Toniolo and Linder, Equation (13)

Average Error: 54.53% → 41.01%

Time: 41.7s

Precision: binary64

Cost: 51340

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t_1 \leq 5 \cdot 10^{-161}:\\ \;\;\;\;\sqrt[3]{n} \cdot \sqrt{U \cdot \left(\sqrt[3]{n} \cdot \left(t + t\right)\right)}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+152}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + -2 \cdot \left(\left(n \cdot \left(U \cdot \ell\right)\right) \cdot \frac{2}{\frac{Om}{\ell}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2 - n \cdot \frac{U*}{Om}}{Om} \cdot \left(\left(U \cdot \ell\right) \cdot \left(\ell \cdot \left(n \cdot -2\right)\right)\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* (* 2.0 n) U)
           (+
            (+ t (* (/ (* l l) Om) -2.0))
            (* (* n (pow (/ l Om) 2.0)) (- U* U)))))))
   (if (<= t_1 5e-161)
     (* (cbrt n) (sqrt (* U (* (cbrt n) (+ t t)))))
     (if (<= t_1 2e+152)
       t_1
       (if (<= t_1 INFINITY)
         (sqrt
          (+
           (* 2.0 (* n (* U t)))
           (* -2.0 (* (* n (* U l)) (/ 2.0 (/ Om l))))))
         (sqrt
          (* (/ (- 2.0 (* n (/ U* Om))) Om) (* (* U l) (* l (* n -2.0))))))))))

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 = sqrt((((2.0 * n) * U) * ((t + (((l * l) / Om) * -2.0)) + ((n * pow((l / Om), 2.0)) * (U_42_ - U)))));
	double tmp;
	if (t_1 <= 5e-161) {
		tmp = cbrt(n) * sqrt((U * (cbrt(n) * (t + t))));
	} else if (t_1 <= 2e+152) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = sqrt(((2.0 * (n * (U * t))) + (-2.0 * ((n * (U * l)) * (2.0 / (Om / l))))));
	} else {
		tmp = sqrt((((2.0 - (n * (U_42_ / Om))) / Om) * ((U * l) * (l * (n * -2.0)))));
	}
	return tmp;
}

Java

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

↓

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = Math.sqrt((((2.0 * n) * U) * ((t + (((l * l) / Om) * -2.0)) + ((n * Math.pow((l / Om), 2.0)) * (U_42_ - U)))));
	double tmp;
	if (t_1 <= 5e-161) {
		tmp = Math.cbrt(n) * Math.sqrt((U * (Math.cbrt(n) * (t + t))));
	} else if (t_1 <= 2e+152) {
		tmp = t_1;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt(((2.0 * (n * (U * t))) + (-2.0 * ((n * (U * l)) * (2.0 / (Om / l))))));
	} else {
		tmp = Math.sqrt((((2.0 - (n * (U_42_ / Om))) / Om) * ((U * l) * (l * (n * -2.0)))));
	}
	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 = sqrt(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 <= 5e-161)
		tmp = Float64(cbrt(n) * sqrt(Float64(U * Float64(cbrt(n) * Float64(t + t)))));
	elseif (t_1 <= 2e+152)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * Float64(U * t))) + Float64(-2.0 * Float64(Float64(n * Float64(U * l)) * Float64(2.0 / Float64(Om / l))))));
	else
		tmp = sqrt(Float64(Float64(Float64(2.0 - Float64(n * Float64(U_42_ / Om))) / Om) * Float64(Float64(U * l) * Float64(l * Float64(n * -2.0)))));
	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[Sqrt[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]], $MachinePrecision]}, If[LessEqual[t$95$1, 5e-161], N[(N[Power[n, 1/3], $MachinePrecision] * N[Sqrt[N[(U * N[(N[Power[n, 1/3], $MachinePrecision] * N[(t + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+152], t$95$1, If[LessEqual[t$95$1, Infinity], N[Sqrt[N[(N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[(n * N[(U * l), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(2.0 - N[(n * N[(U$42$ / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[(N[(U * l), $MachinePrecision] * N[(l * N[(n * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 4.9999999999999999e-161

   1. Initial program 86.82

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

   2. Simplified 86.62

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

      [Start] 86.82	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 86.82	\[ \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 86.83	\[ \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      *-commutative [=>] 86.83	\[ \sqrt{2 \cdot \color{blue}{\left(\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot \left(n \cdot U\right)\right)}} \]

   3. Taylor expanded in l around 0 86.85

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

   4. Applied egg-rr 86.84

      \[\leadsto \color{blue}{{\left({\left(\left(n \cdot U\right) \cdot \left(2 \cdot t\right)\right)}^{0.16666666666666666}\right)}^{3}} \]

   5. Applied egg-rr 66.65

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

   6. Simplified 67.38

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

      [Start] 66.65	\[ \sqrt[3]{n} \cdot \sqrt{\sqrt[3]{n} \cdot \left(U \cdot \left(2 \cdot t\right)\right)} \]
      *-commutative [=>] 66.65	\[ \sqrt[3]{n} \cdot \sqrt{\color{blue}{\left(U \cdot \left(2 \cdot t\right)\right) \cdot \sqrt[3]{n}}} \]
      associate-*l* [=>] 67.38	\[ \sqrt[3]{n} \cdot \sqrt{\color{blue}{U \cdot \left(\left(2 \cdot t\right) \cdot \sqrt[3]{n}\right)}} \]
      count-2 [<=] 67.38	\[ \sqrt[3]{n} \cdot \sqrt{U \cdot \left(\color{blue}{\left(t + t\right)} \cdot \sqrt[3]{n}\right)} \]

   if 4.9999999999999999e-161 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 2.0000000000000001e152

   1. Initial program 2.82

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

   if 2.0000000000000001e152 < (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 99.47

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

   2. Simplified 85.04

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

      [Start] 99.47	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 95.8	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 95.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      sub-neg [=>] 95.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(-\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [<=] 95.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      cancel-sign-sub [<=] 95.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      cancel-sign-sub [=>] 95.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-/l* [=>] 85.66	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 85.04	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in Om around inf 94.47

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

   4. Simplified 94.55

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

      [Start] 94.47	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + -2 \cdot \frac{{\ell}^{2} \cdot U}{Om}\right)} \]
      fma-def [=>] 94.47	\[ \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\mathsf{fma}\left(t, U, -2 \cdot \frac{{\ell}^{2} \cdot U}{Om}\right)}} \]
      associate-/l* [=>] 94.55	\[ \sqrt{\left(2 \cdot n\right) \cdot \mathsf{fma}\left(t, U, -2 \cdot \color{blue}{\frac{{\ell}^{2}}{\frac{Om}{U}}}\right)} \]
      unpow2 [=>] 94.55	\[ \sqrt{\left(2 \cdot n\right) \cdot \mathsf{fma}\left(t, U, -2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{\frac{Om}{U}}\right)} \]

   5. Applied egg-rr 82.92

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

   6. Applied egg-rr 86.1

      \[\leadsto \sqrt{\left(\left(t \cdot U\right) \cdot n\right) \cdot 2 + \color{blue}{\left(e^{\mathsf{log1p}\left(n \cdot \left(\left(U \cdot \ell\right) \cdot \frac{-2 \cdot \ell}{Om}\right)\right)} - 1\right)} \cdot 2} \]

   7. Simplified 80.87

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

      [Start] 86.1	\[ \sqrt{\left(\left(t \cdot U\right) \cdot n\right) \cdot 2 + \left(e^{\mathsf{log1p}\left(n \cdot \left(\left(U \cdot \ell\right) \cdot \frac{-2 \cdot \ell}{Om}\right)\right)} - 1\right) \cdot 2} \]
      expm1-def [=>] 83.72	\[ \sqrt{\left(\left(t \cdot U\right) \cdot n\right) \cdot 2 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(n \cdot \left(\left(U \cdot \ell\right) \cdot \frac{-2 \cdot \ell}{Om}\right)\right)\right)} \cdot 2} \]
      expm1-log1p [=>] 82.92	\[ \sqrt{\left(\left(t \cdot U\right) \cdot n\right) \cdot 2 + \color{blue}{\left(n \cdot \left(\left(U \cdot \ell\right) \cdot \frac{-2 \cdot \ell}{Om}\right)\right)} \cdot 2} \]
      associate-*r* [=>] 80.86	\[ \sqrt{\left(\left(t \cdot U\right) \cdot n\right) \cdot 2 + \color{blue}{\left(\left(n \cdot \left(U \cdot \ell\right)\right) \cdot \frac{-2 \cdot \ell}{Om}\right)} \cdot 2} \]
      associate-/l* [=>] 80.87	\[ \sqrt{\left(\left(t \cdot U\right) \cdot n\right) \cdot 2 + \left(\left(n \cdot \left(U \cdot \ell\right)\right) \cdot \color{blue}{\frac{-2}{\frac{Om}{\ell}}}\right) \cdot 2} \]

   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 90.96

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

      [Start] 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* [=>] 99.96	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 99.96	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      sub-neg [=>] 99.96	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(-\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [<=] 99.96	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      cancel-sign-sub [<=] 99.96	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      cancel-sign-sub [=>] 99.96	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-/l* [=>] 91.56	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 90.96	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in l around inf 93.34

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

   4. Simplified 79.63

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

      [Start] 93.34	\[ \sqrt{-2 \cdot \left(\left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(n \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      *-commutative [=>] 93.34	\[ \sqrt{-2 \cdot \color{blue}{\left(\left(n \cdot \left({\ell}^{2} \cdot U\right)\right) \cdot \left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)\right)}} \]
      associate-*l* [=>] 93.45	\[ \sqrt{-2 \cdot \color{blue}{\left(n \cdot \left(\left({\ell}^{2} \cdot U\right) \cdot \left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)\right)\right)}} \]
      *-commutative [<=] 93.45	\[ \sqrt{-2 \cdot \left(n \cdot \color{blue}{\left(\left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)\right)}\right)} \]
      unpow2 [=>] 93.45	\[ \sqrt{-2 \cdot \left(n \cdot \left(\left(\frac{n \cdot \left(U - U*\right)}{\color{blue}{Om \cdot Om}} + 2 \cdot \frac{1}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      associate-*r/ [=>] 93.45	\[ \sqrt{-2 \cdot \left(n \cdot \left(\left(\frac{n \cdot \left(U - U*\right)}{Om \cdot Om} + \color{blue}{\frac{2 \cdot 1}{Om}}\right) \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      metadata-eval [=>] 93.45	\[ \sqrt{-2 \cdot \left(n \cdot \left(\left(\frac{n \cdot \left(U - U*\right)}{Om \cdot Om} + \frac{\color{blue}{2}}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      unpow2 [=>] 93.45	\[ \sqrt{-2 \cdot \left(n \cdot \left(\left(\frac{n \cdot \left(U - U*\right)}{Om \cdot Om} + \frac{2}{Om}\right) \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot U\right)\right)\right)} \]
      associate-*l* [=>] 79.63	\[ \sqrt{-2 \cdot \left(n \cdot \left(\left(\frac{n \cdot \left(U - U*\right)}{Om \cdot Om} + \frac{2}{Om}\right) \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot U\right)\right)}\right)\right)} \]

   5. Taylor expanded in U around 0 93.53

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

   6. Simplified 72.59

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

      [Start] 93.53	\[ \sqrt{-2 \cdot \left(n \cdot \left({\ell}^{2} \cdot \left(\left(-1 \cdot \frac{n \cdot U*}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot U\right)\right)\right)} \]
      *-commutative [=>] 93.53	\[ \sqrt{-2 \cdot \left(n \cdot \color{blue}{\left(\left(\left(-1 \cdot \frac{n \cdot U*}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot U\right) \cdot {\ell}^{2}\right)}\right)} \]
      associate-*l* [=>] 93.52	\[ \sqrt{-2 \cdot \left(n \cdot \color{blue}{\left(\left(-1 \cdot \frac{n \cdot U*}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(U \cdot {\ell}^{2}\right)\right)}\right)} \]
      +-commutative [=>] 93.52	\[ \sqrt{-2 \cdot \left(n \cdot \left(\color{blue}{\left(2 \cdot \frac{1}{Om} + -1 \cdot \frac{n \cdot U*}{{Om}^{2}}\right)} \cdot \left(U \cdot {\ell}^{2}\right)\right)\right)} \]
      associate-*r/ [=>] 93.52	\[ \sqrt{-2 \cdot \left(n \cdot \left(\left(\color{blue}{\frac{2 \cdot 1}{Om}} + -1 \cdot \frac{n \cdot U*}{{Om}^{2}}\right) \cdot \left(U \cdot {\ell}^{2}\right)\right)\right)} \]
      metadata-eval [=>] 93.52	\[ \sqrt{-2 \cdot \left(n \cdot \left(\left(\frac{\color{blue}{2}}{Om} + -1 \cdot \frac{n \cdot U*}{{Om}^{2}}\right) \cdot \left(U \cdot {\ell}^{2}\right)\right)\right)} \]
      mul-1-neg [=>] 93.52	\[ \sqrt{-2 \cdot \left(n \cdot \left(\left(\frac{2}{Om} + \color{blue}{\left(-\frac{n \cdot U*}{{Om}^{2}}\right)}\right) \cdot \left(U \cdot {\ell}^{2}\right)\right)\right)} \]
      unsub-neg [=>] 93.52	\[ \sqrt{-2 \cdot \left(n \cdot \left(\color{blue}{\left(\frac{2}{Om} - \frac{n \cdot U*}{{Om}^{2}}\right)} \cdot \left(U \cdot {\ell}^{2}\right)\right)\right)} \]
      unpow2 [=>] 93.52	\[ \sqrt{-2 \cdot \left(n \cdot \left(\left(\frac{2}{Om} - \frac{n \cdot U*}{\color{blue}{Om \cdot Om}}\right) \cdot \left(U \cdot {\ell}^{2}\right)\right)\right)} \]
      times-frac [=>] 87.5	\[ \sqrt{-2 \cdot \left(n \cdot \left(\left(\frac{2}{Om} - \color{blue}{\frac{n}{Om} \cdot \frac{U*}{Om}}\right) \cdot \left(U \cdot {\ell}^{2}\right)\right)\right)} \]
      *-commutative [<=] 87.5	\[ \sqrt{-2 \cdot \left(n \cdot \left(\left(\frac{2}{Om} - \frac{n}{Om} \cdot \frac{U*}{Om}\right) \cdot \color{blue}{\left({\ell}^{2} \cdot U\right)}\right)\right)} \]
      unpow2 [=>] 87.5	\[ \sqrt{-2 \cdot \left(n \cdot \left(\left(\frac{2}{Om} - \frac{n}{Om} \cdot \frac{U*}{Om}\right) \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot U\right)\right)\right)} \]
      associate-*l* [=>] 72.59	\[ \sqrt{-2 \cdot \left(n \cdot \left(\left(\frac{2}{Om} - \frac{n}{Om} \cdot \frac{U*}{Om}\right) \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot U\right)\right)}\right)\right)} \]

   7. Applied egg-rr 72.49

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

   8. Applied egg-rr 60.09

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

4. Final simplification 41.01

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 5 \cdot 10^{-161}:\\ \;\;\;\;\sqrt[3]{n} \cdot \sqrt{U \cdot \left(\sqrt[3]{n} \cdot \left(t + t\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 2 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + -2 \cdot \left(\left(n \cdot \left(U \cdot \ell\right)\right) \cdot \frac{2}{\frac{Om}{\ell}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2 - n \cdot \frac{U*}{Om}}{Om} \cdot \left(\left(U \cdot \ell\right) \cdot \left(\ell \cdot \left(n \cdot -2\right)\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	40.17%
Cost	32140

\[\begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+304}:\\ \;\;\;\;\sqrt{t_1}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + -2 \cdot \left(\left(n \cdot \left(U \cdot \ell\right)\right) \cdot \frac{2}{\frac{Om}{\ell}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2 - n \cdot \frac{U*}{Om}}{Om} \cdot \left(\left(U \cdot \ell\right) \cdot \left(\ell \cdot \left(n \cdot -2\right)\right)\right)}\\ \end{array} \]

Alternative 2
Error	52.87%
Cost	15388

\[\begin{array}{l} t_1 := -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := n \cdot \left(U \cdot t\right)\\ \mathbf{if}\;Om \leq -1 \cdot 10^{+115}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \mathsf{fma}\left(t, U, -2 \cdot \left(\ell \cdot \left(U \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;Om \leq -9 \cdot 10^{-136}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\frac{\ell \cdot \left(\ell \cdot U*\right)}{Om} \cdot \frac{n}{Om} + t_1\right)\right)}\\ \mathbf{elif}\;Om \leq -6.4 \cdot 10^{-138}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(\frac{\ell}{\frac{Om}{U \cdot \ell}} \cdot \left(-2 + \frac{n \cdot U*}{Om}\right)\right)\right)}\\ \mathbf{elif}\;Om \leq -1.8 \cdot 10^{-206}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, t_2, -4 \cdot \frac{\ell \cdot \left(U \cdot \ell\right)}{\frac{Om}{n}}\right)}\\ \mathbf{elif}\;Om \leq 5 \cdot 10^{-252}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left(n \cdot \ell\right)}{Om} \cdot \sqrt{U \cdot \left(U* - U\right)}\\ \mathbf{elif}\;Om \leq 9.5 \cdot 10^{-116}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(U \cdot U*\right)}}{\frac{Om}{n \cdot \ell}}\\ \mathbf{elif}\;Om \leq 5 \cdot 10^{+63}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(t_1 + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot t_2 + -2 \cdot \left(\left(n \cdot \left(U \cdot \ell\right)\right) \cdot \frac{2}{\frac{Om}{\ell}}\right)}\\ \end{array} \]

Alternative 3
Error	52.63%
Cost	14416

\[\begin{array}{l} t_1 := -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;Om \leq -6.8 \cdot 10^{+114}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \mathsf{fma}\left(t, U, -2 \cdot \left(\ell \cdot \left(U \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;Om \leq -9 \cdot 10^{-136}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\frac{\ell \cdot \left(\ell \cdot U*\right)}{Om} \cdot \frac{n}{Om} + t_1\right)\right)}\\ \mathbf{elif}\;Om \leq -4.25 \cdot 10^{-137}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(\frac{\ell}{\frac{Om}{U \cdot \ell}} \cdot \left(-2 + \frac{n \cdot U*}{Om}\right)\right)\right)}\\ \mathbf{elif}\;Om \leq -2.15 \cdot 10^{-206}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \frac{\ell \cdot \left(U \cdot \ell\right)}{\frac{Om}{n}}\right)}\\ \mathbf{elif}\;Om \leq 5.3 \cdot 10^{-252}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left(n \cdot \ell\right)}{Om} \cdot \sqrt{U \cdot \left(U* - U\right)}\\ \mathbf{elif}\;Om \leq 3.1 \cdot 10^{-115}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(U \cdot U*\right)}}{\frac{Om}{n \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(t_1 + \left(\frac{n}{Om} \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U* - U\right)\right)\right)\right)}\\ \end{array} \]

Alternative 4
Error	47.63%
Cost	14156

\[\begin{array}{l} \mathbf{if}\;U \leq -8 \cdot 10^{+70}:\\ \;\;\;\;{\left(\frac{-1}{U}\right)}^{-0.5} \cdot \sqrt{n \cdot \left(t \cdot -2\right)}\\ \mathbf{elif}\;U \leq -8 \cdot 10^{-163}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \left(\frac{n}{Om} \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{elif}\;U \leq 7 \cdot 10^{-306}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \mathsf{fma}\left(t, U, -2 \cdot \left(\ell \cdot \left(U \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;U \leq 4.8 \cdot 10^{-265}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(\left(\ell \cdot \left(U \cdot \ell\right)\right) \cdot \left(\frac{U*}{Om} \cdot \frac{n}{Om} + \frac{-2}{Om}\right)\right)\right)}\\ \mathbf{elif}\;U \leq 1.35 \cdot 10^{+148}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + -2 \cdot \left(\left(n \cdot \left(U \cdot \ell\right)\right) \cdot \frac{2}{\frac{Om}{\ell}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot t} \cdot \sqrt{2 \cdot U}\\ \end{array} \]

Alternative 5
Error	47.53%
Cost	13908

\[\begin{array}{l} t_1 := 2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\\ \mathbf{if}\;U \leq -8 \cdot 10^{+70}:\\ \;\;\;\;{\left(\frac{-1}{U}\right)}^{-0.5} \cdot \sqrt{n \cdot \left(t \cdot -2\right)}\\ \mathbf{elif}\;U \leq -5 \cdot 10^{-163}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \left(\frac{n}{Om} \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{elif}\;U \leq 7.2 \cdot 10^{-306}:\\ \;\;\;\;\sqrt{t_1 + 2 \cdot \left(n \cdot \left(-2 \cdot \left(\frac{\ell}{Om} \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \mathbf{elif}\;U \leq 5.8 \cdot 10^{-265}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(\left(\ell \cdot \left(U \cdot \ell\right)\right) \cdot \left(\frac{U*}{Om} \cdot \frac{n}{Om} + \frac{-2}{Om}\right)\right)\right)}\\ \mathbf{elif}\;U \leq 5 \cdot 10^{+149}:\\ \;\;\;\;\sqrt{t_1 + -2 \cdot \left(\left(n \cdot \left(U \cdot \ell\right)\right) \cdot \frac{2}{\frac{Om}{\ell}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot t} \cdot \sqrt{2 \cdot U}\\ \end{array} \]

Alternative 6
Error	52.48%
Cost	8784

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + -2 \cdot \left(\left(n \cdot \left(U \cdot \ell\right)\right) \cdot \frac{2}{\frac{Om}{\ell}}\right)}\\ t_2 := -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;Om \leq -1.2 \cdot 10^{+72}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;Om \leq -2.3 \cdot 10^{-134}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\frac{\ell \cdot \left(\ell \cdot U*\right)}{Om} \cdot \frac{n}{Om} + t_2\right)\right)}\\ \mathbf{elif}\;Om \leq -3.3 \cdot 10^{-166}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;Om \leq 3.2 \cdot 10^{-117}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(U \cdot U*\right)}}{\frac{Om}{n \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(t_2 + \left(\frac{n}{Om} \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U* - U\right)\right)\right)\right)}\\ \end{array} \]

Alternative 7
Error	52.24%
Cost	8392

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + -2 \cdot \left(\left(n \cdot \left(U \cdot \ell\right)\right) \cdot \frac{2}{\frac{Om}{\ell}}\right)}\\ \mathbf{if}\;Om \leq -9.5 \cdot 10^{+67}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;Om \leq -1 \cdot 10^{-135}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\frac{\ell \cdot \left(\ell \cdot U*\right)}{Om} \cdot \frac{n}{Om} + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{elif}\;Om \leq -3.5 \cdot 10^{-166} \lor \neg \left(Om \leq 4.5 \cdot 10^{-117}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(U \cdot U*\right)}}{\frac{Om}{n \cdot \ell}}\\ \end{array} \]

Alternative 8
Error	53.37%
Cost	8009

\[\begin{array}{l} \mathbf{if}\;Om \leq -4.5 \cdot 10^{-166} \lor \neg \left(Om \leq 1.05 \cdot 10^{-116}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \left(n \cdot \left(-2 \cdot \left(\frac{\ell}{Om} \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(U \cdot U*\right)}}{\frac{Om}{n \cdot \ell}}\\ \end{array} \]

Alternative 9
Error	52.22%
Cost	8009

\[\begin{array}{l} \mathbf{if}\;Om \leq -3.3 \cdot 10^{-166} \lor \neg \left(Om \leq 2.1 \cdot 10^{-116}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + -2 \cdot \left(\left(n \cdot \left(U \cdot \ell\right)\right) \cdot \frac{2}{\frac{Om}{\ell}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(U \cdot U*\right)}}{\frac{Om}{n \cdot \ell}}\\ \end{array} \]

Alternative 10
Error	52.75%
Cost	7881

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.96 \cdot 10^{+96} \lor \neg \left(\ell \leq 2 \cdot 10^{+186}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(\frac{\ell}{\frac{Om}{U \cdot \ell}} \cdot \left(-2 + \frac{n \cdot U*}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(-2 \cdot \left(U \cdot \left(\ell \cdot \left(\ell \cdot \frac{2}{Om}\right) - t\right)\right)\right)}\\ \end{array} \]

Alternative 11
Error	52.74%
Cost	7880

\[\begin{array}{l} \mathbf{if}\;\ell \leq -3.7 \cdot 10^{+95}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(\frac{\ell}{\frac{Om}{U \cdot \ell}} \cdot \left(-2 + \frac{n \cdot U*}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{+174}:\\ \;\;\;\;\sqrt{n \cdot \left(-2 \cdot \left(U \cdot \left(\ell \cdot \left(\ell \cdot \frac{2}{Om}\right) - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2 - n \cdot \frac{U*}{Om}}{Om} \cdot \left(\ell \cdot \left(U \cdot \left(\ell \cdot \left(n \cdot -2\right)\right)\right)\right)}\\ \end{array} \]

Alternative 12
Error	52.69%
Cost	7880

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.15 \cdot 10^{+99}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(\frac{\ell}{\frac{Om}{U \cdot \ell}} \cdot \left(-2 + \frac{n \cdot U*}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.22 \cdot 10^{+174}:\\ \;\;\;\;\sqrt{n \cdot \left(-2 \cdot \left(U \cdot \left(\ell \cdot \left(\ell \cdot \frac{2}{Om}\right) - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2 - n \cdot \frac{U*}{Om}}{Om} \cdot \left(\left(U \cdot \ell\right) \cdot \left(\ell \cdot \left(n \cdot -2\right)\right)\right)}\\ \end{array} \]

Alternative 13
Error	58.28%
Cost	7760

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.35 \cdot 10^{+149}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{n}{\frac{\frac{Om}{\ell}}{U \cdot \ell}}}\\ \mathbf{elif}\;\ell \leq -5.5 \cdot 10^{+89}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(\ell \cdot \frac{-n}{Om}\right)\\ \mathbf{elif}\;\ell \leq -1.8 \cdot 10^{+51}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\ell \cdot \left(\ell \cdot \frac{-2}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 3 \cdot 10^{+64}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(-2 \cdot \left(\frac{\ell}{Om} \cdot \left(U \cdot \ell\right)\right)\right)}\\ \end{array} \]

Alternative 14
Error	57.04%
Cost	7760

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.25 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{n}{\frac{\frac{Om}{\ell}}{U \cdot \ell}}}\\ \mathbf{elif}\;\ell \leq -4.2 \cdot 10^{+90}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(\ell \cdot \frac{-n}{Om}\right)\\ \mathbf{elif}\;\ell \leq -1660000000:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\frac{\ell}{Om} \cdot \left(2 \cdot \ell\right) - t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{+65}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(-2 \cdot \left(\frac{\ell}{Om} \cdot \left(U \cdot \ell\right)\right)\right)}\\ \end{array} \]

Alternative 15
Error	58.31%
Cost	7633

\[\begin{array}{l} t_1 := \sqrt{-4 \cdot \frac{n}{\frac{\frac{Om}{\ell}}{U \cdot \ell}}}\\ \mathbf{if}\;\ell \leq -9 \cdot 10^{+152}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -2.65 \cdot 10^{+90}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(\ell \cdot \frac{-n}{Om}\right)\\ \mathbf{elif}\;\ell \leq -6.8 \cdot 10^{+51} \lor \neg \left(\ell \leq 7.8 \cdot 10^{+64}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \end{array} \]

Alternative 16
Error	58.35%
Cost	7632

\[\begin{array}{l} t_1 := \sqrt{-4 \cdot \frac{n}{\frac{\frac{Om}{\ell}}{U \cdot \ell}}}\\ \mathbf{if}\;\ell \leq -9 \cdot 10^{+152}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -1.8 \cdot 10^{+86}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(\ell \cdot \frac{-n}{Om}\right)\\ \mathbf{elif}\;\ell \leq -6.8 \cdot 10^{+51}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\ell \cdot \left(\ell \cdot \frac{-2}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.9 \cdot 10^{+65}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 17
Error	55.13%
Cost	7625

\[\begin{array}{l} \mathbf{if}\;Om \leq -4.5 \cdot 10^{-166} \lor \neg \left(Om \leq 10^{-115}\right):\\ \;\;\;\;\sqrt{n \cdot \left(-2 \cdot \left(U \cdot \left(\ell \cdot \left(\ell \cdot \frac{2}{Om}\right) - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(U \cdot U*\right)}}{\frac{Om}{n \cdot \ell}}\\ \end{array} \]

Alternative 18
Error	55.09%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;Om \leq -3.6 \cdot 10^{-166}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{elif}\;Om \leq 3.2 \cdot 10^{-117}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(U \cdot U*\right)}}{\frac{Om}{n \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(-2 \cdot \left(U \cdot \left(\ell \cdot \left(\ell \cdot \frac{2}{Om}\right) - t\right)\right)\right)}\\ \end{array} \]

Alternative 19
Error	57.61%
Cost	7369

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.4 \cdot 10^{+51} \lor \neg \left(\ell \leq 5.8 \cdot 10^{+65}\right):\\ \;\;\;\;\sqrt{-4 \cdot \frac{n}{\frac{\frac{Om}{\ell}}{U \cdot \ell}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \end{array} \]

Alternative 20
Error	63.28%
Cost	6848

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

Alternative 21
Error	63.1%
Cost	6848

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

Alternative 22
Error	63.11%
Cost	6848

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

Reproduce

herbie shell --seed 2023088 
(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*))))))