Toniolo and Linder, Equation (13)

Average Accuracy: 45.9% → 51.6%

Time: 32.0s

Precision: binary64

Cost: 27668

Math

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

↓

\[\begin{array}{l} t_1 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := {\left({\left(\frac{-1}{U}\right)}^{-0.16666666666666666} \cdot {\left(t \cdot \left(n \cdot -2\right)\right)}^{0.16666666666666666}\right)}^{3}\\ t_3 := \sqrt{2 \cdot \left(\mathsf{fma}\left(t_1, n \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right) \cdot \left(U \cdot n\right)\right)}\\ \mathbf{if}\;U \leq -8.8 \cdot 10^{+189}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;U \leq -5000:\\ \;\;\;\;t_3\\ \mathbf{elif}\;U \leq -2.5 \cdot 10^{-212}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(t_1 \cdot \left(U - U*\right)\right)\right) - t\right)\right)}\\ \mathbf{elif}\;U \leq -6.8 \cdot 10^{-307}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;U \leq 7.6 \cdot 10^{-197}:\\ \;\;\;\;\sqrt{n \cdot \mathsf{fma}\left(\ell, \ell \cdot \frac{-2}{Om}, t\right)} \cdot \sqrt{U \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

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
         (pow
          (*
           (pow (/ -1.0 U) -0.16666666666666666)
           (pow (* t (* n -2.0)) 0.16666666666666666))
          3.0))
        (t_3
         (sqrt
          (*
           2.0
           (* (fma t_1 (* n (- U* U)) (fma (* l (/ l Om)) -2.0 t)) (* U n))))))
   (if (<= U -8.8e+189)
     t_2
     (if (<= U -5000.0)
       t_3
       (if (<= U -2.5e-212)
         (sqrt
          (*
           (* n -2.0)
           (* U (- (+ (* 2.0 (/ l (/ Om l))) (* n (* t_1 (- U U*)))) t))))
         (if (<= U -6.8e-307)
           t_2
           (if (<= U 7.6e-197)
             (* (sqrt (* n (fma l (* l (/ -2.0 Om)) t))) (sqrt (* U 2.0)))
             t_3)))))))

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 = pow((pow((-1.0 / U), -0.16666666666666666) * pow((t * (n * -2.0)), 0.16666666666666666)), 3.0);
	double t_3 = sqrt((2.0 * (fma(t_1, (n * (U_42_ - U)), fma((l * (l / Om)), -2.0, t)) * (U * n))));
	double tmp;
	if (U <= -8.8e+189) {
		tmp = t_2;
	} else if (U <= -5000.0) {
		tmp = t_3;
	} else if (U <= -2.5e-212) {
		tmp = sqrt(((n * -2.0) * (U * (((2.0 * (l / (Om / l))) + (n * (t_1 * (U - U_42_)))) - t))));
	} else if (U <= -6.8e-307) {
		tmp = t_2;
	} else if (U <= 7.6e-197) {
		tmp = sqrt((n * fma(l, (l * (-2.0 / Om)), t))) * sqrt((U * 2.0));
	} else {
		tmp = t_3;
	}
	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 = Float64((Float64(-1.0 / U) ^ -0.16666666666666666) * (Float64(t * Float64(n * -2.0)) ^ 0.16666666666666666)) ^ 3.0
	t_3 = sqrt(Float64(2.0 * Float64(fma(t_1, Float64(n * Float64(U_42_ - U)), fma(Float64(l * Float64(l / Om)), -2.0, t)) * Float64(U * n))))
	tmp = 0.0
	if (U <= -8.8e+189)
		tmp = t_2;
	elseif (U <= -5000.0)
		tmp = t_3;
	elseif (U <= -2.5e-212)
		tmp = sqrt(Float64(Float64(n * -2.0) * Float64(U * Float64(Float64(Float64(2.0 * Float64(l / Float64(Om / l))) + Float64(n * Float64(t_1 * Float64(U - U_42_)))) - t))));
	elseif (U <= -6.8e-307)
		tmp = t_2;
	elseif (U <= 7.6e-197)
		tmp = Float64(sqrt(Float64(n * fma(l, Float64(l * Float64(-2.0 / Om)), t))) * sqrt(Float64(U * 2.0)));
	else
		tmp = t_3;
	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[Power[N[(N[Power[N[(-1.0 / U), $MachinePrecision], -0.16666666666666666], $MachinePrecision] * N[Power[N[(t * N[(n * -2.0), $MachinePrecision]), $MachinePrecision], 0.16666666666666666], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(2.0 * N[(N[(t$95$1 * N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] + N[(N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] * -2.0 + t), $MachinePrecision]), $MachinePrecision] * N[(U * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[U, -8.8e+189], t$95$2, If[LessEqual[U, -5000.0], t$95$3, If[LessEqual[U, -2.5e-212], N[Sqrt[N[(N[(n * -2.0), $MachinePrecision] * N[(U * N[(N[(N[(2.0 * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(n * N[(t$95$1 * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[U, -6.8e-307], t$95$2, If[LessEqual[U, 7.6e-197], N[(N[Sqrt[N[(n * N[(l * N[(l * N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if U < -8.8000000000000002e189 or -2.50000000000000022e-212 < U < -6.79999999999999978e-307

   1. Initial program 38.6%

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

   2. Simplified 37.5%

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

      [Start] 38.6	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 37.9	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 37.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      fma-def [=>] 37.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-*l* [=>] 37.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in t around inf 31.4%

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

   4. Simplified 30.3%

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

      [Start] 31.4	\[ \sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)} \]
      associate-*r* [=>] 30.3	\[ \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot t\right) \cdot U\right)}} \]
      *-commutative [=>] 30.3	\[ \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

   5. Applied egg-rr 28.2%

      \[\leadsto \color{blue}{{\left({\left(\left(n \cdot t\right) \cdot \left(2 \cdot U\right)\right)}^{0.16666666666666666}\right)}^{3}} \]

   6. Taylor expanded in U around -inf 41.8%

      \[\leadsto {\color{blue}{\left(e^{0.16666666666666666 \cdot \left(-1 \cdot \log \left(\frac{-1}{U}\right) + \log \left(-2 \cdot \left(n \cdot t\right)\right)\right)}\right)}}^{3} \]

   7. Simplified 42.1%

      \[\leadsto {\color{blue}{\left({\left(\frac{-1}{U}\right)}^{-0.16666666666666666} \cdot {\left(t \cdot \left(n \cdot -2\right)\right)}^{0.16666666666666666}\right)}}^{3} \]
      Proof

      [Start] 41.8	\[ {\left(e^{0.16666666666666666 \cdot \left(-1 \cdot \log \left(\frac{-1}{U}\right) + \log \left(-2 \cdot \left(n \cdot t\right)\right)\right)}\right)}^{3} \]
      distribute-lft-in [=>] 41.8	\[ {\left(e^{\color{blue}{0.16666666666666666 \cdot \left(-1 \cdot \log \left(\frac{-1}{U}\right)\right) + 0.16666666666666666 \cdot \log \left(-2 \cdot \left(n \cdot t\right)\right)}}\right)}^{3} \]
      *-commutative [<=] 41.8	\[ {\left(e^{0.16666666666666666 \cdot \left(-1 \cdot \log \left(\frac{-1}{U}\right)\right) + \color{blue}{\log \left(-2 \cdot \left(n \cdot t\right)\right) \cdot 0.16666666666666666}}\right)}^{3} \]
      exp-sum [=>] 41.9	\[ {\color{blue}{\left(e^{0.16666666666666666 \cdot \left(-1 \cdot \log \left(\frac{-1}{U}\right)\right)} \cdot e^{\log \left(-2 \cdot \left(n \cdot t\right)\right) \cdot 0.16666666666666666}\right)}}^{3} \]
      *-commutative [=>] 41.9	\[ {\left(e^{\color{blue}{\left(-1 \cdot \log \left(\frac{-1}{U}\right)\right) \cdot 0.16666666666666666}} \cdot e^{\log \left(-2 \cdot \left(n \cdot t\right)\right) \cdot 0.16666666666666666}\right)}^{3} \]
      *-commutative [=>] 41.9	\[ {\left(e^{\color{blue}{\left(\log \left(\frac{-1}{U}\right) \cdot -1\right)} \cdot 0.16666666666666666} \cdot e^{\log \left(-2 \cdot \left(n \cdot t\right)\right) \cdot 0.16666666666666666}\right)}^{3} \]
      associate-*l* [=>] 41.9	\[ {\left(e^{\color{blue}{\log \left(\frac{-1}{U}\right) \cdot \left(-1 \cdot 0.16666666666666666\right)}} \cdot e^{\log \left(-2 \cdot \left(n \cdot t\right)\right) \cdot 0.16666666666666666}\right)}^{3} \]
      metadata-eval [=>] 41.9	\[ {\left(e^{\log \left(\frac{-1}{U}\right) \cdot \color{blue}{-0.16666666666666666}} \cdot e^{\log \left(-2 \cdot \left(n \cdot t\right)\right) \cdot 0.16666666666666666}\right)}^{3} \]
      metadata-eval [<=] 41.9	\[ {\left(e^{\log \left(\frac{-1}{U}\right) \cdot \color{blue}{\left(0.16666666666666666 \cdot -1\right)}} \cdot e^{\log \left(-2 \cdot \left(n \cdot t\right)\right) \cdot 0.16666666666666666}\right)}^{3} \]
      exp-to-pow [=>] 42.0	\[ {\left(\color{blue}{{\left(\frac{-1}{U}\right)}^{\left(0.16666666666666666 \cdot -1\right)}} \cdot e^{\log \left(-2 \cdot \left(n \cdot t\right)\right) \cdot 0.16666666666666666}\right)}^{3} \]
      metadata-eval [=>] 42.0	\[ {\left({\left(\frac{-1}{U}\right)}^{\color{blue}{-0.16666666666666666}} \cdot e^{\log \left(-2 \cdot \left(n \cdot t\right)\right) \cdot 0.16666666666666666}\right)}^{3} \]
      exp-to-pow [=>] 42.1	\[ {\left({\left(\frac{-1}{U}\right)}^{-0.16666666666666666} \cdot \color{blue}{{\left(-2 \cdot \left(n \cdot t\right)\right)}^{0.16666666666666666}}\right)}^{3} \]
      associate-*r* [=>] 42.1	\[ {\left({\left(\frac{-1}{U}\right)}^{-0.16666666666666666} \cdot {\color{blue}{\left(\left(-2 \cdot n\right) \cdot t\right)}}^{0.16666666666666666}\right)}^{3} \]
      *-commutative [=>] 42.1	\[ {\left({\left(\frac{-1}{U}\right)}^{-0.16666666666666666} \cdot {\color{blue}{\left(t \cdot \left(-2 \cdot n\right)\right)}}^{0.16666666666666666}\right)}^{3} \]
      *-commutative [=>] 42.1	\[ {\left({\left(\frac{-1}{U}\right)}^{-0.16666666666666666} \cdot {\left(t \cdot \color{blue}{\left(n \cdot -2\right)}\right)}^{0.16666666666666666}\right)}^{3} \]

   if -8.8000000000000002e189 < U < -5e3 or 7.5999999999999998e-197 < U

   1. Initial program 51.2%

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

   2. Simplified 54.2%

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

      [Start] 51.2	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 51.2	\[ \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 51.2	\[ \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      *-commutative [=>] 51.2	\[ \sqrt{2 \cdot \color{blue}{\left(\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot \left(n \cdot U\right)\right)}} \]

   if -5e3 < U < -2.50000000000000022e-212

   1. Initial program 46.1%

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

   2. Simplified 54.4%

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

      [Start] 46.1	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 50.4	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 50.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      sub-neg [=>] 50.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(-\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [<=] 50.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      cancel-sign-sub [<=] 50.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      cancel-sign-sub [=>] 50.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-/l* [=>] 55.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 54.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   if -6.79999999999999978e-307 < U < 7.5999999999999998e-197

   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. Taylor expanded in n around 0 34.7%

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

   3. Simplified 35.5%

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

      [Start] 34.7	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot U\right)\right)} \]
      associate-*r* [=>] 31.8	\[ \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right)}} \]
      *-commutative [=>] 31.8	\[ \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
      cancel-sign-sub-inv [=>] 31.8	\[ \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{\left(t + \left(-2\right) \cdot \frac{{\ell}^{2}}{Om}\right)}\right)\right)} \]
      metadata-eval [=>] 31.8	\[ \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \color{blue}{-2} \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)} \]
      unpow2 [=>] 31.8	\[ \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + -2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right)\right)\right)} \]
      associate-*r/ [<=] 35.5	\[ \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + -2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)}\right)\right)\right)} \]
      *-commutative [<=] 35.5	\[ \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2}\right)\right)\right)} \]

   4. Applied egg-rr 48.6%

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

   5. Simplified 48.6%

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

      [Start] 48.6	\[ \sqrt{2 \cdot U} \cdot \sqrt{n \cdot \mathsf{fma}\left(\ell, \ell \cdot \frac{-2}{Om}, t\right)} \]
      *-commutative [=>] 48.6	\[ \color{blue}{\sqrt{n \cdot \mathsf{fma}\left(\ell, \ell \cdot \frac{-2}{Om}, t\right)} \cdot \sqrt{2 \cdot U}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -8.8 \cdot 10^{+189}:\\ \;\;\;\;{\left({\left(\frac{-1}{U}\right)}^{-0.16666666666666666} \cdot {\left(t \cdot \left(n \cdot -2\right)\right)}^{0.16666666666666666}\right)}^{3}\\ \mathbf{elif}\;U \leq -5000:\\ \;\;\;\;\sqrt{2 \cdot \left(\mathsf{fma}\left({\left(\frac{\ell}{Om}\right)}^{2}, n \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right) \cdot \left(U \cdot n\right)\right)}\\ \mathbf{elif}\;U \leq -2.5 \cdot 10^{-212}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right) - t\right)\right)}\\ \mathbf{elif}\;U \leq -6.8 \cdot 10^{-307}:\\ \;\;\;\;{\left({\left(\frac{-1}{U}\right)}^{-0.16666666666666666} \cdot {\left(t \cdot \left(n \cdot -2\right)\right)}^{0.16666666666666666}\right)}^{3}\\ \mathbf{elif}\;U \leq 7.6 \cdot 10^{-197}:\\ \;\;\;\;\sqrt{n \cdot \mathsf{fma}\left(\ell, \ell \cdot \frac{-2}{Om}, t\right)} \cdot \sqrt{U \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\mathsf{fma}\left({\left(\frac{\ell}{Om}\right)}^{2}, n \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right) \cdot \left(U \cdot n\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	57.7%
Cost	37900

\[\begin{array}{l} t_1 := \left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - t\right)\right)\right)}\\ \mathbf{elif}\;t_1 \leq 10^{+304}:\\ \;\;\;\;\sqrt{t_1}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\sqrt{-2 \cdot \left(\ell \cdot \left(2 \cdot \left(\frac{\ell}{Om} \cdot \left(U \cdot n\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{U \cdot \frac{n \cdot -2}{Om}}\right)\\ \end{array} \]

Alternative 2
Accuracy	51.3%
Cost	14804

\[\begin{array}{l} t_1 := \sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \frac{n}{Om} \cdot \frac{\ell \cdot \left(U - U*\right)}{\frac{Om}{\ell}}\right) - t\right)\right)}\\ \mathbf{if}\;\ell \leq -1.85 \cdot 10^{+195}:\\ \;\;\;\;\sqrt{\ell \cdot \left(2 \cdot \left(\left(\ell \cdot \left(U \cdot n\right)\right) \cdot \left(\frac{-2}{Om} + n \cdot \frac{U*}{Om \cdot Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -3.8 \cdot 10^{-279}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{-242}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(t \cdot n\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{-81}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 2.75 \cdot 10^{+144}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{-2}{Om} + \frac{n}{Om \cdot Om} \cdot \left(U* - U\right)\right)\right)}\right)\\ \end{array} \]

Alternative 3
Accuracy	51.5%
Cost	14728

\[\begin{array}{l} \mathbf{if}\;\ell \leq -9.5 \cdot 10^{+195}:\\ \;\;\;\;\sqrt{\ell \cdot \left(2 \cdot \left(\left(\ell \cdot \left(U \cdot n\right)\right) \cdot \left(\frac{-2}{Om} + n \cdot \frac{U*}{Om \cdot Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.2 \cdot 10^{-82}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right) - t\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+144}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{-2}{Om} + \frac{n}{Om \cdot Om} \cdot \left(U* - U\right)\right)\right)}\right)\\ \end{array} \]

Alternative 4
Accuracy	49.0%
Cost	8784

\[\begin{array}{l} t_1 := 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - t\\ t_2 := \sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \frac{n}{Om} \cdot \frac{\ell \cdot \left(U - U*\right)}{\frac{Om}{\ell}}\right) - t\right)\right)}\\ \mathbf{if}\;n \leq -1.75 \cdot 10^{+74}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right)}\\ \mathbf{elif}\;n \leq -2.65 \cdot 10^{-154}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;n \leq -6.2 \cdot 10^{-201}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot t_1\right)}\\ \mathbf{elif}\;n \leq 4.4 \cdot 10^{-64}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Accuracy	48.9%
Cost	8784

\[\begin{array}{l} t_1 := 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - t\\ t_3 := \ell \cdot \left(U - U*\right)\\ \mathbf{if}\;n \leq -9.2 \cdot 10^{+73}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right)}\\ \mathbf{elif}\;n \leq -1.18 \cdot 10^{-154}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(t_1 - \frac{-1}{\frac{Om}{t_3 \cdot \left(\ell \cdot \frac{n}{Om}\right)}}\right) - t\right)\right)}\\ \mathbf{elif}\;n \leq -3.5 \cdot 10^{-203}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot t_2\right)}\\ \mathbf{elif}\;n \leq 1.45 \cdot 10^{-64}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot t_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(t_1 + \frac{n}{Om} \cdot \frac{t_3}{\frac{Om}{\ell}}\right) - t\right)\right)}\\ \end{array} \]

Alternative 6
Accuracy	47.0%
Cost	8532

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ t_2 := \ell \cdot \left(n \cdot \ell\right)\\ \mathbf{if}\;\ell \leq -2 \cdot 10^{+195}:\\ \;\;\;\;\sqrt{\ell \cdot \left(2 \cdot \left(\left(\ell \cdot \left(U \cdot n\right)\right) \cdot \left(\frac{-2}{Om} + n \cdot \frac{U*}{Om \cdot Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{-168}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\ell \cdot \left(2 \cdot \frac{\ell}{Om}\right) - t\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.6 \cdot 10^{-40}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(t + \left(-2 \cdot t_1 + \frac{t_2}{Om} \cdot \frac{U*}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 8.5 \cdot 10^{+124}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot t_1 - t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{+273}:\\ \;\;\;\;\sqrt{\left(\frac{2}{Om} + \frac{n}{Om} \cdot \frac{U - U*}{Om}\right) \cdot \left(-2 \cdot \left(U \cdot t_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\ell \cdot \left(2 \cdot \left(\frac{\ell}{Om} \cdot \left(U \cdot n\right)\right)\right)\right)}\\ \end{array} \]

Alternative 7
Accuracy	46.3%
Cost	8400

\[\begin{array}{l} \mathbf{if}\;\ell \leq -3.1 \cdot 10^{+196}:\\ \;\;\;\;\sqrt{\ell \cdot \left(2 \cdot \left(\left(\ell \cdot \left(U \cdot n\right)\right) \cdot \left(\frac{-2}{Om} + n \cdot \frac{U*}{Om \cdot Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{-307}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\ell \cdot \left(2 \cdot \frac{\ell}{Om}\right) - t\right)\right)}\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+119}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{+273}:\\ \;\;\;\;\sqrt{\left(\frac{2}{Om} + \frac{n}{Om} \cdot \frac{U - U*}{Om}\right) \cdot \left(-2 \cdot \left(U \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\ell \cdot \left(2 \cdot \left(\frac{\ell}{Om} \cdot \left(U \cdot n\right)\right)\right)\right)}\\ \end{array} \]

Alternative 8
Accuracy	38.2%
Cost	7628

\[\begin{array}{l} \mathbf{if}\;\ell \leq -6.2 \cdot 10^{+45}:\\ \;\;\;\;\sqrt{-2 \cdot \left(2 \cdot \frac{\ell \cdot \left(U \cdot \ell\right)}{\frac{Om}{n}}\right)}\\ \mathbf{elif}\;\ell \leq -5.2 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 10^{+74}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(t \cdot n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(2 \cdot \frac{U \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}{Om}\right)}\\ \end{array} \]

Alternative 9
Accuracy	38.4%
Cost	7628

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.92 \cdot 10^{+37}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(\frac{-2}{Om} \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.7 \cdot 10^{-291}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 8.5 \cdot 10^{+77}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(t \cdot n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(2 \cdot \frac{U \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}{Om}\right)}\\ \end{array} \]

Alternative 10
Accuracy	45.4%
Cost	7625

\[\begin{array}{l} \mathbf{if}\;\ell \leq -4.6 \cdot 10^{+15} \lor \neg \left(\ell \leq -1.45 \cdot 10^{-305}\right):\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]

Alternative 11
Accuracy	45.9%
Cost	7625

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.3 \cdot 10^{+21} \lor \neg \left(\ell \leq -5.6 \cdot 10^{-305}\right):\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{Om} - t\right)\right)\right)}\\ \end{array} \]

Alternative 12
Accuracy	48.8%
Cost	7625

\[\begin{array}{l} t_1 := 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - t\\ \mathbf{if}\;n \leq -6.2 \cdot 10^{-92} \lor \neg \left(n \leq 4.4 \cdot 10^{+65}\right):\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot t_1\right)\right)}\\ \end{array} \]

Alternative 13
Accuracy	41.7%
Cost	7497

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.75 \cdot 10^{+183} \lor \neg \left(\ell \leq 6.8 \cdot 10^{+77}\right):\\ \;\;\;\;\sqrt{-2 \cdot \left(\ell \cdot \left(2 \cdot \left(\frac{\ell}{Om} \cdot \left(U \cdot n\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(t \cdot n\right)\right)}\\ \end{array} \]

Alternative 14
Accuracy	38.7%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;U \leq -1.8 \cdot 10^{+83} \lor \neg \left(U \leq 0.001\right):\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(t \cdot n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]

Alternative 15
Accuracy	38.7%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;n \leq -4.1 \cdot 10^{-221} \lor \neg \left(n \leq 5 \cdot 10^{+65}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(U \cdot n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(t \cdot n\right)\right)}\\ \end{array} \]

Alternative 16
Accuracy	37.2%
Cost	6848

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

Reproduce

herbie shell --seed 2023122 
(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*))))))