Toniolo and Linder, Equation (13)

Average Accuracy: 45.7% → 53.6%

Time: 43.7s

Precision: binary64

Cost: 27664

Math

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

↓

\[\begin{array}{l} t_1 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;n \leq -3 \cdot 10^{+144}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot t_2 + n \cdot \left(t_2 \cdot \frac{U - U*}{Om}\right)\right) - t\right)\right)}\\ \mathbf{elif}\;n \leq -2.3 \cdot 10^{-113}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \left(\frac{n}{Om} \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq -8.5 \cdot 10^{-204}:\\ \;\;\;\;\sqrt{2 \cdot \left(\mathsf{fma}\left(t_1, n \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right) \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;n \leq 1.45 \cdot 10^{-282}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, t_2, n \cdot \left(\left(U - U*\right) \cdot t_1\right)\right)\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (pow (/ l Om) 2.0)) (t_2 (/ l (/ Om l))))
   (if (<= n -3e+144)
     (sqrt
      (* (* n -2.0) (* U (- (+ (* 2.0 t_2) (* n (* t_2 (/ (- U U*) Om)))) t))))
     (if (<= n -2.3e-113)
       (sqrt (fma 2.0 (* n (* U t)) (* -4.0 (* (/ n Om) (* l (* U l))))))
       (if (<= n -8.5e-204)
         (sqrt
          (*
           2.0
           (* (fma t_1 (* n (- U* U)) (fma (* l (/ l Om)) -2.0 t)) (* n U))))
         (if (<= n 1.45e-282)
           (sqrt (* 2.0 (* U (* n (+ t (* (/ l Om) (* l -2.0)))))))
           (*
            (sqrt (* n 2.0))
            (sqrt (* U (- t (fma 2.0 t_2 (* n (* (- U U*) t_1)))))))))))))

C

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

↓

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = pow((l / Om), 2.0);
	double t_2 = l / (Om / l);
	double tmp;
	if (n <= -3e+144) {
		tmp = sqrt(((n * -2.0) * (U * (((2.0 * t_2) + (n * (t_2 * ((U - U_42_) / Om)))) - t))));
	} else if (n <= -2.3e-113) {
		tmp = sqrt(fma(2.0, (n * (U * t)), (-4.0 * ((n / Om) * (l * (U * l))))));
	} else if (n <= -8.5e-204) {
		tmp = sqrt((2.0 * (fma(t_1, (n * (U_42_ - U)), fma((l * (l / Om)), -2.0, t)) * (n * U))));
	} else if (n <= 1.45e-282) {
		tmp = sqrt((2.0 * (U * (n * (t + ((l / Om) * (l * -2.0)))))));
	} else {
		tmp = sqrt((n * 2.0)) * sqrt((U * (t - fma(2.0, t_2, (n * ((U - U_42_) * t_1))))));
	}
	return tmp;
}

Julia

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

↓

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l / Om) ^ 2.0
	t_2 = Float64(l / Float64(Om / l))
	tmp = 0.0
	if (n <= -3e+144)
		tmp = sqrt(Float64(Float64(n * -2.0) * Float64(U * Float64(Float64(Float64(2.0 * t_2) + Float64(n * Float64(t_2 * Float64(Float64(U - U_42_) / Om)))) - t))));
	elseif (n <= -2.3e-113)
		tmp = sqrt(fma(2.0, Float64(n * Float64(U * t)), Float64(-4.0 * Float64(Float64(n / Om) * Float64(l * Float64(U * l))))));
	elseif (n <= -8.5e-204)
		tmp = sqrt(Float64(2.0 * Float64(fma(t_1, Float64(n * Float64(U_42_ - U)), fma(Float64(l * Float64(l / Om)), -2.0, t)) * Float64(n * U))));
	elseif (n <= 1.45e-282)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t + Float64(Float64(l / Om) * Float64(l * -2.0)))))));
	else
		tmp = Float64(sqrt(Float64(n * 2.0)) * sqrt(Float64(U * Float64(t - fma(2.0, t_2, Float64(n * Float64(Float64(U - U_42_) * t_1)))))));
	end
	return tmp
end

Wolfram

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

↓

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -3e+144], N[Sqrt[N[(N[(n * -2.0), $MachinePrecision] * N[(U * N[(N[(N[(2.0 * t$95$2), $MachinePrecision] + N[(n * N[(t$95$2 * N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, -2.3e-113], N[Sqrt[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * N[(N[(n / Om), $MachinePrecision] * N[(l * N[(U * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, -8.5e-204], N[Sqrt[N[(2.0 * N[(N[(t$95$1 * N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] + N[(N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] * -2.0 + t), $MachinePrecision]), $MachinePrecision] * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 1.45e-282], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t + N[(N[(l / Om), $MachinePrecision] * N[(l * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t - N[(2.0 * t$95$2 + N[(n * N[(N[(U - U$42$), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if n < -2.9999999999999999e144

   1. Initial program 41.7%

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

   2. Simplified 44.6%

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

      [Start] 41.7	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 40.8	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 40.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      sub-neg [=>] 40.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(-\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [<=] 40.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      cancel-sign-sub [<=] 40.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      cancel-sign-sub [=>] 40.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-/l* [=>] 42.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 44.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in l around 0 36.4%

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

   4. Simplified 44.0%

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

      [Start] 36.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \frac{{\ell}^{2} \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]
      unpow2 [=>] 36.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \frac{{\ell}^{2} \cdot \left(U - U*\right)}{\color{blue}{Om \cdot Om}}\right)\right)\right)} \]
      times-frac [=>] 39.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \color{blue}{\left(\frac{{\ell}^{2}}{Om} \cdot \frac{U - U*}{Om}\right)}\right)\right)\right)} \]
      unpow2 [=>] 39.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{Om} \cdot \frac{U - U*}{Om}\right)\right)\right)\right)} \]
      associate-/l* [=>] 44.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(\color{blue}{\frac{\ell}{\frac{Om}{\ell}}} \cdot \frac{U - U*}{Om}\right)\right)\right)\right)} \]

   if -2.9999999999999999e144 < n < -2.30000000000000008e-113

   1. Initial program 52.5%

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

   2. Simplified 57.8%

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

      [Start] 52.5	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 52.8	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 52.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      sub-neg [=>] 52.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(-\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [<=] 52.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      cancel-sign-sub [<=] 52.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      cancel-sign-sub [=>] 52.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-/l* [=>] 58.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 57.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in Om around inf 44.6%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(t \cdot U\right)\right) + -4 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}}} \]

   4. Simplified 48.7%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \left(\frac{n}{Om} \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right)\right)}} \]
      Proof

      [Start] 44.6	\[ \sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right)\right) + -4 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}} \]
      fma-def [=>] 44.6	\[ \sqrt{\color{blue}{\mathsf{fma}\left(2, n \cdot \left(t \cdot U\right), -4 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}} \]
      *-commutative [=>] 44.6	\[ \sqrt{\mathsf{fma}\left(2, n \cdot \color{blue}{\left(U \cdot t\right)}, -4 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)} \]
      associate-/l* [=>] 44.9	\[ \sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \color{blue}{\frac{n}{\frac{Om}{{\ell}^{2} \cdot U}}}\right)} \]
      associate-/r/ [=>] 44.8	\[ \sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \color{blue}{\left(\frac{n}{Om} \cdot \left({\ell}^{2} \cdot U\right)\right)}\right)} \]
      unpow2 [=>] 44.8	\[ \sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \left(\frac{n}{Om} \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot U\right)\right)\right)} \]
      associate-*l* [=>] 48.7	\[ \sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \left(\frac{n}{Om} \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot U\right)\right)}\right)\right)} \]

   if -2.30000000000000008e-113 < n < -8.4999999999999997e-204

   1. Initial program 44.2%

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

   2. Simplified 49.5%

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

      [Start] 44.2	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 44.2	\[ \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 44.2	\[ \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      *-commutative [=>] 44.2	\[ \sqrt{2 \cdot \color{blue}{\left(\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot \left(n \cdot U\right)\right)}} \]

   if -8.4999999999999997e-204 < n < 1.44999999999999999e-282

   1. Initial program 35.7%

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

   2. Taylor expanded in n around 0 35.6%

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

   3. Simplified 47.0%

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

      [Start] 35.6	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot U\right)\right)} \]
      associate-*r* [=>] 40.2	\[ \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right)}} \]
      *-commutative [=>] 40.2	\[ \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
      unpow2 [=>] 40.2	\[ \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right)\right)\right)} \]
      associate-*r/ [<=] 47.0	\[ \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)}\right)\right)\right)} \]
      cancel-sign-sub-inv [=>] 47.0	\[ \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{\left(t + \left(-2\right) \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\right)\right)} \]
      metadata-eval [=>] 47.0	\[ \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \color{blue}{-2} \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)} \]
      *-commutative [<=] 47.0	\[ \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2}\right)\right)\right)} \]
      *-commutative [=>] 47.0	\[ \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot -2\right)\right)\right)} \]
      associate-*l* [=>] 47.0	\[ \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)}\right)\right)\right)} \]

   if 1.44999999999999999e-282 < n

   1. Initial program 46.6%

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

   2. Simplified 50.5%

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

      [Start] 46.6	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 46.9	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 46.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      sub-neg [=>] 46.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(-\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [<=] 46.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      cancel-sign-sub [<=] 46.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      cancel-sign-sub [=>] 46.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-/l* [=>] 51.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 50.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Applied egg-rr 50.3%

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

      [Start] 50.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)} \]
      sqrt-prod [=>] 60.4	\[ \color{blue}{\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)}} \]
      fma-def [=>] 60.4	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \color{blue}{\mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)}\right)} \]
      associate-/r/ [=>] 60.4	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \color{blue}{\frac{\ell}{Om} \cdot \ell}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l/ [=>] 55.6	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \color{blue}{\frac{\ell \cdot \ell}{Om}}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [=>] 55.6	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right) \cdot n}\right)\right)} \]
      associate-*l* [=>] 50.3	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U - U*\right) \cdot n\right)}\right)\right)} \]

   4. Simplified 60.4%

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

      [Start] 50.3	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U - U*\right) \cdot n\right)\right)\right)} \]
      associate-/l* [=>] 54.9	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U - U*\right) \cdot n\right)\right)\right)} \]
      associate-*r* [=>] 60.4	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right) \cdot n}\right)\right)} \]
      *-commutative [=>] 60.4	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)} \]
      *-commutative [=>] 60.4	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, n \cdot \color{blue}{\left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right)\right)} \]

3. Recombined 5 regimes into one program.

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3 \cdot 10^{+144}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(\frac{\ell}{\frac{Om}{\ell}} \cdot \frac{U - U*}{Om}\right)\right) - t\right)\right)}\\ \mathbf{elif}\;n \leq -2.3 \cdot 10^{-113}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \left(\frac{n}{Om} \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq -8.5 \cdot 10^{-204}:\\ \;\;\;\;\sqrt{2 \cdot \left(\mathsf{fma}\left({\left(\frac{\ell}{Om}\right)}^{2}, n \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right) \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;n \leq 1.45 \cdot 10^{-282}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, n \cdot \left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	57.8%
Cost	64524

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := \sqrt{\left(\left(n \cdot 2\right) \cdot U\right) \cdot \left(\left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(U* - U\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)}\\ t_3 := \frac{U - U*}{Om}\\ \mathbf{if}\;t_2 \leq 0:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(t_1 \cdot -2 - n \cdot \frac{\ell}{\frac{Om \cdot \frac{-Om}{\ell}}{U*}}\right)\right)\right)}\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+148}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, t_1, n \cdot \left(\ell \cdot \left(t_3 \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right) \cdot \left(\frac{-2}{Om} - t_3 \cdot \frac{n}{Om}\right)\right)}\\ \end{array} \]

Alternative 2
Accuracy	55.6%
Cost	43528

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

Alternative 3
Accuracy	50.5%
Cost	21721

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;n \leq -1.4 \cdot 10^{+145}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot t_1 + n \cdot \left(t_1 \cdot \frac{U - U*}{Om}\right)\right) - t\right)\right)}\\ \mathbf{elif}\;n \leq -1.2 \cdot 10^{-110}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \left(\frac{n}{Om} \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq -1.65 \cdot 10^{-120}:\\ \;\;\;\;\sqrt{U \cdot \left(U* - U\right)} \cdot \frac{\sqrt{2} \cdot \left(-\ell\right)}{\frac{Om}{n}}\\ \mathbf{elif}\;n \leq 1.3 \cdot 10^{-264}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq 2 \cdot 10^{-63} \lor \neg \left(n \leq 2.05 \cdot 10^{+183}\right):\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, n \cdot \frac{U - U*}{Om \cdot \frac{Om}{\ell \cdot \ell}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(t_1 \cdot -2 - n \cdot \frac{\ell}{\frac{Om \cdot \frac{-Om}{\ell}}{U*}}\right)\right)\right)}\\ \end{array} \]

Alternative 4
Accuracy	51.3%
Cost	14540

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := \frac{U - U*}{Om}\\ t_3 := \frac{-2}{Om} - t_2 \cdot \frac{n}{Om}\\ \mathbf{if}\;\ell \leq -5 \cdot 10^{+150}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right) \cdot t_3\right)}\\ \mathbf{elif}\;\ell \leq -1.95 \cdot 10^{-163}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot t_1 + n \cdot \left(t_1 \cdot t_2\right)\right) - t\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.58 \cdot 10^{+100}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(\left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right) - \frac{n}{\frac{Om}{\frac{\ell \cdot \left(\ell \cdot U*\right)}{Om}}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(n \cdot U\right) \cdot t_3}\right)\\ \end{array} \]

Alternative 5
Accuracy	49.1%
Cost	14152

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;n \leq -6.8 \cdot 10^{+145}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot t_1 + n \cdot \left(t_1 \cdot \frac{U - U*}{Om}\right)\right) - t\right)\right)}\\ \mathbf{elif}\;n \leq -1.2 \cdot 10^{-110}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \left(\frac{n}{Om} \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq -1.65 \cdot 10^{-120}:\\ \;\;\;\;\sqrt{U \cdot \left(U* - U\right)} \cdot \frac{\sqrt{2} \cdot \left(-\ell\right)}{\frac{Om}{n}}\\ \mathbf{elif}\;n \leq 3.45 \cdot 10^{-264}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq 4.2 \cdot 10^{+184}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(t_1 \cdot -2 - n \cdot \frac{\ell}{\frac{Om \cdot \frac{-Om}{\ell}}{U*}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot t}\\ \end{array} \]

Alternative 6
Accuracy	51.0%
Cost	13644

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;n \leq -8 \cdot 10^{-116}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot t_1 + n \cdot \left(t_1 \cdot \frac{U - U*}{Om}\right)\right) - t\right)\right)}\\ \mathbf{elif}\;n \leq 4.6 \cdot 10^{-264}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq 3.8 \cdot 10^{+184}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(t_1 \cdot -2 - n \cdot \frac{\ell}{\frac{Om \cdot \frac{-Om}{\ell}}{U*}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot t}\\ \end{array} \]

Alternative 7
Accuracy	46.3%
Cost	8924

\[\begin{array}{l} t_1 := \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{2 \cdot \ell}{\frac{Om}{\ell}}\right)}\\ t_2 := \sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \frac{n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om}\right) - t\right)\right)\right)}\\ \mathbf{if}\;Om \leq -6.2 \cdot 10^{+139}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;Om \leq -3.7 \cdot 10^{-37}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;Om \leq -2.9 \cdot 10^{-150}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;Om \leq 5.2 \cdot 10^{-215}:\\ \;\;\;\;\ell \cdot \left(\frac{n}{Om} \cdot \sqrt{\left(U - U*\right) \cdot \left(U \cdot -2\right)}\right)\\ \mathbf{elif}\;Om \leq 7.9 \cdot 10^{-171}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;Om \leq 10^{-60}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;Om \leq 1.6 \cdot 10^{+175}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{Om} - \frac{n}{\frac{Om \cdot Om}{U* - U}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)\right)}\\ \end{array} \]

Alternative 8
Accuracy	46.0%
Cost	8720

\[\begin{array}{l} t_1 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{\frac{Om}{\ell}} \cdot -2 - n \cdot \frac{\ell}{\frac{Om \cdot \frac{-Om}{\ell}}{U*}}\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -3.4 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right) \cdot \left(\frac{-2}{Om} - \frac{U - U*}{Om} \cdot \frac{n}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq -9.2 \cdot 10^{-164}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{-134}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{2 \cdot \ell}{\frac{Om}{\ell}}\right)}\\ \mathbf{elif}\;\ell \leq 3.4 \cdot 10^{+164}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\frac{n}{Om} \cdot \sqrt{\left(U - U*\right) \cdot \left(U \cdot -2\right)}\right)\\ \end{array} \]

Alternative 9
Accuracy	45.3%
Cost	8528

\[\begin{array}{l} t_1 := \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{2 \cdot \ell}{\frac{Om}{\ell}}\right)}\\ \mathbf{if}\;Om \leq -9.8 \cdot 10^{-152}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;Om \leq 7.2 \cdot 10^{-206}:\\ \;\;\;\;\ell \cdot \left(\frac{n}{Om} \cdot \sqrt{\left(U - U*\right) \cdot \left(U \cdot -2\right)}\right)\\ \mathbf{elif}\;Om \leq 6.1 \cdot 10^{-61}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;Om \leq 1.08 \cdot 10^{+175}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{Om} - \frac{n}{\frac{Om \cdot Om}{U* - U}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)\right)}\\ \end{array} \]

Alternative 10
Accuracy	46.3%
Cost	8524

\[\begin{array}{l} \mathbf{if}\;\ell \leq -5.1 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right) \cdot \left(\frac{-2}{Om} - \frac{U - U*}{Om} \cdot \frac{n}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq -1.5 \cdot 10^{-163}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{\frac{Om}{\ell}} \cdot -2 - n \cdot \frac{\ell}{\frac{Om \cdot \frac{-Om}{\ell}}{U*}}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.5 \cdot 10^{+133}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(\left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right) - \frac{n}{\frac{Om}{\frac{\ell \cdot \left(\ell \cdot U*\right)}{Om}}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\frac{n}{Om} \cdot \sqrt{\left(U - U*\right) \cdot \left(U \cdot -2\right)}\right)\\ \end{array} \]

Alternative 11
Accuracy	46.8%
Cost	8524

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := \frac{U - U*}{Om}\\ \mathbf{if}\;\ell \leq -1.55 \cdot 10^{+147}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right) \cdot \left(\frac{-2}{Om} - t_2 \cdot \frac{n}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-163}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot t_1 + n \cdot \left(t_1 \cdot t_2\right)\right) - t\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{+130}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(\left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right) - \frac{n}{\frac{Om}{\frac{\ell \cdot \left(\ell \cdot U*\right)}{Om}}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\frac{n}{Om} \cdot \sqrt{\left(U - U*\right) \cdot \left(U \cdot -2\right)}\right)\\ \end{array} \]

Alternative 12
Accuracy	45.6%
Cost	8004

\[\begin{array}{l} \mathbf{if}\;\ell \leq -3.3 \cdot 10^{+115}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right) \cdot \left(\frac{-2}{Om} - \frac{U - U*}{Om} \cdot \frac{n}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq -3.5 \cdot 10^{-164}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \ell \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{2 \cdot \ell}{\frac{Om}{\ell}}\right)}\\ \end{array} \]

Alternative 13
Accuracy	45.3%
Cost	7889

\[\begin{array}{l} t_1 := \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{2 \cdot \ell}{\frac{Om}{\ell}}\right)}\\ \mathbf{if}\;Om \leq -2.15 \cdot 10^{-149}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;Om \leq 1.25 \cdot 10^{-206}:\\ \;\;\;\;\ell \cdot \left(\frac{n}{Om} \cdot \sqrt{\left(U - U*\right) \cdot \left(U \cdot -2\right)}\right)\\ \mathbf{elif}\;Om \leq 7.2 \cdot 10^{+170} \lor \neg \left(Om \leq 1.55 \cdot 10^{+203}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \ell \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \end{array} \]

Alternative 14
Accuracy	45.6%
Cost	7876

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{+214}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(\left(\ell \cdot \left(U \cdot \ell\right)\right) \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -9.2 \cdot 10^{-164}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \ell \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{2 \cdot \ell}{\frac{Om}{\ell}}\right)}\\ \end{array} \]

Alternative 15
Accuracy	39.7%
Cost	7764

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ t_2 := \sqrt{-4 \cdot \left(\frac{n}{Om} \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -3.1 \cdot 10^{+99}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-134}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq -4 \cdot 10^{-237}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{+102}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{+213}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 16
Accuracy	39.4%
Cost	7760

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.8 \cdot 10^{+99}:\\ \;\;\;\;\sqrt{-4 \cdot \left(\frac{n}{Om} \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -2.7 \cdot 10^{-136}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq -4.8 \cdot 10^{-236}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 4 \cdot 10^{+139}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\frac{n}{Om} \cdot \sqrt{\left(U - U*\right) \cdot \left(U \cdot -2\right)}\right)\\ \end{array} \]

Alternative 17
Accuracy	45.0%
Cost	7756

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \ell \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{if}\;Om \leq -2.35 \cdot 10^{-141}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;Om \leq 4.3 \cdot 10^{-203}:\\ \;\;\;\;\ell \cdot \left(\frac{n}{Om} \cdot \sqrt{\left(U - U*\right) \cdot \left(U \cdot -2\right)}\right)\\ \mathbf{elif}\;Om \leq 10^{-24}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)\right)}\\ \end{array} \]

Alternative 18
Accuracy	43.6%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;\ell \leq -8.2 \cdot 10^{+100}:\\ \;\;\;\;\sqrt{-4 \cdot \left(\frac{n}{Om} \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -7.4 \cdot 10^{-136}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)\right)}\\ \end{array} \]

Alternative 19
Accuracy	38.4%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;U \leq 9.2 \cdot 10^{+41}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]

Alternative 20
Accuracy	37.7%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;n \leq -1 \cdot 10^{-111}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 21
Accuracy	37.8%
Cost	6848

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

Reproduce

herbie shell --seed 2023135 
(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*))))))