Toniolo and Linder, Equation (13)

Average Error: 54.94% → 43.4%

Time: 43.1s

Precision: binary64

Cost: 51532

Math

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

↓

\[\begin{array}{l} t_1 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot t_1\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_2 \leq 2 \cdot 10^{-199}:\\ \;\;\;\;{\left({\left(U \cdot \left(\left(n \cdot -2\right) \cdot \left(\mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, t_1 \cdot \left(n \cdot \left(U - U*\right)\right)\right) - t\right)\right)\right)}^{0.25}\right)}^{2}\\ \mathbf{elif}\;t_2 \leq 10^{+294}:\\ \;\;\;\;\sqrt{t_2}\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;\sqrt{t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, n \cdot \left(t_1 \cdot \left(U - U*\right)\right)\right)} \cdot \sqrt{2 \cdot \left(n \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right)}\right)\\ \end{array} \]

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 (pow (/ l Om) 2.0))
        (t_2
         (*
          (* (* 2.0 n) U)
          (+ (+ t (* (/ (* l l) Om) -2.0)) (* (* n t_1) (- U* U))))))
   (if (<= t_2 2e-199)
     (pow
      (pow
       (*
        U
        (* (* n -2.0) (- (fma 2.0 (* l (/ l Om)) (* t_1 (* n (- U U*)))) t)))
       0.25)
      2.0)
     (if (<= t_2 1e+294)
       (sqrt t_2)
       (if (<= t_2 INFINITY)
         (*
          (sqrt (- t (fma 2.0 (/ l (/ Om l)) (* n (* t_1 (- U U*))))))
          (sqrt (* 2.0 (* n U))))
         (*
          (sqrt 2.0)
          (* l (sqrt (* (* n U) (+ (* (/ n Om) (/ 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 = pow((l / Om), 2.0);
	double t_2 = ((2.0 * n) * U) * ((t + (((l * l) / Om) * -2.0)) + ((n * t_1) * (U_42_ - U)));
	double tmp;
	if (t_2 <= 2e-199) {
		tmp = pow(pow((U * ((n * -2.0) * (fma(2.0, (l * (l / Om)), (t_1 * (n * (U - U_42_)))) - t))), 0.25), 2.0);
	} else if (t_2 <= 1e+294) {
		tmp = sqrt(t_2);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt((t - fma(2.0, (l / (Om / l)), (n * (t_1 * (U - U_42_)))))) * sqrt((2.0 * (n * U)));
	} else {
		tmp = sqrt(2.0) * (l * sqrt(((n * U) * (((n / Om) * (U_42_ / 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 / Om) ^ 2.0
	t_2 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t + Float64(Float64(Float64(l * l) / Om) * -2.0)) + Float64(Float64(n * t_1) * Float64(U_42_ - U))))
	tmp = 0.0
	if (t_2 <= 2e-199)
		tmp = (Float64(U * Float64(Float64(n * -2.0) * Float64(fma(2.0, Float64(l * Float64(l / Om)), Float64(t_1 * Float64(n * Float64(U - U_42_)))) - t))) ^ 0.25) ^ 2.0;
	elseif (t_2 <= 1e+294)
		tmp = sqrt(t_2);
	elseif (t_2 <= Inf)
		tmp = Float64(sqrt(Float64(t - fma(2.0, Float64(l / Float64(Om / l)), Float64(n * Float64(t_1 * Float64(U - U_42_)))))) * sqrt(Float64(2.0 * Float64(n * U))));
	else
		tmp = Float64(sqrt(2.0) * Float64(l * sqrt(Float64(Float64(n * U) * Float64(Float64(Float64(n / Om) * Float64(U_42_ / 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[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t + N[(N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] + N[(N[(n * t$95$1), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 2e-199], N[Power[N[Power[N[(U * N[(N[(n * -2.0), $MachinePrecision] * N[(N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[t$95$2, 1e+294], N[Sqrt[t$95$2], $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[Sqrt[N[(t - N[(2.0 * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision] + N[(n * N[(t$95$1 * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(l * N[Sqrt[N[(N[(n * U), $MachinePrecision] * N[(N[(N[(n / Om), $MachinePrecision] * N[(U$42$ / Om), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.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*)))) < 1.99999999999999996e-199

   1. Initial program 68.51

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

   2. Simplified 53.9

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

      [Start] 68.51	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 53.14	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 53.14	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      fma-def [=>] 53.14	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-*l* [=>] 53.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Applied egg-rr 53.19

      \[\leadsto \color{blue}{{\left({\left(U \cdot \left(\left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U - U*\right) \cdot n\right)\right)\right) \cdot \left(2 \cdot n\right)\right)\right)}^{0.25}\right)}^{2}} \]

   if 1.99999999999999996e-199 < (*.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*)))) < 1.00000000000000007e294

   1. Initial program 2.54

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

   if 1.00000000000000007e294 < (*.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 96.24

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

   2. Simplified 83.92

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

      [Start] 96.24	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 94.62	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 94.62	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      sub-neg [=>] 94.62	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(-\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [<=] 94.62	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      cancel-sign-sub [<=] 94.62	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      cancel-sign-sub [=>] 94.62	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-/l* [=>] 84.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 83.92	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Applied egg-rr 87.42

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

   4. Simplified 75.55

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

      [Start] 87.42	\[ \sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U - U*\right) \cdot n\right)\right)} \]
      *-commutative [=>] 87.42	\[ \color{blue}{\sqrt{t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U - U*\right) \cdot n\right)\right)} \cdot \sqrt{2 \cdot \left(n \cdot U\right)}} \]
      associate-/l* [=>] 78.28	\[ \sqrt{t - \mathsf{fma}\left(2, \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U - U*\right) \cdot n\right)\right)} \cdot \sqrt{2 \cdot \left(n \cdot U\right)} \]
      associate-*r* [=>] 75.55	\[ \sqrt{t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right) \cdot n}\right)} \cdot \sqrt{2 \cdot \left(n \cdot U\right)} \]
      *-commutative [=>] 75.55	\[ \sqrt{t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)} \cdot \sqrt{2 \cdot \left(n \cdot U\right)} \]
      *-commutative [=>] 75.55	\[ \sqrt{t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, n \cdot \color{blue}{\left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right)} \cdot \sqrt{2 \cdot \left(n \cdot U\right)} \]

   if +inf.0 < (*.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. Taylor expanded in U around 0 97.74

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

   3. Simplified 99.36

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

      [Start] 97.74	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right)\right)} \]
      associate-*r* [=>] 97.74	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(\left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right)}} \]
      *-commutative [=>] 97.74	\[ \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(U \cdot \left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
      associate-*r* [=>] 99.91	\[ \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]
      associate-*r* [<=] 99.91	\[ \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot \left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]
      +-commutative [=>] 99.91	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} + -1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)} \]
      mul-1-neg [=>] 99.91	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \frac{{\ell}^{2}}{Om} + \color{blue}{\left(-\frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)\right)} \]
      unsub-neg [=>] 99.91	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)} \]
      unpow2 [=>] 99.91	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)} \]
      associate-/l* [=>] 99.91	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)} \]
      associate-/l* [=>] 99.93	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \color{blue}{\frac{n}{\frac{{Om}^{2}}{{\ell}^{2} \cdot U*}}}\right)\right)} \]
      associate-/r* [=>] 99.98	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \frac{n}{\color{blue}{\frac{\frac{{Om}^{2}}{{\ell}^{2}}}{U*}}}\right)\right)} \]

   4. Taylor expanded in l around inf 80.3

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

   5. Simplified 80.19

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

      [Start] 80.3	\[ \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(\left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)} \]
      associate-*l* [=>] 80.31	\[ \color{blue}{\sqrt{2} \cdot \left(\ell \cdot \sqrt{n \cdot \left(\left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)}\right)} \]
      *-commutative [=>] 80.31	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{n \cdot \color{blue}{\left(U \cdot \left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}}\right) \]
      associate-*r* [=>] 83.27	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{\color{blue}{\left(n \cdot U\right) \cdot \left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)}}\right) \]
      cancel-sign-sub-inv [=>] 83.27	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(n \cdot U\right) \cdot \color{blue}{\left(\frac{n \cdot U*}{{Om}^{2}} + \left(-2\right) \cdot \frac{1}{Om}\right)}}\right) \]
      unpow2 [=>] 83.27	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n \cdot U*}{\color{blue}{Om \cdot Om}} + \left(-2\right) \cdot \frac{1}{Om}\right)}\right) \]
      times-frac [=>] 80.19	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\color{blue}{\frac{n}{Om} \cdot \frac{U*}{Om}} + \left(-2\right) \cdot \frac{1}{Om}\right)}\right) \]
      metadata-eval [=>] 80.19	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \color{blue}{-2} \cdot \frac{1}{Om}\right)}\right) \]
      associate-*r/ [=>] 80.19	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \color{blue}{\frac{-2 \cdot 1}{Om}}\right)}\right) \]
      metadata-eval [=>] 80.19	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{\color{blue}{-2}}{Om}\right)}\right) \]

3. Recombined 4 regimes into one program.

4. Final simplification 43.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 2 \cdot 10^{-199}:\\ \;\;\;\;{\left({\left(U \cdot \left(\left(n \cdot -2\right) \cdot \left(\mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right) - t\right)\right)\right)}^{0.25}\right)}^{2}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 10^{+294}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)} \cdot \sqrt{2 \cdot \left(n \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right)}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	42.46%
Cost	51532

\[\begin{array}{l} t_1 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot t_1\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_2 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \left(-2 + \frac{n}{Om} \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{elif}\;t_2 \leq 10^{+294}:\\ \;\;\;\;\sqrt{t_2}\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;\sqrt{t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, n \cdot \left(t_1 \cdot \left(U - U*\right)\right)\right)} \cdot \sqrt{2 \cdot \left(n \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right)}\right)\\ \end{array} \]

Alternative 2
Error	46.08%
Cost	21660

\[\begin{array}{l} t_1 := \frac{\ell \cdot \ell}{Om}\\ t_2 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_3 := \frac{\ell}{\frac{Om}{\ell}}\\ t_4 := \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right)}\\ t_5 := \sqrt{2} \cdot \left(t_4 \cdot \left(-\ell\right)\right)\\ \mathbf{if}\;\ell \leq -2.1 \cdot 10^{+176}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;\ell \leq -8.2 \cdot 10^{+108}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \mathsf{fma}\left(t, U, -2 \cdot \frac{\ell \cdot \left(U \cdot \ell\right)}{Om}\right)}\\ \mathbf{elif}\;\ell \leq -2.3 \cdot 10^{+69}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-152}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + t_1 \cdot -2\right) + \left(t_3 \cdot \frac{n}{Om}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.4 \cdot 10^{-178}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(t_3 \cdot -2 + n \cdot \left(t_2 \cdot \left(U* - U\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.22 \cdot 10^{-38}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(2 \cdot \left(t - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \left(2 - \frac{n}{\frac{Om}{U*}}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{+149}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, t_1, n \cdot \left(t_2 \cdot \left(U - U*\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot t_4\right)\\ \end{array} \]

Alternative 3
Error	46.14%
Cost	15388

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right)}\\ t_3 := \sqrt{2} \cdot \left(t_2 \cdot \left(-\ell\right)\right)\\ t_4 := \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(t_1 \cdot -2 + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -2.5 \cdot 10^{+180}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\ell \leq -8 \cdot 10^{+108}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \mathsf{fma}\left(t, U, -2 \cdot \frac{\ell \cdot \left(U \cdot \ell\right)}{Om}\right)}\\ \mathbf{elif}\;\ell \leq -2.15 \cdot 10^{+69}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\ell \leq -2.55 \cdot 10^{-152}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(t_1 \cdot \frac{n}{Om}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;\ell \leq 5.5 \cdot 10^{-178}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;\ell \leq 8.8 \cdot 10^{-35}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(2 \cdot \left(t - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \left(2 - \frac{n}{\frac{Om}{U*}}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.7 \cdot 10^{+150}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot t_2\right)\\ \end{array} \]

Alternative 4
Error	47.73%
Cost	15072

\[\begin{array}{l} t_1 := U \cdot \left(n \cdot t\right)\\ t_2 := \ell \cdot \left(U \cdot \ell\right)\\ t_3 := \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right)}\\ t_4 := \sqrt{2} \cdot \left(t_3 \cdot \left(-\ell\right)\right)\\ \mathbf{if}\;\ell \leq -6 \cdot 10^{+178}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;\ell \leq -1.3 \cdot 10^{+109}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \mathsf{fma}\left(t, U, -2 \cdot \frac{t_2}{Om}\right)}\\ \mathbf{elif}\;\ell \leq -2.3 \cdot 10^{+69}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;\ell \leq -6.2 \cdot 10^{-142}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(\frac{\ell}{\frac{Om}{\ell}} \cdot \frac{n}{Om}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;\ell \leq -2.1 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{2 \cdot \left|t_1\right|}\\ \mathbf{elif}\;\ell \leq 1.65 \cdot 10^{+131}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(2 \cdot \left(t - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \left(2 - \frac{n}{\frac{Om}{U*}}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 3 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot t_2\right) \cdot \left(\frac{-2}{Om} + \frac{n}{Om} \cdot \frac{U* - U}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 7.4 \cdot 10^{+167}:\\ \;\;\;\;\sqrt{2 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot t_3\right)\\ \end{array} \]

Alternative 5
Error	49.05%
Cost	14808

\[\begin{array}{l} t_1 := \sqrt{\left(n \cdot U\right) \cdot \left(2 \cdot \left(t - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \left(2 - \frac{n}{\frac{Om}{U*}}\right)\right)\right)}\\ t_2 := U \cdot \left(n \cdot t\right)\\ \mathbf{if}\;\ell \leq -2.8 \cdot 10^{+185}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \frac{-2}{Om}\right)} \cdot \left(\ell \cdot \left(-\sqrt{2}\right)\right)\\ \mathbf{elif}\;\ell \leq -4.7 \cdot 10^{-74}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -4 \cdot 10^{-306}:\\ \;\;\;\;\sqrt{2 \cdot \left|t_2\right|}\\ \mathbf{elif}\;\ell \leq 1.65 \cdot 10^{+131}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 2.8 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right) \cdot \left(\frac{-2}{Om} + \frac{n}{Om} \cdot \frac{U* - U}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 7.4 \cdot 10^{+167}:\\ \;\;\;\;\sqrt{2 \cdot t_2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right)}\right)\\ \end{array} \]

Alternative 6
Error	51.61%
Cost	14296

\[\begin{array}{l} t_1 := \ell \cdot \left(U \cdot \ell\right)\\ \mathbf{if}\;\ell \leq -9.4 \cdot 10^{+129}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + t_1 \cdot \left(\frac{-2}{Om} + \frac{n \cdot \left(U* - U\right)}{Om \cdot Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq -5.8 \cdot 10^{-143}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(\frac{\ell}{\frac{Om}{\ell}} \cdot \frac{n}{Om}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;\ell \leq 10^{-307}:\\ \;\;\;\;\sqrt{2 \cdot \left|U \cdot \left(n \cdot t\right)\right|}\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{+130}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(2 \cdot \left(t - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \left(2 - \frac{n}{\frac{Om}{U*}}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot t_1\right) \cdot \left(\frac{-2}{Om} + \frac{n}{Om} \cdot \frac{U* - U}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.35 \cdot 10^{+229}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot \frac{n \cdot -2}{Om}}\\ \end{array} \]

Alternative 7
Error	51.35%
Cost	14296

\[\begin{array}{l} t_1 := \ell \cdot \left(U \cdot \ell\right)\\ \mathbf{if}\;\ell \leq -5.9 \cdot 10^{+131}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + t_1 \cdot \left(\frac{-2}{Om} + \frac{n \cdot \left(U* - U\right)}{Om \cdot Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq -2.6 \cdot 10^{-144}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(\frac{\ell}{\frac{Om}{\ell}} \cdot \frac{n}{Om}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;\ell \leq -5.2 \cdot 10^{-304}:\\ \;\;\;\;\sqrt{2 \cdot \left|U \cdot \left(n \cdot t\right)\right|}\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{+131}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(2 \cdot \left(t - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \left(2 - \frac{n}{\frac{Om}{U*}}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.2 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot t_1\right) \cdot \left(\frac{-2}{Om} + \frac{n}{Om} \cdot \frac{U* - U}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.2 \cdot 10^{+229}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \frac{-2}{Om}}\\ \end{array} \]

Alternative 8
Error	50.25%
Cost	14296

\[\begin{array}{l} t_1 := \sqrt{\left(n \cdot U\right) \cdot \left(2 \cdot \left(t - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \left(2 - \frac{n}{\frac{Om}{U*}}\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -3.1 \cdot 10^{+185}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \frac{-2}{Om}\right)} \cdot \left(\ell \cdot \left(-\sqrt{2}\right)\right)\\ \mathbf{elif}\;\ell \leq -3.7 \cdot 10^{-74}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -1.3 \cdot 10^{-307}:\\ \;\;\;\;\sqrt{2 \cdot \left|U \cdot \left(n \cdot t\right)\right|}\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{+131}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 3.9 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right) \cdot \left(\frac{-2}{Om} + \frac{n}{Om} \cdot \frac{U* - U}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 9.2 \cdot 10^{+230}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \frac{-2}{Om}}\\ \end{array} \]

Alternative 9
Error	52.77%
Cost	13644

\[\begin{array}{l} t_1 := \ell \cdot \left(U \cdot \ell\right)\\ \mathbf{if}\;\ell \leq -4.4 \cdot 10^{+132}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + t_1 \cdot \left(\frac{-2}{Om} + \frac{n \cdot \left(U* - U\right)}{Om \cdot Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq -7.2 \cdot 10^{-143}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(\frac{\ell}{\frac{Om}{\ell}} \cdot \frac{n}{Om}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;\ell \leq -3.8 \cdot 10^{-306}:\\ \;\;\;\;\sqrt{2 \cdot \left|U \cdot \left(n \cdot t\right)\right|}\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{+131}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(2 \cdot \left(t - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \left(2 - \frac{n}{\frac{Om}{U*}}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot t_1\right) \cdot \left(\frac{-2}{Om} + \frac{n}{Om} \cdot \frac{U* - U}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)}\\ \end{array} \]

Alternative 10
Error	50.44%
Cost	8788

\[\begin{array}{l} t_1 := \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\frac{n}{\frac{\frac{Om}{\ell} \cdot \frac{Om}{\ell}}{U*}} + \frac{\ell}{\frac{Om}{\ell}} \cdot -2\right)\right)}\\ \mathbf{if}\;n \leq -2.7 \cdot 10^{-42}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq -5 \cdot 10^{-158}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(\left(\ell \cdot \left(U \cdot \ell\right)\right) \cdot \left(\frac{2}{Om} + \frac{n \cdot \left(U - U*\right)}{Om \cdot Om}\right) - U \cdot t\right)}\\ \mathbf{elif}\;n \leq 3.5 \cdot 10^{-230}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(2 \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \left(-2 + \frac{n}{Om} \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq 1.75 \cdot 10^{-177}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right)\right)\right)}\\ \mathbf{elif}\;n \leq 1.45 \cdot 10^{-93}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	52.73%
Cost	8532

\[\begin{array}{l} t_1 := \ell \cdot \left(U \cdot \ell\right)\\ \mathbf{if}\;\ell \leq -3.6 \cdot 10^{+126}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + t_1 \cdot \left(\frac{-2}{Om} + \frac{n \cdot \left(U* - U\right)}{Om \cdot Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq -4.2 \cdot 10^{-150}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(\frac{\ell}{\frac{Om}{\ell}} \cdot \frac{n}{Om}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{-307}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{+131}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(2 \cdot \left(t - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \left(2 - \frac{n}{\frac{Om}{U*}}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot t_1\right) \cdot \left(\frac{-2}{Om} + \frac{n}{Om} \cdot \frac{U* - U}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)}\\ \end{array} \]

Alternative 12
Error	53%
Cost	8400

\[\begin{array}{l} t_1 := \sqrt{\left(n \cdot U\right) \cdot \left(2 \cdot \left(t - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \left(2 - \frac{n}{\frac{Om}{U*}}\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -7.2 \cdot 10^{-74}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 4 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 8.5 \cdot 10^{+130}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 10^{+152}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right) \cdot \left(\frac{-2}{Om} + \frac{n}{Om} \cdot \frac{U* - U}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)}\\ \end{array} \]

Alternative 13
Error	50.95%
Cost	8272

\[\begin{array}{l} t_1 := \sqrt{\left(n \cdot U\right) \cdot \left(2 \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \left(-2 + \frac{n}{\frac{Om}{U*}}\right)\right)\right)}\\ t_2 := \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{if}\;U \leq -8.8 \cdot 10^{-83}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;U \leq 8.5 \cdot 10^{-211}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;U \leq 3.2 \cdot 10^{-199}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(\left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right) \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \mathbf{elif}\;U \leq 6 \cdot 10^{-100}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Error	54.71%
Cost	8144

\[\begin{array}{l} t_1 := t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\\ \mathbf{if}\;\ell \leq -2.8 \cdot 10^{-43}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t_1\right)}\\ \mathbf{elif}\;\ell \leq -8.6 \cdot 10^{-163}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(2 \cdot \left(t + \frac{n}{\frac{Om}{U*} \cdot \frac{Om}{\ell \cdot \ell}}\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.9 \cdot 10^{-306}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.5 \cdot 10^{+18}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(2 \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \left(U* \cdot \frac{n}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot t_1}\\ \end{array} \]

Alternative 15
Error	51.67%
Cost	8140

\[\begin{array}{l} t_1 := 2 \cdot \left(n \cdot U\right)\\ t_2 := t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\\ t_3 := \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t_2\right)}\\ \mathbf{if}\;U \leq -2 \cdot 10^{-51}:\\ \;\;\;\;\sqrt{t_1 \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)}\\ \mathbf{elif}\;U \leq 1.8 \cdot 10^{-208}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;U \leq 3.2 \cdot 10^{-199}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(\left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right) \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \mathbf{elif}\;U \leq 1.78 \cdot 10^{+140}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t_1 \cdot t_2}\\ \end{array} \]

Alternative 16
Error	58.38%
Cost	7756

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{if}\;t \leq -1.2 \cdot 10^{+133}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2.8 \cdot 10^{-180}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{Om} - t\right)\right)\right)}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{+121}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right)\right)\right)}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+213}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 17
Error	54.92%
Cost	7625

\[\begin{array}{l} \mathbf{if}\;\ell \leq -8 \cdot 10^{-41} \lor \neg \left(\ell \leq 9.5 \cdot 10^{+153}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right)\right)\right)}\\ \end{array} \]

Alternative 18
Error	51.53%
Cost	7625

\[\begin{array}{l} t_1 := t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\\ \mathbf{if}\;U \leq -2 \cdot 10^{-51} \lor \neg \left(U \leq 2 \cdot 10^{+140}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t_1\right)}\\ \end{array} \]

Alternative 19
Error	59.42%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;U \leq -2.55 \cdot 10^{-51}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;U \leq 4.2 \cdot 10^{-98}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{Om} - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]

Alternative 20
Error	51.53%
Cost	7624

\[\begin{array}{l} t_1 := 2 \cdot \left(n \cdot U\right)\\ t_2 := t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\\ \mathbf{if}\;U \leq -2.5 \cdot 10^{-51}:\\ \;\;\;\;\sqrt{t_1 \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)}\\ \mathbf{elif}\;U \leq 1.78 \cdot 10^{+140}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t_1 \cdot t_2}\\ \end{array} \]

Alternative 21
Error	62.4%
Cost	7364

\[\begin{array}{l} \mathbf{if}\;\ell \leq -9.2 \cdot 10^{+25}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(\ell \cdot \left(U \cdot \ell\right)\right) \cdot \frac{n \cdot -2}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]

Alternative 22
Error	61.57%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;U \leq -3.8 \cdot 10^{-52}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;U \leq 10^{-38}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]

Alternative 23
Error	63.78%
Cost	6848

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

Reproduce

herbie shell --seed 2023121 
(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*))))))