Toniolo and Linder, Equation (13)

Percentage Accurate: 50.2% → 59.8%

Time: 39.3s

Precision: binary64

Cost: 47944

Math

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

↓

\[\begin{array}{l} t_1 := \mathsf{fma}\left(\ell, -2, \left(n \cdot \left(U* - U\right)\right) \cdot \frac{\ell}{Om}\right)\\ \mathbf{if}\;\ell \leq -1.08 \cdot 10^{+149}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \left(-\sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om \cdot Om} - \frac{2}{Om}\right)\right)}\right)\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+146}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{2}, \sqrt{\frac{t_1}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}}, \left(0.5 \cdot \left(\sqrt{2} \cdot t\right)\right) \cdot \sqrt{\frac{n}{\frac{\ell \cdot t_1}{U \cdot Om}}}\right)\\ \end{array} \]

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 (fma l -2.0 (* (* n (- U* U)) (/ l Om)))))
   (if (<= l -1.08e+149)
     (*
      (* l (sqrt 2.0))
      (- (sqrt (* n (* U (- (/ (* n U*) (* Om Om)) (/ 2.0 Om)))))))
     (if (<= l 2.6e+146)
       (sqrt
        (*
         (* 2.0 n)
         (* U (+ t (/ (* l (+ (* l -2.0) (/ (* n (* l U*)) Om))) Om)))))
       (fma
        (sqrt 2.0)
        (sqrt (/ t_1 (/ Om (* n (* l U)))))
        (* (* 0.5 (* (sqrt 2.0) t)) (sqrt (/ n (/ (* l t_1) (* U 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 = fma(l, -2.0, ((n * (U_42_ - U)) * (l / Om)));
	double tmp;
	if (l <= -1.08e+149) {
		tmp = (l * sqrt(2.0)) * -sqrt((n * (U * (((n * U_42_) / (Om * Om)) - (2.0 / Om)))));
	} else if (l <= 2.6e+146) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	} else {
		tmp = fma(sqrt(2.0), sqrt((t_1 / (Om / (n * (l * U))))), ((0.5 * (sqrt(2.0) * t)) * sqrt((n / ((l * t_1) / (U * 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 = fma(l, -2.0, Float64(Float64(n * Float64(U_42_ - U)) * Float64(l / Om)))
	tmp = 0.0
	if (l <= -1.08e+149)
		tmp = Float64(Float64(l * sqrt(2.0)) * Float64(-sqrt(Float64(n * Float64(U * Float64(Float64(Float64(n * U_42_) / Float64(Om * Om)) - Float64(2.0 / Om)))))));
	elseif (l <= 2.6e+146)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l * Float64(Float64(l * -2.0) + Float64(Float64(n * Float64(l * U_42_)) / Om))) / Om)))));
	else
		tmp = fma(sqrt(2.0), sqrt(Float64(t_1 / Float64(Om / Float64(n * Float64(l * U))))), Float64(Float64(0.5 * Float64(sqrt(2.0) * t)) * sqrt(Float64(n / Float64(Float64(l * t_1) / Float64(U * 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 * -2.0 + N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.08e+149], N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * (-N[Sqrt[N[(n * N[(U * N[(N[(N[(n * U$42$), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, 2.6e+146], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l * N[(N[(l * -2.0), $MachinePrecision] + N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(t$95$1 / N[(Om / N[(n * N[(l * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[(0.5 * N[(N[Sqrt[2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(n / N[(N[(l * t$95$1), $MachinePrecision] / N[(U * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.08000000000000008e149

   1. Initial program 24.1%

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

   2. Simplified 57.5%

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

      [Start] 24.1%	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 24.2%	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      sub-neg [=>] 24.2%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]
      associate--l+ [=>] 24.2%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      *-commutative [=>] 24.2%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      distribute-rgt-neg-in [=>] 24.2%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l/ [<=] 42.9%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 42.9%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [<=] 42.9%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [=>] 42.9%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 42.9%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]
      unpow2 [=>] 42.9%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]
      associate-*l* [=>] 42.9%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in U around 0 38.4%

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

   4. Applied egg-rr 38.3%

      \[\leadsto \color{blue}{1 \cdot \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \frac{\ell \cdot \mathsf{fma}\left(-2, \ell, \frac{\left(n \cdot \ell\right) \cdot U*}{Om}\right)}{Om}\right) \cdot U\right)}} \]
      Step-by-step derivation

      [Start] 38.4%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right) \cdot U\right)} \]
      *-un-lft-identity [=>] 38.4%	\[ \color{blue}{1 \cdot \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right) \cdot U\right)}} \]
      fma-def [=>] 38.4%	\[ 1 \cdot \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \frac{\ell \cdot \color{blue}{\mathsf{fma}\left(-2, \ell, \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}}{Om}\right) \cdot U\right)} \]
      associate-*r* [=>] 38.3%	\[ 1 \cdot \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \frac{\ell \cdot \mathsf{fma}\left(-2, \ell, \frac{\color{blue}{\left(n \cdot \ell\right) \cdot U*}}{Om}\right)}{Om}\right) \cdot U\right)} \]

   5. Simplified 53.8%

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

      [Start] 38.3%	\[ 1 \cdot \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \frac{\ell \cdot \mathsf{fma}\left(-2, \ell, \frac{\left(n \cdot \ell\right) \cdot U*}{Om}\right)}{Om}\right) \cdot U\right)} \]
      *-lft-identity [=>] 38.3%	\[ \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \frac{\ell \cdot \mathsf{fma}\left(-2, \ell, \frac{\left(n \cdot \ell\right) \cdot U*}{Om}\right)}{Om}\right) \cdot U\right)}} \]
      associate-*r* [=>] 38.3%	\[ \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot \left(t + \frac{\ell \cdot \mathsf{fma}\left(-2, \ell, \frac{\left(n \cdot \ell\right) \cdot U*}{Om}\right)}{Om}\right)\right) \cdot U}} \]
      associate-/l* [=>] 47.7%	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot \left(t + \color{blue}{\frac{\ell}{\frac{Om}{\mathsf{fma}\left(-2, \ell, \frac{\left(n \cdot \ell\right) \cdot U*}{Om}\right)}}}\right)\right) \cdot U} \]
      fma-udef [=>] 47.7%	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot \left(t + \frac{\ell}{\frac{Om}{\color{blue}{-2 \cdot \ell + \frac{\left(n \cdot \ell\right) \cdot U*}{Om}}}}\right)\right) \cdot U} \]
      *-commutative [<=] 47.7%	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot \left(t + \frac{\ell}{\frac{Om}{\color{blue}{\ell \cdot -2} + \frac{\left(n \cdot \ell\right) \cdot U*}{Om}}}\right)\right) \cdot U} \]
      fma-def [=>] 47.7%	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot \left(t + \frac{\ell}{\frac{Om}{\color{blue}{\mathsf{fma}\left(\ell, -2, \frac{\left(n \cdot \ell\right) \cdot U*}{Om}\right)}}}\right)\right) \cdot U} \]
      associate-/l* [=>] 53.8%	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot \left(t + \frac{\ell}{\frac{Om}{\mathsf{fma}\left(\ell, -2, \color{blue}{\frac{n \cdot \ell}{\frac{Om}{U*}}}\right)}}\right)\right) \cdot U} \]

   6. Taylor expanded in l around 0 38.8%

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

   7. Simplified 38.8%

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

      [Start] 38.8%	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot \left(t + \frac{{\ell}^{2} \cdot \left(\frac{n \cdot U*}{Om} - 2\right)}{Om}\right)\right) \cdot U} \]
      associate-/l* [=>] 38.8%	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot \left(t + \color{blue}{\frac{{\ell}^{2}}{\frac{Om}{\frac{n \cdot U*}{Om} - 2}}}\right)\right) \cdot U} \]
      unpow2 [=>] 38.8%	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot \left(t + \frac{\color{blue}{\ell \cdot \ell}}{\frac{Om}{\frac{n \cdot U*}{Om} - 2}}\right)\right) \cdot U} \]
      sub-neg [=>] 38.8%	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot \left(t + \frac{\ell \cdot \ell}{\frac{Om}{\color{blue}{\frac{n \cdot U*}{Om} + \left(-2\right)}}}\right)\right) \cdot U} \]
      associate-/l* [=>] 38.8%	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot \left(t + \frac{\ell \cdot \ell}{\frac{Om}{\color{blue}{\frac{n}{\frac{Om}{U*}}} + \left(-2\right)}}\right)\right) \cdot U} \]
      metadata-eval [=>] 38.8%	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot \left(t + \frac{\ell \cdot \ell}{\frac{Om}{\frac{n}{\frac{Om}{U*}} + \color{blue}{-2}}}\right)\right) \cdot U} \]

   8. Taylor expanded in l around -inf 74.2%

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

   9. Simplified 74.2%

      \[\leadsto \color{blue}{-\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om \cdot Om} - \frac{2}{Om}\right)\right)}} \]
      Step-by-step derivation

      [Start] 74.2%	\[ -1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(\left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)}\right) \]
      mul-1-neg [=>] 74.2%	\[ \color{blue}{-\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(\left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)}} \]
      *-commutative [=>] 74.2%	\[ -\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \color{blue}{\left(U \cdot \left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}} \]
      unpow2 [=>] 74.2%	\[ -\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{\color{blue}{Om \cdot Om}} - 2 \cdot \frac{1}{Om}\right)\right)} \]
      associate-*r/ [=>] 74.2%	\[ -\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om \cdot Om} - \color{blue}{\frac{2 \cdot 1}{Om}}\right)\right)} \]
      metadata-eval [=>] 74.2%	\[ -\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om \cdot Om} - \frac{\color{blue}{2}}{Om}\right)\right)} \]

   if -1.08000000000000008e149 < l < 2.60000000000000014e146

   1. Initial program 56.3%

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

   2. Simplified 58.5%

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

      [Start] 56.3%	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 60.9%	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      sub-neg [=>] 60.9%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]
      associate--l+ [=>] 60.9%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      *-commutative [=>] 60.9%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      distribute-rgt-neg-in [=>] 60.9%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l/ [<=] 60.9%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 60.9%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [<=] 60.9%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [=>] 60.9%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 56.3%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]
      unpow2 [=>] 56.3%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]
      associate-*l* [=>] 57.4%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in U around 0 63.0%

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

   if 2.60000000000000014e146 < l

   1. Initial program 25.8%

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

   2. Simplified 62.2%

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

      [Start] 25.8%	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 26.0%	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      sub-neg [=>] 26.0%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]
      associate--l+ [=>] 26.0%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      *-commutative [=>] 26.0%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      distribute-rgt-neg-in [=>] 26.0%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l/ [<=] 36.5%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 36.5%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [<=] 36.5%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [=>] 36.5%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 36.5%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]
      unpow2 [=>] 36.5%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]
      associate-*l* [=>] 36.6%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in t around 0 65.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{n \cdot \left(Om \cdot U\right)}{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \ell}}\right) + \sqrt{2} \cdot \sqrt{\frac{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(n \cdot \left(\ell \cdot U\right)\right)}{Om}}} \]

   4. Simplified 79.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{2}, \sqrt{\frac{\mathsf{fma}\left(\ell, -2, \left(n \cdot \left(U* - U\right)\right) \cdot \frac{\ell}{Om}\right)}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}}, \left(0.5 \cdot \left(\sqrt{2} \cdot t\right)\right) \cdot \sqrt{\frac{n}{\frac{\ell \cdot \mathsf{fma}\left(\ell, -2, \left(n \cdot \left(U* - U\right)\right) \cdot \frac{\ell}{Om}\right)}{Om \cdot U}}}\right)} \]
      Step-by-step derivation

      [Start] 65.1%	\[ 0.5 \cdot \left(\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{n \cdot \left(Om \cdot U\right)}{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \ell}}\right) + \sqrt{2} \cdot \sqrt{\frac{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(n \cdot \left(\ell \cdot U\right)\right)}{Om}} \]
      +-commutative [=>] 65.1%	\[ \color{blue}{\sqrt{2} \cdot \sqrt{\frac{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(n \cdot \left(\ell \cdot U\right)\right)}{Om}} + 0.5 \cdot \left(\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{n \cdot \left(Om \cdot U\right)}{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \ell}}\right)} \]
      fma-def [=>] 65.1%	\[ \color{blue}{\mathsf{fma}\left(\sqrt{2}, \sqrt{\frac{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(n \cdot \left(\ell \cdot U\right)\right)}{Om}}, 0.5 \cdot \left(\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{n \cdot \left(Om \cdot U\right)}{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \ell}}\right)\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.08 \cdot 10^{+149}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \left(-\sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om \cdot Om} - \frac{2}{Om}\right)\right)}\right)\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+146}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{2}, \sqrt{\frac{\mathsf{fma}\left(\ell, -2, \left(n \cdot \left(U* - U\right)\right) \cdot \frac{\ell}{Om}\right)}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}}, \left(0.5 \cdot \left(\sqrt{2} \cdot t\right)\right) \cdot \sqrt{\frac{n}{\frac{\ell \cdot \mathsf{fma}\left(\ell, -2, \left(n \cdot \left(U* - U\right)\right) \cdot \frac{\ell}{Om}\right)}{U \cdot Om}}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	59.8%
Cost	47944

\[\begin{array}{l} t_1 := \mathsf{fma}\left(\ell, -2, \left(n \cdot \left(U* - U\right)\right) \cdot \frac{\ell}{Om}\right)\\ \mathbf{if}\;\ell \leq -1.08 \cdot 10^{+149}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \left(-\sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om \cdot Om} - \frac{2}{Om}\right)\right)}\right)\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+146}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{2}, \sqrt{\frac{t_1}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}}, \left(0.5 \cdot \left(\sqrt{2} \cdot t\right)\right) \cdot \sqrt{\frac{n}{\frac{\ell \cdot t_1}{U \cdot Om}}}\right)\\ \end{array} \]

Alternative 2
Accuracy	59.8%
Cost	20676

\[\begin{array}{l} \mathbf{if}\;n \leq -3.4 \cdot 10^{-306}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \left(t + \frac{\ell}{\frac{Om}{\mathsf{fma}\left(\ell, -2, \frac{\ell \cdot n}{\frac{Om}{U*}}\right)}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + \frac{\ell \cdot \mathsf{fma}\left(-2, \ell, \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}{Om}\right)}\\ \end{array} \]

Alternative 3
Accuracy	59.1%
Cost	14212

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.05 \cdot 10^{+150}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \left(-\sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om \cdot Om} - \frac{2}{Om}\right)\right)}\right)\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{+38}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \frac{\left(n \cdot \left(\ell \cdot U\right)\right) \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om}\right)}{Om}}\\ \end{array} \]

Alternative 4
Accuracy	57.3%
Cost	8648

\[\begin{array}{l} \mathbf{if}\;Om \leq -4.8 \cdot 10^{+133}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{elif}\;Om \leq 4 \cdot 10^{-165}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \frac{\left(n \cdot \left(\ell \cdot U\right)\right) \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om}\right)}{Om}}\\ \mathbf{elif}\;Om \leq 1.45 \cdot 10^{+245}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \left(t + \frac{\ell \cdot \ell}{\frac{Om}{-2 + \frac{n}{\frac{Om}{U*}}}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\\ \end{array} \]

Alternative 5
Accuracy	49.1%
Cost	8144

\[\begin{array}{l} \mathbf{if}\;Om \leq -4.9 \cdot 10^{-79}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;Om \leq -5.2 \cdot 10^{-181}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \frac{\frac{n \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(U \cdot U*\right)\right)}{Om}}{Om}}\\ \mathbf{elif}\;Om \leq -3.4 \cdot 10^{-220}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;Om \leq 3.6 \cdot 10^{-146}:\\ \;\;\;\;\sqrt{2 \cdot \left(\frac{\left(\ell \cdot n\right) \cdot \left(\ell \cdot n\right)}{Om} \cdot \frac{U \cdot \left(U* - U\right)}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)}\\ \end{array} \]

Alternative 6
Accuracy	49.5%
Cost	8144

\[\begin{array}{l} \mathbf{if}\;Om \leq -5.1 \cdot 10^{-79}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;Om \leq -6.2 \cdot 10^{-179}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \frac{\frac{n \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(U \cdot U*\right)\right)}{Om}}{Om}}\\ \mathbf{elif}\;Om \leq -3.3 \cdot 10^{-220}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;Om \leq 1.95 \cdot 10^{-144}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)}\\ \end{array} \]

Alternative 7
Accuracy	53.4%
Cost	8140

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ \mathbf{if}\;Om \leq -5.4 \cdot 10^{-76}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot t_1\right)\right)}\\ \mathbf{elif}\;Om \leq 6.5 \cdot 10^{-168}:\\ \;\;\;\;\sqrt{2 \cdot \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)\right)\right)}{Om}}\\ \mathbf{elif}\;Om \leq 1.8 \cdot 10^{+244}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \left(t + \frac{\ell \cdot \ell}{\frac{Om}{-2 + \frac{n}{\frac{Om}{U*}}}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(t + -2 \cdot t_1\right)}\\ \end{array} \]

Alternative 8
Accuracy	57.1%
Cost	8140

\[\begin{array}{l} \mathbf{if}\;Om \leq -4.2 \cdot 10^{+133}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{elif}\;Om \leq 4 \cdot 10^{-165}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;Om \leq 6.4 \cdot 10^{+245}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \left(t + \frac{\ell \cdot \ell}{\frac{Om}{-2 + \frac{n}{\frac{Om}{U*}}}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\\ \end{array} \]

Alternative 9
Accuracy	52.5%
Cost	8008

\[\begin{array}{l} \mathbf{if}\;Om \leq -8.5 \cdot 10^{-74}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;Om \leq 9.8 \cdot 10^{-144}:\\ \;\;\;\;\sqrt{2 \cdot \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)}\\ \end{array} \]

Alternative 10
Accuracy	52.9%
Cost	7881

\[\begin{array}{l} \mathbf{if}\;Om \leq -2.65 \cdot 10^{-76} \lor \neg \left(Om \leq 1.4 \cdot 10^{+55}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \left(t + \frac{n}{Om} \cdot \frac{U* \cdot \left(\ell \cdot \ell\right)}{Om}\right)\right)}\\ \end{array} \]

Alternative 11
Accuracy	48.9%
Cost	7752

\[\begin{array}{l} \mathbf{if}\;Om \leq -1.7 \cdot 10^{-155}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;Om \leq 3.4 \cdot 10^{-145}:\\ \;\;\;\;\sqrt{2 \cdot \left(\frac{\left(\ell \cdot \ell\right) \cdot \left(U \cdot U*\right)}{Om} \cdot \frac{n \cdot n}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)}\\ \end{array} \]

Alternative 12
Accuracy	48.7%
Cost	7752

\[\begin{array}{l} \mathbf{if}\;Om \leq -3.8 \cdot 10^{-76}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;Om \leq 6.5 \cdot 10^{-145}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\left(U \cdot U*\right) \cdot \left(\left(\ell \cdot n\right) \cdot \left(\ell \cdot n\right)\right)}{Om \cdot Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)}\\ \end{array} \]

Alternative 13
Accuracy	49.1%
Cost	7752

\[\begin{array}{l} \mathbf{if}\;Om \leq -1.85 \cdot 10^{-78}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;Om \leq 2.5 \cdot 10^{-145}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot \frac{n}{Om}\right) \cdot \frac{U \cdot U*}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)}\\ \end{array} \]

Alternative 14
Accuracy	40.4%
Cost	7497

\[\begin{array}{l} \mathbf{if}\;\ell \leq -3.55 \cdot 10^{+81} \lor \neg \left(\ell \leq 6.4 \cdot 10^{+92}\right):\\ \;\;\;\;\sqrt{-2 \cdot \left(2 \cdot \frac{U \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 15
Accuracy	40.0%
Cost	7496

\[\begin{array}{l} \mathbf{if}\;\ell \leq -4.2 \cdot 10^{+85}:\\ \;\;\;\;\sqrt{-2 \cdot \left(2 \cdot \frac{U \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}{Om}\right)}\\ \mathbf{elif}\;\ell \leq 5.9 \cdot 10^{+93}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(-2 \cdot \frac{\ell \cdot \ell}{\frac{Om}{U}}\right)}\\ \end{array} \]

Alternative 16
Accuracy	39.9%
Cost	7369

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.45 \cdot 10^{+82} \lor \neg \left(\ell \leq 2.65 \cdot 10^{+90}\right):\\ \;\;\;\;\sqrt{-4 \cdot \frac{n}{\frac{Om}{U \cdot \left(\ell \cdot \ell\right)}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 17
Accuracy	40.0%
Cost	7368

\[\begin{array}{l} t_1 := U \cdot \left(\ell \cdot \ell\right)\\ \mathbf{if}\;\ell \leq -5 \cdot 10^{+83}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{n}{\frac{Om}{t_1}}}\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{+90}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-4 \cdot \left(n \cdot t_1\right)}{Om}}\\ \end{array} \]

Alternative 18
Accuracy	47.9%
Cost	7360

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

Alternative 19
Accuracy	37.6%
Cost	6912

\[{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5} \]

Alternative 20
Accuracy	36.2%
Cost	6848

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

Alternative 21
Accuracy	37.1%
Cost	6848

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

Reproduce

herbie shell --seed 2023263 
(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*))))))