Toniolo and Linder, Equation (13)

Average Error: 34.8 → 24.5

Time: 47.0s

Precision: binary64

Cost: 64460

Math

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

↓

\[\begin{array}{l} t_1 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := \sqrt{\left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot t_1\right) \cdot \left(U* - U\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{if}\;t_2 \leq 4 \cdot 10^{-161}:\\ \;\;\;\;{\left(\sqrt[3]{U \cdot t} \cdot \sqrt[3]{2 \cdot n}\right)}^{1.5}\\ \mathbf{elif}\;t_2 \leq 10^{+151}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{t + \left(\frac{\ell}{\frac{Om}{\ell}} \cdot -2 + n \cdot \left(t_1 \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \left(\frac{n}{Om} \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \end{array} \]

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
         (sqrt
          (*
           (+ (+ t (* (/ (* l l) Om) -2.0)) (* (* n t_1) (- U* U)))
           (* (* 2.0 n) U)))))
   (if (<= t_2 4e-161)
     (pow (* (cbrt (* U t)) (cbrt (* 2.0 n))) 1.5)
     (if (<= t_2 1e+151)
       t_2
       (if (<= t_2 INFINITY)
         (*
          (sqrt (* 2.0 (* n U)))
          (sqrt (+ t (+ (* (/ l (/ Om l)) -2.0) (* n (* t_1 (- U* U)))))))
         (sqrt
          (fma 2.0 (* n (* U t)) (* -4.0 (* (/ n Om) (* l (* U l)))))))))))

C

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

↓

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = pow((l / Om), 2.0);
	double t_2 = sqrt((((t + (((l * l) / Om) * -2.0)) + ((n * t_1) * (U_42_ - U))) * ((2.0 * n) * U)));
	double tmp;
	if (t_2 <= 4e-161) {
		tmp = pow((cbrt((U * t)) * cbrt((2.0 * n))), 1.5);
	} else if (t_2 <= 1e+151) {
		tmp = t_2;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt((2.0 * (n * U))) * sqrt((t + (((l / (Om / l)) * -2.0) + (n * (t_1 * (U_42_ - U))))));
	} else {
		tmp = sqrt(fma(2.0, (n * (U * t)), (-4.0 * ((n / Om) * (l * (U * l))))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_)))))
end

↓

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l / Om) ^ 2.0
	t_2 = sqrt(Float64(Float64(Float64(t + Float64(Float64(Float64(l * l) / Om) * -2.0)) + Float64(Float64(n * t_1) * Float64(U_42_ - U))) * Float64(Float64(2.0 * n) * U)))
	tmp = 0.0
	if (t_2 <= 4e-161)
		tmp = Float64(cbrt(Float64(U * t)) * cbrt(Float64(2.0 * n))) ^ 1.5;
	elseif (t_2 <= 1e+151)
		tmp = t_2;
	elseif (t_2 <= Inf)
		tmp = Float64(sqrt(Float64(2.0 * Float64(n * U))) * sqrt(Float64(t + Float64(Float64(Float64(l / Float64(Om / l)) * -2.0) + Float64(n * Float64(t_1 * Float64(U_42_ - U)))))));
	else
		tmp = sqrt(fma(2.0, Float64(n * Float64(U * t)), Float64(-4.0 * Float64(Float64(n / Om) * Float64(l * Float64(U * l))))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

↓

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(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] * N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 4e-161], N[Power[N[(N[Power[N[(U * t), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(2.0 * n), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], If[LessEqual[t$95$2, 1e+151], t$95$2, If[LessEqual[t$95$2, Infinity], N[(N[Sqrt[N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t + N[(N[(N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision] + N[(n * N[(t$95$1 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * N[(N[(n / Om), $MachinePrecision] * N[(l * N[(U * l), $MachinePrecision]), $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*))))) < 4.00000000000000011e-161

   1. Initial program 55.6

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

   2. Simplified 55.6

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

      [Start] 55.6	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 55.6	\[ \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 55.6	\[ \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      *-commutative [=>] 55.6	\[ \sqrt{2 \cdot \color{blue}{\left(\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot \left(n \cdot U\right)\right)}} \]

   3. Taylor expanded in l around 0 42.8

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

   4. Applied egg-rr 42.9

      \[\leadsto \color{blue}{{\left(\sqrt[3]{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)}\right)}^{1.5}} \]

   5. Applied egg-rr 24.6

      \[\leadsto {\color{blue}{\left(\sqrt[3]{2 \cdot n} \cdot \sqrt[3]{t \cdot U}\right)}}^{1.5} \]

   6. Simplified 24.6

      \[\leadsto {\color{blue}{\left(\sqrt[3]{t \cdot U} \cdot \sqrt[3]{2 \cdot n}\right)}}^{1.5} \]
      Proof

      [Start] 24.6	\[ {\left(\sqrt[3]{2 \cdot n} \cdot \sqrt[3]{t \cdot U}\right)}^{1.5} \]
      *-commutative [=>] 24.6	\[ {\color{blue}{\left(\sqrt[3]{t \cdot U} \cdot \sqrt[3]{2 \cdot n}\right)}}^{1.5} \]

   if 4.00000000000000011e-161 < (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*))))) < 1.00000000000000002e151

   1. Initial program 1.6

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

   if 1.00000000000000002e151 < (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 63.4

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

   2. Simplified 53.5

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

      [Start] 63.4	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 61.0	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 61.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      sub-neg [=>] 61.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(-\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [<=] 61.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      cancel-sign-sub [<=] 61.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      cancel-sign-sub [=>] 61.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-/l* [=>] 54.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 53.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Applied egg-rr 49.8

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

   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 64.0

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

   2. Simplified 58.4

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

      [Start] 64.0	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 64.0	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 64.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      sub-neg [=>] 64.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(-\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [<=] 64.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      cancel-sign-sub [<=] 64.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      cancel-sign-sub [=>] 64.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-/l* [=>] 58.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 58.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in Om around inf 58.1

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

   4. Simplified 48.8

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

      [Start] 58.1	\[ \sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right)\right) + -4 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}} \]
      fma-def [=>] 58.1	\[ \sqrt{\color{blue}{\mathsf{fma}\left(2, n \cdot \left(t \cdot U\right), -4 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}} \]
      *-commutative [=>] 58.1	\[ \sqrt{\mathsf{fma}\left(2, n \cdot \color{blue}{\left(U \cdot t\right)}, -4 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)} \]
      associate-/l* [=>] 57.9	\[ \sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \color{blue}{\frac{n}{\frac{Om}{{\ell}^{2} \cdot U}}}\right)} \]
      associate-/r/ [=>] 57.0	\[ \sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \color{blue}{\left(\frac{n}{Om} \cdot \left({\ell}^{2} \cdot U\right)\right)}\right)} \]
      unpow2 [=>] 57.0	\[ \sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \left(\frac{n}{Om} \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot U\right)\right)\right)} \]
      associate-*l* [=>] 48.8	\[ \sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \left(\frac{n}{Om} \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot U\right)\right)}\right)\right)} \]

3. Recombined 4 regimes into one program.

4. Final simplification 24.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)} \leq 4 \cdot 10^{-161}:\\ \;\;\;\;{\left(\sqrt[3]{U \cdot t} \cdot \sqrt[3]{2 \cdot n}\right)}^{1.5}\\ \mathbf{elif}\;\sqrt{\left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)} \leq 10^{+151}:\\ \;\;\;\;\sqrt{\left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)} \leq \infty:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{t + \left(\frac{\ell}{\frac{Om}{\ell}} \cdot -2 + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \left(\frac{n}{Om} \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	27.9
Cost	57484

\[\begin{array}{l} t_1 := \sqrt{\left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\sqrt{n \cdot t} \cdot \sqrt{2 \cdot U}\\ \mathbf{elif}\;t_1 \leq 10^{+151}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{t + \frac{-2}{\frac{\frac{Om}{\ell}}{\ell}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \left(\frac{n}{Om} \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \end{array} \]

Alternative 2
Error	27.0
Cost	57484

\[\begin{array}{l} t_1 := \sqrt{\left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{if}\;t_1 \leq 4 \cdot 10^{-161}:\\ \;\;\;\;\sqrt{U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - n \cdot \left(\left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{U* - U}{Om}\right)\right)\right)} \cdot \sqrt{2 \cdot n}\\ \mathbf{elif}\;t_1 \leq 10^{+151}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{t + \frac{-2}{\frac{\frac{Om}{\ell}}{\ell}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \left(\frac{n}{Om} \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \end{array} \]

Alternative 3
Error	24.8
Cost	57484

\[\begin{array}{l} t_1 := \sqrt{\left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{if}\;t_1 \leq 4 \cdot 10^{-161}:\\ \;\;\;\;{\left(\sqrt[3]{U \cdot t} \cdot \sqrt[3]{2 \cdot n}\right)}^{1.5}\\ \mathbf{elif}\;t_1 \leq 10^{+151}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{t + \frac{-2}{\frac{\frac{Om}{\ell}}{\ell}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \left(\frac{n}{Om} \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \end{array} \]

Alternative 4
Error	33.3
Cost	15520

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ t_2 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_3 := \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{\frac{Om}{\ell}} \cdot -2 + n \cdot \left(t_2 \cdot \left(U* - U\right)\right)\right)\right)\right)}\\ \mathbf{if}\;U \leq -1.85 \cdot 10^{+76}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(t_1, -2, t\right)\right)\right)}\\ \mathbf{elif}\;U \leq -1.8 \cdot 10^{-238}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;U \leq -7.4 \cdot 10^{-277}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left(\ell \cdot \left(n \cdot \sqrt{U \cdot \left(U* - U\right)}\right)\right)}{Om}\\ \mathbf{elif}\;U \leq 2.35 \cdot 10^{-268}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(\frac{n \cdot \ell}{Om} \cdot \frac{n \cdot \ell}{\frac{1}{U - U*}}\right) \cdot \frac{-1}{\frac{Om}{U}}\right)}\\ \mathbf{elif}\;U \leq 9.5 \cdot 10^{-219}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \left(\frac{n}{Om} \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \mathbf{elif}\;U \leq 1.06 \cdot 10^{-64}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;U \leq 3.8 \cdot 10^{-48}:\\ \;\;\;\;\sqrt{\left(\left(U - U*\right) \cdot \frac{n}{Om \cdot Om} + \frac{2}{Om}\right) \cdot \left(-2 \cdot \left(U \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)\right)\right)}\\ \mathbf{elif}\;U \leq 1.9 \cdot 10^{+103}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \left(t - \left(t_2 \cdot \left(n \cdot \left(U - U*\right)\right) + 2 \cdot t_1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot t} \cdot \sqrt{U}\\ \end{array} \]

Alternative 5
Error	29.9
Cost	14860

\[\begin{array}{l} t_1 := \frac{U* - U}{Om}\\ t_2 := \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{Om} \cdot t_1 + \frac{-2}{Om}\right)}\\ t_3 := \ell \cdot \frac{\ell}{Om}\\ \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+76}:\\ \;\;\;\;t_2 \cdot \left(\sqrt{2} \cdot \left(-\ell\right)\right)\\ \mathbf{elif}\;\ell \leq -4 \cdot 10^{-267}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - n \cdot \left(t_3 \cdot t_1\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 10^{-152}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \left(t - \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right) + 2 \cdot t_3\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.9 \cdot 10^{+76}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(t_3 \cdot -2 + \frac{n \cdot \left(\ell \cdot \ell\right)}{Om} \cdot \frac{U*}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 8.5 \cdot 10^{+143}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]

Alternative 6
Error	32.0
Cost	14804

\[\begin{array}{l} t_1 := \frac{U* - U}{Om}\\ t_2 := \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - n \cdot \left(\left(\ell \cdot \frac{\ell}{Om}\right) \cdot t_1\right)\right)\right)\right)}\\ t_3 := \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{Om} \cdot t_1 + \frac{-2}{Om}\right)}\\ \mathbf{if}\;\ell \leq -8 \cdot 10^{+76}:\\ \;\;\;\;t_3 \cdot \left(\sqrt{2} \cdot \left(-\ell\right)\right)\\ \mathbf{elif}\;\ell \leq -1.25 \cdot 10^{-281}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{-188}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{t}\\ \mathbf{elif}\;\ell \leq 4.4 \cdot 10^{+52}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{+143}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \left(\frac{\left(\ell \cdot \ell\right) \cdot -2}{Om} + \frac{U* \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}{Om \cdot Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]

Alternative 7
Error	31.8
Cost	14284

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ t_2 := \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - n \cdot \left(t_1 \cdot \frac{U* - U}{Om}\right)\right)\right)\right)}\\ \mathbf{if}\;n \leq -1.55 \cdot 10^{-37}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;n \leq 4.7 \cdot 10^{-265}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(t_1, -2, t\right)\right)\right)}\\ \mathbf{elif}\;n \leq 4.4 \cdot 10^{-130}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \frac{\ell \cdot \left(n \cdot \ell\right)}{\frac{Om}{U}}\right)}\\ \mathbf{elif}\;n \leq 2.2 \cdot 10^{-98}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \left(\frac{\left(\ell \cdot \ell\right) \cdot -2}{Om} + \frac{U* \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}{Om \cdot Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Error	32.9
Cost	14028

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ t_2 := \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;U \leq -8.2 \cdot 10^{+75}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(t_1, -2, t\right)\right)\right)}\\ \mathbf{elif}\;U \leq -1.8 \cdot 10^{-238}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(t_2 \cdot -2 - n \cdot \frac{\ell}{\frac{Om}{\ell} \cdot \frac{Om}{U - U*}}\right)\right)\right)}\\ \mathbf{elif}\;U \leq 1.3 \cdot 10^{-307}:\\ \;\;\;\;\left(\sqrt{2} \cdot \left(n \cdot \ell\right)\right) \cdot \frac{\sqrt{U \cdot \left(U* - U\right)}}{Om}\\ \mathbf{elif}\;U \leq 8 \cdot 10^{+104}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot t_2 - n \cdot \left(t_1 \cdot \frac{U* - U}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot t} \cdot \sqrt{U}\\ \end{array} \]

Alternative 9
Error	32.1
Cost	14024

\[\begin{array}{l} t_1 := 2 \cdot \left(n \cdot U\right)\\ \mathbf{if}\;t \leq -9.5 \cdot 10^{+157}:\\ \;\;\;\;\sqrt{t \cdot t_1}\\ \mathbf{elif}\;t \leq 4.55 \cdot 10^{+49}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - n \cdot \left(\left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{U* - U}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t_1} \cdot \sqrt{t + \frac{-2}{\frac{\frac{Om}{\ell}}{\ell}}}\\ \end{array} \]

Alternative 10
Error	32.8
Cost	13900

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;U \leq -1.1 \cdot 10^{+123}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)}\\ \mathbf{elif}\;U \leq -2.5 \cdot 10^{-238}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(t_1 \cdot -2 - n \cdot \frac{\ell}{\frac{Om}{\ell} \cdot \frac{Om}{U - U*}}\right)\right)\right)}\\ \mathbf{elif}\;U \leq 1.3 \cdot 10^{-307}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left(n \cdot \ell\right)}{\frac{Om}{\sqrt{U \cdot U*}}}\\ \mathbf{elif}\;U \leq 2.4 \cdot 10^{+107}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot t_1 - n \cdot \left(\left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{U* - U}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot t} \cdot \sqrt{U}\\ \end{array} \]

Alternative 11
Error	32.9
Cost	13900

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ t_2 := \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;U \leq -8.2 \cdot 10^{+76}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(t_1, -2, t\right)\right)\right)}\\ \mathbf{elif}\;U \leq -1.8 \cdot 10^{-238}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(t_2 \cdot -2 - n \cdot \frac{\ell}{\frac{Om}{\ell} \cdot \frac{Om}{U - U*}}\right)\right)\right)}\\ \mathbf{elif}\;U \leq 1.3 \cdot 10^{-307}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left(n \cdot \ell\right)}{\frac{Om}{\sqrt{U \cdot U*}}}\\ \mathbf{elif}\;U \leq 5.5 \cdot 10^{+107}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot t_2 - n \cdot \left(t_1 \cdot \frac{U* - U}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot t} \cdot \sqrt{U}\\ \end{array} \]

Alternative 12
Error	32.2
Cost	13512

\[\begin{array}{l} \mathbf{if}\;t \leq -3.9 \cdot 10^{+161}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;t \leq 1.38 \cdot 10^{+57}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - n \cdot \left(\left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{U* - U}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{t}\\ \end{array} \]

Alternative 13
Error	34.8
Cost	8784

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;Om \leq -1.7 \cdot 10^{+96}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + t_1 \cdot -2\right)}\\ \mathbf{elif}\;Om \leq -9.5 \cdot 10^{+58}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right) \cdot \left(\frac{-2}{Om} + \frac{n \cdot \left(U* - U\right)}{Om \cdot Om}\right)\right)}\\ \mathbf{elif}\;Om \leq -9.5 \cdot 10^{-155}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)}\\ \mathbf{elif}\;Om \leq 1.6 \cdot 10^{-273}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(\frac{n \cdot \ell}{Om} \cdot \frac{n \cdot \ell}{\frac{1}{U - U*}}\right) \cdot \frac{-1}{\frac{Om}{U}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot t_1 - n \cdot \left(\left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{U* - U}{Om}\right)\right)\right)\right)}\\ \end{array} \]

Alternative 14
Error	34.4
Cost	8656

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2 + \frac{n \cdot \left(\ell \cdot \ell\right)}{Om} \cdot \frac{U*}{Om}\right)\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -1.45 \cdot 10^{+94}:\\ \;\;\;\;\sqrt{\left(\left(U - U*\right) \cdot \frac{n}{Om \cdot Om} + \frac{2}{Om}\right) \cdot \left(-2 \cdot \left(U \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-226}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 2.4 \cdot 10^{-155}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \frac{-2}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 7.4 \cdot 10^{+119}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right) \cdot \left(\frac{-2}{Om} + \frac{n \cdot \left(U* - U\right)}{Om \cdot Om}\right)\right)}\\ \end{array} \]

Alternative 15
Error	35.5
Cost	8400

\[\begin{array}{l} \mathbf{if}\;Om \leq -1.7 \cdot 10^{+96}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \frac{\ell}{\frac{Om}{\ell}} \cdot -2\right)}\\ \mathbf{elif}\;Om \leq -6.5 \cdot 10^{+58}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right) \cdot \left(\frac{-2}{Om} + \frac{n \cdot \left(U* - U\right)}{Om \cdot Om}\right)\right)}\\ \mathbf{elif}\;Om \leq -1.65 \cdot 10^{-155}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)}\\ \mathbf{elif}\;Om \leq 6.8 \cdot 10^{-258}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(\frac{n \cdot \ell}{Om} \cdot \frac{n \cdot \ell}{\frac{1}{U - U*}}\right) \cdot \frac{-1}{\frac{Om}{U}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)}\\ \end{array} \]

Alternative 16
Error	36.2
Cost	8144

\[\begin{array}{l} \mathbf{if}\;Om \leq -3.3 \cdot 10^{+96}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \frac{\ell}{\frac{Om}{\ell}} \cdot -2\right)}\\ \mathbf{elif}\;Om \leq -9.5 \cdot 10^{+58}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right) \cdot \left(\frac{-2}{Om} + \frac{n \cdot \left(U* - U\right)}{Om \cdot Om}\right)\right)}\\ \mathbf{elif}\;Om \leq -1.16 \cdot 10^{-152}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)}\\ \mathbf{elif}\;Om \leq -1.3 \cdot 10^{-247}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{n \cdot \ell}{\frac{\frac{Om \cdot \frac{Om}{U}}{U - U*}}{n \cdot \ell}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)}\\ \end{array} \]

Alternative 17
Error	35.3
Cost	7880

\[\begin{array}{l} \mathbf{if}\;U \leq -3.2 \cdot 10^{-238}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \frac{\ell}{\frac{Om}{\ell}} \cdot -2\right)}\\ \mathbf{elif}\;U \leq 1.5 \cdot 10^{-259}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{n \cdot \ell}{\frac{\frac{Om \cdot \frac{Om}{U}}{U - U*}}{n \cdot \ell}}}\\ \mathbf{elif}\;U \leq 4 \cdot 10^{-39}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\frac{2 \cdot \ell}{\frac{Om}{\ell}} - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)}\\ \end{array} \]

Alternative 18
Error	35.5
Cost	7625

\[\begin{array}{l} \mathbf{if}\;t \leq -2.65 \cdot 10^{+146} \lor \neg \left(t \leq 1.72 \cdot 10^{+53}\right):\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\frac{2 \cdot \ell}{\frac{Om}{\ell}} - t\right)\right)\right)}\\ \end{array} \]

Alternative 19
Error	32.9
Cost	7625

\[\begin{array}{l} \mathbf{if}\;U \leq -7.5 \cdot 10^{-66} \lor \neg \left(U \leq 5.2 \cdot 10^{-38}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\frac{2 \cdot \ell}{\frac{Om}{\ell}} - t\right)\right)\right)}\\ \end{array} \]

Alternative 20
Error	32.9
Cost	7625

\[\begin{array}{l} \mathbf{if}\;U \leq -7.5 \cdot 10^{-66} \lor \neg \left(U \leq 2.7 \cdot 10^{-38}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)}\\ \end{array} \]

Alternative 21
Error	32.9
Cost	7624

\[\begin{array}{l} \mathbf{if}\;U \leq -1 \cdot 10^{-65}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \frac{\ell}{\frac{Om}{\ell}} \cdot -2\right)}\\ \mathbf{elif}\;U \leq 5.2 \cdot 10^{-38}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)}\\ \end{array} \]

Alternative 22
Error	39.6
Cost	7113

\[\begin{array}{l} \mathbf{if}\;U \leq -5 \cdot 10^{-70} \lor \neg \left(U \leq 5 \cdot 10^{-172}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]

Alternative 23
Error	39.5
Cost	7113

\[\begin{array}{l} \mathbf{if}\;U \leq -1 \cdot 10^{-65} \lor \neg \left(U \leq 4 \cdot 10^{-50}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(t \cdot \left(2 \cdot U\right)\right)}\\ \end{array} \]

Alternative 24
Error	39.5
Cost	7112

\[\begin{array}{l} \mathbf{if}\;n \leq -2.15 \cdot 10^{-72}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;n \leq 10^{-25}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 25
Error	40.5
Cost	6848

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

Reproduce

herbie shell --seed 2023011 
(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*))))))