Toniolo and Linder, Equation (13)

Average Accuracy: 46.2% → 52.0%

Time: 38.8s

Precision: binary64

Cost: 27796

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 := \ell \cdot \frac{\ell}{Om}\\ t_2 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_3 := n \cdot t_2\\ \mathbf{if}\;\ell \leq -4.6 \cdot 10^{+179}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{\frac{Om \cdot Om}{U* - U}} + \frac{-2}{Om}\right)} \cdot \left(\ell \cdot \left(-\sqrt{2}\right)\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-37}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om \cdot \frac{Om}{\ell}}, n \cdot \left(U* - U\right), \mathsf{fma}\left(t_1, -2, t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -2.05 \cdot 10^{-258}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot \frac{\ell}{\frac{Om}{\ell}} - n \cdot \left(t_2 \cdot \left(U - U*\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -3.2 \cdot 10^{-280}:\\ \;\;\;\;{\left({\left(\frac{-1}{U}\right)}^{-0.16666666666666666} \cdot {\left(n \cdot \left(-2 \cdot t\right)\right)}^{0.16666666666666666}\right)}^{3}\\ \mathbf{elif}\;\ell \leq -3.4 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, t_1, \left(U - U*\right) \cdot t_3\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.4 \cdot 10^{-280}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 5.7 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(U* - U\right) \cdot t_3\right)}\\ \mathbf{elif}\;\ell \leq 2.2 \cdot 10^{+107}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \left(\frac{n}{Om} \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{+110}:\\ \;\;\;\;\left|\sqrt{2} \cdot \left(\sqrt{U \cdot \left(U* - U\right)} \cdot \left(\ell \cdot \frac{n}{Om}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\left(U* - U\right) \cdot \frac{n}{Om \cdot Om} + \frac{-2}{Om}\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 (/ l Om))) (t_2 (pow (/ l Om) 2.0)) (t_3 (* n t_2)))
   (if (<= l -4.6e+179)
     (*
      (sqrt (* (* n U) (+ (/ n (/ (* Om Om) (- U* U))) (/ -2.0 Om))))
      (* l (- (sqrt 2.0))))
     (if (<= l -5e-37)
       (sqrt
        (*
         2.0
         (*
          (* n U)
          (fma (/ l (* Om (/ Om l))) (* n (- U* U)) (fma t_1 -2.0 t)))))
       (if (<= l -2.05e-258)
         (sqrt
          (*
           (* 2.0 n)
           (* U (+ t (- (* -2.0 (/ l (/ Om l))) (* n (* t_2 (- U U*))))))))
         (if (<= l -3.2e-280)
           (pow
            (*
             (pow (/ -1.0 U) -0.16666666666666666)
             (pow (* n (* -2.0 t)) 0.16666666666666666))
            3.0)
           (if (<= l -3.4e-308)
             (*
              (sqrt (* 2.0 n))
              (sqrt (* U (- t (fma 2.0 t_1 (* (- U U*) t_3))))))
             (if (<= l 3.4e-280)
               (sqrt (* 2.0 (* U (* n t))))
               (if (<= l 5.7e+15)
                 (sqrt
                  (*
                   (* U (* 2.0 n))
                   (+ (+ t (* -2.0 (/ (* l l) Om))) (* (- U* U) t_3))))
                 (if (<= l 2.2e+107)
                   (sqrt
                    (fma
                     2.0
                     (* n (* U t))
                     (* -4.0 (* (/ n Om) (* l (* l U))))))
                   (if (<= l 2.7e+110)
                     (fabs
                      (* (sqrt 2.0) (* (sqrt (* U (- U* U))) (* l (/ n Om)))))
                     (*
                      (sqrt 2.0)
                      (*
                       l
                       (sqrt
                        (*
                         (* n U)
                         (+
                          (* (- U* U) (/ n (* Om Om)))
                          (/ -2.0 Om)))))))))))))))))

C

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

↓

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * (l / Om);
	double t_2 = pow((l / Om), 2.0);
	double t_3 = n * t_2;
	double tmp;
	if (l <= -4.6e+179) {
		tmp = sqrt(((n * U) * ((n / ((Om * Om) / (U_42_ - U))) + (-2.0 / Om)))) * (l * -sqrt(2.0));
	} else if (l <= -5e-37) {
		tmp = sqrt((2.0 * ((n * U) * fma((l / (Om * (Om / l))), (n * (U_42_ - U)), fma(t_1, -2.0, t)))));
	} else if (l <= -2.05e-258) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((-2.0 * (l / (Om / l))) - (n * (t_2 * (U - U_42_))))))));
	} else if (l <= -3.2e-280) {
		tmp = pow((pow((-1.0 / U), -0.16666666666666666) * pow((n * (-2.0 * t)), 0.16666666666666666)), 3.0);
	} else if (l <= -3.4e-308) {
		tmp = sqrt((2.0 * n)) * sqrt((U * (t - fma(2.0, t_1, ((U - U_42_) * t_3)))));
	} else if (l <= 3.4e-280) {
		tmp = sqrt((2.0 * (U * (n * t))));
	} else if (l <= 5.7e+15) {
		tmp = sqrt(((U * (2.0 * n)) * ((t + (-2.0 * ((l * l) / Om))) + ((U_42_ - U) * t_3))));
	} else if (l <= 2.2e+107) {
		tmp = sqrt(fma(2.0, (n * (U * t)), (-4.0 * ((n / Om) * (l * (l * U))))));
	} else if (l <= 2.7e+110) {
		tmp = fabs((sqrt(2.0) * (sqrt((U * (U_42_ - U))) * (l * (n / Om)))));
	} else {
		tmp = sqrt(2.0) * (l * sqrt(((n * U) * (((U_42_ - U) * (n / (Om * Om))) + (-2.0 / Om)))));
	}
	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(l / Om))
	t_2 = Float64(l / Om) ^ 2.0
	t_3 = Float64(n * t_2)
	tmp = 0.0
	if (l <= -4.6e+179)
		tmp = Float64(sqrt(Float64(Float64(n * U) * Float64(Float64(n / Float64(Float64(Om * Om) / Float64(U_42_ - U))) + Float64(-2.0 / Om)))) * Float64(l * Float64(-sqrt(2.0))));
	elseif (l <= -5e-37)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * fma(Float64(l / Float64(Om * Float64(Om / l))), Float64(n * Float64(U_42_ - U)), fma(t_1, -2.0, t)))));
	elseif (l <= -2.05e-258)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(-2.0 * Float64(l / Float64(Om / l))) - Float64(n * Float64(t_2 * Float64(U - U_42_))))))));
	elseif (l <= -3.2e-280)
		tmp = Float64((Float64(-1.0 / U) ^ -0.16666666666666666) * (Float64(n * Float64(-2.0 * t)) ^ 0.16666666666666666)) ^ 3.0;
	elseif (l <= -3.4e-308)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * Float64(t - fma(2.0, t_1, Float64(Float64(U - U_42_) * t_3))))));
	elseif (l <= 3.4e-280)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t))));
	elseif (l <= 5.7e+15)
		tmp = sqrt(Float64(Float64(U * Float64(2.0 * n)) * Float64(Float64(t + Float64(-2.0 * Float64(Float64(l * l) / Om))) + Float64(Float64(U_42_ - U) * t_3))));
	elseif (l <= 2.2e+107)
		tmp = sqrt(fma(2.0, Float64(n * Float64(U * t)), Float64(-4.0 * Float64(Float64(n / Om) * Float64(l * Float64(l * U))))));
	elseif (l <= 2.7e+110)
		tmp = abs(Float64(sqrt(2.0) * Float64(sqrt(Float64(U * Float64(U_42_ - U))) * Float64(l * Float64(n / Om)))));
	else
		tmp = Float64(sqrt(2.0) * Float64(l * sqrt(Float64(Float64(n * U) * Float64(Float64(Float64(U_42_ - U) * Float64(n / Float64(Om * Om))) + Float64(-2.0 / Om))))));
	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[(l / Om), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(n * t$95$2), $MachinePrecision]}, If[LessEqual[l, -4.6e+179], N[(N[Sqrt[N[(N[(n * U), $MachinePrecision] * N[(N[(n / N[(N[(Om * Om), $MachinePrecision] / N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5e-37], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(N[(l / N[(Om * N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * -2.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, -2.05e-258], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(-2.0 * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(n * N[(t$95$2 * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, -3.2e-280], N[Power[N[(N[Power[N[(-1.0 / U), $MachinePrecision], -0.16666666666666666], $MachinePrecision] * N[Power[N[(n * N[(-2.0 * t), $MachinePrecision]), $MachinePrecision], 0.16666666666666666], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[l, -3.4e-308], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t - N[(2.0 * t$95$1 + N[(N[(U - U$42$), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.4e-280], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 5.7e+15], N[Sqrt[N[(N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * N[(N[(t + N[(-2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(U$42$ - U), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 2.2e+107], N[Sqrt[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * N[(N[(n / Om), $MachinePrecision] * N[(l * N[(l * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 2.7e+110], N[Abs[N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(U * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(l * N[Sqrt[N[(N[(n * U), $MachinePrecision] * N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(n / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]

Derivation

1. Split input into 10 regimes

2. if l < -4.59999999999999988e179

   1. Initial program 0.0%

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

   2. Simplified 17.3%

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

      [Start] 0.0	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 0.0	\[ \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 0.0	\[ \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      *-commutative [=>] 0.0	\[ \sqrt{2 \cdot \color{blue}{\left(\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot \left(n \cdot U\right)\right)}} \]

   3. Taylor expanded in l around -inf 45.4%

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

   4. Simplified 46.7%

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

      [Start] 45.4	\[ -1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(\left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)}\right) \]
      mul-1-neg [=>] 45.4	\[ \color{blue}{-\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(\left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)}} \]
      *-commutative [<=] 45.4	\[ -\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \color{blue}{\left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}} \]
      associate-*r* [=>] 47.5	\[ -\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\color{blue}{\left(n \cdot U\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)}} \]
      associate-/l* [=>] 46.7	\[ -\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\color{blue}{\frac{n}{\frac{{Om}^{2}}{U* - U}}} - 2 \cdot \frac{1}{Om}\right)} \]
      unpow2 [=>] 46.7	\[ -\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{\frac{\color{blue}{Om \cdot Om}}{U* - U}} - 2 \cdot \frac{1}{Om}\right)} \]
      associate-*r/ [=>] 46.7	\[ -\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{\frac{Om \cdot Om}{U* - U}} - \color{blue}{\frac{2 \cdot 1}{Om}}\right)} \]
      metadata-eval [=>] 46.7	\[ -\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{\frac{Om \cdot Om}{U* - U}} - \frac{\color{blue}{2}}{Om}\right)} \]

   if -4.59999999999999988e179 < l < -4.9999999999999997e-37

   1. Initial program 45.3%

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

   2. Simplified 48.1%

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

      [Start] 45.3	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 45.3	\[ \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 45.3	\[ \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      *-commutative [=>] 45.3	\[ \sqrt{2 \cdot \color{blue}{\left(\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot \left(n \cdot U\right)\right)}} \]

   3. Applied egg-rr 48.0%

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

   if -4.9999999999999997e-37 < l < -2.05e-258

   1. Initial program 59.6%

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

   2. Simplified 57.5%

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

      [Start] 59.6	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 58.5	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 58.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      sub-neg [=>] 58.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(-\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [<=] 58.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      cancel-sign-sub [<=] 58.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      cancel-sign-sub [=>] 58.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-/l* [=>] 58.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 57.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   if -2.05e-258 < l < -3.2000000000000001e-280

   1. Initial program 61.8%

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

   2. Simplified 56.3%

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

      [Start] 61.8	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 61.8	\[ \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 61.8	\[ \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      *-commutative [=>] 61.8	\[ \sqrt{2 \cdot \color{blue}{\left(\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot \left(n \cdot U\right)\right)}} \]

   3. Taylor expanded in l around 0 50.8%

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

   4. Applied egg-rr 46.8%

      \[\leadsto \color{blue}{{\left({\left(\left(t \cdot U\right) \cdot \left(2 \cdot n\right)\right)}^{0.16666666666666666}\right)}^{3}} \]

   5. Taylor expanded in U around -inf 32.4%

      \[\leadsto {\color{blue}{\left(e^{0.16666666666666666 \cdot \left(-1 \cdot \log \left(\frac{-1}{U}\right) + \log \left(-2 \cdot \left(n \cdot t\right)\right)\right)}\right)}}^{3} \]

   6. Simplified 32.6%

      \[\leadsto {\color{blue}{\left({\left(\frac{-1}{U}\right)}^{-0.16666666666666666} \cdot {\left(n \cdot \left(t \cdot -2\right)\right)}^{0.16666666666666666}\right)}}^{3} \]
      Proof

      [Start] 32.4	\[ {\left(e^{0.16666666666666666 \cdot \left(-1 \cdot \log \left(\frac{-1}{U}\right) + \log \left(-2 \cdot \left(n \cdot t\right)\right)\right)}\right)}^{3} \]
      distribute-lft-in [=>] 32.2	\[ {\left(e^{\color{blue}{0.16666666666666666 \cdot \left(-1 \cdot \log \left(\frac{-1}{U}\right)\right) + 0.16666666666666666 \cdot \log \left(-2 \cdot \left(n \cdot t\right)\right)}}\right)}^{3} \]
      exp-sum [=>] 32.3	\[ {\color{blue}{\left(e^{0.16666666666666666 \cdot \left(-1 \cdot \log \left(\frac{-1}{U}\right)\right)} \cdot e^{0.16666666666666666 \cdot \log \left(-2 \cdot \left(n \cdot t\right)\right)}\right)}}^{3} \]
      *-commutative [=>] 32.3	\[ {\left(e^{\color{blue}{\left(-1 \cdot \log \left(\frac{-1}{U}\right)\right) \cdot 0.16666666666666666}} \cdot e^{0.16666666666666666 \cdot \log \left(-2 \cdot \left(n \cdot t\right)\right)}\right)}^{3} \]
      *-commutative [=>] 32.3	\[ {\left(e^{\color{blue}{\left(\log \left(\frac{-1}{U}\right) \cdot -1\right)} \cdot 0.16666666666666666} \cdot e^{0.16666666666666666 \cdot \log \left(-2 \cdot \left(n \cdot t\right)\right)}\right)}^{3} \]
      associate-*l* [=>] 32.3	\[ {\left(e^{\color{blue}{\log \left(\frac{-1}{U}\right) \cdot \left(-1 \cdot 0.16666666666666666\right)}} \cdot e^{0.16666666666666666 \cdot \log \left(-2 \cdot \left(n \cdot t\right)\right)}\right)}^{3} \]
      metadata-eval [=>] 32.3	\[ {\left(e^{\log \left(\frac{-1}{U}\right) \cdot \color{blue}{-0.16666666666666666}} \cdot e^{0.16666666666666666 \cdot \log \left(-2 \cdot \left(n \cdot t\right)\right)}\right)}^{3} \]
      metadata-eval [<=] 32.3	\[ {\left(e^{\log \left(\frac{-1}{U}\right) \cdot \color{blue}{\left(0.16666666666666666 \cdot -1\right)}} \cdot e^{0.16666666666666666 \cdot \log \left(-2 \cdot \left(n \cdot t\right)\right)}\right)}^{3} \]
      exp-to-pow [=>] 32.4	\[ {\left(\color{blue}{{\left(\frac{-1}{U}\right)}^{\left(0.16666666666666666 \cdot -1\right)}} \cdot e^{0.16666666666666666 \cdot \log \left(-2 \cdot \left(n \cdot t\right)\right)}\right)}^{3} \]
      metadata-eval [=>] 32.4	\[ {\left({\left(\frac{-1}{U}\right)}^{\color{blue}{-0.16666666666666666}} \cdot e^{0.16666666666666666 \cdot \log \left(-2 \cdot \left(n \cdot t\right)\right)}\right)}^{3} \]
      *-commutative [=>] 32.4	\[ {\left({\left(\frac{-1}{U}\right)}^{-0.16666666666666666} \cdot e^{\color{blue}{\log \left(-2 \cdot \left(n \cdot t\right)\right) \cdot 0.16666666666666666}}\right)}^{3} \]
      exp-to-pow [=>] 32.6	\[ {\left({\left(\frac{-1}{U}\right)}^{-0.16666666666666666} \cdot \color{blue}{{\left(-2 \cdot \left(n \cdot t\right)\right)}^{0.16666666666666666}}\right)}^{3} \]
      *-commutative [=>] 32.6	\[ {\left({\left(\frac{-1}{U}\right)}^{-0.16666666666666666} \cdot {\color{blue}{\left(\left(n \cdot t\right) \cdot -2\right)}}^{0.16666666666666666}\right)}^{3} \]
      associate-*l* [=>] 32.6	\[ {\left({\left(\frac{-1}{U}\right)}^{-0.16666666666666666} \cdot {\color{blue}{\left(n \cdot \left(t \cdot -2\right)\right)}}^{0.16666666666666666}\right)}^{3} \]

   if -3.2000000000000001e-280 < l < -3.39999999999999999e-308

   1. Initial program 57.9%

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

   2. Applied egg-rr 31.2%

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

   3. Simplified 37.7%

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

      [Start] 31.2	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*r* [=>] 37.7	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U - U*\right)}\right)\right)} \]
      *-commutative [<=] 37.7	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)} \cdot \left(U - U*\right)\right)\right)} \]
      *-commutative [=>] 37.7	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \color{blue}{\left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right)\right)} \]

   if -3.39999999999999999e-308 < l < 3.3999999999999998e-280

   1. Initial program 66.4%

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

   2. Simplified 59.3%

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

      [Start] 66.4	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 66.4	\[ \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 66.3	\[ \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      *-commutative [=>] 66.3	\[ \sqrt{2 \cdot \color{blue}{\left(\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot \left(n \cdot U\right)\right)}} \]

   3. Taylor expanded in l around 0 54.8%

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

   4. Applied egg-rr 29.4%

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

   5. Simplified 60.8%

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

      [Start] 29.4	\[ \sqrt{2 \cdot \left(e^{\mathsf{log1p}\left(n \cdot \left(t \cdot U\right)\right)} + -1\right)} \]
      metadata-eval [<=] 29.4	\[ \sqrt{2 \cdot \left(e^{\mathsf{log1p}\left(n \cdot \left(t \cdot U\right)\right)} + \color{blue}{\left(-1\right)}\right)} \]
      sub-neg [<=] 29.4	\[ \sqrt{2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(n \cdot \left(t \cdot U\right)\right)} - 1\right)}} \]
      expm1-def [=>] 53.3	\[ \sqrt{2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(n \cdot \left(t \cdot U\right)\right)\right)}} \]
      expm1-log1p [=>] 54.8	\[ \sqrt{2 \cdot \color{blue}{\left(n \cdot \left(t \cdot U\right)\right)}} \]
      associate-*r* [=>] 60.8	\[ \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot t\right) \cdot U\right)}} \]
      *-commutative [=>] 60.8	\[ \sqrt{2 \cdot \left(\color{blue}{\left(t \cdot n\right)} \cdot U\right)} \]

   if 3.3999999999999998e-280 < l < 5.7e15

   1. Initial program 57.5%

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

   if 5.7e15 < l < 2.2e107

   1. Initial program 51.4%

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

   2. Simplified 49.9%

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

      [Start] 51.4	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 51.9	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 51.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      sub-neg [=>] 51.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(-\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [<=] 51.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      cancel-sign-sub [<=] 51.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      cancel-sign-sub [=>] 51.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-/l* [=>] 51.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 49.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in Om around inf 45.3%

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

   4. Simplified 46.3%

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

      [Start] 45.3	\[ \sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right)\right) + -4 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}} \]
      fma-def [=>] 45.3	\[ \sqrt{\color{blue}{\mathsf{fma}\left(2, n \cdot \left(t \cdot U\right), -4 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}} \]
      *-commutative [=>] 45.3	\[ \sqrt{\mathsf{fma}\left(2, n \cdot \color{blue}{\left(U \cdot t\right)}, -4 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)} \]
      associate-/l* [=>] 46.8	\[ \sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \color{blue}{\frac{n}{\frac{Om}{{\ell}^{2} \cdot U}}}\right)} \]
      associate-/r/ [=>] 46.2	\[ \sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \color{blue}{\left(\frac{n}{Om} \cdot \left({\ell}^{2} \cdot U\right)\right)}\right)} \]
      unpow2 [=>] 46.2	\[ \sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \left(\frac{n}{Om} \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot U\right)\right)\right)} \]
      associate-*l* [=>] 46.3	\[ \sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \left(\frac{n}{Om} \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot U\right)\right)}\right)\right)} \]

   if 2.2e107 < l < 2.7000000000000001e110

   1. Initial program 36.6%

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

   2. Simplified 45.7%

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

      [Start] 36.6	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 36.6	\[ \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 36.6	\[ \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      *-commutative [=>] 36.6	\[ \sqrt{2 \cdot \color{blue}{\left(\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot \left(n \cdot U\right)\right)}} \]

   3. Taylor expanded in Om around 0 9.0%

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

   4. Simplified 11.5%

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

      [Start] 9.0	\[ \sqrt{2 \cdot \frac{{n}^{2} \cdot \left({\ell}^{2} \cdot \left(\left(U* - U\right) \cdot U\right)\right)}{{Om}^{2}}} \]
      associate-*r* [=>] 9.3	\[ \sqrt{2 \cdot \frac{\color{blue}{\left({n}^{2} \cdot {\ell}^{2}\right) \cdot \left(\left(U* - U\right) \cdot U\right)}}{{Om}^{2}}} \]
      unpow2 [=>] 9.3	\[ \sqrt{2 \cdot \frac{\left({n}^{2} \cdot {\ell}^{2}\right) \cdot \left(\left(U* - U\right) \cdot U\right)}{\color{blue}{Om \cdot Om}}} \]
      times-frac [=>] 9.5	\[ \sqrt{2 \cdot \color{blue}{\left(\frac{{n}^{2} \cdot {\ell}^{2}}{Om} \cdot \frac{\left(U* - U\right) \cdot U}{Om}\right)}} \]
      unpow2 [=>] 9.5	\[ \sqrt{2 \cdot \left(\frac{\color{blue}{\left(n \cdot n\right)} \cdot {\ell}^{2}}{Om} \cdot \frac{\left(U* - U\right) \cdot U}{Om}\right)} \]
      unpow2 [=>] 9.5	\[ \sqrt{2 \cdot \left(\frac{\left(n \cdot n\right) \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{Om} \cdot \frac{\left(U* - U\right) \cdot U}{Om}\right)} \]
      unswap-sqr [=>] 11.5	\[ \sqrt{2 \cdot \left(\frac{\color{blue}{\left(n \cdot \ell\right) \cdot \left(n \cdot \ell\right)}}{Om} \cdot \frac{\left(U* - U\right) \cdot U}{Om}\right)} \]
      *-commutative [=>] 11.5	\[ \sqrt{2 \cdot \left(\frac{\left(n \cdot \ell\right) \cdot \left(n \cdot \ell\right)}{Om} \cdot \frac{\color{blue}{U \cdot \left(U* - U\right)}}{Om}\right)} \]

   5. Applied egg-rr 11.5%

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

   6. Simplified 11.5%

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

      [Start] 11.5	\[ \sqrt{2 \cdot \frac{-n \cdot \ell}{\frac{Om}{U \cdot \left(U* - U\right)} \cdot \left(-\frac{Om}{n \cdot \ell}\right)}} \]
      associate-*l/ [=>] 11.5	\[ \sqrt{2 \cdot \frac{-n \cdot \ell}{\color{blue}{\frac{Om \cdot \left(-\frac{Om}{n \cdot \ell}\right)}{U \cdot \left(U* - U\right)}}}} \]
      distribute-rgt-neg-in [<=] 11.5	\[ \sqrt{2 \cdot \frac{-n \cdot \ell}{\frac{\color{blue}{-Om \cdot \frac{Om}{n \cdot \ell}}}{U \cdot \left(U* - U\right)}}} \]
      distribute-lft-neg-out [<=] 11.5	\[ \sqrt{2 \cdot \frac{-n \cdot \ell}{\frac{\color{blue}{\left(-Om\right) \cdot \frac{Om}{n \cdot \ell}}}{U \cdot \left(U* - U\right)}}} \]
      *-commutative [<=] 11.5	\[ \sqrt{2 \cdot \frac{-n \cdot \ell}{\frac{\color{blue}{\frac{Om}{n \cdot \ell} \cdot \left(-Om\right)}}{U \cdot \left(U* - U\right)}}} \]
      associate-/r/ [=>] 11.5	\[ \sqrt{2 \cdot \color{blue}{\left(\frac{-n \cdot \ell}{\frac{Om}{n \cdot \ell} \cdot \left(-Om\right)} \cdot \left(U \cdot \left(U* - U\right)\right)\right)}} \]
      distribute-lft-neg-in [=>] 11.5	\[ \sqrt{2 \cdot \left(\frac{\color{blue}{\left(-n\right) \cdot \ell}}{\frac{Om}{n \cdot \ell} \cdot \left(-Om\right)} \cdot \left(U \cdot \left(U* - U\right)\right)\right)} \]
      *-commutative [=>] 11.5	\[ \sqrt{2 \cdot \left(\frac{\color{blue}{\ell \cdot \left(-n\right)}}{\frac{Om}{n \cdot \ell} \cdot \left(-Om\right)} \cdot \left(U \cdot \left(U* - U\right)\right)\right)} \]
      associate-*l/ [=>] 11.5	\[ \sqrt{2 \cdot \left(\frac{\ell \cdot \left(-n\right)}{\color{blue}{\frac{Om \cdot \left(-Om\right)}{n \cdot \ell}}} \cdot \left(U \cdot \left(U* - U\right)\right)\right)} \]
      associate-*r/ [<=] 11.5	\[ \sqrt{2 \cdot \left(\frac{\ell \cdot \left(-n\right)}{\color{blue}{Om \cdot \frac{-Om}{n \cdot \ell}}} \cdot \left(U \cdot \left(U* - U\right)\right)\right)} \]
      distribute-neg-frac [<=] 11.5	\[ \sqrt{2 \cdot \left(\frac{\ell \cdot \left(-n\right)}{Om \cdot \color{blue}{\left(-\frac{Om}{n \cdot \ell}\right)}} \cdot \left(U \cdot \left(U* - U\right)\right)\right)} \]
      associate-/r* [=>] 11.5	\[ \sqrt{2 \cdot \left(\frac{\ell \cdot \left(-n\right)}{Om \cdot \left(-\color{blue}{\frac{\frac{Om}{n}}{\ell}}\right)} \cdot \left(U \cdot \left(U* - U\right)\right)\right)} \]
      distribute-neg-frac [=>] 11.5	\[ \sqrt{2 \cdot \left(\frac{\ell \cdot \left(-n\right)}{Om \cdot \color{blue}{\frac{-\frac{Om}{n}}{\ell}}} \cdot \left(U \cdot \left(U* - U\right)\right)\right)} \]

   7. Applied egg-rr 19.1%

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

   if 2.7000000000000001e110 < l

   1. Initial program 11.5%

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

   2. Simplified 30.1%

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

      [Start] 11.5	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 11.5	\[ \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 11.5	\[ \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      *-commutative [=>] 11.5	\[ \sqrt{2 \cdot \color{blue}{\left(\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot \left(n \cdot U\right)\right)}} \]

   3. Applied egg-rr 29.8%

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

   4. Taylor expanded in l around inf 46.0%

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

   5. Simplified 46.6%

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

      [Start] 46.0	\[ \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)} \]
      associate-*l* [=>] 46.1	\[ \color{blue}{\sqrt{2} \cdot \left(\ell \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}\right)} \]
      associate-*r* [=>] 46.2	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{\color{blue}{\left(n \cdot U\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)}}\right) \]
      *-commutative [=>] 46.2	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{\color{blue}{\left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot \left(n \cdot U\right)}}\right) \]
      cancel-sign-sub-inv [=>] 46.2	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{\color{blue}{\left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} + \left(-2\right) \cdot \frac{1}{Om}\right)} \cdot \left(n \cdot U\right)}\right) \]
      associate-/l* [=>] 45.5	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(\color{blue}{\frac{n}{\frac{{Om}^{2}}{U* - U}}} + \left(-2\right) \cdot \frac{1}{Om}\right) \cdot \left(n \cdot U\right)}\right) \]
      associate-/r/ [=>] 46.6	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(\color{blue}{\frac{n}{{Om}^{2}} \cdot \left(U* - U\right)} + \left(-2\right) \cdot \frac{1}{Om}\right) \cdot \left(n \cdot U\right)}\right) \]
      unpow2 [=>] 46.6	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(\frac{n}{\color{blue}{Om \cdot Om}} \cdot \left(U* - U\right) + \left(-2\right) \cdot \frac{1}{Om}\right) \cdot \left(n \cdot U\right)}\right) \]
      metadata-eval [=>] 46.6	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) + \color{blue}{-2} \cdot \frac{1}{Om}\right) \cdot \left(n \cdot U\right)}\right) \]
      associate-*r/ [=>] 46.6	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) + \color{blue}{\frac{-2 \cdot 1}{Om}}\right) \cdot \left(n \cdot U\right)}\right) \]
      metadata-eval [=>] 46.6	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) + \frac{\color{blue}{-2}}{Om}\right) \cdot \left(n \cdot U\right)}\right) \]

3. Recombined 10 regimes into one program.

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.6 \cdot 10^{+179}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{\frac{Om \cdot Om}{U* - U}} + \frac{-2}{Om}\right)} \cdot \left(\ell \cdot \left(-\sqrt{2}\right)\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-37}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om \cdot \frac{Om}{\ell}}, n \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -2.05 \cdot 10^{-258}:\\ \;\;\;\;\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)}\\ \mathbf{elif}\;\ell \leq -3.2 \cdot 10^{-280}:\\ \;\;\;\;{\left({\left(\frac{-1}{U}\right)}^{-0.16666666666666666} \cdot {\left(n \cdot \left(-2 \cdot t\right)\right)}^{0.16666666666666666}\right)}^{3}\\ \mathbf{elif}\;\ell \leq -3.4 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.4 \cdot 10^{-280}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 5.7 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(U* - U\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.2 \cdot 10^{+107}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \left(\frac{n}{Om} \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{+110}:\\ \;\;\;\;\left|\sqrt{2} \cdot \left(\sqrt{U \cdot \left(U* - U\right)} \cdot \left(\ell \cdot \frac{n}{Om}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\left(U* - U\right) \cdot \frac{n}{Om \cdot Om} + \frac{-2}{Om}\right)}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	58.9%
Cost	43528

\[\begin{array}{l} t_1 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(U* - U\right) \cdot \left(n \cdot t_1\right)\right)}\\ \mathbf{if}\;t_2 \leq 10^{-135}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot \frac{\ell}{\frac{Om}{\ell}} - n \cdot \left(t_1 \cdot \left(U - U*\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+145}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \left(\left(\ell \cdot U\right) \cdot \frac{\ell \cdot n}{Om}\right)\right)}\\ \end{array} \]

Alternative 2
Accuracy	52.4%
Cost	14860

\[\begin{array}{l} t_1 := -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;\ell \leq -7.2 \cdot 10^{+178}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{\frac{Om \cdot Om}{U* - U}} + \frac{-2}{Om}\right)} \cdot \left(\ell \cdot \left(-\sqrt{2}\right)\right)\\ \mathbf{elif}\;\ell \leq -5.8 \cdot 10^{-34}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + t_1\right)}\\ \mathbf{elif}\;\ell \leq 1.15 \cdot 10^{+94}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(t_1 - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\left(U* - U\right) \cdot \frac{n}{Om \cdot Om} + \frac{-2}{Om}\right)}\right)\\ \end{array} \]

Alternative 3
Accuracy	51.7%
Cost	14549

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := -2 \cdot t_1\\ \mathbf{if}\;U \leq -180000000:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + t_2\right)}\\ \mathbf{elif}\;U \leq -7.8 \cdot 10^{-224}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(t_2 + n \cdot \left(t_1 \cdot \frac{U* - U}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;U \leq -5 \cdot 10^{-310} \lor \neg \left(U \leq 6.2 \cdot 10^{-250}\right) \land U \leq 2.75 \cdot 10^{+104}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \left(\left(\ell \cdot U\right) \cdot \frac{\ell \cdot n}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot t} \cdot \sqrt{2 \cdot U}\\ \end{array} \]

Alternative 4
Accuracy	51.6%
Cost	14549

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := -2 \cdot t_1\\ t_3 := n \cdot \left(U \cdot t\right)\\ \mathbf{if}\;U \leq -170000000:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + t_2\right)}\\ \mathbf{elif}\;U \leq -8.1 \cdot 10^{-224}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(t_2 + n \cdot \left(t_1 \cdot \frac{U* - U}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;U \leq 2 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, t_3, -4 \cdot \frac{\left(\ell \cdot U\right) \cdot \left(\ell \cdot n\right)}{Om}\right)}\\ \mathbf{elif}\;U \leq 4.3 \cdot 10^{-249} \lor \neg \left(U \leq 1.7 \cdot 10^{+106}\right):\\ \;\;\;\;\sqrt{n \cdot t} \cdot \sqrt{2 \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, t_3, -4 \cdot \left(\left(\ell \cdot U\right) \cdot \frac{\ell \cdot n}{Om}\right)\right)}\\ \end{array} \]

Alternative 5
Accuracy	51.6%
Cost	14549

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := -2 \cdot t_1\\ t_3 := n \cdot \left(U \cdot t\right)\\ \mathbf{if}\;U \leq -170000000:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + t_2\right)}\\ \mathbf{elif}\;U \leq -6.8 \cdot 10^{-224}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(t_2 + n \cdot \left(t_1 \cdot \frac{U* - U}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;U \leq 4.5 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, t_3, -4 \cdot \frac{\left(\ell \cdot U\right) \cdot \left(\ell \cdot n\right)}{Om}\right)}\\ \mathbf{elif}\;U \leq 6.3 \cdot 10^{-249} \lor \neg \left(U \leq 4.8 \cdot 10^{+102}\right):\\ \;\;\;\;\sqrt{n \cdot t} \cdot \sqrt{2 \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, t_3, \left(\ell \cdot n\right) \cdot \left(\left(\ell \cdot U\right) \cdot \frac{-4}{Om}\right)\right)}\\ \end{array} \]

Alternative 6
Accuracy	51.0%
Cost	14549

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := \sqrt{\mathsf{fma}\left(2, \left(n \cdot U\right) \cdot t, -4 \cdot \left(U \cdot \frac{\ell}{\frac{Om}{\ell \cdot n}}\right)\right)}\\ \mathbf{if}\;U \leq -9 \cdot 10^{-65}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;U \leq -2.2 \cdot 10^{-224}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot t_1 + n \cdot \left(t_1 \cdot \frac{U* - U}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;U \leq 2 \cdot 10^{-310}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;U \leq 4.4 \cdot 10^{-250} \lor \neg \left(U \leq 1.45 \cdot 10^{+100}\right):\\ \;\;\;\;\sqrt{n \cdot t} \cdot \sqrt{2 \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), \left(\ell \cdot n\right) \cdot \left(\left(\ell \cdot U\right) \cdot \frac{-4}{Om}\right)\right)}\\ \end{array} \]

Alternative 7
Accuracy	50.9%
Cost	14028

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := -2 \cdot t_1\\ t_3 := \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(t_2 + n \cdot \left(t_1 \cdot \frac{U* - U}{Om}\right)\right)\right)\right)}\\ \mathbf{if}\;U \leq -180000000:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + t_2\right)}\\ \mathbf{elif}\;U \leq -5.6 \cdot 10^{-267}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;U \leq -1.55 \cdot 10^{-286}:\\ \;\;\;\;\sqrt{2} \cdot \frac{\ell \cdot \left(n \cdot \sqrt{U \cdot \left(U* - U\right)}\right)}{Om}\\ \mathbf{elif}\;U \leq 7.2 \cdot 10^{+103}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot t} \cdot \sqrt{2 \cdot U}\\ \end{array} \]

Alternative 8
Accuracy	50.8%
Cost	14028

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := -2 \cdot t_1\\ t_3 := \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(t_2 + n \cdot \left(t_1 \cdot \frac{U* - U}{Om}\right)\right)\right)\right)}\\ \mathbf{if}\;U \leq -175000000:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + t_2\right)}\\ \mathbf{elif}\;U \leq -5.6 \cdot 10^{-267}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;U \leq -1.55 \cdot 10^{-286}:\\ \;\;\;\;\sqrt{2} \cdot \frac{n \cdot \left(\ell \cdot \sqrt{U \cdot \left(U* - U\right)}\right)}{Om}\\ \mathbf{elif}\;U \leq 1.25 \cdot 10^{+100}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot t} \cdot \sqrt{2 \cdot U}\\ \end{array} \]

Alternative 9
Accuracy	46.2%
Cost	8532

\[\begin{array}{l} t_1 := \sqrt{U \cdot \left(-4 \cdot \frac{n \cdot \left(\ell \cdot \ell\right)}{Om} + 2 \cdot \left(n \cdot t\right)\right)}\\ t_2 := \frac{\ell}{\frac{Om}{\ell}}\\ t_3 := -2 \cdot t_2\\ \mathbf{if}\;Om \leq -1.24 \cdot 10^{-30}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(t_3 + n \cdot \left(t_2 \cdot \frac{U* - U}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;Om \leq 2.25 \cdot 10^{-306}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;Om \leq 4.6 \cdot 10^{-250}:\\ \;\;\;\;\ell \cdot \left(\sqrt{\left(U* - U\right) \cdot \left(2 \cdot U\right)} \cdot \frac{-n}{Om}\right)\\ \mathbf{elif}\;Om \leq 7.2 \cdot 10^{-114}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;Om \leq 3.2 \cdot 10^{-43}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(\frac{2}{Om} + \frac{n}{Om} \cdot \frac{U - U*}{Om}\right) \cdot \left(n \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + t_3\right)}\\ \end{array} \]

Alternative 10
Accuracy	44.5%
Cost	8528

\[\begin{array}{l} t_1 := t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := \sqrt{2 \cdot \left(n \cdot \left(U \cdot t_1\right)\right)}\\ \mathbf{if}\;Om \leq -4.8 \cdot 10^{-308}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;Om \leq 1.05 \cdot 10^{-247}:\\ \;\;\;\;\ell \cdot \left(\sqrt{\left(U* - U\right) \cdot \left(2 \cdot U\right)} \cdot \frac{-n}{Om}\right)\\ \mathbf{elif}\;Om \leq 2.3 \cdot 10^{-47}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;Om \leq 1.4 \cdot 10^{-18}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\frac{n}{\frac{Om \cdot Om}{U* - U}} + \frac{-2}{Om}\right) \cdot \left(\ell \cdot \ell\right)\right)\right)}\\ \mathbf{elif}\;Om \leq 2.1 \cdot 10^{+60}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot t_1}\\ \end{array} \]

Alternative 11
Accuracy	44.4%
Cost	7625

\[\begin{array}{l} \mathbf{if}\;U \leq -4.3 \cdot 10^{-200} \lor \neg \left(U \leq 2.1 \cdot 10^{-142}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \ell \cdot \frac{\ell \cdot -2}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]

Alternative 12
Accuracy	45.1%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;Om \leq 2.9 \cdot 10^{-306}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;Om \leq 2.45 \cdot 10^{-251}:\\ \;\;\;\;\ell \cdot \left(\sqrt{\left(U* - U\right) \cdot \left(2 \cdot U\right)} \cdot \frac{-n}{Om}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \ell \cdot \frac{\ell \cdot -2}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 13
Accuracy	48.0%
Cost	7624

\[\begin{array}{l} t_1 := t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;U \leq -200000000:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot t_1}\\ \mathbf{elif}\;U \leq 4 \cdot 10^{+57}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \ell \cdot \frac{\ell \cdot -2}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 14
Accuracy	39.3%
Cost	7497

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{+99} \lor \neg \left(\ell \leq 4.5 \cdot 10^{+118}\right):\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\ell \cdot \left(\ell \cdot \frac{2}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]

Alternative 15
Accuracy	38.4%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;n \leq -4 \cdot 10^{-149} \lor \neg \left(n \leq 10^{-32}\right):\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]

Alternative 16
Accuracy	37.0%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;U \leq 10^{+46}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]

Alternative 17
Accuracy	36.0%
Cost	6848

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

Reproduce

herbie shell --seed 2023125 
(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*))))))