Toniolo and Linder, Equation (13)

Average Error: 54.21% → 42.05%

Time: 36.9s

Precision: binary64

Cost: 38860

Math

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

↓

\[\begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{-94}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;t_1 \leq 10^{-311}:\\ \;\;\;\;\sqrt[3]{n} \cdot \sqrt{\sqrt[3]{n} \cdot \left(t \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{elif}\;t_1 \leq 10^{+306}:\\ \;\;\;\;\sqrt{t_1}\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{2 \cdot \left(U \cdot \left(U* - U\right)\right)} \cdot \frac{n \cdot \ell}{Om}\right|\\ \end{array} \]

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
         (*
          (* (* 2.0 n) U)
          (+
           (+ t (* (/ (* l l) Om) -2.0))
           (* (* n (pow (/ l Om) 2.0)) (- U* U))))))
   (if (<= t_1 -2e-94)
     (sqrt
      (* 2.0 (* n (* U (* (* l l) (+ (* (/ n Om) (/ U* Om)) (/ -2.0 Om)))))))
     (if (<= t_1 1e-311)
       (* (cbrt n) (sqrt (* (cbrt n) (* t (* 2.0 U)))))
       (if (<= t_1 1e+306)
         (sqrt t_1)
         (fabs (* (sqrt (* 2.0 (* U (- U* U)))) (/ (* n l) Om))))))))

C

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

↓

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = ((2.0 * n) * U) * ((t + (((l * l) / Om) * -2.0)) + ((n * pow((l / Om), 2.0)) * (U_42_ - U)));
	double tmp;
	if (t_1 <= -2e-94) {
		tmp = sqrt((2.0 * (n * (U * ((l * l) * (((n / Om) * (U_42_ / Om)) + (-2.0 / Om)))))));
	} else if (t_1 <= 1e-311) {
		tmp = cbrt(n) * sqrt((cbrt(n) * (t * (2.0 * U))));
	} else if (t_1 <= 1e+306) {
		tmp = sqrt(t_1);
	} else {
		tmp = fabs((sqrt((2.0 * (U * (U_42_ - U)))) * ((n * l) / Om)));
	}
	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 = ((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 <= -2e-94) {
		tmp = Math.sqrt((2.0 * (n * (U * ((l * l) * (((n / Om) * (U_42_ / Om)) + (-2.0 / Om)))))));
	} else if (t_1 <= 1e-311) {
		tmp = Math.cbrt(n) * Math.sqrt((Math.cbrt(n) * (t * (2.0 * U))));
	} else if (t_1 <= 1e+306) {
		tmp = Math.sqrt(t_1);
	} else {
		tmp = Math.abs((Math.sqrt((2.0 * (U * (U_42_ - U)))) * ((n * l) / Om)));
	}
	return tmp;
}

Julia

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

↓

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t + Float64(Float64(Float64(l * l) / Om) * -2.0)) + Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U_42_ - U))))
	tmp = 0.0
	if (t_1 <= -2e-94)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * Float64(Float64(l * l) * Float64(Float64(Float64(n / Om) * Float64(U_42_ / Om)) + Float64(-2.0 / Om)))))));
	elseif (t_1 <= 1e-311)
		tmp = Float64(cbrt(n) * sqrt(Float64(cbrt(n) * Float64(t * Float64(2.0 * U)))));
	elseif (t_1 <= 1e+306)
		tmp = sqrt(t_1);
	else
		tmp = abs(Float64(sqrt(Float64(2.0 * Float64(U * Float64(U_42_ - U)))) * Float64(Float64(n * l) / Om)));
	end
	return tmp
end

Wolfram

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

↓

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t + N[(N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-94], N[Sqrt[N[(2.0 * N[(n * N[(U * N[(N[(l * l), $MachinePrecision] * N[(N[(N[(n / Om), $MachinePrecision] * N[(U$42$ / Om), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 1e-311], N[(N[Power[n, 1/3], $MachinePrecision] * N[Sqrt[N[(N[Power[n, 1/3], $MachinePrecision] * N[(t * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+306], N[Sqrt[t$95$1], $MachinePrecision], N[Abs[N[(N[Sqrt[N[(2.0 * N[(U * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(n * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))) < -1.9999999999999999e-94

   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 66.9

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\mathsf{fma}\left({\left(\frac{\ell}{Om}\right)}^{2}, n \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right) \cdot \left(n \cdot U\right)\right)}} \]
      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 inf 87.24

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

   4. Simplified 82.95

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

      [Start] 87.24	\[ \sqrt{2 \cdot \left(n \cdot \left({\ell}^{2} \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)\right)\right)} \]
      *-commutative [=>] 87.24	\[ \sqrt{2 \cdot \left(n \cdot \left({\ell}^{2} \cdot \color{blue}{\left(\left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)}\right)\right)} \]
      associate-*r* [=>] 87.23	\[ \sqrt{2 \cdot \left(n \cdot \color{blue}{\left(\left({\ell}^{2} \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right) \cdot U\right)}\right)} \]
      unpow2 [=>] 87.23	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right) \cdot U\right)\right)} \]
      associate-/l* [=>] 82.95	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot \left(\color{blue}{\frac{n}{\frac{{Om}^{2}}{U* - U}}} - 2 \cdot \frac{1}{Om}\right)\right) \cdot U\right)\right)} \]
      unpow2 [=>] 82.95	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot \left(\frac{n}{\frac{\color{blue}{Om \cdot Om}}{U* - U}} - 2 \cdot \frac{1}{Om}\right)\right) \cdot U\right)\right)} \]
      associate-*r/ [=>] 82.95	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot \left(\frac{n}{\frac{Om \cdot Om}{U* - U}} - \color{blue}{\frac{2 \cdot 1}{Om}}\right)\right) \cdot U\right)\right)} \]
      metadata-eval [=>] 82.95	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot \left(\frac{n}{\frac{Om \cdot Om}{U* - U}} - \frac{\color{blue}{2}}{Om}\right)\right) \cdot U\right)\right)} \]

   5. Taylor expanded in U around 0 87.22

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

   6. Simplified 71.61

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

      [Start] 87.22	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(\left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot {\ell}^{2}\right) \cdot U\right)\right)} \]
      cancel-sign-sub-inv [=>] 87.22	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(\color{blue}{\left(\frac{n \cdot U*}{{Om}^{2}} + \left(-2\right) \cdot \frac{1}{Om}\right)} \cdot {\ell}^{2}\right) \cdot U\right)\right)} \]
      unpow2 [=>] 87.22	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(\left(\frac{n \cdot U*}{\color{blue}{Om \cdot Om}} + \left(-2\right) \cdot \frac{1}{Om}\right) \cdot {\ell}^{2}\right) \cdot U\right)\right)} \]
      times-frac [=>] 71.61	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(\left(\color{blue}{\frac{n}{Om} \cdot \frac{U*}{Om}} + \left(-2\right) \cdot \frac{1}{Om}\right) \cdot {\ell}^{2}\right) \cdot U\right)\right)} \]
      metadata-eval [=>] 71.61	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(\left(\frac{n}{Om} \cdot \frac{U*}{Om} + \color{blue}{-2} \cdot \frac{1}{Om}\right) \cdot {\ell}^{2}\right) \cdot U\right)\right)} \]
      associate-*r/ [=>] 71.61	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(\left(\frac{n}{Om} \cdot \frac{U*}{Om} + \color{blue}{\frac{-2 \cdot 1}{Om}}\right) \cdot {\ell}^{2}\right) \cdot U\right)\right)} \]
      metadata-eval [=>] 71.61	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(\left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{\color{blue}{-2}}{Om}\right) \cdot {\ell}^{2}\right) \cdot U\right)\right)} \]
      unpow2 [=>] 71.61	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(\left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right) \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot U\right)\right)} \]

   if -1.9999999999999999e-94 < (*.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*)))) < 9.99999999999948e-312

   1. Initial program 87.07

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

   2. Simplified 87.09

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

      [Start] 87.07	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 87.07	\[ \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 87.15	\[ \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      *-commutative [=>] 87.15	\[ \sqrt{2 \cdot \color{blue}{\left(\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot \left(n \cdot U\right)\right)}} \]

   3. Taylor expanded in l around 0 67.38

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

   4. Applied egg-rr 88.62

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

   5. Simplified 67.64

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

      [Start] 88.62	\[ \sqrt{2 \cdot \left(e^{\mathsf{log1p}\left(n \cdot \left(t \cdot U\right)\right)} + -1\right)} \]
      metadata-eval [<=] 88.62	\[ \sqrt{2 \cdot \left(e^{\mathsf{log1p}\left(n \cdot \left(t \cdot U\right)\right)} + \color{blue}{\left(-1\right)}\right)} \]
      sub-neg [<=] 88.62	\[ \sqrt{2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(n \cdot \left(t \cdot U\right)\right)} - 1\right)}} \]
      expm1-def [=>] 67.38	\[ \sqrt{2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(n \cdot \left(t \cdot U\right)\right)\right)}} \]
      expm1-log1p [=>] 67.38	\[ \sqrt{2 \cdot \color{blue}{\left(n \cdot \left(t \cdot U\right)\right)}} \]
      associate-*r* [=>] 67.64	\[ \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot t\right) \cdot U\right)}} \]
      *-commutative [=>] 67.64	\[ \sqrt{2 \cdot \left(\color{blue}{\left(t \cdot n\right)} \cdot U\right)} \]

   6. Taylor expanded in t around 0 67.38

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

   7. Simplified 67.28

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

      [Start] 67.38	\[ \sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)} \]
      *-commutative [=>] 67.38	\[ \sqrt{\color{blue}{\left(n \cdot \left(t \cdot U\right)\right) \cdot 2}} \]
      associate-*l* [=>] 67.28	\[ \sqrt{\color{blue}{n \cdot \left(\left(t \cdot U\right) \cdot 2\right)}} \]

   8. Applied egg-rr 66.19

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

   if 9.99999999999948e-312 < (*.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*)))) < 1.00000000000000002e306

   1. Initial program 2.25

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

   if 1.00000000000000002e306 < (*.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 99.61

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

   2. Simplified 89.38

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

      [Start] 99.61	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 99.61	\[ \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 99.58	\[ \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      *-commutative [=>] 99.58	\[ \sqrt{2 \cdot \color{blue}{\left(\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot \left(n \cdot U\right)\right)}} \]

   3. Taylor expanded in Om around 0 88.46

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

   4. Applied egg-rr 88.02

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

   5. Applied egg-rr 90.47

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

   6. Simplified 77.48

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

      [Start] 90.47	\[ \sqrt{{\left(\sqrt{\left(U \cdot \left(U* - U\right)\right) \cdot 2} \cdot \frac{n \cdot \ell}{Om}\right)}^{2}} \]
      unpow2 [=>] 90.47	\[ \sqrt{\color{blue}{\left(\sqrt{\left(U \cdot \left(U* - U\right)\right) \cdot 2} \cdot \frac{n \cdot \ell}{Om}\right) \cdot \left(\sqrt{\left(U \cdot \left(U* - U\right)\right) \cdot 2} \cdot \frac{n \cdot \ell}{Om}\right)}} \]
      rem-sqrt-square [=>] 77.48	\[ \color{blue}{\left|\sqrt{\left(U \cdot \left(U* - U\right)\right) \cdot 2} \cdot \frac{n \cdot \ell}{Om}\right|} \]
      *-commutative [=>] 77.48	\[ \left|\sqrt{\color{blue}{2 \cdot \left(U \cdot \left(U* - U\right)\right)}} \cdot \frac{n \cdot \ell}{Om}\right| \]

3. Recombined 4 regimes into one program.

4. Final simplification 42.05

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq -2 \cdot 10^{-94}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 10^{-311}:\\ \;\;\;\;\sqrt[3]{n} \cdot \sqrt{\sqrt[3]{n} \cdot \left(t \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 10^{+306}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{2 \cdot \left(U \cdot \left(U* - U\right)\right)} \cdot \frac{n \cdot \ell}{Om}\right|\\ \end{array} \]

Alternatives

Alternative 1
Error	42.32%
Cost	43528

\[\begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t_1 \leq 4 \cdot 10^{-156}:\\ \;\;\;\;\sqrt{U \cdot t} \cdot \sqrt{2 \cdot n}\\ \mathbf{elif}\;t_1 \leq 10^{+153}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{2 \cdot \left(U \cdot \left(U* - U\right)\right)} \cdot \frac{n \cdot \ell}{Om}\right|\\ \end{array} \]

Alternative 2
Error	49.04%
Cost	14556

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) - U \cdot \left(\left(n \cdot \ell\right) \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)\right)}\\ t_2 := \frac{\ell}{\frac{Om}{\ell}}\\ t_3 := \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot t_2 + n \cdot \left(t_2 \cdot \frac{U* - U}{Om}\right)\right)\right)\right)}\\ t_4 := \left|\sqrt{2 \cdot \left(U \cdot \left(U* - U\right)\right)} \cdot \left(n \cdot \frac{\ell}{Om}\right)\right|\\ \mathbf{if}\;t \leq -1.05 \cdot 10^{+98}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.1 \cdot 10^{+49}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -2 \cdot 10^{-89}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-195}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{-189}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{-133}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4.7 \cdot 10^{-113}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{-57} \lor \neg \left(t \leq 5 \cdot 10^{+189}\right):\\ \;\;\;\;\sqrt{2 \cdot t} \cdot \sqrt{n \cdot U}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	49.04%
Cost	14556

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := \sqrt{2 \cdot \left(U \cdot \left(U* - U\right)\right)}\\ t_3 := \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) - U \cdot \left(\left(n \cdot \ell\right) \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)\right)}\\ t_4 := \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot t_1 + n \cdot \left(t_1 \cdot \frac{U* - U}{Om}\right)\right)\right)\right)}\\ \mathbf{if}\;t \leq -1.5 \cdot 10^{+98}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -2.15 \cdot 10^{+48}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq -1.25 \cdot 10^{-88}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 8.8 \cdot 10^{-195}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-189}:\\ \;\;\;\;\left|t_2 \cdot \frac{n \cdot \ell}{Om}\right|\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-133}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-113}:\\ \;\;\;\;\left|t_2 \cdot \left(n \cdot \frac{\ell}{Om}\right)\right|\\ \mathbf{elif}\;t \leq 10^{-58} \lor \neg \left(t \leq 4.2 \cdot 10^{+189}\right):\\ \;\;\;\;\sqrt{2 \cdot t} \cdot \sqrt{n \cdot U}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 4
Error	50.04%
Cost	14556

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) - U \cdot \left(\left(n \cdot \ell\right) \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)\right)}\\ t_3 := \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot t_1 + n \cdot \left(t_1 \cdot \frac{U* - U}{Om}\right)\right)\right)\right)}\\ \mathbf{if}\;t \leq -1.3 \cdot 10^{+98}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -3.4 \cdot 10^{+48}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -9.5 \cdot 10^{-89}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.6 \cdot 10^{-283}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-193}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \left(\frac{n}{Om} \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{-136}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-114}:\\ \;\;\;\;\left|\sqrt{2 \cdot \left(U \cdot \left(U* - U\right)\right)} \cdot \left(n \cdot \frac{\ell}{Om}\right)\right|\\ \mathbf{elif}\;t \leq 6.4 \cdot 10^{-59} \lor \neg \left(t \leq 4.8 \cdot 10^{+189}\right):\\ \;\;\;\;\sqrt{2 \cdot t} \cdot \sqrt{n \cdot U}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	50.01%
Cost	13908

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot t_1 + n \cdot \left(t_1 \cdot \frac{U* - U}{Om}\right)\right)\right)\right)}\\ t_3 := U \cdot \left(n \cdot t\right)\\ t_4 := \sqrt{2 \cdot \left(t_3 - U \cdot \left(\left(n \cdot \ell\right) \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -1.3 \cdot 10^{+95}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;\ell \leq -7 \cdot 10^{-227}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 2.55 \cdot 10^{-300}:\\ \;\;\;\;\sqrt{2 \cdot t_3}\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{-250}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot t\right)} \cdot \sqrt{n}\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{-223}:\\ \;\;\;\;\sqrt{2 \cdot \left|t \cdot \left(n \cdot U\right)\right|}\\ \mathbf{elif}\;\ell \leq 0.000105:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{+264}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\ell \cdot \left(n \cdot \left(\left(U \cdot \ell\right) \cdot \left(\frac{2}{Om} - \frac{n}{\frac{-Om}{U} \cdot \left(Om + U* \cdot \frac{Om}{U}\right)}\right)\right)\right)\right)}\\ \end{array} \]

Alternative 6
Error	49.91%
Cost	8720

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) - U \cdot \left(\left(n \cdot \ell\right) \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)\right)}\\ t_2 := \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;\ell \leq -3.2 \cdot 10^{+58}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{-217}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot t_2 + n \cdot \left(t_2 \cdot \frac{U* - U}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.1 \cdot 10^{+269}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\ell \cdot \left(n \cdot \left(\left(U \cdot \ell\right) \cdot \left(\frac{2}{Om} - \frac{n}{\frac{-Om}{U} \cdot \left(Om + U* \cdot \frac{Om}{U}\right)}\right)\right)\right)\right)}\\ \end{array} \]

Alternative 7
Error	50.49%
Cost	8588

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) - U \cdot \left(\left(n \cdot \ell\right) \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -4.2 \cdot 10^{+60}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 0.004:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+263}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\ell \cdot \left(n \cdot \left(\left(U \cdot \ell\right) \cdot \left(\frac{2}{Om} - \frac{n}{\frac{-Om}{U} \cdot \left(Om + U* \cdot \frac{Om}{U}\right)}\right)\right)\right)\right)}\\ \end{array} \]

Alternative 8
Error	52.65%
Cost	7756

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(U \cdot \left(-2 \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \ell\right)\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -3.2 \cdot 10^{+234}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -2.15 \cdot 10^{+35}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\ell \cdot \left(2 \cdot \frac{\ell}{Om}\right) - t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{+99}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	52.65%
Cost	7756

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(U \cdot \left(-2 \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \ell\right)\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -3.2 \cdot 10^{+234}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -4.8 \cdot 10^{+34}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\ell \cdot \left(2 \cdot \frac{\ell}{Om}\right) - t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.08 \cdot 10^{+99}:\\ \;\;\;\;\sqrt{\left(-2 \cdot \left(n \cdot U\right)\right) \cdot \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - t\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	51.36%
Cost	7748

\[\begin{array}{l} \mathbf{if}\;U \leq 1.4 \cdot 10^{-22}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) - U \cdot \left(\left(n \cdot \ell\right) \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;U \leq 3.5 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(U* - U\right)\right)} \cdot \left(n \cdot \frac{\ell}{Om}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(-2 \cdot \left(n \cdot U\right)\right) \cdot \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - t\right)}\\ \end{array} \]

Alternative 11
Error	54.62%
Cost	7628

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(U \cdot \left(-2 \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \ell\right)\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -5 \cdot 10^{+234}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -2.6 \cdot 10^{-48}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\ell \cdot \left(2 \cdot \frac{\ell}{Om}\right) - t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.4 \cdot 10^{+24}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	60.05%
Cost	7497

\[\begin{array}{l} \mathbf{if}\;\ell \leq -5.2 \cdot 10^{+46} \lor \neg \left(\ell \leq 1.95 \cdot 10^{+25}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \left(\frac{n}{Om} \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 13
Error	56.45%
Cost	7497

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.7 \cdot 10^{+47} \lor \neg \left(\ell \leq 9.6 \cdot 10^{+25}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(-2 \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \ell\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 14
Error	55.09%
Cost	7496

\[\begin{array}{l} \mathbf{if}\;\ell \leq -6.5 \cdot 10^{-143}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \ell \cdot \frac{\ell \cdot -2}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{+26}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(-2 \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \ell\right)\right)\right)\right)}\\ \end{array} \]

Alternative 15
Error	62.65%
Cost	7369

\[\begin{array}{l} \mathbf{if}\;t \leq -2.3 \cdot 10^{-254} \lor \neg \left(t \leq 1.35 \cdot 10^{-210}\right):\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-4 \cdot \left(\frac{n}{Om} \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)\right)}\\ \end{array} \]

Alternative 16
Error	63.15%
Cost	7369

\[\begin{array}{l} \mathbf{if}\;\ell \leq -6.5 \cdot 10^{+46} \lor \neg \left(\ell \leq 3.8 \cdot 10^{+52}\right):\\ \;\;\;\;\sqrt{\frac{-4}{\frac{Om}{\left(\ell \cdot \ell\right) \cdot \left(n \cdot U\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 17
Error	62.06%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{+125} \lor \neg \left(t \leq 4.5 \cdot 10^{-196}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]

Alternative 18
Error	62.38%
Cost	6848

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

Alternative 19
Error	62.23%
Cost	6848

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

Reproduce

herbie shell --seed 2023103 
(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*))))))