Toniolo and Linder, Equation (13)

Average Error: 53.11% → 40.41%

Time: 45.8s

Precision: binary64

Cost: 64524

Math

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

↓

\[\begin{array}{l} t_1 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := \ell \cdot \frac{\ell}{Om}\\ t_3 := \frac{\ell \cdot \ell}{Om}\\ t_4 := \sqrt{\left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(\left(n \cdot t_1\right) \cdot \left(U - U*\right) + \left(2 \cdot t_3 - t\right)\right)}\\ t_5 := \frac{U - U*}{Om}\\ \mathbf{if}\;t_4 \leq 0:\\ \;\;\;\;\sqrt{U \cdot \left(t - \mathsf{fma}\left(2, t_3, t_1 \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)} \cdot \sqrt{2 \cdot n}\\ \mathbf{elif}\;t_4 \leq 4 \cdot 10^{+148}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{t - \mathsf{fma}\left(2, t_2, \left(n \cdot t_2\right) \cdot t_5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(n \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right) \cdot \left(\frac{2}{Om} + t_5 \cdot \frac{n}{Om}\right)\right)}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (sqrt
  (*
   (* (* 2.0 n) U)
   (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))

↓

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (pow (/ l Om) 2.0))
        (t_2 (* l (/ l Om)))
        (t_3 (/ (* l l) Om))
        (t_4
         (sqrt
          (* (* U (* n -2.0)) (+ (* (* n t_1) (- U U*)) (- (* 2.0 t_3) t)))))
        (t_5 (/ (- U U*) Om)))
   (if (<= t_4 0.0)
     (*
      (sqrt (* U (- t (fma 2.0 t_3 (* t_1 (* n (- U U*)))))))
      (sqrt (* 2.0 n)))
     (if (<= t_4 4e+148)
       t_4
       (if (<= t_4 INFINITY)
         (*
          (sqrt (* 2.0 (* n U)))
          (sqrt (- t (fma 2.0 t_2 (* (* n t_2) t_5)))))
         (sqrt
          (*
           -2.0
           (* (* n (* l (* U l))) (+ (/ 2.0 Om) (* t_5 (/ n Om)))))))))))

C

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

↓

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = pow((l / Om), 2.0);
	double t_2 = l * (l / Om);
	double t_3 = (l * l) / Om;
	double t_4 = sqrt(((U * (n * -2.0)) * (((n * t_1) * (U - U_42_)) + ((2.0 * t_3) - t))));
	double t_5 = (U - U_42_) / Om;
	double tmp;
	if (t_4 <= 0.0) {
		tmp = sqrt((U * (t - fma(2.0, t_3, (t_1 * (n * (U - U_42_))))))) * sqrt((2.0 * n));
	} else if (t_4 <= 4e+148) {
		tmp = t_4;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt((2.0 * (n * U))) * sqrt((t - fma(2.0, t_2, ((n * t_2) * t_5))));
	} else {
		tmp = sqrt((-2.0 * ((n * (l * (U * l))) * ((2.0 / Om) + (t_5 * (n / Om))))));
	}
	return tmp;
}

Julia

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

↓

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l / Om) ^ 2.0
	t_2 = Float64(l * Float64(l / Om))
	t_3 = Float64(Float64(l * l) / Om)
	t_4 = sqrt(Float64(Float64(U * Float64(n * -2.0)) * Float64(Float64(Float64(n * t_1) * Float64(U - U_42_)) + Float64(Float64(2.0 * t_3) - t))))
	t_5 = Float64(Float64(U - U_42_) / Om)
	tmp = 0.0
	if (t_4 <= 0.0)
		tmp = Float64(sqrt(Float64(U * Float64(t - fma(2.0, t_3, Float64(t_1 * Float64(n * Float64(U - U_42_))))))) * sqrt(Float64(2.0 * n)));
	elseif (t_4 <= 4e+148)
		tmp = t_4;
	elseif (t_4 <= Inf)
		tmp = Float64(sqrt(Float64(2.0 * Float64(n * U))) * sqrt(Float64(t - fma(2.0, t_2, Float64(Float64(n * t_2) * t_5)))));
	else
		tmp = sqrt(Float64(-2.0 * Float64(Float64(n * Float64(l * Float64(U * l))) * Float64(Float64(2.0 / Om) + Float64(t_5 * Float64(n / Om))))));
	end
	return tmp
end

Wolfram

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

↓

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(N[(U * N[(n * -2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(n * t$95$1), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * t$95$3), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision]}, If[LessEqual[t$95$4, 0.0], N[(N[Sqrt[N[(U * N[(t - N[(2.0 * t$95$3 + N[(t$95$1 * N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 4e+148], t$95$4, If[LessEqual[t$95$4, Infinity], N[(N[Sqrt[N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t - N[(2.0 * t$95$2 + N[(N[(n * t$95$2), $MachinePrecision] * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(-2.0 * N[(N[(n * N[(l * N[(U * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / Om), $MachinePrecision] + N[(t$95$5 * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 0.0

   1. Initial program 87.41

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

   2. Simplified 62.07

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

      [Start] 87.41	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 59.43	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 59.43	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      sub-neg [=>] 59.43	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(-\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [<=] 59.43	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      cancel-sign-sub [<=] 59.43	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      cancel-sign-sub [=>] 59.43	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-/l* [=>] 59.43	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 62.07	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Applied egg-rr 62.54

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 4.0000000000000002e148

   1. Initial program 2.72

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

   if 4.0000000000000002e148 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < +inf.0

   1. Initial program 98.63

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

   2. Simplified 83.07

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

      [Start] 98.63	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 96.05	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 96.05	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      sub-neg [=>] 96.05	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(-\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [<=] 96.05	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      cancel-sign-sub [<=] 96.05	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      cancel-sign-sub [=>] 96.05	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-/l* [=>] 84.07	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 83.07	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in U around 0 95.38

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

   4. Simplified 82.67

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

      [Start] 95.38	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(\frac{{\ell}^{2} \cdot U}{{Om}^{2}} + -1 \cdot \frac{{\ell}^{2} \cdot U*}{{Om}^{2}}\right)\right)\right)\right)} \]
      mul-1-neg [=>] 95.38	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(\frac{{\ell}^{2} \cdot U}{{Om}^{2}} + \color{blue}{\left(-\frac{{\ell}^{2} \cdot U*}{{Om}^{2}}\right)}\right)\right)\right)\right)} \]
      unsub-neg [=>] 95.38	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \color{blue}{\left(\frac{{\ell}^{2} \cdot U}{{Om}^{2}} - \frac{{\ell}^{2} \cdot U*}{{Om}^{2}}\right)}\right)\right)\right)} \]
      unpow2 [=>] 95.38	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(\frac{{\ell}^{2} \cdot U}{\color{blue}{Om \cdot Om}} - \frac{{\ell}^{2} \cdot U*}{{Om}^{2}}\right)\right)\right)\right)} \]
      times-frac [=>] 95.45	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(\color{blue}{\frac{{\ell}^{2}}{Om} \cdot \frac{U}{Om}} - \frac{{\ell}^{2} \cdot U*}{{Om}^{2}}\right)\right)\right)\right)} \]
      unpow2 [=>] 95.45	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{Om} \cdot \frac{U}{Om} - \frac{{\ell}^{2} \cdot U*}{{Om}^{2}}\right)\right)\right)\right)} \]
      associate-/l* [=>] 95.47	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(\color{blue}{\frac{\ell}{\frac{Om}{\ell}}} \cdot \frac{U}{Om} - \frac{{\ell}^{2} \cdot U*}{{Om}^{2}}\right)\right)\right)\right)} \]
      associate-/r/ [=>] 95.47	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \frac{U}{Om} - \frac{{\ell}^{2} \cdot U*}{{Om}^{2}}\right)\right)\right)\right)} \]
      unpow2 [=>] 95.47	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(\left(\frac{\ell}{Om} \cdot \ell\right) \cdot \frac{U}{Om} - \frac{{\ell}^{2} \cdot U*}{\color{blue}{Om \cdot Om}}\right)\right)\right)\right)} \]
      times-frac [=>] 94.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(\left(\frac{\ell}{Om} \cdot \ell\right) \cdot \frac{U}{Om} - \color{blue}{\frac{{\ell}^{2}}{Om} \cdot \frac{U*}{Om}}\right)\right)\right)\right)} \]
      unpow2 [=>] 94.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(\left(\frac{\ell}{Om} \cdot \ell\right) \cdot \frac{U}{Om} - \frac{\color{blue}{\ell \cdot \ell}}{Om} \cdot \frac{U*}{Om}\right)\right)\right)\right)} \]
      associate-/l* [=>] 82.67	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(\left(\frac{\ell}{Om} \cdot \ell\right) \cdot \frac{U}{Om} - \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} \cdot \frac{U*}{Om}\right)\right)\right)\right)} \]
      associate-/r/ [=>] 82.67	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(\left(\frac{\ell}{Om} \cdot \ell\right) \cdot \frac{U}{Om} - \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \frac{U*}{Om}\right)\right)\right)\right)} \]

   5. Applied egg-rr 80.33

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

   if +inf.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))))

   1. Initial program 100

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

   2. Simplified 95.81

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

      [Start] 100	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 100	\[ \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 100	\[ \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      *-commutative [=>] 100	\[ \sqrt{2 \cdot \color{blue}{\left(\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot \left(n \cdot U\right)\right)}} \]

   3. Taylor expanded in l around 0 93.45

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

   4. Simplified 96.54

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

      [Start] 93.45	\[ \sqrt{2 \cdot \left(n \cdot \left({\ell}^{2} \cdot \left(\left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)\right) + n \cdot \left(t \cdot U\right)\right)} \]
      distribute-lft-out [=>] 93.45	\[ \sqrt{2 \cdot \color{blue}{\left(n \cdot \left({\ell}^{2} \cdot \left(\left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right) + t \cdot U\right)\right)}} \]
      associate-*r* [=>] 96.64	\[ \sqrt{2 \cdot \left(n \cdot \left(\color{blue}{\left({\ell}^{2} \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right) \cdot U} + t \cdot U\right)\right)} \]
      distribute-rgt-out [=>] 96.64	\[ \sqrt{2 \cdot \left(n \cdot \color{blue}{\left(U \cdot \left({\ell}^{2} \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) + t\right)\right)}\right)} \]
      +-commutative [<=] 96.64	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \color{blue}{\left(t + {\ell}^{2} \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}\right)\right)} \]
      unpow2 [=>] 96.64	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)\right)\right)} \]
      cancel-sign-sub-inv [=>] 96.64	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \color{blue}{\left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} + \left(-2\right) \cdot \frac{1}{Om}\right)}\right)\right)\right)} \]
      associate-/l* [=>] 96.5	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\color{blue}{\frac{n}{\frac{{Om}^{2}}{U* - U}}} + \left(-2\right) \cdot \frac{1}{Om}\right)\right)\right)\right)} \]
      associate-/r/ [=>] 96.54	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\color{blue}{\frac{n}{{Om}^{2}} \cdot \left(U* - U\right)} + \left(-2\right) \cdot \frac{1}{Om}\right)\right)\right)\right)} \]
      unpow2 [=>] 96.54	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n}{\color{blue}{Om \cdot Om}} \cdot \left(U* - U\right) + \left(-2\right) \cdot \frac{1}{Om}\right)\right)\right)\right)} \]
      metadata-eval [=>] 96.54	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) + \color{blue}{-2} \cdot \frac{1}{Om}\right)\right)\right)\right)} \]
      associate-*r/ [=>] 96.54	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) + \color{blue}{\frac{-2 \cdot 1}{Om}}\right)\right)\right)\right)} \]
      metadata-eval [=>] 96.54	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) + \frac{\color{blue}{-2}}{Om}\right)\right)\right)\right)} \]

   5. Taylor expanded in t around 0 94.12

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

   6. Simplified 67.63

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

      [Start] 94.12	\[ \sqrt{2 \cdot \left(n \cdot \left({\ell}^{2} \cdot \left(\left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)\right)\right)} \]
      *-commutative [<=] 94.12	\[ \sqrt{2 \cdot \left(n \cdot \left({\ell}^{2} \cdot \color{blue}{\left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}\right)\right)} \]
      associate-*r* [=>] 94.15	\[ \sqrt{2 \cdot \left(n \cdot \color{blue}{\left(\left({\ell}^{2} \cdot U\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}\right)} \]
      associate-*r* [=>] 93.01	\[ \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot \left({\ell}^{2} \cdot U\right)\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}} \]
      unpow2 [=>] 93.01	\[ \sqrt{2 \cdot \left(\left(n \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot U\right)\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)} \]
      associate-*l* [=>] 75.14	\[ \sqrt{2 \cdot \left(\left(n \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot U\right)\right)}\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)} \]
      cancel-sign-sub-inv [=>] 75.14	\[ \sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right) \cdot \color{blue}{\left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} + \left(-2\right) \cdot \frac{1}{Om}\right)}\right)} \]
      unpow2 [=>] 75.14	\[ \sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{\color{blue}{Om \cdot Om}} + \left(-2\right) \cdot \frac{1}{Om}\right)\right)} \]
      times-frac [=>] 67.63	\[ \sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right) \cdot \left(\color{blue}{\frac{n}{Om} \cdot \frac{U* - U}{Om}} + \left(-2\right) \cdot \frac{1}{Om}\right)\right)} \]
      metadata-eval [=>] 67.63	\[ \sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right) \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \color{blue}{-2} \cdot \frac{1}{Om}\right)\right)} \]
      associate-*r/ [=>] 67.63	\[ \sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right) \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \color{blue}{\frac{-2 \cdot 1}{Om}}\right)\right)} \]
      metadata-eval [=>] 67.63	\[ \sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right) \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{\color{blue}{-2}}{Om}\right)\right)} \]

3. Recombined 4 regimes into one program.

4. Final simplification 40.41

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right) + \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)} \cdot \sqrt{2 \cdot n}\\ \mathbf{elif}\;\sqrt{\left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right) + \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right)} \leq 4 \cdot 10^{+148}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right) + \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right) + \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(n \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{U - U*}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(n \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right) \cdot \left(\frac{2}{Om} + \frac{U - U*}{Om} \cdot \frac{n}{Om}\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	41.02%
Cost	64524

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ t_2 := \frac{U - U*}{Om}\\ t_3 := \sqrt{\left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right) + \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right)}\\ \mathbf{if}\;t_3 \leq 5 \cdot 10^{-158}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot t\right)} \cdot \sqrt{U}\\ \mathbf{elif}\;t_3 \leq 4 \cdot 10^{+148}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{t - \mathsf{fma}\left(2, t_1, \left(n \cdot t_1\right) \cdot t_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(n \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right) \cdot \left(\frac{2}{Om} + t_2 \cdot \frac{n}{Om}\right)\right)}\\ \end{array} \]

Alternative 2
Error	40.42%
Cost	64524

\[\begin{array}{l} t_1 := \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\\ t_2 := \sqrt{\left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(t_1 + \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right)}\\ t_3 := \ell \cdot \frac{\ell}{Om}\\ t_4 := \frac{U - U*}{Om}\\ \mathbf{if}\;t_2 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, t_3, t_1\right)\right)}\\ \mathbf{elif}\;t_2 \leq 4 \cdot 10^{+148}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{t - \mathsf{fma}\left(2, t_3, \left(n \cdot t_3\right) \cdot t_4\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(n \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right) \cdot \left(\frac{2}{Om} + t_4 \cdot \frac{n}{Om}\right)\right)}\\ \end{array} \]

Alternative 3
Error	40.89%
Cost	63628

\[\begin{array}{l} t_1 := \sqrt{\left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right) + \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right)}\\ \mathbf{if}\;t_1 \leq 5 \cdot 10^{-158}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot t\right)} \cdot \sqrt{U}\\ \mathbf{elif}\;t_1 \leq 10^{+150}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot -2, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(n \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right) \cdot \left(\frac{2}{Om} + \frac{U - U*}{Om} \cdot \frac{n}{Om}\right)\right)}\\ \end{array} \]

Alternative 4
Error	41.47%
Cost	51468

\[\begin{array}{l} t_1 := \sqrt{\left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right) + \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right)}\\ \mathbf{if}\;t_1 \leq 5 \cdot 10^{-158}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot t\right)} \cdot \sqrt{U}\\ \mathbf{elif}\;t_1 \leq 10^{+150}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \frac{2 \cdot \ell}{Om} - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(n \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right) \cdot \left(\frac{2}{Om} + \frac{U - U*}{Om} \cdot \frac{n}{Om}\right)\right)}\\ \end{array} \]

Alternative 5
Error	48.3%
Cost	14416

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := \ell \cdot \frac{\ell}{Om}\\ t_3 := t_2 \cdot \frac{U*}{Om}\\ \mathbf{if}\;n \leq -1.3 \cdot 10^{-56}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot t_1 + n \cdot \left(t_3 - \frac{\frac{\ell}{\frac{Om}{U}}}{\frac{Om}{\ell}}\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq 5.6 \cdot 10^{-296}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \frac{2 \cdot \ell}{Om} - t\right)\right)\right)}\\ \mathbf{elif}\;n \leq 5.6 \cdot 10^{-268}:\\ \;\;\;\;\sqrt{U \cdot t} \cdot \sqrt{n + n}\\ \mathbf{elif}\;n \leq 2.9 \cdot 10^{-180}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{n \cdot -4}{\frac{Om}{\ell \cdot \left(U \cdot \ell\right)}}\right)}\\ \mathbf{elif}\;n \leq 1.15 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot t_1 - n \cdot \left(t_3 - t_2 \cdot \frac{U}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot U\right)} \cdot \sqrt{n}\\ \end{array} \]

Alternative 6
Error	47.9%
Cost	13908

\[\begin{array}{l} t_1 := \sqrt{U \cdot t} \cdot \sqrt{n + n}\\ t_2 := \frac{\ell}{\frac{Om}{\ell}}\\ t_3 := \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \frac{2 \cdot \ell}{Om} - t\right)\right)\right)}\\ t_4 := \ell \cdot \frac{\ell}{Om}\\ t_5 := t_4 \cdot \frac{U*}{Om}\\ \mathbf{if}\;n \leq -1.3 \cdot 10^{-56}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot t_2 + n \cdot \left(t_5 - \frac{\frac{\ell}{\frac{Om}{U}}}{\frac{Om}{\ell}}\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq 1.28 \cdot 10^{-297}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;n \leq 1.45 \cdot 10^{-271}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{-160}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;n \leq 4.1 \cdot 10^{+133}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot t_2 - n \cdot \left(t_5 - t_4 \cdot \frac{U}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	47.73%
Cost	13908

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \frac{2 \cdot \ell}{Om} - t\right)\right)\right)}\\ t_3 := \ell \cdot \frac{\ell}{Om}\\ t_4 := t_3 \cdot \frac{U*}{Om}\\ \mathbf{if}\;n \leq -1.3 \cdot 10^{-56}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot t_1 + n \cdot \left(t_4 - \frac{\frac{\ell}{\frac{Om}{U}}}{\frac{Om}{\ell}}\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq 4.45 \cdot 10^{-296}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;n \leq 6.8 \cdot 10^{-270}:\\ \;\;\;\;\sqrt{U \cdot t} \cdot \sqrt{n + n}\\ \mathbf{elif}\;n \leq 5.8 \cdot 10^{-157}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;n \leq 1.06 \cdot 10^{+150}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot t_1 - n \cdot \left(t_4 - t_3 \cdot \frac{U}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot U\right)} \cdot \sqrt{n}\\ \end{array} \]

Alternative 8
Error	48.37%
Cost	13777

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ t_2 := \frac{\ell}{\frac{Om}{\ell}}\\ t_3 := -2 \cdot t_2\\ t_4 := t_1 \cdot \frac{U*}{Om}\\ \mathbf{if}\;U \leq -2.8 \cdot 10^{+61}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + t_3\right)\right)}\\ \mathbf{elif}\;U \leq 2 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot t_2 - n \cdot \left(t_4 - t_1 \cdot \frac{U}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;U \leq 1.5 \cdot 10^{-161} \lor \neg \left(U \leq 1.16 \cdot 10^{+23}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot t\right)} \cdot \sqrt{U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(t_3 + n \cdot \left(t_4 - \frac{\frac{\ell}{\frac{Om}{U}}}{\frac{Om}{\ell}}\right)\right)\right)\right)}\\ \end{array} \]

Alternative 9
Error	53.72%
Cost	13776

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := -2 \cdot t_1\\ t_3 := \ell \cdot \frac{\ell}{Om}\\ t_4 := t_3 \cdot \frac{U*}{Om}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{-273}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(t_2 + n \cdot \left(t_4 - \frac{\frac{\ell}{\frac{Om}{U}}}{\frac{Om}{\ell}}\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 7.6 \cdot 10^{-156}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + t_2\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.9 \cdot 10^{+36}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{Om} + \left(U - U*\right) \cdot \frac{n}{Om \cdot Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.15 \cdot 10^{+94}:\\ \;\;\;\;\sqrt{2 \cdot \left|U \cdot \left(n \cdot t\right)\right|}\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{+119}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(\ell \cdot \frac{-n}{Om}\right)\\ \mathbf{elif}\;\ell \leq 1.52 \cdot 10^{+144}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot t_1 - n \cdot \left(t_4 - t_3 \cdot \frac{U}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(n \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right) \cdot \left(\frac{2}{Om} + \frac{U - U*}{Om} \cdot \frac{n}{Om}\right)\right)}\\ \end{array} \]

Alternative 10
Error	48.36%
Cost	13776

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ t_2 := \frac{\ell}{\frac{Om}{\ell}}\\ t_3 := -2 \cdot t_2\\ t_4 := t_1 \cdot \frac{U*}{Om}\\ \mathbf{if}\;U \leq -1.7 \cdot 10^{+62}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + t_3\right)\right)}\\ \mathbf{elif}\;U \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot t_2 - n \cdot \left(t_4 - t_1 \cdot \frac{U}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;U \leq 4.5 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{n \cdot t} \cdot \sqrt{2 \cdot U}\\ \mathbf{elif}\;U \leq 2.7 \cdot 10^{+23}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(t_3 + n \cdot \left(t_4 - \frac{\frac{\ell}{\frac{Om}{U}}}{\frac{Om}{\ell}}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot t\right)} \cdot \sqrt{U}\\ \end{array} \]

Alternative 11
Error	53.97%
Cost	9560

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := \ell \cdot \frac{\ell}{Om}\\ t_3 := \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot t_1 - n \cdot \left(t_2 \cdot \frac{U*}{Om} - t_2 \cdot \frac{U}{Om}\right)\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -3.35 \cdot 10^{-276}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{-156}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + -2 \cdot t_1\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+38}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{Om} + \left(U - U*\right) \cdot \frac{n}{Om \cdot Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{+93}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{+119}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(\ell \cdot \frac{-n}{Om}\right)\\ \mathbf{elif}\;\ell \leq 1.4 \cdot 10^{+149}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(n \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right) \cdot \left(\frac{2}{Om} + \frac{U - U*}{Om} \cdot \frac{n}{Om}\right)\right)}\\ \end{array} \]

Alternative 12
Error	53.81%
Cost	9560

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := -2 \cdot t_1\\ t_3 := \ell \cdot \frac{\ell}{Om}\\ t_4 := t_3 \cdot \frac{U*}{Om}\\ \mathbf{if}\;\ell \leq -8 \cdot 10^{-271}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(t_2 + n \cdot \left(t_4 - \frac{\frac{\ell}{\frac{Om}{U}}}{\frac{Om}{\ell}}\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 9.5 \cdot 10^{-156}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + t_2\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.9 \cdot 10^{+36}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{Om} + \left(U - U*\right) \cdot \frac{n}{Om \cdot Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.4 \cdot 10^{+94}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{+119}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(\ell \cdot \frac{-n}{Om}\right)\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{+149}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot t_1 - n \cdot \left(t_4 - t_3 \cdot \frac{U}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(n \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right) \cdot \left(\frac{2}{Om} + \frac{U - U*}{Om} \cdot \frac{n}{Om}\right)\right)}\\ \end{array} \]

Alternative 13
Error	55.2%
Cost	8928

\[\begin{array}{l} t_1 := t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{Om} + \left(U - U*\right) \cdot \frac{n}{Om \cdot Om}\right)\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -3.6 \cdot 10^{+165}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t_1\right)}\\ \mathbf{elif}\;\ell \leq -3.7 \cdot 10^{-176}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -2.8 \cdot 10^{-301}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 9.1 \cdot 10^{-156}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot t_1\right)}\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+38}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 2.2 \cdot 10^{+88}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{+119}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(\ell \cdot \frac{-n}{Om}\right)\\ \mathbf{elif}\;\ell \leq 2.8 \cdot 10^{+144}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{-2}{Om} + \frac{n}{\frac{Om \cdot Om}{U*}}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(n \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right) \cdot \left(\frac{2}{Om} + \frac{U - U*}{Om} \cdot \frac{n}{Om}\right)\right)}\\ \end{array} \]

Alternative 14
Error	50.9%
Cost	8648

\[\begin{array}{l} t_1 := \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \frac{2 \cdot \ell}{Om} - t\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -2.75 \cdot 10^{+14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -1.76 \cdot 10^{-265}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\frac{n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om}\right) - U \cdot \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - t\right)\right)}\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{-156}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{+37}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{Om} + \left(U - U*\right) \cdot \frac{n}{Om \cdot Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 15
Error	53.5%
Cost	8532

\[\begin{array}{l} t_1 := t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{-2}{Om} + \frac{n}{\frac{Om \cdot Om}{U*}}\right)\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -3.6 \cdot 10^{+165}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t_1\right)}\\ \mathbf{elif}\;\ell \leq -2.4 \cdot 10^{-174}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -4.4 \cdot 10^{-301}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 8.1 \cdot 10^{-156}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot t_1\right)}\\ \mathbf{elif}\;\ell \leq 10^{+30}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \frac{2 \cdot \ell}{Om} - t\right)\right)\right)}\\ \end{array} \]

Alternative 16
Error	52.67%
Cost	7625

\[\begin{array}{l} \mathbf{if}\;\ell \leq -3.9 \cdot 10^{-68} \lor \neg \left(\ell \leq 5.6 \cdot 10^{-94}\right):\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \frac{2 \cdot \ell}{Om} - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 17
Error	50.31%
Cost	7625

\[\begin{array}{l} \mathbf{if}\;U \leq -1 \cdot 10^{-95} \lor \neg \left(U \leq 2 \cdot 10^{-126}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 18
Error	50.35%
Cost	7625

\[\begin{array}{l} t_1 := t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;U \leq -6 \cdot 10^{-103} \lor \neg \left(U \leq 1.85 \cdot 10^{-126}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t_1\right)}\\ \end{array} \]

Alternative 19
Error	59.25%
Cost	7497

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.4 \cdot 10^{+83} \lor \neg \left(\ell \leq 9 \cdot 10^{+138}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \frac{n}{\frac{Om}{\ell \cdot \left(U \cdot \ell\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 20
Error	53.3%
Cost	7492

\[\begin{array}{l} \mathbf{if}\;U* \leq -4.6 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \frac{2 \cdot \ell}{Om} - t\right)\right)\right)}\\ \end{array} \]

Alternative 21
Error	61.62%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;U \leq -1.2 \cdot 10^{-301} \lor \neg \left(U \leq 2.7 \cdot 10^{-127}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]

Alternative 22
Error	61.37%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;U \leq 4.05 \cdot 10^{-249} \lor \neg \left(U \leq 2.6 \cdot 10^{-126}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \end{array} \]

Alternative 23
Error	61.8%
Cost	6848

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

Reproduce

herbie shell --seed 2023115 
(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*))))))