Toniolo and Linder, Equation (13)

Average Error: 34.8 → 29.3

Time: 36.4s

Precision: binary64

Cost: 26948

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{\frac{Om}{\ell}} \cdot -2 + \left(n \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{U* - U}{Om}\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -4.6 \cdot 10^{+124}:\\ \;\;\;\;-{\left(\sqrt[3]{\sqrt{\frac{\frac{n}{Om} \cdot U* + -2}{Om} \cdot \left(n \cdot U\right)} \cdot \left(\ell \cdot \sqrt{2}\right)}\right)}^{3}\\ \mathbf{elif}\;\ell \leq -7 \cdot 10^{-195}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 4.5 \cdot 10^{-95}:\\ \;\;\;\;\sqrt{2 \cdot \left|U \cdot \left(n \cdot t\right)\right|}\\ \mathbf{elif}\;\ell \leq 1.85 \cdot 10^{+145}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right)\right)}\right)\\ \end{array} \]

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
          (*
           (* n 2.0)
           (*
            U
            (+
             t
             (+
              (* (/ l (/ Om l)) -2.0)
              (* (* n (* l (/ l Om))) (/ (- U* U) Om)))))))))
   (if (<= l -4.6e+124)
     (-
      (pow
       (cbrt
        (*
         (sqrt (* (/ (+ (* (/ n Om) U*) -2.0) Om) (* n U)))
         (* l (sqrt 2.0))))
       3.0))
     (if (<= l -7e-195)
       t_1
       (if (<= l 4.5e-95)
         (sqrt (* 2.0 (fabs (* U (* n t)))))
         (if (<= l 1.85e+145)
           t_1
           (*
            (sqrt 2.0)
            (*
             l
             (sqrt (* U (* n (+ (* (/ n Om) (/ U* Om)) (/ -2.0 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 = sqrt(((n * 2.0) * (U * (t + (((l / (Om / l)) * -2.0) + ((n * (l * (l / Om))) * ((U_42_ - U) / Om)))))));
	double tmp;
	if (l <= -4.6e+124) {
		tmp = -pow(cbrt((sqrt((((((n / Om) * U_42_) + -2.0) / Om) * (n * U))) * (l * sqrt(2.0)))), 3.0);
	} else if (l <= -7e-195) {
		tmp = t_1;
	} else if (l <= 4.5e-95) {
		tmp = sqrt((2.0 * fabs((U * (n * t)))));
	} else if (l <= 1.85e+145) {
		tmp = t_1;
	} else {
		tmp = sqrt(2.0) * (l * sqrt((U * (n * (((n / Om) * (U_42_ / Om)) + (-2.0 / 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 = Math.sqrt(((n * 2.0) * (U * (t + (((l / (Om / l)) * -2.0) + ((n * (l * (l / Om))) * ((U_42_ - U) / Om)))))));
	double tmp;
	if (l <= -4.6e+124) {
		tmp = -Math.pow(Math.cbrt((Math.sqrt((((((n / Om) * U_42_) + -2.0) / Om) * (n * U))) * (l * Math.sqrt(2.0)))), 3.0);
	} else if (l <= -7e-195) {
		tmp = t_1;
	} else if (l <= 4.5e-95) {
		tmp = Math.sqrt((2.0 * Math.abs((U * (n * t)))));
	} else if (l <= 1.85e+145) {
		tmp = t_1;
	} else {
		tmp = Math.sqrt(2.0) * (l * Math.sqrt((U * (n * (((n / Om) * (U_42_ / Om)) + (-2.0 / 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 = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t + Float64(Float64(Float64(l / Float64(Om / l)) * -2.0) + Float64(Float64(n * Float64(l * Float64(l / Om))) * Float64(Float64(U_42_ - U) / Om)))))))
	tmp = 0.0
	if (l <= -4.6e+124)
		tmp = Float64(-(cbrt(Float64(sqrt(Float64(Float64(Float64(Float64(Float64(n / Om) * U_42_) + -2.0) / Om) * Float64(n * U))) * Float64(l * sqrt(2.0)))) ^ 3.0));
	elseif (l <= -7e-195)
		tmp = t_1;
	elseif (l <= 4.5e-95)
		tmp = sqrt(Float64(2.0 * abs(Float64(U * Float64(n * t)))));
	elseif (l <= 1.85e+145)
		tmp = t_1;
	else
		tmp = Float64(sqrt(2.0) * Float64(l * sqrt(Float64(U * Float64(n * Float64(Float64(Float64(n / Om) * Float64(U_42_ / Om)) + Float64(-2.0 / 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[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t + N[(N[(N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision] + N[(N[(n * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -4.6e+124], (-N[Power[N[Power[N[(N[Sqrt[N[(N[(N[(N[(N[(n / Om), $MachinePrecision] * U$42$), $MachinePrecision] + -2.0), $MachinePrecision] / Om), $MachinePrecision] * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), If[LessEqual[l, -7e-195], t$95$1, If[LessEqual[l, 4.5e-95], N[Sqrt[N[(2.0 * N[Abs[N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 1.85e+145], t$95$1, N[(N[Sqrt[2.0], $MachinePrecision] * N[(l * N[Sqrt[N[(U * N[(n * N[(N[(N[(n / Om), $MachinePrecision] * N[(U$42$ / Om), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -4.59999999999999969e124

   1. Initial program 59.0

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

   2. Simplified 47.1

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

      [Start] 59.0	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 59.0	\[ \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 59.0	\[ \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      *-commutative [=>] 59.0	\[ \sqrt{2 \cdot \color{blue}{\left(\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot \left(n \cdot U\right)\right)}} \]

   3. Taylor expanded in l around -inf 33.5

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

   4. Taylor expanded in U around 0 33.9

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

   5. Simplified 33.9

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

      [Start] 33.9	\[ -1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(\left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)}\right) \]
      *-commutative [=>] 33.9	\[ -1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \color{blue}{\left(U \cdot \left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}}\right) \]
      unpow2 [=>] 33.9	\[ -1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{\color{blue}{Om \cdot Om}} - 2 \cdot \frac{1}{Om}\right)\right)}\right) \]
      associate-*r/ [=>] 33.9	\[ -1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om \cdot Om} - \color{blue}{\frac{2 \cdot 1}{Om}}\right)\right)}\right) \]
      metadata-eval [=>] 33.9	\[ -1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om \cdot Om} - \frac{\color{blue}{2}}{Om}\right)\right)}\right) \]

   6. Applied egg-rr 29.6

      \[\leadsto -1 \cdot \color{blue}{{\left(\sqrt[3]{\sqrt{\frac{\frac{n}{Om} \cdot U* - 2}{Om} \cdot \left(n \cdot U\right)} \cdot \left(\sqrt{2} \cdot \ell\right)}\right)}^{3}} \]

   if -4.59999999999999969e124 < l < -7.00000000000000028e-195 or 4.5e-95 < l < 1.84999999999999997e145

   1. Initial program 30.0

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

   2. Simplified 30.5

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

      [Start] 30.0	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 29.6	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 29.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      sub-neg [=>] 29.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(-\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [<=] 29.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      cancel-sign-sub [<=] 29.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      cancel-sign-sub [=>] 29.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-/l* [=>] 29.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 30.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in U around 0 34.6

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

   4. Simplified 29.9

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

      [Start] 34.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(\frac{{\ell}^{2} \cdot U}{{Om}^{2}} + -1 \cdot \frac{{\ell}^{2} \cdot U*}{{Om}^{2}}\right)\right)\right)\right)} \]
      mul-1-neg [=>] 34.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(\frac{{\ell}^{2} \cdot U}{{Om}^{2}} + \color{blue}{\left(-\frac{{\ell}^{2} \cdot U*}{{Om}^{2}}\right)}\right)\right)\right)\right)} \]
      unsub-neg [=>] 34.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \color{blue}{\left(\frac{{\ell}^{2} \cdot U}{{Om}^{2}} - \frac{{\ell}^{2} \cdot U*}{{Om}^{2}}\right)}\right)\right)\right)} \]
      unpow2 [=>] 34.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(\frac{{\ell}^{2} \cdot U}{\color{blue}{Om \cdot Om}} - \frac{{\ell}^{2} \cdot U*}{{Om}^{2}}\right)\right)\right)\right)} \]
      times-frac [=>] 33.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(\color{blue}{\frac{{\ell}^{2}}{Om} \cdot \frac{U}{Om}} - \frac{{\ell}^{2} \cdot U*}{{Om}^{2}}\right)\right)\right)\right)} \]
      unpow2 [=>] 33.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{Om} \cdot \frac{U}{Om} - \frac{{\ell}^{2} \cdot U*}{{Om}^{2}}\right)\right)\right)\right)} \]
      unpow2 [=>] 33.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(\frac{\ell \cdot \ell}{Om} \cdot \frac{U}{Om} - \frac{{\ell}^{2} \cdot U*}{\color{blue}{Om \cdot Om}}\right)\right)\right)\right)} \]
      times-frac [=>] 29.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(\frac{\ell \cdot \ell}{Om} \cdot \frac{U}{Om} - \color{blue}{\frac{{\ell}^{2}}{Om} \cdot \frac{U*}{Om}}\right)\right)\right)\right)} \]
      unpow2 [=>] 29.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(\frac{\ell \cdot \ell}{Om} \cdot \frac{U}{Om} - \frac{\color{blue}{\ell \cdot \ell}}{Om} \cdot \frac{U*}{Om}\right)\right)\right)\right)} \]

   5. Applied egg-rr 29.7

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

   6. Simplified 28.2

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

      [Start] 29.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + {\left(n \cdot \left(\frac{\ell}{\frac{Om}{\ell}} \cdot \frac{U - U*}{Om}\right)\right)}^{1}\right)\right)\right)} \]
      unpow1 [=>] 29.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left(\frac{\ell}{\frac{Om}{\ell}} \cdot \frac{U - U*}{Om}\right)}\right)\right)\right)} \]
      associate-*r* [=>] 28.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{\left(n \cdot \frac{\ell}{\frac{Om}{\ell}}\right) \cdot \frac{U - U*}{Om}}\right)\right)\right)} \]
      *-commutative [=>] 28.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{\frac{U - U*}{Om} \cdot \left(n \cdot \frac{\ell}{\frac{Om}{\ell}}\right)}\right)\right)\right)} \]
      associate-/r/ [=>] 28.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \frac{U - U*}{Om} \cdot \left(n \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)}\right)\right)\right)\right)} \]
      *-commutative [=>] 28.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \frac{U - U*}{Om} \cdot \left(n \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)}\right)\right)\right)\right)} \]

   if -7.00000000000000028e-195 < l < 4.5e-95

   1. Initial program 25.2

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

   2. Simplified 26.2

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

      [Start] 25.2	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 25.6	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 25.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      fma-def [=>] 25.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-*l* [=>] 26.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in t around inf 29.2

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

   4. Simplified 29.7

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

      [Start] 29.2	\[ \sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)} \]
      associate-*r* [=>] 29.7	\[ \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot t\right) \cdot U\right)}} \]
      *-commutative [=>] 29.7	\[ \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

   5. Applied egg-rr 29.6

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

   if 1.84999999999999997e145 < l

   1. Initial program 62.1

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

   2. Simplified 49.7

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

      [Start] 62.1	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 62.1	\[ \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 62.1	\[ \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      *-commutative [=>] 62.1	\[ \sqrt{2 \cdot \color{blue}{\left(\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot \left(n \cdot U\right)\right)}} \]

   3. Taylor expanded in l around inf 34.5

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

   4. Simplified 38.1

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

      [Start] 34.5	\[ \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)} \]
      associate-*l* [=>] 34.4	\[ \color{blue}{\sqrt{2} \cdot \left(\ell \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}\right)} \]
      *-commutative [=>] 34.4	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{n \cdot \color{blue}{\left(\left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)}}\right) \]
      associate-*r* [=>] 37.7	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{\color{blue}{\left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right) \cdot U}}\right) \]
      *-commutative [=>] 37.7	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{\color{blue}{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}}\right) \]
      cancel-sign-sub-inv [=>] 37.7	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{U \cdot \left(n \cdot \color{blue}{\left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} + \left(-2\right) \cdot \frac{1}{Om}\right)}\right)}\right) \]
      associate-/l* [=>] 39.4	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{U \cdot \left(n \cdot \left(\color{blue}{\frac{n}{\frac{{Om}^{2}}{U* - U}}} + \left(-2\right) \cdot \frac{1}{Om}\right)\right)}\right) \]
      associate-/r/ [=>] 38.1	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{U \cdot \left(n \cdot \left(\color{blue}{\frac{n}{{Om}^{2}} \cdot \left(U* - U\right)} + \left(-2\right) \cdot \frac{1}{Om}\right)\right)}\right) \]
      unpow2 [=>] 38.1	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{n}{\color{blue}{Om \cdot Om}} \cdot \left(U* - U\right) + \left(-2\right) \cdot \frac{1}{Om}\right)\right)}\right) \]
      metadata-eval [=>] 38.1	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) + \color{blue}{-2} \cdot \frac{1}{Om}\right)\right)}\right) \]
      associate-*r/ [=>] 38.1	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) + \color{blue}{\frac{-2 \cdot 1}{Om}}\right)\right)}\right) \]
      metadata-eval [=>] 38.1	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) + \frac{\color{blue}{-2}}{Om}\right)\right)}\right) \]

   5. Taylor expanded in U* around inf 38.2

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

   6. Simplified 33.2

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

      [Start] 38.2	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{n \cdot U*}{{Om}^{2}} + \frac{-2}{Om}\right)\right)}\right) \]
      unpow2 [=>] 38.2	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{n \cdot U*}{\color{blue}{Om \cdot Om}} + \frac{-2}{Om}\right)\right)}\right) \]
      times-frac [=>] 33.2	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{U \cdot \left(n \cdot \left(\color{blue}{\frac{n}{Om} \cdot \frac{U*}{Om}} + \frac{-2}{Om}\right)\right)}\right) \]

3. Recombined 4 regimes into one program.

4. Final simplification 29.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.6 \cdot 10^{+124}:\\ \;\;\;\;-{\left(\sqrt[3]{\sqrt{\frac{\frac{n}{Om} \cdot U* + -2}{Om} \cdot \left(n \cdot U\right)} \cdot \left(\ell \cdot \sqrt{2}\right)}\right)}^{3}\\ \mathbf{elif}\;\ell \leq -7 \cdot 10^{-195}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{\frac{Om}{\ell}} \cdot -2 + \left(n \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{U* - U}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.5 \cdot 10^{-95}:\\ \;\;\;\;\sqrt{2 \cdot \left|U \cdot \left(n \cdot t\right)\right|}\\ \mathbf{elif}\;\ell \leq 1.85 \cdot 10^{+145}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{\frac{Om}{\ell}} \cdot -2 + \left(n \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{U* - U}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right)\right)}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	31.9
Cost	14676

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;\ell \leq -8.5 \cdot 10^{+78}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(U \cdot \left(\ell \cdot \left(\ell \cdot n\right)\right)\right)\right) \cdot \left(\frac{-2}{Om} - \frac{n}{Om} \cdot \frac{U - U*}{Om}\right)}\\ \mathbf{elif}\;\ell \leq -1.35 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq -1.65 \cdot 10^{-194}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot t_1 - \frac{n}{\frac{Om}{\ell} \cdot \frac{Om}{\ell \cdot U*}}\right) - t\right)\right)}\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{-95}:\\ \;\;\;\;\sqrt{2 \cdot \left|U \cdot \left(n \cdot t\right)\right|}\\ \mathbf{elif}\;\ell \leq 3.5 \cdot 10^{+146}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(t_1 \cdot -2 + \left(n \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{U* - U}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right)\right)}\right)\\ \end{array} \]

Alternative 2
Error	29.5
Cost	14544

\[\begin{array}{l} t_1 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{\frac{Om}{\ell}} \cdot -2 + \left(n \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{U* - U}{Om}\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -1.55 \cdot 10^{+121}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(\frac{-2}{Om} + \frac{n}{\frac{Om}{\frac{U*}{Om}}}\right)} \cdot \left(\sqrt{2} \cdot \left(-\ell\right)\right)\\ \mathbf{elif}\;\ell \leq -2.8 \cdot 10^{-194}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 4.5 \cdot 10^{-95}:\\ \;\;\;\;\sqrt{2 \cdot \left|U \cdot \left(n \cdot t\right)\right|}\\ \mathbf{elif}\;\ell \leq 7.8 \cdot 10^{+149}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right)\right)}\right)\\ \end{array} \]

Alternative 3
Error	32.4
Cost	14164

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;\ell \leq -2.6 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(U \cdot \left(\ell \cdot \left(\ell \cdot n\right)\right)\right)\right) \cdot \left(\frac{-2}{Om} - \frac{n}{Om} \cdot \frac{U - U*}{Om}\right)}\\ \mathbf{elif}\;\ell \leq -1.25 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq -1.45 \cdot 10^{-194}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot t_1 - \frac{n}{\frac{Om}{\ell} \cdot \frac{Om}{\ell \cdot U*}}\right) - t\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.6 \cdot 10^{-95}:\\ \;\;\;\;\sqrt{2 \cdot \left|U \cdot \left(n \cdot t\right)\right|}\\ \mathbf{elif}\;\ell \leq 4 \cdot 10^{+177}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(t_1 \cdot -2 + \left(n \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{U* - U}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{-2 \cdot \frac{n}{\frac{Om}{U}}}\right)\\ \end{array} \]

Alternative 4
Error	32.4
Cost	14164

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;\ell \leq -7.3 \cdot 10^{+78}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(U \cdot \left(\ell \cdot \left(\ell \cdot n\right)\right)\right)\right) \cdot \left(\frac{-2}{Om} - \frac{n}{Om} \cdot \frac{U - U*}{Om}\right)}\\ \mathbf{elif}\;\ell \leq -1.35 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq -1.95 \cdot 10^{-194}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot t_1 - \frac{n}{\frac{Om}{\ell} \cdot \frac{Om}{\ell \cdot U*}}\right) - t\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.5 \cdot 10^{-95}:\\ \;\;\;\;\sqrt{2 \cdot \left|U \cdot \left(n \cdot t\right)\right|}\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{+178}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(t_1 \cdot -2 + \left(n \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{U* - U}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{n \cdot \left(U \cdot \frac{-2}{Om}\right)}\right)\\ \end{array} \]

Alternative 5
Error	33.7
Cost	13776

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;\ell \leq -1.02 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(U \cdot \left(\ell \cdot \left(\ell \cdot n\right)\right)\right)\right) \cdot \left(\frac{-2}{Om} - \frac{n}{Om} \cdot \frac{U - U*}{Om}\right)}\\ \mathbf{elif}\;\ell \leq -1.35 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-194}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot t_1 - \frac{n}{\frac{Om}{\ell} \cdot \frac{Om}{\ell \cdot U*}}\right) - t\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.5 \cdot 10^{-95}:\\ \;\;\;\;\sqrt{2 \cdot \left|U \cdot \left(n \cdot t\right)\right|}\\ \mathbf{elif}\;\ell \leq 4.8 \cdot 10^{+149}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(t_1 \cdot -2 + \left(n \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{U* - U}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right) \cdot \left(\frac{-2}{Om} + \frac{U* - U}{\frac{Om \cdot Om}{n}}\right)\right)}\\ \end{array} \]

Alternative 6
Error	33.5
Cost	8916

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;\ell \leq -2.3 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(U \cdot \left(\ell \cdot \left(\ell \cdot n\right)\right)\right)\right) \cdot \left(\frac{-2}{Om} - \frac{n}{Om} \cdot \frac{U - U*}{Om}\right)}\\ \mathbf{elif}\;\ell \leq -1.35 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq -1.22 \cdot 10^{-194}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot t_1 - \frac{n}{\frac{Om}{\ell} \cdot \frac{Om}{\ell \cdot U*}}\right) - t\right)\right)}\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{-188}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.4 \cdot 10^{+149}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(t_1 \cdot -2 + \left(n \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{U* - U}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right) \cdot \left(\frac{-2}{Om} + \frac{U* - U}{\frac{Om \cdot Om}{n}}\right)\right)}\\ \end{array} \]

Alternative 7
Error	33.6
Cost	8656

\[\begin{array}{l} t_1 := \sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \frac{n}{\frac{Om}{\ell} \cdot \frac{Om}{\ell \cdot U*}}\right) - t\right)\right)}\\ \mathbf{if}\;\ell \leq -3 \cdot 10^{+97}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(U \cdot \left(\ell \cdot \left(\ell \cdot n\right)\right)\right)\right) \cdot \left(\frac{-2}{Om} - \frac{n}{Om} \cdot \frac{U - U*}{Om}\right)}\\ \mathbf{elif}\;\ell \leq -5.3 \cdot 10^{-194}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.18 \cdot 10^{-271}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.26 \cdot 10^{+94}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(U - U*\right) \cdot \frac{n}{Om \cdot Om} + \frac{2}{Om}\right) \cdot \left(-2 \cdot \left(n \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right)\right)}\\ \end{array} \]

Alternative 8
Error	36.8
Cost	8401

\[\begin{array}{l} \mathbf{if}\;U* \leq -2.9 \cdot 10^{+228}:\\ \;\;\;\;\sqrt{2 \cdot \left(\frac{\ell \cdot \left(\ell \cdot \left(U* \cdot U\right)\right)}{Om} \cdot \frac{n \cdot n}{Om}\right)}\\ \mathbf{elif}\;U* \leq -7 \cdot 10^{-99}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{\frac{Om}{\ell}} \cdot -2\right)\right)\right)}\\ \mathbf{elif}\;U* \leq 4.2 \cdot 10^{+131} \lor \neg \left(U* \leq 4.6 \cdot 10^{+185}\right):\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\ell \cdot \left(2 \cdot \frac{\ell}{Om}\right) - t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right) \cdot \left(\frac{-2}{Om} + \frac{U* - U}{\frac{Om \cdot Om}{n}}\right)\right)}\\ \end{array} \]

Alternative 9
Error	34.7
Cost	8392

\[\begin{array}{l} t_1 := \ell \cdot \left(\ell \cdot n\right)\\ \mathbf{if}\;\ell \leq -5.8 \cdot 10^{+77}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(U \cdot t_1\right)\right) \cdot \left(\frac{-2}{Om} - \frac{n}{Om} \cdot \frac{U - U*}{Om}\right)}\\ \mathbf{elif}\;\ell \leq -5.8 \cdot 10^{-194}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\frac{U*}{Om} \cdot \frac{t_1}{Om} + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.8 \cdot 10^{+143}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{\frac{Om}{\ell}} \cdot -2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right) \cdot \left(\frac{-2}{Om} + \frac{U* - U}{\frac{Om \cdot Om}{n}}\right)\right)}\\ \end{array} \]

Alternative 10
Error	36.0
Cost	7624

\[\begin{array}{l} \mathbf{if}\;U* \leq -2.9 \cdot 10^{+228}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \frac{n}{\frac{\frac{Om}{\ell}}{\ell} \cdot \frac{Om}{U*}}\right)}\\ \mathbf{elif}\;U* \leq -5 \cdot 10^{-98}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{\frac{Om}{\ell}} \cdot -2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\ell \cdot \left(2 \cdot \frac{\ell}{Om}\right) - t\right)\right)}\\ \end{array} \]

Alternative 11
Error	36.1
Cost	7624

\[\begin{array}{l} \mathbf{if}\;U* \leq -2.9 \cdot 10^{+228}:\\ \;\;\;\;\sqrt{2 \cdot \left(\frac{\ell \cdot \left(\ell \cdot \left(U* \cdot U\right)\right)}{Om} \cdot \frac{n \cdot n}{Om}\right)}\\ \mathbf{elif}\;U* \leq -5.5 \cdot 10^{-101}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{\frac{Om}{\ell}} \cdot -2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\ell \cdot \left(2 \cdot \frac{\ell}{Om}\right) - t\right)\right)}\\ \end{array} \]

Alternative 12
Error	35.1
Cost	7620

\[\begin{array}{l} \mathbf{if}\;t \leq 1.9 \cdot 10^{+155}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t + n \cdot \left(\ell \cdot \left(\ell \cdot \frac{-2}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]

Alternative 13
Error	40.7
Cost	7496

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{if}\;t \leq -4.8 \cdot 10^{-198}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-260}:\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \left(\frac{n}{Om} \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)\right)\right)}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+153}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]

Alternative 14
Error	40.6
Cost	7496

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{if}\;t \leq -8.5 \cdot 10^{-201}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.72 \cdot 10^{-258}:\\ \;\;\;\;\sqrt{-2 \cdot \left(2 \cdot \frac{n}{\frac{Om}{U \cdot \left(\ell \cdot \ell\right)}}\right)}\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]

Alternative 15
Error	40.5
Cost	7496

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{if}\;t \leq -4.5 \cdot 10^{-196}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{-257}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(2 \cdot \frac{n}{\frac{Om}{\ell \cdot \ell}}\right)\right)}\\ \mathbf{elif}\;t \leq 7.6 \cdot 10^{+153}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]

Alternative 16
Error	35.1
Cost	7492

\[\begin{array}{l} \mathbf{if}\;t \leq 1.5 \cdot 10^{+155}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]

Alternative 17
Error	40.5
Cost	6980

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.9 \cdot 10^{-194}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]

Alternative 18
Error	40.5
Cost	6980

\[\begin{array}{l} \mathbf{if}\;\ell \leq -3.2 \cdot 10^{-194}:\\ \;\;\;\;\sqrt{n \cdot \left(t \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]

Alternative 19
Error	40.6
Cost	6848

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

Reproduce

herbie shell --seed 2023034 
(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*))))))