Toniolo and Linder, Equation (13)

Average Error: 34.1 → 26.8

Time: 28.8s

Precision: binary64

Cost: 27928

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)}\\ t_2 := \frac{\ell}{Om} \cdot \left(U \cdot \left(n \cdot 2\right)\right)\\ t_3 := \left(n \cdot U\right) \cdot \left(2 \cdot t\right)\\ \mathbf{if}\;n \leq -1 \cdot 10^{-37}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(n, \left(U* - U\right) \cdot \frac{\ell}{Om}, \ell \cdot -2\right) \cdot t_2 + t_3}\\ \mathbf{elif}\;n \leq 4.4 \cdot 10^{-251}:\\ \;\;\;\;{\left(\mathsf{fma}\left(n, t \cdot \left(U \cdot 2\right), \left(-2 - \frac{n}{Om} \cdot \left(U - U*\right)\right) \cdot \left(\ell \cdot \left(\left(n \cdot 2\right) \cdot \frac{U}{\frac{Om}{\ell}}\right)\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;n \leq 2 \cdot 10^{-223}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq 7 \cdot 10^{-172}:\\ \;\;\;\;\sqrt{t_3 + \left(\ell \cdot \left(-2 + \frac{n \cdot \left(U* - U\right)}{Om}\right)\right) \cdot \left(2 \cdot \frac{n \cdot \left(U \cdot \ell\right)}{Om}\right)}\\ \mathbf{elif}\;n \leq 1.1 \cdot 10^{-102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq 10^{+45}:\\ \;\;\;\;\sqrt{t_3 + t_2 \cdot \left(\ell \cdot \left(-2 + \frac{n \cdot U*}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (* n 2.0))
          (sqrt
           (*
            U
            (-
             t
             (fma
              2.0
              (* l (/ l Om))
              (* n (* (pow (/ l Om) 2.0) (- U U*)))))))))
        (t_2 (* (/ l Om) (* U (* n 2.0))))
        (t_3 (* (* n U) (* 2.0 t))))
   (if (<= n -1e-37)
     (sqrt (+ (* (fma n (* (- U* U) (/ l Om)) (* l -2.0)) t_2) t_3))
     (if (<= n 4.4e-251)
       (pow
        (fma
         n
         (* t (* U 2.0))
         (* (- -2.0 (* (/ n Om) (- U U*))) (* l (* (* n 2.0) (/ U (/ Om l))))))
        0.5)
       (if (<= n 2e-223)
         t_1
         (if (<= n 7e-172)
           (sqrt
            (+
             t_3
             (*
              (* l (+ -2.0 (/ (* n (- U* U)) Om)))
              (* 2.0 (/ (* n (* U l)) Om)))))
           (if (<= n 1.1e-102)
             t_1
             (if (<= n 1e+45)
               (sqrt (+ t_3 (* t_2 (* l (+ -2.0 (/ (* n U*) Om))))))
               t_1))))))))

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((n * 2.0)) * sqrt((U * (t - fma(2.0, (l * (l / Om)), (n * (pow((l / Om), 2.0) * (U - U_42_)))))));
	double t_2 = (l / Om) * (U * (n * 2.0));
	double t_3 = (n * U) * (2.0 * t);
	double tmp;
	if (n <= -1e-37) {
		tmp = sqrt(((fma(n, ((U_42_ - U) * (l / Om)), (l * -2.0)) * t_2) + t_3));
	} else if (n <= 4.4e-251) {
		tmp = pow(fma(n, (t * (U * 2.0)), ((-2.0 - ((n / Om) * (U - U_42_))) * (l * ((n * 2.0) * (U / (Om / l)))))), 0.5);
	} else if (n <= 2e-223) {
		tmp = t_1;
	} else if (n <= 7e-172) {
		tmp = sqrt((t_3 + ((l * (-2.0 + ((n * (U_42_ - U)) / Om))) * (2.0 * ((n * (U * l)) / Om)))));
	} else if (n <= 1.1e-102) {
		tmp = t_1;
	} else if (n <= 1e+45) {
		tmp = sqrt((t_3 + (t_2 * (l * (-2.0 + ((n * U_42_) / Om))))));
	} else {
		tmp = t_1;
	}
	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(sqrt(Float64(n * 2.0)) * sqrt(Float64(U * Float64(t - fma(2.0, Float64(l * Float64(l / Om)), Float64(n * Float64((Float64(l / Om) ^ 2.0) * Float64(U - U_42_))))))))
	t_2 = Float64(Float64(l / Om) * Float64(U * Float64(n * 2.0)))
	t_3 = Float64(Float64(n * U) * Float64(2.0 * t))
	tmp = 0.0
	if (n <= -1e-37)
		tmp = sqrt(Float64(Float64(fma(n, Float64(Float64(U_42_ - U) * Float64(l / Om)), Float64(l * -2.0)) * t_2) + t_3));
	elseif (n <= 4.4e-251)
		tmp = fma(n, Float64(t * Float64(U * 2.0)), Float64(Float64(-2.0 - Float64(Float64(n / Om) * Float64(U - U_42_))) * Float64(l * Float64(Float64(n * 2.0) * Float64(U / Float64(Om / l)))))) ^ 0.5;
	elseif (n <= 2e-223)
		tmp = t_1;
	elseif (n <= 7e-172)
		tmp = sqrt(Float64(t_3 + Float64(Float64(l * Float64(-2.0 + Float64(Float64(n * Float64(U_42_ - U)) / Om))) * Float64(2.0 * Float64(Float64(n * Float64(U * l)) / Om)))));
	elseif (n <= 1.1e-102)
		tmp = t_1;
	elseif (n <= 1e+45)
		tmp = sqrt(Float64(t_3 + Float64(t_2 * Float64(l * Float64(-2.0 + Float64(Float64(n * U_42_) / Om))))));
	else
		tmp = t_1;
	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[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + N[(n * N[(N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(l / Om), $MachinePrecision] * N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(n * U), $MachinePrecision] * N[(2.0 * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1e-37], N[Sqrt[N[(N[(N[(n * N[(N[(U$42$ - U), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 4.4e-251], N[Power[N[(n * N[(t * N[(U * 2.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 - N[(N[(n / Om), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * N[(N[(n * 2.0), $MachinePrecision] * N[(U / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[n, 2e-223], t$95$1, If[LessEqual[n, 7e-172], N[Sqrt[N[(t$95$3 + N[(N[(l * N[(-2.0 + N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[(n * N[(U * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 1.1e-102], t$95$1, If[LessEqual[n, 1e+45], N[Sqrt[N[(t$95$3 + N[(t$95$2 * N[(l * N[(-2.0 + N[(N[(n * U$42$), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if n < -1.00000000000000007e-37

   1. Initial program 32.5

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

   2. Simplified 33.2

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

   3. Applied egg-rr 27.1

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

   if -1.00000000000000007e-37 < n < 4.4e-251

   1. Initial program 35.7

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

   2. Simplified 32.0

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

   3. Applied egg-rr 31.4

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

   4. Taylor expanded in l around 0 29.3

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

   5. Applied egg-rr 27.6

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(n, \left(U \cdot 2\right) \cdot t, \left(\frac{n}{Om} \cdot \left(U* - U\right) + -2\right) \cdot \left(\ell \cdot \left(\left(n \cdot 2\right) \cdot \frac{U}{\frac{Om}{\ell}}\right)\right)\right)\right)}^{0.5}} \]

   if 4.4e-251 < n < 1.9999999999999999e-223 or 7.00000000000000057e-172 < n < 1.10000000000000006e-102 or 9.9999999999999993e44 < n

   1. Initial program 34.5

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

   2. Applied egg-rr 25.0

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

   if 1.9999999999999999e-223 < n < 7.00000000000000057e-172

   1. Initial program 37.8

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

   2. Simplified 33.8

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

   3. Applied egg-rr 32.5

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

   4. Taylor expanded in l around 0 31.6

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

   5. Taylor expanded in l around 0 30.6

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

   if 1.10000000000000006e-102 < n < 9.9999999999999993e44

   1. Initial program 30.8

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

   2. Simplified 27.0

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

   3. Applied egg-rr 24.0

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

   4. Taylor expanded in l around 0 26.1

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

   5. Taylor expanded in U* around inf 26.2

      \[\leadsto \sqrt{\left(\left(\color{blue}{\frac{n \cdot U*}{Om}} - 2\right) \cdot \ell\right) \cdot \left(\frac{\ell}{Om} \cdot \left(U \cdot \left(2 \cdot n\right)\right)\right) + \left(n \cdot U\right) \cdot \left(2 \cdot t\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 26.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1 \cdot 10^{-37}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(n, \left(U* - U\right) \cdot \frac{\ell}{Om}, \ell \cdot -2\right) \cdot \left(\frac{\ell}{Om} \cdot \left(U \cdot \left(n \cdot 2\right)\right)\right) + \left(n \cdot U\right) \cdot \left(2 \cdot t\right)}\\ \mathbf{elif}\;n \leq 4.4 \cdot 10^{-251}:\\ \;\;\;\;{\left(\mathsf{fma}\left(n, t \cdot \left(U \cdot 2\right), \left(-2 - \frac{n}{Om} \cdot \left(U - U*\right)\right) \cdot \left(\ell \cdot \left(\left(n \cdot 2\right) \cdot \frac{U}{\frac{Om}{\ell}}\right)\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;n \leq 2 \cdot 10^{-223}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq 7 \cdot 10^{-172}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(2 \cdot t\right) + \left(\ell \cdot \left(-2 + \frac{n \cdot \left(U* - U\right)}{Om}\right)\right) \cdot \left(2 \cdot \frac{n \cdot \left(U \cdot \ell\right)}{Om}\right)}\\ \mathbf{elif}\;n \leq 1.1 \cdot 10^{-102}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq 10^{+45}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(2 \cdot t\right) + \left(\frac{\ell}{Om} \cdot \left(U \cdot \left(n \cdot 2\right)\right)\right) \cdot \left(\ell \cdot \left(-2 + \frac{n \cdot U*}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	24.6
Cost	32780

\[\begin{array}{l} t_1 := U \cdot \left(n \cdot 2\right)\\ t_2 := t_1 \cdot \left(\left(U* - U\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) - \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right)\\ \mathbf{if}\;t_2 \leq 0:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\frac{\ell \cdot \left(\ell \cdot 2 - \frac{n \cdot \left(U* \cdot \ell\right)}{Om}\right)}{Om} - t\right)\right)\right)}\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(\frac{\ell}{Om} \cdot \left(\ell \cdot 2\right) - \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(2 \cdot t\right) + \left(\frac{\ell}{Om} \cdot t_1\right) \cdot \left(\ell \cdot \left(-2 + \frac{n \cdot U*}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \frac{\left(n \cdot \left(U \cdot \ell\right)\right) \cdot \left(\ell \cdot -2 - \frac{n \cdot \left(\ell \cdot \left(U - U*\right)\right)}{Om}\right)}{Om}}\\ \end{array} \]

Alternative 2
Error	26.1
Cost	32780

\[\begin{array}{l} t_1 := U \cdot \left(n \cdot 2\right)\\ t_2 := t_1 \cdot \left(\left(U* - U\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) - \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right)\\ \mathbf{if}\;t_2 \leq 0:\\ \;\;\;\;\sqrt{t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{n}{\frac{\frac{Om}{U*}}{\ell}}\right)} \cdot \sqrt{n \cdot \left(U \cdot 2\right)}\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(\frac{\ell}{Om} \cdot \left(\ell \cdot 2\right) - \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(2 \cdot t\right) + \left(\frac{\ell}{Om} \cdot t_1\right) \cdot \left(\ell \cdot \left(-2 + \frac{n \cdot U*}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \frac{\left(n \cdot \left(U \cdot \ell\right)\right) \cdot \left(\ell \cdot -2 - \frac{n \cdot \left(\ell \cdot \left(U - U*\right)\right)}{Om}\right)}{Om}}\\ \end{array} \]

Alternative 3
Error	30.6
Cost	9048

\[\begin{array}{l} t_1 := \left(n \cdot U\right) \cdot \left(2 \cdot t\right)\\ t_2 := \frac{n \cdot \left(U \cdot \ell\right)}{Om}\\ t_3 := \sqrt{t_1 + \left(\ell \cdot \left(-2 + \frac{n \cdot \left(U* - U\right)}{Om}\right)\right) \cdot \left(2 \cdot t_2\right)}\\ \mathbf{if}\;U \leq -1 \cdot 10^{-120}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;U \leq -2.7 \cdot 10^{-193}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + -2 \cdot \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(t_2 + \ell \cdot 2\right)\right)\right)}{Om}}\\ \mathbf{elif}\;U \leq -1.75 \cdot 10^{-255}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;U \leq 2.6 \cdot 10^{-231}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\frac{\ell \cdot \left(\ell \cdot 2 - \frac{n \cdot \left(U* \cdot \ell\right)}{Om}\right)}{Om} - t\right)\right)\right)}\\ \mathbf{elif}\;U \leq 4.1 \cdot 10^{-160}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;U \leq 100000:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(\frac{\ell}{Om} \cdot \left(\ell \cdot 2\right) - \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t_1 + \left(\frac{\ell}{Om} \cdot \left(U \cdot \left(n \cdot 2\right)\right)\right) \cdot \left(\ell \cdot \left(-2 + \frac{n \cdot U*}{Om}\right)\right)}\\ \end{array} \]

Alternative 4
Error	32.0
Cost	8920

\[\begin{array}{l} t_1 := \ell \cdot 2 - \frac{n \cdot \left(U* \cdot \ell\right)}{Om}\\ t_2 := \sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\frac{\ell \cdot t_1}{Om} - t\right)\right)\right)}\\ t_3 := \sqrt{\left(n \cdot U\right) \cdot \left(2 \cdot t\right) + \left(\frac{\ell}{Om} \cdot \left(U \cdot \left(n \cdot 2\right)\right)\right) \cdot \left(\ell \cdot \left(-2 + \frac{n \cdot U*}{Om}\right)\right)}\\ \mathbf{if}\;U \leq -1 \cdot 10^{-120}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;U \leq -8.2 \cdot 10^{-168}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;U \leq 10^{-273}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;U \leq 8.6 \cdot 10^{-235}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;U \leq 4.4 \cdot 10^{-191}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{n \cdot \left(\ell \cdot \left(U \cdot t_1\right)\right)}{Om}}\\ \mathbf{elif}\;U \leq 100000:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 5
Error	32.3
Cost	8664

\[\begin{array}{l} t_1 := \ell \cdot 2 - \frac{n \cdot \left(U* \cdot \ell\right)}{Om}\\ t_2 := \sqrt{\left(n \cdot U\right) \cdot \left(2 \cdot t\right) - \left(\ell \cdot -2\right) \cdot \left(\frac{\ell}{Om} \cdot \left(U \cdot \left(n \cdot -2\right)\right)\right)}\\ t_3 := \sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\frac{\ell \cdot t_1}{Om} - t\right)\right)\right)}\\ \mathbf{if}\;Om \leq -3.25 \cdot 10^{+90}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;Om \leq -3.7 \cdot 10^{-119}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;Om \leq -1.2 \cdot 10^{-254}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{n \cdot \left(\ell \cdot \left(U \cdot t_1\right)\right)}{Om}}\\ \mathbf{elif}\;Om \leq 2.35 \cdot 10^{+41}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;Om \leq 1.2 \cdot 10^{+139}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;Om \leq 2.5 \cdot 10^{+221}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	29.5
Cost	8520

\[\begin{array}{l} t_1 := \sqrt{\left(n \cdot U\right) \cdot \left(2 \cdot t\right) + \left(\ell \cdot \left(-2 + \frac{n \cdot \left(U* - U\right)}{Om}\right)\right) \cdot \left(2 \cdot \frac{n \cdot \left(U \cdot \ell\right)}{Om}\right)}\\ \mathbf{if}\;\ell \leq -2.9 \cdot 10^{+65}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 7 \cdot 10^{+71}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\frac{\ell \cdot \left(\ell \cdot 2 - \frac{n \cdot \left(U* \cdot \ell\right)}{Om}\right)}{Om} - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	31.5
Cost	8272

\[\begin{array}{l} t_1 := \sqrt{\left(n \cdot U\right) \cdot \left(2 \cdot t\right) - \left(\ell \cdot -2\right) \cdot \left(\frac{\ell}{Om} \cdot \left(U \cdot \left(n \cdot -2\right)\right)\right)}\\ t_2 := \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{-2 + U* \cdot \frac{n}{Om}}{\frac{Om}{\ell \cdot \ell}}\right)}\\ \mathbf{if}\;\ell \leq -7.4 \cdot 10^{+141}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -1.65 \cdot 10^{-172}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 1.7 \cdot 10^{-127}:\\ \;\;\;\;\sqrt{n \cdot \left(t \cdot \left(U \cdot 2\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.9 \cdot 10^{+113}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	34.3
Cost	8008

\[\begin{array}{l} t_1 := \sqrt{\left(-2 \cdot \left(n \cdot U\right)\right) \cdot \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - t\right)}\\ \mathbf{if}\;Om \leq -3.7 \cdot 10^{-119}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;Om \leq 3.6 \cdot 10^{-106}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\ell \cdot 2 - \frac{n \cdot \left(U* \cdot \ell\right)}{Om}\right)\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	32.9
Cost	8008

\[\begin{array}{l} t_1 := \sqrt{\left(n \cdot U\right) \cdot \left(2 \cdot t\right) - \left(\ell \cdot -2\right) \cdot \left(\frac{\ell}{Om} \cdot \left(U \cdot \left(n \cdot -2\right)\right)\right)}\\ \mathbf{if}\;Om \leq -3.7 \cdot 10^{-119}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;Om \leq 1.05 \cdot 10^{-151}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\ell \cdot 2 - \frac{n \cdot \left(U* \cdot \ell\right)}{Om}\right)\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	36.6
Cost	7880

\[\begin{array}{l} t_1 := \sqrt{\left(-2 \cdot \left(n \cdot U\right)\right) \cdot \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - t\right)}\\ \mathbf{if}\;\ell \leq 3.7 \cdot 10^{-100}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\left(-2 + U* \cdot \frac{n}{Om}\right) \cdot \left(n \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	35.9
Cost	7624

\[\begin{array}{l} t_1 := \sqrt{\left(-2 \cdot \left(n \cdot U\right)\right) \cdot \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - t\right)}\\ \mathbf{if}\;n \leq -6.6 \cdot 10^{-205}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq 2.2 \cdot 10^{-61}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	39.0
Cost	7112

\[\begin{array}{l} t_1 := \sqrt{n \cdot \left(t \cdot \left(U \cdot 2\right)\right)}\\ \mathbf{if}\;n \leq -1 \cdot 10^{-140}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq 1.6 \cdot 10^{+39}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	38.8
Cost	7112

\[\begin{array}{l} \mathbf{if}\;n \leq -1 \cdot 10^{+32}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot t\right)}\\ \mathbf{elif}\;n \leq 1.6 \cdot 10^{+39}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(t \cdot \left(U \cdot 2\right)\right)}\\ \end{array} \]

Alternative 14
Error	40.1
Cost	6848

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

Reproduce

herbie shell --seed 2022217 
(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*))))))