Toniolo and Linder, Equation (13)

Average Accuracy: 46.2% → 61.0%

Time: 52.0s

Precision: binary64

Cost: 70540

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\ell \cdot \ell}{Om}\\ t_2 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_3 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + t_1 \cdot -2\right) + \left(n \cdot t_2\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t_3 \leq 0:\\ \;\;\;\;\sqrt{U \cdot \left(t - \mathsf{fma}\left(2, t_1, t_2 \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)} \cdot \sqrt{2 \cdot n}\\ \mathbf{elif}\;t_3 \leq 5 \cdot 10^{+151}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;\sqrt{2} \cdot \frac{\sqrt{\mathsf{fma}\left(\ell, -2, \frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om}\right)}}{\sqrt{\frac{Om}{\ell \cdot \left(n \cdot U\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-2}{\frac{\frac{1}{U \cdot \ell} \cdot \frac{Om}{n \cdot \ell}}{2 + \frac{U - U*}{\frac{Om}{n}}}}}\\ \end{array} \]

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 (/ (* l l) Om))
        (t_2 (pow (/ l Om) 2.0))
        (t_3
         (sqrt
          (* (* (* 2.0 n) U) (+ (+ t (* t_1 -2.0)) (* (* n t_2) (- U* U)))))))
   (if (<= t_3 0.0)
     (*
      (sqrt (* U (- t (fma 2.0 t_1 (* t_2 (* n (- U U*)))))))
      (sqrt (* 2.0 n)))
     (if (<= t_3 5e+151)
       t_3
       (if (<= t_3 INFINITY)
         (*
          (sqrt 2.0)
          (/
           (sqrt (fma l -2.0 (/ (* (- U* U) (* n l)) Om)))
           (sqrt (/ Om (* l (* n U))))))
         (sqrt
          (/
           -2.0
           (/
            (* (/ 1.0 (* U l)) (/ Om (* n l)))
            (+ 2.0 (/ (- U U*) (/ Om n)))))))))))

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 = (l * l) / Om;
	double t_2 = pow((l / Om), 2.0);
	double t_3 = sqrt((((2.0 * n) * U) * ((t + (t_1 * -2.0)) + ((n * t_2) * (U_42_ - U)))));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt((U * (t - fma(2.0, t_1, (t_2 * (n * (U - U_42_))))))) * sqrt((2.0 * n));
	} else if (t_3 <= 5e+151) {
		tmp = t_3;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt(2.0) * (sqrt(fma(l, -2.0, (((U_42_ - U) * (n * l)) / Om))) / sqrt((Om / (l * (n * U)))));
	} else {
		tmp = sqrt((-2.0 / (((1.0 / (U * l)) * (Om / (n * l))) / (2.0 + ((U - U_42_) / (Om / n))))));
	}
	return tmp;
}

Julia

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

↓

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(l * l) / Om)
	t_2 = Float64(l / Om) ^ 2.0
	t_3 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t + Float64(t_1 * -2.0)) + Float64(Float64(n * t_2) * Float64(U_42_ - U)))))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = Float64(sqrt(Float64(U * Float64(t - fma(2.0, t_1, Float64(t_2 * Float64(n * Float64(U - U_42_))))))) * sqrt(Float64(2.0 * n)));
	elseif (t_3 <= 5e+151)
		tmp = t_3;
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(2.0) * Float64(sqrt(fma(l, -2.0, Float64(Float64(Float64(U_42_ - U) * Float64(n * l)) / Om))) / sqrt(Float64(Om / Float64(l * Float64(n * U))))));
	else
		tmp = sqrt(Float64(-2.0 / Float64(Float64(Float64(1.0 / Float64(U * l)) * Float64(Om / Float64(n * l))) / Float64(2.0 + Float64(Float64(U - U_42_) / Float64(Om / n))))));
	end
	return tmp
end

Wolfram

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

↓

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t + N[(t$95$1 * -2.0), $MachinePrecision]), $MachinePrecision] + N[(N[(n * t$95$2), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[(N[Sqrt[N[(U * N[(t - N[(2.0 * t$95$1 + N[(t$95$2 * N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5e+151], t$95$3, If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(l * -2.0 + N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(Om / N[(l * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(-2.0 / N[(N[(N[(1.0 / N[(U * l), $MachinePrecision]), $MachinePrecision] * N[(Om / N[(n * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(N[(U - U$42$), $MachinePrecision] / N[(Om / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (sqrt.f64 (*.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 12.1%

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

   2. Simplified 39.7%

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

      [Start] 12.1	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 42.0	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      sub-neg [=>] 42.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(-\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      associate-+l- [=>] 42.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [=>] 42.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(-\left(-\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)\right)} \]
      associate-/l* [=>] 42.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(-\left(-\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)} \]
      remove-double-neg [=>] 42.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)\right)\right)} \]
      associate-*l* [=>] 39.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Applied egg-rr 39.4%

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

      [Start] 39.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)} \]
      sqrt-prod [=>] 38.2	\[ \color{blue}{\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)}} \]
      *-commutative [=>] 38.2	\[ \color{blue}{\sqrt{U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)} \cdot \sqrt{2 \cdot n}} \]
      fma-def [=>] 38.2	\[ \sqrt{U \cdot \left(t - \color{blue}{\mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)}\right)} \cdot \sqrt{2 \cdot n} \]
      associate-/r/ [=>] 38.2	\[ \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \color{blue}{\frac{\ell}{Om} \cdot \ell}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)} \cdot \sqrt{2 \cdot n} \]
      associate-*l/ [=>] 38.1	\[ \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \color{blue}{\frac{\ell \cdot \ell}{Om}}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)} \cdot \sqrt{2 \cdot n} \]
      *-commutative [=>] 38.1	\[ \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right) \cdot n}\right)\right)} \cdot \sqrt{2 \cdot n} \]
      associate-*l* [=>] 39.4	\[ \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U - U*\right) \cdot n\right)}\right)\right)} \cdot \sqrt{2 \cdot n} \]

   if 0.0 < (sqrt.f64 (*.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*))))) < 5.0000000000000002e151

   1. Initial program 97.3%

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

   if 5.0000000000000002e151 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < +inf.0

   1. Initial program 0.8%

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

   2. Simplified 12.4%

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

      [Start] 0.8	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 3.9	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      sub-neg [=>] 3.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]
      associate--l+ [=>] 3.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      *-commutative [=>] 3.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      distribute-rgt-neg-in [=>] 3.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l/ [<=] 14.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 14.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [<=] 14.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [=>] 14.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 11.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]
      unpow2 [=>] 11.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]
      associate-*l* [=>] 12.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in t around 0 12.2%

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

   4. Applied egg-rr 15.4%

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

      [Start] 12.2	\[ \sqrt{2} \cdot \sqrt{\frac{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(n \cdot \left(\ell \cdot U\right)\right)}{Om}} \]
      associate-/l* [=>] 14.8	\[ \sqrt{2} \cdot \sqrt{\color{blue}{\frac{\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}}} \]
      sqrt-div [=>] 12.0	\[ \sqrt{2} \cdot \color{blue}{\frac{\sqrt{\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell}}{\sqrt{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}}} \]
      +-commutative [=>] 12.0	\[ \sqrt{2} \cdot \frac{\sqrt{\color{blue}{-2 \cdot \ell + \frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om}}}}{\sqrt{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}} \]
      *-commutative [=>] 12.0	\[ \sqrt{2} \cdot \frac{\sqrt{\color{blue}{\ell \cdot -2} + \frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om}}}{\sqrt{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}} \]
      fma-def [=>] 12.0	\[ \sqrt{2} \cdot \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\ell, -2, \frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om}\right)}}}{\sqrt{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}} \]
      associate-*r* [=>] 12.8	\[ \sqrt{2} \cdot \frac{\sqrt{\mathsf{fma}\left(\ell, -2, \frac{\color{blue}{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}}{Om}\right)}}{\sqrt{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}} \]
      *-commutative [=>] 12.8	\[ \sqrt{2} \cdot \frac{\sqrt{\mathsf{fma}\left(\ell, -2, \frac{\color{blue}{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}}{Om}\right)}}{\sqrt{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}} \]
      *-commutative [=>] 12.8	\[ \sqrt{2} \cdot \frac{\sqrt{\mathsf{fma}\left(\ell, -2, \frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om}\right)}}{\sqrt{\frac{Om}{\color{blue}{\left(\ell \cdot U\right) \cdot n}}}} \]
      associate-*l* [=>] 15.4	\[ \sqrt{2} \cdot \frac{\sqrt{\mathsf{fma}\left(\ell, -2, \frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om}\right)}}{\sqrt{\frac{Om}{\color{blue}{\ell \cdot \left(U \cdot n\right)}}}} \]

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

   1. Initial program 0.0%

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

   2. Simplified 17.2%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]
      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/ [<=] 8.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 8.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [<=] 8.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [=>] 8.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 9.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]
      unpow2 [=>] 9.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]
      associate-*l* [=>] 17.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in t around 0 24.5%

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

   4. Simplified 31.5%

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

      [Start] 24.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \frac{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)}{Om}} \]
      associate-/l* [=>] 27.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\frac{\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell}{\frac{Om}{\ell \cdot U}}}} \]
      +-commutative [=>] 27.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \frac{\color{blue}{-2 \cdot \ell + \frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om}}}{\frac{Om}{\ell \cdot U}}} \]
      *-commutative [=>] 27.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \frac{\color{blue}{\ell \cdot -2} + \frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om}}{\frac{Om}{\ell \cdot U}}} \]
      associate-*r* [=>] 31.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \frac{\ell \cdot -2 + \frac{\color{blue}{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}}{Om}}{\frac{Om}{\ell \cdot U}}} \]

   5. Taylor expanded in l around -inf 16.7%

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

   6. Simplified 16.2%

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

      [Start] 16.7	\[ \sqrt{-2 \cdot \frac{n \cdot \left({\ell}^{2} \cdot \left(\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot U\right)\right)}{Om}} \]
      associate-*r/ [=>] 16.7	\[ \sqrt{\color{blue}{\frac{-2 \cdot \left(n \cdot \left({\ell}^{2} \cdot \left(\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot U\right)\right)\right)}{Om}}} \]
      associate-/l* [=>] 16.6	\[ \sqrt{\color{blue}{\frac{-2}{\frac{Om}{n \cdot \left({\ell}^{2} \cdot \left(\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot U\right)\right)}}}} \]
      associate-*r* [=>] 17.4	\[ \sqrt{\frac{-2}{\frac{Om}{\color{blue}{\left(n \cdot {\ell}^{2}\right) \cdot \left(\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot U\right)}}}} \]
      *-commutative [=>] 17.4	\[ \sqrt{\frac{-2}{\frac{Om}{\left(n \cdot {\ell}^{2}\right) \cdot \color{blue}{\left(U \cdot \left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)}}}} \]
      associate-*r* [=>] 15.7	\[ \sqrt{\frac{-2}{\frac{Om}{\color{blue}{\left(\left(n \cdot {\ell}^{2}\right) \cdot U\right) \cdot \left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)}}}} \]
      *-commutative [=>] 15.7	\[ \sqrt{\frac{-2}{\frac{Om}{\color{blue}{\left(U \cdot \left(n \cdot {\ell}^{2}\right)\right)} \cdot \left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)}}} \]
      unpow2 [=>] 15.7	\[ \sqrt{\frac{-2}{\frac{Om}{\left(U \cdot \left(n \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)\right) \cdot \left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)}}} \]
      mul-1-neg [=>] 15.7	\[ \sqrt{\frac{-2}{\frac{Om}{\left(U \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right) \cdot \left(2 + \color{blue}{\left(-\frac{n \cdot \left(U* - U\right)}{Om}\right)}\right)}}} \]
      unsub-neg [=>] 15.7	\[ \sqrt{\frac{-2}{\frac{Om}{\left(U \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right) \cdot \color{blue}{\left(2 - \frac{n \cdot \left(U* - U\right)}{Om}\right)}}}} \]
      associate-/l* [=>] 14.6	\[ \sqrt{\frac{-2}{\frac{Om}{\left(U \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right) \cdot \left(2 - \color{blue}{\frac{n}{\frac{Om}{U* - U}}}\right)}}} \]
      associate-/r/ [=>] 16.2	\[ \sqrt{\frac{-2}{\frac{Om}{\left(U \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right) \cdot \left(2 - \color{blue}{\frac{n}{Om} \cdot \left(U* - U\right)}\right)}}} \]

   7. Applied egg-rr 0.6%

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

      [Start] 16.2	\[ \sqrt{\frac{-2}{\frac{Om}{\left(U \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right) \cdot \left(2 - \frac{n}{Om} \cdot \left(U* - U\right)\right)}}} \]
      expm1-log1p-u [=>] 10.7	\[ \sqrt{\frac{-2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{Om}{\left(U \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right) \cdot \left(2 - \frac{n}{Om} \cdot \left(U* - U\right)\right)}\right)\right)}}} \]
      expm1-udef [=>] 0.5	\[ \sqrt{\frac{-2}{\color{blue}{e^{\mathsf{log1p}\left(\frac{Om}{\left(U \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right) \cdot \left(2 - \frac{n}{Om} \cdot \left(U* - U\right)\right)}\right)} - 1}}} \]
      associate-*l* [=>] 0.6	\[ \sqrt{\frac{-2}{e^{\mathsf{log1p}\left(\frac{Om}{\color{blue}{U \cdot \left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(2 - \frac{n}{Om} \cdot \left(U* - U\right)\right)\right)}}\right)} - 1}} \]
      associate-*l* [=>] 0.6	\[ \sqrt{\frac{-2}{e^{\mathsf{log1p}\left(\frac{Om}{U \cdot \color{blue}{\left(n \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(2 - \frac{n}{Om} \cdot \left(U* - U\right)\right)\right)\right)}}\right)} - 1}} \]
      *-commutative [=>] 0.6	\[ \sqrt{\frac{-2}{e^{\mathsf{log1p}\left(\frac{Om}{U \cdot \left(n \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(2 - \color{blue}{\left(U* - U\right) \cdot \frac{n}{Om}}\right)\right)\right)}\right)} - 1}} \]
      clear-num [=>] 0.6	\[ \sqrt{\frac{-2}{e^{\mathsf{log1p}\left(\frac{Om}{U \cdot \left(n \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(2 - \left(U* - U\right) \cdot \color{blue}{\frac{1}{\frac{Om}{n}}}\right)\right)\right)}\right)} - 1}} \]
      un-div-inv [=>] 0.6	\[ \sqrt{\frac{-2}{e^{\mathsf{log1p}\left(\frac{Om}{U \cdot \left(n \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(2 - \color{blue}{\frac{U* - U}{\frac{Om}{n}}}\right)\right)\right)}\right)} - 1}} \]

   8. Simplified 35.5%

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

      [Start] 0.6	\[ \sqrt{\frac{-2}{e^{\mathsf{log1p}\left(\frac{Om}{U \cdot \left(n \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(2 - \frac{U* - U}{\frac{Om}{n}}\right)\right)\right)}\right)} - 1}} \]
      expm1-def [=>] 9.5	\[ \sqrt{\frac{-2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{Om}{U \cdot \left(n \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(2 - \frac{U* - U}{\frac{Om}{n}}\right)\right)\right)}\right)\right)}}} \]
      expm1-log1p [=>] 15.8	\[ \sqrt{\frac{-2}{\color{blue}{\frac{Om}{U \cdot \left(n \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(2 - \frac{U* - U}{\frac{Om}{n}}\right)\right)\right)}}}} \]
      rem-square-sqrt [<=] 7.4	\[ \sqrt{\frac{-2}{\frac{\color{blue}{\sqrt{Om} \cdot \sqrt{Om}}}{U \cdot \left(n \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(2 - \frac{U* - U}{\frac{Om}{n}}\right)\right)\right)}}} \]
      times-frac [=>] 8.8	\[ \sqrt{\frac{-2}{\color{blue}{\frac{\sqrt{Om}}{U} \cdot \frac{\sqrt{Om}}{n \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(2 - \frac{U* - U}{\frac{Om}{n}}\right)\right)}}}} \]
      associate-*r/ [=>] 8.8	\[ \sqrt{\frac{-2}{\color{blue}{\frac{\frac{\sqrt{Om}}{U} \cdot \sqrt{Om}}{n \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(2 - \frac{U* - U}{\frac{Om}{n}}\right)\right)}}}} \]
      unpow2 [<=] 8.8	\[ \sqrt{\frac{-2}{\frac{\frac{\sqrt{Om}}{U} \cdot \sqrt{Om}}{n \cdot \left(\color{blue}{{\ell}^{2}} \cdot \left(2 - \frac{U* - U}{\frac{Om}{n}}\right)\right)}}} \]
      associate-*r* [=>] 9.7	\[ \sqrt{\frac{-2}{\frac{\frac{\sqrt{Om}}{U} \cdot \sqrt{Om}}{\color{blue}{\left(n \cdot {\ell}^{2}\right) \cdot \left(2 - \frac{U* - U}{\frac{Om}{n}}\right)}}}} \]
      unpow2 [=>] 9.7	\[ \sqrt{\frac{-2}{\frac{\frac{\sqrt{Om}}{U} \cdot \sqrt{Om}}{\left(n \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \left(2 - \frac{U* - U}{\frac{Om}{n}}\right)}}} \]
      times-frac [=>] 9.1	\[ \sqrt{\frac{-2}{\color{blue}{\frac{\frac{\sqrt{Om}}{U}}{n \cdot \left(\ell \cdot \ell\right)} \cdot \frac{\sqrt{Om}}{2 - \frac{U* - U}{\frac{Om}{n}}}}}} \]
      associate-/r* [<=] 7.8	\[ \sqrt{\frac{-2}{\color{blue}{\frac{\sqrt{Om}}{U \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}} \cdot \frac{\sqrt{Om}}{2 - \frac{U* - U}{\frac{Om}{n}}}}} \]
      *-commutative [<=] 7.8	\[ \sqrt{\frac{-2}{\color{blue}{\frac{\sqrt{Om}}{2 - \frac{U* - U}{\frac{Om}{n}}} \cdot \frac{\sqrt{Om}}{U \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}}}} \]
      associate-*l/ [=>] 8.0	\[ \sqrt{\frac{-2}{\color{blue}{\frac{\sqrt{Om} \cdot \frac{\sqrt{Om}}{U \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}}{2 - \frac{U* - U}{\frac{Om}{n}}}}}} \]

   9. Applied egg-rr 48.8%

      \[\leadsto \sqrt{\frac{-2}{\frac{\color{blue}{\frac{1}{U \cdot \ell} \cdot \frac{Om}{\ell \cdot n}}}{2 - \frac{U* - U}{\frac{Om}{n}}}}} \]
      Proof

      [Start] 35.5	\[ \sqrt{\frac{-2}{\frac{\frac{Om}{U \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)}}{2 - \frac{U* - U}{\frac{Om}{n}}}}} \]
      *-un-lft-identity [=>] 35.5	\[ \sqrt{\frac{-2}{\frac{\frac{\color{blue}{1 \cdot Om}}{U \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)}}{2 - \frac{U* - U}{\frac{Om}{n}}}}} \]
      associate-*r* [=>] 43.6	\[ \sqrt{\frac{-2}{\frac{\frac{1 \cdot Om}{\color{blue}{\left(U \cdot \ell\right) \cdot \left(n \cdot \ell\right)}}}{2 - \frac{U* - U}{\frac{Om}{n}}}}} \]
      times-frac [=>] 48.8	\[ \sqrt{\frac{-2}{\frac{\color{blue}{\frac{1}{U \cdot \ell} \cdot \frac{Om}{n \cdot \ell}}}{2 - \frac{U* - U}{\frac{Om}{n}}}}} \]
      *-commutative [=>] 48.8	\[ \sqrt{\frac{-2}{\frac{\frac{1}{U \cdot \ell} \cdot \frac{Om}{\color{blue}{\ell \cdot n}}}{2 - \frac{U* - U}{\frac{Om}{n}}}}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)} \cdot \sqrt{2 \cdot n}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 5 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{2} \cdot \frac{\sqrt{\mathsf{fma}\left(\ell, -2, \frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om}\right)}}{\sqrt{\frac{Om}{\ell \cdot \left(n \cdot U\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-2}{\frac{\frac{1}{U \cdot \ell} \cdot \frac{Om}{n \cdot \ell}}{2 + \frac{U - U*}{\frac{Om}{n}}}}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	59.8%
Cost	70540

\[\begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot t\right)} \cdot \sqrt{n}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+151}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\sqrt{2} \cdot \frac{\sqrt{\mathsf{fma}\left(\ell, -2, \frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om}\right)}}{\sqrt{\frac{Om}{\ell \cdot \left(n \cdot U\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-2}{\frac{\frac{1}{U \cdot \ell} \cdot \frac{Om}{n \cdot \ell}}{2 + \frac{U - U*}{\frac{Om}{n}}}}}\\ \end{array} \]

Alternative 2
Accuracy	60.9%
Cost	70540

\[\begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right), t\right)}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+151}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\sqrt{2} \cdot \frac{\sqrt{\mathsf{fma}\left(\ell, -2, \frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om}\right)}}{\sqrt{\frac{Om}{\ell \cdot \left(n \cdot U\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-2}{\frac{\frac{1}{U \cdot \ell} \cdot \frac{Om}{n \cdot \ell}}{2 + \frac{U - U*}{\frac{Om}{n}}}}}\\ \end{array} \]

Alternative 3
Accuracy	61.0%
Cost	70540

\[\begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right), t\right)}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+151}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\sqrt{2} \cdot \frac{\sqrt{\mathsf{fma}\left(\ell, -2, \frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om}\right)}}{\sqrt{\frac{Om}{\ell \cdot \left(n \cdot U\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-2}{\frac{\frac{1}{U \cdot \ell} \cdot \frac{Om}{n \cdot \ell}}{2 + \frac{U - U*}{\frac{Om}{n}}}}}\\ \end{array} \]

Alternative 4
Accuracy	54.7%
Cost	14992

\[\begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;\ell \leq -2.5 \cdot 10^{+51}:\\ \;\;\;\;\sqrt{\frac{-2}{\frac{\frac{Om}{\left(n \cdot \ell\right) \cdot \left(U \cdot \ell\right)}}{2 + \frac{U - U*}{\frac{Om}{n}}}}}\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{-257}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{-228}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{elif}\;\ell \leq 10^{-70}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 9.6 \cdot 10^{-66}:\\ \;\;\;\;\left(\sqrt{2} \cdot \frac{n}{\frac{Om}{\ell}}\right) \cdot \left(-\sqrt{U \cdot U*}\right)\\ \mathbf{elif}\;\ell \leq 3.8 \cdot 10^{+99}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 - \frac{\left(U - U*\right) \cdot \left(n \cdot \ell\right)}{Om}}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{Om} \cdot \left(U \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)}\right)\\ \end{array} \]

Alternative 5
Accuracy	51.7%
Cost	14284

\[\begin{array}{l} \mathbf{if}\;\ell \leq -7.2 \cdot 10^{+33}:\\ \;\;\;\;\sqrt{\frac{-2}{\frac{\frac{Om}{\left(n \cdot \ell\right) \cdot \left(U \cdot \ell\right)}}{2 + \frac{U - U*}{\frac{Om}{n}}}}}\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{-260}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(2 \cdot t\right)}\\ \mathbf{elif}\;\ell \leq 6.9 \cdot 10^{+100}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \frac{\ell \cdot \left(2 \cdot \ell - \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{Om} \cdot \left(U \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)}\right)\\ \end{array} \]

Alternative 6
Accuracy	53.6%
Cost	13908

\[\begin{array}{l} t_1 := \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\\ t_2 := \frac{Om}{n \cdot \left(U \cdot \ell\right)}\\ t_3 := n \cdot \left(U \cdot t\right)\\ t_4 := \sqrt{2 \cdot \left(t_3 + \frac{\ell \cdot -2 - \frac{\left(U - U*\right) \cdot \left(n \cdot \ell\right)}{Om}}{t_2}\right)}\\ \mathbf{if}\;U \leq -1 \cdot 10^{+39}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)}\\ \mathbf{elif}\;U \leq -9.5 \cdot 10^{-166}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;U \leq -3.1 \cdot 10^{-247}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \frac{\ell \cdot \left(2 \cdot \ell - t_1\right)}{Om}\right)\right)}\\ \mathbf{elif}\;U \leq 4 \cdot 10^{-233}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_3 + \frac{\ell \cdot -2 + t_1}{t_2}\right)}\\ \mathbf{elif}\;U \leq 90000000000000:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U + U} \cdot \sqrt{n \cdot t}\\ \end{array} \]

Alternative 7
Accuracy	46.7%
Cost	8272

\[\begin{array}{l} t_1 := \sqrt{\left(\ell \cdot \left(-2 + \left(U* - U\right) \cdot \frac{n}{Om}\right)\right) \cdot \left(2 \cdot \left(n \cdot \frac{\ell}{\frac{Om}{U}}\right)\right)}\\ \mathbf{if}\;\ell \leq -2.05 \cdot 10^{+36}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{-257}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(2 \cdot t\right)}\\ \mathbf{elif}\;\ell \leq 5.4 \cdot 10^{+98}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \ell \cdot \left(\frac{\ell}{Om} \cdot -2\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.8 \cdot 10^{+115}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Accuracy	49.4%
Cost	8268

\[\begin{array}{l} \mathbf{if}\;\ell \leq -3.4 \cdot 10^{+34}:\\ \;\;\;\;\sqrt{\frac{-2}{\frac{\frac{Om}{\left(n \cdot \ell\right) \cdot \left(U \cdot \ell\right)}}{2 + \frac{U - U*}{\frac{Om}{n}}}}}\\ \mathbf{elif}\;\ell \leq 3.5 \cdot 10^{-260}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(2 \cdot t\right)}\\ \mathbf{elif}\;\ell \leq 5.6 \cdot 10^{+99}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \frac{\ell \cdot \left(2 \cdot \ell - \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot \ell\right) \cdot \frac{n \cdot -2}{\frac{-Om}{\ell \cdot \left(-2 + \left(U* - U\right) \cdot \frac{n}{Om}\right)}}}\\ \end{array} \]

Alternative 9
Accuracy	48.2%
Cost	8268

\[\begin{array}{l} \mathbf{if}\;\ell \leq -9.6 \cdot 10^{+31}:\\ \;\;\;\;\sqrt{\frac{-2}{\frac{\frac{Om}{\left(n \cdot \ell\right) \cdot \left(U \cdot \ell\right)}}{2 + \frac{U - U*}{\frac{Om}{n}}}}}\\ \mathbf{elif}\;\ell \leq 3.4 \cdot 10^{-260}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(2 \cdot t\right)}\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{+132}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \frac{\ell \cdot \left(2 \cdot \ell - \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \frac{\ell \cdot -2 - \frac{n \cdot \left(U \cdot \ell\right)}{Om}}{\frac{Om}{U \cdot \ell}}}\\ \end{array} \]

Alternative 10
Accuracy	49.7%
Cost	8268

\[\begin{array}{l} t_1 := 2 + \frac{U - U*}{\frac{Om}{n}}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+32}:\\ \;\;\;\;\sqrt{\frac{-2}{\frac{\frac{Om}{\left(n \cdot \ell\right) \cdot \left(U \cdot \ell\right)}}{t_1}}}\\ \mathbf{elif}\;\ell \leq 7 \cdot 10^{-258}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(2 \cdot t\right)}\\ \mathbf{elif}\;\ell \leq 4.5 \cdot 10^{+99}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \frac{\ell \cdot \left(2 \cdot \ell - \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-2}{\frac{\frac{1}{U \cdot \ell} \cdot \frac{Om}{n \cdot \ell}}{t_1}}}\\ \end{array} \]

Alternative 11
Accuracy	45.3%
Cost	8072

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.85 \cdot 10^{+35}:\\ \;\;\;\;\sqrt{\frac{-2}{\frac{\frac{Om}{\left(n \cdot \ell\right) \cdot \left(U \cdot \ell\right)}}{2 + \frac{U - U*}{\frac{Om}{n}}}}}\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{-26}:\\ \;\;\;\;\sqrt{\left(n \cdot t\right) \cdot \left(2 \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot \ell\right) \cdot \frac{n \cdot -2}{\frac{-Om}{\ell \cdot \left(-2 + \left(U* - U\right) \cdot \frac{n}{Om}\right)}}}\\ \end{array} \]

Alternative 12
Accuracy	45.4%
Cost	8009

\[\begin{array}{l} \mathbf{if}\;\ell \leq -9.5 \cdot 10^{+36} \lor \neg \left(\ell \leq 2 \cdot 10^{-26}\right):\\ \;\;\;\;\sqrt{\frac{-2}{\frac{\frac{Om}{\left(n \cdot \ell\right) \cdot \left(U \cdot \ell\right)}}{2 + \frac{U - U*}{\frac{Om}{n}}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot t\right) \cdot \left(2 \cdot U\right)}\\ \end{array} \]

Alternative 13
Accuracy	44.6%
Cost	7876

\[\begin{array}{l} \mathbf{if}\;\ell \leq -9 \cdot 10^{+33}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{2 + \frac{U - U*}{\frac{Om}{n}}}{\frac{Om}{U \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)}}}\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{-259}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(2 \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \ell \cdot \left(\frac{\ell}{Om} \cdot -2\right)\right)\right)\right)}\\ \end{array} \]

Alternative 14
Accuracy	45.1%
Cost	7748

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.2 \cdot 10^{+125}:\\ \;\;\;\;\sqrt{\frac{-2}{\frac{\frac{Om}{U \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)}}{2 + n \cdot \frac{U}{Om}}}}\\ \mathbf{elif}\;\ell \leq 3.2 \cdot 10^{-257}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \ell \cdot \left(\frac{\ell}{Om} \cdot -2\right)\right)\right)\right)}\\ \end{array} \]

Alternative 15
Accuracy	43.5%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{+37}:\\ \;\;\;\;\sqrt{\frac{-2}{\frac{Om}{U \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)} \cdot 0.5}}\\ \mathbf{elif}\;\ell \leq 1.02 \cdot 10^{-259}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(2 \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \ell \cdot \left(\frac{\ell}{Om} \cdot -2\right)\right)\right)\right)}\\ \end{array} \]

Alternative 16
Accuracy	45.1%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;\ell \leq -6.2 \cdot 10^{+126}:\\ \;\;\;\;\sqrt{\frac{-2}{\frac{Om}{U \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)} \cdot 0.5}}\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{-260}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \ell \cdot \left(\frac{\ell}{Om} \cdot -2\right)\right)\right)\right)}\\ \end{array} \]

Alternative 17
Accuracy	41.8%
Cost	7497

\[\begin{array}{l} \mathbf{if}\;\ell \leq -4.1 \cdot 10^{+36} \lor \neg \left(\ell \leq 5.2 \cdot 10^{+62}\right):\\ \;\;\;\;\sqrt{\frac{-2}{\frac{Om}{U \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)} \cdot 0.5}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot t\right) \cdot \left(2 \cdot U\right)}\\ \end{array} \]

Alternative 18
Accuracy	37.8%
Cost	7432

\[\begin{array}{l} \mathbf{if}\;Om \leq -4.3 \cdot 10^{-142}:\\ \;\;\;\;\sqrt{\left(n \cdot t\right) \cdot \left(2 \cdot U\right)}\\ \mathbf{elif}\;Om \leq 1.85 \cdot 10^{-235}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(U \cdot U*\right)}}{Om} \cdot \left(n \cdot \left(-\ell\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(2 \cdot t\right)}\\ \end{array} \]

Alternative 19
Accuracy	39.9%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;U \leq -5.5 \cdot 10^{-59} \lor \neg \left(U \leq 2.3 \cdot 10^{-97}\right):\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(2 \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \end{array} \]

Alternative 20
Accuracy	38.3%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;U \leq 7.2 \cdot 10^{-289} \lor \neg \left(U \leq 3.9 \cdot 10^{+54}\right):\\ \;\;\;\;\sqrt{\left(n \cdot t\right) \cdot \left(2 \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \end{array} \]

Alternative 21
Accuracy	37.4%
Cost	6848

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

Reproduce

herbie shell --seed 2023136 
(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*))))))