Toniolo and Linder, Equation (13)

Percentage Accurate: 20.5% → 71.0%

Time: 35.6s

Precision: binary64

Cost: 51592

• 66.9% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} t_1 := U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\\ t_2 := \frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om} + \ell \cdot -2\\ t_3 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_3 \leq -20000000000:\\ \;\;\;\;{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\left(2 \cdot n\right) \cdot t_1\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;t_3 \leq -2 \cdot 10^{-303}:\\ \;\;\;\;0.5 \cdot \left(\left(t \cdot \sqrt{2}\right) \cdot \sqrt[3]{{\left(\frac{n}{\frac{\ell \cdot t_2}{U \cdot Om}}\right)}^{1.5}}\right) + \sqrt{2} \cdot \sqrt{\frac{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + \ell \cdot -2\right) \cdot \left(n \cdot \left(U \cdot \ell\right)\right)}{Om}}\\ \mathbf{elif}\;t_3 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{t_1}\\ \mathbf{elif}\;t_3 \leq 10^{+285}:\\ \;\;\;\;\sqrt{t_3}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{2 \cdot \left(n \cdot \left(\left(U \cdot \ell\right) \cdot t_2\right)\right)}{Om}\right)}^{0.5}\\ \end{array} \]

FPCore

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

↓

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* U (+ t (* (/ l Om) (fma l -2.0 (* (/ l Om) (* n (- U* U))))))))
        (t_2 (+ (/ (* (- U* U) (* n l)) Om) (* l -2.0)))
        (t_3
         (*
          (* (* 2.0 n) U)
          (+
           (- t (* 2.0 (/ (* l l) Om)))
           (* (* n (pow (/ l Om) 2.0)) (- U* U))))))
   (if (<= t_3 -20000000000.0)
     (pow (log1p (expm1 (* (* 2.0 n) t_1))) 0.5)
     (if (<= t_3 -2e-303)
       (+
        (*
         0.5
         (* (* t (sqrt 2.0)) (cbrt (pow (/ n (/ (* l t_2) (* U Om))) 1.5))))
        (*
         (sqrt 2.0)
         (sqrt
          (/
           (* (+ (/ (* n (* l (- U* U))) Om) (* l -2.0)) (* n (* U l)))
           Om))))
       (if (<= t_3 0.0)
         (* (sqrt (* 2.0 n)) (sqrt t_1))
         (if (<= t_3 1e+285)
           (sqrt t_3)
           (pow (/ (* 2.0 (* n (* (* U l) t_2))) Om) 0.5)))))))

C

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

↓

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = U * (t + ((l / Om) * fma(l, -2.0, ((l / Om) * (n * (U_42_ - U))))));
	double t_2 = (((U_42_ - U) * (n * l)) / Om) + (l * -2.0);
	double t_3 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + ((n * pow((l / Om), 2.0)) * (U_42_ - U)));
	double tmp;
	if (t_3 <= -20000000000.0) {
		tmp = pow(log1p(expm1(((2.0 * n) * t_1))), 0.5);
	} else if (t_3 <= -2e-303) {
		tmp = (0.5 * ((t * sqrt(2.0)) * cbrt(pow((n / ((l * t_2) / (U * Om))), 1.5)))) + (sqrt(2.0) * sqrt((((((n * (l * (U_42_ - U))) / Om) + (l * -2.0)) * (n * (U * l))) / Om)));
	} else if (t_3 <= 0.0) {
		tmp = sqrt((2.0 * n)) * sqrt(t_1);
	} else if (t_3 <= 1e+285) {
		tmp = sqrt(t_3);
	} else {
		tmp = pow(((2.0 * (n * ((U * l) * t_2))) / Om), 0.5);
	}
	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(U * Float64(t + Float64(Float64(l / Om) * fma(l, -2.0, Float64(Float64(l / Om) * Float64(n * Float64(U_42_ - U)))))))
	t_2 = Float64(Float64(Float64(Float64(U_42_ - U) * Float64(n * l)) / Om) + Float64(l * -2.0))
	t_3 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U_42_ - U))))
	tmp = 0.0
	if (t_3 <= -20000000000.0)
		tmp = log1p(expm1(Float64(Float64(2.0 * n) * t_1))) ^ 0.5;
	elseif (t_3 <= -2e-303)
		tmp = Float64(Float64(0.5 * Float64(Float64(t * sqrt(2.0)) * cbrt((Float64(n / Float64(Float64(l * t_2) / Float64(U * Om))) ^ 1.5)))) + Float64(sqrt(2.0) * sqrt(Float64(Float64(Float64(Float64(Float64(n * Float64(l * Float64(U_42_ - U))) / Om) + Float64(l * -2.0)) * Float64(n * Float64(U * l))) / Om))));
	elseif (t_3 <= 0.0)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(t_1));
	elseif (t_3 <= 1e+285)
		tmp = sqrt(t_3);
	else
		tmp = Float64(Float64(2.0 * Float64(n * Float64(Float64(U * l) * t_2))) / Om) ^ 0.5;
	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[(U * N[(t + N[(N[(l / Om), $MachinePrecision] * N[(l * -2.0 + N[(N[(l / Om), $MachinePrecision] * N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -20000000000.0], N[Power[N[Log[1 + N[(Exp[N[(N[(2.0 * n), $MachinePrecision] * t$95$1), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[t$95$3, -2e-303], N[(N[(0.5 * N[(N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[N[(n / N[(N[(l * t$95$2), $MachinePrecision] / N[(U * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(N[(N[(N[(N[(n * N[(l * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision] * N[(n * N[(U * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e+285], N[Sqrt[t$95$3], $MachinePrecision], N[Power[N[(N[(2.0 * N[(n * N[(N[(U * l), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision], 0.5], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 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*)))) < -2e10

   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 1.9%

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

      [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/ [<=] 1.0%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 1.0%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [<=] 1.0%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [=>] 1.0%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 1.0%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]
      unpow2 [=>] 1.0%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]
      associate-*l* [=>] 1.9%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Applied egg-rr 76.2%

      \[\leadsto \color{blue}{{\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)\right)}^{0.5}} \]
      Step-by-step derivation

      [Start] 1.9%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)} \]
      pow1/2 [=>] 76.2%	\[ \color{blue}{{\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)\right)}^{0.5}} \]

   4. Applied egg-rr 94.3%

      \[\leadsto {\color{blue}{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)\right)\right)\right)}}^{0.5} \]
      Step-by-step derivation

      [Start] 76.2%	\[ {\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)\right)}^{0.5} \]
      log1p-expm1-u [=>] 94.3%	\[ {\color{blue}{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)\right)\right)\right)}}^{0.5} \]

   if -2e10 < (*.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*)))) < -1.99999999999999986e-303

   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 0.2%

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

      [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* [=>] 0.2%	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      sub-neg [=>] 0.2%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]
      associate--l+ [=>] 0.2%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      *-commutative [=>] 0.2%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      distribute-rgt-neg-in [=>] 0.2%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l/ [<=] 0.2%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 0.2%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [<=] 0.2%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [=>] 0.2%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 0.2%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]
      unpow2 [=>] 0.2%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]
      associate-*l* [=>] 0.2%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in t around 0 9.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{n \cdot \left(Om \cdot U\right)}{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \ell}}\right) + \sqrt{2} \cdot \sqrt{\frac{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(n \cdot \left(\ell \cdot U\right)\right)}{Om}}} \]

   4. Applied egg-rr 14.0%

      \[\leadsto 0.5 \cdot \left(\left(\sqrt{2} \cdot t\right) \cdot \color{blue}{\sqrt[3]{\left(\frac{n}{\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}} + \ell \cdot -2} \cdot \frac{U \cdot Om}{\ell}\right) \cdot \sqrt{\frac{n}{\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}} + \ell \cdot -2} \cdot \frac{U \cdot Om}{\ell}}}}\right) + \sqrt{2} \cdot \sqrt{\frac{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(n \cdot \left(\ell \cdot U\right)\right)}{Om}} \]
      Step-by-step derivation

      [Start] 9.5%	\[ 0.5 \cdot \left(\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{n \cdot \left(Om \cdot U\right)}{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \ell}}\right) + \sqrt{2} \cdot \sqrt{\frac{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(n \cdot \left(\ell \cdot U\right)\right)}{Om}} \]
      add-cbrt-cube [=>] 9.5%	\[ 0.5 \cdot \left(\left(\sqrt{2} \cdot t\right) \cdot \color{blue}{\sqrt[3]{\left(\sqrt{\frac{n \cdot \left(Om \cdot U\right)}{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \ell}} \cdot \sqrt{\frac{n \cdot \left(Om \cdot U\right)}{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \ell}}\right) \cdot \sqrt{\frac{n \cdot \left(Om \cdot U\right)}{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \ell}}}}\right) + \sqrt{2} \cdot \sqrt{\frac{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(n \cdot \left(\ell \cdot U\right)\right)}{Om}} \]
      add-sqr-sqrt [<=] 9.5%	\[ 0.5 \cdot \left(\left(\sqrt{2} \cdot t\right) \cdot \sqrt[3]{\color{blue}{\frac{n \cdot \left(Om \cdot U\right)}{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \ell}} \cdot \sqrt{\frac{n \cdot \left(Om \cdot U\right)}{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \ell}}}\right) + \sqrt{2} \cdot \sqrt{\frac{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(n \cdot \left(\ell \cdot U\right)\right)}{Om}} \]
      times-frac [=>] 13.9%	\[ 0.5 \cdot \left(\left(\sqrt{2} \cdot t\right) \cdot \sqrt[3]{\color{blue}{\left(\frac{n}{\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell} \cdot \frac{Om \cdot U}{\ell}\right)} \cdot \sqrt{\frac{n \cdot \left(Om \cdot U\right)}{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \ell}}}\right) + \sqrt{2} \cdot \sqrt{\frac{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(n \cdot \left(\ell \cdot U\right)\right)}{Om}} \]
      associate-/l* [=>] 13.9%	\[ 0.5 \cdot \left(\left(\sqrt{2} \cdot t\right) \cdot \sqrt[3]{\left(\frac{n}{\color{blue}{\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}}} + -2 \cdot \ell} \cdot \frac{Om \cdot U}{\ell}\right) \cdot \sqrt{\frac{n \cdot \left(Om \cdot U\right)}{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \ell}}}\right) + \sqrt{2} \cdot \sqrt{\frac{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(n \cdot \left(\ell \cdot U\right)\right)}{Om}} \]
      *-commutative [=>] 13.9%	\[ 0.5 \cdot \left(\left(\sqrt{2} \cdot t\right) \cdot \sqrt[3]{\left(\frac{n}{\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}} + \color{blue}{\ell \cdot -2}} \cdot \frac{Om \cdot U}{\ell}\right) \cdot \sqrt{\frac{n \cdot \left(Om \cdot U\right)}{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \ell}}}\right) + \sqrt{2} \cdot \sqrt{\frac{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(n \cdot \left(\ell \cdot U\right)\right)}{Om}} \]
      *-commutative [=>] 13.9%	\[ 0.5 \cdot \left(\left(\sqrt{2} \cdot t\right) \cdot \sqrt[3]{\left(\frac{n}{\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}} + \ell \cdot -2} \cdot \frac{\color{blue}{U \cdot Om}}{\ell}\right) \cdot \sqrt{\frac{n \cdot \left(Om \cdot U\right)}{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \ell}}}\right) + \sqrt{2} \cdot \sqrt{\frac{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(n \cdot \left(\ell \cdot U\right)\right)}{Om}} \]

   5. Simplified 32.1%

      \[\leadsto 0.5 \cdot \left(\left(\sqrt{2} \cdot t\right) \cdot \color{blue}{\sqrt[3]{{\left(\frac{n}{\frac{\ell \cdot \left(\ell \cdot -2 + \frac{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}{Om}\right)}{Om \cdot U}}\right)}^{1.5}}}\right) + \sqrt{2} \cdot \sqrt{\frac{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(n \cdot \left(\ell \cdot U\right)\right)}{Om}} \]
      Step-by-step derivation

      [Start] 14.0%	\[ 0.5 \cdot \left(\left(\sqrt{2} \cdot t\right) \cdot \sqrt[3]{\left(\frac{n}{\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}} + \ell \cdot -2} \cdot \frac{U \cdot Om}{\ell}\right) \cdot \sqrt{\frac{n}{\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}} + \ell \cdot -2} \cdot \frac{U \cdot Om}{\ell}}}\right) + \sqrt{2} \cdot \sqrt{\frac{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(n \cdot \left(\ell \cdot U\right)\right)}{Om}} \]
      *-commutative [=>] 14.0%	\[ 0.5 \cdot \left(\left(\sqrt{2} \cdot t\right) \cdot \sqrt[3]{\color{blue}{\sqrt{\frac{n}{\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}} + \ell \cdot -2} \cdot \frac{U \cdot Om}{\ell}} \cdot \left(\frac{n}{\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}} + \ell \cdot -2} \cdot \frac{U \cdot Om}{\ell}\right)}}\right) + \sqrt{2} \cdot \sqrt{\frac{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(n \cdot \left(\ell \cdot U\right)\right)}{Om}} \]
      unpow1/2 [<=] 18.5%	\[ 0.5 \cdot \left(\left(\sqrt{2} \cdot t\right) \cdot \sqrt[3]{\color{blue}{{\left(\frac{n}{\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}} + \ell \cdot -2} \cdot \frac{U \cdot Om}{\ell}\right)}^{0.5}} \cdot \left(\frac{n}{\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}} + \ell \cdot -2} \cdot \frac{U \cdot Om}{\ell}\right)}\right) + \sqrt{2} \cdot \sqrt{\frac{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(n \cdot \left(\ell \cdot U\right)\right)}{Om}} \]
      pow-plus [=>] 32.2%	\[ 0.5 \cdot \left(\left(\sqrt{2} \cdot t\right) \cdot \sqrt[3]{\color{blue}{{\left(\frac{n}{\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}} + \ell \cdot -2} \cdot \frac{U \cdot Om}{\ell}\right)}^{\left(0.5 + 1\right)}}}\right) + \sqrt{2} \cdot \sqrt{\frac{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(n \cdot \left(\ell \cdot U\right)\right)}{Om}} \]

   if -1.99999999999999986e-303 < (*.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 4.1%

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

   2. Simplified 14.7%

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

      [Start] 4.1%	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 14.7%	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      sub-neg [=>] 14.7%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]
      associate--l+ [=>] 14.7%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      *-commutative [=>] 14.7%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      distribute-rgt-neg-in [=>] 14.7%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l/ [<=] 14.7%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 14.7%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [<=] 14.7%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [=>] 14.7%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 14.7%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]
      unpow2 [=>] 14.7%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]
      associate-*l* [=>] 14.7%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Applied egg-rr 33.2%

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

      [Start] 14.7%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)} \]
      sqrt-prod [=>] 33.2%	\[ \color{blue}{\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)}} \]

   4. Simplified 33.2%

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

      [Start] 33.2%	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)} \]
      *-commutative [=>] 33.2%	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + \color{blue}{\mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}}\right)} \]
      *-commutative [=>] 33.2%	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + \mathsf{fma}\left(\ell, -2, \color{blue}{\left(n \cdot \left(U* - U\right)\right) \cdot \frac{\ell}{Om}}\right) \cdot \frac{\ell}{Om}\right)} \]

   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*)))) < 9.9999999999999998e284

   1. Initial program 97.1%

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

   if 9.9999999999999998e284 < (*.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 15.7%

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

   2. Simplified 18.4%

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

      [Start] 15.7%	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 14.5%	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      sub-neg [=>] 14.5%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]
      associate--l+ [=>] 14.5%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      *-commutative [=>] 14.5%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      distribute-rgt-neg-in [=>] 14.5%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l/ [<=] 15.9%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 15.9%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [<=] 15.9%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [=>] 15.9%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 11.3%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]
      unpow2 [=>] 11.3%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]
      associate-*l* [=>] 14.1%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Applied egg-rr 51.6%

      \[\leadsto \color{blue}{{\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)\right)}^{0.5}} \]
      Step-by-step derivation

      [Start] 18.4%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)} \]
      pow1/2 [=>] 51.6%	\[ \color{blue}{{\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)\right)}^{0.5}} \]

   4. Applied egg-rr 51.6%

      \[\leadsto {\color{blue}{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)\right)\right)\right)}}^{0.5} \]
      Step-by-step derivation

      [Start] 51.6%	\[ {\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)\right)}^{0.5} \]
      log1p-expm1-u [=>] 51.6%	\[ {\color{blue}{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)\right)\right)\right)}}^{0.5} \]

   5. Taylor expanded in t around 0 56.5%

      \[\leadsto {\color{blue}{\left(2 \cdot \frac{n \cdot \left(\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)\right)}{Om}\right)}}^{0.5} \]

   6. Simplified 56.4%

      \[\leadsto {\color{blue}{\left(\frac{2 \cdot \left(n \cdot \left(\left(-2 \cdot \ell + \frac{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}{Om}\right) \cdot \left(\ell \cdot U\right)\right)\right)}{Om}\right)}}^{0.5} \]
      Step-by-step derivation

      [Start] 56.5%	\[ {\left(2 \cdot \frac{n \cdot \left(\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)\right)}{Om}\right)}^{0.5} \]
      associate-*r/ [=>] 56.5%	\[ {\color{blue}{\left(\frac{2 \cdot \left(n \cdot \left(\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)\right)\right)}{Om}\right)}}^{0.5} \]
      +-commutative [=>] 56.5%	\[ {\left(\frac{2 \cdot \left(n \cdot \left(\color{blue}{\left(-2 \cdot \ell + \frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om}\right)} \cdot \left(\ell \cdot U\right)\right)\right)}{Om}\right)}^{0.5} \]
      associate-*r* [=>] 56.4%	\[ {\left(\frac{2 \cdot \left(n \cdot \left(\left(-2 \cdot \ell + \frac{\color{blue}{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}}{Om}\right) \cdot \left(\ell \cdot U\right)\right)\right)}{Om}\right)}^{0.5} \]

3. Recombined 5 regimes into one program.

4. Final simplification 74.3%

   \[\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 -20000000000:\\ \;\;\;\;{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\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)\right)\right)}^{0.5}\\ \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 -2 \cdot 10^{-303}:\\ \;\;\;\;0.5 \cdot \left(\left(t \cdot \sqrt{2}\right) \cdot \sqrt[3]{{\left(\frac{n}{\frac{\ell \cdot \left(\frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om} + \ell \cdot -2\right)}{U \cdot Om}}\right)}^{1.5}}\right) + \sqrt{2} \cdot \sqrt{\frac{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + \ell \cdot -2\right) \cdot \left(n \cdot \left(U \cdot \ell\right)\right)}{Om}}\\ \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 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{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)}\\ \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^{+285}:\\ \;\;\;\;\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{else}:\\ \;\;\;\;{\left(\frac{2 \cdot \left(n \cdot \left(\left(U \cdot \ell\right) \cdot \left(\frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om} + \ell \cdot -2\right)\right)\right)}{Om}\right)}^{0.5}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	71.0%
Cost	51592

\[\begin{array}{l} t_1 := U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\\ t_2 := \frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om} + \ell \cdot -2\\ t_3 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_3 \leq -20000000000:\\ \;\;\;\;{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\left(2 \cdot n\right) \cdot t_1\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;t_3 \leq -2 \cdot 10^{-303}:\\ \;\;\;\;0.5 \cdot \left(\left(t \cdot \sqrt{2}\right) \cdot \sqrt[3]{{\left(\frac{n}{\frac{\ell \cdot t_2}{U \cdot Om}}\right)}^{1.5}}\right) + \sqrt{2} \cdot \sqrt{\frac{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + \ell \cdot -2\right) \cdot \left(n \cdot \left(U \cdot \ell\right)\right)}{Om}}\\ \mathbf{elif}\;t_3 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{t_1}\\ \mathbf{elif}\;t_3 \leq 10^{+285}:\\ \;\;\;\;\sqrt{t_3}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{2 \cdot \left(n \cdot \left(\left(U \cdot \ell\right) \cdot t_2\right)\right)}{Om}\right)}^{0.5}\\ \end{array} \]

Alternative 2
Accuracy	70.9%
Cost	70604

\[\begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+142}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;{\left(\frac{2 \cdot \left(n \cdot \left(\left(U \cdot \ell\right) \cdot \left(\frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om} + \ell \cdot -2\right)\right)\right)}{Om}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)\right)\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 3
Accuracy	65.6%
Cost	55124

\[\begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+296}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{\frac{Om}{\ell \cdot -2 - \frac{n \cdot \left(U \cdot \ell\right)}{Om}}}\right)\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;t_1 \leq -1 \cdot 10^{+35}:\\ \;\;\;\;{\left(2 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(n \cdot \left(U \cdot t\right)\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{-303}:\\ \;\;\;\;{\left(\frac{2 \cdot \left(\left(n \cdot \ell\right) \cdot \left(U \cdot \left(\ell \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)\right)\right)}{Om}\right)}^{0.5}\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{elif}\;t_1 \leq 10^{+285}:\\ \;\;\;\;\sqrt{t_1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{2 \cdot \left(n \cdot \left(\left(U \cdot \ell\right) \cdot \left(\frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om} + \ell \cdot -2\right)\right)\right)}{Om}\right)}^{0.5}\\ \end{array} \]

Alternative 4
Accuracy	66.5%
Cost	55124

\[\begin{array}{l} t_1 := \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\\ t_2 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_2 \leq -2 \cdot 10^{+296}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \sqrt[3]{t_1 \cdot \left(t_1 \cdot t_1\right)}\right)\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;t_2 \leq -1 \cdot 10^{+35}:\\ \;\;\;\;{\left(2 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(n \cdot \left(U \cdot t\right)\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;t_2 \leq -2 \cdot 10^{-303}:\\ \;\;\;\;{\left(\frac{2 \cdot \left(\left(n \cdot \ell\right) \cdot \left(U \cdot \left(\ell \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)\right)\right)}{Om}\right)}^{0.5}\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, t_1\right)\right)}\\ \mathbf{elif}\;t_2 \leq 10^{+285}:\\ \;\;\;\;\sqrt{t_2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{2 \cdot \left(n \cdot \left(\left(U \cdot \ell\right) \cdot \left(\frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om} + \ell \cdot -2\right)\right)\right)}{Om}\right)}^{0.5}\\ \end{array} \]

Alternative 5
Accuracy	62.3%
Cost	46992

\[\begin{array}{l} t_1 := n \cdot \left(U \cdot t\right)\\ t_2 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_2 \leq -2 \cdot 10^{+296}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{\frac{Om}{\ell \cdot -2 - \frac{n \cdot \left(U \cdot \ell\right)}{Om}}}\right)\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;t_2 \leq -2 \cdot 10^{+116}:\\ \;\;\;\;{\left(2 \cdot \sqrt[3]{t_1 \cdot \left(t_1 \cdot t_1\right)}\right)}^{0.5}\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;{\left(\frac{2 \cdot \left(\left(n \cdot \ell\right) \cdot \left(U \cdot \left(\ell \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)\right)\right)}{Om}\right)}^{0.5}\\ \mathbf{elif}\;t_2 \leq 10^{+285}:\\ \;\;\;\;\sqrt{t_2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{2 \cdot \left(n \cdot \left(\left(U \cdot \ell\right) \cdot \left(\frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om} + \ell \cdot -2\right)\right)\right)}{Om}\right)}^{0.5}\\ \end{array} \]

Alternative 6
Accuracy	64.6%
Cost	46992

\[\begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+296}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{\frac{Om}{\ell \cdot -2 - \frac{n \cdot \left(U \cdot \ell\right)}{Om}}}\right)\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;t_1 \leq -1 \cdot 10^{+35}:\\ \;\;\;\;{\left(2 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(n \cdot \left(U \cdot t\right)\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;{\left(\frac{2 \cdot \left(\left(n \cdot \ell\right) \cdot \left(U \cdot \left(\ell \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)\right)\right)}{Om}\right)}^{0.5}\\ \mathbf{elif}\;t_1 \leq 10^{+285}:\\ \;\;\;\;\sqrt{t_1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{2 \cdot \left(n \cdot \left(\left(U \cdot \ell\right) \cdot \left(\frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om} + \ell \cdot -2\right)\right)\right)}{Om}\right)}^{0.5}\\ \end{array} \]

Alternative 7
Accuracy	72.6%
Cost	43272

\[\begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_1 \leq -20000000000:\\ \;\;\;\;{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)\right)\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;t_1 \leq 10^{+285}:\\ \;\;\;\;\sqrt{t_1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{2 \cdot \left(n \cdot \left(\left(U \cdot \ell\right) \cdot \left(\frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om} + \ell \cdot -2\right)\right)\right)}{Om}\right)}^{0.5}\\ \end{array} \]

Alternative 8
Accuracy	57.2%
Cost	8072

\[\begin{array}{l} \mathbf{if}\;\ell \leq -7.8 \cdot 10^{+69}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{-11}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{\frac{\frac{Om \cdot Om}{n}}{\ell \cdot U*}}\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)\right)\right)}{Om}\right)}^{0.5}\\ \end{array} \]

Alternative 9
Accuracy	57.3%
Cost	7944

\[\begin{array}{l} \mathbf{if}\;\ell \leq -8 \cdot 10^{+69}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 6.6 \cdot 10^{+35}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{\frac{\frac{Om \cdot Om}{n}}{\ell \cdot U*}}\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(-2 \cdot \frac{\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U \cdot \left(2 - \frac{n}{\frac{Om}{U*}}\right)\right)}{Om}\right)}^{0.5}\\ \end{array} \]

Alternative 10
Accuracy	59.7%
Cost	7808

\[{\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{\frac{Om}{\ell \cdot \left(\frac{n \cdot U*}{Om} - 2\right)}}\right)\right)\right)}^{0.5} \]

Alternative 11
Accuracy	49.3%
Cost	7433

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.55 \cdot 10^{+38} \lor \neg \left(\ell \leq 2.05 \cdot 10^{-14}\right):\\ \;\;\;\;{\left(-4 \cdot \frac{n}{\frac{Om}{U \cdot \left(\ell \cdot \ell\right)}}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 12
Accuracy	52.7%
Cost	7424

\[{\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{-0.5 \cdot \frac{Om}{\ell}}\right)\right)\right)}^{0.5} \]

Alternative 13
Accuracy	53.1%
Cost	7424

\[{\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right)\right)\right)}^{0.5} \]

Alternative 14
Accuracy	30.2%
Cost	6912

\[{\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.5} \]

Alternative 15
Accuracy	29.6%
Cost	6912

\[{\left(\left(2 \cdot n\right) \cdot \left(U \cdot t\right)\right)}^{0.5} \]

Alternative 16
Accuracy	20.7%
Cost	6848

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

Alternative 17
Accuracy	20.0%
Cost	6848

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

Reproduce

herbie shell --seed 2023255 
(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*))))))