Toniolo and Linder, Equation (13)

Average Error: 34.4 → 27.6

Time: 1.3min

Precision: binary64

Cost: 33944

Math

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

↓

\[\begin{array}{l} t_1 := \frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\\ \mathbf{if}\;\ell \leq -1.45 \cdot 10^{+191}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} + 2 \cdot \frac{-1}{Om}\right)\right)} \cdot \left(\ell \cdot \left(-\sqrt{2}\right)\right)\\ \mathbf{elif}\;\ell \leq -2.9 \cdot 10^{+75}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq -6100:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(U* - U\right) \cdot \left(\frac{\ell}{\frac{Om}{\ell}} \cdot \frac{n}{Om}\right) - \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right)}\\ \mathbf{elif}\;\ell \leq -3.3 \cdot 10^{-142}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{\ell \cdot t_1}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.55 \cdot 10^{-201}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right) - \mathsf{fma}\left(\ell, -2, n \cdot \left(\left(U* - U\right) \cdot \frac{\ell}{Om}\right)\right) \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U \cdot -2\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{+69}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \frac{n \cdot \left(\ell \cdot \left(U \cdot t_1\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n}{Om} \cdot \left(U \cdot \mathsf{fma}\left(\frac{n}{Om}, U* - U, -2\right)\right)}}\right)}^{2}\\ \end{array} \]

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 (* l U*)) Om) (* l -2.0))))
   (if (<= l -1.45e+191)
     (*
      (sqrt
       (* n (* U (+ (/ (* n (- U* U)) (pow Om 2.0)) (* 2.0 (/ -1.0 Om))))))
      (* l (- (sqrt 2.0))))
     (if (<= l -2.9e+75)
       (pow
        (*
         (* 2.0 n)
         (*
          U
          (-
           t
           (fma 2.0 (* l (/ l Om)) (* n (* (pow (/ l Om) 2.0) (- U U*)))))))
        0.5)
       (if (<= l -6100.0)
         (sqrt
          (*
           (* U (* 2.0 n))
           (-
            (* (- U* U) (* (/ l (/ Om l)) (/ n Om)))
            (- (* 2.0 (/ (* l l) Om)) t))))
         (if (<= l -3.3e-142)
           (sqrt (* 2.0 (* n (* U (+ t (/ (* l t_1) Om))))))
           (if (<= l 1.55e-201)
             (sqrt
              (-
               (* 2.0 (* t (* n U)))
               (*
                (fma l -2.0 (* n (* (- U* U) (/ l Om))))
                (* (/ l Om) (* n (* U -2.0))))))
             (if (<= l 6e+69)
               (sqrt
                (+ (* 2.0 (* n (* U t))) (* 2.0 (/ (* n (* l (* U t_1))) Om))))
               (pow
                (sqrt
                 (*
                  (* l (sqrt 2.0))
                  (sqrt (* (/ n Om) (* U (fma (/ n Om) (- U* U) -2.0))))))
                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 = ((n * (l * U_42_)) / Om) + (l * -2.0);
	double tmp;
	if (l <= -1.45e+191) {
		tmp = sqrt((n * (U * (((n * (U_42_ - U)) / pow(Om, 2.0)) + (2.0 * (-1.0 / Om)))))) * (l * -sqrt(2.0));
	} else if (l <= -2.9e+75) {
		tmp = pow(((2.0 * n) * (U * (t - fma(2.0, (l * (l / Om)), (n * (pow((l / Om), 2.0) * (U - U_42_))))))), 0.5);
	} else if (l <= -6100.0) {
		tmp = sqrt(((U * (2.0 * n)) * (((U_42_ - U) * ((l / (Om / l)) * (n / Om))) - ((2.0 * ((l * l) / Om)) - t))));
	} else if (l <= -3.3e-142) {
		tmp = sqrt((2.0 * (n * (U * (t + ((l * t_1) / Om))))));
	} else if (l <= 1.55e-201) {
		tmp = sqrt(((2.0 * (t * (n * U))) - (fma(l, -2.0, (n * ((U_42_ - U) * (l / Om)))) * ((l / Om) * (n * (U * -2.0))))));
	} else if (l <= 6e+69) {
		tmp = sqrt(((2.0 * (n * (U * t))) + (2.0 * ((n * (l * (U * t_1))) / Om))));
	} else {
		tmp = pow(sqrt(((l * sqrt(2.0)) * sqrt(((n / Om) * (U * fma((n / Om), (U_42_ - U), -2.0)))))), 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(Float64(Float64(n * Float64(l * U_42_)) / Om) + Float64(l * -2.0))
	tmp = 0.0
	if (l <= -1.45e+191)
		tmp = Float64(sqrt(Float64(n * Float64(U * Float64(Float64(Float64(n * Float64(U_42_ - U)) / (Om ^ 2.0)) + Float64(2.0 * Float64(-1.0 / Om)))))) * Float64(l * Float64(-sqrt(2.0))));
	elseif (l <= -2.9e+75)
		tmp = Float64(Float64(2.0 * n) * Float64(U * Float64(t - fma(2.0, Float64(l * Float64(l / Om)), Float64(n * Float64((Float64(l / Om) ^ 2.0) * Float64(U - U_42_))))))) ^ 0.5;
	elseif (l <= -6100.0)
		tmp = sqrt(Float64(Float64(U * Float64(2.0 * n)) * Float64(Float64(Float64(U_42_ - U) * Float64(Float64(l / Float64(Om / l)) * Float64(n / Om))) - Float64(Float64(2.0 * Float64(Float64(l * l) / Om)) - t))));
	elseif (l <= -3.3e-142)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * Float64(t + Float64(Float64(l * t_1) / Om))))));
	elseif (l <= 1.55e-201)
		tmp = sqrt(Float64(Float64(2.0 * Float64(t * Float64(n * U))) - Float64(fma(l, -2.0, Float64(n * Float64(Float64(U_42_ - U) * Float64(l / Om)))) * Float64(Float64(l / Om) * Float64(n * Float64(U * -2.0))))));
	elseif (l <= 6e+69)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * Float64(U * t))) + Float64(2.0 * Float64(Float64(n * Float64(l * Float64(U * t_1))) / Om))));
	else
		tmp = sqrt(Float64(Float64(l * sqrt(2.0)) * sqrt(Float64(Float64(n / Om) * Float64(U * fma(Float64(n / Om), Float64(U_42_ - U), -2.0)))))) ^ 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[(N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.45e+191], N[(N[Sqrt[N[(n * N[(U * N[(N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(-1.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2.9e+75], N[Power[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + N[(n * N[(N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l, -6100.0], N[Sqrt[N[(N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, -3.3e-142], N[Sqrt[N[(2.0 * N[(n * N[(U * N[(t + N[(N[(l * t$95$1), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 1.55e-201], N[Sqrt[N[(N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(l * -2.0 + N[(n * N[(N[(U$42$ - U), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / Om), $MachinePrecision] * N[(n * N[(U * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 6e+69], N[Sqrt[N[(N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[(n * N[(l * N[(U * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[Sqrt[N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(n / Om), $MachinePrecision] * N[(U * N[(N[(n / Om), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if l < -1.4500000000000001e191

   1. Initial program 64.0

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

   2. Simplified 59.3

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, n \cdot \frac{\ell \cdot \left(U* - U\right)}{Om}\right)\right)}} \]
      Proof
      (sqrt.f64 (*.f64 (*.f64 2 (*.f64 n U)) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l -2 (*.f64 n (/.f64 (*.f64 l (-.f64 U* U)) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 2 n) U)) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l -2 (*.f64 n (/.f64 (*.f64 l (-.f64 U* U)) Om))))))): 0 points increase in error, 2 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (Rewrite<= metadata-eval (neg.f64 2)) (*.f64 n (/.f64 (*.f64 l (-.f64 U* U)) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (Rewrite=> sub-neg_binary64 (+.f64 U* (neg.f64 U)))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (+.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 U*))) (neg.f64 U))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 (neg.f64 U*) U)))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 U (neg.f64 U*))))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 U U*)))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 l Om) (neg.f64 (-.f64 U U*)))))))))): 2 points increase in error, 17 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (Rewrite=> distribute-rgt-neg-out_binary64 (neg.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)))))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 n (*.f64 (/.f64 l Om) (-.f64 U U*)))))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 l (neg.f64 2)) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 (neg.f64 2) l)) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (Rewrite<= distribute-lft-out--_binary64 (-.f64 (*.f64 (/.f64 l Om) (*.f64 (neg.f64 2) l)) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 l (*.f64 (neg.f64 2) l)) Om)) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 15 points increase in error, 1 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (neg.f64 2) l) l)) Om) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (neg.f64 2) (*.f64 l l))) Om) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (Rewrite<= associate-*r/_binary64 (*.f64 (neg.f64 2) (/.f64 (*.f64 l l) Om))) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om)))) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om))) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 l Om) (*.f64 (/.f64 l Om) (-.f64 U U*))) n)))))): 8 points increase in error, 1 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 l Om) (/.f64 l Om)) (-.f64 U U*))) n))))): 7 points increase in error, 1 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)) n))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om))) (Rewrite<= *-commutative_binary64 (*.f64 n (*.f64 (pow.f64 (/.f64 l Om) 2) (-.f64 U U*)))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om))) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))))): 0 points increase in error, 7 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 t (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om)))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (Rewrite<= sub-neg_binary64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om)))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in l around inf 64.0

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

   4. Simplified 53.7

      \[\leadsto \sqrt{\color{blue}{n \cdot \left(\left(\mathsf{fma}\left(\frac{n}{Om}, \frac{U* - U}{Om}, \frac{-2}{Om}\right) \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right) \cdot 2\right)}} \]
      Proof
      (*.f64 n (*.f64 (*.f64 (fma.f64 (/.f64 n Om) (/.f64 (-.f64 U* U) Om) (/.f64 -2 Om)) (*.f64 l (*.f64 l U))) 2)): 0 points increase in error, 0 points decrease in error
      (*.f64 n (*.f64 (*.f64 (fma.f64 (/.f64 n Om) (/.f64 (-.f64 U* U) Om) (/.f64 (Rewrite<= metadata-eval (neg.f64 2)) Om)) (*.f64 l (*.f64 l U))) 2)): 0 points increase in error, 0 points decrease in error
      (*.f64 n (*.f64 (*.f64 (fma.f64 (/.f64 n Om) (/.f64 (-.f64 U* U) Om) (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 2 Om)))) (*.f64 l (*.f64 l U))) 2)): 0 points increase in error, 0 points decrease in error
      (*.f64 n (*.f64 (*.f64 (fma.f64 (/.f64 n Om) (/.f64 (-.f64 U* U) Om) (neg.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 2 1)) Om))) (*.f64 l (*.f64 l U))) 2)): 0 points increase in error, 0 points decrease in error
      (*.f64 n (*.f64 (*.f64 (fma.f64 (/.f64 n Om) (/.f64 (-.f64 U* U) Om) (neg.f64 (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 1 Om))))) (*.f64 l (*.f64 l U))) 2)): 0 points increase in error, 0 points decrease in error
      (*.f64 n (*.f64 (*.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 (/.f64 n Om) (/.f64 (-.f64 U* U) Om)) (*.f64 2 (/.f64 1 Om)))) (*.f64 l (*.f64 l U))) 2)): 0 points increase in error, 0 points decrease in error
      (*.f64 n (*.f64 (*.f64 (-.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 n (-.f64 U* U)) (*.f64 Om Om))) (*.f64 2 (/.f64 1 Om))) (*.f64 l (*.f64 l U))) 2)): 33 points increase in error, 5 points decrease in error
      (*.f64 n (*.f64 (*.f64 (-.f64 (/.f64 (*.f64 n (-.f64 U* U)) (Rewrite<= unpow2_binary64 (pow.f64 Om 2))) (*.f64 2 (/.f64 1 Om))) (*.f64 l (*.f64 l U))) 2)): 0 points increase in error, 0 points decrease in error
      (*.f64 n (*.f64 (*.f64 (-.f64 (/.f64 (*.f64 n (-.f64 U* U)) (pow.f64 Om 2)) (*.f64 2 (/.f64 1 Om))) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 l l) U))) 2)): 17 points increase in error, 8 points decrease in error
      (*.f64 n (*.f64 (*.f64 (-.f64 (/.f64 (*.f64 n (-.f64 U* U)) (pow.f64 Om 2)) (*.f64 2 (/.f64 1 Om))) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) U)) 2)): 0 points increase in error, 0 points decrease in error
      (*.f64 n (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (pow.f64 l 2) U) (-.f64 (/.f64 (*.f64 n (-.f64 U* U)) (pow.f64 Om 2)) (*.f64 2 (/.f64 1 Om))))) 2)): 0 points increase in error, 0 points decrease in error
      (*.f64 n (*.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (pow.f64 l 2) (*.f64 U (-.f64 (/.f64 (*.f64 n (-.f64 U* U)) (pow.f64 Om 2)) (*.f64 2 (/.f64 1 Om)))))) 2)): 13 points increase in error, 9 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 n (*.f64 (pow.f64 l 2) (*.f64 U (-.f64 (/.f64 (*.f64 n (-.f64 U* U)) (pow.f64 Om 2)) (*.f64 2 (/.f64 1 Om)))))) 2)): 0 points increase in error, 2 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 2 (*.f64 n (*.f64 (pow.f64 l 2) (*.f64 U (-.f64 (/.f64 (*.f64 n (-.f64 U* U)) (pow.f64 Om 2)) (*.f64 2 (/.f64 1 Om)))))))): 0 points increase in error, 0 points decrease in error

   5. Taylor expanded in l around -inf 33.3

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

   if -1.4500000000000001e191 < l < -2.8999999999999998e75

   1. Initial program 41.6

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

   2. Applied egg-rr 35.9

      \[\leadsto \color{blue}{{\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)\right)}^{0.5}} \]

   if -2.8999999999999998e75 < l < -6100

   1. Initial program 30.3

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

   2. Taylor expanded in n around 0 31.8

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

   3. Simplified 27.6

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \color{blue}{\left(\frac{\ell}{\frac{Om}{\ell}} \cdot \frac{n}{Om}\right)} \cdot \left(U - U*\right)\right)} \]
      Proof
      (*.f64 (/.f64 l (/.f64 Om l)) (/.f64 n Om)): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) Om)) (/.f64 n Om)): 40 points increase in error, 14 points decrease in error
      (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) Om) (/.f64 n Om)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (pow.f64 l 2) n) (*.f64 Om Om))): 47 points increase in error, 19 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 n (pow.f64 l 2))) (*.f64 Om Om)): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 n (pow.f64 l 2)) (Rewrite<= unpow2_binary64 (pow.f64 Om 2))): 0 points increase in error, 0 points decrease in error

   if -6100 < l < -3.2999999999999997e-142

   1. Initial program 27.5

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

   2. Simplified 26.0

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, n \cdot \frac{\ell \cdot \left(U* - U\right)}{Om}\right)\right)}} \]
      Proof
      (sqrt.f64 (*.f64 (*.f64 2 (*.f64 n U)) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l -2 (*.f64 n (/.f64 (*.f64 l (-.f64 U* U)) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 2 n) U)) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l -2 (*.f64 n (/.f64 (*.f64 l (-.f64 U* U)) Om))))))): 0 points increase in error, 2 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (Rewrite<= metadata-eval (neg.f64 2)) (*.f64 n (/.f64 (*.f64 l (-.f64 U* U)) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (Rewrite=> sub-neg_binary64 (+.f64 U* (neg.f64 U)))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (+.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 U*))) (neg.f64 U))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 (neg.f64 U*) U)))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 U (neg.f64 U*))))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 U U*)))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 l Om) (neg.f64 (-.f64 U U*)))))))))): 2 points increase in error, 17 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (Rewrite=> distribute-rgt-neg-out_binary64 (neg.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)))))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 n (*.f64 (/.f64 l Om) (-.f64 U U*)))))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 l (neg.f64 2)) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 (neg.f64 2) l)) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (Rewrite<= distribute-lft-out--_binary64 (-.f64 (*.f64 (/.f64 l Om) (*.f64 (neg.f64 2) l)) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 l (*.f64 (neg.f64 2) l)) Om)) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 15 points increase in error, 1 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (neg.f64 2) l) l)) Om) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (neg.f64 2) (*.f64 l l))) Om) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (Rewrite<= associate-*r/_binary64 (*.f64 (neg.f64 2) (/.f64 (*.f64 l l) Om))) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om)))) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om))) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 l Om) (*.f64 (/.f64 l Om) (-.f64 U U*))) n)))))): 8 points increase in error, 1 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 l Om) (/.f64 l Om)) (-.f64 U U*))) n))))): 7 points increase in error, 1 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)) n))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om))) (Rewrite<= *-commutative_binary64 (*.f64 n (*.f64 (pow.f64 (/.f64 l Om) 2) (-.f64 U U*)))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om))) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))))): 0 points increase in error, 7 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 t (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om)))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (Rewrite<= sub-neg_binary64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om)))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in U around 0 26.7

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

   if -3.2999999999999997e-142 < l < 1.5499999999999999e-201

   1. Initial program 24.1

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

   2. Simplified 24.2

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, n \cdot \frac{\ell \cdot \left(U* - U\right)}{Om}\right)\right)}} \]
      Proof
      (sqrt.f64 (*.f64 (*.f64 2 (*.f64 n U)) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l -2 (*.f64 n (/.f64 (*.f64 l (-.f64 U* U)) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 2 n) U)) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l -2 (*.f64 n (/.f64 (*.f64 l (-.f64 U* U)) Om))))))): 0 points increase in error, 2 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (Rewrite<= metadata-eval (neg.f64 2)) (*.f64 n (/.f64 (*.f64 l (-.f64 U* U)) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (Rewrite=> sub-neg_binary64 (+.f64 U* (neg.f64 U)))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (+.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 U*))) (neg.f64 U))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 (neg.f64 U*) U)))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 U (neg.f64 U*))))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 U U*)))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 l Om) (neg.f64 (-.f64 U U*)))))))))): 2 points increase in error, 17 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (Rewrite=> distribute-rgt-neg-out_binary64 (neg.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)))))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 n (*.f64 (/.f64 l Om) (-.f64 U U*)))))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 l (neg.f64 2)) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 (neg.f64 2) l)) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (Rewrite<= distribute-lft-out--_binary64 (-.f64 (*.f64 (/.f64 l Om) (*.f64 (neg.f64 2) l)) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 l (*.f64 (neg.f64 2) l)) Om)) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 15 points increase in error, 1 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (neg.f64 2) l) l)) Om) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (neg.f64 2) (*.f64 l l))) Om) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (Rewrite<= associate-*r/_binary64 (*.f64 (neg.f64 2) (/.f64 (*.f64 l l) Om))) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om)))) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om))) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 l Om) (*.f64 (/.f64 l Om) (-.f64 U U*))) n)))))): 8 points increase in error, 1 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 l Om) (/.f64 l Om)) (-.f64 U U*))) n))))): 7 points increase in error, 1 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)) n))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om))) (Rewrite<= *-commutative_binary64 (*.f64 n (*.f64 (pow.f64 (/.f64 l Om) 2) (-.f64 U U*)))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om))) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))))): 0 points increase in error, 7 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 t (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om)))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (Rewrite<= sub-neg_binary64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om)))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 23.7

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

   if 1.5499999999999999e-201 < l < 5.99999999999999967e69

   1. Initial program 28.5

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

   2. Simplified 27.2

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, n \cdot \frac{\ell \cdot \left(U* - U\right)}{Om}\right)\right)}} \]
      Proof
      (sqrt.f64 (*.f64 (*.f64 2 (*.f64 n U)) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l -2 (*.f64 n (/.f64 (*.f64 l (-.f64 U* U)) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 2 n) U)) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l -2 (*.f64 n (/.f64 (*.f64 l (-.f64 U* U)) Om))))))): 0 points increase in error, 2 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (Rewrite<= metadata-eval (neg.f64 2)) (*.f64 n (/.f64 (*.f64 l (-.f64 U* U)) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (Rewrite=> sub-neg_binary64 (+.f64 U* (neg.f64 U)))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (+.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 U*))) (neg.f64 U))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 (neg.f64 U*) U)))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 U (neg.f64 U*))))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 U U*)))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 l Om) (neg.f64 (-.f64 U U*)))))))))): 2 points increase in error, 17 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (Rewrite=> distribute-rgt-neg-out_binary64 (neg.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)))))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 n (*.f64 (/.f64 l Om) (-.f64 U U*)))))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 l (neg.f64 2)) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 (neg.f64 2) l)) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (Rewrite<= distribute-lft-out--_binary64 (-.f64 (*.f64 (/.f64 l Om) (*.f64 (neg.f64 2) l)) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 l (*.f64 (neg.f64 2) l)) Om)) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 15 points increase in error, 1 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (neg.f64 2) l) l)) Om) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (neg.f64 2) (*.f64 l l))) Om) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (Rewrite<= associate-*r/_binary64 (*.f64 (neg.f64 2) (/.f64 (*.f64 l l) Om))) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om)))) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om))) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 l Om) (*.f64 (/.f64 l Om) (-.f64 U U*))) n)))))): 8 points increase in error, 1 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 l Om) (/.f64 l Om)) (-.f64 U U*))) n))))): 7 points increase in error, 1 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)) n))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om))) (Rewrite<= *-commutative_binary64 (*.f64 n (*.f64 (pow.f64 (/.f64 l Om) 2) (-.f64 U U*)))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om))) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))))): 0 points increase in error, 7 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 t (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om)))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (Rewrite<= sub-neg_binary64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om)))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in U* around inf 27.2

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

   4. Simplified 27.0

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, n \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot U*\right)}\right)\right)} \]
      Proof
      (*.f64 (/.f64 l Om) U*): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r/_binary64 (/.f64 l (/.f64 Om U*))): 46 points increase in error, 42 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l U*) Om)): 48 points increase in error, 46 points decrease in error

   5. Taylor expanded in t around 0 27.9

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

   if 5.99999999999999967e69 < l

   1. Initial program 52.7

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

   2. Simplified 48.7

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, n \cdot \frac{\ell \cdot \left(U* - U\right)}{Om}\right)\right)}} \]
      Proof
      (sqrt.f64 (*.f64 (*.f64 2 (*.f64 n U)) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l -2 (*.f64 n (/.f64 (*.f64 l (-.f64 U* U)) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 2 n) U)) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l -2 (*.f64 n (/.f64 (*.f64 l (-.f64 U* U)) Om))))))): 0 points increase in error, 2 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (Rewrite<= metadata-eval (neg.f64 2)) (*.f64 n (/.f64 (*.f64 l (-.f64 U* U)) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (Rewrite=> sub-neg_binary64 (+.f64 U* (neg.f64 U)))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (+.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 U*))) (neg.f64 U))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 (neg.f64 U*) U)))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 U (neg.f64 U*))))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 U U*)))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 l Om) (neg.f64 (-.f64 U U*)))))))))): 2 points increase in error, 17 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (Rewrite=> distribute-rgt-neg-out_binary64 (neg.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)))))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 n (*.f64 (/.f64 l Om) (-.f64 U U*)))))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 l (neg.f64 2)) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 (neg.f64 2) l)) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (Rewrite<= distribute-lft-out--_binary64 (-.f64 (*.f64 (/.f64 l Om) (*.f64 (neg.f64 2) l)) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 l (*.f64 (neg.f64 2) l)) Om)) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 15 points increase in error, 1 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (neg.f64 2) l) l)) Om) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (neg.f64 2) (*.f64 l l))) Om) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (Rewrite<= associate-*r/_binary64 (*.f64 (neg.f64 2) (/.f64 (*.f64 l l) Om))) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om)))) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om))) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 l Om) (*.f64 (/.f64 l Om) (-.f64 U U*))) n)))))): 8 points increase in error, 1 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 l Om) (/.f64 l Om)) (-.f64 U U*))) n))))): 7 points increase in error, 1 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)) n))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om))) (Rewrite<= *-commutative_binary64 (*.f64 n (*.f64 (pow.f64 (/.f64 l Om) 2) (-.f64 U U*)))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om))) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))))): 0 points increase in error, 7 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 t (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om)))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (Rewrite<= sub-neg_binary64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om)))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 36.8

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

   4. Taylor expanded in l around inf 29.5

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

   5. Applied egg-rr 28.6

      \[\leadsto \color{blue}{{\left(\sqrt{\sqrt{\frac{n}{Om} \cdot \left(U \cdot \mathsf{fma}\left(\frac{n}{Om}, U* - U, -2\right)\right)} \cdot \left(\sqrt{2} \cdot \ell\right)}\right)}^{2}} \]
3. Recombined 7 regimes into one program.

4. Final simplification 27.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.45 \cdot 10^{+191}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} + 2 \cdot \frac{-1}{Om}\right)\right)} \cdot \left(\ell \cdot \left(-\sqrt{2}\right)\right)\\ \mathbf{elif}\;\ell \leq -2.9 \cdot 10^{+75}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq -6100:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(U* - U\right) \cdot \left(\frac{\ell}{\frac{Om}{\ell}} \cdot \frac{n}{Om}\right) - \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right)}\\ \mathbf{elif}\;\ell \leq -3.3 \cdot 10^{-142}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\right)}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.55 \cdot 10^{-201}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right) - \mathsf{fma}\left(\ell, -2, n \cdot \left(\left(U* - U\right) \cdot \frac{\ell}{Om}\right)\right) \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U \cdot -2\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{+69}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\right)\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n}{Om} \cdot \left(U \cdot \mathsf{fma}\left(\frac{n}{Om}, U* - U, -2\right)\right)}}\right)}^{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	26.4
Cost	60820

\[\begin{array}{l} t_1 := \frac{\ell \cdot \ell}{Om}\\ t_2 := U \cdot \left(2 \cdot n\right)\\ t_3 := t_2 \cdot \left(\left(t + t_1 \cdot -2\right) + \left(U* - U\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\\ \mathbf{if}\;t_3 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\right)}{Om}\right)\right)\right)}\\ \mathbf{elif}\;t_3 \leq 5 \cdot 10^{-127}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell \cdot n}{\frac{Om}{U* - U}}\right)\right)}\\ \mathbf{elif}\;t_3 \leq 4 \cdot 10^{+27}:\\ \;\;\;\;\sqrt{t_2 \cdot \left(\left(U* - U\right) \cdot \left(\frac{\ell}{\frac{Om}{\ell}} \cdot \frac{n}{Om}\right) - \left(2 \cdot t_1 - t\right)\right)}\\ \mathbf{elif}\;t_3 \leq 4 \cdot 10^{+301}:\\ \;\;\;\;\sqrt{t_3}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(-2 + \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{n}{Om} \cdot \left(U \cdot \mathsf{fma}\left(\frac{n}{Om}, U*, -2\right)\right)}\right)\\ \end{array} \]

Alternative 2
Error	26.7
Cost	52816

\[\begin{array}{l} t_1 := \frac{\ell \cdot \ell}{Om}\\ t_2 := U \cdot \left(2 \cdot n\right)\\ t_3 := t_2 \cdot \left(\left(t + t_1 \cdot -2\right) + \left(U* - U\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\\ \mathbf{if}\;t_3 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\right)}{Om}\right)\right)\right)}\\ \mathbf{elif}\;t_3 \leq 5 \cdot 10^{-127}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell \cdot n}{\frac{Om}{U* - U}}\right)\right)}\\ \mathbf{elif}\;t_3 \leq 4 \cdot 10^{+27}:\\ \;\;\;\;\sqrt{t_2 \cdot \left(\left(U* - U\right) \cdot \left(\frac{\ell}{\frac{Om}{\ell}} \cdot \frac{n}{Om}\right) - \left(2 \cdot t_1 - t\right)\right)}\\ \mathbf{elif}\;t_3 \leq 4 \cdot 10^{+301}:\\ \;\;\;\;\sqrt{t_3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{Om} \cdot \left(U \cdot \mathsf{fma}\left(\frac{n}{Om}, U* - U, -2\right)\right)}\right)\\ \end{array} \]

Alternative 3
Error	27.6
Cost	21080

\[\begin{array}{l} t_1 := \frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\\ \mathbf{if}\;\ell \leq -1.45 \cdot 10^{+191}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} + 2 \cdot \frac{-1}{Om}\right)\right)} \cdot \left(\ell \cdot \left(-\sqrt{2}\right)\right)\\ \mathbf{elif}\;\ell \leq -2.9 \cdot 10^{+75}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq -6100:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(U* - U\right) \cdot \left(\frac{\ell}{\frac{Om}{\ell}} \cdot \frac{n}{Om}\right) - \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right)}\\ \mathbf{elif}\;\ell \leq -3.3 \cdot 10^{-142}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{\ell \cdot t_1}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.55 \cdot 10^{-201}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right) - \mathsf{fma}\left(\ell, -2, n \cdot \left(\left(U* - U\right) \cdot \frac{\ell}{Om}\right)\right) \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U \cdot -2\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{+69}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \frac{n \cdot \left(\ell \cdot \left(U \cdot t_1\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{Om} \cdot \left(U \cdot \mathsf{fma}\left(\frac{n}{Om}, U* - U, -2\right)\right)}\right)\\ \end{array} \]

Alternative 4
Error	28.1
Cost	15052

\[\begin{array}{l} t_1 := \mathsf{fma}\left(\ell, -2, n \cdot \left(\left(U* - U\right) \cdot \frac{\ell}{Om}\right)\right)\\ t_2 := \frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\\ t_3 := 2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\\ \mathbf{if}\;\ell \leq -6100:\\ \;\;\;\;\sqrt{t_3 + t_1 \cdot \frac{\ell \cdot \left(U \cdot \left(2 \cdot n\right)\right)}{Om}}\\ \mathbf{elif}\;\ell \leq -3.3 \cdot 10^{-142}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{\ell \cdot t_2}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.55 \cdot 10^{-201}:\\ \;\;\;\;\sqrt{t_3 - t_1 \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U \cdot -2\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \frac{n \cdot \left(\ell \cdot \left(U \cdot t_2\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(-2 + \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)}{Om}}\\ \end{array} \]

Alternative 5
Error	28.3
Cost	14788

\[\begin{array}{l} t_1 := \frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\\ \mathbf{if}\;\ell \leq -6100:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right) + \mathsf{fma}\left(\ell, -2, n \cdot \left(\left(U* - U\right) \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell \cdot \left(U \cdot \left(2 \cdot n\right)\right)}{Om}}\\ \mathbf{elif}\;\ell \leq -3.3 \cdot 10^{-142}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{\ell \cdot t_1}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.55 \cdot 10^{-201}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, n \cdot \frac{\ell \cdot \left(U* - U\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \frac{n \cdot \left(\ell \cdot \left(U \cdot t_1\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(-2 + \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)}{Om}}\\ \end{array} \]

Alternative 6
Error	28.7
Cost	14668

\[\begin{array}{l} t_1 := \frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\\ \mathbf{if}\;\ell \leq -6100:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right) + \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(2 \cdot U\right)\right)\right) \cdot \left(\ell \cdot -2 - U \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq -3.3 \cdot 10^{-142}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{\ell \cdot t_1}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.55 \cdot 10^{-201}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, n \cdot \frac{\ell \cdot \left(U* - U\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \frac{n \cdot \left(\ell \cdot \left(U \cdot t_1\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(-2 + \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)}{Om}}\\ \end{array} \]

Alternative 7
Error	28.2
Cost	14668

\[\begin{array}{l} t_1 := \frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\\ \mathbf{if}\;\ell \leq -1:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right) + \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(2 \cdot U\right)\right)\right) \cdot \mathsf{fma}\left(\ell, -2, n \cdot \left(U* \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq -3.3 \cdot 10^{-142}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{\ell \cdot t_1}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.55 \cdot 10^{-201}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, n \cdot \frac{\ell \cdot \left(U* - U\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \frac{n \cdot \left(\ell \cdot \left(U \cdot t_1\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(-2 + \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)}{Om}}\\ \end{array} \]

Alternative 8
Error	28.6
Cost	14544

\[\begin{array}{l} t_1 := \frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\\ \mathbf{if}\;\ell \leq -6100:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right) + \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(2 \cdot U\right)\right)\right) \cdot \left(\ell \cdot -2 - U \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq -2.9 \cdot 10^{-149}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{\ell \cdot t_1}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.55 \cdot 10^{-201}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, n \cdot \left(U* \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \frac{n \cdot \left(\ell \cdot \left(U \cdot t_1\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(-2 + \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)}{Om}}\\ \end{array} \]

Alternative 9
Error	29.3
Cost	14412

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{+125}:\\ \;\;\;\;{\left(U \cdot \left(\ell \cdot \left(\left(2 \cdot n\right) \cdot \frac{\ell \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)}{Om}\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 10^{-200}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(U* - U\right) \cdot \left(\frac{\ell}{\frac{Om}{\ell}} \cdot \frac{n}{Om}\right) - \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\right)\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(-2 + \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)}{Om}}\\ \end{array} \]

Alternative 10
Error	29.4
Cost	14284

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{+125}:\\ \;\;\;\;{\left(U \cdot \left(\ell \cdot \left(\left(2 \cdot n\right) \cdot \frac{\ell \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)}{Om}\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 10^{-200}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(U* - U\right) \cdot \left(\frac{\ell}{\frac{Om}{\ell}} \cdot \frac{n}{Om}\right) - \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\right)\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(-2 + \frac{n \cdot U*}{Om}\right)\right)}{Om}}\\ \end{array} \]

Alternative 11
Error	30.2
Cost	14160

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := U \cdot \left(2 \cdot n\right)\\ t_3 := \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\right)\right)\right)}{Om}}\\ \mathbf{if}\;U \leq -6.8 \cdot 10^{-63}:\\ \;\;\;\;\sqrt{t_2 \cdot \left(\left(U* - U\right) \cdot \left(t_1 \cdot \frac{n}{Om}\right) - \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right)}\\ \mathbf{elif}\;U \leq -2.6 \cdot 10^{-133}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-4, \ell \cdot \left(\ell \cdot \left(U \cdot \frac{n}{Om}\right)\right), 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\\ \mathbf{elif}\;U \leq -9.5 \cdot 10^{-185}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;U \leq 2.1 \cdot 10^{-307}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \mathsf{fma}\left(-2, t_1, t\right)\right)}\\ \mathbf{elif}\;U \leq 6.2 \cdot 10^{+18}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t_2 \cdot \left(t + \frac{\ell \cdot -2 - U \cdot \left(n \cdot \frac{\ell}{Om}\right)}{\frac{Om}{\ell}}\right)}\\ \end{array} \]

Alternative 12
Error	30.6
Cost	13776

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right) + \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(2 \cdot U\right)\right)\right) \cdot \left(\ell \cdot -2 - U \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{if}\;Om \leq -5800000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;Om \leq 7 \cdot 10^{+123}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\right)\right)\right)}{Om}}\\ \mathbf{elif}\;Om \leq 3 \cdot 10^{+174}:\\ \;\;\;\;{\left(U \cdot \left(\ell \cdot \left(\left(2 \cdot n\right) \cdot \frac{\ell \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)}{Om}\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;Om \leq 6.5 \cdot 10^{+180}:\\ \;\;\;\;\sqrt{2 \cdot t} \cdot \sqrt{n \cdot U}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	36.0
Cost	8664

\[\begin{array}{l} t_1 := U \cdot \left(2 \cdot n\right)\\ t_2 := \sqrt{t_1 \cdot \left(t + \frac{\ell \cdot -2 - U \cdot \left(n \cdot \frac{\ell}{Om}\right)}{\frac{Om}{\ell}}\right)}\\ t_3 := \sqrt{2 \cdot \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\right)\right)\right)}{Om}}\\ \mathbf{if}\;Om \leq -1.15 \cdot 10^{-160}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;Om \leq 4.7 \cdot 10^{-200}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;Om \leq 7.6 \cdot 10^{-87}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;Om \leq 1.02 \cdot 10^{-54}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;Om \leq 1.6 \cdot 10^{+125}:\\ \;\;\;\;\sqrt{t_1 \cdot \left(t + \frac{-2 - \frac{n \cdot U}{Om}}{\frac{Om}{\ell \cdot \ell}}\right)}\\ \mathbf{elif}\;Om \leq 3 \cdot 10^{+174}:\\ \;\;\;\;{\left(U \cdot \left(\ell \cdot \left(\left(2 \cdot n\right) \cdot \frac{\ell \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)}{Om}\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 14
Error	36.3
Cost	8664

\[\begin{array}{l} t_1 := U \cdot \left(2 \cdot n\right)\\ t_2 := \sqrt{t_1 \cdot \left(t + \frac{\ell \cdot -2 - U \cdot \left(n \cdot \frac{\ell}{Om}\right)}{\frac{Om}{\ell}}\right)}\\ \mathbf{if}\;Om \leq -1.15 \cdot 10^{-160}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;Om \leq 4.7 \cdot 10^{-200}:\\ \;\;\;\;\sqrt{2 \cdot \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\right)\right)\right)}{Om}}\\ \mathbf{elif}\;Om \leq 7.6 \cdot 10^{-87}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;Om \leq 5.4 \cdot 10^{-27}:\\ \;\;\;\;\sqrt{\left(\frac{2}{Om} + \frac{U - U*}{\frac{Om \cdot Om}{n}}\right) \cdot \left(-2 \cdot \left(n \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)\right)\right)}\\ \mathbf{elif}\;Om \leq 1.6 \cdot 10^{+125}:\\ \;\;\;\;\sqrt{t_1 \cdot \left(t + \frac{-2 - \frac{n \cdot U}{Om}}{\frac{Om}{\ell \cdot \ell}}\right)}\\ \mathbf{elif}\;Om \leq 3 \cdot 10^{+174}:\\ \;\;\;\;{\left(U \cdot \left(\ell \cdot \left(\left(2 \cdot n\right) \cdot \frac{\ell \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)}{Om}\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 15
Error	36.7
Cost	8664

\[\begin{array}{l} t_1 := U \cdot \left(2 \cdot n\right)\\ t_2 := \sqrt{t_1 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 - \frac{n}{Om} \cdot \left(\ell \cdot U\right)\right)\right)}\\ \mathbf{if}\;Om \leq -1.15 \cdot 10^{-160}:\\ \;\;\;\;\sqrt{t_1 \cdot \left(t + \frac{\ell \cdot -2 - U \cdot \left(n \cdot \frac{\ell}{Om}\right)}{\frac{Om}{\ell}}\right)}\\ \mathbf{elif}\;Om \leq 4.7 \cdot 10^{-200}:\\ \;\;\;\;\sqrt{2 \cdot \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\right)\right)\right)}{Om}}\\ \mathbf{elif}\;Om \leq 7.6 \cdot 10^{-87}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;Om \leq 5.4 \cdot 10^{-27}:\\ \;\;\;\;\sqrt{\left(\frac{2}{Om} + \frac{U - U*}{\frac{Om \cdot Om}{n}}\right) \cdot \left(-2 \cdot \left(n \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)\right)\right)}\\ \mathbf{elif}\;Om \leq 1.6 \cdot 10^{+125}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;Om \leq 3 \cdot 10^{+174}:\\ \;\;\;\;{\left(U \cdot \left(\ell \cdot \left(\left(2 \cdot n\right) \cdot \frac{\ell \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)}{Om}\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 16
Error	35.3
Cost	8536

\[\begin{array}{l} t_1 := U* \cdot \frac{n}{Om}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+125}:\\ \;\;\;\;{\left(U \cdot \left(\ell \cdot \left(\left(2 \cdot n\right) \cdot \frac{\ell \cdot \left(-2 + t_1\right)}{Om}\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 10^{-200}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{\left(\ell \cdot \ell\right) \cdot -2}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{-31}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.9 \cdot 10^{+38}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \frac{\left(2 - t_1\right) \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)}{Om}}\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{+69}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \mathbf{elif}\;\ell \leq 10^{+125}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{n \cdot \left(\ell \cdot \ell\right)}{\frac{\frac{Om}{U}}{2 + \left(U - U*\right) \cdot \frac{n}{Om}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2 \cdot \left(\left(U \cdot \left(\ell \cdot \left(\ell \cdot n\right)\right)\right) \cdot \left(-2 - U \cdot \frac{n}{Om}\right)\right)}{Om}}\\ \end{array} \]

Alternative 17
Error	31.4
Cost	8520

\[\begin{array}{l} t_1 := U \cdot \left(2 \cdot n\right)\\ \mathbf{if}\;Om \leq -2.9 \cdot 10^{+19}:\\ \;\;\;\;\sqrt{t_1 \cdot \left(t + \frac{\ell \cdot -2 - U \cdot \left(n \cdot \frac{\ell}{Om}\right)}{\frac{Om}{\ell}}\right)}\\ \mathbf{elif}\;Om \leq 7 \cdot 10^{+123}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\right)\right)\right)}{Om}}\\ \mathbf{elif}\;Om \leq 3 \cdot 10^{+174}:\\ \;\;\;\;{\left(U \cdot \left(\ell \cdot \left(\left(2 \cdot n\right) \cdot \frac{\ell \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)}{Om}\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t_1 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 - \frac{n}{Om} \cdot \left(\ell \cdot U\right)\right)\right)}\\ \end{array} \]

Alternative 18
Error	29.3
Cost	8520

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right) + \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(2 \cdot U\right)\right)\right) \cdot \left(\ell \cdot -2 - U \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{if}\;Om \leq -5800000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;Om \leq 1.75 \cdot 10^{+137}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\right)\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 19
Error	29.2
Cost	8520

\[\begin{array}{l} t_1 := U \cdot \left(2 \cdot n\right)\\ \mathbf{if}\;U \leq -5 \cdot 10^{-63}:\\ \;\;\;\;\sqrt{t_1 \cdot \left(\left(U* - U\right) \cdot \left(\frac{\ell}{\frac{Om}{\ell}} \cdot \frac{n}{Om}\right) - \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right)}\\ \mathbf{elif}\;U \leq 6.2 \cdot 10^{+18}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\right)\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t_1 \cdot \left(t + \frac{\ell \cdot -2 - U \cdot \left(n \cdot \frac{\ell}{Om}\right)}{\frac{Om}{\ell}}\right)}\\ \end{array} \]

Alternative 20
Error	34.4
Cost	8472

\[\begin{array}{l} t_1 := U* \cdot \frac{n}{Om}\\ t_2 := \sqrt{\left(n \cdot -2\right) \cdot \frac{\left(2 - t_1\right) \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)}{Om}}\\ t_3 := {\left(U \cdot \left(\ell \cdot \left(\left(2 \cdot n\right) \cdot \frac{\ell \cdot \left(-2 + t_1\right)}{Om}\right)\right)\right)}^{0.5}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+125}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\ell \leq 10^{-200}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{\left(\ell \cdot \ell\right) \cdot -2}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{-31}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.9 \cdot 10^{+38}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{+69}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{+77}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 21
Error	32.0
Cost	8400

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\right)}{Om}\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+125}:\\ \;\;\;\;{\left(U \cdot \left(\ell \cdot \left(\left(2 \cdot n\right) \cdot \frac{\ell \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)}{Om}\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-249}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 10^{-200}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{+69}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{2}{Om} - \frac{n}{Om} \cdot \frac{U* - U}{Om}\right) \cdot \left(-2 \cdot \left(U \cdot \left(\ell \cdot \left(\ell \cdot n\right)\right)\right)\right)}\\ \end{array} \]

Alternative 22
Error	37.6
Cost	8276

\[\begin{array}{l} t_1 := \sqrt{\left(n \cdot -2\right) \cdot \frac{\left(2 - U* \cdot \frac{n}{Om}\right) \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)}{Om}}\\ \mathbf{if}\;\ell \leq 10^{-200}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{\left(\ell \cdot \ell\right) \cdot -2}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{-31}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.9 \cdot 10^{+38}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{+69}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \mathbf{elif}\;\ell \leq 10^{+80}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(\frac{-2}{Om} \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right)\right)}\\ \end{array} \]

Alternative 23
Error	37.4
Cost	8276

\[\begin{array}{l} t_1 := \sqrt{\left(n \cdot -2\right) \cdot \frac{\left(2 - U* \cdot \frac{n}{Om}\right) \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)}{Om}}\\ \mathbf{if}\;\ell \leq 10^{-200}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{\left(\ell \cdot \ell\right) \cdot -2}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{-31}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.9 \cdot 10^{+38}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{+69}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \mathbf{elif}\;\ell \leq 10^{+80}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2 \cdot \left(\left(U \cdot \left(\ell \cdot \left(\ell \cdot n\right)\right)\right) \cdot \left(-2 - U \cdot \frac{n}{Om}\right)\right)}{Om}}\\ \end{array} \]

Alternative 24
Error	37.3
Cost	8020

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{\left(\ell \cdot \ell\right) \cdot -2}{Om}\right)\right)}\\ \mathbf{if}\;U \leq -6.8 \cdot 10^{-63}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;U \leq -1.7 \cdot 10^{-250}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \mathbf{elif}\;U \leq -3.4 \cdot 10^{-273}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \frac{\left(\ell \cdot \ell\right) \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)}{Om}\right)}\\ \mathbf{elif}\;U \leq -1.95 \cdot 10^{-293}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(\frac{-2}{Om} \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right)\right)}\\ \mathbf{elif}\;U \leq 2.2 \cdot 10^{-171}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 25
Error	36.3
Cost	7756

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{\left(\ell \cdot \ell\right) \cdot -2}{Om}\right)\right)}\\ \mathbf{if}\;\ell \leq 10^{-200}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{-33}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 10^{+120}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(\frac{-2}{Om} \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right)\right)}\\ \end{array} \]

Alternative 26
Error	40.7
Cost	7632

\[\begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{+50}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \mathbf{elif}\;t \leq -3.5 \cdot 10^{-288}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{-259}:\\ \;\;\;\;\sqrt{-4 \cdot \left(\frac{n}{Om} \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right)}\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{-89}:\\ \;\;\;\;\sqrt{-4 \cdot \left(n \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{U}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]

Alternative 27
Error	38.0
Cost	7628

\[\begin{array}{l} t_1 := \sqrt{n \cdot \left(2 \cdot \left(\frac{-2}{Om} \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -1.55 \cdot 10^{+46}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 10^{-200}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{+69}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 28
Error	39.5
Cost	7368

\[\begin{array}{l} \mathbf{if}\;n \leq -7.6 \cdot 10^{-82}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;n \leq -2.3 \cdot 10^{-131}:\\ \;\;\;\;\sqrt{\frac{n \cdot -4}{\frac{Om}{U \cdot \left(\ell \cdot \ell\right)}}}\\ \mathbf{elif}\;n \leq 10^{-120}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]

Alternative 29
Error	39.3
Cost	7112

\[\begin{array}{l} t_1 := \sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{if}\;U \leq -6 \cdot 10^{-63}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;U \leq 1.45 \cdot 10^{-170}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 30
Error	39.2
Cost	6980

\[\begin{array}{l} \mathbf{if}\;U \leq -1 \cdot 10^{-25}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]

Alternative 31
Error	39.8
Cost	6848

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

Reproduce

herbie shell --seed 2022300 
(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*))))))