Toniolo and Linder, Equation (13)

Average Error: 54.12% → 41.69%

Time: 38.7s

Precision: binary64

Cost: 51468

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\ell \cdot \ell}{Om}\\ t_2 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_3 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + t_1 \cdot -2\right) + \left(n \cdot t_2\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t_3 \leq 10^{-156}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(n, t_2 \cdot \left(U - U*\right), 2 \cdot t_1\right)\right)}\right)\\ \mathbf{elif}\;t_3 \leq 5 \cdot 10^{+147}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \frac{2 \cdot \ell}{Om} - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right) \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)\right)}\\ \end{array} \]

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
         (sqrt
          (* (* (* 2.0 n) U) (+ (+ t (* t_1 -2.0)) (* (* n t_2) (- U* U)))))))
   (if (<= t_3 1e-156)
     (*
      (sqrt 2.0)
      (* (sqrt n) (sqrt (* U (- t (fma n (* t_2 (- U U*)) (* 2.0 t_1)))))))
     (if (<= t_3 5e+147)
       t_3
       (if (<= t_3 INFINITY)
         (sqrt (* -2.0 (* U (* n (- (* l (/ (* 2.0 l) Om)) t)))))
         (sqrt
          (*
           2.0
           (*
            (* n (* l (* U l)))
            (+ (* (/ n Om) (/ (- U* U) 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 = sqrt((((2.0 * n) * U) * ((t + (t_1 * -2.0)) + ((n * t_2) * (U_42_ - U)))));
	double tmp;
	if (t_3 <= 1e-156) {
		tmp = sqrt(2.0) * (sqrt(n) * sqrt((U * (t - fma(n, (t_2 * (U - U_42_)), (2.0 * t_1))))));
	} else if (t_3 <= 5e+147) {
		tmp = t_3;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((-2.0 * (U * (n * ((l * ((2.0 * l) / Om)) - t)))));
	} else {
		tmp = sqrt((2.0 * ((n * (l * (U * l))) * (((n / Om) * ((U_42_ - U) / 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(Float64(l * l) / Om)
	t_2 = Float64(l / Om) ^ 2.0
	t_3 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t + Float64(t_1 * -2.0)) + Float64(Float64(n * t_2) * Float64(U_42_ - U)))))
	tmp = 0.0
	if (t_3 <= 1e-156)
		tmp = Float64(sqrt(2.0) * Float64(sqrt(n) * sqrt(Float64(U * Float64(t - fma(n, Float64(t_2 * Float64(U - U_42_)), Float64(2.0 * t_1)))))));
	elseif (t_3 <= 5e+147)
		tmp = t_3;
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(-2.0 * Float64(U * Float64(n * Float64(Float64(l * Float64(Float64(2.0 * l) / Om)) - t)))));
	else
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * Float64(l * Float64(U * l))) * Float64(Float64(Float64(n / Om) * Float64(Float64(U_42_ - U) / 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[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t + N[(t$95$1 * -2.0), $MachinePrecision]), $MachinePrecision] + N[(N[(n * t$95$2), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 1e-156], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(U * N[(t - N[(n * N[(t$95$2 * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] + N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5e+147], t$95$3, If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(-2.0 * N[(U * N[(n * N[(N[(l * N[(N[(2.0 * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(N[(n * N[(l * N[(U * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(n / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 1.00000000000000004e-156

   1. Initial program 86.86

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

   2. Simplified 64.07

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

      [Start] 86.86	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 61.67	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 61.67	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      sub-neg [=>] 61.67	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(-\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [<=] 61.67	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      cancel-sign-sub [<=] 61.67	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      cancel-sign-sub [=>] 61.67	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-/l* [=>] 61.67	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 64.07	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Applied egg-rr 86.83

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

   4. Simplified 86.83

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

      [Start] 86.83	\[ \sqrt{2} \cdot \sqrt{\left(n \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U - U*\right) \cdot n\right)\right)\right)} \]
      associate-/l* [=>] 86.83	\[ \sqrt{2} \cdot \sqrt{\left(n \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U - U*\right) \cdot n\right)\right)\right)} \]
      *-commutative [<=] 86.83	\[ \sqrt{2} \cdot \sqrt{\left(n \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \color{blue}{\left(n \cdot \left(U - U*\right)\right)}\right)\right)} \]
      *-commutative [=>] 86.83	\[ \sqrt{2} \cdot \sqrt{\left(n \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \color{blue}{\left(n \cdot \left(U - U*\right)\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}}\right)\right)} \]

   5. Applied egg-rr 61.19

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

   6. Simplified 61.27

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

      [Start] 61.19	\[ \sqrt{2} \cdot \left(\sqrt{n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, n \cdot \left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right)}\right) \]
      fma-def [<=] 61.19	\[ \sqrt{2} \cdot \left(\sqrt{n} \cdot \sqrt{U \cdot \left(t - \color{blue}{\left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) + n \cdot \left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)}\right)}\right) \]
      +-commutative [<=] 61.19	\[ \sqrt{2} \cdot \left(\sqrt{n} \cdot \sqrt{U \cdot \left(t - \color{blue}{\left(n \cdot \left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) + 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\right)}\right) \]
      fma-def [=>] 61.19	\[ \sqrt{2} \cdot \left(\sqrt{n} \cdot \sqrt{U \cdot \left(t - \color{blue}{\mathsf{fma}\left(n, \left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}, 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\right)}\right) \]
      associate-*r/ [=>] 61.27	\[ \sqrt{2} \cdot \left(\sqrt{n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(n, \left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}, 2 \cdot \color{blue}{\frac{\ell \cdot \ell}{Om}}\right)\right)}\right) \]

   if 1.00000000000000004e-156 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 5.0000000000000002e147

   1. Initial program 2.37

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

   if 5.0000000000000002e147 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < +inf.0

   1. Initial program 97.43

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

   2. Taylor expanded in n around 0 93.32

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

   3. Simplified 82.49

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

      [Start] 93.32	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot U\right)\right)} \]
      associate-*r* [=>] 93.94	\[ \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right)}} \]
      *-commutative [=>] 93.94	\[ \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
      cancel-sign-sub-inv [=>] 93.94	\[ \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{\left(t + \left(-2\right) \cdot \frac{{\ell}^{2}}{Om}\right)}\right)\right)} \]
      metadata-eval [=>] 93.94	\[ \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \color{blue}{-2} \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)} \]
      unpow2 [=>] 93.94	\[ \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + -2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right)\right)\right)} \]
      associate-*r/ [<=] 82.49	\[ \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + -2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)}\right)\right)\right)} \]
      *-commutative [<=] 82.49	\[ \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2}\right)\right)\right)} \]
      associate-*l* [=>] 82.49	\[ \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \color{blue}{\ell \cdot \left(\frac{\ell}{Om} \cdot -2\right)}\right)\right)\right)} \]
      associate-*l/ [=>] 82.49	\[ \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \ell \cdot \color{blue}{\frac{\ell \cdot -2}{Om}}\right)\right)\right)} \]

   if +inf.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))))

   1. Initial program 100

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

   2. Simplified 89.65

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

      [Start] 100	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 99.95	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 99.95	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      sub-neg [=>] 99.95	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(-\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [<=] 99.95	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      cancel-sign-sub [<=] 99.95	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      cancel-sign-sub [=>] 99.95	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-/l* [=>] 89.91	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 89.65	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in l around inf 92.92

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

   4. Simplified 67.77

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

      [Start] 92.92	\[ \sqrt{-2 \cdot \left(\left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(n \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      *-commutative [=>] 92.92	\[ \sqrt{-2 \cdot \left(\left(\frac{\color{blue}{\left(U - U*\right) \cdot n}}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(n \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      unpow2 [=>] 92.92	\[ \sqrt{-2 \cdot \left(\left(\frac{\left(U - U*\right) \cdot n}{\color{blue}{Om \cdot Om}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(n \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      times-frac [=>] 85.79	\[ \sqrt{-2 \cdot \left(\left(\color{blue}{\frac{U - U*}{Om} \cdot \frac{n}{Om}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(n \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      associate-*r/ [=>] 85.79	\[ \sqrt{-2 \cdot \left(\left(\frac{U - U*}{Om} \cdot \frac{n}{Om} + \color{blue}{\frac{2 \cdot 1}{Om}}\right) \cdot \left(n \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      metadata-eval [=>] 85.79	\[ \sqrt{-2 \cdot \left(\left(\frac{U - U*}{Om} \cdot \frac{n}{Om} + \frac{\color{blue}{2}}{Om}\right) \cdot \left(n \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      unpow2 [=>] 85.79	\[ \sqrt{-2 \cdot \left(\left(\frac{U - U*}{Om} \cdot \frac{n}{Om} + \frac{2}{Om}\right) \cdot \left(n \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot U\right)\right)\right)} \]
      associate-*l* [=>] 67.77	\[ \sqrt{-2 \cdot \left(\left(\frac{U - U*}{Om} \cdot \frac{n}{Om} + \frac{2}{Om}\right) \cdot \left(n \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot U\right)\right)}\right)\right)} \]

3. Recombined 4 regimes into one program.

4. Final simplification 41.69

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 10^{-156}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right), 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\right)\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 5 \cdot 10^{+147}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \frac{2 \cdot \ell}{Om} - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right) \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	41.89%
Cost	51468

\[\begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t_1 \leq 10^{-156}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \frac{n}{Om} \cdot \frac{\ell}{\frac{Om}{U \cdot \ell}}\right)\right)}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+147}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \frac{2 \cdot \ell}{Om} - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right) \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)\right)}\\ \end{array} \]

Alternative 2
Error	41.67%
Cost	51468

\[\begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t_1 \leq 10^{-156}:\\ \;\;\;\;\sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, n \cdot \left(\left(U - U*\right) \cdot \frac{\frac{\ell}{Om}}{\frac{Om}{\ell}}\right)\right)\right)} \cdot \sqrt{2 \cdot n}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+147}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \frac{2 \cdot \ell}{Om} - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right) \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)\right)}\\ \end{array} \]

Alternative 3
Error	43.89%
Cost	38604

\[\begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + \ell \cdot \left(\frac{\ell}{Om} \cdot -2\right)\right)}\\ \mathbf{elif}\;t_1 \leq 10^{+295}:\\ \;\;\;\;\sqrt{t_1}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \frac{2 \cdot \ell}{Om} - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) + \frac{-2}{Om}\right) \cdot \left(n \cdot U\right)} \cdot \left(\sqrt{2} \cdot \left(-\ell\right)\right)\\ \end{array} \]

Alternative 4
Error	43.9%
Cost	38604

\[\begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{n} \cdot \sqrt{U \cdot \left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)}\right)\\ \mathbf{elif}\;t_1 \leq 10^{+295}:\\ \;\;\;\;\sqrt{t_1}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \frac{2 \cdot \ell}{Om} - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) + \frac{-2}{Om}\right) \cdot \left(n \cdot U\right)} \cdot \left(\sqrt{2} \cdot \left(-\ell\right)\right)\\ \end{array} \]

Alternative 5
Error	54.28%
Cost	14816

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := 2 \cdot \left(n \cdot U\right)\\ t_3 := \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \frac{2 \cdot \ell}{Om} - t\right)\right)\right)}\\ t_4 := 2 \cdot t_1\\ \mathbf{if}\;Om \leq -4.8 \cdot 10^{-80}:\\ \;\;\;\;\sqrt{\left(t - t_4\right) \cdot t_2}\\ \mathbf{elif}\;Om \leq -4.1 \cdot 10^{-299}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(n \cdot \left(\left(U - U*\right) \cdot \frac{\frac{\ell}{Om}}{\frac{Om}{\ell}}\right) + t_4\right) - t\right)\right)}\\ \mathbf{elif}\;Om \leq 7.4 \cdot 10^{-149}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;Om \leq 3.1 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot t_1 + n \cdot \frac{\ell \cdot \left(U* - U\right)}{Om \cdot \frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;Om \leq 5.5 \cdot 10^{+18}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(\ell \cdot \left(n \cdot \left(U \cdot \ell\right)\right)\right) \cdot \left(U* \cdot \left(n \cdot {Om}^{-2}\right) + \frac{-2}{Om}\right)\right)}\\ \mathbf{elif}\;Om \leq 1.4 \cdot 10^{+83}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + \ell \cdot \left(\frac{\ell}{Om} \cdot -2\right)\right)}\\ \mathbf{elif}\;Om \leq 2.25 \cdot 10^{+145}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \left(\frac{n}{Om} \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \mathbf{elif}\;Om \leq 3.4 \cdot 10^{+190}:\\ \;\;\;\;\sqrt{t_2 \cdot \left(t + -2 \cdot \left|\ell \cdot \frac{\ell}{Om}\right|\right)}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 6
Error	50.19%
Cost	14804

\[\begin{array}{l} t_1 := 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := \sqrt{\left(t - t_1\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{if}\;\ell \leq -1.5 \cdot 10^{+111}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 8.5 \cdot 10^{-148}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(n \cdot \left(\left(U - U*\right) \cdot \frac{\frac{\ell}{Om}}{\frac{Om}{\ell}}\right) + t_1\right) - t\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.1 \cdot 10^{-36}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 9.2 \cdot 10^{-24}:\\ \;\;\;\;\sqrt{2 \cdot \left(\frac{\ell \cdot \ell}{\frac{Om}{n}} \cdot \frac{U \cdot \left(U* - U\right)}{\frac{Om}{n}}\right)}\\ \mathbf{elif}\;\ell \leq 8.5 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \frac{2 \cdot \ell}{Om} - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) + \frac{-2}{Om}\right) \cdot \left(n \cdot U\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]

Alternative 7
Error	47.8%
Cost	14804

\[\begin{array}{l} t_1 := \sqrt{\left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) + \frac{-2}{Om}\right) \cdot \left(n \cdot U\right)}\\ t_2 := 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;\ell \leq -3.6 \cdot 10^{+90}:\\ \;\;\;\;t_1 \cdot \left(\sqrt{2} \cdot \left(-\ell\right)\right)\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{-152}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(n \cdot \left(\left(U - U*\right) \cdot \frac{\frac{\ell}{Om}}{\frac{Om}{\ell}}\right) + t_2\right) - t\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.1 \cdot 10^{-36}:\\ \;\;\;\;\sqrt{\left(t - t_2\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;\ell \leq 9.2 \cdot 10^{-24}:\\ \;\;\;\;\sqrt{2 \cdot \left(\frac{\ell \cdot \ell}{\frac{Om}{n}} \cdot \frac{U \cdot \left(U* - U\right)}{\frac{Om}{n}}\right)}\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \frac{2 \cdot \ell}{Om} - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]

Alternative 8
Error	48.06%
Cost	13644

\[\begin{array}{l} t_1 := 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;U \leq -50:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \frac{2 \cdot \ell}{Om} - t\right)\right)\right)}\\ \mathbf{elif}\;U \leq 1120:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(n \cdot \left(\left(U - U*\right) \cdot \frac{\frac{\ell}{Om}}{\frac{Om}{\ell}}\right) + t_1\right) - t\right)\right)}\\ \mathbf{elif}\;U \leq 4 \cdot 10^{+220}:\\ \;\;\;\;\sqrt{\left(t - t_1\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot t} \cdot \sqrt{2 \cdot U}\\ \end{array} \]

Alternative 9
Error	49.24%
Cost	13512

\[\begin{array}{l} \mathbf{if}\;n \leq -6.6 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(n \cdot \left(\left(U - U*\right) \cdot \frac{\frac{\ell}{Om}}{\frac{Om}{\ell}}\right) + 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - t\right)\right)}\\ \mathbf{elif}\;n \leq 1.3 \cdot 10^{+89}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \frac{2 \cdot \ell}{Om} - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{2 \cdot \left(U \cdot t\right)}\\ \end{array} \]

Alternative 10
Error	49.25%
Cost	13512

\[\begin{array}{l} \mathbf{if}\;n \leq -4.8 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(n \cdot \left(\left(U - U*\right) \cdot \frac{\frac{\ell}{Om}}{\frac{Om}{\ell}}\right) + 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - t\right)\right)}\\ \mathbf{elif}\;n \leq 2.35 \cdot 10^{+89}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \frac{2 \cdot \ell}{Om} - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t}\\ \end{array} \]

Alternative 11
Error	52.15%
Cost	8784

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := 2 \cdot t_1\\ t_3 := \sqrt{\left(t - t_2\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ t_4 := \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \frac{2 \cdot \ell}{Om} - t\right)\right)\right)}\\ \mathbf{if}\;Om \leq -1.6 \cdot 10^{-69}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;Om \leq -2 \cdot 10^{-292}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(n \cdot \left(\left(U - U*\right) \cdot \frac{\frac{\ell}{Om}}{\frac{Om}{\ell}}\right) + t_2\right) - t\right)\right)}\\ \mathbf{elif}\;Om \leq 1.6 \cdot 10^{-149}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;Om \leq 6.2 \cdot 10^{+57}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot t_1 + n \cdot \frac{\ell \cdot \left(U* - U\right)}{Om \cdot \frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;Om \leq 1.4 \cdot 10^{+191}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 12
Error	54.95%
Cost	8520

\[\begin{array}{l} t_1 := 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := \sqrt{\left(t - t_1\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ t_3 := \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \frac{2 \cdot \ell}{Om} - t\right)\right)\right)}\\ \mathbf{if}\;Om \leq -3.5 \cdot 10^{-60}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;Om \leq -4.2 \cdot 10^{-289}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(n \cdot \left(\left(U - U*\right) \cdot \frac{\frac{\ell}{Om}}{\frac{Om}{\ell}}\right) + t_1\right) - t\right)\right)}\\ \mathbf{elif}\;Om \leq 1.45 \cdot 10^{-46}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;Om \leq 6.2 \cdot 10^{+55}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(U \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)\right) \cdot \left(\frac{n}{\frac{Om}{\frac{U*}{Om}}} + \frac{-2}{Om}\right)\right)}\\ \mathbf{elif}\;Om \leq 3.7 \cdot 10^{+190}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 13
Error	53.62%
Cost	8408

\[\begin{array}{l} t_1 := 2 \cdot \left(n \cdot U\right)\\ t_2 := 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\\ t_3 := \sqrt{\left(t - t_2\right) \cdot t_1}\\ t_4 := \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \frac{2 \cdot \ell}{Om} - t\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -8.5 \cdot 10^{+108}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\ell \leq -5.4 \cdot 10^{-127}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(t_2 - n \cdot \frac{\frac{U*}{Om}}{\frac{\frac{Om}{\ell}}{\ell}}\right) - t\right)\right)}\\ \mathbf{elif}\;\ell \leq -6 \cdot 10^{-296}:\\ \;\;\;\;\sqrt{t \cdot t_1}\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{-132}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;\ell \leq 4.1 \cdot 10^{-36}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\ell \leq 1.02 \cdot 10^{-23}:\\ \;\;\;\;\sqrt{2 \cdot \left(\frac{\ell \cdot \ell}{\frac{Om}{n}} \cdot \frac{U \cdot \left(U* - U\right)}{\frac{Om}{n}}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 14
Error	51.32%
Cost	8392

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := 2 \cdot t_1\\ \mathbf{if}\;U \leq -20:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \frac{2 \cdot \ell}{Om} - t\right)\right)\right)}\\ \mathbf{elif}\;U \leq 1.2 \cdot 10^{-73}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(t_2 + n \cdot \left(t_1 \cdot \frac{U}{Om}\right)\right) - t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t - t_2\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 15
Error	65.1%
Cost	7760

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(-2 \cdot \frac{n}{\frac{\frac{Om}{\ell}}{U \cdot \ell}}\right)}\\ t_2 := \sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{if}\;t \leq -7.2 \cdot 10^{+122}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.15 \cdot 10^{+31}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -9.6 \cdot 10^{-175}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{-202}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]

Alternative 16
Error	53.5%
Cost	7625

\[\begin{array}{l} \mathbf{if}\;\ell \leq -3 \cdot 10^{-123} \lor \neg \left(\ell \leq -4.7 \cdot 10^{-296}\right):\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \frac{2 \cdot \ell}{Om} - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 17
Error	53.91%
Cost	7625

\[\begin{array}{l} \mathbf{if}\;Om \leq -5.6 \cdot 10^{-24} \lor \neg \left(Om \leq -5.7 \cdot 10^{-294}\right):\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \frac{2 \cdot \ell}{Om} - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \end{array} \]

Alternative 18
Error	53.9%
Cost	7625

\[\begin{array}{l} \mathbf{if}\;Om \leq -1.2 \cdot 10^{-25} \lor \neg \left(Om \leq -1.8 \cdot 10^{-285}\right):\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \frac{2 \cdot \ell}{Om} - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \end{array} \]

Alternative 19
Error	51.3%
Cost	7624

\[\begin{array}{l} t_1 := t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;U \leq -8.6 \cdot 10^{+84}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \frac{2 \cdot \ell}{Om} - t\right)\right)\right)}\\ \mathbf{elif}\;U \leq 5.2 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t_1 \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 20
Error	61.22%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;U \leq -7.2 \cdot 10^{+83} \lor \neg \left(U \leq 5 \cdot 10^{-109}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]

Alternative 21
Error	62.48%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;t \leq 2 \cdot 10^{-201}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{+296}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]

Alternative 22
Error	63.12%
Cost	6848

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

Reproduce

herbie shell --seed 2023090 
(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*))))))