Toniolo and Linder, Equation (13)

Average Error: 34.6 → 23.1

Time: 48.1s

Precision: binary64

Cost: 70732

Math

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

↓

\[\begin{array}{l} t_1 := \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\\ t_2 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) - t_1\right)}\\ \mathbf{if}\;t_2 \leq 0:\\ \;\;\;\;{\left(\sqrt[3]{U \cdot t} \cdot \sqrt[3]{2 \cdot n}\right)}^{1.5}\\ \mathbf{elif}\;t_2 \leq 4.1 \cdot 10^{+148}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;\sqrt{t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, t_1\right)} \cdot \sqrt{2 \cdot \left(n \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{-2}{Om} + \frac{n}{Om} \cdot \frac{U* - U}{Om}\right) \cdot \left(2 \cdot \left(n \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* n (pow (/ l Om) 2.0)) (- U U*)))
        (t_2 (sqrt (* (* (* 2.0 n) U) (- (+ t (* (/ (* l l) Om) -2.0)) t_1)))))
   (if (<= t_2 0.0)
     (pow (* (cbrt (* U t)) (cbrt (* 2.0 n))) 1.5)
     (if (<= t_2 4.1e+148)
       t_2
       (if (<= t_2 INFINITY)
         (* (sqrt (- t (fma 2.0 (/ l (/ Om l)) t_1))) (sqrt (* 2.0 (* n U))))
         (sqrt
          (*
           (+ (/ -2.0 Om) (* (/ n Om) (/ (- U* U) Om)))
           (* 2.0 (* n (* l (* U l)))))))))))

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 = (n * pow((l / Om), 2.0)) * (U - U_42_);
	double t_2 = sqrt((((2.0 * n) * U) * ((t + (((l * l) / Om) * -2.0)) - t_1)));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = pow((cbrt((U * t)) * cbrt((2.0 * n))), 1.5);
	} else if (t_2 <= 4.1e+148) {
		tmp = t_2;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt((t - fma(2.0, (l / (Om / l)), t_1))) * sqrt((2.0 * (n * U)));
	} else {
		tmp = sqrt((((-2.0 / Om) + ((n / Om) * ((U_42_ - U) / Om))) * (2.0 * (n * (l * (U * l))))));
	}
	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(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_))
	t_2 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t + Float64(Float64(Float64(l * l) / Om) * -2.0)) - t_1)))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(cbrt(Float64(U * t)) * cbrt(Float64(2.0 * n))) ^ 1.5;
	elseif (t_2 <= 4.1e+148)
		tmp = t_2;
	elseif (t_2 <= Inf)
		tmp = Float64(sqrt(Float64(t - fma(2.0, Float64(l / Float64(Om / l)), t_1))) * sqrt(Float64(2.0 * Float64(n * U))));
	else
		tmp = sqrt(Float64(Float64(Float64(-2.0 / Om) + Float64(Float64(n / Om) * Float64(Float64(U_42_ - U) / Om))) * Float64(2.0 * Float64(n * Float64(l * Float64(U * l))))));
	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[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t + N[(N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[Power[N[(N[Power[N[(U * t), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(2.0 * n), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], If[LessEqual[t$95$2, 4.1e+148], t$95$2, If[LessEqual[t$95$2, Infinity], N[(N[Sqrt[N[(t - N[(2.0 * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(N[(-2.0 / Om), $MachinePrecision] + N[(N[(n / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(n * N[(l * N[(U * l), $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 56.1

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

   2. Simplified 40.3

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

      [Start] 56.1	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 38.3	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      cancel-sign-sub-inv [=>] 38.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)} \]
      +-commutative [=>] 38.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(\left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\right)} \]
      distribute-lft-neg-in [<=] 38.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(-\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)} \]
      associate-*l* [=>] 40.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(-\color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)} \]
      distribute-rgt-neg-in [=>] 40.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{n \cdot \left(-{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)} \]
      fma-def [=>] 40.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\mathsf{fma}\left(n, -{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right), t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)}\right)} \]

   3. Taylor expanded in l around 0 42.1

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

   4. Simplified 56.1

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

      [Start] 42.1	\[ \sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)} \]
      associate-*r* [=>] 42.1	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(t \cdot U\right)}} \]
      *-commutative [=>] 42.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(U \cdot t\right)}} \]
      associate-*r* [=>] 56.1	\[ \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}} \]
      *-commutative [=>] 56.1	\[ \sqrt{\left(\color{blue}{\left(n \cdot 2\right)} \cdot U\right) \cdot t} \]
      associate-*l* [=>] 56.1	\[ \sqrt{\color{blue}{\left(n \cdot \left(2 \cdot U\right)\right)} \cdot t} \]

   5. Applied egg-rr 42.3

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

   6. Applied egg-rr 23.4

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

   7. Simplified 23.4

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

      [Start] 23.4	\[ {\left(\sqrt[3]{n \cdot 2} \cdot \sqrt[3]{U \cdot t}\right)}^{1.5} \]
      *-commutative [=>] 23.4	\[ {\color{blue}{\left(\sqrt[3]{U \cdot t} \cdot \sqrt[3]{n \cdot 2}\right)}}^{1.5} \]
      *-commutative [<=] 23.4	\[ {\left(\sqrt[3]{\color{blue}{t \cdot U}} \cdot \sqrt[3]{n \cdot 2}\right)}^{1.5} \]
      *-commutative [=>] 23.4	\[ {\left(\sqrt[3]{t \cdot U} \cdot \sqrt[3]{\color{blue}{2 \cdot n}}\right)}^{1.5} \]

   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*))))) < 4.0999999999999998e148

   1. Initial program 1.8

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

   if 4.0999999999999998e148 < (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 62.7

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

   2. Simplified 52.9

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

      [Start] 62.7	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 61.4	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 61.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      sub-neg [=>] 61.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(-\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [<=] 61.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      cancel-sign-sub [<=] 61.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      cancel-sign-sub [=>] 61.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-/l* [=>] 53.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 52.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Applied egg-rr 56.6

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

   4. Simplified 48.0

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

      [Start] 56.6	\[ \sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U - U*\right) \cdot n\right)\right)} \]
      *-commutative [=>] 56.6	\[ \color{blue}{\sqrt{t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U - U*\right) \cdot n\right)\right)} \cdot \sqrt{2 \cdot \left(n \cdot U\right)}} \]
      associate-/l* [=>] 49.8	\[ \sqrt{t - \mathsf{fma}\left(2, \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U - U*\right) \cdot n\right)\right)} \cdot \sqrt{2 \cdot \left(n \cdot U\right)} \]
      *-commutative [<=] 49.8	\[ \sqrt{t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \color{blue}{\left(n \cdot \left(U - U*\right)\right)}\right)} \cdot \sqrt{2 \cdot \left(n \cdot U\right)} \]
      associate-*r* [=>] 48.0	\[ \sqrt{t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U - U*\right)}\right)} \cdot \sqrt{2 \cdot \left(n \cdot U\right)} \]
      *-commutative [<=] 48.0	\[ \sqrt{t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)} \cdot \left(U - U*\right)\right)} \cdot \sqrt{2 \cdot \left(n \cdot U\right)} \]
      *-commutative [=>] 48.0	\[ \sqrt{t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \color{blue}{\left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right)} \cdot \sqrt{2 \cdot \left(n \cdot U\right)} \]

   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 64.0

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

   2. Simplified 58.1

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

      [Start] 64.0	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 64.0	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 64.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      sub-neg [=>] 64.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(-\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [<=] 64.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      cancel-sign-sub [<=] 64.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      cancel-sign-sub [=>] 64.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-/l* [=>] 58.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 58.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in l around inf 59.0

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

   4. Simplified 44.4

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

      [Start] 59.0	\[ \sqrt{-2 \cdot \left(\left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(n \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      *-commutative [=>] 59.0	\[ \sqrt{-2 \cdot \color{blue}{\left(\left(n \cdot \left({\ell}^{2} \cdot U\right)\right) \cdot \left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)\right)}} \]
      associate-*r* [=>] 59.0	\[ \sqrt{\color{blue}{\left(-2 \cdot \left(n \cdot \left({\ell}^{2} \cdot U\right)\right)\right) \cdot \left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)}} \]
      unpow2 [=>] 59.0	\[ \sqrt{\left(-2 \cdot \left(n \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot U\right)\right)\right) \cdot \left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)} \]
      associate-*l* [=>] 48.2	\[ \sqrt{\left(-2 \cdot \left(n \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot U\right)\right)}\right)\right) \cdot \left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)} \]
      *-commutative [=>] 48.2	\[ \sqrt{\left(-2 \cdot \left(n \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right)\right) \cdot \left(\frac{\color{blue}{\left(U - U*\right) \cdot n}}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)} \]
      unpow2 [=>] 48.2	\[ \sqrt{\left(-2 \cdot \left(n \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right)\right) \cdot \left(\frac{\left(U - U*\right) \cdot n}{\color{blue}{Om \cdot Om}} + 2 \cdot \frac{1}{Om}\right)} \]
      times-frac [=>] 44.4	\[ \sqrt{\left(-2 \cdot \left(n \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right)\right) \cdot \left(\color{blue}{\frac{U - U*}{Om} \cdot \frac{n}{Om}} + 2 \cdot \frac{1}{Om}\right)} \]
      associate-*r/ [=>] 44.4	\[ \sqrt{\left(-2 \cdot \left(n \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right)\right) \cdot \left(\frac{U - U*}{Om} \cdot \frac{n}{Om} + \color{blue}{\frac{2 \cdot 1}{Om}}\right)} \]
      metadata-eval [=>] 44.4	\[ \sqrt{\left(-2 \cdot \left(n \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right)\right) \cdot \left(\frac{U - U*}{Om} \cdot \frac{n}{Om} + \frac{\color{blue}{2}}{Om}\right)} \]

3. Recombined 4 regimes into one program.

4. Final simplification 23.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq 0:\\ \;\;\;\;{\left(\sqrt[3]{U \cdot t} \cdot \sqrt[3]{2 \cdot n}\right)}^{1.5}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq 4.1 \cdot 10^{+148}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \cdot \sqrt{2 \cdot \left(n \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{-2}{Om} + \frac{n}{Om} \cdot \frac{U* - U}{Om}\right) \cdot \left(2 \cdot \left(n \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	22.9
Cost	63500

\[\begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;{\left(\sqrt[3]{U \cdot t} \cdot \sqrt[3]{2 \cdot n}\right)}^{1.5}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+152}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\left|\frac{n \cdot \sqrt{2}}{\frac{\frac{Om}{\ell}}{\sqrt{U \cdot U*}}}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{-2}{Om} + \frac{n}{Om} \cdot \frac{U* - U}{Om}\right) \cdot \left(2 \cdot \left(n \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \end{array} \]

Alternative 2
Error	24.3
Cost	57996

\[\begin{array}{l} t_1 := \frac{-2}{Om} + \frac{n}{Om} \cdot \frac{U* - U}{Om}\\ t_2 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t_2 \leq 0:\\ \;\;\;\;{\left(\sqrt[3]{U \cdot \left(2 \cdot t\right)} \cdot \sqrt[3]{n}\right)}^{1.5}\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+152}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + \ell \cdot \left(\ell \cdot t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t_1 \cdot \left(2 \cdot \left(n \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \end{array} \]

Alternative 3
Error	24.3
Cost	57996

\[\begin{array}{l} t_1 := \frac{-2}{Om} + \frac{n}{Om} \cdot \frac{U* - U}{Om}\\ t_2 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t_2 \leq 0:\\ \;\;\;\;{\left(\sqrt[3]{U \cdot t} \cdot \sqrt[3]{2 \cdot n}\right)}^{1.5}\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+152}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + \ell \cdot \left(\ell \cdot t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t_1 \cdot \left(2 \cdot \left(n \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \end{array} \]

Alternative 4
Error	26.4
Cost	38796

\[\begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ t_2 := \frac{-2}{Om} + \frac{n}{Om} \cdot \frac{U* - U}{Om}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \left(\frac{\ell}{Om} \cdot \frac{\ell}{\frac{Om}{n}}\right) \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+304}:\\ \;\;\;\;\sqrt{t_1}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + \ell \cdot \left(\ell \cdot t_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t_2 \cdot \left(2 \cdot \left(n \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \end{array} \]

Alternative 5
Error	27.7
Cost	30728

\[\begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \left(\frac{\ell}{Om} \cdot \frac{\ell}{\frac{Om}{n}}\right) \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+304}:\\ \;\;\;\;\sqrt{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{-2}{Om} + \frac{n}{Om} \cdot \frac{U* - U}{Om}\right) \cdot \left(2 \cdot \left(n \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \end{array} \]

Alternative 6
Error	29.1
Cost	14992

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := \sqrt{-2 \cdot \left(\left(n \cdot U\right) \cdot \left(\left(2 \cdot t_1 - \frac{\ell \cdot \left(n \cdot \ell\right)}{Om} \cdot \frac{U*}{Om}\right) - t\right)\right)}\\ t_3 := t_1 \cdot -2\\ \mathbf{if}\;\ell \leq -2.1 \cdot 10^{+183}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} + \frac{-2}{Om}\right)\right)} \cdot \left(\ell \cdot \left(-\sqrt{2}\right)\right)\\ \mathbf{elif}\;\ell \leq -9.8 \cdot 10^{+22}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(t_3 + n \cdot \left(t_1 \cdot \frac{U* - U}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -3.9 \cdot 10^{-206}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 2:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) + t_3\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{+122}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{\frac{Om \cdot Om}{U* - U}} + \frac{-2}{Om}\right)}\\ \end{array} \]

Alternative 7
Error	30.2
Cost	13776

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := 2 \cdot t_1\\ \mathbf{if}\;U \leq -2.1 \cdot 10^{-51}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + t_1 \cdot -2\right)}\\ \mathbf{elif}\;U \leq 2.25 \cdot 10^{+89}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(t_2 - \left(\frac{\ell}{Om} \cdot \frac{\ell}{\frac{Om}{n}}\right) \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{elif}\;U \leq 2 \cdot 10^{+174}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(n \cdot U\right) \cdot \left(\left(t_2 - \frac{\ell \cdot \left(n \cdot \ell\right)}{Om} \cdot \frac{U*}{Om}\right) - t\right)\right)}\\ \mathbf{elif}\;U \leq 7 \cdot 10^{+227}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \frac{2 \cdot \ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot t} \cdot \sqrt{2 \cdot U}\\ \end{array} \]

Alternative 8
Error	30.6
Cost	13512

\[\begin{array}{l} \mathbf{if}\;t \leq -7.4 \cdot 10^{+73}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \ell \cdot \frac{\ell \cdot -2}{Om}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 8.4 \cdot 10^{+91}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \left(\frac{\ell}{Om} \cdot \frac{\ell}{\frac{Om}{n}}\right) \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{t}\\ \end{array} \]

Alternative 9
Error	33.4
Cost	8920

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := 2 \cdot t_1\\ t_3 := \sqrt{-2 \cdot \left(\left(n \cdot U\right) \cdot \left(\left(t_2 - \frac{\ell \cdot \left(n \cdot \ell\right)}{Om} \cdot \frac{U*}{Om}\right) - t\right)\right)}\\ t_4 := \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \ell \cdot \frac{\ell \cdot -2}{Om}\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -2.4 \cdot 10^{+45}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \frac{2 \cdot \ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -3.6 \cdot 10^{-206}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\ell \leq 1.9 \cdot 10^{-231}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{-220}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(t_2 - \frac{n}{\frac{\frac{\frac{Om}{\frac{\ell}{Om}}}{\ell}}{U*}}\right) - t\right)\right)}\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{-120}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(t_1 \cdot -2 + n \cdot \left(t_1 \cdot \frac{U* - U}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 7 \cdot 10^{+107}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\ell \leq 4.5 \cdot 10^{+202}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right) \cdot \left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) + \frac{-2}{Om}\right)}\\ \end{array} \]

Alternative 10
Error	33.7
Cost	8656

\[\begin{array}{l} t_1 := \sqrt{-2 \cdot \left(\left(n \cdot U\right) \cdot \left(\left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \frac{\ell \cdot \left(n \cdot \ell\right)}{Om} \cdot \frac{U*}{Om}\right) - t\right)\right)}\\ \mathbf{if}\;\ell \leq -2.3 \cdot 10^{+45}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \frac{2 \cdot \ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -3.7 \cdot 10^{-206}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 4.9 \cdot 10^{-175}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \ell \cdot \frac{\ell \cdot -2}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.8 \cdot 10^{+136}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right) \cdot \left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) + \frac{-2}{Om}\right)}\\ \end{array} \]

Alternative 11
Error	33.0
Cost	8656

\[\begin{array}{l} t_1 := 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := \sqrt{-2 \cdot \left(\left(n \cdot U\right) \cdot \left(\left(t_1 - \frac{\ell \cdot \left(n \cdot \ell\right)}{Om} \cdot \frac{U*}{Om}\right) - t\right)\right)}\\ \mathbf{if}\;\ell \leq -2.4 \cdot 10^{+45}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \frac{2 \cdot \ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -3.8 \cdot 10^{-206}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{-166}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(t_1 - \frac{n}{\frac{\frac{\frac{Om}{\frac{\ell}{Om}}}{\ell}}{U*}}\right) - t\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.1 \cdot 10^{+136}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right) \cdot \left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) + \frac{-2}{Om}\right)}\\ \end{array} \]

Alternative 12
Error	33.0
Cost	8652

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := 2 \cdot t_1\\ \mathbf{if}\;\ell \leq -2.3 \cdot 10^{+45}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \frac{2 \cdot \ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -3.85 \cdot 10^{-206}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(n \cdot U\right) \cdot \left(\left(t_2 - \frac{\ell \cdot \left(n \cdot \ell\right)}{Om} \cdot \frac{U*}{Om}\right) - t\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{-231}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \ell \cdot \frac{\ell \cdot -2}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(t_2 - \left(t_1 \cdot \frac{n}{Om}\right) \cdot \left(U* - U\right)\right) - t\right)\right)}\\ \end{array} \]

Alternative 13
Error	30.2
Cost	8520

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := 2 \cdot t_1\\ \mathbf{if}\;U \leq -3.7 \cdot 10^{-51}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + t_1 \cdot -2\right)}\\ \mathbf{elif}\;U \leq 3.4 \cdot 10^{+89}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(t_2 - \left(\frac{\ell}{Om} \cdot \frac{\ell}{\frac{Om}{n}}\right) \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(n \cdot U\right) \cdot \left(\left(t_2 - \frac{\ell \cdot \left(n \cdot \ell\right)}{Om} \cdot \frac{U*}{Om}\right) - t\right)\right)}\\ \end{array} \]

Alternative 14
Error	36.1
Cost	8400

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \frac{2 \cdot \ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{if}\;n \leq -1 \cdot 10^{-223}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \ell \cdot \frac{\ell \cdot -2}{Om}\right)\right)\right)}\\ \mathbf{elif}\;n \leq 7.6 \cdot 10^{-222}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{\frac{Om}{\ell}} \cdot -2\right)}\\ \mathbf{elif}\;n \leq 4.8 \cdot 10^{-138}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{-85}:\\ \;\;\;\;\sqrt{\left(\frac{-2}{Om} + \frac{n}{Om} \cdot \frac{U* - U}{Om}\right) \cdot \left(2 \cdot \left(n \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq 5.4 \cdot 10^{+256}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(n \cdot \left(2 \cdot U\right)\right)}\\ \end{array} \]

Alternative 15
Error	39.9
Cost	7893

\[\begin{array}{l} t_1 := \sqrt{t \cdot \left(n \cdot \left(2 \cdot U\right)\right)}\\ t_2 := \sqrt{\left(n \cdot -2\right) \cdot \left(2 \cdot \frac{\ell \cdot \ell}{\frac{Om}{U}}\right)}\\ \mathbf{if}\;\ell \leq -2.2 \cdot 10^{+45}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 1.95 \cdot 10^{-231}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 6.6:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{+89} \lor \neg \left(\ell \leq 3.4 \cdot 10^{+136}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 16
Error	37.9
Cost	7760

\[\begin{array}{l} t_1 := \sqrt{t \cdot \left(n \cdot \left(2 \cdot U\right)\right)}\\ t_2 := \sqrt{\left(\ell \cdot \left(\ell \cdot \left(U \cdot \frac{2}{Om}\right)\right)\right) \cdot \left(n \cdot -2\right)}\\ \mathbf{if}\;\ell \leq -1.95 \cdot 10^{+45}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 1.95 \cdot 10^{-231}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 0.0024:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{elif}\;\ell \leq 4.1 \cdot 10^{+89}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 17
Error	34.7
Cost	7625

\[\begin{array}{l} \mathbf{if}\;\ell \leq 1.95 \cdot 10^{-231} \lor \neg \left(\ell \leq 1150\right):\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \ell \cdot \frac{\ell \cdot -2}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \end{array} \]

Alternative 18
Error	34.5
Cost	7625

\[\begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{+76} \lor \neg \left(t \leq 5 \cdot 10^{+157}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \ell \cdot \frac{\ell \cdot -2}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \frac{2 \cdot \ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \end{array} \]

Alternative 19
Error	32.3
Cost	7625

\[\begin{array}{l} \mathbf{if}\;U \leq -2 \cdot 10^{-51} \lor \neg \left(U \leq 2.9 \cdot 10^{-62}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{\frac{Om}{\ell}} \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \frac{2 \cdot \ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \end{array} \]

Alternative 20
Error	39.2
Cost	7113

\[\begin{array}{l} \mathbf{if}\;U \leq -100000000 \lor \neg \left(U \leq 2.4 \cdot 10^{-257}\right):\\ \;\;\;\;\sqrt{t \cdot \left(n \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \end{array} \]

Alternative 21
Error	40.3
Cost	6848

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

Reproduce

herbie shell --seed 2023057 
(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*))))))