Toniolo and Linder, Equation (13)

Average Error: 34.4 → 29.8

Time: 32.5s

Precision: binary64

Cost: 14796

Math

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

↓

\[\begin{array}{l} t_1 := 2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\\ t_2 := \sqrt{t_1 + -2 \cdot \frac{\left(n \cdot \left(U \cdot \ell\right)\right) \cdot \left(2 \cdot \ell - \frac{\ell \cdot \left(n \cdot U*\right)}{Om}\right)}{Om}}\\ t_3 := \frac{Om}{U \cdot \ell}\\ t_4 := \sqrt{\left(-2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(\frac{\ell}{Om} \cdot \left(\left(U - U*\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) + \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right) - t\right)}\\ \mathbf{if}\;U \leq -5.2 \cdot 10^{+64}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;U \leq -1.3 \cdot 10^{-299}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;U \leq 2.4 \cdot 10^{-277}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \mathsf{fma}\left(\frac{n}{Om}, \frac{U* - U}{Om}, \frac{-2}{Om}\right) \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)}\\ \mathbf{elif}\;U \leq 1.4 \cdot 10^{-180}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;U \leq 10^{-141}:\\ \;\;\;\;\sqrt{t_1 - -2 \cdot \frac{n}{\frac{t_3}{\ell \cdot -2 - \frac{n}{t_3}}}}\\ \mathbf{elif}\;U \leq 4.8 \cdot 10^{+103}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot U} \cdot \sqrt{n \cdot t}\\ \end{array} \]

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))))
        (t_2
         (sqrt
          (+
           t_1
           (*
            -2.0
            (/ (* (* n (* U l)) (- (* 2.0 l) (/ (* l (* n U*)) Om))) Om)))))
        (t_3 (/ Om (* U l)))
        (t_4
         (sqrt
          (*
           (* -2.0 (* n U))
           (-
            (+ (* (/ l Om) (* (- U U*) (* n (/ l Om)))) (* (/ l Om) (* 2.0 l)))
            t)))))
   (if (<= U -5.2e+64)
     t_4
     (if (<= U -1.3e-299)
       t_2
       (if (<= U 2.4e-277)
         (sqrt
          (*
           (* 2.0 n)
           (+
            (* U t)
            (* (fma (/ n Om) (/ (- U* U) Om) (/ -2.0 Om)) (* l (* U l))))))
         (if (<= U 1.4e-180)
           t_4
           (if (<= U 1e-141)
             (sqrt (- t_1 (* -2.0 (/ n (/ t_3 (- (* l -2.0) (/ n t_3)))))))
             (if (<= U 4.8e+103)
               t_2
               (* (sqrt (* 2.0 U)) (sqrt (* n t)))))))))))

C

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

↓

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = 2.0 * (n * (U * t));
	double t_2 = sqrt((t_1 + (-2.0 * (((n * (U * l)) * ((2.0 * l) - ((l * (n * U_42_)) / Om))) / Om))));
	double t_3 = Om / (U * l);
	double t_4 = sqrt(((-2.0 * (n * U)) * ((((l / Om) * ((U - U_42_) * (n * (l / Om)))) + ((l / Om) * (2.0 * l))) - t)));
	double tmp;
	if (U <= -5.2e+64) {
		tmp = t_4;
	} else if (U <= -1.3e-299) {
		tmp = t_2;
	} else if (U <= 2.4e-277) {
		tmp = sqrt(((2.0 * n) * ((U * t) + (fma((n / Om), ((U_42_ - U) / Om), (-2.0 / Om)) * (l * (U * l))))));
	} else if (U <= 1.4e-180) {
		tmp = t_4;
	} else if (U <= 1e-141) {
		tmp = sqrt((t_1 - (-2.0 * (n / (t_3 / ((l * -2.0) - (n / t_3)))))));
	} else if (U <= 4.8e+103) {
		tmp = t_2;
	} else {
		tmp = sqrt((2.0 * U)) * sqrt((n * t));
	}
	return tmp;
}

Julia

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

↓

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(2.0 * Float64(n * Float64(U * t)))
	t_2 = sqrt(Float64(t_1 + Float64(-2.0 * Float64(Float64(Float64(n * Float64(U * l)) * Float64(Float64(2.0 * l) - Float64(Float64(l * Float64(n * U_42_)) / Om))) / Om))))
	t_3 = Float64(Om / Float64(U * l))
	t_4 = sqrt(Float64(Float64(-2.0 * Float64(n * U)) * Float64(Float64(Float64(Float64(l / Om) * Float64(Float64(U - U_42_) * Float64(n * Float64(l / Om)))) + Float64(Float64(l / Om) * Float64(2.0 * l))) - t)))
	tmp = 0.0
	if (U <= -5.2e+64)
		tmp = t_4;
	elseif (U <= -1.3e-299)
		tmp = t_2;
	elseif (U <= 2.4e-277)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(Float64(U * t) + Float64(fma(Float64(n / Om), Float64(Float64(U_42_ - U) / Om), Float64(-2.0 / Om)) * Float64(l * Float64(U * l))))));
	elseif (U <= 1.4e-180)
		tmp = t_4;
	elseif (U <= 1e-141)
		tmp = sqrt(Float64(t_1 - Float64(-2.0 * Float64(n / Float64(t_3 / Float64(Float64(l * -2.0) - Float64(n / t_3)))))));
	elseif (U <= 4.8e+103)
		tmp = t_2;
	else
		tmp = Float64(sqrt(Float64(2.0 * U)) * sqrt(Float64(n * t)));
	end
	return tmp
end

Wolfram

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

↓

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(t$95$1 + N[(-2.0 * N[(N[(N[(n * N[(U * l), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * l), $MachinePrecision] - N[(N[(l * N[(n * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(Om / N[(U * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(N[(-2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(l / Om), $MachinePrecision] * N[(N[(U - U$42$), $MachinePrecision] * N[(n * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(l / Om), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[U, -5.2e+64], t$95$4, If[LessEqual[U, -1.3e-299], t$95$2, If[LessEqual[U, 2.4e-277], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(N[(U * t), $MachinePrecision] + N[(N[(N[(n / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision] * N[(l * N[(U * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[U, 1.4e-180], t$95$4, If[LessEqual[U, 1e-141], N[Sqrt[N[(t$95$1 - N[(-2.0 * N[(n / N[(t$95$3 / N[(N[(l * -2.0), $MachinePrecision] - N[(n / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[U, 4.8e+103], t$95$2, N[(N[Sqrt[N[(2.0 * U), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(n * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if U < -5.19999999999999994e64 or 2.4e-277 < U < 1.39999999999999999e-180

   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 33.5

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

   3. Applied egg-rr 31.8

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

   if -5.19999999999999994e64 < U < -1.2999999999999999e-299 or 1e-141 < U < 4.7999999999999997e103

   1. Initial program 34.3

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

   2. Simplified 32.2

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

   3. Taylor expanded in t around inf 29.4

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

   4. Taylor expanded in U* around inf 29.5

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

   5. Simplified 29.2

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

   if -1.2999999999999999e-299 < U < 2.4e-277

   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. Simplified 43.9

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

   3. Taylor expanded in l around 0 43.8

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

   4. Simplified 32.1

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

   if 1.39999999999999999e-180 < U < 1e-141

   1. Initial program 36.5

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

   2. Simplified 35.5

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

   3. Taylor expanded in t around inf 32.7

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

   4. Taylor expanded in U* around 0 36.0

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

   5. Simplified 33.7

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

   if 4.7999999999999997e103 < U

   1. Initial program 29.3

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

   2. Simplified 25.9

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

   3. Taylor expanded in U* around 0 33.2

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

   4. Simplified 33.1

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

   5. Taylor expanded in t around inf 41.7

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

   6. Simplified 35.5

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

   7. Applied egg-rr 25.2

      \[\leadsto \color{blue}{\sqrt{2 \cdot U} \cdot \sqrt{n \cdot t}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 29.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -5.2 \cdot 10^{+64}:\\ \;\;\;\;\sqrt{\left(-2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(\frac{\ell}{Om} \cdot \left(\left(U - U*\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) + \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right) - t\right)}\\ \mathbf{elif}\;U \leq -1.3 \cdot 10^{-299}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + -2 \cdot \frac{\left(n \cdot \left(U \cdot \ell\right)\right) \cdot \left(2 \cdot \ell - \frac{\ell \cdot \left(n \cdot U*\right)}{Om}\right)}{Om}}\\ \mathbf{elif}\;U \leq 2.4 \cdot 10^{-277}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \mathsf{fma}\left(\frac{n}{Om}, \frac{U* - U}{Om}, \frac{-2}{Om}\right) \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)}\\ \mathbf{elif}\;U \leq 1.4 \cdot 10^{-180}:\\ \;\;\;\;\sqrt{\left(-2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(\frac{\ell}{Om} \cdot \left(\left(U - U*\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) + \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right) - t\right)}\\ \mathbf{elif}\;U \leq 10^{-141}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) - -2 \cdot \frac{n}{\frac{\frac{Om}{U \cdot \ell}}{\ell \cdot -2 - \frac{n}{\frac{Om}{U \cdot \ell}}}}}\\ \mathbf{elif}\;U \leq 4.8 \cdot 10^{+103}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + -2 \cdot \frac{\left(n \cdot \left(U \cdot \ell\right)\right) \cdot \left(2 \cdot \ell - \frac{\ell \cdot \left(n \cdot U*\right)}{Om}\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot U} \cdot \sqrt{n \cdot t}\\ \end{array} \]

Alternatives

Alternative 1
Error	26.2
Cost	43528

\[\begin{array}{l} t_1 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot t_1\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t_2 \leq 2 \cdot 10^{-145}:\\ \;\;\;\;{\left({\left(2 \cdot \left(n \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, n \cdot \left(t_1 \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right)}^{0.25}\right)}^{2}\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+153}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + -2 \cdot \frac{\left(n \cdot \left(U \cdot \ell\right)\right) \cdot \left(2 \cdot \ell - \frac{\ell \cdot \left(n \cdot U*\right)}{Om}\right)}{Om}}\\ \end{array} \]

Alternative 2
Error	28.7
Cost	13644

\[\begin{array}{l} t_1 := \sqrt{\left(-2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(\frac{\ell}{Om} \cdot \left(\left(U - U*\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) + \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right) - t\right)}\\ \mathbf{if}\;n \leq -1.8 \cdot 10^{-131}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{-159}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + -2 \cdot \frac{\left(n \cdot \left(U \cdot \ell\right)\right) \cdot \left(2 \cdot \ell - \frac{\ell \cdot \left(n \cdot U*\right)}{Om}\right)}{Om}}\\ \mathbf{elif}\;n \leq 8 \cdot 10^{+124}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{t \cdot \left(2 \cdot U\right)}\\ \end{array} \]

Alternative 3
Error	29.6
Cost	13512

\[\begin{array}{l} \mathbf{if}\;U \leq -5.2 \cdot 10^{+64}:\\ \;\;\;\;\sqrt{\left(-2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(\frac{\ell}{Om} \cdot \left(\left(U - U*\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) + \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right) - t\right)}\\ \mathbf{elif}\;U \leq 4.8 \cdot 10^{+103}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + -2 \cdot \frac{\left(n \cdot \left(U \cdot \ell\right)\right) \cdot \left(2 \cdot \ell - \frac{\ell \cdot \left(n \cdot U*\right)}{Om}\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot U} \cdot \sqrt{n \cdot t}\\ \end{array} \]

Alternative 4
Error	30.1
Cost	8652

\[\begin{array}{l} t_1 := t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\\ t_2 := \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + -2 \cdot \frac{\left(n \cdot \left(U \cdot \ell\right)\right) \cdot \left(2 \cdot \ell - \frac{\ell \cdot \left(n \cdot U*\right)}{Om}\right)}{Om}}\\ \mathbf{if}\;\ell \leq -8.4 \cdot 10^{+58}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-264}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t_1\right)\right)}\\ \mathbf{elif}\;\ell \leq 10^{-62}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	29.9
Cost	8652

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + -2 \cdot \frac{\left(n \cdot \left(U \cdot \ell\right)\right) \cdot \left(2 \cdot \ell - \frac{\ell \cdot \left(n \cdot U*\right)}{Om}\right)}{Om}}\\ \mathbf{if}\;\ell \leq -8.4 \cdot 10^{+58}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-264}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.15 \cdot 10^{-63}:\\ \;\;\;\;\sqrt{\left(-2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(\frac{\ell}{Om} \cdot \left(\left(U - U*\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) + \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right) - t\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	32.8
Cost	8400

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -3.4 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{\left(-2 \cdot \left(n \cdot U\right)\right) \cdot \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - t\right)}\\ \mathbf{elif}\;\ell \leq 1.18 \cdot 10^{-206}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.15 \cdot 10^{-63}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{n}{Om} \cdot \frac{\left(\ell \cdot \ell\right) \cdot \left(U* - U\right)}{Om}\right)}\\ \mathbf{elif}\;\ell \leq 10^{+130}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2 \cdot n}{\frac{-0.5}{U} \cdot \frac{\frac{Om}{\ell}}{\ell}}}\\ \end{array} \]

Alternative 7
Error	33.0
Cost	8268

\[\begin{array}{l} t_1 := t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\\ \mathbf{if}\;\ell \leq -3.4 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{\left(-2 \cdot \left(n \cdot U\right)\right) \cdot \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - t\right)}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-264}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t_1\right)\right)}\\ \mathbf{elif}\;\ell \leq 4 \cdot 10^{-62}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) - 2 \cdot \left(2 \cdot \left(\frac{n}{Om} \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \end{array} \]

Alternative 8
Error	35.9
Cost	8208

\[\begin{array}{l} t_1 := \sqrt{\left(-2 \cdot \left(n \cdot U\right)\right) \cdot \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - t\right)}\\ \mathbf{if}\;\ell \leq -1.66 \cdot 10^{-12}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-264}:\\ \;\;\;\;\sqrt{n \cdot \left(t \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{elif}\;\ell \leq 5.5 \cdot 10^{+39}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 10^{+230}:\\ \;\;\;\;\sqrt{\frac{2 \cdot n}{\frac{\frac{-Om}{\ell \cdot \left(U \cdot \ell\right)}}{2 - U* \cdot \frac{n}{Om}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2 \cdot n}{\frac{-0.5}{U} \cdot \frac{\frac{Om}{\ell}}{\ell}}}\\ \end{array} \]

Alternative 9
Error	33.6
Cost	8076

\[\begin{array}{l} t_1 := \ell \cdot \left(U \cdot \ell\right)\\ t_2 := \sqrt{\left(-2 \cdot \left(n \cdot U\right)\right) \cdot \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - t\right)}\\ \mathbf{if}\;U \leq -1 \cdot 10^{+30}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;U \leq 4.8 \cdot 10^{-291}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) - 2 \cdot \left(2 \cdot \left(\frac{n}{Om} \cdot t_1\right)\right)}\\ \mathbf{elif}\;U \leq 2.4 \cdot 10^{-277}:\\ \;\;\;\;\sqrt{\frac{2 \cdot n}{\frac{\frac{-Om}{t_1}}{2 - U* \cdot \frac{n}{Om}}}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 10
Error	38.4
Cost	7628

\[\begin{array}{l} t_1 := \sqrt{\frac{2 \cdot n}{\frac{-0.5}{U} \cdot \frac{\frac{Om}{\ell}}{\ell}}}\\ \mathbf{if}\;\ell \leq -660000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -5.2 \cdot 10^{-212}:\\ \;\;\;\;\sqrt{n \cdot \left(t \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.4 \cdot 10^{+67}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	35.4
Cost	7624

\[\begin{array}{l} t_1 := \sqrt{\left(-2 \cdot \left(n \cdot U\right)\right) \cdot \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - t\right)}\\ \mathbf{if}\;\ell \leq -1.66 \cdot 10^{-12}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-264}:\\ \;\;\;\;\sqrt{n \cdot \left(t \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	39.7
Cost	7500

\[\begin{array}{l} t_1 := \sqrt{n \cdot \left(t \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{if}\;\ell \leq -1.3 \cdot 10^{-206}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.4 \cdot 10^{+67}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 7.4 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{\frac{-4}{Om} \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	39.8
Cost	7500

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.3 \cdot 10^{-206}:\\ \;\;\;\;\sqrt{n \cdot \left(t \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.4 \cdot 10^{+67}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 7.4 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{\frac{-4}{Om} \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 1\right) + -1}\\ \end{array} \]

Alternative 14
Error	40.2
Cost	6980

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.3 \cdot 10^{-206}:\\ \;\;\;\;\sqrt{n \cdot \left(t \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]

Alternative 15
Error	40.4
Cost	6848

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

Reproduce

herbie shell --seed 2022221 
(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*))))))