Toniolo and Linder, Equation (13)

Average Error: 34.3 → 29.0

Time: 50.0s

Precision: binary64

Cost: 34132

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\frac{-2 \cdot \ell}{\frac{Om}{\ell}} - \frac{\left(\left(n \cdot \ell\right) \cdot \frac{\ell}{Om}\right) \cdot U*}{-Om}\right)\right)\right)}\\ t_2 := \sqrt{\left(n \cdot 2\right) \cdot \left(t + \ell \cdot \left(n \cdot \frac{U*}{\frac{Om}{\frac{\ell}{Om}}} + -2 \cdot \frac{\ell}{Om}\right)\right)} \cdot \sqrt{U}\\ \mathbf{if}\;U \leq -1.28 \cdot 10^{+97}:\\ \;\;\;\;{\left({\left(\frac{-1}{U}\right)}^{-0.25} \cdot {\left(t \cdot \left(n \cdot -2\right)\right)}^{0.25}\right)}^{2}\\ \mathbf{elif}\;U \leq -3.1 \cdot 10^{-240}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;U \leq 4.4 \cdot 10^{-277}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, U \cdot \left(t \cdot n\right), \frac{-4}{Om} \cdot \left(\left(n \cdot \ell\right) \cdot \left(U \cdot \ell\right)\right)\right)}\\ \mathbf{elif}\;U \leq 8 \cdot 10^{-176}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;U \leq 2 \cdot 10^{-154}:\\ \;\;\;\;\frac{\sqrt{2}}{\left|\frac{Om}{n \cdot \ell}\right|} \cdot \sqrt{U \cdot \left(U* - U\right)} - \sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{U}{U* - U}}\right)\\ \mathbf{elif}\;U \leq 1.95 \cdot 10^{-72} \lor \neg \left(U \leq 4.4 \cdot 10^{+41}\right):\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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
         (sqrt
          (*
           (* n 2.0)
           (*
            U
            (+
             t
             (-
              (/ (* -2.0 l) (/ Om l))
              (/ (* (* (* n l) (/ l Om)) U*) (- Om))))))))
        (t_2
         (*
          (sqrt
           (*
            (* n 2.0)
            (+ t (* l (+ (* n (/ U* (/ Om (/ l Om)))) (* -2.0 (/ l Om)))))))
          (sqrt U))))
   (if (<= U -1.28e+97)
     (pow (* (pow (/ -1.0 U) -0.25) (pow (* t (* n -2.0)) 0.25)) 2.0)
     (if (<= U -3.1e-240)
       t_1
       (if (<= U 4.4e-277)
         (sqrt (fma 2.0 (* U (* t n)) (* (/ -4.0 Om) (* (* n l) (* U l)))))
         (if (<= U 8e-176)
           t_2
           (if (<= U 2e-154)
             (-
              (* (/ (sqrt 2.0) (fabs (/ Om (* n l)))) (sqrt (* U (- U* U))))
              (* (sqrt 2.0) (* l (sqrt (/ U (- U* U))))))
             (if (or (<= U 1.95e-72) (not (<= U 4.4e+41))) t_2 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 = sqrt(((n * 2.0) * (U * (t + (((-2.0 * l) / (Om / l)) - ((((n * l) * (l / Om)) * U_42_) / -Om))))));
	double t_2 = sqrt(((n * 2.0) * (t + (l * ((n * (U_42_ / (Om / (l / Om)))) + (-2.0 * (l / Om))))))) * sqrt(U);
	double tmp;
	if (U <= -1.28e+97) {
		tmp = pow((pow((-1.0 / U), -0.25) * pow((t * (n * -2.0)), 0.25)), 2.0);
	} else if (U <= -3.1e-240) {
		tmp = t_1;
	} else if (U <= 4.4e-277) {
		tmp = sqrt(fma(2.0, (U * (t * n)), ((-4.0 / Om) * ((n * l) * (U * l)))));
	} else if (U <= 8e-176) {
		tmp = t_2;
	} else if (U <= 2e-154) {
		tmp = ((sqrt(2.0) / fabs((Om / (n * l)))) * sqrt((U * (U_42_ - U)))) - (sqrt(2.0) * (l * sqrt((U / (U_42_ - U)))));
	} else if ((U <= 1.95e-72) || !(U <= 4.4e+41)) {
		tmp = t_2;
	} else {
		tmp = 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 = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t + Float64(Float64(Float64(-2.0 * l) / Float64(Om / l)) - Float64(Float64(Float64(Float64(n * l) * Float64(l / Om)) * U_42_) / Float64(-Om)))))))
	t_2 = Float64(sqrt(Float64(Float64(n * 2.0) * Float64(t + Float64(l * Float64(Float64(n * Float64(U_42_ / Float64(Om / Float64(l / Om)))) + Float64(-2.0 * Float64(l / Om))))))) * sqrt(U))
	tmp = 0.0
	if (U <= -1.28e+97)
		tmp = Float64((Float64(-1.0 / U) ^ -0.25) * (Float64(t * Float64(n * -2.0)) ^ 0.25)) ^ 2.0;
	elseif (U <= -3.1e-240)
		tmp = t_1;
	elseif (U <= 4.4e-277)
		tmp = sqrt(fma(2.0, Float64(U * Float64(t * n)), Float64(Float64(-4.0 / Om) * Float64(Float64(n * l) * Float64(U * l)))));
	elseif (U <= 8e-176)
		tmp = t_2;
	elseif (U <= 2e-154)
		tmp = Float64(Float64(Float64(sqrt(2.0) / abs(Float64(Om / Float64(n * l)))) * sqrt(Float64(U * Float64(U_42_ - U)))) - Float64(sqrt(2.0) * Float64(l * sqrt(Float64(U / Float64(U_42_ - U))))));
	elseif ((U <= 1.95e-72) || !(U <= 4.4e+41))
		tmp = t_2;
	else
		tmp = 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[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t + N[(N[(N[(-2.0 * l), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(n * l), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision] * U$42$), $MachinePrecision] / (-Om)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(t + N[(l * N[(N[(n * N[(U$42$ / N[(Om / N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[U, -1.28e+97], N[Power[N[(N[Power[N[(-1.0 / U), $MachinePrecision], -0.25], $MachinePrecision] * N[Power[N[(t * N[(n * -2.0), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[U, -3.1e-240], t$95$1, If[LessEqual[U, 4.4e-277], N[Sqrt[N[(2.0 * N[(U * N[(t * n), $MachinePrecision]), $MachinePrecision] + N[(N[(-4.0 / Om), $MachinePrecision] * N[(N[(n * l), $MachinePrecision] * N[(U * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[U, 8e-176], t$95$2, If[LessEqual[U, 2e-154], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[Abs[N[(Om / N[(n * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(U * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[2.0], $MachinePrecision] * N[(l * N[Sqrt[N[(U / N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[U, 1.95e-72], N[Not[LessEqual[U, 4.4e+41]], $MachinePrecision]], t$95$2, t$95$1]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if U < -1.28000000000000003e97

   1. Initial program 29.6

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

   2. Simplified 27.0

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

      [Start] 29.6	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 29.6	\[ \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 29.6	\[ \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      *-commutative [=>] 29.6	\[ \sqrt{2 \cdot \color{blue}{\left(\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot \left(n \cdot U\right)\right)}} \]

   3. Taylor expanded in l around 0 34.6

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

   4. Applied egg-rr 34.8

      \[\leadsto \color{blue}{{\left({\left(\left(n \cdot U\right) \cdot \left(2 \cdot t\right)\right)}^{0.25}\right)}^{2}} \]

   5. Taylor expanded in U around -inf 28.8

      \[\leadsto {\color{blue}{\left(e^{0.25 \cdot \left(-1 \cdot \log \left(\frac{-1}{U}\right) + \log \left(-2 \cdot \left(n \cdot t\right)\right)\right)}\right)}}^{2} \]

   6. Simplified 25.7

      \[\leadsto {\color{blue}{\left({\left(\frac{-1}{U}\right)}^{-0.25} \cdot {\left(t \cdot \left(n \cdot -2\right)\right)}^{0.25}\right)}}^{2} \]
      Proof

      [Start] 28.8	\[ {\left(e^{0.25 \cdot \left(-1 \cdot \log \left(\frac{-1}{U}\right) + \log \left(-2 \cdot \left(n \cdot t\right)\right)\right)}\right)}^{2} \]
      distribute-lft-in [=>] 28.8	\[ {\left(e^{\color{blue}{0.25 \cdot \left(-1 \cdot \log \left(\frac{-1}{U}\right)\right) + 0.25 \cdot \log \left(-2 \cdot \left(n \cdot t\right)\right)}}\right)}^{2} \]
      *-commutative [<=] 28.8	\[ {\left(e^{0.25 \cdot \left(-1 \cdot \log \left(\frac{-1}{U}\right)\right) + \color{blue}{\log \left(-2 \cdot \left(n \cdot t\right)\right) \cdot 0.25}}\right)}^{2} \]
      exp-sum [=>] 28.6	\[ {\color{blue}{\left(e^{0.25 \cdot \left(-1 \cdot \log \left(\frac{-1}{U}\right)\right)} \cdot e^{\log \left(-2 \cdot \left(n \cdot t\right)\right) \cdot 0.25}\right)}}^{2} \]
      *-commutative [=>] 28.6	\[ {\left(e^{\color{blue}{\left(-1 \cdot \log \left(\frac{-1}{U}\right)\right) \cdot 0.25}} \cdot e^{\log \left(-2 \cdot \left(n \cdot t\right)\right) \cdot 0.25}\right)}^{2} \]
      *-commutative [=>] 28.6	\[ {\left(e^{\color{blue}{\left(\log \left(\frac{-1}{U}\right) \cdot -1\right)} \cdot 0.25} \cdot e^{\log \left(-2 \cdot \left(n \cdot t\right)\right) \cdot 0.25}\right)}^{2} \]
      associate-*l* [=>] 28.6	\[ {\left(e^{\color{blue}{\log \left(\frac{-1}{U}\right) \cdot \left(-1 \cdot 0.25\right)}} \cdot e^{\log \left(-2 \cdot \left(n \cdot t\right)\right) \cdot 0.25}\right)}^{2} \]
      metadata-eval [=>] 28.6	\[ {\left(e^{\log \left(\frac{-1}{U}\right) \cdot \color{blue}{-0.25}} \cdot e^{\log \left(-2 \cdot \left(n \cdot t\right)\right) \cdot 0.25}\right)}^{2} \]
      metadata-eval [<=] 28.6	\[ {\left(e^{\log \left(\frac{-1}{U}\right) \cdot \color{blue}{\left(0.25 \cdot -1\right)}} \cdot e^{\log \left(-2 \cdot \left(n \cdot t\right)\right) \cdot 0.25}\right)}^{2} \]
      exp-to-pow [=>] 27.9	\[ {\left(\color{blue}{{\left(\frac{-1}{U}\right)}^{\left(0.25 \cdot -1\right)}} \cdot e^{\log \left(-2 \cdot \left(n \cdot t\right)\right) \cdot 0.25}\right)}^{2} \]
      metadata-eval [=>] 27.9	\[ {\left({\left(\frac{-1}{U}\right)}^{\color{blue}{-0.25}} \cdot e^{\log \left(-2 \cdot \left(n \cdot t\right)\right) \cdot 0.25}\right)}^{2} \]
      exp-to-pow [=>] 25.7	\[ {\left({\left(\frac{-1}{U}\right)}^{-0.25} \cdot \color{blue}{{\left(-2 \cdot \left(n \cdot t\right)\right)}^{0.25}}\right)}^{2} \]
      associate-*r* [=>] 25.7	\[ {\left({\left(\frac{-1}{U}\right)}^{-0.25} \cdot {\color{blue}{\left(\left(-2 \cdot n\right) \cdot t\right)}}^{0.25}\right)}^{2} \]
      *-commutative [=>] 25.7	\[ {\left({\left(\frac{-1}{U}\right)}^{-0.25} \cdot {\color{blue}{\left(t \cdot \left(-2 \cdot n\right)\right)}}^{0.25}\right)}^{2} \]
      *-commutative [=>] 25.7	\[ {\left({\left(\frac{-1}{U}\right)}^{-0.25} \cdot {\left(t \cdot \color{blue}{\left(n \cdot -2\right)}\right)}^{0.25}\right)}^{2} \]

   if -1.28000000000000003e97 < U < -3.10000000000000017e-240 or 1.95e-72 < U < 4.3999999999999998e41

   1. Initial program 33.5

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

   2. Simplified 29.5

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

      [Start] 33.5	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 32.1	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 32.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      sub-neg [=>] 32.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(-\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [<=] 32.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      cancel-sign-sub [<=] 32.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      cancel-sign-sub [=>] 32.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-/l* [=>] 28.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 29.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in U around 0 37.7

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

   4. Simplified 32.3

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

      [Start] 37.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right)} \]
      *-commutative [=>] 37.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(U \cdot \left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
      +-commutative [=>] 37.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} + -1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)\right)} \]
      mul-1-neg [=>] 37.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{{\ell}^{2}}{Om} + \color{blue}{\left(-\frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)\right)\right)} \]
      unsub-neg [=>] 37.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)\right)} \]
      unpow2 [=>] 37.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)\right)} \]
      associate-/l* [=>] 37.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)\right)} \]
      associate-*r/ [=>] 37.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\color{blue}{\frac{2 \cdot \ell}{\frac{Om}{\ell}}} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)\right)} \]
      associate-*r* [=>] 37.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\frac{2 \cdot \ell}{\frac{Om}{\ell}} - \frac{\color{blue}{\left(n \cdot {\ell}^{2}\right) \cdot U*}}{{Om}^{2}}\right)\right)\right)} \]
      unpow2 [=>] 37.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\frac{2 \cdot \ell}{\frac{Om}{\ell}} - \frac{\left(n \cdot {\ell}^{2}\right) \cdot U*}{\color{blue}{Om \cdot Om}}\right)\right)\right)} \]
      times-frac [=>] 34.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\frac{2 \cdot \ell}{\frac{Om}{\ell}} - \color{blue}{\frac{n \cdot {\ell}^{2}}{Om} \cdot \frac{U*}{Om}}\right)\right)\right)} \]
      unpow2 [=>] 34.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\frac{2 \cdot \ell}{\frac{Om}{\ell}} - \frac{n \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{Om} \cdot \frac{U*}{Om}\right)\right)\right)} \]
      associate-*r* [=>] 32.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\frac{2 \cdot \ell}{\frac{Om}{\ell}} - \frac{\color{blue}{\left(n \cdot \ell\right) \cdot \ell}}{Om} \cdot \frac{U*}{Om}\right)\right)\right)} \]

   5. Applied egg-rr 28.6

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

   if -3.10000000000000017e-240 < U < 4.39999999999999991e-277

   1. Initial program 43.5

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

   2. Simplified 37.1

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

      [Start] 43.5	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 40.5	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 40.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      sub-neg [=>] 40.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(-\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [<=] 40.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      cancel-sign-sub [<=] 40.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      cancel-sign-sub [=>] 40.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-/l* [=>] 37.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 37.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in U around 0 45.4

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

   4. Simplified 41.4

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

      [Start] 45.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right)} \]
      *-commutative [=>] 45.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(U \cdot \left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
      +-commutative [=>] 45.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} + -1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)\right)} \]
      mul-1-neg [=>] 45.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{{\ell}^{2}}{Om} + \color{blue}{\left(-\frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)\right)\right)} \]
      unsub-neg [=>] 45.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)\right)} \]
      unpow2 [=>] 45.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)\right)} \]
      associate-/l* [=>] 45.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)\right)} \]
      associate-*r/ [=>] 45.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\color{blue}{\frac{2 \cdot \ell}{\frac{Om}{\ell}}} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)\right)} \]
      associate-*r* [=>] 45.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\frac{2 \cdot \ell}{\frac{Om}{\ell}} - \frac{\color{blue}{\left(n \cdot {\ell}^{2}\right) \cdot U*}}{{Om}^{2}}\right)\right)\right)} \]
      unpow2 [=>] 45.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\frac{2 \cdot \ell}{\frac{Om}{\ell}} - \frac{\left(n \cdot {\ell}^{2}\right) \cdot U*}{\color{blue}{Om \cdot Om}}\right)\right)\right)} \]
      times-frac [=>] 42.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\frac{2 \cdot \ell}{\frac{Om}{\ell}} - \color{blue}{\frac{n \cdot {\ell}^{2}}{Om} \cdot \frac{U*}{Om}}\right)\right)\right)} \]
      unpow2 [=>] 42.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\frac{2 \cdot \ell}{\frac{Om}{\ell}} - \frac{n \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{Om} \cdot \frac{U*}{Om}\right)\right)\right)} \]
      associate-*r* [=>] 41.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\frac{2 \cdot \ell}{\frac{Om}{\ell}} - \frac{\color{blue}{\left(n \cdot \ell\right) \cdot \ell}}{Om} \cdot \frac{U*}{Om}\right)\right)\right)} \]

   5. Applied egg-rr 38.1

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

   6. Taylor expanded in Om around inf 43.7

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

   7. Simplified 38.7

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

      [Start] 43.7	\[ \sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right)\right) + -4 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}} \]
      fma-def [=>] 43.7	\[ \sqrt{\color{blue}{\mathsf{fma}\left(2, n \cdot \left(t \cdot U\right), -4 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}} \]
      associate-*r* [=>] 44.5	\[ \sqrt{\mathsf{fma}\left(2, \color{blue}{\left(n \cdot t\right) \cdot U}, -4 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)} \]
      *-commutative [=>] 44.5	\[ \sqrt{\mathsf{fma}\left(2, \color{blue}{U \cdot \left(n \cdot t\right)}, -4 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)} \]
      associate-*r/ [=>] 44.5	\[ \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \color{blue}{\frac{-4 \cdot \left(n \cdot \left({\ell}^{2} \cdot U\right)\right)}{Om}}\right)} \]
      associate-/l* [=>] 44.5	\[ \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \color{blue}{\frac{-4}{\frac{Om}{n \cdot \left({\ell}^{2} \cdot U\right)}}}\right)} \]
      associate-/r/ [=>] 44.5	\[ \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \color{blue}{\frac{-4}{Om} \cdot \left(n \cdot \left({\ell}^{2} \cdot U\right)\right)}\right)} \]
      associate-*r* [=>] 45.1	\[ \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{-4}{Om} \cdot \color{blue}{\left(\left(n \cdot {\ell}^{2}\right) \cdot U\right)}\right)} \]
      unpow2 [=>] 45.1	\[ \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{-4}{Om} \cdot \left(\left(n \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot U\right)\right)} \]
      associate-*r* [=>] 42.2	\[ \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{-4}{Om} \cdot \left(\color{blue}{\left(\left(n \cdot \ell\right) \cdot \ell\right)} \cdot U\right)\right)} \]
      associate-*l* [=>] 38.7	\[ \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{-4}{Om} \cdot \color{blue}{\left(\left(n \cdot \ell\right) \cdot \left(\ell \cdot U\right)\right)}\right)} \]

   if 4.39999999999999991e-277 < U < 8e-176 or 1.9999999999999999e-154 < U < 1.95e-72 or 4.3999999999999998e41 < U

   1. Initial program 33.6

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

   2. Simplified 32.1

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

      [Start] 33.6	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 34.3	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 34.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      sub-neg [=>] 34.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(-\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [<=] 34.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      cancel-sign-sub [<=] 34.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      cancel-sign-sub [=>] 34.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-/l* [=>] 32.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 32.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in U around 0 38.9

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

   4. Simplified 35.3

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

      [Start] 38.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right)} \]
      *-commutative [=>] 38.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(U \cdot \left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
      +-commutative [=>] 38.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} + -1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)\right)} \]
      mul-1-neg [=>] 38.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{{\ell}^{2}}{Om} + \color{blue}{\left(-\frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)\right)\right)} \]
      unsub-neg [=>] 38.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)\right)} \]
      unpow2 [=>] 38.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)\right)} \]
      associate-/l* [=>] 38.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)\right)} \]
      associate-*r/ [=>] 38.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\color{blue}{\frac{2 \cdot \ell}{\frac{Om}{\ell}}} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)\right)} \]
      associate-*r* [=>] 38.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\frac{2 \cdot \ell}{\frac{Om}{\ell}} - \frac{\color{blue}{\left(n \cdot {\ell}^{2}\right) \cdot U*}}{{Om}^{2}}\right)\right)\right)} \]
      unpow2 [=>] 38.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\frac{2 \cdot \ell}{\frac{Om}{\ell}} - \frac{\left(n \cdot {\ell}^{2}\right) \cdot U*}{\color{blue}{Om \cdot Om}}\right)\right)\right)} \]
      times-frac [=>] 36.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\frac{2 \cdot \ell}{\frac{Om}{\ell}} - \color{blue}{\frac{n \cdot {\ell}^{2}}{Om} \cdot \frac{U*}{Om}}\right)\right)\right)} \]
      unpow2 [=>] 36.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\frac{2 \cdot \ell}{\frac{Om}{\ell}} - \frac{n \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{Om} \cdot \frac{U*}{Om}\right)\right)\right)} \]
      associate-*r* [=>] 35.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\frac{2 \cdot \ell}{\frac{Om}{\ell}} - \frac{\color{blue}{\left(n \cdot \ell\right) \cdot \ell}}{Om} \cdot \frac{U*}{Om}\right)\right)\right)} \]

   5. Applied egg-rr 48.6

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

   6. Simplified 32.9

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

      [Start] 48.6	\[ e^{\mathsf{log1p}\left(\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\frac{n \cdot \ell}{\frac{Om}{\ell} \cdot \frac{Om}{U*}} + \left(t + \left(\ell \cdot -2\right) \cdot \frac{\ell}{Om}\right)\right)\right)}\right)} - 1 \]
      expm1-def [=>] 34.3	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\frac{n \cdot \ell}{\frac{Om}{\ell} \cdot \frac{Om}{U*}} + \left(t + \left(\ell \cdot -2\right) \cdot \frac{\ell}{Om}\right)\right)\right)}\right)\right)} \]
      expm1-log1p [=>] 33.4	\[ \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\frac{n \cdot \ell}{\frac{Om}{\ell} \cdot \frac{Om}{U*}} + \left(t + \left(\ell \cdot -2\right) \cdot \frac{\ell}{Om}\right)\right)\right)}} \]
      *-commutative [=>] 33.4	\[ \sqrt{\color{blue}{\left(U \cdot \left(\frac{n \cdot \ell}{\frac{Om}{\ell} \cdot \frac{Om}{U*}} + \left(t + \left(\ell \cdot -2\right) \cdot \frac{\ell}{Om}\right)\right)\right) \cdot \left(2 \cdot n\right)}} \]
      associate-*l* [=>] 33.1	\[ \sqrt{\color{blue}{U \cdot \left(\left(\frac{n \cdot \ell}{\frac{Om}{\ell} \cdot \frac{Om}{U*}} + \left(t + \left(\ell \cdot -2\right) \cdot \frac{\ell}{Om}\right)\right) \cdot \left(2 \cdot n\right)\right)}} \]

   7. Applied egg-rr 26.3

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

   if 8e-176 < U < 1.9999999999999999e-154

   1. Initial program 40.3

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

   2. Simplified 39.5

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

      [Start] 40.3	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 40.3	\[ \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 40.3	\[ \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      *-commutative [=>] 40.3	\[ \sqrt{2 \cdot \color{blue}{\left(\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot \left(n \cdot U\right)\right)}} \]

   3. Taylor expanded in Om around 0 58.7

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

   4. Simplified 58.7

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

      [Start] 58.7	\[ -1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{U}{U* - U}}\right) + \frac{\sqrt{2} \cdot \left(n \cdot \ell\right)}{Om} \cdot \sqrt{\left(U* - U\right) \cdot U} \]
      +-commutative [=>] 58.7	\[ \color{blue}{\frac{\sqrt{2} \cdot \left(n \cdot \ell\right)}{Om} \cdot \sqrt{\left(U* - U\right) \cdot U} + -1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{U}{U* - U}}\right)} \]
      mul-1-neg [=>] 58.7	\[ \frac{\sqrt{2} \cdot \left(n \cdot \ell\right)}{Om} \cdot \sqrt{\left(U* - U\right) \cdot U} + \color{blue}{\left(-\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{U}{U* - U}}\right)} \]
      unsub-neg [=>] 58.7	\[ \color{blue}{\frac{\sqrt{2} \cdot \left(n \cdot \ell\right)}{Om} \cdot \sqrt{\left(U* - U\right) \cdot U} - \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{U}{U* - U}}} \]
      associate-/l* [=>] 58.7	\[ \color{blue}{\frac{\sqrt{2}}{\frac{Om}{n \cdot \ell}}} \cdot \sqrt{\left(U* - U\right) \cdot U} - \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{U}{U* - U}} \]
      *-commutative [=>] 58.7	\[ \frac{\sqrt{2}}{\frac{Om}{n \cdot \ell}} \cdot \sqrt{\color{blue}{U \cdot \left(U* - U\right)}} - \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{U}{U* - U}} \]
      associate-*l* [=>] 58.7	\[ \frac{\sqrt{2}}{\frac{Om}{n \cdot \ell}} \cdot \sqrt{U \cdot \left(U* - U\right)} - \color{blue}{\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{U}{U* - U}}\right)} \]

   5. Applied egg-rr 57.6

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

   6. Simplified 49.9

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

      [Start] 57.6	\[ \frac{\sqrt{2}}{\sqrt{{\left(\frac{\frac{Om}{n}}{\ell}\right)}^{2}}} \cdot \sqrt{U \cdot \left(U* - U\right)} - \sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{U}{U* - U}}\right) \]
      unpow2 [=>] 57.6	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{\frac{Om}{n}}{\ell} \cdot \frac{\frac{Om}{n}}{\ell}}}} \cdot \sqrt{U \cdot \left(U* - U\right)} - \sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{U}{U* - U}}\right) \]
      rem-sqrt-square [=>] 52.0	\[ \frac{\sqrt{2}}{\color{blue}{\left|\frac{\frac{Om}{n}}{\ell}\right|}} \cdot \sqrt{U \cdot \left(U* - U\right)} - \sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{U}{U* - U}}\right) \]
      associate-/r* [<=] 49.9	\[ \frac{\sqrt{2}}{\left|\color{blue}{\frac{Om}{n \cdot \ell}}\right|} \cdot \sqrt{U \cdot \left(U* - U\right)} - \sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{U}{U* - U}}\right) \]

3. Recombined 5 regimes into one program.

4. Final simplification 29.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -1.28 \cdot 10^{+97}:\\ \;\;\;\;{\left({\left(\frac{-1}{U}\right)}^{-0.25} \cdot {\left(t \cdot \left(n \cdot -2\right)\right)}^{0.25}\right)}^{2}\\ \mathbf{elif}\;U \leq -3.1 \cdot 10^{-240}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\frac{-2 \cdot \ell}{\frac{Om}{\ell}} - \frac{\left(\left(n \cdot \ell\right) \cdot \frac{\ell}{Om}\right) \cdot U*}{-Om}\right)\right)\right)}\\ \mathbf{elif}\;U \leq 4.4 \cdot 10^{-277}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, U \cdot \left(t \cdot n\right), \frac{-4}{Om} \cdot \left(\left(n \cdot \ell\right) \cdot \left(U \cdot \ell\right)\right)\right)}\\ \mathbf{elif}\;U \leq 8 \cdot 10^{-176}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(t + \ell \cdot \left(n \cdot \frac{U*}{\frac{Om}{\frac{\ell}{Om}}} + -2 \cdot \frac{\ell}{Om}\right)\right)} \cdot \sqrt{U}\\ \mathbf{elif}\;U \leq 2 \cdot 10^{-154}:\\ \;\;\;\;\frac{\sqrt{2}}{\left|\frac{Om}{n \cdot \ell}\right|} \cdot \sqrt{U \cdot \left(U* - U\right)} - \sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{U}{U* - U}}\right)\\ \mathbf{elif}\;U \leq 1.95 \cdot 10^{-72} \lor \neg \left(U \leq 4.4 \cdot 10^{+41}\right):\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(t + \ell \cdot \left(n \cdot \frac{U*}{\frac{Om}{\frac{\ell}{Om}}} + -2 \cdot \frac{\ell}{Om}\right)\right)} \cdot \sqrt{U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\frac{-2 \cdot \ell}{\frac{Om}{\ell}} - \frac{\left(\left(n \cdot \ell\right) \cdot \frac{\ell}{Om}\right) \cdot U*}{-Om}\right)\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	30.1
Cost	20364

\[\begin{array}{l} t_1 := -2 \cdot \frac{\ell}{Om}\\ t_2 := {\left({\left(U \cdot \left(n \cdot -2\right)\right)}^{0.25} \cdot {\left(\frac{-1}{t}\right)}^{-0.25}\right)}^{2}\\ t_3 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\frac{-2 \cdot \ell}{\frac{Om}{\ell}} - \frac{\left(\left(n \cdot \ell\right) \cdot \frac{\ell}{Om}\right) \cdot U*}{-Om}\right)\right)\right)}\\ \mathbf{if}\;t \leq -1.3 \cdot 10^{+178}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.32 \cdot 10^{+94}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \mathsf{fma}\left(t, U, \frac{-2}{\frac{Om}{\ell \cdot \left(U \cdot \ell\right)}}\right)}\\ \mathbf{elif}\;t \leq -5.8 \cdot 10^{+89}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -5 \cdot 10^{-114}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -2.1 \cdot 10^{-274}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(n \cdot 2\right) \cdot \left(t + \ell \cdot \left(n \cdot \left(\frac{\ell}{Om} \cdot \frac{U*}{Om}\right) + t_1\right)\right)\right)}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-279}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(t + \ell \cdot \left(n \cdot \frac{U*}{\frac{Om}{\frac{\ell}{Om}}} + t_1\right)\right)} \cdot \sqrt{U}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-206}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot 2\right)} \cdot \sqrt{\frac{n \cdot \ell}{\frac{Om}{\ell} \cdot \frac{Om}{U*}} + \left(t + \frac{\ell}{Om} \cdot \left(-2 \cdot \ell\right)\right)}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{+107}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot n\right)} \cdot \sqrt{t}\\ \end{array} \]

Alternative 2
Error	30.9
Cost	15192

\[\begin{array}{l} t_1 := \left(\sqrt{2} \cdot \frac{n}{\frac{Om}{\ell}}\right) \cdot \sqrt{U \cdot U*}\\ t_2 := \sqrt{\left(n \cdot 2\right) \cdot \left(t + \ell \cdot \left(n \cdot \frac{U*}{\frac{Om}{\frac{\ell}{Om}}} + -2 \cdot \frac{\ell}{Om}\right)\right)} \cdot \sqrt{U}\\ \mathbf{if}\;U \leq -4.05 \cdot 10^{-240}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\frac{-2 \cdot \ell}{\frac{Om}{\ell}} - \frac{\left(\left(n \cdot \ell\right) \cdot \frac{\ell}{Om}\right) \cdot U*}{-Om}\right)\right)\right)}\\ \mathbf{elif}\;U \leq 4.1 \cdot 10^{-278}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, U \cdot \left(t \cdot n\right), \frac{-4}{Om} \cdot \left(\left(n \cdot \ell\right) \cdot \left(U \cdot \ell\right)\right)\right)}\\ \mathbf{elif}\;U \leq 1.5 \cdot 10^{-206}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;U \leq 8.6 \cdot 10^{-197}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;U \leq 8 \cdot 10^{-176}:\\ \;\;\;\;\sqrt{t \cdot n} \cdot \sqrt{U \cdot 2}\\ \mathbf{elif}\;U \leq 3.6 \cdot 10^{-163}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	27.9
Cost	14664

\[\begin{array}{l} \mathbf{if}\;n \leq -5.5 \cdot 10^{-42}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\frac{-2 \cdot \ell}{\frac{Om}{\ell}} - \frac{\left(\left(n \cdot \ell\right) \cdot \frac{\ell}{Om}\right) \cdot U*}{-Om}\right)\right)\right)}\\ \mathbf{elif}\;n \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, U \cdot \left(t \cdot n\right), \frac{-4}{Om} \cdot \left(\left(n \cdot \ell\right) \cdot \left(U \cdot \ell\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t - \ell \cdot \left(2 \cdot \frac{\ell}{Om} - \frac{U*}{Om} \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 4
Error	32.9
Cost	13908

\[\begin{array}{l} t_1 := \frac{-2 \cdot \ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;Om \leq -2.6 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(n \cdot 2\right) \cdot \left(\left(t - \frac{2}{\frac{\frac{Om}{\ell}}{\ell}}\right) + \ell \cdot \frac{n}{\frac{Om}{\frac{U*}{\frac{Om}{\ell}}}}\right)\right)}\\ \mathbf{elif}\;Om \leq -3.9 \cdot 10^{-203}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(t_1 - \frac{\left(\left(n \cdot \ell\right) \cdot \frac{\ell}{Om}\right) \cdot U*}{-Om}\right)\right)\right)}\\ \mathbf{elif}\;Om \leq 2 \cdot 10^{-190}:\\ \;\;\;\;\left(\sqrt{2} \cdot \frac{n}{\frac{Om}{\ell}}\right) \cdot \sqrt{U \cdot U*}\\ \mathbf{elif}\;Om \leq 2.5 \cdot 10^{-101}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\frac{\left(n \cdot \ell\right) \cdot U*}{Om \cdot \frac{Om}{\ell}} + t_1\right)\right)\right)}\\ \mathbf{elif}\;Om \leq 7 \cdot 10^{-69}:\\ \;\;\;\;\sqrt{t \cdot n} \cdot \sqrt{U \cdot 2}\\ \mathbf{elif}\;Om \leq 2.05 \cdot 10^{-30}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(\left(\ell \cdot \left(U \cdot \ell\right)\right) \cdot \left(\frac{-2}{Om} - \frac{U - U*}{Om} \cdot \frac{n}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(n \cdot 2\right) \cdot \left(t + \ell \cdot \left(n \cdot \left(\frac{\ell}{Om} \cdot \frac{U*}{Om}\right) + -2 \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 5
Error	32.6
Cost	13908

\[\begin{array}{l} t_1 := \frac{-2 \cdot \ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;Om \leq -2.6 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(n \cdot 2\right) \cdot \left(\left(t - \frac{2}{\frac{\frac{Om}{\ell}}{\ell}}\right) + \ell \cdot \frac{n}{\frac{Om}{\frac{U*}{\frac{Om}{\ell}}}}\right)\right)}\\ \mathbf{elif}\;Om \leq -6.8 \cdot 10^{-203}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(t_1 - \frac{\left(\left(n \cdot \ell\right) \cdot \frac{\ell}{Om}\right) \cdot U*}{-Om}\right)\right)\right)}\\ \mathbf{elif}\;Om \leq 3.3 \cdot 10^{-194}:\\ \;\;\;\;\sqrt{U \cdot U*} \cdot \frac{\sqrt{2}}{\frac{Om}{n \cdot \ell}}\\ \mathbf{elif}\;Om \leq 5.4 \cdot 10^{-101}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\frac{\left(n \cdot \ell\right) \cdot U*}{Om \cdot \frac{Om}{\ell}} + t_1\right)\right)\right)}\\ \mathbf{elif}\;Om \leq 7 \cdot 10^{-69}:\\ \;\;\;\;\sqrt{t \cdot n} \cdot \sqrt{U \cdot 2}\\ \mathbf{elif}\;Om \leq 2.05 \cdot 10^{-30}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(\left(\ell \cdot \left(U \cdot \ell\right)\right) \cdot \left(\frac{-2}{Om} - \frac{U - U*}{Om} \cdot \frac{n}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(n \cdot 2\right) \cdot \left(t + \ell \cdot \left(n \cdot \left(\frac{\ell}{Om} \cdot \frac{U*}{Om}\right) + -2 \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 6
Error	32.6
Cost	13908

\[\begin{array}{l} t_1 := \frac{-2 \cdot \ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;Om \leq -2.6 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(n \cdot 2\right) \cdot \left(\left(t - \frac{2}{\frac{\frac{Om}{\ell}}{\ell}}\right) + \ell \cdot \frac{n}{\frac{Om}{\frac{U*}{\frac{Om}{\ell}}}}\right)\right)}\\ \mathbf{elif}\;Om \leq -2.8 \cdot 10^{-203}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(t_1 - \frac{\left(\left(n \cdot \ell\right) \cdot \frac{\ell}{Om}\right) \cdot U*}{-Om}\right)\right)\right)}\\ \mathbf{elif}\;Om \leq 10^{-194}:\\ \;\;\;\;\sqrt{U \cdot U*} \cdot \frac{\left(n \cdot \ell\right) \cdot \sqrt{2}}{Om}\\ \mathbf{elif}\;Om \leq 5.4 \cdot 10^{-101}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\frac{\left(n \cdot \ell\right) \cdot U*}{Om \cdot \frac{Om}{\ell}} + t_1\right)\right)\right)}\\ \mathbf{elif}\;Om \leq 3.2 \cdot 10^{-69}:\\ \;\;\;\;\sqrt{t \cdot n} \cdot \sqrt{U \cdot 2}\\ \mathbf{elif}\;Om \leq 2.05 \cdot 10^{-30}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(\left(\ell \cdot \left(U \cdot \ell\right)\right) \cdot \left(\frac{-2}{Om} - \frac{U - U*}{Om} \cdot \frac{n}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(n \cdot 2\right) \cdot \left(t + \ell \cdot \left(n \cdot \left(\frac{\ell}{Om} \cdot \frac{U*}{Om}\right) + -2 \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 7
Error	31.0
Cost	13644

\[\begin{array}{l} t_1 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\frac{-2 \cdot \ell}{\frac{Om}{\ell}} - \frac{\left(\left(n \cdot \ell\right) \cdot \frac{\ell}{Om}\right) \cdot U*}{-Om}\right)\right)\right)}\\ \mathbf{if}\;t \leq -4.7 \cdot 10^{-114}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.95 \cdot 10^{-228}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(n \cdot 2\right) \cdot \left(t + \ell \cdot \left(n \cdot \left(\frac{\ell}{Om} \cdot \frac{U*}{Om}\right) + -2 \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{+107}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot n\right)} \cdot \sqrt{t}\\ \end{array} \]

Alternative 8
Error	33.8
Cost	8660

\[\begin{array}{l} t_1 := \frac{U*}{Om} \cdot \frac{n}{Om}\\ t_2 := \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + -2 \cdot \left(n \cdot \left(2 \cdot \left(\frac{\ell}{Om} \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -3.5 \cdot 10^{+145}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -2.4 \cdot 10^{+41}:\\ \;\;\;\;\sqrt{\left(\ell \cdot \left(U \cdot \ell\right)\right) \cdot \left(\left(n \cdot 2\right) \cdot \left(t_1 + \frac{-2}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-128}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \ell \cdot \left(\ell \cdot \left(\frac{2}{Om} - t_1\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 10^{-167}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.6 \cdot 10^{-69}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n}{\frac{Om \cdot Om}{U* - U}} + \frac{-2}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 9
Error	33.3
Cost	8660

\[\begin{array}{l} \mathbf{if}\;Om \leq -2.6 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(n \cdot 2\right) \cdot \left(\left(t - \frac{2}{\frac{\frac{Om}{\ell}}{\ell}}\right) + \ell \cdot \frac{n}{\frac{Om}{\frac{U*}{\frac{Om}{\ell}}}}\right)\right)}\\ \mathbf{elif}\;Om \leq -2.1 \cdot 10^{-134}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \left(\frac{2 \cdot \ell}{\frac{Om}{\ell}} - \frac{U*}{Om} \cdot \frac{\ell \cdot \left(n \cdot \ell\right)}{Om}\right)\right)\right)}\\ \mathbf{elif}\;Om \leq -1.5 \cdot 10^{-191}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(t \cdot n\right)\right)}\\ \mathbf{elif}\;Om \leq 6 \cdot 10^{-240}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\frac{\left(U - U*\right) \cdot \left(\ell \cdot \ell\right)}{\frac{Om}{n}} \cdot \frac{U}{\frac{Om}{n}}\right)}\\ \mathbf{elif}\;Om \leq 1.05 \cdot 10^{-23}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + -2 \cdot \left(n \cdot \left(2 \cdot \left(\frac{\ell}{Om} \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(n \cdot 2\right) \cdot \left(t + \ell \cdot \left(n \cdot \left(\frac{\ell}{Om} \cdot \frac{U*}{Om}\right) + -2 \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 10
Error	31.7
Cost	8589

\[\begin{array}{l} \mathbf{if}\;\ell \leq -8.5 \cdot 10^{+147}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + -2 \cdot \left(n \cdot \left(2 \cdot \left(\frac{\ell}{Om} \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -1.7 \cdot 10^{+41} \lor \neg \left(\ell \leq 1.4 \cdot 10^{+148}\right):\\ \;\;\;\;\sqrt{\left(\ell \cdot \left(U \cdot \ell\right)\right) \cdot \left(\left(n \cdot 2\right) \cdot \left(\frac{U*}{Om} \cdot \frac{n}{Om} + \frac{-2}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\frac{-2 \cdot \ell}{\frac{Om}{\ell}} - \frac{\left(\left(n \cdot \ell\right) \cdot \frac{\ell}{Om}\right) \cdot U*}{-Om}\right)\right)\right)}\\ \end{array} \]

Alternative 11
Error	33.6
Cost	8532

\[\begin{array}{l} t_1 := \frac{U*}{Om} \cdot \frac{n}{Om}\\ t_2 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \ell \cdot \left(\ell \cdot \left(\frac{2}{Om} - t_1\right)\right)\right)\right)}\\ t_3 := \sqrt{\left(\ell \cdot \left(U \cdot \ell\right)\right) \cdot \left(\left(n \cdot 2\right) \cdot \left(t_1 + \frac{-2}{Om}\right)\right)}\\ \mathbf{if}\;\ell \leq -2.35 \cdot 10^{+145}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + -2 \cdot \left(n \cdot \left(2 \cdot \left(\frac{\ell}{Om} \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -2.4 \cdot 10^{+41}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\ell \leq -1.3 \cdot 10^{-165}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 1.4 \cdot 10^{-226}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(U \cdot n\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.55 \cdot 10^{+149}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 12
Error	33.3
Cost	8528

\[\begin{array}{l} t_1 := \sqrt{U \cdot \left(\left(n \cdot 2\right) \cdot \left(t + \ell \cdot \left(n \cdot \left(\frac{\ell}{Om} \cdot \frac{U*}{Om}\right) + -2 \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{if}\;Om \leq -2.3 \cdot 10^{-36}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;Om \leq -2.8 \cdot 10^{-133}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + -2 \cdot \left(n \cdot \left(2 \cdot \left(\frac{\ell}{Om} \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \mathbf{elif}\;Om \leq -1.2 \cdot 10^{-191}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(t \cdot n\right)\right)}\\ \mathbf{elif}\;Om \leq 1.05 \cdot 10^{-239}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\frac{\left(U - U*\right) \cdot \left(\ell \cdot \ell\right)}{\frac{Om}{n}} \cdot \frac{U}{\frac{Om}{n}}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	32.4
Cost	8524

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + -2 \cdot \left(n \cdot \left(2 \cdot \left(\frac{\ell}{Om} \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -3 \cdot 10^{+147}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -7.5 \cdot 10^{+40}:\\ \;\;\;\;\sqrt{\left(\ell \cdot \left(U \cdot \ell\right)\right) \cdot \left(\left(n \cdot 2\right) \cdot \left(\frac{U*}{Om} \cdot \frac{n}{Om} + \frac{-2}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.6 \cdot 10^{-27}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\frac{\left(n \cdot \ell\right) \cdot U*}{Om \cdot \frac{Om}{\ell}} + \frac{-2 \cdot \ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Error	35.8
Cost	8144

\[\begin{array}{l} t_1 := \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \frac{2 \cdot \ell}{Om} - t\right)\right)\right)}\\ \mathbf{if}\;Om \leq -7.6 \cdot 10^{+25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;Om \leq -6 \cdot 10^{-37}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(\frac{-1}{Om} \cdot \frac{U \cdot \left(\left(n \cdot \ell\right) \cdot \left(\ell \cdot \left(U - U*\right)\right)\right)}{Om}\right)}\\ \mathbf{elif}\;Om \leq -1.4 \cdot 10^{-191}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(U \cdot n\right)\right)}\\ \mathbf{elif}\;Om \leq 2 \cdot 10^{-239}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\frac{\left(U - U*\right) \cdot \left(\ell \cdot \ell\right)}{\frac{Om}{n}} \cdot \frac{U}{\frac{Om}{n}}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 15
Error	34.7
Cost	7880

\[\begin{array}{l} \mathbf{if}\;Om \leq -6.2 \cdot 10^{-198}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{elif}\;Om \leq 8.8 \cdot 10^{-220}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\frac{U \cdot \left(U - U*\right)}{\frac{Om}{n}} \cdot \frac{\ell \cdot \ell}{\frac{Om}{n}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \frac{2 \cdot \ell}{Om} - t\right)\right)\right)}\\ \end{array} \]

Alternative 16
Error	34.7
Cost	7880

\[\begin{array}{l} \mathbf{if}\;Om \leq -3.7 \cdot 10^{-190}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{elif}\;Om \leq 5.6 \cdot 10^{-240}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\frac{\left(U - U*\right) \cdot \left(\ell \cdot \ell\right)}{\frac{Om}{n}} \cdot \frac{U}{\frac{Om}{n}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \frac{2 \cdot \ell}{Om} - t\right)\right)\right)}\\ \end{array} \]

Alternative 17
Error	34.8
Cost	7880

\[\begin{array}{l} \mathbf{if}\;Om \leq -6.2 \cdot 10^{-198}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{elif}\;Om \leq 7.5 \cdot 10^{-240}:\\ \;\;\;\;\sqrt{\frac{n}{\frac{Om}{\ell \cdot \left(\ell \cdot \left(U - U*\right)\right)} \cdot \frac{Om}{U}} \cdot \left(n \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \frac{2 \cdot \ell}{Om} - t\right)\right)\right)}\\ \end{array} \]

Alternative 18
Error	32.6
Cost	7876

\[\begin{array}{l} \mathbf{if}\;U \leq 1.2 \cdot 10^{+101}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + -2 \cdot \left(n \cdot \left(2 \cdot \left(\frac{\ell}{Om} \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \frac{2 \cdot \ell}{Om} - t\right)\right)\right)}\\ \end{array} \]

Alternative 19
Error	34.8
Cost	7625

\[\begin{array}{l} \mathbf{if}\;Om \leq -2.6 \cdot 10^{+79} \lor \neg \left(Om \leq -2.25 \cdot 10^{-131}\right):\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \frac{2 \cdot \ell}{Om} - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot t\right)}\\ \end{array} \]

Alternative 20
Error	33.9
Cost	7492

\[\begin{array}{l} \mathbf{if}\;U \leq 10^{+98}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \frac{2 \cdot \ell}{Om} - t\right)\right)\right)}\\ \end{array} \]

Alternative 21
Error	40.0
Cost	7369

\[\begin{array}{l} \mathbf{if}\;\ell \leq 1.4 \cdot 10^{+27} \lor \neg \left(\ell \leq 5.4 \cdot 10^{+143}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(t \cdot n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-4 \cdot \left(n \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{U}{Om}\right)\right)}\\ \end{array} \]

Alternative 22
Error	40.1
Cost	6980

\[\begin{array}{l} \mathbf{if}\;U* \leq 7.5 \cdot 10^{-49}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(t \cdot n\right)\right)}\\ \end{array} \]

Alternative 23
Error	40.2
Cost	6848

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

Reproduce

herbie shell --seed 2023083 
(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*))))))