Toniolo and Linder, Equation (13)

Average Error: 35.0 → 25.5

Time: 43.2s

Precision: binary64

Cost: 44300

Math

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

↓

\[\begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_1 \leq 2 \cdot 10^{-321}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{U \cdot \mathsf{fma}\left(\ell, \frac{\ell \cdot -2}{Om}, t\right)} \cdot \sqrt{n}\right)\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{+301}:\\ \;\;\;\;\sqrt{t_1}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\left|\frac{\left(\ell \cdot \sqrt{U \cdot U*}\right) \cdot \left(n \cdot \sqrt{2}\right)}{Om}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{n \cdot \ell}{\frac{Om}{U \cdot \ell}}}\\ \end{array} \]

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
         (*
          (* (* 2.0 n) U)
          (+
           (+ t (* (/ (* l l) Om) -2.0))
           (* (* n (pow (/ l Om) 2.0)) (- U* U))))))
   (if (<= t_1 2e-321)
     (* (sqrt 2.0) (* (sqrt (* U (fma l (/ (* l -2.0) Om) t))) (sqrt n)))
     (if (<= t_1 4e+301)
       (sqrt t_1)
       (if (<= t_1 INFINITY)
         (fabs (/ (* (* l (sqrt (* U U*))) (* n (sqrt 2.0))) Om))
         (sqrt (* -4.0 (/ (* n l) (/ Om (* 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 = ((2.0 * n) * U) * ((t + (((l * l) / Om) * -2.0)) + ((n * pow((l / Om), 2.0)) * (U_42_ - U)));
	double tmp;
	if (t_1 <= 2e-321) {
		tmp = sqrt(2.0) * (sqrt((U * fma(l, ((l * -2.0) / Om), t))) * sqrt(n));
	} else if (t_1 <= 4e+301) {
		tmp = sqrt(t_1);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fabs((((l * sqrt((U * U_42_))) * (n * sqrt(2.0))) / Om));
	} else {
		tmp = sqrt((-4.0 * ((n * l) / (Om / (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(Float64(Float64(2.0 * n) * U) * Float64(Float64(t + Float64(Float64(Float64(l * l) / Om) * -2.0)) + Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U_42_ - U))))
	tmp = 0.0
	if (t_1 <= 2e-321)
		tmp = Float64(sqrt(2.0) * Float64(sqrt(Float64(U * fma(l, Float64(Float64(l * -2.0) / Om), t))) * sqrt(n)));
	elseif (t_1 <= 4e+301)
		tmp = sqrt(t_1);
	elseif (t_1 <= Inf)
		tmp = abs(Float64(Float64(Float64(l * sqrt(Float64(U * U_42_))) * Float64(n * sqrt(2.0))) / Om));
	else
		tmp = sqrt(Float64(-4.0 * Float64(Float64(n * l) / Float64(Om / 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[(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 * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-321], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(U * N[(l * N[(N[(l * -2.0), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+301], N[Sqrt[t$95$1], $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[Abs[N[(N[(N[(l * N[Sqrt[N[(U * U$42$), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(n * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-4.0 * N[(N[(n * l), $MachinePrecision] / N[(Om / N[(U * l), $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*)))) < 2.00097e-321

   1. Initial program 57.1

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

   2. Simplified 41.7

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

      [Start] 57.1	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 41.9	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 41.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      sub-neg [=>] 41.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(-\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [<=] 41.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      cancel-sign-sub [<=] 41.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      cancel-sign-sub [=>] 41.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-/l* [=>] 40.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 41.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in U* around 0 44.4

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

   4. Simplified 41.8

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

      [Start] 44.4	\[ \sqrt{2} \cdot \sqrt{n \cdot \left(\left(t - \left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{{Om}^{2}}\right)\right) \cdot U\right)} \]
      *-commutative [=>] 44.4	\[ \sqrt{2} \cdot \sqrt{n \cdot \color{blue}{\left(U \cdot \left(t - \left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{{Om}^{2}}\right)\right)\right)}} \]
      fma-def [=>] 44.4	\[ \sqrt{2} \cdot \sqrt{n \cdot \left(U \cdot \left(t - \color{blue}{\mathsf{fma}\left(2, \frac{{\ell}^{2}}{Om}, \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{{Om}^{2}}\right)}\right)\right)} \]
      unpow2 [=>] 44.4	\[ \sqrt{2} \cdot \sqrt{n \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{{Om}^{2}}\right)\right)\right)} \]
      *-commutative [=>] 44.4	\[ \sqrt{2} \cdot \sqrt{n \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \frac{\color{blue}{\left({\ell}^{2} \cdot U\right) \cdot n}}{{Om}^{2}}\right)\right)\right)} \]
      unpow2 [=>] 44.4	\[ \sqrt{2} \cdot \sqrt{n \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \frac{\left({\ell}^{2} \cdot U\right) \cdot n}{\color{blue}{Om \cdot Om}}\right)\right)\right)} \]
      times-frac [=>] 41.8	\[ \sqrt{2} \cdot \sqrt{n \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \color{blue}{\frac{{\ell}^{2} \cdot U}{Om} \cdot \frac{n}{Om}}\right)\right)\right)} \]
      unpow2 [=>] 41.8	\[ \sqrt{2} \cdot \sqrt{n \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot U}{Om} \cdot \frac{n}{Om}\right)\right)\right)} \]
      associate-*l* [=>] 41.8	\[ \sqrt{2} \cdot \sqrt{n \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \frac{\color{blue}{\ell \cdot \left(\ell \cdot U\right)}}{Om} \cdot \frac{n}{Om}\right)\right)\right)} \]

   5. Taylor expanded in Om around inf 42.4

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

   6. Simplified 40.8

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

      [Start] 42.4	\[ \sqrt{2} \cdot \sqrt{-2 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om} + n \cdot \left(t \cdot U\right)} \]
      +-commutative [=>] 42.4	\[ \sqrt{2} \cdot \sqrt{\color{blue}{n \cdot \left(t \cdot U\right) + -2 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}}} \]
      fma-def [=>] 42.4	\[ \sqrt{2} \cdot \sqrt{\color{blue}{\mathsf{fma}\left(n, t \cdot U, -2 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}} \]
      *-commutative [=>] 42.4	\[ \sqrt{2} \cdot \sqrt{\mathsf{fma}\left(n, \color{blue}{U \cdot t}, -2 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)} \]
      *-commutative [=>] 42.4	\[ \sqrt{2} \cdot \sqrt{\mathsf{fma}\left(n, U \cdot t, -2 \cdot \frac{\color{blue}{\left({\ell}^{2} \cdot U\right) \cdot n}}{Om}\right)} \]
      unpow2 [=>] 42.4	\[ \sqrt{2} \cdot \sqrt{\mathsf{fma}\left(n, U \cdot t, -2 \cdot \frac{\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot U\right) \cdot n}{Om}\right)} \]
      *-commutative [<=] 42.4	\[ \sqrt{2} \cdot \sqrt{\mathsf{fma}\left(n, U \cdot t, -2 \cdot \frac{\color{blue}{\left(U \cdot \left(\ell \cdot \ell\right)\right)} \cdot n}{Om}\right)} \]
      associate-*l/ [<=] 42.0	\[ \sqrt{2} \cdot \sqrt{\mathsf{fma}\left(n, U \cdot t, -2 \cdot \color{blue}{\left(\frac{U \cdot \left(\ell \cdot \ell\right)}{Om} \cdot n\right)}\right)} \]
      associate-*r* [=>] 41.3	\[ \sqrt{2} \cdot \sqrt{\mathsf{fma}\left(n, U \cdot t, -2 \cdot \left(\frac{\color{blue}{\left(U \cdot \ell\right) \cdot \ell}}{Om} \cdot n\right)\right)} \]
      *-commutative [<=] 41.3	\[ \sqrt{2} \cdot \sqrt{\mathsf{fma}\left(n, U \cdot t, -2 \cdot \left(\frac{\color{blue}{\left(\ell \cdot U\right)} \cdot \ell}{Om} \cdot n\right)\right)} \]
      associate-*r/ [<=] 41.0	\[ \sqrt{2} \cdot \sqrt{\mathsf{fma}\left(n, U \cdot t, -2 \cdot \left(\color{blue}{\left(\left(\ell \cdot U\right) \cdot \frac{\ell}{Om}\right)} \cdot n\right)\right)} \]
      associate-*l* [=>] 40.8	\[ \sqrt{2} \cdot \sqrt{\mathsf{fma}\left(n, U \cdot t, -2 \cdot \color{blue}{\left(\left(\ell \cdot U\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)}\right)} \]
      *-commutative [<=] 40.8	\[ \sqrt{2} \cdot \sqrt{\mathsf{fma}\left(n, U \cdot t, -2 \cdot \left(\left(\ell \cdot U\right) \cdot \color{blue}{\left(n \cdot \frac{\ell}{Om}\right)}\right)\right)} \]

   7. Taylor expanded in n around 0 42.0

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

   8. Simplified 41.0

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

      [Start] 42.0	\[ \sqrt{2} \cdot \sqrt{n \cdot \left(t \cdot U + -2 \cdot \frac{{\ell}^{2} \cdot U}{Om}\right)} \]
      *-commutative [=>] 42.0	\[ \sqrt{2} \cdot \sqrt{n \cdot \left(\color{blue}{U \cdot t} + -2 \cdot \frac{{\ell}^{2} \cdot U}{Om}\right)} \]
      fma-def [=>] 42.0	\[ \sqrt{2} \cdot \sqrt{n \cdot \color{blue}{\mathsf{fma}\left(U, t, -2 \cdot \frac{{\ell}^{2} \cdot U}{Om}\right)}} \]
      unpow2 [=>] 42.0	\[ \sqrt{2} \cdot \sqrt{n \cdot \mathsf{fma}\left(U, t, -2 \cdot \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot U}{Om}\right)} \]
      associate-*r* [<=] 41.3	\[ \sqrt{2} \cdot \sqrt{n \cdot \mathsf{fma}\left(U, t, -2 \cdot \frac{\color{blue}{\ell \cdot \left(\ell \cdot U\right)}}{Om}\right)} \]
      *-commutative [<=] 41.3	\[ \sqrt{2} \cdot \sqrt{n \cdot \mathsf{fma}\left(U, t, -2 \cdot \frac{\ell \cdot \color{blue}{\left(U \cdot \ell\right)}}{Om}\right)} \]
      associate-*l/ [<=] 41.0	\[ \sqrt{2} \cdot \sqrt{n \cdot \mathsf{fma}\left(U, t, -2 \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \left(U \cdot \ell\right)\right)}\right)} \]
      *-commutative [<=] 41.0	\[ \sqrt{2} \cdot \sqrt{n \cdot \mathsf{fma}\left(U, t, -2 \cdot \color{blue}{\left(\left(U \cdot \ell\right) \cdot \frac{\ell}{Om}\right)}\right)} \]

   9. Applied egg-rr 42.0

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

   10. Simplified 42.0

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

       [Start] 42.0	\[ \sqrt{2} \cdot \left(\sqrt{\mathsf{fma}\left(U, t, U \cdot \left(\left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)} \cdot \sqrt{n}\right) \]
       fma-udef [=>] 42.0	\[ \sqrt{2} \cdot \left(\sqrt{\color{blue}{U \cdot t + U \cdot \left(\left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)}} \cdot \sqrt{n}\right) \]
       distribute-lft-out [=>] 42.0	\[ \sqrt{2} \cdot \left(\sqrt{\color{blue}{U \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)}} \cdot \sqrt{n}\right) \]
       associate-*r/ [=>] 42.9	\[ \sqrt{2} \cdot \left(\sqrt{U \cdot \left(t + \color{blue}{\frac{\ell \cdot \ell}{Om}} \cdot -2\right)} \cdot \sqrt{n}\right) \]
       unpow2 [<=] 42.9	\[ \sqrt{2} \cdot \left(\sqrt{U \cdot \left(t + \frac{\color{blue}{{\ell}^{2}}}{Om} \cdot -2\right)} \cdot \sqrt{n}\right) \]
       *-commutative [<=] 42.9	\[ \sqrt{2} \cdot \left(\sqrt{U \cdot \left(t + \color{blue}{-2 \cdot \frac{{\ell}^{2}}{Om}}\right)} \cdot \sqrt{n}\right) \]
       +-commutative [=>] 42.9	\[ \sqrt{2} \cdot \left(\sqrt{U \cdot \color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om} + t\right)}} \cdot \sqrt{n}\right) \]
       *-commutative [=>] 42.9	\[ \sqrt{2} \cdot \left(\sqrt{U \cdot \left(\color{blue}{\frac{{\ell}^{2}}{Om} \cdot -2} + t\right)} \cdot \sqrt{n}\right) \]
       unpow2 [=>] 42.9	\[ \sqrt{2} \cdot \left(\sqrt{U \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{Om} \cdot -2 + t\right)} \cdot \sqrt{n}\right) \]
       associate-*r/ [<=] 42.0	\[ \sqrt{2} \cdot \left(\sqrt{U \cdot \left(\color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)} \cdot -2 + t\right)} \cdot \sqrt{n}\right) \]
       associate-*l* [=>] 42.0	\[ \sqrt{2} \cdot \left(\sqrt{U \cdot \left(\color{blue}{\ell \cdot \left(\frac{\ell}{Om} \cdot -2\right)} + t\right)} \cdot \sqrt{n}\right) \]
       fma-def [=>] 42.0	\[ \sqrt{2} \cdot \left(\sqrt{U \cdot \color{blue}{\mathsf{fma}\left(\ell, \frac{\ell}{Om} \cdot -2, t\right)}} \cdot \sqrt{n}\right) \]
       associate-*l/ [=>] 42.0	\[ \sqrt{2} \cdot \left(\sqrt{U \cdot \mathsf{fma}\left(\ell, \color{blue}{\frac{\ell \cdot -2}{Om}}, t\right)} \cdot \sqrt{n}\right) \]

   if 2.00097e-321 < (*.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*)))) < 4.00000000000000021e301

   1. Initial program 1.7

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

   if 4.00000000000000021e301 < (*.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 63.0

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

   2. Taylor expanded in U* around inf 62.7

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

   3. Simplified 62.5

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

      [Start] 62.7	\[ \sqrt{2 \cdot \frac{{n}^{2} \cdot \left({\ell}^{2} \cdot \left(U \cdot U*\right)\right)}{{Om}^{2}}} \]
      associate-*r/ [=>] 62.7	\[ \sqrt{\color{blue}{\frac{2 \cdot \left({n}^{2} \cdot \left({\ell}^{2} \cdot \left(U \cdot U*\right)\right)\right)}{{Om}^{2}}}} \]
      associate-*r* [=>] 62.7	\[ \sqrt{\frac{\color{blue}{\left(2 \cdot {n}^{2}\right) \cdot \left({\ell}^{2} \cdot \left(U \cdot U*\right)\right)}}{{Om}^{2}}} \]
      unpow2 [=>] 62.7	\[ \sqrt{\frac{\left(2 \cdot \color{blue}{\left(n \cdot n\right)}\right) \cdot \left({\ell}^{2} \cdot \left(U \cdot U*\right)\right)}{{Om}^{2}}} \]
      unpow2 [=>] 62.7	\[ \sqrt{\frac{\left(2 \cdot \left(n \cdot n\right)\right) \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(U \cdot U*\right)\right)}{{Om}^{2}}} \]
      associate-*l* [=>] 62.5	\[ \sqrt{\frac{\left(2 \cdot \left(n \cdot n\right)\right) \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \left(U \cdot U*\right)\right)\right)}}{{Om}^{2}}} \]
      unpow2 [=>] 62.5	\[ \sqrt{\frac{\left(2 \cdot \left(n \cdot n\right)\right) \cdot \left(\ell \cdot \left(\ell \cdot \left(U \cdot U*\right)\right)\right)}{\color{blue}{Om \cdot Om}}} \]

   4. Applied egg-rr 48.3

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

   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 59.6

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}} \]
      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	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      cancel-sign-sub-inv [=>] 64.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)} \]
      +-commutative [=>] 64.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(\left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\right)} \]
      distribute-lft-neg-in [<=] 64.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(-\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)} \]
      associate-*l* [=>] 64.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(-\color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)} \]
      distribute-rgt-neg-in [=>] 64.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{n \cdot \left(-{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)} \]
      fma-def [=>] 64.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\mathsf{fma}\left(n, -{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right), t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)}\right)} \]

   3. Taylor expanded in l around inf 61.0

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

   4. Simplified 47.6

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

      [Start] 61.0	\[ \sqrt{2 \cdot \left(n \cdot \left({\ell}^{2} \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)\right)\right)} \]
      associate-*r* [=>] 61.0	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left({\ell}^{2} \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)\right)}} \]
      associate-*r* [=>] 61.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(\left({\ell}^{2} \cdot U\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}} \]
      associate-*r* [=>] 60.4	\[ \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot \left({\ell}^{2} \cdot U\right)\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)}} \]
      unpow2 [=>] 60.4	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot U\right)\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)} \]
      associate-*l* [=>] 47.6	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot U\right)\right)}\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)} \]
      cancel-sign-sub-inv [=>] 47.6	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right) \cdot \color{blue}{\left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} + \left(-2\right) \cdot \frac{1}{Om}\right)}} \]
      associate-/l* [=>] 49.0	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right) \cdot \left(\color{blue}{\frac{n}{\frac{{Om}^{2}}{U* - U}}} + \left(-2\right) \cdot \frac{1}{Om}\right)} \]
      associate-/r/ [=>] 47.6	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right) \cdot \left(\color{blue}{\frac{n}{{Om}^{2}} \cdot \left(U* - U\right)} + \left(-2\right) \cdot \frac{1}{Om}\right)} \]
      unpow2 [=>] 47.6	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right) \cdot \left(\frac{n}{\color{blue}{Om \cdot Om}} \cdot \left(U* - U\right) + \left(-2\right) \cdot \frac{1}{Om}\right)} \]
      metadata-eval [=>] 47.6	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right) \cdot \left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) + \color{blue}{-2} \cdot \frac{1}{Om}\right)} \]
      associate-*r/ [=>] 47.6	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right) \cdot \left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) + \color{blue}{\frac{-2 \cdot 1}{Om}}\right)} \]
      metadata-eval [=>] 47.6	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right) \cdot \left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) + \frac{\color{blue}{-2}}{Om}\right)} \]

   5. Taylor expanded in n around 0 60.0

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

   6. Simplified 48.9

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

      [Start] 60.0	\[ \sqrt{-4 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}} \]
      associate-/l* [=>] 59.2	\[ \sqrt{-4 \cdot \color{blue}{\frac{n}{\frac{Om}{{\ell}^{2} \cdot U}}}} \]
      unpow2 [=>] 59.2	\[ \sqrt{-4 \cdot \frac{n}{\frac{Om}{\color{blue}{\left(\ell \cdot \ell\right)} \cdot U}}} \]
      associate-*r* [<=] 49.0	\[ \sqrt{-4 \cdot \frac{n}{\frac{Om}{\color{blue}{\ell \cdot \left(\ell \cdot U\right)}}}} \]
      associate-/r/ [=>] 48.9	\[ \sqrt{-4 \cdot \color{blue}{\left(\frac{n}{Om} \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right)}} \]

   7. Applied egg-rr 39.2

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

4. Final simplification 25.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 2 \cdot 10^{-321}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{U \cdot \mathsf{fma}\left(\ell, \frac{\ell \cdot -2}{Om}, t\right)} \cdot \sqrt{n}\right)\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 4 \cdot 10^{+301}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\left|\frac{\left(\ell \cdot \sqrt{U \cdot U*}\right) \cdot \left(n \cdot \sqrt{2}\right)}{Om}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{n \cdot \ell}{\frac{Om}{U \cdot \ell}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	25.3
Cost	44300

\[\begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \ell \cdot \left(\frac{\ell}{Om} \cdot -2\right)\right)\right)\right)}\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{+301}:\\ \;\;\;\;\sqrt{t_1}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\left|\frac{\left(\ell \cdot \sqrt{U \cdot U*}\right) \cdot \left(n \cdot \sqrt{2}\right)}{Om}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{n \cdot \ell}{\frac{Om}{U \cdot \ell}}}\\ \end{array} \]

Alternative 2
Error	25.5
Cost	44300

\[\begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_1 \leq 2 \cdot 10^{-321}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{n} \cdot \sqrt{U \cdot \left(t + \ell \cdot \left(\ell \cdot \left(\frac{-2}{Om} - \frac{n}{Om} \cdot \frac{U}{Om}\right)\right)\right)}\right)\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{+301}:\\ \;\;\;\;\sqrt{t_1}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\left|\frac{\left(\ell \cdot \sqrt{U \cdot U*}\right) \cdot \left(n \cdot \sqrt{2}\right)}{Om}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{n \cdot \ell}{\frac{Om}{U \cdot \ell}}}\\ \end{array} \]

Alternative 3
Error	26.9
Cost	30728

\[\begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \ell \cdot \left(\frac{\ell}{Om} \cdot -2\right)\right)\right)\right)}\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{+301}:\\ \;\;\;\;\sqrt{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|\ell \cdot \left(\frac{-4}{Om} \cdot \left(U \cdot \left(n \cdot \ell\right)\right)\right)\right|}\\ \end{array} \]

Alternative 4
Error	30.5
Cost	14672

\[\begin{array}{l} t_1 := \ell \cdot \sqrt{2}\\ \mathbf{if}\;\ell \leq -1.1 \cdot 10^{+154}:\\ \;\;\;\;t_1 \cdot \left(-\sqrt{n \cdot \left(U \cdot \left(n \cdot \frac{\frac{U*}{Om}}{Om} + \frac{-2}{Om}\right)\right)}\right)\\ \mathbf{elif}\;\ell \leq -3 \cdot 10^{-82}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{Om} + \frac{n \cdot \left(U - U*\right)}{Om \cdot Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 9.5 \cdot 10^{-174}:\\ \;\;\;\;\sqrt{\left|\left(n \cdot U\right) \cdot \left(2 \cdot t\right)\right|}\\ \mathbf{elif}\;\ell \leq 2.45 \cdot 10^{+120}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(\left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - n \cdot \left(\frac{\ell \cdot \ell}{Om} \cdot \frac{U* - U}{Om}\right)\right) - t\right)\right) \cdot \left(n \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\left(U* - U\right) \cdot \frac{n}{Om \cdot Om} + \frac{-2}{Om}\right)}\\ \end{array} \]

Alternative 5
Error	30.2
Cost	14672

\[\begin{array}{l} t_1 := \frac{U* - U}{Om}\\ \mathbf{if}\;\ell \leq -4.1 \cdot 10^{+133}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(\frac{n}{Om} \cdot t_1 + \frac{-2}{Om}\right)\right)} \cdot \left(\ell \cdot \left(-\sqrt{2}\right)\right)\\ \mathbf{elif}\;\ell \leq -2.35 \cdot 10^{-87}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{Om} + \frac{n \cdot \left(U - U*\right)}{Om \cdot Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 10^{-176}:\\ \;\;\;\;\sqrt{\left|\left(n \cdot U\right) \cdot \left(2 \cdot t\right)\right|}\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{+116}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(\left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - n \cdot \left(\frac{\ell \cdot \ell}{Om} \cdot t_1\right)\right) - t\right)\right) \cdot \left(n \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\left(U* - U\right) \cdot \frac{n}{Om \cdot Om} + \frac{-2}{Om}\right)}\\ \end{array} \]

Alternative 6
Error	31.1
Cost	14212

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \left(-\sqrt{n \cdot \left(U \cdot \left(n \cdot \frac{\frac{U*}{Om}}{Om} + \frac{-2}{Om}\right)\right)}\right)\\ \mathbf{elif}\;\ell \leq -6.2 \cdot 10^{-91}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{Om} + \frac{n \cdot \left(U - U*\right)}{Om \cdot Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{-176}:\\ \;\;\;\;\sqrt{\left|\left(n \cdot U\right) \cdot \left(2 \cdot t\right)\right|}\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{+150}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(\left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - n \cdot \left(\frac{\ell \cdot \ell}{Om} \cdot \frac{U* - U}{Om}\right)\right) - t\right)\right) \cdot \left(n \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\ell \cdot \left(\frac{-4}{Om} \cdot \left(U \cdot \left(n \cdot \ell\right)\right)\right)}\\ \end{array} \]

Alternative 7
Error	32.1
Cost	14152

\[\begin{array}{l} t_1 := 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := \sqrt{\ell \cdot \left(\frac{-4}{Om} \cdot \left(U \cdot \left(n \cdot \ell\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -1.75 \cdot 10^{+146}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -8 \cdot 10^{+122}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \left(\frac{n}{Om} \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -9.5 \cdot 10^{+29}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \ell \cdot \frac{2 \cdot \ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -3.3 \cdot 10^{-84}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(\left(t_1 - \frac{n}{Om \cdot Om} \cdot \left(\ell \cdot \left(\ell \cdot \left(U* - U\right)\right)\right)\right) - t\right)\right) \cdot \left(n \cdot -2\right)}\\ \mathbf{elif}\;\ell \leq 6.8 \cdot 10^{-178}:\\ \;\;\;\;\sqrt{\left|\left(n \cdot U\right) \cdot \left(2 \cdot t\right)\right|}\\ \mathbf{elif}\;\ell \leq 3 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(\left(t_1 - n \cdot \left(\frac{\ell \cdot \ell}{Om} \cdot \frac{U* - U}{Om}\right)\right) - t\right)\right) \cdot \left(n \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Error	31.9
Cost	13644

\[\begin{array}{l} t_1 := \sqrt{\ell \cdot \left(\frac{-4}{Om} \cdot \left(U \cdot \left(n \cdot \ell\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -2.8 \cdot 10^{+151}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -5.1 \cdot 10^{-82}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{Om} + \frac{n \cdot \left(U - U*\right)}{Om \cdot Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.2 \cdot 10^{-175}:\\ \;\;\;\;\sqrt{\left|\left(n \cdot U\right) \cdot \left(2 \cdot t\right)\right|}\\ \mathbf{elif}\;\ell \leq 1.32 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(\left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - n \cdot \left(\frac{\ell \cdot \ell}{Om} \cdot \frac{U* - U}{Om}\right)\right) - t\right)\right) \cdot \left(n \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	31.9
Cost	8784

\[\begin{array}{l} t_1 := \sqrt{\ell \cdot \left(\frac{-4}{Om} \cdot \left(U \cdot \left(n \cdot \ell\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -3.2 \cdot 10^{+146}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -2.9 \cdot 10^{-80}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{Om} + \frac{n \cdot \left(U - U*\right)}{Om \cdot Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.8 \cdot 10^{-176}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.8 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(\left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - n \cdot \left(\frac{\ell \cdot \ell}{Om} \cdot \frac{U* - U}{Om}\right)\right) - t\right)\right) \cdot \left(n \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	32.8
Cost	8264

\[\begin{array}{l} t_1 := \sqrt{\ell \cdot \left(\frac{-4}{Om} \cdot \left(U \cdot \left(n \cdot \ell\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -4.6 \cdot 10^{+151}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -8.8 \cdot 10^{-88}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{Om} + \frac{n \cdot \left(U - U*\right)}{Om \cdot Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.15 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	33.2
Cost	7625

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.2 \cdot 10^{+152} \lor \neg \left(\ell \leq 9.2 \cdot 10^{+154}\right):\\ \;\;\;\;\sqrt{\ell \cdot \left(\frac{-4}{Om} \cdot \left(U \cdot \left(n \cdot \ell\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \ell \cdot \left(\frac{\ell}{Om} \cdot -2\right)\right)\right)\right)}\\ \end{array} \]

Alternative 12
Error	33.7
Cost	7624

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.45 \cdot 10^{-82}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \ell \cdot \frac{2 \cdot \ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.55 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\ell \cdot \left(\frac{-4}{Om} \cdot \left(U \cdot \left(n \cdot \ell\right)\right)\right)}\\ \end{array} \]

Alternative 13
Error	34.8
Cost	7492

\[\begin{array}{l} \mathbf{if}\;\ell \leq -5.9 \cdot 10^{-83}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \ell \cdot \frac{2 \cdot \ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+81}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\ell \cdot \left(\frac{-4}{Om} \cdot \left(U \cdot \left(n \cdot \ell\right)\right)\right)}\\ \end{array} \]

Alternative 14
Error	37.1
Cost	7369

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.05 \cdot 10^{+105} \lor \neg \left(\ell \leq 6.2 \cdot 10^{+148}\right):\\ \;\;\;\;\sqrt{-4 \cdot \frac{\ell}{\frac{\frac{Om}{n}}{U \cdot \ell}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 15
Error	36.8
Cost	7369

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.9 \cdot 10^{+105} \lor \neg \left(\ell \leq 1.46 \cdot 10^{+147}\right):\\ \;\;\;\;\sqrt{-4 \cdot \frac{n}{\frac{\frac{\frac{Om}{\ell}}{U}}{\ell}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 16
Error	36.1
Cost	7369

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.3 \cdot 10^{+105} \lor \neg \left(\ell \leq 1.32 \cdot 10^{+149}\right):\\ \;\;\;\;\sqrt{-4 \cdot \frac{n \cdot \ell}{\frac{Om}{U \cdot \ell}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 17
Error	35.4
Cost	7369

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.6 \cdot 10^{+105} \lor \neg \left(\ell \leq 2 \cdot 10^{+78}\right):\\ \;\;\;\;\sqrt{\ell \cdot \left(\frac{-4}{Om} \cdot \left(U \cdot \left(n \cdot \ell\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 18
Error	39.3
Cost	7236

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.9 \cdot 10^{+105}:\\ \;\;\;\;\sqrt{-4 \cdot \left(\frac{n}{Om} \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 19
Error	40.1
Cost	6848

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

Reproduce

herbie shell --seed 2023012 
(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*))))))