Toniolo and Linder, Equation (13)

Average Error: 34.1 → 23.4

Time: 35.1s

Precision: binary64

Cost: 38860

Math

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

↓

\[\begin{array}{l} t_1 := \left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right) + \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right)\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{-24}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(\left(\ell \cdot \left(U \cdot \ell\right)\right) \cdot \left(\frac{U* - U}{\frac{Om}{n}} \cdot \frac{-1}{Om} + \frac{2}{Om}\right)\right)\right)}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{-310}:\\ \;\;\;\;{\left(\sqrt[3]{U \cdot t} \cdot \sqrt[3]{2 \cdot n}\right)}^{1.5}\\ \mathbf{elif}\;t_1 \leq 8 \cdot 10^{+298}:\\ \;\;\;\;\sqrt{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \left(\frac{n \cdot \ell}{Om} \cdot \frac{\ell}{\frac{1}{U}}\right)\right)}\\ \end{array} \]

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
         (*
          (* U (* n -2.0))
          (+
           (* (* n (pow (/ l Om) 2.0)) (- U U*))
           (- (* 2.0 (/ (* l l) Om)) t)))))
   (if (<= t_1 -5e-24)
     (sqrt
      (*
       -2.0
       (*
        n
        (*
         (* l (* U l))
         (+ (* (/ (- U* U) (/ Om n)) (/ -1.0 Om)) (/ 2.0 Om))))))
     (if (<= t_1 5e-310)
       (pow (* (cbrt (* U t)) (cbrt (* 2.0 n))) 1.5)
       (if (<= t_1 8e+298)
         (sqrt t_1)
         (sqrt
          (fma
           2.0
           (* n (* U t))
           (* -4.0 (* (/ (* n l) Om) (/ l (/ 1.0 U)))))))))))

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)) * (((n * pow((l / Om), 2.0)) * (U - U_42_)) + ((2.0 * ((l * l) / Om)) - t));
	double tmp;
	if (t_1 <= -5e-24) {
		tmp = sqrt((-2.0 * (n * ((l * (U * l)) * ((((U_42_ - U) / (Om / n)) * (-1.0 / Om)) + (2.0 / Om))))));
	} else if (t_1 <= 5e-310) {
		tmp = pow((cbrt((U * t)) * cbrt((2.0 * n))), 1.5);
	} else if (t_1 <= 8e+298) {
		tmp = sqrt(t_1);
	} else {
		tmp = sqrt(fma(2.0, (n * (U * t)), (-4.0 * (((n * l) / Om) * (l / (1.0 / U))))));
	}
	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(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_)) + Float64(Float64(2.0 * Float64(Float64(l * l) / Om)) - t)))
	tmp = 0.0
	if (t_1 <= -5e-24)
		tmp = sqrt(Float64(-2.0 * Float64(n * Float64(Float64(l * Float64(U * l)) * Float64(Float64(Float64(Float64(U_42_ - U) / Float64(Om / n)) * Float64(-1.0 / Om)) + Float64(2.0 / Om))))));
	elseif (t_1 <= 5e-310)
		tmp = Float64(cbrt(Float64(U * t)) * cbrt(Float64(2.0 * n))) ^ 1.5;
	elseif (t_1 <= 8e+298)
		tmp = sqrt(t_1);
	else
		tmp = sqrt(fma(2.0, Float64(n * Float64(U * t)), Float64(-4.0 * Float64(Float64(Float64(n * l) / Om) * Float64(l / Float64(1.0 / U))))));
	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[(U * N[(n * -2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-24], N[Sqrt[N[(-2.0 * N[(n * N[(N[(l * N[(U * l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(U$42$ - U), $MachinePrecision] / N[(Om / n), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / Om), $MachinePrecision]), $MachinePrecision] + N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 5e-310], N[Power[N[(N[Power[N[(U * t), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(2.0 * n), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], If[LessEqual[t$95$1, 8e+298], N[Sqrt[t$95$1], $MachinePrecision], N[Sqrt[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * N[(N[(N[(n * l), $MachinePrecision] / Om), $MachinePrecision] * N[(l / N[(1.0 / U), $MachinePrecision]), $MachinePrecision]), $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*)))) < -4.9999999999999998e-24

   1. Initial program 64.0

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

   2. Simplified 43.6

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

      [Start] 64.0	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 64.0	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 64.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      sub-neg [=>] 64.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(-\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [<=] 64.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      cancel-sign-sub [<=] 64.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      cancel-sign-sub [=>] 64.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-/l* [=>] 48.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 43.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in l around inf 56.9

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

   4. Simplified 56.5

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

      [Start] 56.9	\[ \sqrt{-2 \cdot \left(\left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(n \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      *-commutative [=>] 56.9	\[ \sqrt{-2 \cdot \color{blue}{\left(\left(n \cdot \left({\ell}^{2} \cdot U\right)\right) \cdot \left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)\right)}} \]
      associate-*l* [=>] 56.9	\[ \sqrt{-2 \cdot \color{blue}{\left(n \cdot \left(\left({\ell}^{2} \cdot U\right) \cdot \left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)\right)\right)}} \]
      unpow2 [=>] 56.9	\[ \sqrt{-2 \cdot \left(n \cdot \left(\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot U\right) \cdot \left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)\right)\right)} \]
      associate-*l* [=>] 56.5	\[ \sqrt{-2 \cdot \left(n \cdot \left(\color{blue}{\left(\ell \cdot \left(\ell \cdot U\right)\right)} \cdot \left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)\right)\right)} \]
      unpow2 [=>] 56.5	\[ \sqrt{-2 \cdot \left(n \cdot \left(\left(\ell \cdot \left(\ell \cdot U\right)\right) \cdot \left(\frac{n \cdot \left(U - U*\right)}{\color{blue}{Om \cdot Om}} + 2 \cdot \frac{1}{Om}\right)\right)\right)} \]
      associate-*r/ [=>] 56.5	\[ \sqrt{-2 \cdot \left(n \cdot \left(\left(\ell \cdot \left(\ell \cdot U\right)\right) \cdot \left(\frac{n \cdot \left(U - U*\right)}{Om \cdot Om} + \color{blue}{\frac{2 \cdot 1}{Om}}\right)\right)\right)} \]
      metadata-eval [=>] 56.5	\[ \sqrt{-2 \cdot \left(n \cdot \left(\left(\ell \cdot \left(\ell \cdot U\right)\right) \cdot \left(\frac{n \cdot \left(U - U*\right)}{Om \cdot Om} + \frac{\color{blue}{2}}{Om}\right)\right)\right)} \]

   5. Applied egg-rr 42.0

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

   if -4.9999999999999998e-24 < (*.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*)))) < 4.999999999999985e-310

   1. Initial program 55.8

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

   2. Simplified 42.1

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

      [Start] 55.8	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 40.8	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      cancel-sign-sub-inv [=>] 40.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)} \]
      +-commutative [=>] 40.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(\left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\right)} \]
      distribute-lft-neg-in [<=] 40.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(-\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)} \]
      associate-*l* [=>] 42.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(-\color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)} \]
      distribute-rgt-neg-in [=>] 42.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{n \cdot \left(-{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)} \]
      fma-def [=>] 42.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\mathsf{fma}\left(n, -{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right), t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)}\right)} \]

   3. Taylor expanded in l around 0 44.7

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

   4. Applied egg-rr 44.9

      \[\leadsto \color{blue}{{\left(\sqrt[3]{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\right)}^{1.5}} \]

   5. Applied egg-rr 26.9

      \[\leadsto {\color{blue}{\left(\sqrt[3]{2 \cdot n} \cdot \sqrt[3]{U \cdot t}\right)}}^{1.5} \]

   6. Simplified 26.9

      \[\leadsto {\color{blue}{\left(\sqrt[3]{t \cdot U} \cdot \sqrt[3]{2 \cdot n}\right)}}^{1.5} \]
      Proof

      [Start] 26.9	\[ {\left(\sqrt[3]{2 \cdot n} \cdot \sqrt[3]{U \cdot t}\right)}^{1.5} \]
      *-commutative [=>] 26.9	\[ {\color{blue}{\left(\sqrt[3]{U \cdot t} \cdot \sqrt[3]{2 \cdot n}\right)}}^{1.5} \]
      *-commutative [<=] 26.9	\[ {\left(\sqrt[3]{\color{blue}{t \cdot U}} \cdot \sqrt[3]{2 \cdot n}\right)}^{1.5} \]

   if 4.999999999999985e-310 < (*.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*)))) < 7.9999999999999997e298

   1. Initial program 1.4

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

   if 7.9999999999999997e298 < (*.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 63.2

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

   2. Simplified 54.6

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

      [Start] 63.2	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 62.2	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      cancel-sign-sub-inv [=>] 62.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)} \]
      +-commutative [=>] 62.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(\left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\right)} \]
      distribute-lft-neg-in [<=] 62.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(-\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)} \]
      associate-*l* [=>] 61.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(-\color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)} \]
      distribute-rgt-neg-in [=>] 61.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{n \cdot \left(-{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)} \]
      fma-def [=>] 61.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\mathsf{fma}\left(n, -{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right), t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)}\right)} \]

   3. Taylor expanded in Om around inf 60.1

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(t \cdot U\right)\right) + -4 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}}} \]

   4. Simplified 59.8

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \frac{n \cdot \left(\ell \cdot \ell\right)}{\frac{Om}{U}}\right)}} \]
      Proof

      [Start] 60.1	\[ \sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right)\right) + -4 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}} \]
      fma-def [=>] 60.1	\[ \sqrt{\color{blue}{\mathsf{fma}\left(2, n \cdot \left(t \cdot U\right), -4 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}} \]
      *-commutative [=>] 60.1	\[ \sqrt{\mathsf{fma}\left(2, n \cdot \color{blue}{\left(U \cdot t\right)}, -4 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)} \]
      associate-*r* [=>] 60.2	\[ \sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \frac{\color{blue}{\left(n \cdot {\ell}^{2}\right) \cdot U}}{Om}\right)} \]
      associate-/l* [=>] 59.8	\[ \sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \color{blue}{\frac{n \cdot {\ell}^{2}}{\frac{Om}{U}}}\right)} \]
      unpow2 [=>] 59.8	\[ \sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \frac{n \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{\frac{Om}{U}}\right)} \]

   5. Applied egg-rr 47.5

      \[\leadsto \sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \color{blue}{\left(\frac{n \cdot \ell}{Om} \cdot \frac{\ell}{\frac{1}{U}}\right)}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 23.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right) + \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right) \leq -5 \cdot 10^{-24}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(\left(\ell \cdot \left(U \cdot \ell\right)\right) \cdot \left(\frac{U* - U}{\frac{Om}{n}} \cdot \frac{-1}{Om} + \frac{2}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right) + \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right) \leq 5 \cdot 10^{-310}:\\ \;\;\;\;{\left(\sqrt[3]{U \cdot t} \cdot \sqrt[3]{2 \cdot n}\right)}^{1.5}\\ \mathbf{elif}\;\left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right) + \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right) \leq 8 \cdot 10^{+298}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right) + \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \left(\frac{n \cdot \ell}{Om} \cdot \frac{\ell}{\frac{1}{U}}\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	26.0
Cost	30728

\[\begin{array}{l} t_1 := \frac{\ell \cdot \ell}{Om}\\ t_2 := \left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right) + \left(2 \cdot t_1 - t\right)\right)\\ \mathbf{if}\;t_2 \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(t_1 \cdot \frac{U* - U}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;t_2 \leq 8 \cdot 10^{+298}:\\ \;\;\;\;\sqrt{t_2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \left(\frac{n \cdot \ell}{Om} \cdot \frac{\ell}{\frac{1}{U}}\right)\right)}\\ \end{array} \]

Alternative 2
Error	32.3
Cost	15072

\[\begin{array}{l} t_1 := n \cdot \left(U \cdot t\right)\\ t_2 := \left(U* - U\right) \cdot \frac{n}{Om \cdot Om} + \frac{-2}{Om}\\ t_3 := \sqrt{2 \cdot \left|U \cdot \left(n \cdot t\right)\right|}\\ \mathbf{if}\;\ell \leq -1.32 \cdot 10^{+231}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \left(-\sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} + \frac{-2}{Om}\right)\right)}\right)\right)\\ \mathbf{elif}\;\ell \leq -4.8 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, t_1, -4 \cdot \left(\ell \cdot \frac{n}{\frac{Om}{U \cdot \ell}}\right)\right)}\\ \mathbf{elif}\;\ell \leq -2.6 \cdot 10^{+73}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \left(-\sqrt{U \cdot \left(n \cdot t_2\right)}\right)\\ \mathbf{elif}\;\ell \leq -3.05 \cdot 10^{-266}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\ell \leq 4.4 \cdot 10^{-273}:\\ \;\;\;\;\sqrt{2 \cdot \left|t_1\right|}\\ \mathbf{elif}\;\ell \leq 2.2 \cdot 10^{-78}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\ell \leq 1:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \ell \cdot \left(\ell \cdot t_2\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.55 \cdot 10^{+100}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(\frac{\ell \cdot \ell}{Om} \cdot \frac{U* - U}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, t_1, -4 \cdot \left(\frac{n \cdot \ell}{Om} \cdot \frac{\ell}{\frac{1}{U}}\right)\right)}\\ \end{array} \]

Alternative 3
Error	31.6
Cost	15072

\[\begin{array}{l} t_1 := \frac{\ell \cdot \ell}{Om}\\ t_2 := n \cdot \left(U \cdot t\right)\\ t_3 := \left(U* - U\right) \cdot \frac{n}{Om \cdot Om} + \frac{-2}{Om}\\ \mathbf{if}\;\ell \leq -1.12 \cdot 10^{+230}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \left(-\sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} + \frac{-2}{Om}\right)\right)}\right)\right)\\ \mathbf{elif}\;\ell \leq -7 \cdot 10^{+75}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, t_2, -4 \cdot \left(\ell \cdot \frac{n}{\frac{Om}{U \cdot \ell}}\right)\right)}\\ \mathbf{elif}\;\ell \leq -2.2 \cdot 10^{+74}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \left(-\sqrt{U \cdot \left(n \cdot t_3\right)}\right)\\ \mathbf{elif}\;\ell \leq -2.6 \cdot 10^{-130}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \mathsf{fma}\left(2, t_1, n \cdot \frac{\ell \cdot \left(\ell \cdot \left(-U*\right)\right)}{Om \cdot Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{-272}:\\ \;\;\;\;\sqrt{2 \cdot \left|t_2\right|}\\ \mathbf{elif}\;\ell \leq 9.5 \cdot 10^{-77}:\\ \;\;\;\;\sqrt{2 \cdot \left|U \cdot \left(n \cdot t\right)\right|}\\ \mathbf{elif}\;\ell \leq 0.75:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \ell \cdot \left(\ell \cdot t_3\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{+99}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(t_1 \cdot \frac{U* - U}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, t_2, -4 \cdot \left(\frac{n \cdot \ell}{Om} \cdot \frac{\ell}{\frac{1}{U}}\right)\right)}\\ \end{array} \]

Alternative 4
Error	31.2
Cost	14860

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.25 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \left(-\sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} + \frac{-2}{Om}\right)\right)}\right)\right)\\ \mathbf{elif}\;\ell \leq -8.8 \cdot 10^{-123}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right)}\\ \mathbf{elif}\;\ell \leq 1.26 \cdot 10^{-271}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 9.5 \cdot 10^{-76}:\\ \;\;\;\;\sqrt{2 \cdot \left|U \cdot \left(n \cdot t\right)\right|}\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{+132}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \ell \cdot \left(\ell \cdot \left(\left(U* - U\right) \cdot \frac{n}{Om \cdot Om} + \frac{-2}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \left(\ell \cdot \frac{n}{\frac{Om}{U \cdot \ell}}\right)\right)}\\ \end{array} \]

Alternative 5
Error	31.9
Cost	14676

\[\begin{array}{l} t_1 := \sqrt{2 \cdot n}\\ t_2 := n \cdot \left(U \cdot t\right)\\ \mathbf{if}\;n \leq -5.6 \cdot 10^{+247}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(\frac{\ell \cdot \ell}{Om} \cdot \frac{U* - U}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq -9.6 \cdot 10^{+119}:\\ \;\;\;\;\sqrt{n \cdot U} \cdot \sqrt{2 \cdot t}\\ \mathbf{elif}\;n \leq 7 \cdot 10^{-268}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, t_2, -4 \cdot \left(\ell \cdot \frac{n}{\frac{Om}{U \cdot \ell}}\right)\right)}\\ \mathbf{elif}\;n \leq 2.45 \cdot 10^{-133}:\\ \;\;\;\;t_1 \cdot \sqrt{U \cdot \left(t + \ell \cdot \left(\frac{\ell}{Om} \cdot -2\right)\right)}\\ \mathbf{elif}\;n \leq 4.1 \cdot 10^{+125}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, t_2, -4 \cdot \left(\ell \cdot \frac{n \cdot \left(U \cdot \ell\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \sqrt{U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om} \cdot \frac{n}{Om}\right)}\\ \end{array} \]

Alternative 6
Error	32.7
Cost	14548

\[\begin{array}{l} t_1 := \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(\frac{\ell \cdot \ell}{Om} \cdot \frac{U* - U}{Om}\right)\right)\right)\right)}\\ t_2 := \sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \left(\ell \cdot \frac{n}{\frac{Om}{U \cdot \ell}}\right)\right)}\\ \mathbf{if}\;n \leq -5.8 \cdot 10^{+247}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq -2.5 \cdot 10^{+119}:\\ \;\;\;\;\sqrt{n \cdot U} \cdot \sqrt{2 \cdot t}\\ \mathbf{elif}\;n \leq 1.3 \cdot 10^{-267}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;n \leq 1.52 \cdot 10^{-133}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + \ell \cdot \left(\frac{\ell}{Om} \cdot -2\right)\right)}\\ \mathbf{elif}\;n \leq 0.00041:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	33.0
Cost	14020

\[\begin{array}{l} \mathbf{if}\;t \leq -1300000000000:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \left(n \cdot \left(\ell \cdot \left(\ell \cdot \frac{U}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;t \leq 5.9 \cdot 10^{-288}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + U \cdot \left(\ell \cdot \left(\ell \cdot \left(\left(U* - U\right) \cdot \frac{n}{Om \cdot Om} + \frac{-2}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{-121} \lor \neg \left(t \leq 2.6 \cdot 10^{+78}\right):\\ \;\;\;\;\sqrt{t} \cdot \sqrt{n \cdot \left(2 \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \end{array} \]

Alternative 8
Error	33.4
Cost	13909

\[\begin{array}{l} t_1 := -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;t \leq -1.85 \cdot 10^{+143}:\\ \;\;\;\;\sqrt{2 \cdot \left|U \cdot \left(n \cdot t\right)\right|}\\ \mathbf{elif}\;t \leq -2000000000000:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(t_1 + n \cdot \left(\frac{\ell \cdot \ell}{Om} \cdot \frac{U* - U}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;t \leq 1.12 \cdot 10^{-287}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + U \cdot \left(\ell \cdot \left(\ell \cdot \left(\left(U* - U\right) \cdot \frac{n}{Om \cdot Om} + \frac{-2}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-120} \lor \neg \left(t \leq 3.2 \cdot 10^{+86}\right):\\ \;\;\;\;\sqrt{t} \cdot \sqrt{n \cdot \left(2 \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + t_1\right)\right)\right)}\\ \end{array} \]

Alternative 9
Error	33.6
Cost	13644

\[\begin{array}{l} t_1 := \frac{\ell \cdot \ell}{Om}\\ \mathbf{if}\;U \leq -3.1 \cdot 10^{-40}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + t_1 \cdot -2\right)}\\ \mathbf{elif}\;U \leq 2.05 \cdot 10^{-187}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(t_1 \cdot \frac{U* - U}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;U \leq 1.05 \cdot 10^{+93}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \ell \cdot \left(\ell \cdot \left(\left(U* - U\right) \cdot \frac{n}{Om \cdot Om} + \frac{-2}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{2 \cdot \left(n \cdot t\right)}\\ \end{array} \]

Alternative 10
Error	35.3
Cost	13512

\[\begin{array}{l} \mathbf{if}\;\ell \leq 6.4 \cdot 10^{-275}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right)}\\ \mathbf{elif}\;\ell \leq 1.25 \cdot 10^{-77}:\\ \;\;\;\;\sqrt{2 \cdot \left|U \cdot \left(n \cdot t\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \ell \cdot \left(\ell \cdot \left(\left(U* - U\right) \cdot \frac{n}{Om \cdot Om} + \frac{-2}{Om}\right)\right)\right)\right)}\\ \end{array} \]

Alternative 11
Error	35.3
Cost	8264

\[\begin{array}{l} \mathbf{if}\;\ell \leq 7.3 \cdot 10^{-271}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right)}\\ \mathbf{elif}\;\ell \leq 7.8 \cdot 10^{-78}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \ell \cdot \left(\ell \cdot \left(\left(U* - U\right) \cdot \frac{n}{Om \cdot Om} + \frac{-2}{Om}\right)\right)\right)\right)}\\ \end{array} \]

Alternative 12
Error	35.9
Cost	8140

\[\begin{array}{l} \mathbf{if}\;\ell \leq 1.6 \cdot 10^{-271}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right)}\\ \mathbf{elif}\;\ell \leq 4.2 \cdot 10^{-76}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 6.4 \cdot 10^{+125}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(\left(\ell \cdot \left(U \cdot \ell\right)\right) \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 13
Error	35.8
Cost	7889

\[\begin{array}{l} \mathbf{if}\;\ell \leq 4.3 \cdot 10^{-274}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right)}\\ \mathbf{elif}\;\ell \leq 1.25 \cdot 10^{-79}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+128} \lor \neg \left(\ell \leq 3.7 \cdot 10^{+149}\right):\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(-2 \cdot \frac{\ell \cdot \ell}{\frac{Om}{U}}\right)\right)}\\ \end{array} \]

Alternative 14
Error	35.0
Cost	7888

\[\begin{array}{l} t_1 := \sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - t\right)\right)\right)}\\ t_2 := \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{if}\;\ell \leq -4.4 \cdot 10^{+66}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -2.85 \cdot 10^{-234}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 2.4 \cdot 10^{-270}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{elif}\;\ell \leq 5.2 \cdot 10^{-85}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 15
Error	40.1
Cost	7496

\[\begin{array}{l} \mathbf{if}\;t \leq -3.3 \cdot 10^{-190}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-287}:\\ \;\;\;\;\sqrt{-2 \cdot \left(2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \left(n \cdot U\right)}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(n \cdot \left(2 \cdot U\right)\right)}\\ \end{array} \]

Alternative 16
Error	39.9
Cost	6848

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

Reproduce

herbie shell --seed 2023047 
(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*))))))