Toniolo and Linder, Equation (13)

Average Accuracy: 45.2% → 62.4%

Time: 46.9s

Precision: binary64

Cost: 64076

Math

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

↓

\[\begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := t_1 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_2 \leq 0:\\ \;\;\;\;{\left(\sqrt[3]{n \cdot t} \cdot \sqrt[3]{U + U}\right)}^{1.5}\\ \mathbf{elif}\;t_2 \leq 10^{+301}:\\ \;\;\;\;\sqrt{t_2}\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right), t\right)} \cdot \sqrt{t_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{{\left(\sqrt[3]{n}\right)}^{2} \cdot \left(\sqrt[3]{n} \cdot \frac{\mathsf{fma}\left(\frac{n}{Om}, U* - U, -2\right)}{\frac{Om}{U}}\right)}\\ \end{array} \]

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 (* (* 2.0 n) U))
        (t_2
         (*
          t_1
          (+
           (- t (* 2.0 (/ (* l l) Om)))
           (* (* n (pow (/ l Om) 2.0)) (- U* U))))))
   (if (<= t_2 0.0)
     (pow (* (cbrt (* n t)) (cbrt (+ U U))) 1.5)
     (if (<= t_2 1e+301)
       (sqrt t_2)
       (if (<= t_2 INFINITY)
         (*
          (sqrt (fma (/ l Om) (fma l -2.0 (* n (* (/ l Om) (- U* U)))) t))
          (sqrt t_1))
         (*
          (* l (sqrt 2.0))
          (sqrt
           (*
            (pow (cbrt n) 2.0)
            (* (cbrt n) (/ (fma (/ n Om) (- U* U) -2.0) (/ Om U)))))))))))

C

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

↓

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = t_1 * ((t - (2.0 * ((l * l) / Om))) + ((n * pow((l / Om), 2.0)) * (U_42_ - U)));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = pow((cbrt((n * t)) * cbrt((U + U))), 1.5);
	} else if (t_2 <= 1e+301) {
		tmp = sqrt(t_2);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(fma((l / Om), fma(l, -2.0, (n * ((l / Om) * (U_42_ - U)))), t)) * sqrt(t_1);
	} else {
		tmp = (l * sqrt(2.0)) * sqrt((pow(cbrt(n), 2.0) * (cbrt(n) * (fma((n / Om), (U_42_ - U), -2.0) / (Om / U)))));
	}
	return tmp;
}

Julia

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

↓

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U_42_ - U))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(cbrt(Float64(n * t)) * cbrt(Float64(U + U))) ^ 1.5;
	elseif (t_2 <= 1e+301)
		tmp = sqrt(t_2);
	elseif (t_2 <= Inf)
		tmp = Float64(sqrt(fma(Float64(l / Om), fma(l, -2.0, Float64(n * Float64(Float64(l / Om) * Float64(U_42_ - U)))), t)) * sqrt(t_1));
	else
		tmp = Float64(Float64(l * sqrt(2.0)) * sqrt(Float64((cbrt(n) ^ 2.0) * Float64(cbrt(n) * Float64(fma(Float64(n / Om), Float64(U_42_ - U), -2.0) / Float64(Om / U))))));
	end
	return tmp
end

Wolfram

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

↓

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[Power[N[(N[Power[N[(n * t), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(U + U), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], If[LessEqual[t$95$2, 1e+301], N[Sqrt[t$95$2], $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[Sqrt[N[(N[(l / Om), $MachinePrecision] * N[(l * -2.0 + N[(n * N[(N[(l / Om), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision], N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Power[N[Power[n, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[n, 1/3], $MachinePrecision] * N[(N[(N[(n / Om), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision] + -2.0), $MachinePrecision] / N[(Om / U), $MachinePrecision]), $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 10.7%

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

   2. Simplified 36.4%

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

      [Start] 10.7	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 36.0	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      sub-neg [=>] 36.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(-\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      associate-+l- [=>] 36.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [=>] 36.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(-\left(-\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)\right)} \]
      associate-/l* [=>] 38.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(-\left(-\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)} \]
      remove-double-neg [=>] 38.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)\right)\right)} \]
      associate-*l* [=>] 36.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in t around inf 32.2%

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

   4. Simplified 32.6%

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

      [Start] 32.2	\[ \sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)} \]
      *-commutative [=>] 32.2	\[ \sqrt{2 \cdot \color{blue}{\left(\left(t \cdot U\right) \cdot n\right)}} \]
      *-commutative [=>] 32.2	\[ \sqrt{2 \cdot \left(\color{blue}{\left(U \cdot t\right)} \cdot n\right)} \]
      associate-*l* [=>] 32.6	\[ \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(t \cdot n\right)\right)}} \]

   5. Taylor expanded in U around 0 32.2%

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

   6. Simplified 13.9%

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

      [Start] 32.2	\[ \sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)} \]
      *-commutative [=>] 32.2	\[ \sqrt{\color{blue}{\left(n \cdot \left(t \cdot U\right)\right) \cdot 2}} \]
      associate-*r* [=>] 32.6	\[ \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]
      *-commutative [<=] 32.6	\[ \sqrt{\left(\color{blue}{\left(t \cdot n\right)} \cdot U\right) \cdot 2} \]
      associate-*r* [<=] 13.9	\[ \sqrt{\color{blue}{\left(t \cdot \left(n \cdot U\right)\right)} \cdot 2} \]
      associate-*l* [=>] 13.9	\[ \sqrt{\color{blue}{t \cdot \left(\left(n \cdot U\right) \cdot 2\right)}} \]

   7. Applied egg-rr 13.9%

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

      [Start] 13.9	\[ \sqrt{t \cdot \left(\left(n \cdot U\right) \cdot 2\right)} \]
      pow1/2 [=>] 13.9	\[ \color{blue}{{\left(t \cdot \left(\left(n \cdot U\right) \cdot 2\right)\right)}^{0.5}} \]
      add-cube-cbrt [=>] 13.9	\[ {\color{blue}{\left(\left(\sqrt[3]{t \cdot \left(\left(n \cdot U\right) \cdot 2\right)} \cdot \sqrt[3]{t \cdot \left(\left(n \cdot U\right) \cdot 2\right)}\right) \cdot \sqrt[3]{t \cdot \left(\left(n \cdot U\right) \cdot 2\right)}\right)}}^{0.5} \]
      pow3 [=>] 13.9	\[ {\color{blue}{\left({\left(\sqrt[3]{t \cdot \left(\left(n \cdot U\right) \cdot 2\right)}\right)}^{3}\right)}}^{0.5} \]
      metadata-eval [<=] 13.9	\[ {\left({\left(\sqrt[3]{t \cdot \left(\left(n \cdot U\right) \cdot 2\right)}\right)}^{\color{blue}{\left(1 + 2\right)}}\right)}^{0.5} \]
      pow-pow [=>] 13.9	\[ \color{blue}{{\left(\sqrt[3]{t \cdot \left(\left(n \cdot U\right) \cdot 2\right)}\right)}^{\left(\left(1 + 2\right) \cdot 0.5\right)}} \]
      associate-*l* [=>] 13.9	\[ {\left(\sqrt[3]{t \cdot \color{blue}{\left(n \cdot \left(U \cdot 2\right)\right)}}\right)}^{\left(\left(1 + 2\right) \cdot 0.5\right)} \]
      metadata-eval [=>] 13.9	\[ {\left(\sqrt[3]{t \cdot \left(n \cdot \left(U \cdot 2\right)\right)}\right)}^{\left(\color{blue}{3} \cdot 0.5\right)} \]
      metadata-eval [=>] 13.9	\[ {\left(\sqrt[3]{t \cdot \left(n \cdot \left(U \cdot 2\right)\right)}\right)}^{\color{blue}{1.5}} \]

   8. Applied egg-rr 54.1%

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

      [Start] 13.9	\[ {\left(\sqrt[3]{t \cdot \left(n \cdot \left(U \cdot 2\right)\right)}\right)}^{1.5} \]
      associate-*r* [=>] 32.4	\[ {\left(\sqrt[3]{\color{blue}{\left(t \cdot n\right) \cdot \left(U \cdot 2\right)}}\right)}^{1.5} \]
      cbrt-prod [=>] 54.1	\[ {\color{blue}{\left(\sqrt[3]{t \cdot n} \cdot \sqrt[3]{U \cdot 2}\right)}}^{1.5} \]
      add-log-exp [=>] 10.6	\[ {\left(\sqrt[3]{t \cdot n} \cdot \sqrt[3]{\color{blue}{\log \left(e^{U \cdot 2}\right)}}\right)}^{1.5} \]
      exp-lft-sqr [=>] 10.6	\[ {\left(\sqrt[3]{t \cdot n} \cdot \sqrt[3]{\log \color{blue}{\left(e^{U} \cdot e^{U}\right)}}\right)}^{1.5} \]
      log-prod [=>] 10.6	\[ {\left(\sqrt[3]{t \cdot n} \cdot \sqrt[3]{\color{blue}{\log \left(e^{U}\right) + \log \left(e^{U}\right)}}\right)}^{1.5} \]
      add-log-exp [<=] 17.5	\[ {\left(\sqrt[3]{t \cdot n} \cdot \sqrt[3]{\color{blue}{U} + \log \left(e^{U}\right)}\right)}^{1.5} \]
      add-log-exp [<=] 54.1	\[ {\left(\sqrt[3]{t \cdot n} \cdot \sqrt[3]{U + \color{blue}{U}}\right)}^{1.5} \]

   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*)))) < 1.00000000000000005e301

   1. Initial program 97.3%

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

   if 1.00000000000000005e301 < (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))) < +inf.0

   1. Initial program 1.7%

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

   2. Simplified 12.1%

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

      [Start] 1.7	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 4.3	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      sub-neg [=>] 4.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]
      associate--l+ [=>] 4.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      *-commutative [=>] 4.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      distribute-rgt-neg-in [=>] 4.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l/ [<=] 13.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 13.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [<=] 13.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [=>] 13.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 11.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]
      unpow2 [=>] 11.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]
      associate-*l* [=>] 12.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Applied egg-rr 22.0%

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

      [Start] 12.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)} \]
      associate-*r* [=>] 16.6	\[ \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)}} \]
      sqrt-prod [=>] 20.2	\[ \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)}} \]
      associate-*l* [=>] 20.2	\[ \sqrt{\color{blue}{2 \cdot \left(n \cdot U\right)}} \cdot \sqrt{t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)} \]
      +-commutative [=>] 20.2	\[ \sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{\color{blue}{\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) + t}} \]
      fma-def [=>] 20.2	\[ \sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right), t\right)}} \]
      *-commutative [=>] 20.2	\[ \sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, \color{blue}{\left(n \cdot \left(U* - U\right)\right) \cdot \frac{\ell}{Om}}\right), t\right)} \]
      associate-*l* [=>] 22.0	\[ \sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, \color{blue}{n \cdot \left(\left(U* - U\right) \cdot \frac{\ell}{Om}\right)}\right), t\right)} \]

   4. Simplified 22.0%

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

      [Start] 22.0	\[ \sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, n \cdot \left(\left(U* - U\right) \cdot \frac{\ell}{Om}\right)\right), t\right)} \]
      unpow1/2 [<=] 22.0	\[ \color{blue}{{\left(2 \cdot \left(n \cdot U\right)\right)}^{0.5}} \cdot \sqrt{\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, n \cdot \left(\left(U* - U\right) \cdot \frac{\ell}{Om}\right)\right), t\right)} \]
      *-commutative [=>] 22.0	\[ \color{blue}{\sqrt{\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, n \cdot \left(\left(U* - U\right) \cdot \frac{\ell}{Om}\right)\right), t\right)} \cdot {\left(2 \cdot \left(n \cdot U\right)\right)}^{0.5}} \]
      unpow1/2 [=>] 22.0	\[ \sqrt{\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, n \cdot \left(\left(U* - U\right) \cdot \frac{\ell}{Om}\right)\right), t\right)} \cdot \color{blue}{\sqrt{2 \cdot \left(n \cdot U\right)}} \]
      associate-*r* [=>] 22.0	\[ \sqrt{\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, n \cdot \left(\left(U* - U\right) \cdot \frac{\ell}{Om}\right)\right), t\right)} \cdot \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot U}} \]

   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 12.6%

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

      [Start] 0.0	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 0.0	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      sub-neg [=>] 0.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]
      associate--l+ [=>] 0.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      *-commutative [=>] 0.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      distribute-rgt-neg-in [=>] 0.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l/ [<=] 7.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 7.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [<=] 7.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [=>] 7.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 8.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]
      unpow2 [=>] 8.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]
      associate-*l* [=>] 12.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in t around inf 32.0%

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

   4. Simplified 33.8%

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

      [Start] 32.0	\[ \sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right)\right) + 2 \cdot \frac{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(n \cdot \left(\ell \cdot U\right)\right)}{Om}} \]
      distribute-lft-out [=>] 32.0	\[ \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(n \cdot \left(\ell \cdot U\right)\right)}{Om}\right)}} \]
      *-commutative [<=] 32.0	\[ \sqrt{2 \cdot \left(n \cdot \color{blue}{\left(U \cdot t\right)} + \frac{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(n \cdot \left(\ell \cdot U\right)\right)}{Om}\right)} \]
      associate-/l* [=>] 28.9	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \color{blue}{\frac{\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}}\right)} \]
      +-commutative [=>] 28.9	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\color{blue}{-2 \cdot \ell + \frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om}}}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}\right)} \]
      *-commutative [=>] 28.9	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\color{blue}{\ell \cdot -2} + \frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om}}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}\right)} \]
      associate-*r* [=>] 33.8	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{\color{blue}{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}}{Om}}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}\right)} \]

   5. Applied egg-rr 11.9%

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

      [Start] 33.8	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}{Om}}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}\right)} \]
      expm1-log1p-u [=>] 20.9	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}{Om}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{Om}{n \cdot \left(\ell \cdot U\right)}\right)\right)}}\right)} \]
      expm1-udef [=>] 11.9	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}{Om}}{\color{blue}{e^{\mathsf{log1p}\left(\frac{Om}{n \cdot \left(\ell \cdot U\right)}\right)} - 1}}\right)} \]
      *-commutative [=>] 11.9	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}{Om}}{e^{\mathsf{log1p}\left(\frac{Om}{n \cdot \color{blue}{\left(U \cdot \ell\right)}}\right)} - 1}\right)} \]

   6. Simplified 14.3%

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

      [Start] 11.9	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}{Om}}{e^{\mathsf{log1p}\left(\frac{Om}{n \cdot \left(U \cdot \ell\right)}\right)} - 1}\right)} \]
      expm1-def [=>] 20.9	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}{Om}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{Om}{n \cdot \left(U \cdot \ell\right)}\right)\right)}}\right)} \]
      expm1-log1p [=>] 33.8	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}{Om}}{\color{blue}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}}\right)} \]
      associate-*r* [=>] 14.3	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}{Om}}{\frac{Om}{\color{blue}{\left(n \cdot U\right) \cdot \ell}}}\right)} \]

   7. Taylor expanded in l around inf 27.8%

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

   8. Simplified 31.6%

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

      [Start] 27.8	\[ \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n \cdot \left(\left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right) \cdot U\right)}{Om}} \]
      associate-/l* [=>] 35.4	\[ \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\color{blue}{\frac{n}{\frac{Om}{\left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right) \cdot U}}}} \]
      *-commutative [=>] 35.4	\[ \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{\color{blue}{U \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right)}}}} \]
      sub-neg [=>] 35.4	\[ \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \color{blue}{\left(\frac{n \cdot \left(U* - U\right)}{Om} + \left(-2\right)\right)}}}} \]
      associate-/l* [=>] 31.6	\[ \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(\color{blue}{\frac{n}{\frac{Om}{U* - U}}} + \left(-2\right)\right)}}} \]
      metadata-eval [=>] 31.6	\[ \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(\frac{n}{\frac{Om}{U* - U}} + \color{blue}{-2}\right)}}} \]

   9. Applied egg-rr 35.0%

      \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\color{blue}{{\left(\sqrt[3]{n}\right)}^{2} \cdot \left(\sqrt[3]{n} \cdot \frac{\mathsf{fma}\left(\frac{n}{Om}, U* - U, -2\right)}{\frac{Om}{U}}\right)}} \]
      Proof

      [Start] 31.6	\[ \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(\frac{n}{\frac{Om}{U* - U}} + -2\right)}}} \]
      div-inv [=>] 31.5	\[ \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\color{blue}{n \cdot \frac{1}{\frac{Om}{U \cdot \left(\frac{n}{\frac{Om}{U* - U}} + -2\right)}}}} \]
      add-cube-cbrt [=>] 31.3	\[ \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\color{blue}{\left(\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \sqrt[3]{n}\right)} \cdot \frac{1}{\frac{Om}{U \cdot \left(\frac{n}{\frac{Om}{U* - U}} + -2\right)}}} \]
      clear-num [<=] 31.4	\[ \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \sqrt[3]{n}\right) \cdot \color{blue}{\frac{U \cdot \left(\frac{n}{\frac{Om}{U* - U}} + -2\right)}{Om}}} \]
      associate-*l* [=>] 31.4	\[ \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\color{blue}{\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \left(\sqrt[3]{n} \cdot \frac{U \cdot \left(\frac{n}{\frac{Om}{U* - U}} + -2\right)}{Om}\right)}} \]
      pow2 [=>] 31.4	\[ \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\color{blue}{{\left(\sqrt[3]{n}\right)}^{2}} \cdot \left(\sqrt[3]{n} \cdot \frac{U \cdot \left(\frac{n}{\frac{Om}{U* - U}} + -2\right)}{Om}\right)} \]
      *-commutative [=>] 31.4	\[ \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{{\left(\sqrt[3]{n}\right)}^{2} \cdot \left(\sqrt[3]{n} \cdot \frac{\color{blue}{\left(\frac{n}{\frac{Om}{U* - U}} + -2\right) \cdot U}}{Om}\right)} \]
      associate-/l* [=>] 30.9	\[ \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{{\left(\sqrt[3]{n}\right)}^{2} \cdot \left(\sqrt[3]{n} \cdot \color{blue}{\frac{\frac{n}{\frac{Om}{U* - U}} + -2}{\frac{Om}{U}}}\right)} \]
      associate-/r/ [=>] 35.0	\[ \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{{\left(\sqrt[3]{n}\right)}^{2} \cdot \left(\sqrt[3]{n} \cdot \frac{\color{blue}{\frac{n}{Om} \cdot \left(U* - U\right)} + -2}{\frac{Om}{U}}\right)} \]
      fma-def [=>] 35.0	\[ \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{{\left(\sqrt[3]{n}\right)}^{2} \cdot \left(\sqrt[3]{n} \cdot \frac{\color{blue}{\mathsf{fma}\left(\frac{n}{Om}, U* - U, -2\right)}}{\frac{Om}{U}}\right)} \]

3. Recombined 4 regimes into one program.

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 0:\\ \;\;\;\;{\left(\sqrt[3]{n \cdot t} \cdot \sqrt[3]{U + U}\right)}^{1.5}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 10^{+301}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right), t\right)} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{{\left(\sqrt[3]{n}\right)}^{2} \cdot \left(\sqrt[3]{n} \cdot \frac{\mathsf{fma}\left(\frac{n}{Om}, U* - U, -2\right)}{\frac{Om}{U}}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	61.9%
Cost	51340

\[\begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := t_1 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_2 \leq 0:\\ \;\;\;\;{\left(\sqrt[3]{n \cdot t} \cdot \sqrt[3]{U + U}\right)}^{1.5}\\ \mathbf{elif}\;t_2 \leq 10^{+301}:\\ \;\;\;\;\sqrt{t_2}\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right), t\right)} \cdot \sqrt{t_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(-2 + \frac{n}{\frac{Om}{U*}}\right)}}}\\ \end{array} \]

Alternative 2
Accuracy	60.2%
Cost	44108

\[\begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om}}{\frac{1}{\frac{\ell \cdot \left(n \cdot U\right)}{Om}}}\right)}\\ \mathbf{elif}\;t_1 \leq 10^{+301}:\\ \;\;\;\;\sqrt{t_1}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;{\left(\sqrt[3]{n \cdot \left(U + U\right)} \cdot \sqrt[3]{t}\right)}^{1.5}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(-2 + \frac{n}{\frac{Om}{U*}}\right)}}}\\ \end{array} \]

Alternative 3
Accuracy	63.4%
Cost	44108

\[\begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;{\left(\sqrt[3]{n \cdot t} \cdot \sqrt[3]{U + U}\right)}^{1.5}\\ \mathbf{elif}\;t_1 \leq 10^{+301}:\\ \;\;\;\;\sqrt{t_1}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;{\left(\sqrt[3]{n \cdot \left(U + U\right)} \cdot \sqrt[3]{t}\right)}^{1.5}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(-2 + \frac{n}{\frac{Om}{U*}}\right)}}}\\ \end{array} \]

Alternative 4
Accuracy	54.3%
Cost	13777

\[\begin{array}{l} t_1 := n \cdot \left(U \cdot t\right)\\ t_2 := \ell \cdot -2 + \frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om}\\ \mathbf{if}\;U \leq -7.8 \cdot 10^{+23}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) + \left(n \cdot U\right) \cdot \frac{\ell \cdot U*}{\frac{Om}{\ell} \cdot \frac{Om}{n}}\right)}\\ \mathbf{elif}\;U \leq 4 \cdot 10^{-273}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_1 + \frac{t_2}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)}\\ \mathbf{elif}\;U \leq 4.2 \cdot 10^{-212} \lor \neg \left(U \leq 1.45 \cdot 10^{+126}\right):\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot t\right)} \cdot \sqrt{U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_1 + \frac{t_2}{\frac{-Om}{U \cdot \left(n \cdot \left(-\ell\right)\right)}}\right)}\\ \end{array} \]

Alternative 5
Accuracy	54.3%
Cost	13776

\[\begin{array}{l} t_1 := n \cdot \left(U \cdot t\right)\\ t_2 := \ell \cdot -2 + \frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om}\\ \mathbf{if}\;U \leq -4 \cdot 10^{+23}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) + \left(n \cdot U\right) \cdot \frac{\ell \cdot U*}{\frac{Om}{\ell} \cdot \frac{Om}{n}}\right)}\\ \mathbf{elif}\;U \leq 6.5 \cdot 10^{-273}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_1 + \frac{t_2}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)}\\ \mathbf{elif}\;U \leq 3.5 \cdot 10^{-212}:\\ \;\;\;\;\sqrt{U + U} \cdot \sqrt{n \cdot t}\\ \mathbf{elif}\;U \leq 4.3 \cdot 10^{+122}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_1 + \frac{t_2}{\frac{-Om}{U \cdot \left(n \cdot \left(-\ell\right)\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot t\right)} \cdot \sqrt{U}\\ \end{array} \]

Alternative 6
Accuracy	49.0%
Cost	9316

\[\begin{array}{l} t_1 := \ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\\ t_2 := \frac{Om}{\ell \cdot \left(n \cdot U\right)}\\ t_3 := \sqrt{\left(2 \cdot n\right) \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(n \cdot \left(U \cdot \left(U* - U\right)\right)\right)\right)}\\ t_4 := n \cdot \left(U \cdot t\right)\\ t_5 := \sqrt{2 \cdot \left(t_4 + \frac{t_1}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)}\\ t_6 := \sqrt{2 \cdot \left(t_4 + \frac{\ell \cdot -2 - \frac{n}{\frac{\frac{Om}{U}}{\ell}}}{t_2}\right)}\\ \mathbf{if}\;Om \leq -5 \cdot 10^{+106}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;Om \leq -2.15 \cdot 10^{-57}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\frac{n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om} - \frac{2}{\frac{Om}{\ell \cdot \ell}}\right)\right)\right)}\\ \mathbf{elif}\;Om \leq -1.35 \cdot 10^{-179}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_4 + \frac{t_1}{t_2}\right)}\\ \mathbf{elif}\;Om \leq -1.55 \cdot 10^{-260}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;Om \leq -8.5 \cdot 10^{-270}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;Om \leq -9 \cdot 10^{-287}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;Om \leq -8.5 \cdot 10^{-306}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;Om \leq 1.15 \cdot 10^{-278}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{n}{\frac{\frac{\frac{Om}{U}}{2 + \left(U - U*\right) \cdot \frac{n}{Om}}}{\ell \cdot \ell}}}\\ \mathbf{elif}\;Om \leq 5 \cdot 10^{+138}:\\ \;\;\;\;t_5\\ \mathbf{else}:\\ \;\;\;\;t_6\\ \end{array} \]

Alternative 7
Accuracy	51.9%
Cost	8780

\[\begin{array}{l} t_1 := n \cdot \left(U \cdot t\right)\\ t_2 := \ell \cdot -2 + \frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om}\\ t_3 := \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) + \left(n \cdot U\right) \cdot \frac{\ell \cdot U*}{\frac{Om}{\ell} \cdot \frac{Om}{n}}\right)}\\ \mathbf{if}\;U \leq -2.5 \cdot 10^{+23}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;U \leq 2.9 \cdot 10^{-291}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_1 + \frac{t_2}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)}\\ \mathbf{elif}\;U \leq 3.6 \cdot 10^{-213}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_1 + \frac{t_2}{\frac{-Om}{U \cdot \left(n \cdot \left(-\ell\right)\right)}}\right)}\\ \end{array} \]

Alternative 8
Accuracy	48.1%
Cost	8525

\[\begin{array}{l} \mathbf{if}\;U \leq -2.35 \cdot 10^{+75}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\frac{n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om} - \frac{2}{\frac{Om}{\ell \cdot \ell}}\right)\right)\right)}\\ \mathbf{elif}\;U \leq 1.9 \cdot 10^{-247} \lor \neg \left(U \leq 3.75 \cdot 10^{-181}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \frac{\frac{\left(n \cdot U\right) \cdot \left(U + U*\right)}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}}\\ \end{array} \]

Alternative 9
Accuracy	47.8%
Cost	8524

\[\begin{array}{l} t_1 := n \cdot \left(U \cdot t\right)\\ t_2 := \ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\\ \mathbf{if}\;U \leq -2 \cdot 10^{+75}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\frac{n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om} - \frac{2}{\frac{Om}{\ell \cdot \ell}}\right)\right)\right)}\\ \mathbf{elif}\;U \leq 2.8 \cdot 10^{-250}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_1 + \frac{t_2}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)}\\ \mathbf{elif}\;U \leq 3.75 \cdot 10^{-181}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \frac{\frac{\left(n \cdot U\right) \cdot \left(U + U*\right)}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_1 + \frac{t_2}{\frac{Om}{\ell \cdot \left(n \cdot U\right)}}\right)}\\ \end{array} \]

Alternative 10
Accuracy	53.1%
Cost	8516

\[\begin{array}{l} t_1 := n \cdot \left(U \cdot t\right)\\ \mathbf{if}\;Om \leq -4 \cdot 10^{+106}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_1 + \frac{\ell \cdot -2 - \frac{n}{\frac{\frac{Om}{U}}{\ell}}}{\frac{Om}{\ell \cdot \left(n \cdot U\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_1 + \frac{\ell \cdot -2 + \frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om}}{\frac{-Om}{U \cdot \left(n \cdot \left(-\ell\right)\right)}}\right)}\\ \end{array} \]

Alternative 11
Accuracy	48.5%
Cost	8392

\[\begin{array}{l} \mathbf{if}\;\ell \leq -5.6 \cdot 10^{+90}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -1.15 \cdot 10^{-271}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\frac{n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om} - \frac{2}{\frac{Om}{\ell \cdot \ell}}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \frac{n \cdot \left(\left(\ell \cdot -2 + \frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om}\right) \cdot \left(U \cdot \ell\right)\right)}{Om}}\\ \end{array} \]

Alternative 12
Accuracy	50.9%
Cost	8388

\[\begin{array}{l} t_1 := n \cdot \left(U \cdot t\right)\\ \mathbf{if}\;Om \leq -3 \cdot 10^{+53}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_1 + \frac{\ell \cdot -2 - \frac{n}{\frac{\frac{Om}{U}}{\ell}}}{\frac{Om}{\ell \cdot \left(n \cdot U\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_1 + \frac{\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}}{\frac{-Om}{U \cdot \left(n \cdot \left(-\ell\right)\right)}}\right)}\\ \end{array} \]

Alternative 13
Accuracy	50.5%
Cost	8388

\[\begin{array}{l} t_1 := n \cdot \left(U \cdot t\right)\\ \mathbf{if}\;U \leq 5 \cdot 10^{-182}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_1 + \frac{\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}}{\frac{-Om}{U \cdot \left(n \cdot \left(-\ell\right)\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_1 + \frac{\ell \cdot -2 + \frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om}}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)}\\ \end{array} \]

Alternative 14
Accuracy	50.1%
Cost	8388

\[\begin{array}{l} t_1 := n \cdot \left(U \cdot t\right)\\ \mathbf{if}\;U \leq 5 \cdot 10^{-208}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_1 + \frac{\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}}{\frac{-Om}{U \cdot \left(n \cdot \left(-\ell\right)\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_1 + \frac{\ell \cdot -2 + \frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om}}{\frac{Om}{\ell \cdot \left(n \cdot U\right)}}\right)}\\ \end{array} \]

Alternative 15
Accuracy	49.3%
Cost	8136

\[\begin{array}{l} \mathbf{if}\;\ell \leq -7 \cdot 10^{+92}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{+150}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \frac{n \cdot \left(\left(\ell \cdot -2 + \frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om}\right) \cdot \left(U \cdot \ell\right)\right)}{Om}}\\ \end{array} \]

Alternative 16
Accuracy	43.1%
Cost	7752

\[\begin{array}{l} \mathbf{if}\;U* \leq 2.9 \cdot 10^{+108}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;U* \leq 5.8 \cdot 10^{+237}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\frac{n \cdot n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot \left(U \cdot U*\right)\right)}{Om}\right)}\\ \end{array} \]

Alternative 17
Accuracy	43.5%
Cost	7752

\[\begin{array}{l} \mathbf{if}\;U* \leq 3.1 \cdot 10^{+108}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;U* \leq 9.2 \cdot 10^{+244}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\frac{n \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)}{Om} \cdot \frac{U \cdot U*}{Om}\right)}\\ \end{array} \]

Alternative 18
Accuracy	43.2%
Cost	7752

\[\begin{array}{l} \mathbf{if}\;U* \leq 3.5 \cdot 10^{+108}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;U* \leq 5.8 \cdot 10^{+237}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\frac{n}{Om} \cdot \frac{\left(\ell \cdot \ell\right) \cdot \left(U \cdot U*\right)}{Om}\right)}\\ \end{array} \]

Alternative 19
Accuracy	43.3%
Cost	7752

\[\begin{array}{l} \mathbf{if}\;U* \leq 1.1 \cdot 10^{+107}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;U* \leq 5.8 \cdot 10^{+237}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{Om} \cdot \frac{\left(\left(n \cdot \ell\right) \cdot \left(n \cdot \ell\right)\right) \cdot \left(U \cdot U*\right)}{Om}}\\ \end{array} \]

Alternative 20
Accuracy	44.7%
Cost	7625

\[\begin{array}{l} \mathbf{if}\;U* \leq 4.9 \cdot 10^{+107} \lor \neg \left(U* \leq 7.2 \cdot 10^{+247}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 21
Accuracy	38.4%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;n \leq -4.6 \cdot 10^{+174} \lor \neg \left(n \leq 5 \cdot 10^{+15}\right):\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]

Alternative 22
Accuracy	38.4%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;n \leq -3.8 \cdot 10^{+124}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{elif}\;n \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 23
Accuracy	36.9%
Cost	6848

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

Reproduce

herbie shell --seed 2023146 
(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*))))))