Toniolo and Linder, Equation (13)

Average Accuracy: 49.4% → 65.1%

Time: 40.0s

Precision: binary64

Cost: 43528

• Unsound rule application detected in e-graph. Results may not be sound.

• Unsound rule application detected in e-graph. Results may not be sound.

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \mathsf{fma}\left(\ell \cdot \left(-2 + \frac{n \cdot \left(U* - U\right)}{Om}\right), \frac{\ell}{Om}, t\right)}\\ \mathbf{elif}\;t_1 \leq 6.5 \cdot 10^{+139}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\mathsf{fma}\left(\ell, -2, n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right) \cdot \left(U \cdot \frac{\ell}{Om}\right) + U \cdot t\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* (* 2.0 n) U)
           (+
            (- t (* 2.0 (/ (* l l) Om)))
            (* (* n (pow (/ l Om) 2.0)) (- U* U)))))))
   (if (<= t_1 0.0)
     (*
      (sqrt (* 2.0 n))
      (sqrt (* U (fma (* l (+ -2.0 (/ (* n (- U* U)) Om))) (/ l Om) t))))
     (if (<= t_1 6.5e+139)
       t_1
       (sqrt
        (*
         (* 2.0 n)
         (+
          (* (fma l -2.0 (* n (* (/ l Om) (- U* U)))) (* U (/ l Om)))
          (* U t))))))))

C

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

↓

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + ((n * pow((l / Om), 2.0)) * (U_42_ - U)))));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = sqrt((2.0 * n)) * sqrt((U * fma((l * (-2.0 + ((n * (U_42_ - U)) / Om))), (l / Om), t)));
	} else if (t_1 <= 6.5e+139) {
		tmp = t_1;
	} else {
		tmp = sqrt(((2.0 * n) * ((fma(l, -2.0, (n * ((l / Om) * (U_42_ - U)))) * (U * (l / Om))) + (U * t))));
	}
	return tmp;
}

Julia

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

↓

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U_42_ - U)))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * fma(Float64(l * Float64(-2.0 + Float64(Float64(n * Float64(U_42_ - U)) / Om))), Float64(l / Om), t))));
	elseif (t_1 <= 6.5e+139)
		tmp = t_1;
	else
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(Float64(fma(l, -2.0, Float64(n * Float64(Float64(l / Om) * Float64(U_42_ - U)))) * Float64(U * Float64(l / Om))) + Float64(U * t))));
	end
	return tmp
end

Wolfram

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

↓

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(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$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(N[(l * N[(-2.0 + N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 6.5e+139], t$95$1, N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(N[(N[(l * -2.0 + N[(n * N[(N[(l / Om), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 11.0%

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

   2. Simplified 35.9%

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

      [Start] 11.0	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 35.9	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      sub-neg [=>] 35.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]
      associate--l+ [=>] 35.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      *-commutative [=>] 35.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      distribute-rgt-neg-in [=>] 35.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l/ [<=] 35.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 35.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [<=] 35.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [=>] 35.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 35.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]
      unpow2 [=>] 35.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]
      associate-*l* [=>] 35.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Applied egg-rr 35.8%

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

      [Start] 35.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)} \]
      +-commutative [=>] 35.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) + t\right)}\right)} \]
      distribute-lft-in [=>] 35.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(U \cdot \left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) + U \cdot t\right)}} \]
      *-commutative [=>] 35.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot U} + U \cdot t\right)} \]
      *-commutative [=>] 35.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\color{blue}{\left(\mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}\right)} \cdot U + U \cdot t\right)} \]
      associate-*l* [=>] 35.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\color{blue}{\mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \left(\frac{\ell}{Om} \cdot U\right)} + U \cdot t\right)} \]
      *-commutative [=>] 35.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\mathsf{fma}\left(\ell, -2, \color{blue}{\left(n \cdot \left(U* - U\right)\right) \cdot \frac{\ell}{Om}}\right) \cdot \left(\frac{\ell}{Om} \cdot U\right) + U \cdot t\right)} \]
      associate-*l* [=>] 35.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\mathsf{fma}\left(\ell, -2, \color{blue}{n \cdot \left(\left(U* - U\right) \cdot \frac{\ell}{Om}\right)}\right) \cdot \left(\frac{\ell}{Om} \cdot U\right) + U \cdot t\right)} \]

   4. Applied egg-rr 50.6%

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

      [Start] 35.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\mathsf{fma}\left(\ell, -2, n \cdot \left(\left(U* - U\right) \cdot \frac{\ell}{Om}\right)\right) \cdot \left(\frac{\ell}{Om} \cdot U\right) + U \cdot t\right)} \]
      sqrt-prod [=>] 50.5	\[ \color{blue}{\sqrt{2 \cdot n} \cdot \sqrt{\mathsf{fma}\left(\ell, -2, n \cdot \left(\left(U* - U\right) \cdot \frac{\ell}{Om}\right)\right) \cdot \left(\frac{\ell}{Om} \cdot U\right) + U \cdot t}} \]
      associate-*r* [=>] 50.6	\[ \sqrt{2 \cdot n} \cdot \sqrt{\color{blue}{\left(\mathsf{fma}\left(\ell, -2, n \cdot \left(\left(U* - U\right) \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om}\right) \cdot U} + U \cdot t} \]
      *-commutative [=>] 50.6	\[ \sqrt{2 \cdot n} \cdot \sqrt{\left(\mathsf{fma}\left(\ell, -2, n \cdot \left(\left(U* - U\right) \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om}\right) \cdot U + \color{blue}{t \cdot U}} \]
      distribute-rgt-out [=>] 50.6	\[ \sqrt{2 \cdot n} \cdot \sqrt{\color{blue}{U \cdot \left(\mathsf{fma}\left(\ell, -2, n \cdot \left(\left(U* - U\right) \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + t\right)}} \]
      *-commutative [=>] 50.6	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(\color{blue}{\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, n \cdot \left(\left(U* - U\right) \cdot \frac{\ell}{Om}\right)\right)} + t\right)} \]
      *-commutative [=>] 50.6	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \color{blue}{\left(\left(U* - U\right) \cdot \frac{\ell}{Om}\right) \cdot n}\right) + t\right)} \]
      associate-*l* [=>] 50.6	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \color{blue}{\left(U* - U\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)}\right) + t\right)} \]

   5. Simplified 50.6%

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

      [Start] 50.6	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \left(U* - U\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) + t\right)} \]
      *-commutative [=>] 50.6	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(\color{blue}{\mathsf{fma}\left(\ell, -2, \left(U* - U\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) \cdot \frac{\ell}{Om}} + t\right)} \]
      fma-def [=>] 50.6	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\ell, -2, \left(U* - U\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right), \frac{\ell}{Om}, t\right)}} \]
      fma-udef [=>] 50.6	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot -2 + \left(U* - U\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)}, \frac{\ell}{Om}, t\right)} \]
      *-commutative [<=] 50.6	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \mathsf{fma}\left(\color{blue}{-2 \cdot \ell} + \left(U* - U\right) \cdot \left(\frac{\ell}{Om} \cdot n\right), \frac{\ell}{Om}, t\right)} \]
      *-commutative [=>] 50.6	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \mathsf{fma}\left(-2 \cdot \ell + \color{blue}{\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right)}, \frac{\ell}{Om}, t\right)} \]
      associate-*l* [=>] 50.6	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \mathsf{fma}\left(-2 \cdot \ell + \color{blue}{\frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)}, \frac{\ell}{Om}, t\right)} \]
      associate-*l/ [=>] 50.6	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \mathsf{fma}\left(-2 \cdot \ell + \color{blue}{\frac{\ell \cdot \left(n \cdot \left(U* - U\right)\right)}{Om}}, \frac{\ell}{Om}, t\right)} \]
      *-commutative [<=] 50.6	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \mathsf{fma}\left(-2 \cdot \ell + \frac{\color{blue}{\left(n \cdot \left(U* - U\right)\right) \cdot \ell}}{Om}, \frac{\ell}{Om}, t\right)} \]
      associate-*l/ [<=] 50.6	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \mathsf{fma}\left(-2 \cdot \ell + \color{blue}{\frac{n \cdot \left(U* - U\right)}{Om} \cdot \ell}, \frac{\ell}{Om}, t\right)} \]
      distribute-rgt-out [=>] 50.6	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \left(-2 + \frac{n \cdot \left(U* - U\right)}{Om}\right)}, \frac{\ell}{Om}, t\right)} \]

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 6.5000000000000003e139

   1. Initial program 99.3%

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

   if 6.5000000000000003e139 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))))

   1. Initial program 20.2%

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

   2. Simplified 39.8%

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

      [Start] 20.2	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 23.8	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      sub-neg [=>] 23.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]
      associate--l+ [=>] 23.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      *-commutative [=>] 23.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      distribute-rgt-neg-in [=>] 23.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l/ [<=] 34.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 34.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [<=] 34.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [=>] 34.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 28.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]
      unpow2 [=>] 28.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]
      associate-*l* [=>] 31.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Applied egg-rr 44.2%

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

      [Start] 39.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)} \]
      +-commutative [=>] 39.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) + t\right)}\right)} \]
      distribute-lft-in [=>] 37.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(U \cdot \left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) + U \cdot t\right)}} \]
      *-commutative [=>] 37.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot U} + U \cdot t\right)} \]
      *-commutative [=>] 37.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\color{blue}{\left(\mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}\right)} \cdot U + U \cdot t\right)} \]
      associate-*l* [=>] 41.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\color{blue}{\mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \left(\frac{\ell}{Om} \cdot U\right)} + U \cdot t\right)} \]
      *-commutative [=>] 41.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\mathsf{fma}\left(\ell, -2, \color{blue}{\left(n \cdot \left(U* - U\right)\right) \cdot \frac{\ell}{Om}}\right) \cdot \left(\frac{\ell}{Om} \cdot U\right) + U \cdot t\right)} \]
      associate-*l* [=>] 44.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\mathsf{fma}\left(\ell, -2, \color{blue}{n \cdot \left(\left(U* - U\right) \cdot \frac{\ell}{Om}\right)}\right) \cdot \left(\frac{\ell}{Om} \cdot U\right) + U \cdot t\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \mathsf{fma}\left(\ell \cdot \left(-2 + \frac{n \cdot \left(U* - U\right)}{Om}\right), \frac{\ell}{Om}, t\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 6.5 \cdot 10^{+139}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\mathsf{fma}\left(\ell, -2, n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right) \cdot \left(U \cdot \frac{\ell}{Om}\right) + U \cdot t\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	65.0%
Cost	30728

\[\begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;t_1 \leq 4.3 \cdot 10^{+279}:\\ \;\;\;\;\sqrt{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\mathsf{fma}\left(\ell, -2, n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right) \cdot \left(U \cdot \frac{\ell}{Om}\right) + U \cdot t\right)}\\ \end{array} \]

Alternative 2
Accuracy	59.5%
Cost	14665

\[\begin{array}{l} \mathbf{if}\;Om \leq -4.6 \cdot 10^{-146} \lor \neg \left(Om \leq 2.75 \cdot 10^{-219}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\mathsf{fma}\left(\ell, -2, n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right) \cdot \left(U \cdot \frac{\ell}{Om}\right) + U \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\sqrt{2 \cdot \left(n \cdot \left(\ell \cdot \left(\left(U \cdot \ell\right) \cdot \left(n \cdot U*\right)\right)\right)\right)}}{Om}\right|\\ \end{array} \]

Alternative 3
Accuracy	54.2%
Cost	14536

\[\begin{array}{l} \mathbf{if}\;Om \leq -8.6 \cdot 10^{-146}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \frac{\left(U \cdot \ell\right) \cdot \left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\right)}{Om}\right)}\\ \mathbf{elif}\;Om \leq 8.2 \cdot 10^{-218}:\\ \;\;\;\;\left|\frac{\sqrt{2 \cdot \left(n \cdot \left(\ell \cdot \left(\left(U \cdot \ell\right) \cdot \left(n \cdot U*\right)\right)\right)\right)}}{Om}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}\\ \end{array} \]

Alternative 4
Accuracy	49.4%
Cost	8140

\[\begin{array}{l} \mathbf{if}\;Om \leq -8 \cdot 10^{-22}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \frac{\ell \cdot \left(\ell \cdot \left(U \cdot -2\right)\right)}{Om}\right)}\\ \mathbf{elif}\;Om \leq -5.5 \cdot 10^{-153}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \frac{n \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(U \cdot U*\right)\right)}{Om \cdot Om}\right)}\\ \mathbf{elif}\;Om \leq 1.05 \cdot 10^{-211}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(n \cdot \ell\right) \cdot \left(\ell \cdot \left(\left(2 + \left(U - U*\right) \cdot \frac{n}{Om}\right) \cdot \frac{U}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \frac{2 \cdot \ell}{\frac{Om}{\ell}}\right)\right)}\\ \end{array} \]

Alternative 5
Accuracy	59.3%
Cost	8136

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.15 \cdot 10^{+133}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\left(U \cdot \ell\right) \cdot \left(n \cdot \left(\ell \cdot -2 + \frac{n \cdot \ell}{\frac{Om}{U* - U}}\right)\right)}{Om}}\\ \mathbf{elif}\;\ell \leq 8.5 \cdot 10^{+150}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \frac{\ell \cdot -2 + \frac{n \cdot U*}{\frac{Om}{\ell}}}{\frac{Om}{U \cdot \ell}}}\\ \end{array} \]

Alternative 6
Accuracy	55.7%
Cost	8132

\[\begin{array}{l} t_1 := \frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\\ \mathbf{if}\;U* \leq 3.2 \cdot 10^{-15}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \frac{\left(U \cdot \ell\right) \cdot t_1}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot t_1}{Om}\right)\right)}\\ \end{array} \]

Alternative 7
Accuracy	48.8%
Cost	8008

\[\begin{array}{l} \mathbf{if}\;Om \leq -1.12 \cdot 10^{-144}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \frac{\ell \cdot \left(\ell \cdot \left(U \cdot -2\right)\right)}{Om}\right)}\\ \mathbf{elif}\;Om \leq 2.4 \cdot 10^{-210}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(n \cdot \ell\right) \cdot \left(\ell \cdot \left(\left(2 + \left(U - U*\right) \cdot \frac{n}{Om}\right) \cdot \frac{U}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \frac{2 \cdot \ell}{\frac{Om}{\ell}}\right)\right)}\\ \end{array} \]

Alternative 8
Accuracy	49.5%
Cost	7880

\[\begin{array}{l} \mathbf{if}\;Om \leq -1.15 \cdot 10^{-75}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \frac{\ell \cdot \left(\ell \cdot \left(U \cdot -2\right)\right)}{Om}\right)}\\ \mathbf{elif}\;Om \leq 3 \cdot 10^{-211}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\frac{n}{Om} \cdot \left(\ell \cdot \left(\ell \cdot \left(U \cdot \left(2 - U* \cdot \frac{n}{Om}\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \frac{2 \cdot \ell}{\frac{Om}{\ell}}\right)\right)}\\ \end{array} \]

Alternative 9
Accuracy	48.9%
Cost	7876

\[\begin{array}{l} \mathbf{if}\;n \leq 4.9 \cdot 10^{-22}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \frac{\ell \cdot \left(\ell \cdot \left(U \cdot -2\right)\right)}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \frac{\frac{n}{Om} \cdot \left(\ell \cdot \left(U* \cdot \left(U \cdot \ell\right)\right)\right)}{Om}\right)}\\ \end{array} \]

Alternative 10
Accuracy	44.2%
Cost	7760

\[\begin{array}{l} t_1 := \sqrt{\left(2 \cdot n\right) \cdot \left(-2 \cdot \frac{\ell}{\frac{Om}{U \cdot \ell}}\right)}\\ \mathbf{if}\;\ell \leq -1.4 \cdot 10^{+75}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 4.7 \cdot 10^{-299}:\\ \;\;\;\;\sqrt{\left(n \cdot t\right) \cdot \left(2 \cdot U\right)}\\ \mathbf{elif}\;\ell \leq 7.2 \cdot 10^{-76}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(2 \cdot t\right)}\\ \mathbf{elif}\;\ell \leq 7.6 \cdot 10^{+119}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Accuracy	48.0%
Cost	7752

\[\begin{array}{l} \mathbf{if}\;Om \leq -5.5 \cdot 10^{-145}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \frac{\ell \cdot \left(\ell \cdot \left(U \cdot -2\right)\right)}{Om}\right)}\\ \mathbf{elif}\;Om \leq 1.32 \cdot 10^{-210}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(n \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell \cdot \left(U \cdot U*\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \frac{2 \cdot \ell}{\frac{Om}{\ell}}\right)\right)}\\ \end{array} \]

Alternative 12
Accuracy	46.2%
Cost	7752

\[\begin{array}{l} \mathbf{if}\;Om \leq -3.7 \cdot 10^{-77}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \frac{\ell \cdot \left(\ell \cdot \left(U \cdot -2\right)\right)}{Om}\right)}\\ \mathbf{elif}\;Om \leq 1.26 \cdot 10^{-213}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \frac{n \cdot \left(U \cdot \left(\ell \cdot \left(\ell \cdot U*\right)\right)\right)}{Om \cdot Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \frac{2 \cdot \ell}{\frac{Om}{\ell}}\right)\right)}\\ \end{array} \]

Alternative 13
Accuracy	48.4%
Cost	7752

\[\begin{array}{l} \mathbf{if}\;Om \leq -2.1 \cdot 10^{-116}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \frac{\ell \cdot \left(\ell \cdot \left(U \cdot -2\right)\right)}{Om}\right)}\\ \mathbf{elif}\;Om \leq 2.2 \cdot 10^{-210}:\\ \;\;\;\;\sqrt{\frac{\left(2 \cdot n\right) \cdot \frac{\ell}{\frac{Om}{\left(U \cdot \ell\right) \cdot \left(n \cdot U*\right)}}}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \frac{2 \cdot \ell}{\frac{Om}{\ell}}\right)\right)}\\ \end{array} \]

Alternative 14
Accuracy	37.4%
Cost	7496

\[\begin{array}{l} \mathbf{if}\;t \leq -3.6 \cdot 10^{-198}:\\ \;\;\;\;\sqrt{\left(n \cdot t\right) \cdot \left(2 \cdot U\right)}\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-139}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\frac{U}{Om} \cdot \left(2 \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 15
Accuracy	47.3%
Cost	7360

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

Alternative 16
Accuracy	35.2%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;Om \leq -1.25 \cdot 10^{-148} \lor \neg \left(Om \leq 4.4 \cdot 10^{-202}\right):\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(t + t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(t + t\right)}^{6}\\ \end{array} \]

Alternative 17
Accuracy	35.2%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;U \leq -1 \cdot 10^{-296}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(t + t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(2 \cdot t\right)}\\ \end{array} \]

Alternative 18
Accuracy	35.5%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;U \leq -2 \cdot 10^{-283}:\\ \;\;\;\;\sqrt{\left(n \cdot t\right) \cdot \left(2 \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(2 \cdot t\right)}\\ \end{array} \]

Alternative 19
Accuracy	36.9%
Cost	6912

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

Alternative 20
Accuracy	36.9%
Cost	6912

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

Alternative 21
Accuracy	10.2%
Cost	6656

\[{\left(t + t\right)}^{4} \]

Alternative 22
Accuracy	10.5%
Cost	6656

\[{\left(t + t\right)}^{6} \]

Alternative 23
Accuracy	9.0%
Cost	320

\[4 \cdot \left(t \cdot t\right) \]

Alternative 24
Accuracy	3.5%
Cost	192

\[t + t \]

Reproduce

herbie shell --seed 2023158 
(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*))))))