Toniolo and Linder, Equation (13)

Percentage Accurate: 49.8% → 64.7%

Time: 32.7s

Precision: binary64

Cost: 36808

Math

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

↓

\[\begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_1 \leq 10^{-293}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\right)}{Om}\right)\right)}\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{+307}:\\ \;\;\;\;\sqrt{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{\frac{\mathsf{fma}\left(n \cdot \left(U* - U\right), \frac{\ell}{Om}, \ell \cdot -2\right)}{\frac{\frac{Om}{U \cdot \ell}}{n}}}\\ \end{array} \]

FPCore

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

↓

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (*
          (* (* 2.0 n) U)
          (+
           (- t (* 2.0 (/ (* l l) Om)))
           (* (* n (pow (/ l Om) 2.0)) (- U* U))))))
   (if (<= t_1 1e-293)
     (sqrt
      (*
       (* 2.0 n)
       (* U (+ t (/ (* l (+ (/ (* n (* l U*)) Om) (* l -2.0))) Om)))))
     (if (<= t_1 4e+307)
       (sqrt t_1)
       (*
        (sqrt 2.0)
        (sqrt
         (/
          (fma (* n (- U* U)) (/ l Om) (* l -2.0))
          (/ (/ Om (* U l)) n))))))))

C

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

↓

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + ((n * pow((l / Om), 2.0)) * (U_42_ - U)));
	double tmp;
	if (t_1 <= 1e-293) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * (((n * (l * U_42_)) / Om) + (l * -2.0))) / Om)))));
	} else if (t_1 <= 4e+307) {
		tmp = sqrt(t_1);
	} else {
		tmp = sqrt(2.0) * sqrt((fma((n * (U_42_ - U)), (l / Om), (l * -2.0)) / ((Om / (U * l)) / n)));
	}
	return tmp;
}

Julia

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

↓

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(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 <= 1e-293)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l * Float64(Float64(Float64(n * Float64(l * U_42_)) / Om) + Float64(l * -2.0))) / Om)))));
	elseif (t_1 <= 4e+307)
		tmp = sqrt(t_1);
	else
		tmp = Float64(sqrt(2.0) * sqrt(Float64(fma(Float64(n * Float64(U_42_ - U)), Float64(l / Om), Float64(l * -2.0)) / Float64(Float64(Om / Float64(U * l)) / n))));
	end
	return tmp
end

Wolfram

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

↓

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(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]}, If[LessEqual[t$95$1, 1e-293], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l * N[(N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 4e+307], N[Sqrt[t$95$1], $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] * N[(l / Om), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(Om / N[(U * l), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))) < 1.0000000000000001e-293

   1. Initial program 12.9%

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

   2. Simplified 42.9%

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

      [Start] 12.9%	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 40.2%	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      sub-neg [=>] 40.2%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]
      associate--l+ [=>] 40.2%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      *-commutative [=>] 40.2%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      distribute-rgt-neg-in [=>] 40.2%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l/ [<=] 40.3%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 40.3%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [<=] 40.3%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [=>] 40.3%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 40.3%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]
      unpow2 [=>] 40.3%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]
      associate-*l* [=>] 42.9%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in U around 0 47.1%

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

   if 1.0000000000000001e-293 < (*.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*)))) < 3.99999999999999994e307

   1. Initial program 97.1%

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

   if 3.99999999999999994e307 < (*.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 19.4%

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

   2. Simplified 29.9%

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

      [Start] 19.4%	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 21.2%	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      sub-neg [=>] 21.2%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]
      associate--l+ [=>] 21.2%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      *-commutative [=>] 21.2%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      distribute-rgt-neg-in [=>] 21.2%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l/ [<=] 28.7%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 28.7%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [<=] 28.7%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [=>] 28.7%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 25.7%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]
      unpow2 [=>] 25.7%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]
      associate-*l* [=>] 26.7%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in t around 0 41.1%

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

   4. Simplified 43.6%

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

      [Start] 41.1%	\[ \sqrt{2} \cdot \sqrt{\frac{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(n \cdot \left(\ell \cdot U\right)\right)}{Om}} \]
      associate-/l* [=>] 41.8%	\[ \sqrt{2} \cdot \sqrt{\color{blue}{\frac{\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}}} \]
      associate-*r* [=>] 42.0%	\[ \sqrt{2} \cdot \sqrt{\frac{\frac{\color{blue}{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}}{Om} + -2 \cdot \ell}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}} \]
      *-commutative [=>] 42.0%	\[ \sqrt{2} \cdot \sqrt{\frac{\frac{\color{blue}{\left(\ell \cdot n\right)} \cdot \left(U* - U\right)}{Om} + -2 \cdot \ell}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}} \]
      associate-*r* [<=] 43.7%	\[ \sqrt{2} \cdot \sqrt{\frac{\frac{\color{blue}{\ell \cdot \left(n \cdot \left(U* - U\right)\right)}}{Om} + -2 \cdot \ell}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}} \]
      associate-*l/ [<=] 42.4%	\[ \sqrt{2} \cdot \sqrt{\frac{\color{blue}{\frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)} + -2 \cdot \ell}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}} \]
      *-commutative [=>] 42.4%	\[ \sqrt{2} \cdot \sqrt{\frac{\color{blue}{\left(n \cdot \left(U* - U\right)\right) \cdot \frac{\ell}{Om}} + -2 \cdot \ell}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}} \]
      fma-def [=>] 42.4%	\[ \sqrt{2} \cdot \sqrt{\frac{\color{blue}{\mathsf{fma}\left(n \cdot \left(U* - U\right), \frac{\ell}{Om}, -2 \cdot \ell\right)}}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}} \]
      *-commutative [=>] 42.4%	\[ \sqrt{2} \cdot \sqrt{\frac{\mathsf{fma}\left(n \cdot \left(U* - U\right), \frac{\ell}{Om}, \color{blue}{\ell \cdot -2}\right)}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}} \]
      *-commutative [=>] 42.4%	\[ \sqrt{2} \cdot \sqrt{\frac{\mathsf{fma}\left(n \cdot \left(U* - U\right), \frac{\ell}{Om}, \ell \cdot -2\right)}{\frac{Om}{\color{blue}{\left(\ell \cdot U\right) \cdot n}}}} \]
      associate-/r* [=>] 43.6%	\[ \sqrt{2} \cdot \sqrt{\frac{\mathsf{fma}\left(n \cdot \left(U* - U\right), \frac{\ell}{Om}, \ell \cdot -2\right)}{\color{blue}{\frac{\frac{Om}{\ell \cdot U}}{n}}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 10^{-293}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 4 \cdot 10^{+307}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{\frac{\mathsf{fma}\left(n \cdot \left(U* - U\right), \frac{\ell}{Om}, \ell \cdot -2\right)}{\frac{\frac{Om}{U \cdot \ell}}{n}}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	64.7%
Cost	36808

\[\begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_1 \leq 10^{-293}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\right)}{Om}\right)\right)}\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{+307}:\\ \;\;\;\;\sqrt{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{\frac{\mathsf{fma}\left(n \cdot \left(U* - U\right), \frac{\ell}{Om}, \ell \cdot -2\right)}{\frac{\frac{Om}{U \cdot \ell}}{n}}}\\ \end{array} \]

Alternative 2
Accuracy	63.9%
Cost	30728

\[\begin{array}{l} t_1 := \frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\\ t_2 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_2 \leq 10^{-293}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot t_1}{Om}\right)\right)}\\ \mathbf{elif}\;t_2 \leq 4 \cdot 10^{+307}:\\ \;\;\;\;\sqrt{t_2}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{t_1}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)\right)}^{0.5}\\ \end{array} \]

Alternative 3
Accuracy	60.7%
Cost	14280

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.6 \cdot 10^{-51}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}}}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 3 \cdot 10^{+97}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(-2 + \left(U* - U\right) \cdot \frac{n}{Om}\right)}}}\right)\\ \end{array} \]

Alternative 4
Accuracy	60.4%
Cost	14152

\[\begin{array}{l} \mathbf{if}\;\ell \leq -3.5 \cdot 10^{-51}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}}}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 1.06 \cdot 10^{+99}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(-2 + \frac{n}{\frac{Om}{U*}}\right)}}}\right)\\ \end{array} \]

Alternative 5
Accuracy	57.3%
Cost	13897

\[\begin{array}{l} \mathbf{if}\;U \leq -2.65 \cdot 10^{+126} \lor \neg \left(U \leq 3.4 \cdot 10^{+122}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}}}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)\right)}^{0.5}\\ \end{array} \]

Alternative 6
Accuracy	56.2%
Cost	8452

\[\begin{array}{l} \mathbf{if}\;U \leq -2 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}}}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)\right)}^{0.5}\\ \end{array} \]

Alternative 7
Accuracy	55.6%
Cost	8192

\[{\left(2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)\right)}^{0.5} \]

Alternative 8
Accuracy	56.7%
Cost	8137

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.15 \cdot 10^{+64} \lor \neg \left(\ell \leq 5.4 \cdot 10^{+90}\right):\\ \;\;\;\;\sqrt{-2 \cdot \left(\frac{n}{Om} \cdot \left(\ell \cdot \left(\ell \cdot \left(U \cdot \left(2 + \left(U - U*\right) \cdot \frac{n}{Om}\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\right)}{Om}\right)\right)}\\ \end{array} \]

Alternative 9
Accuracy	51.1%
Cost	8009

\[\begin{array}{l} \mathbf{if}\;Om \leq -4.5 \cdot 10^{-120} \lor \neg \left(Om \leq 8.5 \cdot 10^{-138}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\frac{n}{Om} \cdot \left(\ell \cdot \left(\ell \cdot \left(U \cdot \left(2 + \left(U - U*\right) \cdot \frac{n}{Om}\right)\right)\right)\right)\right)}\\ \end{array} \]

Alternative 10
Accuracy	49.4%
Cost	7881

\[\begin{array}{l} \mathbf{if}\;Om \leq -8.2 \cdot 10^{-112} \lor \neg \left(Om \leq 9 \cdot 10^{-138}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{n \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(U \cdot \left(2 - \frac{n \cdot U*}{Om}\right)\right)\right)}{Om}}\\ \end{array} \]

Alternative 11
Accuracy	47.5%
Cost	7620

\[\begin{array}{l} \mathbf{if}\;n \leq 1.65 \cdot 10^{+239}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\frac{\ell}{Om} \cdot \frac{n}{\frac{\frac{Om}{\ell}}{U \cdot U*}}\right)}\\ \end{array} \]

Alternative 12
Accuracy	43.8%
Cost	7497

\[\begin{array}{l} \mathbf{if}\;\ell \leq -3.6 \cdot 10^{+69} \lor \neg \left(\ell \leq 1.9 \cdot 10^{+42}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\frac{\ell}{Om} \cdot \left(U \cdot \left(\ell \cdot -2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]

Alternative 13
Accuracy	39.7%
Cost	7369

\[\begin{array}{l} \mathbf{if}\;\ell \leq -4.7 \cdot 10^{+43} \lor \neg \left(\ell \leq 3.1 \cdot 10^{+102}\right):\\ \;\;\;\;\sqrt{-4 \cdot \frac{n}{\frac{Om}{U \cdot \left(\ell \cdot \ell\right)}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 14
Accuracy	39.8%
Cost	7368

\[\begin{array}{l} t_1 := U \cdot \left(\ell \cdot \ell\right)\\ \mathbf{if}\;\ell \leq -1.08 \cdot 10^{+44}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{n}{\frac{Om}{t_1}}}\\ \mathbf{elif}\;\ell \leq 2.2 \cdot 10^{+102}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{n \cdot t_1}{Om}}\\ \end{array} \]

Alternative 15
Accuracy	47.3%
Cost	7360

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

Alternative 16
Accuracy	37.2%
Cost	6912

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

Alternative 17
Accuracy	36.2%
Cost	6848

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

Reproduce

herbie shell --seed 2023269 
(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*))))))