Toniolo and Linder, Equation (13)

Average Accuracy: 46.4% → 51.9%

Time: 47.0s

Precision: binary64

Cost: 28325

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_3 := t - \mathsf{fma}\left(2, t_1, n \cdot \left(t_2 \cdot \left(U - U*\right)\right)\right)\\ t_4 := \sqrt{2} \cdot \sqrt{U \cdot \left(n \cdot t_3\right)}\\ \mathbf{if}\;n \leq -2.4 \cdot 10^{+26}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot 2\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot t_2\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;n \leq -1.7 \cdot 10^{-51}:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{2 \cdot \left(n \cdot \mathsf{fma}\left(\ell, \ell \cdot \frac{-2}{Om}, t\right)\right)}\\ \mathbf{elif}\;n \leq -2.9 \cdot 10^{-129}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;n \leq -4.5 \cdot 10^{-145}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{n \cdot \left(U \cdot \frac{-2}{Om}\right)}\\ \mathbf{elif}\;n \leq -6 \cdot 10^{-183}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \mathbf{elif}\;n \leq -1.35 \cdot 10^{-256}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;n \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot t_1\right)\right)\right)}\\ \mathbf{elif}\;n \leq 4.5 \cdot 10^{-140} \lor \neg \left(n \leq 2.15 \cdot 10^{-63}\right):\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot t_3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{-2}{\frac{\frac{Om}{\ell}}{\ell}}\right)\right)}\\ \end{array} \]

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 (/ l (/ Om l)))
        (t_2 (pow (/ l Om) 2.0))
        (t_3 (- t (fma 2.0 t_1 (* n (* t_2 (- U U*))))))
        (t_4 (* (sqrt 2.0) (sqrt (* U (* n t_3))))))
   (if (<= n -2.4e+26)
     (sqrt
      (*
       (* (* n 2.0) U)
       (+ (- t (* 2.0 (/ (* l l) Om))) (* (* n t_2) (- U* U)))))
     (if (<= n -1.7e-51)
       (* (sqrt U) (sqrt (* 2.0 (* n (fma l (* l (/ -2.0 Om)) t)))))
       (if (<= n -2.9e-129)
         t_4
         (if (<= n -4.5e-145)
           (* (* l (sqrt 2.0)) (sqrt (* n (* U (/ -2.0 Om)))))
           (if (<= n -6e-183)
             (sqrt (* (* 2.0 U) (* n t)))
             (if (<= n -1.35e-256)
               t_4
               (if (<= n -4e-310)
                 (sqrt (* 2.0 (* n (* U (+ t (* -2.0 t_1))))))
                 (if (or (<= n 4.5e-140) (not (<= n 2.15e-63)))
                   (* (sqrt (* n 2.0)) (sqrt (* U t_3)))
                   (sqrt
                    (*
                     2.0
                     (* (* n U) (+ t (/ -2.0 (/ (/ Om l) 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 = l / (Om / l);
	double t_2 = pow((l / Om), 2.0);
	double t_3 = t - fma(2.0, t_1, (n * (t_2 * (U - U_42_))));
	double t_4 = sqrt(2.0) * sqrt((U * (n * t_3)));
	double tmp;
	if (n <= -2.4e+26) {
		tmp = sqrt((((n * 2.0) * U) * ((t - (2.0 * ((l * l) / Om))) + ((n * t_2) * (U_42_ - U)))));
	} else if (n <= -1.7e-51) {
		tmp = sqrt(U) * sqrt((2.0 * (n * fma(l, (l * (-2.0 / Om)), t))));
	} else if (n <= -2.9e-129) {
		tmp = t_4;
	} else if (n <= -4.5e-145) {
		tmp = (l * sqrt(2.0)) * sqrt((n * (U * (-2.0 / Om))));
	} else if (n <= -6e-183) {
		tmp = sqrt(((2.0 * U) * (n * t)));
	} else if (n <= -1.35e-256) {
		tmp = t_4;
	} else if (n <= -4e-310) {
		tmp = sqrt((2.0 * (n * (U * (t + (-2.0 * t_1))))));
	} else if ((n <= 4.5e-140) || !(n <= 2.15e-63)) {
		tmp = sqrt((n * 2.0)) * sqrt((U * t_3));
	} else {
		tmp = sqrt((2.0 * ((n * U) * (t + (-2.0 / ((Om / l) / 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(l / Float64(Om / l))
	t_2 = Float64(l / Om) ^ 2.0
	t_3 = Float64(t - fma(2.0, t_1, Float64(n * Float64(t_2 * Float64(U - U_42_)))))
	t_4 = Float64(sqrt(2.0) * sqrt(Float64(U * Float64(n * t_3))))
	tmp = 0.0
	if (n <= -2.4e+26)
		tmp = sqrt(Float64(Float64(Float64(n * 2.0) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + Float64(Float64(n * t_2) * Float64(U_42_ - U)))));
	elseif (n <= -1.7e-51)
		tmp = Float64(sqrt(U) * sqrt(Float64(2.0 * Float64(n * fma(l, Float64(l * Float64(-2.0 / Om)), t)))));
	elseif (n <= -2.9e-129)
		tmp = t_4;
	elseif (n <= -4.5e-145)
		tmp = Float64(Float64(l * sqrt(2.0)) * sqrt(Float64(n * Float64(U * Float64(-2.0 / Om)))));
	elseif (n <= -6e-183)
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * t)));
	elseif (n <= -1.35e-256)
		tmp = t_4;
	elseif (n <= -4e-310)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * Float64(t + Float64(-2.0 * t_1))))));
	elseif ((n <= 4.5e-140) || !(n <= 2.15e-63))
		tmp = Float64(sqrt(Float64(n * 2.0)) * sqrt(Float64(U * t_3)));
	else
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * Float64(t + Float64(-2.0 / Float64(Float64(Om / l) / 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[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(t - N[(2.0 * t$95$1 + N[(n * N[(t$95$2 * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(U * N[(n * t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -2.4e+26], N[Sqrt[N[(N[(N[(n * 2.0), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(n * t$95$2), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, -1.7e-51], N[(N[Sqrt[U], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(n * N[(l * N[(l * N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -2.9e-129], t$95$4, If[LessEqual[n, -4.5e-145], N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(n * N[(U * N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -6e-183], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, -1.35e-256], t$95$4, If[LessEqual[n, -4e-310], N[Sqrt[N[(2.0 * N[(n * N[(U * N[(t + N[(-2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[n, 4.5e-140], N[Not[LessEqual[n, 2.15e-63]], $MachinePrecision]], N[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t + N[(-2.0 / N[(N[(Om / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]]]]]]

Derivation

1. Split input into 8 regimes

2. if n < -2.40000000000000005e26

   1. Initial program 49.2%

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

   if -2.40000000000000005e26 < n < -1.70000000000000001e-51

   1. Initial program 53.9%

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

   2. Taylor expanded in n around 0 45.8%

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

   3. Simplified 50.5%

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

      [Start] 45.8	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot U\right)\right)} \]
      associate-*r* [=>] 46.4	\[ \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right)}} \]
      unpow2 [=>] 46.4	\[ \sqrt{2 \cdot \left(\left(n \cdot \left(t - 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right)\right) \cdot U\right)} \]
      associate-*r/ [<=] 50.5	\[ \sqrt{2 \cdot \left(\left(n \cdot \left(t - 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)}\right)\right) \cdot U\right)} \]
      cancel-sign-sub-inv [=>] 50.5	\[ \sqrt{2 \cdot \left(\left(n \cdot \color{blue}{\left(t + \left(-2\right) \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\right) \cdot U\right)} \]
      metadata-eval [=>] 50.5	\[ \sqrt{2 \cdot \left(\left(n \cdot \left(t + \color{blue}{-2} \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right) \cdot U\right)} \]
      *-commutative [<=] 50.5	\[ \sqrt{2 \cdot \left(\left(n \cdot \left(t + \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2}\right)\right) \cdot U\right)} \]
      associate-*l* [=>] 50.5	\[ \sqrt{2 \cdot \left(\left(n \cdot \left(t + \color{blue}{\ell \cdot \left(\frac{\ell}{Om} \cdot -2\right)}\right)\right) \cdot U\right)} \]

   4. Applied egg-rr 34.2%

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

      [Start] 50.5	\[ \sqrt{2 \cdot \left(\left(n \cdot \left(t + \ell \cdot \left(\frac{\ell}{Om} \cdot -2\right)\right)\right) \cdot U\right)} \]
      associate-*r* [=>] 50.5	\[ \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot \left(t + \ell \cdot \left(\frac{\ell}{Om} \cdot -2\right)\right)\right)\right) \cdot U}} \]
      sqrt-prod [=>] 34.2	\[ \color{blue}{\sqrt{2 \cdot \left(n \cdot \left(t + \ell \cdot \left(\frac{\ell}{Om} \cdot -2\right)\right)\right)} \cdot \sqrt{U}} \]
      +-commutative [=>] 34.2	\[ \sqrt{2 \cdot \left(n \cdot \color{blue}{\left(\ell \cdot \left(\frac{\ell}{Om} \cdot -2\right) + t\right)}\right)} \cdot \sqrt{U} \]
      fma-def [=>] 34.2	\[ \sqrt{2 \cdot \left(n \cdot \color{blue}{\mathsf{fma}\left(\ell, \frac{\ell}{Om} \cdot -2, t\right)}\right)} \cdot \sqrt{U} \]
      div-inv [=>] 34.2	\[ \sqrt{2 \cdot \left(n \cdot \mathsf{fma}\left(\ell, \color{blue}{\left(\ell \cdot \frac{1}{Om}\right)} \cdot -2, t\right)\right)} \cdot \sqrt{U} \]
      associate-*l* [=>] 34.2	\[ \sqrt{2 \cdot \left(n \cdot \mathsf{fma}\left(\ell, \color{blue}{\ell \cdot \left(\frac{1}{Om} \cdot -2\right)}, t\right)\right)} \cdot \sqrt{U} \]
      associate-*l/ [=>] 34.2	\[ \sqrt{2 \cdot \left(n \cdot \mathsf{fma}\left(\ell, \ell \cdot \color{blue}{\frac{1 \cdot -2}{Om}}, t\right)\right)} \cdot \sqrt{U} \]
      metadata-eval [=>] 34.2	\[ \sqrt{2 \cdot \left(n \cdot \mathsf{fma}\left(\ell, \ell \cdot \frac{\color{blue}{-2}}{Om}, t\right)\right)} \cdot \sqrt{U} \]

   if -1.70000000000000001e-51 < n < -2.90000000000000017e-129 or -5.9999999999999996e-183 < n < -1.3500000000000001e-256

   1. Initial program 45.7%

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

   2. Simplified 47.9%

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

      [Start] 45.7	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 46.3	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 46.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      sub-neg [=>] 46.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(-\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [<=] 46.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      cancel-sign-sub [<=] 46.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      cancel-sign-sub [=>] 46.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-/l* [=>] 49.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 47.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Applied egg-rr 45.7%

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

      [Start] 47.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)} \]
      associate-*l* [=>] 47.9	\[ \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)\right)}} \]
      sqrt-prod [=>] 47.7	\[ \color{blue}{\sqrt{2} \cdot \sqrt{n \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)}} \]
      associate-*r* [=>] 47.0	\[ \sqrt{2} \cdot \sqrt{\color{blue}{\left(n \cdot U\right) \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)}} \]
      fma-def [=>] 47.0	\[ \sqrt{2} \cdot \sqrt{\left(n \cdot U\right) \cdot \left(t - \color{blue}{\mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)}\right)} \]
      associate-/r/ [=>] 47.0	\[ \sqrt{2} \cdot \sqrt{\left(n \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \color{blue}{\frac{\ell}{Om} \cdot \ell}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l/ [=>] 43.8	\[ \sqrt{2} \cdot \sqrt{\left(n \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \color{blue}{\frac{\ell \cdot \ell}{Om}}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [=>] 43.8	\[ \sqrt{2} \cdot \sqrt{\left(n \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right) \cdot n}\right)\right)} \]
      associate-*l* [=>] 45.7	\[ \sqrt{2} \cdot \sqrt{\left(n \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U - U*\right) \cdot n\right)}\right)\right)} \]

   4. Simplified 51.6%

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

      [Start] 45.7	\[ \sqrt{2} \cdot \sqrt{\left(n \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U - U*\right) \cdot n\right)\right)\right)} \]
      *-commutative [=>] 45.7	\[ \sqrt{2} \cdot \sqrt{\color{blue}{\left(U \cdot n\right)} \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U - U*\right) \cdot n\right)\right)\right)} \]
      associate-*l* [=>] 49.9	\[ \sqrt{2} \cdot \sqrt{\color{blue}{U \cdot \left(n \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U - U*\right) \cdot n\right)\right)\right)\right)}} \]
      associate-/l* [=>] 54.6	\[ \sqrt{2} \cdot \sqrt{U \cdot \left(n \cdot \left(t - \mathsf{fma}\left(2, \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U - U*\right) \cdot n\right)\right)\right)\right)} \]
      associate-*r* [=>] 51.6	\[ \sqrt{2} \cdot \sqrt{U \cdot \left(n \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right) \cdot n}\right)\right)\right)} \]
      *-commutative [=>] 51.6	\[ \sqrt{2} \cdot \sqrt{U \cdot \left(n \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]
      *-commutative [=>] 51.6	\[ \sqrt{2} \cdot \sqrt{U \cdot \left(n \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, n \cdot \color{blue}{\left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right)\right)\right)} \]

   if -2.90000000000000017e-129 < n < -4.5000000000000001e-145

   1. Initial program 41.8%

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

   2. Simplified 47.6%

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

      [Start] 41.8	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 41.8	\[ \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 41.8	\[ \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      *-commutative [=>] 41.8	\[ \sqrt{2 \cdot \color{blue}{\left(\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot \left(n \cdot U\right)\right)}} \]

   3. Taylor expanded in l around inf 11.2%

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

   4. Taylor expanded in n around 0 8.9%

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

   5. Simplified 10.7%

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

      [Start] 8.9	\[ \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{-2 \cdot \frac{n \cdot U}{Om}} \]
      associate-*r/ [=>] 8.9	\[ \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\color{blue}{\frac{-2 \cdot \left(n \cdot U\right)}{Om}}} \]
      associate-*l/ [<=] 8.9	\[ \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\color{blue}{\frac{-2}{Om} \cdot \left(n \cdot U\right)}} \]
      *-commutative [<=] 8.9	\[ \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\color{blue}{\left(n \cdot U\right) \cdot \frac{-2}{Om}}} \]
      associate-*l* [=>] 10.7	\[ \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\color{blue}{n \cdot \left(U \cdot \frac{-2}{Om}\right)}} \]

   if -4.5000000000000001e-145 < n < -5.9999999999999996e-183

   1. Initial program 45.6%

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

   2. Simplified 51.8%

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

      [Start] 45.6	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 45.6	\[ \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 45.6	\[ \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      *-commutative [=>] 45.6	\[ \sqrt{2 \cdot \color{blue}{\left(\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot \left(n \cdot U\right)\right)}} \]

   3. Taylor expanded in l around 0 40.5%

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

   4. Applied egg-rr 12.9%

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

      [Start] 40.5	\[ \sqrt{2} \cdot \sqrt{n \cdot \left(t \cdot U\right)} \]
      expm1-log1p-u [=>] 40.0	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2} \cdot \sqrt{n \cdot \left(t \cdot U\right)}\right)\right)} \]
      expm1-udef [=>] 12.9	\[ \color{blue}{e^{\mathsf{log1p}\left(\sqrt{2} \cdot \sqrt{n \cdot \left(t \cdot U\right)}\right)} - 1} \]
      sub-neg [=>] 12.9	\[ \color{blue}{e^{\mathsf{log1p}\left(\sqrt{2} \cdot \sqrt{n \cdot \left(t \cdot U\right)}\right)} + \left(-1\right)} \]
      sqrt-unprod [=>] 12.9	\[ e^{\mathsf{log1p}\left(\color{blue}{\sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)}}\right)} + \left(-1\right) \]
      *-commutative [=>] 12.9	\[ e^{\mathsf{log1p}\left(\sqrt{2 \cdot \color{blue}{\left(\left(t \cdot U\right) \cdot n\right)}}\right)} + \left(-1\right) \]
      associate-*r* [=>] 12.9	\[ e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(2 \cdot \left(t \cdot U\right)\right) \cdot n}}\right)} + \left(-1\right) \]
      *-commutative [=>] 12.9	\[ e^{\mathsf{log1p}\left(\sqrt{\color{blue}{n \cdot \left(2 \cdot \left(t \cdot U\right)\right)}}\right)} + \left(-1\right) \]
      *-commutative [=>] 12.9	\[ e^{\mathsf{log1p}\left(\sqrt{n \cdot \color{blue}{\left(\left(t \cdot U\right) \cdot 2\right)}}\right)} + \left(-1\right) \]
      metadata-eval [=>] 12.9	\[ e^{\mathsf{log1p}\left(\sqrt{n \cdot \left(\left(t \cdot U\right) \cdot 2\right)}\right)} + \color{blue}{-1} \]

   5. Simplified 44.5%

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

      [Start] 12.9	\[ e^{\mathsf{log1p}\left(\sqrt{n \cdot \left(\left(t \cdot U\right) \cdot 2\right)}\right)} + -1 \]
      metadata-eval [<=] 12.9	\[ e^{\mathsf{log1p}\left(\sqrt{n \cdot \left(\left(t \cdot U\right) \cdot 2\right)}\right)} + \color{blue}{\left(-1\right)} \]
      sub-neg [<=] 12.9	\[ \color{blue}{e^{\mathsf{log1p}\left(\sqrt{n \cdot \left(\left(t \cdot U\right) \cdot 2\right)}\right)} - 1} \]
      expm1-def [=>] 40.1	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{n \cdot \left(\left(t \cdot U\right) \cdot 2\right)}\right)\right)} \]
      expm1-log1p [=>] 40.6	\[ \color{blue}{\sqrt{n \cdot \left(\left(t \cdot U\right) \cdot 2\right)}} \]
      associate-*l* [=>] 40.6	\[ \sqrt{n \cdot \color{blue}{\left(t \cdot \left(U \cdot 2\right)\right)}} \]
      associate-*r* [=>] 44.5	\[ \sqrt{\color{blue}{\left(n \cdot t\right) \cdot \left(U \cdot 2\right)}} \]

   if -1.3500000000000001e-256 < n < -3.999999999999988e-310

   1. Initial program 34.6%

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

   2. Simplified 33.5%

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

      [Start] 34.6	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 33.1	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 33.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      sub-neg [=>] 33.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(-\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [<=] 33.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      cancel-sign-sub [<=] 33.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      cancel-sign-sub [=>] 33.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-/l* [=>] 37.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 33.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in n around 0 33.2%

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

   4. Simplified 37.0%

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

      [Start] 33.2	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot U\right)\right)} \]
      *-commutative [=>] 33.2	\[ \sqrt{2 \cdot \left(n \cdot \color{blue}{\left(U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\right)} \]
      cancel-sign-sub-inv [=>] 33.2	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \color{blue}{\left(t + \left(-2\right) \cdot \frac{{\ell}^{2}}{Om}\right)}\right)\right)} \]
      metadata-eval [=>] 33.2	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \color{blue}{-2} \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)} \]
      unpow2 [=>] 33.2	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right)\right)\right)} \]
      associate-/l* [=>] 37.0	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}\right)\right)\right)} \]

   if -3.999999999999988e-310 < n < 4.50000000000000004e-140 or 2.1499999999999999e-63 < n

   1. Initial program 45.1%

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

   2. Simplified 47.5%

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

      [Start] 45.1	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 44.5	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 44.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      sub-neg [=>] 44.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(-\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [<=] 44.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      cancel-sign-sub [<=] 44.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      cancel-sign-sub [=>] 44.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-/l* [=>] 48.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 47.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Applied egg-rr 49.7%

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

      [Start] 47.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)} \]
      sqrt-prod [=>] 60.2	\[ \color{blue}{\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)}} \]
      fma-def [=>] 60.2	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \color{blue}{\mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)}\right)} \]
      associate-/r/ [=>] 60.2	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \color{blue}{\frac{\ell}{Om} \cdot \ell}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l/ [=>] 55.9	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \color{blue}{\frac{\ell \cdot \ell}{Om}}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [=>] 55.9	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right) \cdot n}\right)\right)} \]
      associate-*l* [=>] 49.7	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U - U*\right) \cdot n\right)}\right)\right)} \]

   4. Simplified 60.2%

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

      [Start] 49.7	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U - U*\right) \cdot n\right)\right)\right)} \]
      associate-/l* [=>] 54.0	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U - U*\right) \cdot n\right)\right)\right)} \]
      associate-*r* [=>] 60.2	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right) \cdot n}\right)\right)} \]
      *-commutative [=>] 60.2	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)} \]
      *-commutative [=>] 60.2	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, n \cdot \color{blue}{\left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right)\right)} \]

   if 4.50000000000000004e-140 < n < 2.1499999999999999e-63

   1. Initial program 52.9%

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

   2. Taylor expanded in n around 0 50.2%

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

   3. Simplified 56.9%

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

      [Start] 50.2	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot U\right)\right)} \]
      associate-*r* [=>] 53.0	\[ \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right)}} \]
      unpow2 [=>] 53.0	\[ \sqrt{2 \cdot \left(\left(n \cdot \left(t - 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right)\right) \cdot U\right)} \]
      associate-*r/ [<=] 56.9	\[ \sqrt{2 \cdot \left(\left(n \cdot \left(t - 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)}\right)\right) \cdot U\right)} \]
      cancel-sign-sub-inv [=>] 56.9	\[ \sqrt{2 \cdot \left(\left(n \cdot \color{blue}{\left(t + \left(-2\right) \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\right) \cdot U\right)} \]
      metadata-eval [=>] 56.9	\[ \sqrt{2 \cdot \left(\left(n \cdot \left(t + \color{blue}{-2} \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right) \cdot U\right)} \]
      *-commutative [<=] 56.9	\[ \sqrt{2 \cdot \left(\left(n \cdot \left(t + \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2}\right)\right) \cdot U\right)} \]
      associate-*l* [=>] 56.9	\[ \sqrt{2 \cdot \left(\left(n \cdot \left(t + \color{blue}{\ell \cdot \left(\frac{\ell}{Om} \cdot -2\right)}\right)\right) \cdot U\right)} \]

   4. Taylor expanded in n around 0 48.8%

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

   5. Simplified 52.2%

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

      [Start] 48.8	\[ \sqrt{2 \cdot \left(\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot \left(n \cdot U\right)\right)} \]
      *-commutative [=>] 48.8	\[ \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot U\right) \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]
      associate-*r/ [=>] 48.8	\[ \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \color{blue}{\frac{-2 \cdot {\ell}^{2}}{Om}}\right)\right)} \]
      unpow2 [=>] 48.8	\[ \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{-2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{Om}\right)\right)} \]
      *-commutative [<=] 48.8	\[ \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{\color{blue}{\left(\ell \cdot \ell\right) \cdot -2}}{Om}\right)\right)} \]
      associate-*r* [<=] 48.8	\[ \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{\color{blue}{\ell \cdot \left(\ell \cdot -2\right)}}{Om}\right)\right)} \]
      associate-/l* [=>] 52.2	\[ \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \color{blue}{\frac{\ell}{\frac{Om}{\ell \cdot -2}}}\right)\right)} \]
      associate-/r/ [=>] 52.2	\[ \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)}\right)\right)} \]

   6. Applied egg-rr 52.2%

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

      [Start] 52.2	\[ \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)} \]
      clear-num [=>] 52.2	\[ \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \color{blue}{\frac{1}{\frac{Om}{\ell}}} \cdot \left(\ell \cdot -2\right)\right)\right)} \]
      associate-*l/ [=>] 52.2	\[ \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \color{blue}{\frac{1 \cdot \left(\ell \cdot -2\right)}{\frac{Om}{\ell}}}\right)\right)} \]
      *-un-lft-identity [<=] 52.2	\[ \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{\color{blue}{\ell \cdot -2}}{\frac{Om}{\ell}}\right)\right)} \]
      *-commutative [=>] 52.2	\[ \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{\color{blue}{-2 \cdot \ell}}{\frac{Om}{\ell}}\right)\right)} \]
      associate-/l* [=>] 52.2	\[ \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \color{blue}{\frac{-2}{\frac{\frac{Om}{\ell}}{\ell}}}\right)\right)} \]

3. Recombined 8 regimes into one program.

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.4 \cdot 10^{+26}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot 2\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;n \leq -1.7 \cdot 10^{-51}:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{2 \cdot \left(n \cdot \mathsf{fma}\left(\ell, \ell \cdot \frac{-2}{Om}, t\right)\right)}\\ \mathbf{elif}\;n \leq -2.9 \cdot 10^{-129}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{U \cdot \left(n \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq -4.5 \cdot 10^{-145}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{n \cdot \left(U \cdot \frac{-2}{Om}\right)}\\ \mathbf{elif}\;n \leq -6 \cdot 10^{-183}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \mathbf{elif}\;n \leq -1.35 \cdot 10^{-256}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{U \cdot \left(n \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;n \leq 4.5 \cdot 10^{-140} \lor \neg \left(n \leq 2.15 \cdot 10^{-63}\right):\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{-2}{\frac{\frac{Om}{\ell}}{\ell}}\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	53.3%
Cost	33800

\[\begin{array}{l} t_1 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := n \cdot \left(t_1 \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;n \leq -2.4 \cdot 10^{+26}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot 2\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot t_1\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;n \leq -4.8 \cdot 10^{-71}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{n \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, t_2\right)\right)} \cdot \sqrt{U}\right)\\ \mathbf{elif}\;n \leq 5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \ell \cdot \left(\frac{\ell}{Om} \cdot -2\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq 4.5 \cdot 10^{-140} \lor \neg \left(n \leq 4.3 \cdot 10^{-63}\right):\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, t_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{-2}{\frac{\frac{Om}{\ell}}{\ell}}\right)\right)}\\ \end{array} \]

Alternative 2
Accuracy	56.1%
Cost	30728

\[\begin{array}{l} t_1 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := \left(\left(n \cdot 2\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot t_1\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_2 \leq 0:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(t_1 \cdot \left(U* - U\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;t_2 \leq 10^{+307}:\\ \;\;\;\;\sqrt{t_2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{-2}{Om} + \frac{n}{Om \cdot Om} \cdot \left(U* - U\right)\right)}\\ \end{array} \]

Alternative 3
Accuracy	54.2%
Cost	14672

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \left(2 \cdot t_1 + n \cdot \left(t_1 \cdot \frac{U - U*}{Om}\right)\right)\right)\right)}\\ t_3 := \ell \cdot \sqrt{2}\\ \mathbf{if}\;\ell \leq -2.95 \cdot 10^{+158}:\\ \;\;\;\;t_3 \cdot \left(-\sqrt{\left(n \cdot U\right) \cdot \left(\frac{-2}{Om} + \frac{n}{\frac{Om \cdot Om}{U* - U}}\right)}\right)\\ \mathbf{elif}\;\ell \leq -1300:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 1.7 \cdot 10^{-28}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot 2\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \frac{n}{\frac{Om}{\frac{\ell \cdot \left(\ell \cdot U*\right)}{Om}}}\right)}\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{+120}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{-2}{Om} + \frac{n}{Om \cdot Om} \cdot \left(U* - U\right)\right)}\\ \end{array} \]

Alternative 4
Accuracy	54.3%
Cost	14672

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \left(2 \cdot t_1 + n \cdot \left(t_1 \cdot \frac{U - U*}{Om}\right)\right)\right)\right)}\\ t_3 := \ell \cdot \sqrt{2}\\ t_4 := \sqrt{\left(n \cdot U\right) \cdot \left(\frac{-2}{Om} + \frac{n}{Om \cdot Om} \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;\ell \leq -1.6 \cdot 10^{+164}:\\ \;\;\;\;t_3 \cdot \left(-t_4\right)\\ \mathbf{elif}\;\ell \leq -10500:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 3.7 \cdot 10^{-34}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot 2\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \frac{n}{\frac{Om}{\frac{\ell \cdot \left(\ell \cdot U*\right)}{Om}}}\right)}\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{+120}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot t_4\\ \end{array} \]

Alternative 5
Accuracy	52.0%
Cost	14540

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \left(2 \cdot t_1 + n \cdot \left(t_1 \cdot \frac{U - U*}{Om}\right)\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -140:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{-32}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot 2\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \frac{n}{\frac{Om}{\frac{\ell \cdot \left(\ell \cdot U*\right)}{Om}}}\right)}\\ \mathbf{elif}\;\ell \leq 2.45 \cdot 10^{+120}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{-2}{Om} + \frac{n}{Om \cdot Om} \cdot \left(U* - U\right)\right)}\\ \end{array} \]

Alternative 6
Accuracy	52.8%
Cost	14412

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \left(2 \cdot t_1 + n \cdot \left(t_1 \cdot \frac{U - U*}{Om}\right)\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -1400:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 1.15 \cdot 10^{-35}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot 2\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \frac{n}{\frac{Om}{\frac{\ell \cdot \left(\ell \cdot U*\right)}{Om}}}\right)}\\ \mathbf{elif}\;\ell \leq 2.1 \cdot 10^{+128}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{n \cdot \left(\frac{U}{Om} \cdot \left(-2 + n \cdot \frac{U* - U}{Om}\right)\right)}\\ \end{array} \]

Alternative 7
Accuracy	51.3%
Cost	13908

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot t_1 + n \cdot \left(t_1 \cdot \frac{U* - U}{Om}\right)\right)\right)\right)}\\ t_3 := \sqrt{n \cdot t} \cdot \sqrt{2 \cdot U}\\ \mathbf{if}\;U \leq -1.12 \cdot 10^{-28}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)}\\ \mathbf{elif}\;U \leq 5.3 \cdot 10^{-250}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;U \leq 7.9 \cdot 10^{-119}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;U \leq 2.8 \cdot 10^{-15}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;U \leq 1.15 \cdot 10^{+208}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \ell \cdot \left(\frac{\ell}{Om} \cdot -2\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 8
Accuracy	51.3%
Cost	13908

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot t_1 + n \cdot \left(t_1 \cdot \frac{U* - U}{Om}\right)\right)\right)\right)}\\ \mathbf{if}\;U \leq -8.6 \cdot 10^{-29}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)}\\ \mathbf{elif}\;U \leq 1.5 \cdot 10^{-249}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;U \leq 8.2 \cdot 10^{-115}:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{\left(n \cdot 2\right) \cdot t}\\ \mathbf{elif}\;U \leq 6.6 \cdot 10^{-15}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;U \leq 1.6 \cdot 10^{+208}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \ell \cdot \left(\frac{\ell}{Om} \cdot -2\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot t} \cdot \sqrt{2 \cdot U}\\ \end{array} \]

Alternative 9
Accuracy	51.1%
Cost	13900

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \left(2 \cdot t_1 + n \cdot \left(t_1 \cdot \frac{U - U*}{Om}\right)\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -230000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 1.9 \cdot 10^{-33}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot 2\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \frac{n}{\frac{Om}{\frac{\ell \cdot \left(\ell \cdot U*\right)}{Om}}}\right)}\\ \mathbf{elif}\;\ell \leq 3.8 \cdot 10^{+208}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{-2 \cdot \frac{n \cdot U}{Om}}\\ \end{array} \]

Alternative 10
Accuracy	51.9%
Cost	8520

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;U \leq -1.12 \cdot 10^{-28}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)}\\ \mathbf{elif}\;U \leq 1.18 \cdot 10^{-14}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot t_1 + n \cdot \left(t_1 \cdot \frac{U* - U}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{-2}{\frac{\frac{Om}{\ell}}{\ell}}\right)\right)}\\ \end{array} \]

Alternative 11
Accuracy	48.2%
Cost	8400

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{-2}{Om} + \frac{n}{\frac{Om}{\frac{U*}{Om}}}\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -1.86 \cdot 10^{+118}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(-2 \cdot \frac{n \cdot \ell}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{elif}\;\ell \leq -2.6 \cdot 10^{-118}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 10^{-86}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.16 \cdot 10^{+147}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right) \cdot \left(\frac{-2}{Om} + \frac{U* - U}{Om} \cdot \frac{n}{Om}\right)\right)}\\ \end{array} \]

Alternative 12
Accuracy	49.0%
Cost	8392

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.9 \cdot 10^{+113}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(-2 \cdot \frac{n \cdot \ell}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.4 \cdot 10^{+86}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot 2\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \frac{n}{\frac{Om}{\frac{\ell \cdot \left(\ell \cdot U*\right)}{Om}}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right) \cdot \left(\frac{-2}{Om} + \frac{U* - U}{Om} \cdot \frac{n}{Om}\right)\right)}\\ \end{array} \]

Alternative 13
Accuracy	47.1%
Cost	8136

\[\begin{array}{l} \mathbf{if}\;U \leq -1.75 \cdot 10^{-157}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)}\\ \mathbf{elif}\;U \leq -5.2 \cdot 10^{-230}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right) \cdot \left(\frac{-2}{Om} + \frac{U* - U}{Om} \cdot \frac{n}{Om}\right)\right)}\\ \mathbf{elif}\;U \leq 9.5 \cdot 10^{-88}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{-2}{\frac{\frac{Om}{\ell}}{\ell}}\right)\right)}\\ \end{array} \]

Alternative 14
Accuracy	47.5%
Cost	7876

\[\begin{array}{l} \mathbf{if}\;n \leq -4 \cdot 10^{+101}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(n \cdot \left(\ell \cdot \frac{U* - U}{Om \cdot Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq 4 \cdot 10^{-216}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \ell \cdot \left(\frac{\ell}{Om} \cdot -2\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{-2}{\frac{\frac{Om}{\ell}}{\ell}}\right)\right)}\\ \end{array} \]

Alternative 15
Accuracy	47.1%
Cost	7748

\[\begin{array}{l} \mathbf{if}\;n \leq -4.9 \cdot 10^{+98}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{n}{Om \cdot Om} \cdot \left(\ell \cdot \left(\ell \cdot U*\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq 3.4 \cdot 10^{-214}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \ell \cdot \left(\frac{\ell}{Om} \cdot -2\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{-2}{\frac{\frac{Om}{\ell}}{\ell}}\right)\right)}\\ \end{array} \]

Alternative 16
Accuracy	37.1%
Cost	7628

\[\begin{array}{l} t_1 := \sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{if}\;t \leq -8 \cdot 10^{+189}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -7.2 \cdot 10^{-182}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \mathbf{elif}\;t \leq 5.1 \cdot 10^{-283}:\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \frac{n}{\frac{Om}{U \cdot \left(\ell \cdot \ell\right)}}\right)}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-190} \lor \neg \left(t \leq 2.5 \cdot 10^{+226}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]

Alternative 17
Accuracy	37.2%
Cost	7628

\[\begin{array}{l} t_1 := \sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{if}\;t \leq -6 \cdot 10^{+188}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -3.2 \cdot 10^{-183}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-282}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(-2 \cdot \frac{n}{\frac{Om}{\ell \cdot \ell}}\right)\right)}\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-164} \lor \neg \left(t \leq 3 \cdot 10^{+226}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]

Alternative 18
Accuracy	45.4%
Cost	7625

\[\begin{array}{l} \mathbf{if}\;t \leq -4.2 \cdot 10^{+183} \lor \neg \left(t \leq 6.2 \cdot 10^{+226}\right):\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \end{array} \]

Alternative 19
Accuracy	45.7%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{+184}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;t \leq -7.8 \cdot 10^{-81}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)}\\ \end{array} \]

Alternative 20
Accuracy	37.5%
Cost	7509

\[\begin{array}{l} t_1 := \sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{if}\;t \leq -7 \cdot 10^{+189}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2.2 \cdot 10^{-180}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{-284}:\\ \;\;\;\;\sqrt{\left(\frac{n}{Om} \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right) \cdot -4}\\ \mathbf{elif}\;t \leq 3.7 \cdot 10^{-190} \lor \neg \left(t \leq 1.7 \cdot 10^{+226}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]

Alternative 21
Accuracy	43.8%
Cost	7497

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.65 \cdot 10^{+24} \lor \neg \left(\ell \leq 1.65 \cdot 10^{+90}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(-2 \cdot \frac{n \cdot \ell}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(n \cdot \left(2 \cdot U\right)\right)}\\ \end{array} \]

Alternative 22
Accuracy	37.9%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{+201} \lor \neg \left(t \leq -5 \cdot 10^{-77}\right):\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]

Alternative 23
Accuracy	37.8%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;t \leq -1.12 \cdot 10^{+201}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-76}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(n \cdot \left(2 \cdot U\right)\right)}\\ \end{array} \]

Alternative 24
Accuracy	37.8%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{+189}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-76}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(n \cdot \left(2 \cdot U\right)\right)}\\ \end{array} \]

Alternative 25
Accuracy	37.2%
Cost	6848

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

Reproduce

herbie shell --seed 2023138 
(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*))))))