2 \cdot \left(\frac{n \cdot \left(U - U*\right)}{Om \cdot Om} \cdot \color{blue}{\left(\left(n \cdot \ell\right) \cdot \left(\ell \cdot U\right)\right)} + \frac{2}{Om} \cdot \left(n \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right)\right)} \]
      *-commutative [=>] 24.6	\[ \sqrt{-2 \cdot \left(\frac{n \cdot \left(U - U*\right)}{Om \cdot Om} \cdot \color{blue}{\left(\left(\ell \cdot U\right) \cdot \left(n \cdot \ell\right)\right)} + \frac{2}{Om} \cdot \left(n \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right)\right)} \]
      *-commutative [=>] 24.6	\[ \sqrt{-2 \cdot \left(\frac{n \cdot \left(U - U*\right)}{Om \cdot Om} \cdot \left(\color{blue}{\left(U \cdot \ell\right)} \cdot \left(n \cdot \ell\right)\right) + \frac{2}{Om} \cdot \left(n \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right)\right)} \]
      associate-*r* [=>] 35.8	\[ \sqrt{-2 \cdot \left(\frac{n \cdot \left(U - U*\right)}{Om \cdot Om} \cdot \left(\left(U \cdot \ell\right) \cdot \left(n \cdot \ell\right)\right) + \frac{2}{Om} \cdot \color{blue}{\left(\left(n \cdot \ell\right) \cdot \left(\ell \cdot U\right)\right)}\right)} \]
      *-commutative [=>] 35.8	\[ \sqrt{-2 \cdot \left(\frac{n \cdot \left(U - U*\right)}{Om \cdot Om} \cdot \left(\left(U \cdot \ell\right) \cdot \left(n \cdot \ell\right)\right) + \frac{2}{Om} \cdot \color{blue}{\left(\left(\ell \cdot U\right) \cdot \left(n \cdot \ell\right)\right)}\right)} \]
      *-commutative [=>] 35.8	\[ \sqrt{-2 \cdot \left(\frac{n \cdot \left(U - U*\right)}{Om \cdot Om} \cdot \left(\left(U \cdot \ell\right) \cdot \left(n \cdot \ell\right)\right) + \frac{2}{Om} \cdot \left(\color{blue}{\left(U \cdot \ell\right)} \cdot \left(n \cdot \ell\right)\right)\right)} \]

3. Recombined 4 regimes into one program.

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 0:\\ \;\;\;\;\sqrt{n \cdot \mathsf{fma}\left(\ell, \frac{\ell}{Om} \cdot -2, t\right)} \cdot \sqrt{2 \cdot U}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 4 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)} \cdot \sqrt{2 \cdot \left(n \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(\left(U \cdot \ell\right) \cdot \left(n \cdot \ell\right)\right) \cdot \frac{n \cdot \left(U* - U\right)}{Om \cdot Om} + \left(\left(U \cdot \ell\right) \cdot \left(n \cdot \ell\right)\right) \cdot \frac{-2}{Om}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	58.6%
Cost	51148

\[\begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ t_2 := \left(U \cdot \ell\right) \cdot \left(n \cdot \ell\right)\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\sqrt{n \cdot \mathsf{fma}\left(\ell, \frac{\ell}{Om} \cdot -2, t\right)} \cdot \sqrt{2 \cdot U}\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{t_1}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;e^{\left(\log \left(2 \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, -t\right)\right)\right) - \log \left(\frac{-1}{U}\right)\right) \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_2 \cdot \frac{n \cdot \left(U* - U\right)}{Om \cdot Om} + t_2 \cdot \frac{-2}{Om}\right)}\\ \end{array} \]

Alternative 2
Accuracy	58.8%
Cost	44428

\[\begin{array}{l} t_1 := \left(U \cdot \ell\right) \cdot \left(n \cdot \ell\right)\\ t_2 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_2 \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot t} \cdot \sqrt{U}\\ \mathbf{elif}\;t_2 \leq 4 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{t_2}\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;\left|\sqrt{2} \cdot \left(\ell \cdot \left(\sqrt{U \cdot \left(U* - U\right)} \cdot \frac{n}{Om}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_1 \cdot \frac{n \cdot \left(U* - U\right)}{Om \cdot Om} + t_1 \cdot \frac{-2}{Om}\right)}\\ \end{array} \]

Alternative 3
Accuracy	60.0%
Cost	44428

\[\begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ t_2 := \left(U \cdot \ell\right) \cdot \left(n \cdot \ell\right)\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\sqrt{n \cdot \mathsf{fma}\left(\ell, \frac{\ell}{Om} \cdot -2, t\right)} \cdot \sqrt{2 \cdot U}\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{t_1}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\left|\sqrt{2} \cdot \left(\ell \cdot \left(\sqrt{U \cdot \left(U* - U\right)} \cdot \frac{n}{Om}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_2 \cdot \frac{n \cdot \left(U* - U\right)}{Om \cdot Om} + t_2 \cdot \frac{-2}{Om}\right)}\\ \end{array} \]

Alternative 4
Accuracy	56.1%
Cost	38028

\[\begin{array}{l} t_1 := \left(U \cdot \ell\right) \cdot \left(n \cdot \ell\right)\\ t_2 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_2 \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot t} \cdot \sqrt{U}\\ \mathbf{elif}\;t_2 \leq 4 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{t_2}\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{Om}{\left(n \cdot \ell\right) \cdot \sqrt{U \cdot \left(U* - U\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_1 \cdot \frac{n \cdot \left(U* - U\right)}{Om \cdot Om} + t_1 \cdot \frac{-2}{Om}\right)}\\ \end{array} \]

Alternative 5
Accuracy	56.9%
Cost	31112

\[\begin{array}{l} t_1 := \left(U \cdot \ell\right) \cdot \left(n \cdot \ell\right)\\ t_2 := n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\\ t_3 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + t_2 \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_3 \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot t} \cdot \sqrt{U}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) + \left(t_2 \cdot \left(U - U*\right)\right) \cdot \left(-2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_1 \cdot \frac{n \cdot \left(U* - U\right)}{Om \cdot Om} + t_1 \cdot \frac{-2}{Om}\right)}\\ \end{array} \]

Alternative 6
Accuracy	50.6%
Cost	15196

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ \mathbf{if}\;\ell \leq -9.5 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{\left(n \cdot U\right) \cdot \left(n \cdot \frac{U* - U}{Om \cdot Om} + \frac{-2}{Om}\right)} \cdot \left(-\ell\right)\right)\\ \mathbf{elif}\;\ell \leq -4.7 \cdot 10^{-110}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot t_1 - \frac{\ell \cdot \left(n \cdot \ell\right)}{Om} \cdot \frac{U*}{Om}\right) - t\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.3 \cdot 10^{-234}:\\ \;\;\;\;\sqrt{\left|\left(n \cdot U\right) \cdot \left(2 \cdot t\right)\right|}\\ \mathbf{elif}\;\ell \leq 2.55 \cdot 10^{-109}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(t_1 \cdot \frac{U}{Om} - t_1 \cdot \frac{U*}{Om}\right)\right) - t\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.1 \cdot 10^{+168}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{Om \cdot \frac{-0.5}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.8 \cdot 10^{+218}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\left(U* - U\right) \cdot \frac{n}{Om \cdot Om} + \frac{-2}{Om}\right)}\\ \mathbf{elif}\;\ell \leq 3.2 \cdot 10^{+272}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(-2 \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \ell\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{\mathsf{fma}\left(-4, U \cdot \frac{n}{Om}, \frac{2 \cdot \left(n \cdot n\right)}{\frac{Om \cdot Om}{U \cdot \left(U* - U\right)}}\right)}\\ \end{array} \]

Alternative 7
Accuracy	49.8%
Cost	14940

\[\begin{array}{l} t_1 := \frac{U* - U}{Om \cdot Om}\\ t_2 := \ell \cdot \frac{\ell}{Om}\\ \mathbf{if}\;\ell \leq -1.08 \cdot 10^{+114}:\\ \;\;\;\;\sqrt{\left(\ell \cdot \left(2 \cdot U\right)\right) \cdot \left(\left(n \cdot \ell\right) \cdot \mathsf{fma}\left(n, t_1, \frac{-2}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq -5.2 \cdot 10^{-105}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot t_2 - \frac{\ell \cdot \left(n \cdot \ell\right)}{Om} \cdot \frac{U*}{Om}\right) - t\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{-234}:\\ \;\;\;\;\sqrt{\left|\left(n \cdot U\right) \cdot \left(2 \cdot t\right)\right|}\\ \mathbf{elif}\;\ell \leq 8.4 \cdot 10^{-110}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(t_2 \cdot \frac{U}{Om} - t_2 \cdot \frac{U*}{Om}\right)\right) - t\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.95 \cdot 10^{+169}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{Om \cdot \frac{-0.5}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.1 \cdot 10^{+219}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(n \cdot U\right) \cdot \left(n \cdot t_1 + \frac{-2}{Om}\right)}\right)\\ \mathbf{elif}\;\ell \leq 1.95 \cdot 10^{+272}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(-2 \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \ell\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{-2}{Om} - \frac{n \cdot U}{Om \cdot Om}\right)\right)}\right)\\ \end{array} \]

Alternative 8
Accuracy	49.7%
Cost	14940

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ \mathbf{if}\;\ell \leq -1.08 \cdot 10^{+114}:\\ \;\;\;\;\sqrt{\left(\ell \cdot \left(2 \cdot U\right)\right) \cdot \left(\left(n \cdot \ell\right) \cdot \mathsf{fma}\left(n, \frac{U* - U}{Om \cdot Om}, \frac{-2}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq -3.8 \cdot 10^{-108}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot t_1 - \frac{\ell \cdot \left(n \cdot \ell\right)}{Om} \cdot \frac{U*}{Om}\right) - t\right)\right)}\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{-235}:\\ \;\;\;\;\sqrt{\left|\left(n \cdot U\right) \cdot \left(2 \cdot t\right)\right|}\\ \mathbf{elif}\;\ell \leq 7.8 \cdot 10^{-109}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(t_1 \cdot \frac{U}{Om} - t_1 \cdot \frac{U*}{Om}\right)\right) - t\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.15 \cdot 10^{+169}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{Om \cdot \frac{-0.5}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{+215}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{\frac{Om \cdot Om}{U* - U}} + \frac{-2}{Om}\right)}\\ \mathbf{elif}\;\ell \leq 3.7 \cdot 10^{+272}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(-2 \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \ell\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{-2}{Om} - \frac{n \cdot U}{Om \cdot Om}\right)\right)}\right)\\ \end{array} \]

Alternative 9
Accuracy	49.8%
Cost	14940

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ \mathbf{if}\;\ell \leq -1.36 \cdot 10^{+113}:\\ \;\;\;\;\sqrt{\left(\ell \cdot \left(2 \cdot U\right)\right) \cdot \left(\left(n \cdot \ell\right) \cdot \mathsf{fma}\left(n, \frac{U* - U}{Om \cdot Om}, \frac{-2}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq -2.7 \cdot 10^{-110}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot t_1 - \frac{\ell \cdot \left(n \cdot \ell\right)}{Om} \cdot \frac{U*}{Om}\right) - t\right)\right)}\\ \mathbf{elif}\;\ell \leq 8.8 \cdot 10^{-239}:\\ \;\;\;\;\sqrt{\left|\left(n \cdot U\right) \cdot \left(2 \cdot t\right)\right|}\\ \mathbf{elif}\;\ell \leq 1.92 \cdot 10^{-109}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(t_1 \cdot \frac{U}{Om} - t_1 \cdot \frac{U*}{Om}\right)\right) - t\right)\right)}\\ \mathbf{elif}\;\ell \leq 10^{+169}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{Om \cdot \frac{-0.5}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 5.4 \cdot 10^{+219}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\left(U* - U\right) \cdot \frac{n}{Om \cdot Om} + \frac{-2}{Om}\right)}\\ \mathbf{elif}\;\ell \leq 4.5 \cdot 10^{+272}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(-2 \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \ell\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{-2}{Om} - \frac{n \cdot U}{Om \cdot Om}\right)\right)}\right)\\ \end{array} \]

Alternative 10
Accuracy	50.7%
Cost	14940

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ \mathbf{if}\;\ell \leq -1.1 \cdot 10^{+80}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{\left(n \cdot U\right) \cdot \left(n \cdot \frac{U* - U}{Om \cdot Om} + \frac{-2}{Om}\right)} \cdot \left(-\ell\right)\right)\\ \mathbf{elif}\;\ell \leq -6 \cdot 10^{-106}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot t_1 - \frac{\ell \cdot \left(n \cdot \ell\right)}{Om} \cdot \frac{U*}{Om}\right) - t\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.6 \cdot 10^{-237}:\\ \;\;\;\;\sqrt{\left|\left(n \cdot U\right) \cdot \left(2 \cdot t\right)\right|}\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{-110}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(t_1 \cdot \frac{U}{Om} - t_1 \cdot \frac{U*}{Om}\right)\right) - t\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{+169}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{Om \cdot \frac{-0.5}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{+220}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\left(U* - U\right) \cdot \frac{n}{Om \cdot Om} + \frac{-2}{Om}\right)}\\ \mathbf{elif}\;\ell \leq 1.85 \cdot 10^{+272}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(-2 \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \ell\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{-2}{Om} - \frac{n \cdot U}{Om \cdot Om}\right)\right)}\right)\\ \end{array} \]

Alternative 11
Accuracy	49.3%
Cost	14808

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ t_2 := \left(U \cdot \ell\right) \cdot \left(n \cdot \ell\right)\\ \mathbf{if}\;\ell \leq -1.35 \cdot 10^{+112}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_2 \cdot \frac{n \cdot \left(U* - U\right)}{Om \cdot Om} + t_2 \cdot \frac{-2}{Om}\right)}\\ \mathbf{elif}\;\ell \leq -8.2 \cdot 10^{-106}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot t_1 - \frac{\ell \cdot \left(n \cdot \ell\right)}{Om} \cdot \frac{U*}{Om}\right) - t\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.6 \cdot 10^{-234}:\\ \;\;\;\;\sqrt{\left|\left(n \cdot U\right) \cdot \left(2 \cdot t\right)\right|}\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{+91}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(t_1 \cdot \frac{U}{Om} - t_1 \cdot \frac{U*}{Om}\right)\right) - t\right)\right)}\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{+272}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(-2 \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \ell\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.85 \cdot 10^{+307}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{-2}{Om} - \frac{n \cdot U}{Om \cdot Om}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{n}{\frac{\frac{Om}{\ell}}{U \cdot \ell}}}\\ \end{array} \]

Alternative 12
Accuracy	49.7%
Cost	14808

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ \mathbf{if}\;\ell \leq -2.4 \cdot 10^{+109}:\\ \;\;\;\;\sqrt{\left(\ell \cdot \left(2 \cdot U\right)\right) \cdot \left(\left(n \cdot \ell\right) \cdot \mathsf{fma}\left(n, \frac{U* - U}{Om \cdot Om}, \frac{-2}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq -8 \cdot 10^{-106}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot t_1 - \frac{\ell \cdot \left(n \cdot \ell\right)}{Om} \cdot \frac{U*}{Om}\right) - t\right)\right)}\\ \mathbf{elif}\;\ell \leq 10^{-239}:\\ \;\;\;\;\sqrt{\left|\left(n \cdot U\right) \cdot \left(2 \cdot t\right)\right|}\\ \mathbf{elif}\;\ell \leq 2.1 \cdot 10^{+92}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(t_1 \cdot \frac{U}{Om} - t_1 \cdot \frac{U*}{Om}\right)\right) - t\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.65 \cdot 10^{+272}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(-2 \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \ell\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{+307}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{-2}{Om} - \frac{n \cdot U}{Om \cdot Om}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{n}{\frac{\frac{Om}{\ell}}{U \cdot \ell}}}\\ \end{array} \]

Alternative 13
Accuracy	53.2%
Cost	14732

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.15 \cdot 10^{+80}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{\left(n \cdot U\right) \cdot \left(n \cdot \frac{U* - U}{Om \cdot Om} + \frac{-2}{Om}\right)} \cdot \left(-\ell\right)\right)\\ \mathbf{elif}\;\ell \leq -1.5 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - \frac{\ell \cdot \left(n \cdot \ell\right)}{Om} \cdot \frac{U*}{Om}\right) - t\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.15 \cdot 10^{-64}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot \left(-2 \cdot \left(n \cdot U\right)\right) + t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{+168}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{Om \cdot \frac{-0.5}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\left(U* - U\right) \cdot \frac{n}{Om \cdot Om} + \frac{-2}{Om}\right)}\\ \end{array} \]

Alternative 14
Accuracy	48.9%
Cost	13644

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ t_2 := \left(U \cdot \ell\right) \cdot \left(n \cdot \ell\right)\\ \mathbf{if}\;\ell \leq -1.08 \cdot 10^{+114}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_2 \cdot \frac{n \cdot \left(U* - U\right)}{Om \cdot Om} + t_2 \cdot \frac{-2}{Om}\right)}\\ \mathbf{elif}\;\ell \leq -8 \cdot 10^{-104}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot t_1 - \frac{\ell \cdot \left(n \cdot \ell\right)}{Om} \cdot \frac{U*}{Om}\right) - t\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.22 \cdot 10^{-235}:\\ \;\;\;\;\sqrt{\left|\left(n \cdot U\right) \cdot \left(2 \cdot t\right)\right|}\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{+93}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(t_1 \cdot \frac{U}{Om} - t_1 \cdot \frac{U*}{Om}\right)\right) - t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(-2 \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \ell\right)\right)\right)\right)}\\ \end{array} \]

Alternative 15
Accuracy	48.9%
Cost	9296

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ t_2 := \left(U \cdot \ell\right) \cdot \left(n \cdot \ell\right)\\ \mathbf{if}\;\ell \leq -1.4 \cdot 10^{+114}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_2 \cdot \frac{n \cdot \left(U* - U\right)}{Om \cdot Om} + t_2 \cdot \frac{-2}{Om}\right)}\\ \mathbf{elif}\;\ell \leq -8 \cdot 10^{-104}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot t_1 - \frac{\ell \cdot \left(n \cdot \ell\right)}{Om} \cdot \frac{U*}{Om}\right) - t\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.15 \cdot 10^{-234}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.1 \cdot 10^{+93}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(t_1 \cdot \frac{U}{Om} - t_1 \cdot \frac{U*}{Om}\right)\right) - t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(-2 \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \ell\right)\right)\right)\right)}\\ \end{array} \]

Alternative 16
Accuracy	49.9%
Cost	8656

\[\begin{array}{l} t_1 := \sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - \frac{\ell \cdot \left(n \cdot \ell\right)}{Om} \cdot \frac{U*}{Om}\right) - t\right)\right)}\\ t_2 := \sqrt{2 \cdot \left(U \cdot \left(-2 \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \ell\right)\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+122}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -1.32 \cdot 10^{-107}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 3.6 \cdot 10^{-238}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.15 \cdot 10^{+106}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 17
Accuracy	49.3%
Cost	8656

\[\begin{array}{l} t_1 := \sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - \frac{\ell \cdot \left(n \cdot \ell\right)}{Om} \cdot \frac{U*}{Om}\right) - t\right)\right)}\\ t_2 := \left(U \cdot \ell\right) \cdot \left(n \cdot \ell\right)\\ \mathbf{if}\;\ell \leq -2.4 \cdot 10^{+108}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_2 \cdot \frac{n \cdot \left(U* - U\right)}{Om \cdot Om} + t_2 \cdot \frac{-2}{Om}\right)}\\ \mathbf{elif}\;\ell \leq -1.2 \cdot 10^{-109}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.52 \cdot 10^{-235}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;\ell \leq 7.8 \cdot 10^{+106}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(-2 \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \ell\right)\right)\right)\right)}\\ \end{array} \]

Alternative 18
Accuracy	46.9%
Cost	8264

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(U \cdot \left(-2 \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \ell\right)\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -7.5 \cdot 10^{+122}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -4.7 \cdot 10^{-30}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n}{\frac{Om \cdot Om}{U* - U}} + \frac{-2}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.2 \cdot 10^{+93}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{Om \cdot \frac{-0.5}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 19
Accuracy	45.4%
Cost	7756

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(U \cdot \left(-2 \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \ell\right)\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -5.6 \cdot 10^{+38}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{-220}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.2 \cdot 10^{+93}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \ell \cdot \frac{2 \cdot \ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 20
Accuracy	45.4%
Cost	7756

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(U \cdot \left(-2 \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \ell\right)\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -2.65 \cdot 10^{+39}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{-218}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{+91}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{Om \cdot \frac{-0.5}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 21
Accuracy	44.1%
Cost	7497

\[\begin{array}{l} \mathbf{if}\;\ell \leq -3.5 \cdot 10^{+39} \lor \neg \left(\ell \leq 10^{+89}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(-2 \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \ell\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 22
Accuracy	42.6%
Cost	7369

\[\begin{array}{l} \mathbf{if}\;\ell \leq -3 \cdot 10^{+39} \lor \neg \left(\ell \leq 5.1 \cdot 10^{+92}\right):\\ \;\;\;\;\sqrt{-4 \cdot \frac{n}{\frac{\frac{Om}{\ell}}{U \cdot \ell}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 23
Accuracy	36.4%
Cost	6848

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

Alternative 24
Accuracy	37.1%
Cost	6848

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

Reproduce

herbie shell --seed 2023140 
(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*))))))