Toniolo and Linder, Equation (13)

Average Error: 34.4 → 26.1

Time: 41.5s

Precision: binary64

Cost: 49096

Math

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

↓

\[\begin{array}{l} t_1 := n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + t_1 \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t_2 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, t_1 \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+149}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\left|n \cdot \left(\frac{\ell}{Om} \cdot \sqrt{2}\right)\right| \cdot \sqrt{U \cdot \left(U* - U\right)}\\ \end{array} \]

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 (* n (pow (/ l Om) 2.0)))
        (t_2
         (sqrt
          (*
           (* (* 2.0 n) U)
           (+ (+ t (* (/ (* l l) Om) -2.0)) (* t_1 (- U* U)))))))
   (if (<= t_2 0.0)
     (*
      (sqrt (* 2.0 n))
      (sqrt (* U (- t (fma 2.0 (* l (/ l Om)) (* t_1 (- U U*)))))))
     (if (<= t_2 2e+149)
       t_2
       (* (fabs (* n (* (/ l Om) (sqrt 2.0)))) (sqrt (* U (- 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 = n * pow((l / Om), 2.0);
	double t_2 = sqrt((((2.0 * n) * U) * ((t + (((l * l) / Om) * -2.0)) + (t_1 * (U_42_ - U)))));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt((2.0 * n)) * sqrt((U * (t - fma(2.0, (l * (l / Om)), (t_1 * (U - U_42_))))));
	} else if (t_2 <= 2e+149) {
		tmp = t_2;
	} else {
		tmp = fabs((n * ((l / Om) * sqrt(2.0)))) * sqrt((U * (U_42_ - U)));
	}
	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(n * (Float64(l / Om) ^ 2.0))
	t_2 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t + Float64(Float64(Float64(l * l) / Om) * -2.0)) + Float64(t_1 * Float64(U_42_ - U)))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * Float64(t - fma(2.0, Float64(l * Float64(l / Om)), Float64(t_1 * Float64(U - U_42_)))))));
	elseif (t_2 <= 2e+149)
		tmp = t_2;
	else
		tmp = Float64(abs(Float64(n * Float64(Float64(l / Om) * sqrt(2.0)))) * sqrt(Float64(U * Float64(U_42_ - U))));
	end
	return tmp
end

Wolfram

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

↓

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t + N[(N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+149], t$95$2, N[(N[Abs[N[(n * N[(N[(l / Om), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 0.0

   1. Initial program 56.6

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

   2. Applied egg-rr 40.1

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

   3. Simplified 40.2

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

      [Start] 40.1	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*r* [=>] 40.2	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U - U*\right)}\right)\right)} \]
      *-commutative [<=] 40.2	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)} \cdot \left(U - U*\right)\right)\right)} \]
      *-commutative [=>] 40.2	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \color{blue}{\left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right)\right)} \]

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 2.0000000000000001e149

   1. Initial program 1.8

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

   if 2.0000000000000001e149 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))))

   1. Initial program 63.3

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

   2. Simplified 55.6

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

      [Start] 63.3	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 63.3	\[ \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 63.3	\[ \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      *-commutative [=>] 63.3	\[ \sqrt{2 \cdot \color{blue}{\left(\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot \left(n \cdot U\right)\right)}} \]

   3. Taylor expanded in Om around 0 55.4

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

   4. Applied egg-rr 56.2

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

   5. Simplified 48.6

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

      [Start] 56.2	\[ \sqrt{{\left(\frac{n \cdot \ell}{Om} \cdot \sqrt{2}\right)}^{2}} \cdot \sqrt{\left(U* - U\right) \cdot U} \]
      unpow2 [=>] 56.2	\[ \sqrt{\color{blue}{\left(\frac{n \cdot \ell}{Om} \cdot \sqrt{2}\right) \cdot \left(\frac{n \cdot \ell}{Om} \cdot \sqrt{2}\right)}} \cdot \sqrt{\left(U* - U\right) \cdot U} \]
      rem-sqrt-square [=>] 48.2	\[ \color{blue}{\left|\frac{n \cdot \ell}{Om} \cdot \sqrt{2}\right|} \cdot \sqrt{\left(U* - U\right) \cdot U} \]
      *-lft-identity [<=] 48.2	\[ \left|\frac{\color{blue}{1 \cdot \left(n \cdot \ell\right)}}{Om} \cdot \sqrt{2}\right| \cdot \sqrt{\left(U* - U\right) \cdot U} \]
      associate-*l/ [<=] 48.2	\[ \left|\color{blue}{\left(\frac{1}{Om} \cdot \left(n \cdot \ell\right)\right)} \cdot \sqrt{2}\right| \cdot \sqrt{\left(U* - U\right) \cdot U} \]
      *-commutative [<=] 48.2	\[ \left|\color{blue}{\left(\left(n \cdot \ell\right) \cdot \frac{1}{Om}\right)} \cdot \sqrt{2}\right| \cdot \sqrt{\left(U* - U\right) \cdot U} \]
      associate-*l* [=>] 48.6	\[ \left|\color{blue}{\left(n \cdot \left(\ell \cdot \frac{1}{Om}\right)\right)} \cdot \sqrt{2}\right| \cdot \sqrt{\left(U* - U\right) \cdot U} \]
      associate-*l* [=>] 48.6	\[ \left|\color{blue}{n \cdot \left(\left(\ell \cdot \frac{1}{Om}\right) \cdot \sqrt{2}\right)}\right| \cdot \sqrt{\left(U* - U\right) \cdot U} \]
      associate-*r/ [=>] 48.6	\[ \left|n \cdot \left(\color{blue}{\frac{\ell \cdot 1}{Om}} \cdot \sqrt{2}\right)\right| \cdot \sqrt{\left(U* - U\right) \cdot U} \]
      associate-/l* [=>] 48.6	\[ \left|n \cdot \left(\color{blue}{\frac{\ell}{\frac{Om}{1}}} \cdot \sqrt{2}\right)\right| \cdot \sqrt{\left(U* - U\right) \cdot U} \]
      /-rgt-identity [=>] 48.6	\[ \left|n \cdot \left(\frac{\ell}{\color{blue}{Om}} \cdot \sqrt{2}\right)\right| \cdot \sqrt{\left(U* - U\right) \cdot U} \]

3. Recombined 3 regimes into one program.

4. Final simplification 26.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 2 \cdot 10^{+149}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|n \cdot \left(\frac{\ell}{Om} \cdot \sqrt{2}\right)\right| \cdot \sqrt{U \cdot \left(U* - U\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	26.5
Cost	38924

\[\begin{array}{l} t_1 := t - \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - n \cdot \frac{\ell}{Om \cdot \frac{\frac{Om}{U*}}{\ell}}\right)\\ t_2 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_2 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t_1}\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+298}:\\ \;\;\;\;\sqrt{t_2}\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;\sqrt{t_1} \cdot \sqrt{2 \cdot \left(n \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(\left(\ell \cdot \left(U \cdot \ell\right)\right) \cdot \left(\frac{1}{Om} \cdot \frac{U* - U}{\frac{Om}{n}} + \frac{-2}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 2
Error	26.5
Cost	36296

\[\begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - n \cdot \frac{\ell}{Om \cdot \frac{\frac{Om}{U*}}{\ell}}\right)\right)}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+298}:\\ \;\;\;\;\sqrt{t_1}\\ \mathbf{else}:\\ \;\;\;\;\left|n \cdot \left(\frac{\ell}{Om} \cdot \sqrt{2}\right)\right| \cdot \sqrt{U \cdot \left(U* - U\right)}\\ \end{array} \]

Alternative 3
Error	28.0
Cost	30728

\[\begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - n \cdot \frac{\ell}{Om \cdot \frac{\frac{Om}{U*}}{\ell}}\right)\right)}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+298}:\\ \;\;\;\;\sqrt{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(2 \cdot \left(\ell \cdot \frac{U}{\frac{Om}{\ell}}\right)\right)}\\ \end{array} \]

Alternative 4
Error	32.1
Cost	14860

\[\begin{array}{l} t_1 := \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot \frac{\ell}{\frac{Om}{\ell}} - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)}\\ t_2 := \sqrt{t \cdot \left(2 \cdot U\right)} \cdot \sqrt{n}\\ \mathbf{if}\;n \leq -6 \cdot 10^{-213}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq -5.2 \cdot 10^{-284}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\frac{\ell}{Om} \cdot \left(2 \cdot \ell\right) - t\right)\right)\right)}\\ \mathbf{elif}\;n \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq 4.1 \cdot 10^{-151}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;n \leq 1.22 \cdot 10^{-92}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \frac{n}{\frac{Om}{\frac{\ell \cdot \left(\ell \cdot U*\right)}{Om}}}\right)}\\ \mathbf{elif}\;n \leq 6.8 \cdot 10^{-81}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{Om}{n \cdot \ell}} \cdot \left(-\sqrt{U \cdot U*}\right)\\ \mathbf{elif}\;n \leq 4.2 \cdot 10^{+213}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - n \cdot \frac{\ell}{Om \cdot \frac{\frac{Om}{U*}}{\ell}}\right)\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	32.6
Cost	13644

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om}\\ \mathbf{if}\;t \leq -1 \cdot 10^{-195}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot t_1 - t\right)\right)\right)}\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-255}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \frac{n}{\frac{Om}{t_2}}\right)}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{+93}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(t_2 \cdot \frac{n}{Om} + -2 \cdot t_1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{t}\\ \end{array} \]

Alternative 6
Error	32.6
Cost	13644

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om}\\ \mathbf{if}\;t \leq -1.8 \cdot 10^{-195}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot t_1 - t\right)\right)\right)}\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{-255}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \frac{n}{\frac{Om}{t_2}}\right)}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+95}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(t_2 \cdot \frac{n}{Om} + -2 \cdot t_1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot U} \cdot \sqrt{2 \cdot t}\\ \end{array} \]

Alternative 7
Error	32.8
Cost	8393

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.15 \cdot 10^{+111} \lor \neg \left(\ell \leq 6.5 \cdot 10^{+129}\right):\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(2 \cdot \left(\ell \cdot \frac{U}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \frac{n}{\frac{Om}{\frac{\ell \cdot \left(\ell \cdot U*\right)}{Om}}}\right)}\\ \end{array} \]

Alternative 8
Error	33.8
Cost	8392

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;t \leq -2 \cdot 10^{-196}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot t_1 - t\right)\right)\right)}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{+94}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell \cdot \left(\ell \cdot U*\right)}{Om} \cdot \frac{n}{Om} + -2 \cdot t_1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{-2}{\frac{\frac{Om}{\ell}}{\ell}}\right)\right)}\\ \end{array} \]

Alternative 9
Error	36.1
Cost	8264

\[\begin{array}{l} \mathbf{if}\;t \leq -1.75 \cdot 10^{-205}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - t\right)\right)\right)}\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{-229}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(\left(\ell \cdot \left(U \cdot \ell\right)\right) \cdot \left(\frac{1}{Om} \cdot \frac{U* - U}{\frac{Om}{n}} + \frac{-2}{Om}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-80}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{-2}{\frac{\frac{Om}{\ell}}{\ell}}\right)\right)}\\ \end{array} \]

Alternative 10
Error	34.0
Cost	7625

\[\begin{array}{l} \mathbf{if}\;\ell \leq -6 \cdot 10^{+209} \lor \neg \left(\ell \leq 4.8 \cdot 10^{+135}\right):\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(2 \cdot \left(\ell \cdot \frac{U}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{-2}{\frac{\frac{Om}{\ell}}{\ell}}\right)\right)}\\ \end{array} \]

Alternative 11
Error	37.1
Cost	7497

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.1 \cdot 10^{+101} \lor \neg \left(\ell \leq 8.5 \cdot 10^{+124}\right):\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(2 \cdot \left(\ell \cdot \frac{U}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 12
Error	40.7
Cost	7368

\[\begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{-197}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{-271}:\\ \;\;\;\;\sqrt{-4 \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(U \cdot \frac{n}{Om}\right)\right)}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{+173}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 13
Error	40.1
Cost	7368

\[\begin{array}{l} \mathbf{if}\;t \leq -3.4 \cdot 10^{-186}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-295}:\\ \;\;\;\;\sqrt{U \cdot \frac{n \cdot -4}{\frac{\frac{Om}{\ell}}{\ell}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 14
Error	39.0
Cost	7113

\[\begin{array}{l} \mathbf{if}\;n \leq -1.05 \cdot 10^{+71} \lor \neg \left(n \leq 1.12 \cdot 10^{-138}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]

Alternative 15
Error	39.3
Cost	7112

\[\begin{array}{l} \mathbf{if}\;n \leq -1.34 \cdot 10^{+98}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;n \leq 6.8 \cdot 10^{+101}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \end{array} \]

Alternative 16
Error	40.2
Cost	6848

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

Reproduce

herbie shell --seed 2023237 
(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*))))))