Toniolo and Linder, Equation (13)

Average Error: 34.4 → 27.6

Time: 42.0s

Precision: binary64

Cost: 27528

Math

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

↓

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ t_2 := \frac{\ell \cdot \ell}{Om}\\ t_3 := {\left(\frac{\ell}{Om}\right)}^{2}\\ \mathbf{if}\;n \leq -3.4 \cdot 10^{+47}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{\frac{Om}{\ell}} \cdot -2 + n \cdot \left(t_2 \cdot \frac{U*}{Om} - t_2 \cdot \frac{U}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq 1.9 \cdot 10^{-301}:\\ \;\;\;\;{\left({\left(U \cdot \left(\left(n \cdot 2\right) \cdot \left(t - \mathsf{fma}\left(2, t_1, t_3 \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)\right)}^{0.25}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, t_1, \left(U - U*\right) \cdot \left(n \cdot t_3\right)\right)\right)}\\ \end{array} \]

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 (/ l Om))) (t_2 (/ (* l l) Om)) (t_3 (pow (/ l Om) 2.0)))
   (if (<= n -3.4e+47)
     (sqrt
      (*
       (* n 2.0)
       (*
        U
        (+
         t
         (+
          (* (/ l (/ Om l)) -2.0)
          (* n (- (* t_2 (/ U* Om)) (* t_2 (/ U Om)))))))))
     (if (<= n 1.9e-301)
       (pow
        (pow
         (* U (* (* n 2.0) (- t (fma 2.0 t_1 (* t_3 (* n (- U U*)))))))
         0.25)
        2.0)
       (*
        (sqrt (* n 2.0))
        (sqrt (* U (- t (fma 2.0 t_1 (* (- U U*) (* n t_3)))))))))))

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 * (l / Om);
	double t_2 = (l * l) / Om;
	double t_3 = pow((l / Om), 2.0);
	double tmp;
	if (n <= -3.4e+47) {
		tmp = sqrt(((n * 2.0) * (U * (t + (((l / (Om / l)) * -2.0) + (n * ((t_2 * (U_42_ / Om)) - (t_2 * (U / Om)))))))));
	} else if (n <= 1.9e-301) {
		tmp = pow(pow((U * ((n * 2.0) * (t - fma(2.0, t_1, (t_3 * (n * (U - U_42_))))))), 0.25), 2.0);
	} else {
		tmp = sqrt((n * 2.0)) * sqrt((U * (t - fma(2.0, t_1, ((U - U_42_) * (n * t_3))))));
	}
	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(l / Om))
	t_2 = Float64(Float64(l * l) / Om)
	t_3 = Float64(l / Om) ^ 2.0
	tmp = 0.0
	if (n <= -3.4e+47)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t + Float64(Float64(Float64(l / Float64(Om / l)) * -2.0) + Float64(n * Float64(Float64(t_2 * Float64(U_42_ / Om)) - Float64(t_2 * Float64(U / Om)))))))));
	elseif (n <= 1.9e-301)
		tmp = (Float64(U * Float64(Float64(n * 2.0) * Float64(t - fma(2.0, t_1, Float64(t_3 * Float64(n * Float64(U - U_42_))))))) ^ 0.25) ^ 2.0;
	else
		tmp = Float64(sqrt(Float64(n * 2.0)) * sqrt(Float64(U * Float64(t - fma(2.0, t_1, Float64(Float64(U - U_42_) * Float64(n * t_3)))))));
	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[(l / Om), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[n, -3.4e+47], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t + N[(N[(N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision] + N[(n * N[(N[(t$95$2 * N[(U$42$ / Om), $MachinePrecision]), $MachinePrecision] - N[(t$95$2 * N[(U / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 1.9e-301], N[Power[N[Power[N[(U * N[(N[(n * 2.0), $MachinePrecision] * N[(t - N[(2.0 * t$95$1 + N[(t$95$3 * N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision], 2.0], $MachinePrecision], N[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t - N[(2.0 * t$95$1 + N[(N[(U - U$42$), $MachinePrecision] * N[(n * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.3999999999999998e47

   1. Initial program 34.1

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

   2. Simplified 31.9

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

      [Start] 34.1	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 34.7	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 34.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      sub-neg [=>] 34.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(-\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [<=] 34.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      cancel-sign-sub [<=] 34.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      cancel-sign-sub [=>] 34.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-/l* [=>] 32.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 31.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in U around 0 37.2

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

   4. Simplified 35.0

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

      [Start] 37.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(\frac{{\ell}^{2} \cdot U}{{Om}^{2}} + -1 \cdot \frac{{\ell}^{2} \cdot U*}{{Om}^{2}}\right)\right)\right)\right)} \]
      mul-1-neg [=>] 37.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(\frac{{\ell}^{2} \cdot U}{{Om}^{2}} + \color{blue}{\left(-\frac{{\ell}^{2} \cdot U*}{{Om}^{2}}\right)}\right)\right)\right)\right)} \]
      unsub-neg [=>] 37.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \color{blue}{\left(\frac{{\ell}^{2} \cdot U}{{Om}^{2}} - \frac{{\ell}^{2} \cdot U*}{{Om}^{2}}\right)}\right)\right)\right)} \]
      unpow2 [=>] 37.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(\frac{{\ell}^{2} \cdot U}{\color{blue}{Om \cdot Om}} - \frac{{\ell}^{2} \cdot U*}{{Om}^{2}}\right)\right)\right)\right)} \]
      times-frac [=>] 37.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(\color{blue}{\frac{{\ell}^{2}}{Om} \cdot \frac{U}{Om}} - \frac{{\ell}^{2} \cdot U*}{{Om}^{2}}\right)\right)\right)\right)} \]
      unpow2 [=>] 37.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{Om} \cdot \frac{U}{Om} - \frac{{\ell}^{2} \cdot U*}{{Om}^{2}}\right)\right)\right)\right)} \]
      unpow2 [=>] 37.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(\frac{\ell \cdot \ell}{Om} \cdot \frac{U}{Om} - \frac{{\ell}^{2} \cdot U*}{\color{blue}{Om \cdot Om}}\right)\right)\right)\right)} \]
      times-frac [=>] 35.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(\frac{\ell \cdot \ell}{Om} \cdot \frac{U}{Om} - \color{blue}{\frac{{\ell}^{2}}{Om} \cdot \frac{U*}{Om}}\right)\right)\right)\right)} \]
      unpow2 [=>] 35.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(\frac{\ell \cdot \ell}{Om} \cdot \frac{U}{Om} - \frac{\color{blue}{\ell \cdot \ell}}{Om} \cdot \frac{U*}{Om}\right)\right)\right)\right)} \]

   if -3.3999999999999998e47 < n < 1.89999999999999998e-301

   1. Initial program 35.0

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

   2. Simplified 35.4

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

      [Start] 35.0	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 34.5	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 34.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      fma-def [=>] 34.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-*l* [=>] 35.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Applied egg-rr 29.4

      \[\leadsto \color{blue}{{\left({\left(U \cdot \left(\left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U - U*\right) \cdot n\right)\right)\right) \cdot \left(2 \cdot n\right)\right)\right)}^{0.25}\right)}^{2}} \]

   if 1.89999999999999998e-301 < n

   1. Initial program 34.2

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

   2. Applied egg-rr 27.0

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

   3. Simplified 23.9

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

      [Start] 27.0	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*r* [=>] 23.9	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U - U*\right)}\right)\right)} \]
      *-commutative [<=] 23.9	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)} \cdot \left(U - U*\right)\right)\right)} \]
      *-commutative [=>] 23.9	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \color{blue}{\left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right)\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 27.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.4 \cdot 10^{+47}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{\frac{Om}{\ell}} \cdot -2 + n \cdot \left(\frac{\ell \cdot \ell}{Om} \cdot \frac{U*}{Om} - \frac{\ell \cdot \ell}{Om} \cdot \frac{U}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq 1.9 \cdot 10^{-301}:\\ \;\;\;\;{\left({\left(U \cdot \left(\left(n \cdot 2\right) \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)\right)}^{0.25}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	25.4
Cost	51532

\[\begin{array}{l} t_1 := \left(U \cdot \ell\right) \cdot \left(n \cdot \ell\right)\\ t_2 := n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\\ t_3 := \sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(U - U*\right) \cdot t_2\right)\right)}\\ t_4 := \left(\left(n \cdot 2\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + t_2 \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_4 \leq 0:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_4 \leq 2 \cdot 10^{+296}:\\ \;\;\;\;\sqrt{t_4}\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_1 \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om}\right) + t_1 \cdot \frac{-2}{Om}\right)}\\ \end{array} \]

Alternative 2
Error	24.4
Cost	37900

\[\begin{array}{l} t_1 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := \left(\left(n \cdot 2\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot t_1\right) \cdot \left(U* - U\right)\right)\\ t_3 := \left(U \cdot \ell\right) \cdot \left(n \cdot \ell\right)\\ \mathbf{if}\;t_2 \leq 0:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(n \cdot \left(t_1 \cdot \left(U* - U\right)\right) - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+296}:\\ \;\;\;\;\sqrt{t_2}\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;\left|\frac{\left(n \cdot \ell\right) \cdot \sqrt{2 \cdot \left(U \cdot U*\right)}}{Om}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_3 \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om}\right) + t_3 \cdot \frac{-2}{Om}\right)}\\ \end{array} \]

Alternative 3
Error	24.4
Cost	37900

\[\begin{array}{l} t_1 := \left(U \cdot \ell\right) \cdot \left(n \cdot \ell\right)\\ t_2 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_3 := \left(\left(n \cdot 2\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot t_2\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_3 \leq 0:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \mathsf{fma}\left(n, t_2 \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}\\ \mathbf{elif}\;t_3 \leq 2 \cdot 10^{+296}:\\ \;\;\;\;\sqrt{t_3}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;\left|\frac{\left(n \cdot \ell\right) \cdot \sqrt{2 \cdot \left(U \cdot U*\right)}}{Om}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_1 \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om}\right) + t_1 \cdot \frac{-2}{Om}\right)}\\ \end{array} \]

Alternative 4
Error	29.5
Cost	14992

\[\begin{array}{l} t_1 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right) - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{if}\;U \leq -2.5 \cdot 10^{+157}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;U \leq -8 \cdot 10^{-193}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;U \leq -2.4 \cdot 10^{-269}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \frac{\ell \cdot \left(U \cdot \ell\right)}{\frac{Om}{n}}\right)}\\ \mathbf{elif}\;U \leq 280:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot U} \cdot \sqrt{n \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)}\\ \end{array} \]

Alternative 5
Error	31.5
Cost	14420

\[\begin{array}{l} t_1 := \frac{\ell \cdot \ell}{Om}\\ t_2 := \ell \cdot \frac{\ell}{Om}\\ t_3 := \sqrt{2 \cdot U} \cdot \sqrt{n \cdot \left(t + t_2 \cdot -2\right)}\\ \mathbf{if}\;U \leq -8.5 \cdot 10^{-193}:\\ \;\;\;\;\sqrt{2 \cdot \left(\mathsf{fma}\left(t_2, -2, t\right) \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;U \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \frac{\ell \cdot \left(U \cdot \ell\right)}{\frac{Om}{n}}\right)}\\ \mathbf{elif}\;U \leq 3.6 \cdot 10^{-122}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;U \leq 2.5 \cdot 10^{-91}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{\frac{Om}{\ell}} \cdot -2 + n \cdot \left(t_1 \cdot \frac{U*}{Om} - t_1 \cdot \frac{U}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;U \leq 8.6 \cdot 10^{-24}:\\ \;\;\;\;\left|\frac{\left(n \cdot \ell\right) \cdot \sqrt{2 \cdot \left(U \cdot U*\right)}}{Om}\right|\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 6
Error	30.0
Cost	14164

\[\begin{array}{l} t_1 := \frac{\ell \cdot \ell}{Om}\\ t_2 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{\frac{Om}{\ell}} \cdot -2 + n \cdot \left(t_1 \cdot \frac{U*}{Om} - t_1 \cdot \frac{U}{Om}\right)\right)\right)\right)}\\ \mathbf{if}\;n \leq -1.18 \cdot 10^{+35}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;n \leq 8 \cdot 10^{+49}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\left(n \cdot \ell\right) \cdot \left(\frac{\ell}{Om} \cdot -2\right) + n \cdot t\right)\right)}\\ \mathbf{elif}\;n \leq 1.6 \cdot 10^{+106}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;n \leq 10^{+127}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \ell \cdot \frac{2 \cdot \ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;n \leq 7.2 \cdot 10^{+160}:\\ \;\;\;\;\left|\frac{\left(n \cdot \ell\right) \cdot \sqrt{2 \cdot \left(U \cdot U*\right)}}{Om}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot t} \cdot \sqrt{n + n}\\ \end{array} \]

Alternative 7
Error	29.5
Cost	14160

\[\begin{array}{l} t_1 := \frac{\ell \cdot \ell}{Om}\\ t_2 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{\frac{Om}{\ell}} \cdot -2 + n \cdot \left(t_1 \cdot \frac{U*}{Om} - t_1 \cdot \frac{U}{Om}\right)\right)\right)\right)}\\ \mathbf{if}\;n \leq -2.95 \cdot 10^{+33}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;n \leq 8 \cdot 10^{+49}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\left(n \cdot \ell\right) \cdot \left(\frac{\ell}{Om} \cdot -2\right) + n \cdot t\right)\right)}\\ \mathbf{elif}\;n \leq 7.8 \cdot 10^{+93}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{+170}:\\ \;\;\;\;\sqrt{2 \cdot \left(\mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right) \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot t} \cdot \sqrt{n + n}\\ \end{array} \]

Alternative 8
Error	29.9
Cost	13512

\[\begin{array}{l} t_1 := \frac{\ell \cdot \ell}{Om}\\ \mathbf{if}\;n \leq -3.5 \cdot 10^{+34}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{\frac{Om}{\ell}} \cdot -2 + n \cdot \left(t_1 \cdot \frac{U*}{Om} - t_1 \cdot \frac{U}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq 1.12 \cdot 10^{-41}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\left(n \cdot \ell\right) \cdot \left(\frac{\ell}{Om} \cdot -2\right) + n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot t} \cdot \sqrt{n + n}\\ \end{array} \]

Alternative 9
Error	29.9
Cost	13512

\[\begin{array}{l} t_1 := \frac{\ell \cdot \ell}{Om}\\ \mathbf{if}\;n \leq -1.15 \cdot 10^{+34}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{\frac{Om}{\ell}} \cdot -2 + n \cdot \left(t_1 \cdot \frac{U*}{Om} - t_1 \cdot \frac{U}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq 5.2 \cdot 10^{-42}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\left(n \cdot \ell\right) \cdot \left(\frac{\ell}{Om} \cdot -2\right) + n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot U\right)} \cdot \sqrt{n}\\ \end{array} \]

Alternative 10
Error	31.2
Cost	9296

\[\begin{array}{l} t_1 := \frac{\ell \cdot \ell}{Om}\\ t_2 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{\frac{Om}{\ell}} \cdot -2 + n \cdot \left(t_1 \cdot \frac{U*}{Om} - t_1 \cdot \frac{U}{Om}\right)\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -2.3 \cdot 10^{+148}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\left(n \cdot \ell\right) \cdot \left(\frac{\ell}{Om} \cdot -2\right) + n \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq -4.5 \cdot 10^{-86}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 4.5 \cdot 10^{-210}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(t + t_1 \cdot -2\right) \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.1 \cdot 10^{+142}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t - \frac{2 \cdot \ell}{\frac{Om}{n \cdot \ell}}\right)\right)}\\ \end{array} \]

Alternative 11
Error	32.7
Cost	7753

\[\begin{array}{l} \mathbf{if}\;Om \leq -4 \cdot 10^{-181} \lor \neg \left(Om \leq 1.4 \cdot 10^{-145}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\left(n \cdot \ell\right) \cdot \left(\frac{\ell}{Om} \cdot -2\right) + n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(U \cdot U*\right)}}{\frac{Om}{n \cdot \ell}}\\ \end{array} \]

Alternative 12
Error	34.3
Cost	7625

\[\begin{array}{l} \mathbf{if}\;Om \leq -4 \cdot 10^{-181} \lor \neg \left(Om \leq 8.6 \cdot 10^{-157}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \ell \cdot \frac{2 \cdot \ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(U \cdot U*\right)}}{\frac{Om}{n \cdot \ell}}\\ \end{array} \]

Alternative 13
Error	33.2
Cost	7624

\[\begin{array}{l} \mathbf{if}\;U \leq -2.3 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;U \leq 2 \cdot 10^{+42}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \ell \cdot \left(\frac{\ell}{Om} \cdot -2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \ell \cdot \frac{2 \cdot \ell}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 14
Error	40.3
Cost	7496

\[\begin{array}{l} \mathbf{if}\;t \leq -3.65 \cdot 10^{-233}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-69}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(-2 \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \ell\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]

Alternative 15
Error	39.9
Cost	7368

\[\begin{array}{l} \mathbf{if}\;Om \leq -1.9 \cdot 10^{-244}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;Om \leq 4.2 \cdot 10^{-173}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \frac{n}{\frac{Om}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]

Alternative 16
Error	39.8
Cost	7368

\[\begin{array}{l} \mathbf{if}\;Om \leq -2.5 \cdot 10^{-255}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;Om \leq 2.5 \cdot 10^{-156}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(U \cdot U*\right)}}{\frac{Om}{n \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]

Alternative 17
Error	39.5
Cost	6980

\[\begin{array}{l} \mathbf{if}\;U \leq -1.9 \cdot 10^{-187}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]

Alternative 18
Error	39.8
Cost	6848

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

Reproduce

herbie shell --seed 2023045 
(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*))))))