Toniolo and Linder, Equation (13)

Average Accuracy: 47.0% → 54.9%

Time: 1.3min

Precision: binary64

Cost: 27468

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.8 \cdot 10^{+198}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U* - U}{Om}, \frac{-2}{Om}\right)\right)} \cdot \left(\sqrt{2} \cdot \left(-\ell\right)\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-148}:\\ \;\;\;\;\sqrt{n \cdot \left(\left(\mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right) + n \cdot \left(\left(U* - U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right) \cdot \left(U \cdot 2\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{+79}:\\ \;\;\;\;{\left({\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, n \cdot \left(\left(U* - U\right) \cdot \frac{\ell}{Om}\right)\right), t\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)}^{0.25}\right)}^{2}\\ \mathbf{elif}\;\ell \leq 6.4 \cdot 10^{+191} \lor \neg \left(\ell \leq 7 \cdot 10^{+223}\right):\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(-2 + \frac{n}{\frac{Om}{U*}}\right)}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \ell \cdot \left(-2 \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l -1.8e+198)
   (*
    (sqrt (* n (* U (fma (/ n Om) (/ (- U* U) Om) (/ -2.0 Om)))))
    (* (sqrt 2.0) (- l)))
   (if (<= l -1e-148)
     (sqrt
      (*
       n
       (*
        (+ (fma (* l (/ l Om)) -2.0 t) (* n (* (- U* U) (pow (/ l Om) 2.0))))
        (* U 2.0))))
     (if (<= l 2.3e+79)
       (pow
        (pow
         (*
          (fma (/ l Om) (fma l -2.0 (* n (* (- U* U) (/ l Om)))) t)
          (* 2.0 (* n U)))
         0.25)
        2.0)
       (if (or (<= l 6.4e+191) (not (<= l 7e+223)))
         (*
          (sqrt 2.0)
          (* l (sqrt (/ n (/ Om (* U (+ -2.0 (/ n (/ Om U*)))))))))
         (sqrt (* 2.0 (* n (* U (+ t (* l (* -2.0 (/ l 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 tmp;
	if (l <= -1.8e+198) {
		tmp = sqrt((n * (U * fma((n / Om), ((U_42_ - U) / Om), (-2.0 / Om))))) * (sqrt(2.0) * -l);
	} else if (l <= -1e-148) {
		tmp = sqrt((n * ((fma((l * (l / Om)), -2.0, t) + (n * ((U_42_ - U) * pow((l / Om), 2.0)))) * (U * 2.0))));
	} else if (l <= 2.3e+79) {
		tmp = pow(pow((fma((l / Om), fma(l, -2.0, (n * ((U_42_ - U) * (l / Om)))), t) * (2.0 * (n * U))), 0.25), 2.0);
	} else if ((l <= 6.4e+191) || !(l <= 7e+223)) {
		tmp = sqrt(2.0) * (l * sqrt((n / (Om / (U * (-2.0 + (n / (Om / U_42_))))))));
	} else {
		tmp = sqrt((2.0 * (n * (U * (t + (l * (-2.0 * (l / 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_)
	tmp = 0.0
	if (l <= -1.8e+198)
		tmp = Float64(sqrt(Float64(n * Float64(U * fma(Float64(n / Om), Float64(Float64(U_42_ - U) / Om), Float64(-2.0 / Om))))) * Float64(sqrt(2.0) * Float64(-l)));
	elseif (l <= -1e-148)
		tmp = sqrt(Float64(n * Float64(Float64(fma(Float64(l * Float64(l / Om)), -2.0, t) + Float64(n * Float64(Float64(U_42_ - U) * (Float64(l / Om) ^ 2.0)))) * Float64(U * 2.0))));
	elseif (l <= 2.3e+79)
		tmp = (Float64(fma(Float64(l / Om), fma(l, -2.0, Float64(n * Float64(Float64(U_42_ - U) * Float64(l / Om)))), t) * Float64(2.0 * Float64(n * U))) ^ 0.25) ^ 2.0;
	elseif ((l <= 6.4e+191) || !(l <= 7e+223))
		tmp = Float64(sqrt(2.0) * Float64(l * sqrt(Float64(n / Float64(Om / Float64(U * Float64(-2.0 + Float64(n / Float64(Om / U_42_)))))))));
	else
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * Float64(t + Float64(l * Float64(-2.0 * Float64(l / 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$42$_] := If[LessEqual[l, -1.8e+198], N[(N[Sqrt[N[(n * N[(U * N[(N[(n / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * (-l)), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1e-148], N[Sqrt[N[(n * N[(N[(N[(N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] * -2.0 + t), $MachinePrecision] + N[(n * N[(N[(U$42$ - U), $MachinePrecision] * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 2.3e+79], N[Power[N[Power[N[(N[(N[(l / Om), $MachinePrecision] * N[(l * -2.0 + N[(n * N[(N[(U$42$ - U), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision], 2.0], $MachinePrecision], If[Or[LessEqual[l, 6.4e+191], N[Not[LessEqual[l, 7e+223]], $MachinePrecision]], N[(N[Sqrt[2.0], $MachinePrecision] * N[(l * N[Sqrt[N[(n / N[(Om / N[(U * N[(-2.0 + N[(n / N[(Om / U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(2.0 * N[(n * N[(U * N[(t + N[(l * N[(-2.0 * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if l < -1.8000000000000001e198

   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 20.0%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]
      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/ [<=] 20.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 20.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [<=] 20.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [=>] 20.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 20.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]
      unpow2 [=>] 20.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]
      associate-*l* [=>] 20.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in l around inf 0.0%

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

   4. Simplified 0.0%

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

      [Start] 0.0	\[ \sqrt{2 \cdot \left(n \cdot \left({\ell}^{2} \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)\right)\right)} \]
      associate-*r* [=>] 0.0	\[ \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot {\ell}^{2}\right) \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)\right)}} \]
      unpow2 [=>] 0.0	\[ \sqrt{2 \cdot \left(\left(n \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)\right)} \]
      sub-neg [=>] 0.0	\[ \sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U \cdot \color{blue}{\left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} + \left(-2 \cdot \frac{1}{Om}\right)\right)}\right)\right)} \]
      unpow2 [=>] 0.0	\[ \sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{\color{blue}{Om \cdot Om}} + \left(-2 \cdot \frac{1}{Om}\right)\right)\right)\right)} \]
      times-frac [=>] 0.0	\[ \sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U \cdot \left(\color{blue}{\frac{n}{Om} \cdot \frac{U* - U}{Om}} + \left(-2 \cdot \frac{1}{Om}\right)\right)\right)\right)} \]
      associate-*r/ [=>] 0.0	\[ \sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \left(-\color{blue}{\frac{2 \cdot 1}{Om}}\right)\right)\right)\right)} \]
      metadata-eval [=>] 0.0	\[ \sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \left(-\frac{\color{blue}{2}}{Om}\right)\right)\right)\right)} \]
      distribute-neg-frac [=>] 0.0	\[ \sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \color{blue}{\frac{-2}{Om}}\right)\right)\right)} \]
      metadata-eval [=>] 0.0	\[ \sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{\color{blue}{-2}}{Om}\right)\right)\right)} \]

   5. Taylor expanded in l around -inf 47.5%

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

   6. Simplified 54.2%

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

      [Start] 47.5	\[ -1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}\right) \]
      mul-1-neg [=>] 47.5	\[ \color{blue}{-\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}} \]
      *-commutative [=>] 47.5	\[ -\color{blue}{\sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)} \cdot \left(\sqrt{2} \cdot \ell\right)} \]
      distribute-rgt-neg-in [=>] 47.5	\[ \color{blue}{\sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)} \cdot \left(-\sqrt{2} \cdot \ell\right)} \]

   if -1.8000000000000001e198 < l < -9.99999999999999936e-149

   1. Initial program 48.1%

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

   2. Simplified 50.3%

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

      [Start] 48.1	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      *-commutative [=>] 48.1	\[ \sqrt{\left(\color{blue}{\left(n \cdot 2\right)} \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 48.1	\[ \sqrt{\color{blue}{\left(n \cdot \left(2 \cdot U\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 47.3	\[ \sqrt{\color{blue}{n \cdot \left(\left(2 \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate-*l* [=>] 45.8	\[ \sqrt{n \cdot \left(\left(2 \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)} \]
      sub-neg [=>] 45.8	\[ \sqrt{n \cdot \left(\left(2 \cdot U\right) \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)} \]
      +-commutative [=>] 45.8	\[ \sqrt{n \cdot \left(\left(2 \cdot U\right) \cdot \left(\color{blue}{\left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t\right)} - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [=>] 45.8	\[ \sqrt{n \cdot \left(\left(2 \cdot U\right) \cdot \left(\left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) + t\right) - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)} \]
      distribute-rgt-neg-in [=>] 45.8	\[ \sqrt{n \cdot \left(\left(2 \cdot U\right) \cdot \left(\left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} + t\right) - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)} \]
      fma-def [=>] 45.8	\[ \sqrt{n \cdot \left(\left(2 \cdot U\right) \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t\right)} - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*r/ [<=] 50.3	\[ \sqrt{n \cdot \left(\left(2 \cdot U\right) \cdot \left(\mathsf{fma}\left(\color{blue}{\ell \cdot \frac{\ell}{Om}}, -2, t\right) - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)} \]
      metadata-eval [=>] 50.3	\[ \sqrt{n \cdot \left(\left(2 \cdot U\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, \color{blue}{-2}, t\right) - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)} \]

   if -9.99999999999999936e-149 < l < 2.3e79

   1. Initial program 59.0%

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

   2. Simplified 54.8%

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

      [Start] 59.0	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 58.5	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      sub-neg [=>] 58.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]
      associate--l+ [=>] 58.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      *-commutative [=>] 58.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      distribute-rgt-neg-in [=>] 58.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l/ [<=] 58.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 58.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [<=] 58.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [=>] 58.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 53.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]
      unpow2 [=>] 53.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]
      associate-*l* [=>] 54.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Applied egg-rr 60.2%

      \[\leadsto \color{blue}{{\left({\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, n \cdot \left(\left(U* - U\right) \cdot \frac{\ell}{Om}\right)\right), t\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)}^{0.25}\right)}^{2}} \]
      Proof

      [Start] 54.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)} \]
      add-sqr-sqrt [=>] 54.6	\[ \color{blue}{\sqrt{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \cdot \sqrt{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}}} \]
      pow2 [=>] 54.6	\[ \color{blue}{{\left(\sqrt{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}}\right)}^{2}} \]

   if 2.3e79 < l < 6.4000000000000005e191 or 7.0000000000000002e223 < l

   1. Initial program 23.9%

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

   2. Simplified 34.4%

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

      [Start] 23.9	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 23.3	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      sub-neg [=>] 23.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]
      associate--l+ [=>] 23.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      *-commutative [=>] 23.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      distribute-rgt-neg-in [=>] 23.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l/ [<=] 34.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 34.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [<=] 34.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [=>] 34.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 33.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]
      unpow2 [=>] 33.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]
      associate-*l* [=>] 34.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in t around inf 35.3%

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

   4. Simplified 39.3%

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

      [Start] 35.3	\[ \sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right)\right) + 2 \cdot \frac{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(n \cdot \left(\ell \cdot U\right)\right)}{Om}} \]
      distribute-lft-out [=>] 35.3	\[ \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(n \cdot \left(\ell \cdot U\right)\right)}{Om}\right)}} \]
      *-commutative [<=] 35.3	\[ \sqrt{2 \cdot \left(n \cdot \color{blue}{\left(U \cdot t\right)} + \frac{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(n \cdot \left(\ell \cdot U\right)\right)}{Om}\right)} \]
      associate-/l* [=>] 35.2	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \color{blue}{\frac{\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}}\right)} \]
      +-commutative [=>] 35.2	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\color{blue}{-2 \cdot \ell + \frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om}}}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}\right)} \]
      *-commutative [=>] 35.2	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\color{blue}{\ell \cdot -2} + \frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om}}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}\right)} \]
      associate-*r* [=>] 39.3	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{\color{blue}{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}}{Om}}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}\right)} \]

   5. Taylor expanded in U* around inf 35.0%

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

   6. Taylor expanded in l around inf 48.6%

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

   7. Simplified 48.9%

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

      [Start] 48.6	\[ \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n \cdot \left(\left(\frac{n \cdot U*}{Om} - 2\right) \cdot U\right)}{Om}} \]
      associate-*l* [=>] 48.6	\[ \color{blue}{\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n \cdot \left(\left(\frac{n \cdot U*}{Om} - 2\right) \cdot U\right)}{Om}}\right)} \]
      associate-/l* [=>] 47.9	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{\color{blue}{\frac{n}{\frac{Om}{\left(\frac{n \cdot U*}{Om} - 2\right) \cdot U}}}}\right) \]
      *-commutative [=>] 47.9	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{\frac{Om}{\color{blue}{U \cdot \left(\frac{n \cdot U*}{Om} - 2\right)}}}}\right) \]
      sub-neg [=>] 47.9	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \color{blue}{\left(\frac{n \cdot U*}{Om} + \left(-2\right)\right)}}}}\right) \]
      associate-/l* [=>] 48.9	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(\color{blue}{\frac{n}{\frac{Om}{U*}}} + \left(-2\right)\right)}}}\right) \]
      metadata-eval [=>] 48.9	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(\frac{n}{\frac{Om}{U*}} + \color{blue}{-2}\right)}}}\right) \]

   if 6.4000000000000005e191 < l < 7.0000000000000002e223

   1. Initial program 0.0%

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

   2. Taylor expanded in n around 0 0.0%

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

   3. Simplified 21.1%

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

      [Start] 0.0	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot U\right)\right)} \]
      *-commutative [=>] 0.0	\[ \sqrt{2 \cdot \left(n \cdot \color{blue}{\left(U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\right)} \]
      cancel-sign-sub-inv [=>] 0.0	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \color{blue}{\left(t + \left(-2\right) \cdot \frac{{\ell}^{2}}{Om}\right)}\right)\right)} \]
      metadata-eval [=>] 0.0	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \color{blue}{-2} \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)} \]
      unpow2 [=>] 0.0	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right)\right)\right)} \]
      associate-*r/ [<=] 21.1	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)}\right)\right)\right)} \]
      *-commutative [<=] 21.1	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2}\right)\right)\right)} \]
      associate-*l* [=>] 21.1	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \color{blue}{\ell \cdot \left(\frac{\ell}{Om} \cdot -2\right)}\right)\right)\right)} \]

3. Recombined 5 regimes into one program.

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.8 \cdot 10^{+198}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U* - U}{Om}, \frac{-2}{Om}\right)\right)} \cdot \left(\sqrt{2} \cdot \left(-\ell\right)\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-148}:\\ \;\;\;\;\sqrt{n \cdot \left(\left(\mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right) + n \cdot \left(\left(U* - U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right) \cdot \left(U \cdot 2\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{+79}:\\ \;\;\;\;{\left({\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, n \cdot \left(\left(U* - U\right) \cdot \frac{\ell}{Om}\right)\right), t\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)}^{0.25}\right)}^{2}\\ \mathbf{elif}\;\ell \leq 6.4 \cdot 10^{+191} \lor \neg \left(\ell \leq 7 \cdot 10^{+223}\right):\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(-2 + \frac{n}{\frac{Om}{U*}}\right)}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \ell \cdot \left(-2 \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	53.8%
Cost	26500

\[\begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{+233}:\\ \;\;\;\;{\left(e^{0.25 \cdot \left(\log \left(-2 \cdot \left(n \cdot U\right)\right) - \log \left(\frac{-1}{t}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-282}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) - \frac{\frac{\left(U - U*\right) \cdot \left(\ell \cdot n\right)}{Om} - \ell \cdot -2}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}\right)}\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{+99}:\\ \;\;\;\;\sqrt{n \cdot \left(\left(\mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right) + n \cdot \left(\left(U* - U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right) \cdot \left(U \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot 2\right)} \cdot \sqrt{t}\\ \end{array} \]

Alternative 2
Accuracy	53.9%
Cost	21132

\[\begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{+230}:\\ \;\;\;\;{\left({\left(U \cdot \left(n \cdot -2\right)\right)}^{0.16666666666666666} \cdot {\left(\frac{-1}{t}\right)}^{-0.16666666666666666}\right)}^{3}\\ \mathbf{elif}\;t \leq 1.22 \cdot 10^{-281}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) - \frac{\frac{\left(U - U*\right) \cdot \left(\ell \cdot n\right)}{Om} - \ell \cdot -2}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}\right)}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+100}:\\ \;\;\;\;\sqrt{n \cdot \left(\left(\mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right) + n \cdot \left(\left(U* - U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right) \cdot \left(U \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot 2\right)} \cdot \sqrt{t}\\ \end{array} \]

Alternative 3
Accuracy	53.9%
Cost	20100

\[\begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{+233}:\\ \;\;\;\;{\left({\left(U \cdot \left(n \cdot -2\right)\right)}^{0.16666666666666666} \cdot {\left(\frac{-1}{t}\right)}^{-0.16666666666666666}\right)}^{3}\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{-281}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) - \frac{\frac{\left(U - U*\right) \cdot \left(\ell \cdot n\right)}{Om} - \ell \cdot -2}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}\right)}\\ \mathbf{elif}\;t \leq 7 \cdot 10^{+101}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(n \cdot \left(\left(U* - U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot 2\right)} \cdot \sqrt{t}\\ \end{array} \]

Alternative 4
Accuracy	52.7%
Cost	14992

\[\begin{array}{l} t_1 := \left(n \cdot U\right) \cdot \left(2 \cdot t\right)\\ t_2 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(n \cdot \left(\left(U* - U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -5.3 \cdot 10^{+200}:\\ \;\;\;\;\left(\sqrt{2} \cdot \left(-\ell\right)\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{\frac{Om \cdot Om}{U* - U}} - \frac{2}{Om}\right)}\\ \mathbf{elif}\;\ell \leq -1.5 \cdot 10^{-189}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 8.2 \cdot 10^{-138}:\\ \;\;\;\;\sqrt{t_1}\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+32}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+107}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-4, \frac{\left(n \cdot U\right) \cdot \left(\ell \cdot \ell\right)}{Om}, t_1\right)}\\ \mathbf{elif}\;\ell \leq 7 \cdot 10^{+223}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) - \frac{\frac{\left(U - U*\right) \cdot \left(\ell \cdot n\right)}{Om} - \ell \cdot -2}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(-2 + \frac{n}{\frac{Om}{U*}}\right)}}}\right)\\ \end{array} \]

Alternative 5
Accuracy	54.7%
Cost	14860

\[\begin{array}{l} t_1 := {\left(\frac{\ell}{Om}\right)}^{2}\\ \mathbf{if}\;\ell \leq -5.8 \cdot 10^{+200}:\\ \;\;\;\;\left(\sqrt{2} \cdot \left(-\ell\right)\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{\frac{Om \cdot Om}{U* - U}} - \frac{2}{Om}\right)}\\ \mathbf{elif}\;\ell \leq -4.5 \cdot 10^{-176}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(n \cdot \left(\left(U* - U\right) \cdot t_1\right) - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.4 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(U* - U\right) \cdot \left(n \cdot t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(-2 + \frac{n}{\frac{Om}{U*}}\right)}}}\right)\\ \end{array} \]

Alternative 6
Accuracy	52.7%
Cost	14681

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.1 \cdot 10^{+232}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \left(-\sqrt{n \cdot \left(U \cdot \frac{-2}{Om}\right)}\right)\\ \mathbf{elif}\;\ell \leq -2.3 \cdot 10^{+88}:\\ \;\;\;\;\sqrt{\left(\frac{2}{Om} + \frac{n}{Om} \cdot \frac{U - U*}{Om}\right) \cdot \left(-2 \cdot \left(n \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.9 \cdot 10^{+32}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.8 \cdot 10^{+81}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-4, \frac{\left(n \cdot U\right) \cdot \left(\ell \cdot \ell\right)}{Om}, \left(n \cdot U\right) \cdot \left(2 \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.7 \cdot 10^{+190} \lor \neg \left(\ell \leq 7 \cdot 10^{+235}\right):\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(-2 + \frac{n}{\frac{Om}{U*}}\right)}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) - \frac{\frac{n}{\frac{\frac{Om}{U}}{\ell}} - \ell \cdot -2}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}\right)}\\ \end{array} \]

Alternative 7
Accuracy	53.1%
Cost	14681

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.25 \cdot 10^{+219}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \left(-\sqrt{\left(n \cdot U\right) \cdot \left(\frac{-2}{Om} - \frac{n}{Om} \cdot \frac{U}{Om}\right)}\right)\\ \mathbf{elif}\;\ell \leq -7.8 \cdot 10^{+96}:\\ \;\;\;\;\sqrt{\left(\frac{2}{Om} + \frac{n}{Om} \cdot \frac{U - U*}{Om}\right) \cdot \left(-2 \cdot \left(n \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{+30}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.25 \cdot 10^{+84}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-4, \frac{\left(n \cdot U\right) \cdot \left(\ell \cdot \ell\right)}{Om}, \left(n \cdot U\right) \cdot \left(2 \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.9 \cdot 10^{+192} \lor \neg \left(\ell \leq 7.6 \cdot 10^{+235}\right):\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(-2 + \frac{n}{\frac{Om}{U*}}\right)}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) - \frac{\frac{n}{\frac{\frac{Om}{U}}{\ell}} - \ell \cdot -2}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}\right)}\\ \end{array} \]

Alternative 8
Accuracy	53.5%
Cost	14681

\[\begin{array}{l} \mathbf{if}\;\ell \leq -4.7 \cdot 10^{+200}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \left(-\sqrt{\left(n \cdot U\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} - \frac{2}{Om}\right)}\right)\right)\\ \mathbf{elif}\;\ell \leq -5.2 \cdot 10^{+91}:\\ \;\;\;\;\sqrt{\left(\frac{2}{Om} + \frac{n}{Om} \cdot \frac{U - U*}{Om}\right) \cdot \left(-2 \cdot \left(n \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 6.4 \cdot 10^{+33}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.55 \cdot 10^{+83}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-4, \frac{\left(n \cdot U\right) \cdot \left(\ell \cdot \ell\right)}{Om}, \left(n \cdot U\right) \cdot \left(2 \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.7 \cdot 10^{+187} \lor \neg \left(\ell \leq 7 \cdot 10^{+235}\right):\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(-2 + \frac{n}{\frac{Om}{U*}}\right)}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) - \frac{\frac{n}{\frac{\frac{Om}{U}}{\ell}} - \ell \cdot -2}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}\right)}\\ \end{array} \]

Alternative 9
Accuracy	53.5%
Cost	14681

\[\begin{array}{l} \mathbf{if}\;\ell \leq -4.7 \cdot 10^{+200}:\\ \;\;\;\;\left(\sqrt{2} \cdot \left(-\ell\right)\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{\frac{Om \cdot Om}{U* - U}} - \frac{2}{Om}\right)}\\ \mathbf{elif}\;\ell \leq -4.4 \cdot 10^{+90}:\\ \;\;\;\;\sqrt{\left(\frac{2}{Om} + \frac{n}{Om} \cdot \frac{U - U*}{Om}\right) \cdot \left(-2 \cdot \left(n \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 7 \cdot 10^{+33}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.9 \cdot 10^{+83}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-4, \frac{\left(n \cdot U\right) \cdot \left(\ell \cdot \ell\right)}{Om}, \left(n \cdot U\right) \cdot \left(2 \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.9 \cdot 10^{+192} \lor \neg \left(\ell \leq 1.42 \cdot 10^{+236}\right):\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(-2 + \frac{n}{\frac{Om}{U*}}\right)}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) - \frac{\frac{n}{\frac{\frac{Om}{U}}{\ell}} - \ell \cdot -2}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}\right)}\\ \end{array} \]

Alternative 10
Accuracy	52.5%
Cost	14164

\[\begin{array}{l} t_1 := \ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\\ t_2 := \ell \cdot \sqrt{2}\\ t_3 := n \cdot \left(U \cdot t\right)\\ \mathbf{if}\;\ell \leq -3.4 \cdot 10^{+232}:\\ \;\;\;\;t_2 \cdot \left(-\sqrt{U \cdot \left(n \cdot \frac{-2}{Om}\right)}\right)\\ \mathbf{elif}\;\ell \leq -6.5 \cdot 10^{+95}:\\ \;\;\;\;\sqrt{\left(\frac{2}{Om} + \frac{n}{Om} \cdot \frac{U - U*}{Om}\right) \cdot \left(-2 \cdot \left(n \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.1 \cdot 10^{-55}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot t_1}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.15 \cdot 10^{+243}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_3 - \frac{\frac{\left(U - U*\right) \cdot \left(\ell \cdot n\right)}{Om} - \ell \cdot -2}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}\right)}\\ \mathbf{elif}\;\ell \leq 1.55 \cdot 10^{+305}:\\ \;\;\;\;t_2 \cdot \sqrt{n \cdot \frac{U \cdot -2}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_3 + \frac{t_1}{\frac{Om}{n} \cdot \frac{1}{\ell \cdot U}}\right)}\\ \end{array} \]

Alternative 11
Accuracy	52.6%
Cost	14164

\[\begin{array}{l} t_1 := \ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\\ t_2 := \ell \cdot \sqrt{2}\\ t_3 := n \cdot \left(U \cdot t\right)\\ \mathbf{if}\;\ell \leq -2.1 \cdot 10^{+232}:\\ \;\;\;\;t_2 \cdot \left(-\sqrt{n \cdot \left(-2 \cdot \frac{U}{Om}\right)}\right)\\ \mathbf{elif}\;\ell \leq -1.9 \cdot 10^{+88}:\\ \;\;\;\;\sqrt{\left(\frac{2}{Om} + \frac{n}{Om} \cdot \frac{U - U*}{Om}\right) \cdot \left(-2 \cdot \left(n \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{-55}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot t_1}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 7 \cdot 10^{+235}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_3 - \frac{\frac{\left(U - U*\right) \cdot \left(\ell \cdot n\right)}{Om} - \ell \cdot -2}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}\right)}\\ \mathbf{elif}\;\ell \leq 1.55 \cdot 10^{+305}:\\ \;\;\;\;t_2 \cdot \sqrt{n \cdot \frac{U \cdot -2}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_3 + \frac{t_1}{\frac{Om}{n} \cdot \frac{1}{\ell \cdot U}}\right)}\\ \end{array} \]

Alternative 12
Accuracy	52.6%
Cost	14164

\[\begin{array}{l} t_1 := \ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\\ t_2 := \ell \cdot \sqrt{2}\\ t_3 := n \cdot \left(U \cdot t\right)\\ \mathbf{if}\;\ell \leq -2.1 \cdot 10^{+232}:\\ \;\;\;\;t_2 \cdot \left(-\sqrt{n \cdot \left(U \cdot \frac{-2}{Om}\right)}\right)\\ \mathbf{elif}\;\ell \leq -3.9 \cdot 10^{+89}:\\ \;\;\;\;\sqrt{\left(\frac{2}{Om} + \frac{n}{Om} \cdot \frac{U - U*}{Om}\right) \cdot \left(-2 \cdot \left(n \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.7 \cdot 10^{-55}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot t_1}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 7 \cdot 10^{+240}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_3 - \frac{\frac{\left(U - U*\right) \cdot \left(\ell \cdot n\right)}{Om} - \ell \cdot -2}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}\right)}\\ \mathbf{elif}\;\ell \leq 1.55 \cdot 10^{+305}:\\ \;\;\;\;t_2 \cdot \sqrt{n \cdot \frac{U \cdot -2}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_3 + \frac{t_1}{\frac{Om}{n} \cdot \frac{1}{\ell \cdot U}}\right)}\\ \end{array} \]

Alternative 13
Accuracy	51.3%
Cost	13644

\[\begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{+262}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(2 \cdot t\right)}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-280}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) - \frac{\frac{\left(U - U*\right) \cdot \left(\ell \cdot n\right)}{Om} - \ell \cdot -2}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}\right)}\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{+97}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \ell \cdot \left(-2 \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot 2\right)} \cdot \sqrt{t}\\ \end{array} \]

Alternative 14
Accuracy	47.7%
Cost	8520

\[\begin{array}{l} t_1 := \sqrt{\left(n \cdot U\right) \cdot \left(2 \cdot t\right)}\\ \mathbf{if}\;t \leq -6.6 \cdot 10^{+262}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{-282}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}}{\frac{Om}{n} \cdot \frac{1}{\ell \cdot U}}\right)}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{+101}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \ell \cdot \left(-2 \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 15
Accuracy	48.4%
Cost	8520

\[\begin{array}{l} t_1 := \sqrt{\left(n \cdot U\right) \cdot \left(2 \cdot t\right)}\\ \mathbf{if}\;t \leq -6.5 \cdot 10^{+262}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-281}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) - \frac{\frac{\left(U - U*\right) \cdot \left(\ell \cdot n\right)}{Om} - \ell \cdot -2}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}\right)}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{+99}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \ell \cdot \left(-2 \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 16
Accuracy	48.0%
Cost	8392

\[\begin{array}{l} t_1 := \sqrt{\left(n \cdot U\right) \cdot \left(2 \cdot t\right)}\\ \mathbf{if}\;t \leq -6.5 \cdot 10^{+262}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-280}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}\right)}\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{+100}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \ell \cdot \left(-2 \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 17
Accuracy	45.5%
Cost	8268

\[\begin{array}{l} \mathbf{if}\;Om \leq -390:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \ell \cdot \left(-2 \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;Om \leq -2 \cdot 10^{-256}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(2 \cdot t\right)}\\ \mathbf{elif}\;Om \leq 2.2 \cdot 10^{-119}:\\ \;\;\;\;\sqrt{2 \cdot \frac{n \cdot \left(\left(\ell \cdot -2 + \frac{\left(U* - U\right) \cdot \left(\ell \cdot n\right)}{Om}\right) \cdot \left(\ell \cdot U\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)\right)}\\ \end{array} \]

Alternative 18
Accuracy	45.6%
Cost	8136

\[\begin{array}{l} t_1 := \sqrt{\left(n \cdot U\right) \cdot \left(2 \cdot t\right)}\\ \mathbf{if}\;t \leq -4.8 \cdot 10^{+262}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -4.4 \cdot 10^{-154}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{+99}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \ell \cdot \left(-2 \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 19
Accuracy	48.5%
Cost	8136

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.3 \cdot 10^{-52}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \ell \cdot \left(-2 \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 450000:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \frac{\ell \cdot -2 + \frac{\left(U* - U\right) \cdot \left(\ell \cdot n\right)}{Om}}{\frac{Om}{\ell \cdot U}}}\\ \end{array} \]

Alternative 20
Accuracy	47.9%
Cost	8136

\[\begin{array}{l} \mathbf{if}\;\ell \leq -7 \cdot 10^{+84}:\\ \;\;\;\;\sqrt{\left(\frac{2}{Om} + \frac{n}{Om} \cdot \frac{U - U*}{Om}\right) \cdot \left(-2 \cdot \left(n \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 450000:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \frac{\ell \cdot -2 + \frac{\left(U* - U\right) \cdot \left(\ell \cdot n\right)}{Om}}{\frac{Om}{\ell \cdot U}}}\\ \end{array} \]

Alternative 21
Accuracy	41.0%
Cost	7628

\[\begin{array}{l} t_1 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(\ell \cdot \left(-2 \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -2.35 \cdot 10^{+34}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -1.22 \cdot 10^{-169}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{+49}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(2 \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 22
Accuracy	46.1%
Cost	7625

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-145} \lor \neg \left(\ell \leq 10^{+43}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \ell \cdot \left(-2 \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(2 \cdot t\right)}\\ \end{array} \]

Alternative 23
Accuracy	46.2%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;\ell \leq -7.1 \cdot 10^{-146}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \ell \cdot \left(-2 \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.15 \cdot 10^{+38}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(2 \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)\right)}\\ \end{array} \]

Alternative 24
Accuracy	39.5%
Cost	7364

\[\begin{array}{l} \mathbf{if}\;\ell \leq -3 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(\ell \cdot \left(\ell \cdot \frac{-2}{\frac{Om}{U}}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -3.15 \cdot 10^{-142}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(2 \cdot t\right)}\\ \end{array} \]

Alternative 25
Accuracy	39.8%
Cost	7364

\[\begin{array}{l} \mathbf{if}\;\ell \leq -4.2 \cdot 10^{+87}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(-2 \cdot \frac{\ell \cdot \left(\ell \cdot U\right)}{Om}\right)}\\ \mathbf{elif}\;\ell \leq -1.9 \cdot 10^{-147}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(2 \cdot t\right)}\\ \end{array} \]

Alternative 26
Accuracy	38.2%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;U \leq 1.2 \cdot 10^{-172} \lor \neg \left(U \leq 3.2 \cdot 10^{+17}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]

Alternative 27
Accuracy	37.9%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;U \leq 1.2 \cdot 10^{-174}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;U \leq 5 \cdot 10^{+192}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(2 \cdot t\right)}\\ \end{array} \]

Alternative 28
Accuracy	37.7%
Cost	6848

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

Reproduce

herbie shell --seed 2023147 
(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*))))))