Toniolo and Linder, Equation (13)

Average Error: 34.1 → 24.8

Time: 36.9s

Precision: binary64

Cost: 41732

Math

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

↓

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_4 := \left(n \cdot t_3\right) \cdot \left(U* - U\right)\\ t_5 := t_2 \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + t_4\right)\\ \mathbf{if}\;t_5 \leq 0:\\ \;\;\;\;\sqrt{{\left(\sqrt[3]{U \cdot \left(\left(n \cdot -2\right) \cdot \left(\mathsf{fma}\left(2, t_1, t_3 \cdot \left(n \cdot \left(U - U*\right)\right)\right) - t\right)\right)}\right)}^{3}}\\ \mathbf{elif}\;t_5 \leq 10^{+299}:\\ \;\;\;\;\sqrt{t_2 \cdot \left(\left(t + t_1 \cdot -2\right) + t_4\right)}\\ \mathbf{elif}\;t_5 \leq \infty:\\ \;\;\;\;\left|\frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{Om}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot \ell\right) \cdot \left(-2 \cdot \frac{U \cdot \ell}{Om}\right)\right)}\\ \end{array} \]

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 (* (* 2.0 n) U))
        (t_3 (pow (/ l Om) 2.0))
        (t_4 (* (* n t_3) (- U* U)))
        (t_5 (* t_2 (+ (+ t (* (/ (* l l) Om) -2.0)) t_4))))
   (if (<= t_5 0.0)
     (sqrt
      (pow
       (cbrt (* U (* (* n -2.0) (- (fma 2.0 t_1 (* t_3 (* n (- U U*)))) t))))
       3.0))
     (if (<= t_5 1e+299)
       (sqrt (* t_2 (+ (+ t (* t_1 -2.0)) t_4)))
       (if (<= t_5 INFINITY)
         (fabs (/ (* (sqrt (* 2.0 (* U U*))) (* n l)) Om))
         (sqrt (* 2.0 (* (* n l) (* -2.0 (/ (* U l) 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 = (2.0 * n) * U;
	double t_3 = pow((l / Om), 2.0);
	double t_4 = (n * t_3) * (U_42_ - U);
	double t_5 = t_2 * ((t + (((l * l) / Om) * -2.0)) + t_4);
	double tmp;
	if (t_5 <= 0.0) {
		tmp = sqrt(pow(cbrt((U * ((n * -2.0) * (fma(2.0, t_1, (t_3 * (n * (U - U_42_)))) - t)))), 3.0));
	} else if (t_5 <= 1e+299) {
		tmp = sqrt((t_2 * ((t + (t_1 * -2.0)) + t_4)));
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = fabs(((sqrt((2.0 * (U * U_42_))) * (n * l)) / Om));
	} else {
		tmp = sqrt((2.0 * ((n * l) * (-2.0 * ((U * l) / Om)))));
	}
	return tmp;
}

Julia

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

↓

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l * Float64(l / Om))
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(l / Om) ^ 2.0
	t_4 = Float64(Float64(n * t_3) * Float64(U_42_ - U))
	t_5 = Float64(t_2 * Float64(Float64(t + Float64(Float64(Float64(l * l) / Om) * -2.0)) + t_4))
	tmp = 0.0
	if (t_5 <= 0.0)
		tmp = sqrt((cbrt(Float64(U * Float64(Float64(n * -2.0) * Float64(fma(2.0, t_1, Float64(t_3 * Float64(n * Float64(U - U_42_)))) - t)))) ^ 3.0));
	elseif (t_5 <= 1e+299)
		tmp = sqrt(Float64(t_2 * Float64(Float64(t + Float64(t_1 * -2.0)) + t_4)));
	elseif (t_5 <= Inf)
		tmp = abs(Float64(Float64(sqrt(Float64(2.0 * Float64(U * U_42_))) * Float64(n * l)) / Om));
	else
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * l) * Float64(-2.0 * Float64(Float64(U * l) / Om)))));
	end
	return tmp
end

Wolfram

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

↓

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[(n * t$95$3), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$2 * N[(N[(t + N[(N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, 0.0], N[Sqrt[N[Power[N[Power[N[(U * N[(N[(n * -2.0), $MachinePrecision] * N[(N[(2.0 * t$95$1 + N[(t$95$3 * N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$5, 1e+299], N[Sqrt[N[(t$95$2 * N[(N[(t + N[(t$95$1 * -2.0), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$5, Infinity], N[Abs[N[(N[(N[Sqrt[N[(2.0 * N[(U * U$42$), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(N[(n * l), $MachinePrecision] * N[(-2.0 * N[(N[(U * l), $MachinePrecision] / Om), $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*)))) < 0.0

   1. Initial program 57.8

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

   2. Simplified 43.3

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

      [Start] 57.8	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 42.6	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 42.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      fma-def [=>] 42.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-*l* [=>] 43.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Applied egg-rr 41.7

      \[\leadsto \sqrt{\color{blue}{{\left(\sqrt[3]{U \cdot \left(\left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U - U*\right) \cdot n\right)\right)\right) \cdot \left(2 \cdot n\right)\right)}\right)}^{3}}} \]

   if 0.0 < (*.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*)))) < 1.0000000000000001e299

   1. Initial program 1.8

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

   2. Applied egg-rr 1.8

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

   if 1.0000000000000001e299 < (*.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 62.9

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

   2. Simplified 60.9

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

      [Start] 62.9	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 61.2	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 61.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      fma-def [=>] 61.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-*l* [=>] 60.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in U* around inf 62.6

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

   4. Simplified 61.8

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

      [Start] 62.6	\[ \sqrt{2 \cdot \frac{{n}^{2} \cdot \left({\ell}^{2} \cdot \left(U \cdot U*\right)\right)}{{Om}^{2}}} \]
      associate-*r/ [=>] 62.6	\[ \sqrt{\color{blue}{\frac{2 \cdot \left({n}^{2} \cdot \left({\ell}^{2} \cdot \left(U \cdot U*\right)\right)\right)}{{Om}^{2}}}} \]
      associate-*r* [=>] 62.5	\[ \sqrt{\frac{2 \cdot \color{blue}{\left(\left({n}^{2} \cdot {\ell}^{2}\right) \cdot \left(U \cdot U*\right)\right)}}{{Om}^{2}}} \]
      *-commutative [=>] 62.5	\[ \sqrt{\frac{2 \cdot \color{blue}{\left(\left(U \cdot U*\right) \cdot \left({n}^{2} \cdot {\ell}^{2}\right)\right)}}{{Om}^{2}}} \]
      unpow2 [=>] 62.5	\[ \sqrt{\frac{2 \cdot \left(\left(U \cdot U*\right) \cdot \left(\color{blue}{\left(n \cdot n\right)} \cdot {\ell}^{2}\right)\right)}{{Om}^{2}}} \]
      unpow2 [=>] 62.5	\[ \sqrt{\frac{2 \cdot \left(\left(U \cdot U*\right) \cdot \left(\left(n \cdot n\right) \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)}{{Om}^{2}}} \]
      unswap-sqr [=>] 61.8	\[ \sqrt{\frac{2 \cdot \left(\left(U \cdot U*\right) \cdot \color{blue}{\left(\left(n \cdot \ell\right) \cdot \left(n \cdot \ell\right)\right)}\right)}{{Om}^{2}}} \]
      unpow2 [=>] 61.8	\[ \sqrt{\frac{2 \cdot \left(\left(U \cdot U*\right) \cdot \left(\left(n \cdot \ell\right) \cdot \left(n \cdot \ell\right)\right)\right)}{\color{blue}{Om \cdot Om}}} \]

   5. Applied egg-rr 47.7

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

   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 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 64.0

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

      [Start] 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	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      fma-def [=>] 64.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-*l* [=>] 64.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in l around inf 60.2

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

   4. Simplified 48.5

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

      [Start] 60.2	\[ \sqrt{-2 \cdot \left(\left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(n \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      associate-/l* [=>] 60.8	\[ \sqrt{-2 \cdot \left(\left(\color{blue}{\frac{n}{\frac{{Om}^{2}}{U - U*}}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(n \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      associate-/r/ [=>] 60.2	\[ \sqrt{-2 \cdot \left(\left(\color{blue}{\frac{n}{{Om}^{2}} \cdot \left(U - U*\right)} + 2 \cdot \frac{1}{Om}\right) \cdot \left(n \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      unpow2 [=>] 60.2	\[ \sqrt{-2 \cdot \left(\left(\frac{n}{\color{blue}{Om \cdot Om}} \cdot \left(U - U*\right) + 2 \cdot \frac{1}{Om}\right) \cdot \left(n \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      associate-*r/ [=>] 60.2	\[ \sqrt{-2 \cdot \left(\left(\frac{n}{Om \cdot Om} \cdot \left(U - U*\right) + \color{blue}{\frac{2 \cdot 1}{Om}}\right) \cdot \left(n \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      metadata-eval [=>] 60.2	\[ \sqrt{-2 \cdot \left(\left(\frac{n}{Om \cdot Om} \cdot \left(U - U*\right) + \frac{\color{blue}{2}}{Om}\right) \cdot \left(n \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      unpow2 [=>] 60.2	\[ \sqrt{-2 \cdot \left(\left(\frac{n}{Om \cdot Om} \cdot \left(U - U*\right) + \frac{2}{Om}\right) \cdot \left(n \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot U\right)\right)\right)} \]
      associate-*l* [=>] 48.5	\[ \sqrt{-2 \cdot \left(\left(\frac{n}{Om \cdot Om} \cdot \left(U - U*\right) + \frac{2}{Om}\right) \cdot \left(n \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot U\right)\right)}\right)\right)} \]

   5. Taylor expanded in l around 0 60.2

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

   6. Simplified 42.1

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

      [Start] 60.2	\[ \sqrt{-2 \cdot \left(\left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(n \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      *-commutative [=>] 60.2	\[ \sqrt{-2 \cdot \color{blue}{\left(\left(n \cdot \left({\ell}^{2} \cdot U\right)\right) \cdot \left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)\right)}} \]
      unpow2 [=>] 60.2	\[ \sqrt{-2 \cdot \left(\left(n \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot U\right)\right) \cdot \left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)\right)} \]
      associate-*l* [=>] 48.3	\[ \sqrt{-2 \cdot \left(\left(n \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot U\right)\right)}\right) \cdot \left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)\right)} \]
      *-commutative [<=] 48.3	\[ \sqrt{-2 \cdot \left(\left(n \cdot \left(\ell \cdot \color{blue}{\left(U \cdot \ell\right)}\right)\right) \cdot \left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)\right)} \]
      associate-*l* [<=] 41.6	\[ \sqrt{-2 \cdot \left(\color{blue}{\left(\left(n \cdot \ell\right) \cdot \left(U \cdot \ell\right)\right)} \cdot \left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)\right)} \]
      unpow2 [=>] 41.6	\[ \sqrt{-2 \cdot \left(\left(\left(n \cdot \ell\right) \cdot \left(U \cdot \ell\right)\right) \cdot \left(\frac{n \cdot \left(U - U*\right)}{\color{blue}{Om \cdot Om}} + 2 \cdot \frac{1}{Om}\right)\right)} \]
      associate-*l/ [<=] 41.7	\[ \sqrt{-2 \cdot \left(\left(\left(n \cdot \ell\right) \cdot \left(U \cdot \ell\right)\right) \cdot \left(\color{blue}{\frac{n}{Om \cdot Om} \cdot \left(U - U*\right)} + 2 \cdot \frac{1}{Om}\right)\right)} \]
      fma-udef [<=] 41.7	\[ \sqrt{-2 \cdot \left(\left(\left(n \cdot \ell\right) \cdot \left(U \cdot \ell\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{n}{Om \cdot Om}, U - U*, 2 \cdot \frac{1}{Om}\right)}\right)} \]
      associate-*r/ [=>] 41.7	\[ \sqrt{-2 \cdot \left(\left(\left(n \cdot \ell\right) \cdot \left(U \cdot \ell\right)\right) \cdot \mathsf{fma}\left(\frac{n}{Om \cdot Om}, U - U*, \color{blue}{\frac{2 \cdot 1}{Om}}\right)\right)} \]
      metadata-eval [=>] 41.7	\[ \sqrt{-2 \cdot \left(\left(\left(n \cdot \ell\right) \cdot \left(U \cdot \ell\right)\right) \cdot \mathsf{fma}\left(\frac{n}{Om \cdot Om}, U - U*, \frac{\color{blue}{2}}{Om}\right)\right)} \]
      associate-*l* [=>] 41.8	\[ \sqrt{-2 \cdot \color{blue}{\left(\left(n \cdot \ell\right) \cdot \left(\left(U \cdot \ell\right) \cdot \mathsf{fma}\left(\frac{n}{Om \cdot Om}, U - U*, \frac{2}{Om}\right)\right)\right)}} \]
      *-commutative [<=] 41.8	\[ \sqrt{-2 \cdot \left(\left(n \cdot \ell\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\frac{n}{Om \cdot Om}, U - U*, \frac{2}{Om}\right) \cdot \left(U \cdot \ell\right)\right)}\right)} \]

   7. Taylor expanded in n around 0 40.5

      \[\leadsto \sqrt{-2 \cdot \left(\left(n \cdot \ell\right) \cdot \color{blue}{\left(2 \cdot \frac{\ell \cdot U}{Om}\right)}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 24.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 0:\\ \;\;\;\;\sqrt{{\left(\sqrt[3]{U \cdot \left(\left(n \cdot -2\right) \cdot \left(\mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right) - t\right)\right)}\right)}^{3}}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 10^{+299}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\left|\frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{Om}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot \ell\right) \cdot \left(-2 \cdot \frac{U \cdot \ell}{Om}\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	24.5
Cost	37900

\[\begin{array}{l} t_1 := \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := t_2 \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + t_1\right)\\ t_4 := \ell \cdot \frac{\ell}{Om}\\ \mathbf{if}\;t_3 \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \frac{\left(n \cdot t_4\right) \cdot \left(U* - U\right)}{Om}\right)\right)\right)}\\ \mathbf{elif}\;t_3 \leq 10^{+299}:\\ \;\;\;\;\sqrt{t_2 \cdot \left(\left(t + t_4 \cdot -2\right) + t_1\right)}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;\left|\frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{Om}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot \ell\right) \cdot \left(-2 \cdot \frac{U \cdot \ell}{Om}\right)\right)}\\ \end{array} \]

Alternative 2
Error	29.0
Cost	14936

\[\begin{array}{l} t_1 := \sqrt{n \cdot \left(U \cdot \left(\frac{n}{\frac{Om \cdot Om}{U* - U}} + \frac{-2}{Om}\right)\right)} \cdot \left(\sqrt{2} \cdot \left(-\ell\right)\right)\\ \mathbf{if}\;\ell \leq -1.8 \cdot 10^{+171}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -3.55 \cdot 10^{+114}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \left(\frac{n}{Om} \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -1.95 \cdot 10^{+89}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 5.7 \cdot 10^{+125}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \frac{\left(n \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U* - U\right)}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 9.5 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{2 \cdot \left(\ell \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \frac{-2}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 5.5 \cdot 10^{+157}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\sqrt{2} \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) + \frac{-2}{Om}\right)\right)}\right)\\ \end{array} \]

Alternative 3
Error	29.8
Cost	14672

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(\ell \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \frac{-2}{Om}\right)\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -1.46 \cdot 10^{+94}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{+122}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \frac{\left(n \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U* - U\right)}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 7 \cdot 10^{+155}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 5.5 \cdot 10^{+157}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\sqrt{2} \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) + \frac{-2}{Om}\right)\right)}\right)\\ \end{array} \]

Alternative 4
Error	32.8
Cost	14284

\[\begin{array}{l} t_1 := \left|\frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{Om}\right|\\ t_2 := -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;Om \leq -1.48 \cdot 10^{-207}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(t_2 - n \cdot \frac{\ell}{\frac{Om \cdot \frac{Om}{\ell}}{U - U*}}\right)\right)\right)}\\ \mathbf{elif}\;Om \leq 1.55 \cdot 10^{-212}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;Om \leq 10^{-146}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \left(\frac{n}{Om} \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \mathbf{elif}\;Om \leq 6 \cdot 10^{-124}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;Om \leq 2.2 \cdot 10^{-46}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(t_2 + \frac{\left(n \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U* - U\right)}{Om}\right)\right)\right)}\\ \mathbf{elif}\;Om \leq 1.45 \cdot 10^{+30}:\\ \;\;\;\;\sqrt{\left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) + \frac{-2}{Om}\right) \cdot \left(2 \cdot \left(U \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + t_2\right)}\\ \end{array} \]

Alternative 5
Error	32.5
Cost	8784

\[\begin{array}{l} \mathbf{if}\;\ell \leq -8.5 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{2 \cdot \left(\ell \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \frac{-2}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-27}:\\ \;\;\;\;\sqrt{\left(-2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(\ell \cdot \left(2 \cdot \frac{\ell}{Om}\right) - \frac{n}{\frac{\frac{Om \cdot Om}{U*}}{\ell \cdot \ell}}\right) - t\right)}\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{+127}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot \frac{\ell}{\frac{Om}{\ell}} - n \cdot \frac{\ell}{\frac{Om \cdot \frac{Om}{\ell}}{U - U*}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot \ell\right) \cdot \left(-2 \cdot \frac{U \cdot \ell}{Om}\right)\right)}\\ \end{array} \]

Alternative 6
Error	31.8
Cost	8652

\[\begin{array}{l} t_1 := \ell \cdot \left(2 \cdot \frac{\ell}{Om}\right)\\ \mathbf{if}\;\ell \leq -1.1 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{2 \cdot \left(\ell \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \frac{-2}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -1.05 \cdot 10^{-148}:\\ \;\;\;\;\sqrt{\left(-2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t_1 - \frac{n}{\frac{\frac{Om \cdot Om}{U*}}{\ell \cdot \ell}}\right) - t\right)}\\ \mathbf{elif}\;\ell \leq 1.85 \cdot 10^{-56}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \frac{\ell \cdot \left(U* - U\right)}{Om \cdot \frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.65 \cdot 10^{+122}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(t_1 - t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot \ell\right) \cdot \left(-2 \cdot \frac{U \cdot \ell}{Om}\right)\right)}\\ \end{array} \]

Alternative 7
Error	30.3
Cost	8520

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.25 \cdot 10^{+94}:\\ \;\;\;\;\sqrt{2 \cdot \left(\ell \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \frac{-2}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.8 \cdot 10^{+125}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \frac{\left(n \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U* - U\right)}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot \ell\right) \cdot \left(-2 \cdot \frac{U \cdot \ell}{Om}\right)\right)}\\ \end{array} \]

Alternative 8
Error	33.1
Cost	8392

\[\begin{array}{l} t_1 := \ell \cdot \left(2 \cdot \frac{\ell}{Om}\right)\\ \mathbf{if}\;\ell \leq -2.3 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{2 \cdot \left(\ell \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \frac{-2}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -2.1 \cdot 10^{-27}:\\ \;\;\;\;\sqrt{\left(-2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t_1 - \frac{n}{\frac{\frac{Om \cdot Om}{U*}}{\ell \cdot \ell}}\right) - t\right)}\\ \mathbf{elif}\;\ell \leq 7.8 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 5.5 \cdot 10^{+121}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(t_1 - t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot \ell\right) \cdot \left(-2 \cdot \frac{U \cdot \ell}{Om}\right)\right)}\\ \end{array} \]

Alternative 9
Error	33.2
Cost	8264

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.9 \cdot 10^{+94}:\\ \;\;\;\;\sqrt{2 \cdot \left(\ell \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \frac{-2}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -2.15 \cdot 10^{-27}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n}{\frac{Om \cdot Om}{U* - U}} + \frac{-2}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.4 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.42 \cdot 10^{+122}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\ell \cdot \left(2 \cdot \frac{\ell}{Om}\right) - t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot \ell\right) \cdot \left(-2 \cdot \frac{U \cdot \ell}{Om}\right)\right)}\\ \end{array} \]

Alternative 10
Error	36.6
Cost	8025

\[\begin{array}{l} t_1 := \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ t_2 := \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{if}\;\ell \leq -3.8 \cdot 10^{+148}:\\ \;\;\;\;\sqrt{2 \cdot \left(\ell \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \frac{-2}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 8.2 \cdot 10^{-308}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 1.26 \cdot 10^{-194}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.65 \cdot 10^{+33}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 2.4 \cdot 10^{+66} \lor \neg \left(\ell \leq 2.4 \cdot 10^{+121}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot \ell\right) \cdot \left(-2 \cdot \frac{U \cdot \ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	36.4
Cost	8024

\[\begin{array}{l} t_1 := \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ t_2 := \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ t_3 := \sqrt{2 \cdot \left(\ell \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \frac{-2}{Om}\right)\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -1.9 \cdot 10^{+153}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{-308}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 1.22 \cdot 10^{-194}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.75 \cdot 10^{+33}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 3.4 \cdot 10^{+66}:\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \left(\frac{n}{Om} \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.1 \cdot 10^{+120}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 12
Error	39.7
Cost	7893

\[\begin{array}{l} t_1 := \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ t_2 := \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{if}\;\ell \leq 1.8 \cdot 10^{-308}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 1.12 \cdot 10^{-194}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.24 \cdot 10^{+33}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{+64} \lor \neg \left(\ell \leq 1.65 \cdot 10^{+122}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \left(\frac{n}{Om} \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	33.4
Cost	7888

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -1.15 \cdot 10^{+94}:\\ \;\;\;\;\sqrt{2 \cdot \left(\ell \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \frac{-2}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -4.4 \cdot 10^{-46}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.15 \cdot 10^{-307}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{+125}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot \ell\right) \cdot \left(-2 \cdot \frac{U \cdot \ell}{Om}\right)\right)}\\ \end{array} \]

Alternative 14
Error	33.4
Cost	7888

\[\begin{array}{l} \mathbf{if}\;\ell \leq -6.4 \cdot 10^{+91}:\\ \;\;\;\;\sqrt{2 \cdot \left(\ell \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \frac{-2}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -7.4 \cdot 10^{-46}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{+127}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\ell \cdot \left(2 \cdot \frac{\ell}{Om}\right) - t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot \ell\right) \cdot \left(-2 \cdot \frac{U \cdot \ell}{Om}\right)\right)}\\ \end{array} \]

Alternative 15
Error	40.2
Cost	7368

\[\begin{array}{l} \mathbf{if}\;Om \leq -5.4 \cdot 10^{-221}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{elif}\;Om \leq 2.2 \cdot 10^{-307}:\\ \;\;\;\;\left(n \cdot \ell\right) \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)}}{Om}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]

Alternative 16
Error	40.2
Cost	7368

\[\begin{array}{l} \mathbf{if}\;Om \leq -1.65 \cdot 10^{-226}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{elif}\;Om \leq -2 \cdot 10^{-308}:\\ \;\;\;\;\frac{n \cdot \ell}{\frac{Om}{\sqrt{2 \cdot \left(U \cdot U*\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]

Alternative 17
Error	40.2
Cost	7113

\[\begin{array}{l} \mathbf{if}\;\ell \leq 8.6 \cdot 10^{-308} \lor \neg \left(\ell \leq 1.52 \cdot 10^{-194}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \end{array} \]

Alternative 18
Error	40.2
Cost	6848

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

Reproduce

herbie shell --seed 2023066 
(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*))))))