Toniolo and Linder, Equation (13)

Average Accuracy: 49.0% → 66.2%

Time: 37.4s

Precision: binary64

Cost: 32268

Math

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

↓

\[\begin{array}{l} t_1 := \left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(U* - U\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + U* \cdot \frac{n \cdot \ell}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+297}:\\ \;\;\;\;\sqrt{t_1}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell}{\frac{Om}{\ell}} \cdot \left(-2 + \frac{U*}{\frac{Om}{n}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot 2\right) \cdot \left(\ell \cdot \left(-2 + \left(U* - U\right) \cdot \frac{n}{Om}\right)\right)\right) \cdot \left(\left(\ell \cdot U\right) \cdot \frac{1}{Om}\right)}\\ \end{array} \]

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
         (*
          (* U (* n 2.0))
          (+
           (- t (* 2.0 (/ (* l l) Om)))
           (* (- U* U) (* n (pow (/ l Om) 2.0)))))))
   (if (<= t_1 0.0)
     (sqrt
      (*
       (* n 2.0)
       (* U (+ t (/ (* l (+ (* l -2.0) (* U* (/ (* n l) Om)))) Om)))))
     (if (<= t_1 5e+297)
       (sqrt t_1)
       (if (<= t_1 INFINITY)
         (sqrt
          (*
           (* n 2.0)
           (* U (+ t (* (/ l (/ Om l)) (+ -2.0 (/ U* (/ Om n))))))))
         (sqrt
          (*
           (* (* n 2.0) (* l (+ -2.0 (* (- U* U) (/ n Om)))))
           (* (* l U) (/ 1.0 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 = (U * (n * 2.0)) * ((t - (2.0 * ((l * l) / Om))) + ((U_42_ - U) * (n * pow((l / Om), 2.0))));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = sqrt(((n * 2.0) * (U * (t + ((l * ((l * -2.0) + (U_42_ * ((n * l) / Om)))) / Om)))));
	} else if (t_1 <= 5e+297) {
		tmp = sqrt(t_1);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = sqrt(((n * 2.0) * (U * (t + ((l / (Om / l)) * (-2.0 + (U_42_ / (Om / n))))))));
	} else {
		tmp = sqrt((((n * 2.0) * (l * (-2.0 + ((U_42_ - U) * (n / Om))))) * ((l * U) * (1.0 / 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 = (U * (n * 2.0)) * ((t - (2.0 * ((l * l) / Om))) + ((U_42_ - U) * (n * Math.pow((l / Om), 2.0))));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = Math.sqrt(((n * 2.0) * (U * (t + ((l * ((l * -2.0) + (U_42_ * ((n * l) / Om)))) / Om)))));
	} else if (t_1 <= 5e+297) {
		tmp = Math.sqrt(t_1);
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt(((n * 2.0) * (U * (t + ((l / (Om / l)) * (-2.0 + (U_42_ / (Om / n))))))));
	} else {
		tmp = Math.sqrt((((n * 2.0) * (l * (-2.0 + ((U_42_ - U) * (n / Om))))) * ((l * U) * (1.0 / Om))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, 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_)))))

↓

def code(n, U, t, l, Om, U_42_):
	t_1 = (U * (n * 2.0)) * ((t - (2.0 * ((l * l) / Om))) + ((U_42_ - U) * (n * math.pow((l / Om), 2.0))))
	tmp = 0
	if t_1 <= 0.0:
		tmp = math.sqrt(((n * 2.0) * (U * (t + ((l * ((l * -2.0) + (U_42_ * ((n * l) / Om)))) / Om)))))
	elif t_1 <= 5e+297:
		tmp = math.sqrt(t_1)
	elif t_1 <= math.inf:
		tmp = math.sqrt(((n * 2.0) * (U * (t + ((l / (Om / l)) * (-2.0 + (U_42_ / (Om / n))))))))
	else:
		tmp = math.sqrt((((n * 2.0) * (l * (-2.0 + ((U_42_ - U) * (n / Om))))) * ((l * U) * (1.0 / 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(U * Float64(n * 2.0)) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + Float64(Float64(U_42_ - U) * Float64(n * (Float64(l / Om) ^ 2.0)))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t + Float64(Float64(l * Float64(Float64(l * -2.0) + Float64(U_42_ * Float64(Float64(n * l) / Om)))) / Om)))));
	elseif (t_1 <= 5e+297)
		tmp = sqrt(t_1);
	elseif (t_1 <= Inf)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t + Float64(Float64(l / Float64(Om / l)) * Float64(-2.0 + Float64(U_42_ / Float64(Om / n))))))));
	else
		tmp = sqrt(Float64(Float64(Float64(n * 2.0) * Float64(l * Float64(-2.0 + Float64(Float64(U_42_ - U) * Float64(n / Om))))) * Float64(Float64(l * U) * Float64(1.0 / Om))));
	end
	return tmp
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * ((l / Om) ^ 2.0)) * (U - U_42_)))));
end

↓

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (U * (n * 2.0)) * ((t - (2.0 * ((l * l) / Om))) + ((U_42_ - U) * (n * ((l / Om) ^ 2.0))));
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = sqrt(((n * 2.0) * (U * (t + ((l * ((l * -2.0) + (U_42_ * ((n * l) / Om)))) / Om)))));
	elseif (t_1 <= 5e+297)
		tmp = sqrt(t_1);
	elseif (t_1 <= Inf)
		tmp = sqrt(((n * 2.0) * (U * (t + ((l / (Om / l)) * (-2.0 + (U_42_ / (Om / n))))))));
	else
		tmp = sqrt((((n * 2.0) * (l * (-2.0 + ((U_42_ - U) * (n / Om))))) * ((l * U) * (1.0 / Om))));
	end
	tmp_2 = 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[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(U$42$ - U), $MachinePrecision] * N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t + N[(N[(l * N[(N[(l * -2.0), $MachinePrecision] + N[(U$42$ * N[(N[(n * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 5e+297], N[Sqrt[t$95$1], $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t + N[(N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision] * N[(-2.0 + N[(U$42$ / N[(Om / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(n * 2.0), $MachinePrecision] * N[(l * N[(-2.0 + N[(N[(U$42$ - U), $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * U), $MachinePrecision] * N[(1.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 16.3%

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

   2. Simplified 54.3%

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

      [Start] 16.3	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 48.3	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      sub-neg [=>] 48.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]
      associate--l+ [=>] 48.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      *-commutative [=>] 48.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      distribute-rgt-neg-in [=>] 48.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l/ [<=] 48.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 48.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [<=] 48.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [=>] 48.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 51.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]
      unpow2 [=>] 51.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]
      associate-*l* [=>] 54.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in U around 0 54.2%

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

   4. Applied egg-rr 47.7%

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

      [Start] 54.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right) \cdot U\right)} \]
      expm1-log1p-u [=>] 50.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)\right)}\right)}{Om}\right) \cdot U\right)} \]
      expm1-udef [=>] 48.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)} - 1\right)}\right)}{Om}\right) \cdot U\right)} \]
      associate-*r* [=>] 48.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\left(n \cdot \ell\right) \cdot U*}}{Om}\right)} - 1\right)\right)}{Om}\right) \cdot U\right)} \]
      associate-/l* [=>] 47.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{n \cdot \ell}{\frac{Om}{U*}}}\right)} - 1\right)\right)}{Om}\right) \cdot U\right)} \]

   5. Simplified 54.3%

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

      [Start] 47.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \left(e^{\mathsf{log1p}\left(\frac{n \cdot \ell}{\frac{Om}{U*}}\right)} - 1\right)\right)}{Om}\right) \cdot U\right)} \]
      expm1-def [=>] 50.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{n \cdot \ell}{\frac{Om}{U*}}\right)\right)}\right)}{Om}\right) \cdot U\right)} \]
      expm1-log1p [=>] 50.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \color{blue}{\frac{n \cdot \ell}{\frac{Om}{U*}}}\right)}{Om}\right) \cdot U\right)} \]
      associate-/r/ [=>] 54.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \color{blue}{\frac{n \cdot \ell}{Om} \cdot U*}\right)}{Om}\right) \cdot U\right)} \]

   if 0.0 < (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))) < 4.9999999999999998e297

   1. Initial program 98.9%

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

   if 4.9999999999999998e297 < (*.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 37.2%

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

   2. Simplified 42.6%

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

      [Start] 37.2	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 42.1	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      sub-neg [=>] 42.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]
      associate--l+ [=>] 42.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      *-commutative [=>] 42.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      distribute-rgt-neg-in [=>] 42.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l/ [<=] 47.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 47.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [<=] 47.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [=>] 47.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 41.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]
      unpow2 [=>] 41.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]
      associate-*l* [=>] 42.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in U around 0 45.7%

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

   4. Taylor expanded in l around 0 40.2%

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

   5. Simplified 50.5%

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

      [Start] 40.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \frac{{\ell}^{2} \cdot \left(\frac{n \cdot U*}{Om} - 2\right)}{Om}\right) \cdot U\right)} \]
      associate-/l* [=>] 38.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \color{blue}{\frac{{\ell}^{2}}{\frac{Om}{\frac{n \cdot U*}{Om} - 2}}}\right) \cdot U\right)} \]
      associate-/r/ [=>] 38.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \color{blue}{\frac{{\ell}^{2}}{Om} \cdot \left(\frac{n \cdot U*}{Om} - 2\right)}\right) \cdot U\right)} \]
      unpow2 [=>] 38.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \frac{\color{blue}{\ell \cdot \ell}}{Om} \cdot \left(\frac{n \cdot U*}{Om} - 2\right)\right) \cdot U\right)} \]
      associate-/l* [=>] 46.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} \cdot \left(\frac{n \cdot U*}{Om} - 2\right)\right) \cdot U\right)} \]
      sub-neg [=>] 46.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \frac{\ell}{\frac{Om}{\ell}} \cdot \color{blue}{\left(\frac{n \cdot U*}{Om} + \left(-2\right)\right)}\right) \cdot U\right)} \]
      *-commutative [=>] 46.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \frac{\ell}{\frac{Om}{\ell}} \cdot \left(\frac{\color{blue}{U* \cdot n}}{Om} + \left(-2\right)\right)\right) \cdot U\right)} \]
      associate-/l* [=>] 50.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \frac{\ell}{\frac{Om}{\ell}} \cdot \left(\color{blue}{\frac{U*}{\frac{Om}{n}}} + \left(-2\right)\right)\right) \cdot U\right)} \]
      metadata-eval [=>] 50.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \frac{\ell}{\frac{Om}{\ell}} \cdot \left(\frac{U*}{\frac{Om}{n}} + \color{blue}{-2}\right)\right) \cdot U\right)} \]

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

   1. Initial program 0.0%

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

   2. Simplified 39.7%

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

      [Start] 0.0	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 3.8	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      sub-neg [=>] 3.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]
      associate--l+ [=>] 3.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      *-commutative [=>] 3.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      distribute-rgt-neg-in [=>] 3.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l/ [<=] 11.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 11.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [<=] 11.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [=>] 11.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 11.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]
      unpow2 [=>] 11.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]
      associate-*l* [=>] 16.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in t around 0 49.9%

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

   4. Simplified 62.5%

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

      [Start] 49.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \frac{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)}{Om}} \]
      associate-/l* [=>] 52.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\frac{\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell}{\frac{Om}{\ell \cdot U}}}} \]
      +-commutative [=>] 52.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \frac{\color{blue}{-2 \cdot \ell + \frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om}}}{\frac{Om}{\ell \cdot U}}} \]
      *-commutative [=>] 52.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \frac{\color{blue}{\ell \cdot -2} + \frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om}}{\frac{Om}{\ell \cdot U}}} \]
      associate-*r* [=>] 62.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \frac{\ell \cdot -2 + \frac{\color{blue}{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}}{Om}}{\frac{Om}{\ell \cdot U}}} \]

   5. Applied egg-rr 54.7%

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

      [Start] 62.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \frac{\ell \cdot -2 + \frac{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}{Om}}{\frac{Om}{\ell \cdot U}}} \]
      associate-*r/ [=>] 75.0	\[ \sqrt{\color{blue}{\frac{\left(2 \cdot n\right) \cdot \left(\ell \cdot -2 + \frac{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}{Om}\right)}{\frac{Om}{\ell \cdot U}}}} \]
      clear-num [=>] 75.1	\[ \sqrt{\color{blue}{\frac{1}{\frac{\frac{Om}{\ell \cdot U}}{\left(2 \cdot n\right) \cdot \left(\ell \cdot -2 + \frac{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}{Om}\right)}}}} \]
      fma-def [=>] 75.1	\[ \sqrt{\frac{1}{\frac{\frac{Om}{\ell \cdot U}}{\left(2 \cdot n\right) \cdot \color{blue}{\mathsf{fma}\left(\ell, -2, \frac{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}{Om}\right)}}}} \]
      associate-/l* [=>] 72.6	\[ \sqrt{\frac{1}{\frac{\frac{Om}{\ell \cdot U}}{\left(2 \cdot n\right) \cdot \mathsf{fma}\left(\ell, -2, \color{blue}{\frac{n \cdot \ell}{\frac{Om}{U* - U}}}\right)}}} \]
      associate-/l* [=>] 54.7	\[ \sqrt{\frac{1}{\frac{\frac{Om}{\ell \cdot U}}{\left(2 \cdot n\right) \cdot \mathsf{fma}\left(\ell, -2, \color{blue}{\frac{n}{\frac{\frac{Om}{U* - U}}{\ell}}}\right)}}} \]

   6. Simplified 75.1%

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

      [Start] 54.7	\[ \sqrt{\frac{1}{\frac{\frac{Om}{\ell \cdot U}}{\left(2 \cdot n\right) \cdot \mathsf{fma}\left(\ell, -2, \frac{n}{\frac{\frac{Om}{U* - U}}{\ell}}\right)}}} \]
      associate-/r/ [=>] 54.8	\[ \sqrt{\color{blue}{\frac{1}{\frac{Om}{\ell \cdot U}} \cdot \left(\left(2 \cdot n\right) \cdot \mathsf{fma}\left(\ell, -2, \frac{n}{\frac{\frac{Om}{U* - U}}{\ell}}\right)\right)}} \]
      *-commutative [=>] 54.8	\[ \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot \mathsf{fma}\left(\ell, -2, \frac{n}{\frac{\frac{Om}{U* - U}}{\ell}}\right)\right) \cdot \frac{1}{\frac{Om}{\ell \cdot U}}}} \]
      *-commutative [=>] 54.8	\[ \sqrt{\color{blue}{\left(\mathsf{fma}\left(\ell, -2, \frac{n}{\frac{\frac{Om}{U* - U}}{\ell}}\right) \cdot \left(2 \cdot n\right)\right)} \cdot \frac{1}{\frac{Om}{\ell \cdot U}}} \]
      fma-udef [=>] 54.8	\[ \sqrt{\left(\color{blue}{\left(\ell \cdot -2 + \frac{n}{\frac{\frac{Om}{U* - U}}{\ell}}\right)} \cdot \left(2 \cdot n\right)\right) \cdot \frac{1}{\frac{Om}{\ell \cdot U}}} \]
      *-commutative [<=] 54.8	\[ \sqrt{\left(\left(\color{blue}{-2 \cdot \ell} + \frac{n}{\frac{\frac{Om}{U* - U}}{\ell}}\right) \cdot \left(2 \cdot n\right)\right) \cdot \frac{1}{\frac{Om}{\ell \cdot U}}} \]
      associate-/r/ [=>] 72.6	\[ \sqrt{\left(\left(-2 \cdot \ell + \color{blue}{\frac{n}{\frac{Om}{U* - U}} \cdot \ell}\right) \cdot \left(2 \cdot n\right)\right) \cdot \frac{1}{\frac{Om}{\ell \cdot U}}} \]
      associate-/l* [<=] 75.1	\[ \sqrt{\left(\left(-2 \cdot \ell + \color{blue}{\frac{n \cdot \left(U* - U\right)}{Om}} \cdot \ell\right) \cdot \left(2 \cdot n\right)\right) \cdot \frac{1}{\frac{Om}{\ell \cdot U}}} \]
      distribute-rgt-out [=>] 75.1	\[ \sqrt{\left(\color{blue}{\left(\ell \cdot \left(-2 + \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)} \cdot \left(2 \cdot n\right)\right) \cdot \frac{1}{\frac{Om}{\ell \cdot U}}} \]
      associate-/l* [=>] 72.6	\[ \sqrt{\left(\left(\ell \cdot \left(-2 + \color{blue}{\frac{n}{\frac{Om}{U* - U}}}\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot \frac{1}{\frac{Om}{\ell \cdot U}}} \]
      associate-/r/ [=>] 75.1	\[ \sqrt{\left(\left(\ell \cdot \left(-2 + \color{blue}{\frac{n}{Om} \cdot \left(U* - U\right)}\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot \frac{1}{\frac{Om}{\ell \cdot U}}} \]
      associate-/r/ [=>] 75.1	\[ \sqrt{\left(\left(\ell \cdot \left(-2 + \frac{n}{Om} \cdot \left(U* - U\right)\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot \color{blue}{\left(\frac{1}{Om} \cdot \left(\ell \cdot U\right)\right)}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(U* - U\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right) \leq 0:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + U* \cdot \frac{n \cdot \ell}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(U* - U\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right) \leq 5 \cdot 10^{+297}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(U* - U\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(U* - U\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell}{\frac{Om}{\ell}} \cdot \left(-2 + \frac{U*}{\frac{Om}{n}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot 2\right) \cdot \left(\ell \cdot \left(-2 + \left(U* - U\right) \cdot \frac{n}{Om}\right)\right)\right) \cdot \left(\left(\ell \cdot U\right) \cdot \frac{1}{Om}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	64.3%
Cost	27208

\[\begin{array}{l} t_1 := \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right), t\right)\\ \mathbf{if}\;n \leq -820000000000:\\ \;\;\;\;{\left({\left(t_1 \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)}^{0.25}\right)}^{2}\\ \mathbf{elif}\;n \leq 3.2 \cdot 10^{-225}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om}}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot t_1} \cdot \sqrt{n \cdot 2}\\ \end{array} \]

Alternative 2
Accuracy	57.1%
Cost	8272

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(\left(\ell \cdot \left(-2 + \left(U* - U\right) \cdot \frac{n}{Om}\right)\right) \cdot \left(n \cdot \frac{\ell \cdot U}{Om}\right)\right)}\\ \mathbf{if}\;\ell \leq -2.9 \cdot 10^{+93}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -6.5 \cdot 10^{+32}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq -26000000:\\ \;\;\;\;\sqrt{\frac{n \cdot -2}{\frac{\frac{\frac{Om}{U}}{2 - \frac{U*}{\frac{Om}{n}}}}{\ell \cdot \ell}}}\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+53}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Accuracy	58.7%
Cost	8272

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(\left(\ell \cdot \left(-2 + \left(U* - U\right) \cdot \frac{n}{Om}\right)\right) \cdot \left(n \cdot \frac{\ell \cdot U}{Om}\right)\right)}\\ t_2 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell}{\frac{Om}{\ell}} \cdot \left(-2 + \frac{U*}{\frac{Om}{n}}\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+83}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -7 \cdot 10^{-225}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 5.2 \cdot 10^{-133}:\\ \;\;\;\;{\left(U \cdot \left(n \cdot \left(t \cdot 2\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{+82}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Accuracy	61.0%
Cost	8137

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.6 \cdot 10^{+96} \lor \neg \left(\ell \leq 1.32 \cdot 10^{+82}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(\left(\ell \cdot \left(-2 + \left(U* - U\right) \cdot \frac{n}{Om}\right)\right) \cdot \left(n \cdot \frac{\ell \cdot U}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + U* \cdot \frac{n \cdot \ell}{Om}\right)}{Om}\right)\right)}\\ \end{array} \]

Alternative 5
Accuracy	61.2%
Cost	8137

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.32 \cdot 10^{+97} \lor \neg \left(\ell \leq 2.45 \cdot 10^{+79}\right):\\ \;\;\;\;\sqrt{\left(\left(n \cdot 2\right) \cdot \left(\ell \cdot \left(-2 + \left(U* - U\right) \cdot \frac{n}{Om}\right)\right)\right) \cdot \left(\left(\ell \cdot U\right) \cdot \frac{1}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + U* \cdot \frac{n \cdot \ell}{Om}\right)}{Om}\right)\right)}\\ \end{array} \]

Alternative 6
Accuracy	49.3%
Cost	8016

\[\begin{array}{l} t_1 := \ell \cdot \left(\ell \cdot U\right)\\ t_2 := \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \ell \cdot \left(\frac{\ell}{Om} \cdot -2\right)\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -180000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 1.46 \cdot 10^{-58}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{+162}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 3.2 \cdot 10^{+273}:\\ \;\;\;\;\sqrt{\frac{\left(n \cdot -4\right) \cdot t_1}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(\frac{n}{Om \cdot Om} \cdot \left(U* \cdot t_1\right)\right)}\\ \end{array} \]

Alternative 7
Accuracy	49.9%
Cost	8012

\[\begin{array}{l} \mathbf{if}\;\ell \leq -180000000:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \ell \cdot \left(\frac{\ell}{Om} \cdot -2\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{+36}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{+138}:\\ \;\;\;\;\sqrt{\frac{n \cdot -2}{\frac{\frac{\frac{Om}{U}}{2 - \frac{U*}{\frac{Om}{n}}}}{\ell \cdot \ell}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(n \cdot -4\right) \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)}{Om}}\\ \end{array} \]

Alternative 8
Accuracy	48.4%
Cost	7881

\[\begin{array}{l} \mathbf{if}\;Om \leq -3.3 \cdot 10^{-202} \lor \neg \left(Om \leq 1.2 \cdot 10^{-137}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \ell \cdot \left(\frac{\ell}{Om} \cdot -2\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(\left(\ell \cdot \left(\ell \cdot \frac{n}{Om}\right)\right) \cdot \left(\left(U - U*\right) \cdot \frac{U}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 9
Accuracy	47.4%
Cost	7817

\[\begin{array}{l} \mathbf{if}\;Om \leq -3.6 \cdot 10^{-221} \lor \neg \left(Om \leq 7.3 \cdot 10^{-270}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \ell \cdot \left(\frac{\ell}{Om} \cdot -2\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{n \cdot U}{\frac{-Om}{\ell \cdot \left(n \cdot \ell\right)} \cdot \frac{Om}{U*}}}\\ \end{array} \]

Alternative 10
Accuracy	47.4%
Cost	7753

\[\begin{array}{l} \mathbf{if}\;Om \leq -8.6 \cdot 10^{-221} \lor \neg \left(Om \leq 7.6 \cdot 10^{-218}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \ell \cdot \left(\frac{\ell}{Om} \cdot -2\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\frac{\ell \cdot \left(\ell \cdot \left(n \cdot n\right)\right)}{Om} \cdot \frac{U* \cdot U}{Om}\right)}\\ \end{array} \]

Alternative 11
Accuracy	45.9%
Cost	7625

\[\begin{array}{l} \mathbf{if}\;\ell \leq 1.5 \cdot 10^{+164} \lor \neg \left(\ell \leq 5 \cdot 10^{+272}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \ell \cdot \left(\frac{\ell}{Om} \cdot -2\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(n \cdot -4\right) \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)}{Om}}\\ \end{array} \]

Alternative 12
Accuracy	38.8%
Cost	7369

\[\begin{array}{l} \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+64} \lor \neg \left(\ell \leq 2 \cdot 10^{+35}\right):\\ \;\;\;\;\sqrt{-4 \cdot \frac{n}{\frac{Om}{U \cdot \left(\ell \cdot \ell\right)}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 13
Accuracy	41.4%
Cost	7369

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.8 \cdot 10^{+67} \lor \neg \left(\ell \leq 2.2 \cdot 10^{+36}\right):\\ \;\;\;\;\sqrt{\frac{\left(n \cdot -4\right) \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 14
Accuracy	36.4%
Cost	7177

\[\begin{array}{l} \mathbf{if}\;\ell \leq -3.5 \cdot 10^{+101} \lor \neg \left(\ell \leq -5 \cdot 10^{-223}\right):\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]

Alternative 15
Accuracy	35.6%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-58}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;n \leq 2.5 \cdot 10^{-208}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(t + t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]

Alternative 16
Accuracy	35.8%
Cost	6912

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

Alternative 17
Accuracy	34.3%
Cost	6848

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

Reproduce

herbie shell --seed 2023160 
(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*))))))