Toniolo and Linder, Equation (13)

Average Error: 54.15% → 41.11%

Time: 40.7s

Precision: binary64

Cost: 48968

Math

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

↓

\[\begin{array}{l} t_1 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := \sqrt{\left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(\left(n \cdot t_1\right) \cdot \left(U - U*\right) + \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right)}\\ \mathbf{if}\;t_2 \leq 2 \cdot 10^{-159}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, n \cdot \left(t_1 \cdot \left(U - U*\right)\right)\right)\right)}\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+150}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{n \cdot \left(\ell \cdot \sqrt{U \cdot U*}\right)}{Om} \cdot \sqrt{2}\right|\\ \end{array} \]

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 (pow (/ l Om) 2.0))
        (t_2
         (sqrt
          (*
           (* U (* n -2.0))
           (+ (* (* n t_1) (- U U*)) (- (* 2.0 (/ (* l l) Om)) t))))))
   (if (<= t_2 2e-159)
     (*
      (sqrt (* 2.0 n))
      (sqrt (* U (- t (fma 2.0 (/ l (/ Om l)) (* n (* t_1 (- U U*))))))))
     (if (<= t_2 2e+150)
       t_2
       (fabs (* (/ (* n (* l (sqrt (* U U*)))) Om) (sqrt 2.0)))))))

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 = pow((l / Om), 2.0);
	double t_2 = sqrt(((U * (n * -2.0)) * (((n * t_1) * (U - U_42_)) + ((2.0 * ((l * l) / Om)) - t))));
	double tmp;
	if (t_2 <= 2e-159) {
		tmp = sqrt((2.0 * n)) * sqrt((U * (t - fma(2.0, (l / (Om / l)), (n * (t_1 * (U - U_42_)))))));
	} else if (t_2 <= 2e+150) {
		tmp = t_2;
	} else {
		tmp = fabs((((n * (l * sqrt((U * U_42_)))) / Om) * sqrt(2.0)));
	}
	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 / Om) ^ 2.0
	t_2 = sqrt(Float64(Float64(U * Float64(n * -2.0)) * Float64(Float64(Float64(n * t_1) * Float64(U - U_42_)) + Float64(Float64(2.0 * Float64(Float64(l * l) / Om)) - t))))
	tmp = 0.0
	if (t_2 <= 2e-159)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * Float64(t - fma(2.0, Float64(l / Float64(Om / l)), Float64(n * Float64(t_1 * Float64(U - U_42_))))))));
	elseif (t_2 <= 2e+150)
		tmp = t_2;
	else
		tmp = abs(Float64(Float64(Float64(n * Float64(l * sqrt(Float64(U * U_42_)))) / Om) * sqrt(2.0)));
	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[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(U * N[(n * -2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(n * t$95$1), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 2e-159], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t - N[(2.0 * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision] + N[(n * N[(t$95$1 * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+150], t$95$2, N[Abs[N[(N[(N[(n * N[(l * N[Sqrt[N[(U * U$42$), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[Sqrt[2.0], $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*))))) < 1.99999999999999998e-159

   1. Initial program 88.84

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

   2. Simplified 63.59

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

      [Start] 88.84	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 61.76	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 61.76	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      sub-neg [=>] 61.76	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(-\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [<=] 61.76	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      cancel-sign-sub [<=] 61.76	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      cancel-sign-sub [=>] 61.76	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-/l* [=>] 61.76	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 63.59	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Applied egg-rr 60.27

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

   4. Simplified 61.46

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

      [Start] 60.27	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U - U*\right) \cdot n\right)\right)\right)} \]
      associate-/l* [=>] 59.94	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U - U*\right) \cdot n\right)\right)\right)} \]
      associate-*r* [=>] 61.46	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right) \cdot n}\right)\right)} \]
      *-commutative [=>] 61.46	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)} \]
      *-commutative [=>] 61.46	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, n \cdot \color{blue}{\left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right)\right)} \]

   if 1.99999999999999998e-159 < (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*))))) < 1.99999999999999996e150

   1. Initial program 1.99

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

   if 1.99999999999999996e150 < (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 98.63

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

   2. Simplified 84.83

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

      [Start] 98.63	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 96.34	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 96.34	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      sub-neg [=>] 96.34	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(-\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [<=] 96.34	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      cancel-sign-sub [<=] 96.34	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      cancel-sign-sub [=>] 96.34	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-/l* [=>] 85.39	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 84.83	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in U* around inf 98.31

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

   4. Simplified 98.31

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

      [Start] 98.31	\[ \sqrt{2 \cdot \frac{{n}^{2} \cdot \left({\ell}^{2} \cdot \left(U \cdot U*\right)\right)}{{Om}^{2}}} \]
      unpow2 [=>] 98.31	\[ \sqrt{2 \cdot \frac{\color{blue}{\left(n \cdot n\right)} \cdot \left({\ell}^{2} \cdot \left(U \cdot U*\right)\right)}{{Om}^{2}}} \]
      unpow2 [=>] 98.31	\[ \sqrt{2 \cdot \frac{\left(n \cdot n\right) \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(U \cdot U*\right)\right)}{{Om}^{2}}} \]
      unpow2 [=>] 98.31	\[ \sqrt{2 \cdot \frac{\left(n \cdot n\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(U \cdot U*\right)\right)}{\color{blue}{Om \cdot Om}}} \]

   5. Applied egg-rr 76.5

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

4. Final simplification 41.11

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right) + \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right)} \leq 2 \cdot 10^{-159}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right) + \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right)} \leq 2 \cdot 10^{+150}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right) + \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{n \cdot \left(\ell \cdot \sqrt{U \cdot U*}\right)}{Om} \cdot \sqrt{2}\right|\\ \end{array} \]

Alternatives

Alternative 1
Error	41.2%
Cost	36168

\[\begin{array}{l} t_1 := \left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right) + \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right)\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \left(\frac{n}{Om} \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+300}:\\ \;\;\;\;\sqrt{t_1}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{n \cdot \left(\ell \cdot \sqrt{U \cdot U*}\right)}{Om} \cdot \sqrt{2}\right|\\ \end{array} \]

Alternative 2
Error	40.93%
Cost	36168

\[\begin{array}{l} t_1 := \left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right) + \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right)\\ \mathbf{if}\;t_1 \leq 5 \cdot 10^{-318}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \ell \cdot \frac{n}{\frac{Om}{\frac{\ell}{Om} \cdot \left(U - U*\right)}}\right)\right)}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+300}:\\ \;\;\;\;\sqrt{t_1}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{n \cdot \left(\ell \cdot \sqrt{U \cdot U*}\right)}{Om} \cdot \sqrt{2}\right|\\ \end{array} \]

Alternative 3
Error	46.39%
Cost	14992

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}} \cdot -2\\ \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+78}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(\frac{1}{Om} \cdot -2 - \frac{n}{Om} \cdot \frac{U - U*}{Om}\right)\right)} \cdot \left(\ell \cdot \left(-\sqrt{2}\right)\right)\\ \mathbf{elif}\;\ell \leq -4.2 \cdot 10^{-163}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\frac{n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om} + t_1\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.75 \cdot 10^{-228}:\\ \;\;\;\;\sqrt{\left|t \cdot \left(U \cdot \left(n \cdot -2\right)\right)\right|}\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{+183}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(t_1 + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(\frac{-2}{Om} - \left(U - U*\right) \cdot \frac{n}{Om \cdot Om}\right) \cdot \left(n \cdot U\right)}\\ \end{array} \]

Alternative 4
Error	44.96%
Cost	14860

\[\begin{array}{l} \mathbf{if}\;\ell \leq -6.5 \cdot 10^{+83}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(\frac{1}{Om} \cdot -2 - \frac{n}{Om} \cdot \frac{U - U*}{Om}\right)\right)} \cdot \left(\ell \cdot \left(-\sqrt{2}\right)\right)\\ \mathbf{elif}\;\ell \leq -1.25 \cdot 10^{-163}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\frac{n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om} + \frac{\ell}{\frac{Om}{\ell}} \cdot -2\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.15 \cdot 10^{+72}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right) + \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right)}\\ \mathbf{elif}\;\ell \leq 5.6 \cdot 10^{+148}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \left(\frac{n}{Om} \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(\frac{-2}{Om} - \left(U - U*\right) \cdot \frac{n}{Om \cdot Om}\right) \cdot \left(n \cdot U\right)}\\ \end{array} \]

Alternative 5
Error	48.59%
Cost	14672

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;\ell \leq -3.2 \cdot 10^{+178}:\\ \;\;\;\;\left(\ell \cdot \left(-\sqrt{2}\right)\right) \cdot \sqrt{U \cdot \frac{-2}{\frac{Om}{n}}}\\ \mathbf{elif}\;\ell \leq -6.6 \cdot 10^{-163}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\frac{n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om} + t_1 \cdot -2\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 10^{-226}:\\ \;\;\;\;\sqrt{\left|t \cdot \left(U \cdot \left(n \cdot -2\right)\right)\right|}\\ \mathbf{elif}\;\ell \leq 7.8 \cdot 10^{+183}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot t_1 - n \cdot \left(t_1 \cdot \frac{U* - U}{Om}\right)\right) - t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(\frac{-2}{Om} - \left(U - U*\right) \cdot \frac{n}{Om \cdot Om}\right) \cdot \left(n \cdot U\right)}\\ \end{array} \]

Alternative 6
Error	47.36%
Cost	14672

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;\ell \leq -2.4 \cdot 10^{+76}:\\ \;\;\;\;\left(\ell \cdot \left(-\sqrt{2}\right)\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{\frac{Om \cdot Om}{U* - U}} + \frac{-2}{Om}\right)}\\ \mathbf{elif}\;\ell \leq -6.5 \cdot 10^{-164}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\frac{n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om} + t_1 \cdot -2\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.5 \cdot 10^{-226}:\\ \;\;\;\;\sqrt{\left|t \cdot \left(U \cdot \left(n \cdot -2\right)\right)\right|}\\ \mathbf{elif}\;\ell \leq 2.1 \cdot 10^{+184}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot t_1 - n \cdot \left(t_1 \cdot \frac{U* - U}{Om}\right)\right) - t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(\frac{-2}{Om} - \left(U - U*\right) \cdot \frac{n}{Om \cdot Om}\right) \cdot \left(n \cdot U\right)}\\ \end{array} \]

Alternative 7
Error	47.1%
Cost	14672

\[\begin{array}{l} t_1 := \ell \cdot \sqrt{2}\\ t_2 := \frac{\ell}{\frac{Om}{\ell}}\\ t_3 := \sqrt{\left(\frac{-2}{Om} - \left(U - U*\right) \cdot \frac{n}{Om \cdot Om}\right) \cdot \left(n \cdot U\right)}\\ \mathbf{if}\;\ell \leq -4.2 \cdot 10^{+75}:\\ \;\;\;\;t_1 \cdot \left(-t_3\right)\\ \mathbf{elif}\;\ell \leq -1.26 \cdot 10^{-163}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\frac{n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om} + t_2 \cdot -2\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 5.5 \cdot 10^{-227}:\\ \;\;\;\;\sqrt{\left|t \cdot \left(U \cdot \left(n \cdot -2\right)\right)\right|}\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{+182}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot t_2 - n \cdot \left(t_2 \cdot \frac{U* - U}{Om}\right)\right) - t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot t_3\\ \end{array} \]

Alternative 8
Error	46.4%
Cost	14672

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;\ell \leq -1.1 \cdot 10^{+74}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(\frac{1}{Om} \cdot -2 - \frac{n}{Om} \cdot \frac{U - U*}{Om}\right)\right)} \cdot \left(\ell \cdot \left(-\sqrt{2}\right)\right)\\ \mathbf{elif}\;\ell \leq -2.25 \cdot 10^{-163}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\frac{n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om} + t_1 \cdot -2\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{-228}:\\ \;\;\;\;\sqrt{\left|t \cdot \left(U \cdot \left(n \cdot -2\right)\right)\right|}\\ \mathbf{elif}\;\ell \leq 3.7 \cdot 10^{+183}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot t_1 - n \cdot \left(t_1 \cdot \frac{U* - U}{Om}\right)\right) - t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(\frac{-2}{Om} - \left(U - U*\right) \cdot \frac{n}{Om \cdot Om}\right) \cdot \left(n \cdot U\right)}\\ \end{array} \]

Alternative 9
Error	47.17%
Cost	13644

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := \sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot t_1 - n \cdot \left(t_1 \cdot \frac{U* - U}{Om}\right)\right) - t\right)\right)}\\ \mathbf{if}\;n \leq -1.9 \cdot 10^{-87}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;n \leq 2.75 \cdot 10^{-50}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \ell \cdot \frac{\ell \cdot -2}{Om}\right)\right)\right)}\\ \mathbf{elif}\;n \leq 9.5 \cdot 10^{+205}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot t\right)} \cdot \sqrt{n}\\ \end{array} \]

Alternative 10
Error	52.09%
Cost	8656

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\frac{n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om} + \frac{\ell}{\frac{Om}{\ell}} \cdot -2\right)\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -5 \cdot 10^{+72}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right) \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq -1.7 \cdot 10^{-163}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 4.1 \cdot 10^{-225}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)}\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{+22}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \ell \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \end{array} \]

Alternative 11
Error	47.95%
Cost	8521

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;n \leq -6 \cdot 10^{-90} \lor \neg \left(n \leq 1.15 \cdot 10^{-49}\right):\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot t_1 - n \cdot \left(t_1 \cdot \frac{U* - U}{Om}\right)\right) - t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \ell \cdot \frac{\ell \cdot -2}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 12
Error	48.07%
Cost	8457

\[\begin{array}{l} \mathbf{if}\;n \leq -2.55 \cdot 10^{-42} \lor \neg \left(n \leq 1.18 \cdot 10^{-48}\right):\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \frac{\ell}{\frac{Om \cdot \frac{-Om}{\ell}}{U*}}\right) - t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \ell \cdot \frac{\ell \cdot -2}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 13
Error	62.71%
Cost	7629

\[\begin{array}{l} \mathbf{if}\;U \leq -3.5 \cdot 10^{-172}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;U \leq 1.3 \cdot 10^{-108} \lor \neg \left(U \leq 2.5 \cdot 10^{-96}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \left(n \cdot \ell\right)\right) \cdot \frac{-2}{Om}\right)\right)}\\ \end{array} \]

Alternative 14
Error	52.4%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;n \leq -2 \cdot 10^{+193}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;n \leq 1.8 \cdot 10^{+41}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \ell \cdot \frac{\ell \cdot -2}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 15
Error	51.81%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-50}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \ell \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq 2.15 \cdot 10^{+42}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \ell \cdot \frac{\ell \cdot -2}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 16
Error	50.84%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-50}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \ell \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq 5.3 \cdot 10^{+42}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \ell \cdot \frac{\ell \cdot -2}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)}\\ \end{array} \]

Alternative 17
Error	61.23%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;n \leq -2 \cdot 10^{-50}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;n \leq 5 \cdot 10^{+112}:\\ \;\;\;\;\sqrt{\left(n \cdot t\right) \cdot \left(2 \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 18
Error	62.54%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;U \leq -3 \cdot 10^{-172}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]

Alternative 19
Error	63.35%
Cost	6848

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

Reproduce

herbie shell --seed 2023102 
(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*))))))