Toniolo and Linder, Equation (13)

Average Accuracy: 46.2% → 58.6%

Time: 48.1s

Precision: binary64

Cost: 27400

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := {\left(\frac{\ell}{Om}\right)}^{2}\\ \mathbf{if}\;n \leq -3.6 \cdot 10^{-137}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot t_1 - n \cdot \left(t_2 \cdot \left(U* - U\right)\right)\right) - t\right)\right)}\\ \mathbf{elif}\;n \leq 4 \cdot 10^{-83}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\frac{n \cdot \ell}{\frac{\frac{Om}{\ell}}{-2}} + n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, t_1, n \cdot \left(t_2 \cdot \left(U - U*\right)\right)\right)\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ l (/ Om l))) (t_2 (pow (/ l Om) 2.0)))
   (if (<= n -3.6e-137)
     (sqrt (* (* n -2.0) (* U (- (- (* 2.0 t_1) (* n (* t_2 (- U* U)))) t))))
     (if (<= n 4e-83)
       (sqrt (* 2.0 (* U (+ (/ (* n l) (/ (/ Om l) -2.0)) (* n t)))))
       (*
        (sqrt (* n 2.0))
        (sqrt (* U (- t (fma 2.0 t_1 (* n (* t_2 (- U 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 = l / (Om / l);
	double t_2 = pow((l / Om), 2.0);
	double tmp;
	if (n <= -3.6e-137) {
		tmp = sqrt(((n * -2.0) * (U * (((2.0 * t_1) - (n * (t_2 * (U_42_ - U)))) - t))));
	} else if (n <= 4e-83) {
		tmp = sqrt((2.0 * (U * (((n * l) / ((Om / l) / -2.0)) + (n * t)))));
	} else {
		tmp = sqrt((n * 2.0)) * sqrt((U * (t - fma(2.0, t_1, (n * (t_2 * (U - U_42_)))))));
	}
	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(l / Float64(Om / l))
	t_2 = Float64(l / Om) ^ 2.0
	tmp = 0.0
	if (n <= -3.6e-137)
		tmp = sqrt(Float64(Float64(n * -2.0) * Float64(U * Float64(Float64(Float64(2.0 * t_1) - Float64(n * Float64(t_2 * Float64(U_42_ - U)))) - t))));
	elseif (n <= 4e-83)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(Float64(Float64(n * l) / Float64(Float64(Om / l) / -2.0)) + Float64(n * t)))));
	else
		tmp = Float64(sqrt(Float64(n * 2.0)) * sqrt(Float64(U * Float64(t - fma(2.0, t_1, Float64(n * Float64(t_2 * Float64(U - U_42_))))))));
	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[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[n, -3.6e-137], N[Sqrt[N[(N[(n * -2.0), $MachinePrecision] * N[(U * N[(N[(N[(2.0 * t$95$1), $MachinePrecision] - N[(n * N[(t$95$2 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 4e-83], N[Sqrt[N[(2.0 * N[(U * N[(N[(N[(n * l), $MachinePrecision] / N[(N[(Om / l), $MachinePrecision] / -2.0), $MachinePrecision]), $MachinePrecision] + N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t - N[(2.0 * t$95$1 + N[(n * N[(t$95$2 * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.60000000000000006e-137

   1. Initial program 50.2%

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

   2. Simplified 54.4%

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

      [Start] 50.2	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 50.1	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 50.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      sub-neg [=>] 50.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(-\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [<=] 50.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      cancel-sign-sub [<=] 50.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      cancel-sign-sub [=>] 50.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-/l* [=>] 54.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 54.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   if -3.60000000000000006e-137 < n < 4.0000000000000001e-83

   1. Initial program 41.8%

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

   2. Taylor expanded in n around 0 40.5%

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

   3. Simplified 51.8%

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

      [Start] 40.5	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot U\right)\right)} \]
      associate-*r* [=>] 45.5	\[ \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right)}} \]
      *-commutative [=>] 45.5	\[ \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
      unpow2 [=>] 45.5	\[ \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right)\right)\right)} \]
      associate-*r/ [<=] 51.8	\[ \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)}\right)\right)\right)} \]
      cancel-sign-sub-inv [=>] 51.8	\[ \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{\left(t + \left(-2\right) \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\right)\right)} \]
      metadata-eval [=>] 51.8	\[ \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \color{blue}{-2} \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)} \]
      *-commutative [<=] 51.8	\[ \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2}\right)\right)\right)} \]
      *-commutative [=>] 51.8	\[ \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot -2\right)\right)\right)} \]
      associate-*l* [=>] 51.8	\[ \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)}\right)\right)\right)} \]

   4. Applied egg-rr 58.8%

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

      [Start] 51.8	\[ \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)\right)} \]
      distribute-lft-in [=>] 51.8	\[ \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t + n \cdot \left(\frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)}\right)} \]
      +-commutative [=>] 51.8	\[ \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot \left(\frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right) + n \cdot t\right)}\right)} \]
      *-commutative [=>] 51.8	\[ \sqrt{2 \cdot \left(U \cdot \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right) \cdot n} + n \cdot t\right)\right)} \]
      associate-*l* [=>] 58.8	\[ \sqrt{2 \cdot \left(U \cdot \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\left(\ell \cdot -2\right) \cdot n\right)} + n \cdot t\right)\right)} \]

   5. Applied egg-rr 59.0%

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

      [Start] 58.8	\[ \sqrt{2 \cdot \left(U \cdot \left(\frac{\ell}{Om} \cdot \left(\left(\ell \cdot -2\right) \cdot n\right) + n \cdot t\right)\right)} \]
      associate-*l/ [=>] 57.1	\[ \sqrt{2 \cdot \left(U \cdot \left(\color{blue}{\frac{\ell \cdot \left(\left(\ell \cdot -2\right) \cdot n\right)}{Om}} + n \cdot t\right)\right)} \]
      *-commutative [=>] 57.1	\[ \sqrt{2 \cdot \left(U \cdot \left(\frac{\color{blue}{\left(\left(\ell \cdot -2\right) \cdot n\right) \cdot \ell}}{Om} + n \cdot t\right)\right)} \]
      associate-/l* [=>] 59.0	\[ \sqrt{2 \cdot \left(U \cdot \left(\color{blue}{\frac{\left(\ell \cdot -2\right) \cdot n}{\frac{Om}{\ell}}} + n \cdot t\right)\right)} \]
      *-commutative [=>] 59.0	\[ \sqrt{2 \cdot \left(U \cdot \left(\frac{\color{blue}{n \cdot \left(\ell \cdot -2\right)}}{\frac{Om}{\ell}} + n \cdot t\right)\right)} \]
      associate-*r* [=>] 59.0	\[ \sqrt{2 \cdot \left(U \cdot \left(\frac{\color{blue}{\left(n \cdot \ell\right) \cdot -2}}{\frac{Om}{\ell}} + n \cdot t\right)\right)} \]
      associate-/l* [=>] 59.0	\[ \sqrt{2 \cdot \left(U \cdot \left(\color{blue}{\frac{n \cdot \ell}{\frac{\frac{Om}{\ell}}{-2}}} + n \cdot t\right)\right)} \]

   if 4.0000000000000001e-83 < n

   1. Initial program 48.5%

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

   2. Simplified 52.3%

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

      [Start] 48.5	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 47.4	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 47.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      sub-neg [=>] 47.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(-\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [<=] 47.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      cancel-sign-sub [<=] 47.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      cancel-sign-sub [=>] 47.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-/l* [=>] 51.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 52.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Applied egg-rr 45.8%

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

      [Start] 52.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)} \]
      sqrt-prod [=>] 62.9	\[ \color{blue}{\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)}} \]
      fma-def [=>] 62.9	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \color{blue}{\mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)}\right)} \]
      associate-/r/ [=>] 62.9	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \color{blue}{\frac{\ell}{Om} \cdot \ell}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l/ [=>] 57.3	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \color{blue}{\frac{\ell \cdot \ell}{Om}}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [=>] 57.3	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right) \cdot n}\right)\right)} \]
      associate-*l* [=>] 45.8	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U - U*\right) \cdot n\right)}\right)\right)} \]

   4. Simplified 62.9%

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

      [Start] 45.8	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U - U*\right) \cdot n\right)\right)\right)} \]
      associate-/l* [=>] 51.0	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U - U*\right) \cdot n\right)\right)\right)} \]
      associate-*r* [=>] 62.9	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right) \cdot n}\right)\right)} \]
      *-commutative [=>] 62.9	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)} \]
      *-commutative [=>] 62.9	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, n \cdot \color{blue}{\left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right)\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.6 \cdot 10^{-137}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right) - t\right)\right)}\\ \mathbf{elif}\;n \leq 4 \cdot 10^{-83}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\frac{n \cdot \ell}{\frac{\frac{Om}{\ell}}{-2}} + n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	58.9%
Cost	51468

\[\begin{array}{l} t_1 := \sqrt{\left(\left(n \cdot 2\right) \cdot U\right) \cdot \left(\left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\sqrt{n \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)} \cdot \sqrt{2 \cdot U}\\ \mathbf{elif}\;t_1 \leq 10^{+133}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t + \frac{\ell}{Om} \cdot \left(n \cdot \left(\ell \cdot -2\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(\frac{U - U*}{Om} \cdot \frac{n}{Om} + \frac{2}{Om}\right) \cdot \left(n \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \end{array} \]

Alternative 2
Accuracy	55.0%
Cost	14596

\[\begin{array}{l} t_1 := 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;n \leq -2.15 \cdot 10^{-137}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(t_1 - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right) - t\right)\right)}\\ \mathbf{elif}\;n \leq 7.2 \cdot 10^{-70}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\frac{n \cdot \ell}{\frac{\frac{Om}{\ell}}{-2}} + n \cdot t\right)\right)}\\ \mathbf{elif}\;n \leq 5.8 \cdot 10^{+203}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\left(t_1 - \frac{n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om}\right) - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot t\right)} \cdot \sqrt{n}\\ \end{array} \]

Alternative 3
Accuracy	54.9%
Cost	13644

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;n \leq -6800000000000:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(t_1 \cdot -2 + n \cdot \left(t_1 \cdot \frac{U* - U}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{-62}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\frac{n \cdot \ell}{\frac{\frac{Om}{\ell}}{-2}} + n \cdot t\right)\right)}\\ \mathbf{elif}\;n \leq 3.8 \cdot 10^{+215}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\left(2 \cdot t_1 - \frac{n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om}\right) - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot t\right)} \cdot \sqrt{n}\\ \end{array} \]

Alternative 4
Accuracy	49.3%
Cost	8660

\[\begin{array}{l} t_1 := \sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \frac{n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om}\right) - t\right)\right)\right)}\\ \mathbf{if}\;Om \leq -1.78 \cdot 10^{-180}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\frac{n \cdot \ell}{\frac{\frac{Om}{\ell}}{-2}} + n \cdot t\right)\right)}\\ \mathbf{elif}\;Om \leq 3 \cdot 10^{-151}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;Om \leq 1.6 \cdot 10^{-90}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(\frac{U - U*}{Om} \cdot \frac{n}{Om} + \frac{2}{Om}\right) \cdot \left(n \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \mathbf{elif}\;Om \leq 2.2 \cdot 10^{-26}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;Om \leq 1.6 \cdot 10^{+89}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(t - \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{Om} - \frac{n}{Om \cdot Om} \cdot \left(U* - U\right)\right)\right) \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t + \frac{\ell}{\frac{Om}{\ell \cdot \left(n \cdot -2\right)}}\right)\right)}\\ \end{array} \]

Alternative 5
Accuracy	54.0%
Cost	8392

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;n \leq -7200000000000:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(t_1 \cdot -2 + n \cdot \left(t_1 \cdot \frac{U* - U}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq 1.1 \cdot 10^{-63}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\frac{n \cdot \ell}{\frac{\frac{Om}{\ell}}{-2}} + n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\left(2 \cdot t_1 - \frac{n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om}\right) - t\right)\right)\right)}\\ \end{array} \]

Alternative 6
Accuracy	46.8%
Cost	7756

\[\begin{array}{l} t_1 := \sqrt{-2 \cdot \left(\frac{\ell}{Om} \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \ell\right)\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -8.5 \cdot 10^{+18}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -5.5 \cdot 10^{-15}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.65 \cdot 10^{+131}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\frac{\ell}{Om} \cdot \left(2 \cdot \ell\right) - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Accuracy	48.6%
Cost	7756

\[\begin{array}{l} t_1 := \sqrt{-2 \cdot \left(\frac{\ell}{Om} \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \ell\right)\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -1.15 \cdot 10^{+187}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -2.05 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)}\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{+131}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\frac{2}{\frac{\frac{Om}{\ell}}{\ell}} - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Accuracy	44.7%
Cost	7753

\[\begin{array}{l} \mathbf{if}\;U \leq -3.6 \cdot 10^{-126} \lor \neg \left(U \leq -1.32 \cdot 10^{-288}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t + \frac{\ell}{Om} \cdot \left(n \cdot \left(\ell \cdot -2\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \frac{n}{Om}\right) \cdot \sqrt{-2 \cdot \left(U \cdot \left(U - U*\right)\right)}\\ \end{array} \]

Alternative 9
Accuracy	44.8%
Cost	7753

\[\begin{array}{l} \mathbf{if}\;U \leq -3.6 \cdot 10^{-126} \lor \neg \left(U \leq -1.32 \cdot 10^{-288}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\frac{n \cdot \ell}{\frac{\frac{Om}{\ell}}{-2}} + n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \frac{n}{Om}\right) \cdot \sqrt{-2 \cdot \left(U \cdot \left(U - U*\right)\right)}\\ \end{array} \]

Alternative 10
Accuracy	36.9%
Cost	7497

\[\begin{array}{l} \mathbf{if}\;t \leq -8.6 \cdot 10^{-304} \lor \neg \left(t \leq 7 \cdot 10^{-184}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \frac{n \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)}{Om}\right)}\\ \end{array} \]

Alternative 11
Accuracy	42.6%
Cost	7497

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.75 \cdot 10^{+16} \lor \neg \left(\ell \leq 4 \cdot 10^{+82}\right):\\ \;\;\;\;\sqrt{-2 \cdot \left(\ell \cdot \left(2 \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot U\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]

Alternative 12
Accuracy	44.5%
Cost	7497

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.75 \cdot 10^{+16} \lor \neg \left(\ell \leq 5.8 \cdot 10^{+86}\right):\\ \;\;\;\;\sqrt{-2 \cdot \left(\frac{\ell}{Om} \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \ell\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]

Alternative 13
Accuracy	39.2%
Cost	7364

\[\begin{array}{l} \mathbf{if}\;\ell \leq -9.2 \cdot 10^{+18}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \frac{\ell}{\frac{Om}{\frac{\ell}{-0.5}}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]

Alternative 14
Accuracy	37.6%
Cost	6848

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

Reproduce

herbie shell --seed 2023133 
(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*))))))