Toniolo and Linder, Equation (13)

Average Accuracy: 40.5% → 48.2%

Time: 42.8s

Precision: binary64

Cost: 27340

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt{n \cdot 2}\\ \mathbf{if}\;n \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \left(n \cdot \ell\right) \cdot \frac{U* - U}{Om}}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)}\\ \mathbf{elif}\;n \leq 1.7 \cdot 10^{-267}:\\ \;\;\;\;\sqrt{U \cdot \left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\frac{\ell}{Om}, n \cdot \left(U* - U\right), \ell \cdot -2\right)\right) + U \cdot t} \cdot t_1\\ \mathbf{elif}\;n \leq 3.3 \cdot 10^{-217}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \sqrt{U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, n \cdot \left(\left(U* - U\right) \cdot \frac{\ell}{Om}\right)\right), t\right)}\\ \end{array} \]

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 (sqrt (* n 2.0))))
   (if (<= n -1e-310)
     (sqrt
      (*
       2.0
       (+
        (* n (* U t))
        (/ (+ (* l -2.0) (* (* n l) (/ (- U* U) Om))) (/ Om (* n (* U l)))))))
     (if (<= n 1.7e-267)
       (*
        (sqrt
         (+
          (* U (* (/ l Om) (fma (/ l Om) (* n (- U* U)) (* l -2.0))))
          (* U t)))
        t_1)
       (if (<= n 3.3e-217)
         (sqrt (* 2.0 (* U (* n t))))
         (*
          t_1
          (sqrt
           (*
            U
            (fma (/ l Om) (fma l -2.0 (* n (* (- U* U) (/ l Om)))) 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 = sqrt((n * 2.0));
	double tmp;
	if (n <= -1e-310) {
		tmp = sqrt((2.0 * ((n * (U * t)) + (((l * -2.0) + ((n * l) * ((U_42_ - U) / Om))) / (Om / (n * (U * l)))))));
	} else if (n <= 1.7e-267) {
		tmp = sqrt(((U * ((l / Om) * fma((l / Om), (n * (U_42_ - U)), (l * -2.0)))) + (U * t))) * t_1;
	} else if (n <= 3.3e-217) {
		tmp = sqrt((2.0 * (U * (n * t))));
	} else {
		tmp = t_1 * sqrt((U * fma((l / Om), fma(l, -2.0, (n * ((U_42_ - U) * (l / Om)))), 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 = sqrt(Float64(n * 2.0))
	tmp = 0.0
	if (n <= -1e-310)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * Float64(U * t)) + Float64(Float64(Float64(l * -2.0) + Float64(Float64(n * l) * Float64(Float64(U_42_ - U) / Om))) / Float64(Om / Float64(n * Float64(U * l)))))));
	elseif (n <= 1.7e-267)
		tmp = Float64(sqrt(Float64(Float64(U * Float64(Float64(l / Om) * fma(Float64(l / Om), Float64(n * Float64(U_42_ - U)), Float64(l * -2.0)))) + Float64(U * t))) * t_1);
	elseif (n <= 3.3e-217)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t))));
	else
		tmp = Float64(t_1 * sqrt(Float64(U * fma(Float64(l / Om), fma(l, -2.0, Float64(n * Float64(Float64(U_42_ - U) * Float64(l / Om)))), 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[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -1e-310], N[Sqrt[N[(2.0 * N[(N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(l * -2.0), $MachinePrecision] + N[(N[(n * l), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om / N[(n * N[(U * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 1.7e-267], N[(N[Sqrt[N[(N[(U * N[(N[(l / Om), $MachinePrecision] * N[(N[(l / Om), $MachinePrecision] * N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[n, 3.3e-217], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t$95$1 * N[Sqrt[N[(U * N[(N[(l / Om), $MachinePrecision] * N[(l * -2.0 + N[(n * N[(N[(U$42$ - U), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -9.999999999999969e-311

   1. Initial program 33.1%

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

   2. Simplified 38.4%

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

   3. Taylor expanded in t around inf 43.7%

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

   4. Simplified 46.3%

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

   5. Applied egg-rr 47.9%

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

   if -9.999999999999969e-311 < n < 1.7000000000000001e-267

   1. Initial program 24.7%

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

   2. Simplified 32.7%

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

   3. Applied egg-rr 45.9%

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

   4. Applied egg-rr 61.3%

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

   if 1.7000000000000001e-267 < n < 3.29999999999999993e-217

   1. Initial program 63.9%

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

   2. Simplified 36.1%

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

   3. Taylor expanded in t around inf 36.8%

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

   4. Simplified 73.1%

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

   if 3.29999999999999993e-217 < n

   1. Initial program 46.3%

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

   2. Simplified 49.2%

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

   3. Applied egg-rr 62.7%

      \[\leadsto \color{blue}{\sqrt{U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, n \cdot \left(\left(U* - U\right) \cdot \frac{\ell}{Om}\right)\right), t\right)} \cdot \sqrt{2 \cdot n}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \left(n \cdot \ell\right) \cdot \frac{U* - U}{Om}}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)}\\ \mathbf{elif}\;n \leq 1.7 \cdot 10^{-267}:\\ \;\;\;\;\sqrt{U \cdot \left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\frac{\ell}{Om}, n \cdot \left(U* - U\right), \ell \cdot -2\right)\right) + U \cdot t} \cdot \sqrt{n \cdot 2}\\ \mathbf{elif}\;n \leq 3.3 \cdot 10^{-217}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, n \cdot \left(\left(U* - U\right) \cdot \frac{\ell}{Om}\right)\right), t\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	46.7%
Cost	15124

\[\begin{array}{l} t_1 := U \cdot \left(n \cdot 2\right)\\ \mathbf{if}\;\ell \leq -7.6 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) - \frac{\frac{\ell \cdot \left(n \cdot U\right)}{Om} - \ell \cdot -2}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)}\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-127}:\\ \;\;\;\;\sqrt{t_1 \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} - \frac{2}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq -1.25 \cdot 10^{-285}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.06 \cdot 10^{-168}:\\ \;\;\;\;\sqrt{t \cdot t_1}\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{+118}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(n \cdot \left(\left(U* - U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(-2 + \left(U* - U\right) \cdot \frac{n}{Om}\right)}}}\right)\\ \end{array} \]

Alternative 2
Accuracy	48.2%
Cost	15124

\[\begin{array}{l} t_1 := U \cdot \left(n \cdot 2\right)\\ \mathbf{if}\;\ell \leq -1.9 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) - \frac{\frac{\ell \cdot \left(n \cdot U\right)}{Om} - \ell \cdot -2}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)}\\ \mathbf{elif}\;\ell \leq -7.5 \cdot 10^{-128}:\\ \;\;\;\;\sqrt{t_1 \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} - \frac{2}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq -2.9 \cdot 10^{-289}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq -9.5 \cdot 10^{-294}:\\ \;\;\;\;\sqrt{U + U} \cdot \sqrt{n \cdot t}\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{-120}:\\ \;\;\;\;\sqrt{t_1 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(U* - U\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;\ell \leq 3 \cdot 10^{+119}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(-2 + \left(U* - U\right) \cdot \frac{n}{Om}\right)}}}\right)\\ \end{array} \]

Alternative 3
Accuracy	47.5%
Cost	14676

\[\begin{array}{l} t_1 := U \cdot \left(n \cdot 2\right)\\ t_2 := \frac{Om}{n \cdot \left(U \cdot \ell\right)}\\ t_3 := n \cdot \left(U \cdot t\right)\\ \mathbf{if}\;\ell \leq -2.1 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_3 - \frac{\frac{\ell \cdot \left(n \cdot U\right)}{Om} - \ell \cdot -2}{t_2}\right)}\\ \mathbf{elif}\;\ell \leq -2.65 \cdot 10^{-129}:\\ \;\;\;\;\sqrt{t_1 \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} - \frac{2}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq -4.2 \cdot 10^{-285}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{-159}:\\ \;\;\;\;\sqrt{t \cdot t_1}\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{+120}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_3 + \frac{\ell \cdot -2 + \left(U* - U\right) \cdot \frac{n}{\frac{Om}{\ell}}}{t_2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(-2 + \left(U* - U\right) \cdot \frac{n}{Om}\right)}}}\right)\\ \end{array} \]

Alternative 4
Accuracy	38.6%
Cost	8800

\[\begin{array}{l} t_1 := \sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{if}\;U* \leq -2 \cdot 10^{-73}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;U* \leq -6.6 \cdot 10^{-204}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;U* \leq -3.2 \cdot 10^{-226}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \frac{\left(U \cdot \ell\right) \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om}\right)}{Om}}\\ \mathbf{elif}\;U* \leq -3.1 \cdot 10^{-244}:\\ \;\;\;\;\sqrt{\frac{-4}{\frac{Om}{\ell \cdot \left(n \cdot \left(U \cdot \ell\right)\right)}}}\\ \mathbf{elif}\;U* \leq 3.6 \cdot 10^{-260}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \frac{2 \cdot \ell}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{elif}\;U* \leq 9.6 \cdot 10^{-226}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \frac{\ell \cdot -2 + \frac{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}{Om}}{\frac{Om}{U \cdot \ell}}}\\ \mathbf{elif}\;U* \leq 1.1 \cdot 10^{-173}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;U* \leq 1.2 \cdot 10^{+120}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(n \cdot \ell\right)}{Om} \cdot \frac{U* - U}{Om}\right)\right)}\\ \end{array} \]

Alternative 5
Accuracy	38.6%
Cost	8800

\[\begin{array}{l} t_1 := \sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\\ t_2 := \frac{U* - U}{Om}\\ t_3 := \ell \cdot \left(n \cdot \left(U \cdot \ell\right)\right)\\ \mathbf{if}\;U* \leq -8.5 \cdot 10^{-73}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;U* \leq -6.6 \cdot 10^{-204}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;U* \leq -8.4 \cdot 10^{-225}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \frac{\left(U \cdot \ell\right) \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om}\right)}{Om}}\\ \mathbf{elif}\;U* \leq -3.1 \cdot 10^{-244}:\\ \;\;\;\;\sqrt{\frac{-4}{\frac{Om}{t_3}}}\\ \mathbf{elif}\;U* \leq 3.6 \cdot 10^{-260}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \frac{2 \cdot \ell}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{elif}\;U* \leq 3.1 \cdot 10^{-226}:\\ \;\;\;\;\sqrt{\left(-2 \cdot t_3\right) \cdot \left(\frac{2}{Om} - \frac{n}{Om} \cdot t_2\right)}\\ \mathbf{elif}\;U* \leq 2.65 \cdot 10^{-166}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;U* \leq 5.5 \cdot 10^{+120}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(n \cdot \ell\right)}{Om} \cdot t_2\right)\right)}\\ \end{array} \]

Alternative 6
Accuracy	38.5%
Cost	8800

\[\begin{array}{l} t_1 := \sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\\ t_2 := \frac{U* - U}{Om}\\ t_3 := \ell \cdot \left(n \cdot \ell\right)\\ \mathbf{if}\;U* \leq -5.8 \cdot 10^{-73}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;U* \leq -6.6 \cdot 10^{-204}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;U* \leq -2.7 \cdot 10^{-224}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \frac{\left(U \cdot \ell\right) \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om}\right)}{Om}}\\ \mathbf{elif}\;U* \leq -3.1 \cdot 10^{-244}:\\ \;\;\;\;\sqrt{\frac{-4}{\frac{Om}{\ell \cdot \left(n \cdot \left(U \cdot \ell\right)\right)}}}\\ \mathbf{elif}\;U* \leq 3.6 \cdot 10^{-260}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \frac{2 \cdot \ell}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{elif}\;U* \leq 2.8 \cdot 10^{-225}:\\ \;\;\;\;\sqrt{\left(-2 \cdot \left(U \cdot t_3\right)\right) \cdot \left(\frac{2}{Om} - \frac{n}{Om} \cdot t_2\right)}\\ \mathbf{elif}\;U* \leq 8.2 \cdot 10^{-174}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;U* \leq 1.7 \cdot 10^{+120}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{t_3}{Om} \cdot t_2\right)\right)}\\ \end{array} \]

Alternative 7
Accuracy	45.2%
Cost	8788

\[\begin{array}{l} t_1 := U \cdot \left(n \cdot 2\right)\\ t_2 := \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{n \cdot \ell}{\frac{Om}{U*}}}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)}\\ \mathbf{if}\;\ell \leq -2.1 \cdot 10^{+62}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-127}:\\ \;\;\;\;\sqrt{t_1 \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} - \frac{2}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq -3.1 \cdot 10^{-285}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 5.9 \cdot 10^{-158}:\\ \;\;\;\;\sqrt{t \cdot t_1}\\ \mathbf{elif}\;\ell \leq 2.2 \cdot 10^{-41}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{n}{Om} \cdot \frac{\ell \cdot \ell}{\frac{Om}{U*}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Accuracy	45.8%
Cost	8784

\[\begin{array}{l} t_1 := U \cdot \left(n \cdot 2\right)\\ t_2 := \frac{Om}{n \cdot \left(U \cdot \ell\right)}\\ t_3 := n \cdot \left(U \cdot t\right)\\ \mathbf{if}\;\ell \leq -1.95 \cdot 10^{+147}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_3 - \frac{\frac{\ell \cdot \left(n \cdot U\right)}{Om} - \ell \cdot -2}{t_2}\right)}\\ \mathbf{elif}\;\ell \leq -2.05 \cdot 10^{-129}:\\ \;\;\;\;\sqrt{t_1 \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} - \frac{2}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq -1.9 \cdot 10^{-284}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 6.1 \cdot 10^{-170}:\\ \;\;\;\;\sqrt{t \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_3 + \frac{\ell \cdot -2 + \left(U* - U\right) \cdot \frac{n}{\frac{Om}{\ell}}}{t_2}\right)}\\ \end{array} \]

Alternative 9
Accuracy	45.0%
Cost	8656

\[\begin{array}{l} t_1 := U \cdot \left(n \cdot 2\right)\\ t_2 := \frac{Om}{n \cdot \left(U \cdot \ell\right)}\\ t_3 := n \cdot \left(U \cdot t\right)\\ \mathbf{if}\;\ell \leq -7.5 \cdot 10^{+148}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_3 - \frac{\frac{\ell \cdot \left(n \cdot U\right)}{Om} - \ell \cdot -2}{t_2}\right)}\\ \mathbf{elif}\;\ell \leq -1.1 \cdot 10^{-127}:\\ \;\;\;\;\sqrt{t_1 \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} - \frac{2}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq -8 \cdot 10^{-285}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{t \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_3 + \frac{\ell \cdot -2 + \frac{n \cdot \ell}{\frac{Om}{U*}}}{t_2}\right)}\\ \end{array} \]

Alternative 10
Accuracy	43.7%
Cost	8400

\[\begin{array}{l} t_1 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{if}\;Om \leq -4.3 \cdot 10^{+136}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \frac{2 \cdot \ell}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{elif}\;Om \leq -7.8 \cdot 10^{-150}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;Om \leq -2.3 \cdot 10^{-218}:\\ \;\;\;\;\sqrt{2 \cdot \frac{n \cdot \left(\left(U \cdot \ell\right) \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om}\right)\right)}{Om}}\\ \mathbf{elif}\;Om \leq 1.5 \cdot 10^{+114}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\\ \end{array} \]

Alternative 11
Accuracy	46.0%
Cost	8268

\[\begin{array}{l} t_1 := \frac{n}{Om} \cdot \frac{U* - U}{Om}\\ t_2 := \sqrt{\left(-2 \cdot \left(\ell \cdot \left(n \cdot \left(U \cdot \ell\right)\right)\right)\right) \cdot \left(\frac{2}{Om} - t_1\right)}\\ \mathbf{if}\;\ell \leq -9 \cdot 10^{+148}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -2.9 \cdot 10^{-129}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(t_1 - \frac{2}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.5 \cdot 10^{+118}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 12
Accuracy	40.7%
Cost	8136

\[\begin{array}{l} t_1 := U \cdot \left(n \cdot 2\right)\\ \mathbf{if}\;U \leq -5.6 \cdot 10^{-250}:\\ \;\;\;\;\sqrt{t_1 \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{elif}\;U \leq -1.52 \cdot 10^{-282}:\\ \;\;\;\;\sqrt{2 \cdot \frac{n \cdot \left(\left(U \cdot \ell\right) \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om}\right)\right)}{Om}}\\ \mathbf{elif}\;U \leq 1.25 \cdot 10^{+61}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \frac{2 \cdot \ell}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot t_1}\\ \end{array} \]

Alternative 13
Accuracy	40.9%
Cost	7880

\[\begin{array}{l} t_1 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \frac{2 \cdot \ell}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{if}\;\ell \leq -5000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{-25}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{n}{Om} \cdot \frac{\ell \cdot \ell}{\frac{Om}{U*}}\right)\right)}\\ \mathbf{elif}\;\ell \leq 7 \cdot 10^{+156}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-4}{\frac{Om}{\ell \cdot \left(n \cdot \left(U \cdot \ell\right)\right)}}}\\ \end{array} \]

Alternative 14
Accuracy	39.9%
Cost	7625

\[\begin{array}{l} \mathbf{if}\;n \leq 7.5 \cdot 10^{-269} \lor \neg \left(n \leq 4 \cdot 10^{-205}\right):\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \frac{2 \cdot \ell}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]

Alternative 15
Accuracy	36.8%
Cost	7369

\[\begin{array}{l} \mathbf{if}\;\ell \leq -4.3 \cdot 10^{+135} \lor \neg \left(\ell \leq 3.9 \cdot 10^{+155}\right):\\ \;\;\;\;\sqrt{\frac{-4}{\frac{Om}{\ell \cdot \left(n \cdot \left(U \cdot \ell\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \end{array} \]

Alternative 16
Accuracy	32.5%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;U* \leq -1.8 \cdot 10^{-166}:\\ \;\;\;\;\sqrt{t \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{elif}\;U* \leq -1.15 \cdot 10^{-192}:\\ \;\;\;\;\sqrt{-4 \cdot \left(\frac{n}{Om} \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]

Alternative 17
Accuracy	33.2%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;n \leq 1.2 \cdot 10^{-200}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot t\right) \cdot \left(n \cdot 2\right)}\\ \end{array} \]

Alternative 18
Accuracy	32.9%
Cost	6848

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

Alternative 19
Accuracy	5.7%
Cost	6528

\[\mathsf{hypot}\left(-2, -2\right) \]

Alternative 20
Accuracy	5.0%
Cost	448

\[t \cdot \left(U \cdot \left(n + n\right)\right) \]

Alternative 21
Accuracy	4.5%
Cost	320

\[\left(n \cdot n\right) \cdot 4 \]

Alternative 22
Accuracy	2.5%
Cost	192

\[n + -2 \]

Alternative 23
Accuracy	3.7%
Cost	192

\[n + n \]

Reproduce

herbie shell --seed 2023157 -o generate:proofs
(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*))))))