Toniolo and Linder, Equation (13)

Average Error: 34.2 → 28.4

Time: 41.5s

Precision: binary64

Cost: 34136

Math

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

↓

\[\begin{array}{l} t_1 := n \cdot \left(U \cdot \ell\right)\\ t_2 := \frac{n}{\frac{Om}{\ell}}\\ t_3 := \frac{\ell}{Om} \cdot \left(n \cdot \left(U \cdot 2\right)\right)\\ t_4 := 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\\ t_5 := \sqrt{\mathsf{fma}\left(\ell, -2, t_2 \cdot U* - U \cdot t_2\right) \cdot t_3 + t_4}\\ t_6 := 2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\\ t_7 := \sqrt{\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right), t\right)} \cdot \left(\sqrt{U \cdot 2} \cdot \sqrt{n}\right)\\ \mathbf{if}\;U \leq -4.5 \cdot 10^{-267}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;U \leq 5.2 \cdot 10^{-301}:\\ \;\;\;\;\sqrt{t_6 + -2 \cdot \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\frac{t_1}{Om} + \ell \cdot 2\right)\right)\right)}{Om}}\\ \mathbf{elif}\;U \leq 8.5 \cdot 10^{-279}:\\ \;\;\;\;t_7\\ \mathbf{elif}\;U \leq 2.3 \cdot 10^{-261}:\\ \;\;\;\;\sqrt{U \cdot \left(U* - U\right)} \cdot \left(\left(\ell \cdot n\right) \cdot \frac{-\sqrt{2}}{Om}\right)\\ \mathbf{elif}\;U \leq 1.5 \cdot 10^{-215}:\\ \;\;\;\;\sqrt{t_4 + t_3 \cdot \mathsf{fma}\left(\ell, -2, n \cdot \frac{U* - U}{\frac{Om}{\ell}}\right)}\\ \mathbf{elif}\;U \leq 4 \cdot 10^{-187}:\\ \;\;\;\;t_7\\ \mathbf{elif}\;U \leq 10^{-58}:\\ \;\;\;\;\sqrt{t_6 + -2 \cdot \frac{t_1 \cdot \left(\frac{n \cdot \left(\ell \cdot \left(U - U*\right)\right)}{Om} + \ell \cdot 2\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;t_5\\ \end{array} \]

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 (* U l)))
        (t_2 (/ n (/ Om l)))
        (t_3 (* (/ l Om) (* n (* U 2.0))))
        (t_4 (* 2.0 (* (* U n) t)))
        (t_5 (sqrt (+ (* (fma l -2.0 (- (* t_2 U*) (* U t_2))) t_3) t_4)))
        (t_6 (* 2.0 (* n (* U t))))
        (t_7
         (*
          (sqrt (fma (/ l Om) (fma l -2.0 (* n (* (/ l Om) (- U* U)))) t))
          (* (sqrt (* U 2.0)) (sqrt n)))))
   (if (<= U -4.5e-267)
     t_5
     (if (<= U 5.2e-301)
       (sqrt
        (+ t_6 (* -2.0 (/ (* n (* l (* U (+ (/ t_1 Om) (* l 2.0))))) Om))))
       (if (<= U 8.5e-279)
         t_7
         (if (<= U 2.3e-261)
           (* (sqrt (* U (- U* U))) (* (* l n) (/ (- (sqrt 2.0)) Om)))
           (if (<= U 1.5e-215)
             (sqrt (+ t_4 (* t_3 (fma l -2.0 (* n (/ (- U* U) (/ Om l)))))))
             (if (<= U 4e-187)
               t_7
               (if (<= U 1e-58)
                 (sqrt
                  (+
                   t_6
                   (*
                    -2.0
                    (/ (* t_1 (+ (/ (* n (* l (- U U*))) Om) (* l 2.0))) Om))))
                 t_5)))))))))

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 * (U * l);
	double t_2 = n / (Om / l);
	double t_3 = (l / Om) * (n * (U * 2.0));
	double t_4 = 2.0 * ((U * n) * t);
	double t_5 = sqrt(((fma(l, -2.0, ((t_2 * U_42_) - (U * t_2))) * t_3) + t_4));
	double t_6 = 2.0 * (n * (U * t));
	double t_7 = sqrt(fma((l / Om), fma(l, -2.0, (n * ((l / Om) * (U_42_ - U)))), t)) * (sqrt((U * 2.0)) * sqrt(n));
	double tmp;
	if (U <= -4.5e-267) {
		tmp = t_5;
	} else if (U <= 5.2e-301) {
		tmp = sqrt((t_6 + (-2.0 * ((n * (l * (U * ((t_1 / Om) + (l * 2.0))))) / Om))));
	} else if (U <= 8.5e-279) {
		tmp = t_7;
	} else if (U <= 2.3e-261) {
		tmp = sqrt((U * (U_42_ - U))) * ((l * n) * (-sqrt(2.0) / Om));
	} else if (U <= 1.5e-215) {
		tmp = sqrt((t_4 + (t_3 * fma(l, -2.0, (n * ((U_42_ - U) / (Om / l)))))));
	} else if (U <= 4e-187) {
		tmp = t_7;
	} else if (U <= 1e-58) {
		tmp = sqrt((t_6 + (-2.0 * ((t_1 * (((n * (l * (U - U_42_))) / Om) + (l * 2.0))) / Om))));
	} else {
		tmp = t_5;
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_)))))
end

↓

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(n * Float64(U * l))
	t_2 = Float64(n / Float64(Om / l))
	t_3 = Float64(Float64(l / Om) * Float64(n * Float64(U * 2.0)))
	t_4 = Float64(2.0 * Float64(Float64(U * n) * t))
	t_5 = sqrt(Float64(Float64(fma(l, -2.0, Float64(Float64(t_2 * U_42_) - Float64(U * t_2))) * t_3) + t_4))
	t_6 = Float64(2.0 * Float64(n * Float64(U * t)))
	t_7 = Float64(sqrt(fma(Float64(l / Om), fma(l, -2.0, Float64(n * Float64(Float64(l / Om) * Float64(U_42_ - U)))), t)) * Float64(sqrt(Float64(U * 2.0)) * sqrt(n)))
	tmp = 0.0
	if (U <= -4.5e-267)
		tmp = t_5;
	elseif (U <= 5.2e-301)
		tmp = sqrt(Float64(t_6 + Float64(-2.0 * Float64(Float64(n * Float64(l * Float64(U * Float64(Float64(t_1 / Om) + Float64(l * 2.0))))) / Om))));
	elseif (U <= 8.5e-279)
		tmp = t_7;
	elseif (U <= 2.3e-261)
		tmp = Float64(sqrt(Float64(U * Float64(U_42_ - U))) * Float64(Float64(l * n) * Float64(Float64(-sqrt(2.0)) / Om)));
	elseif (U <= 1.5e-215)
		tmp = sqrt(Float64(t_4 + Float64(t_3 * fma(l, -2.0, Float64(n * Float64(Float64(U_42_ - U) / Float64(Om / l)))))));
	elseif (U <= 4e-187)
		tmp = t_7;
	elseif (U <= 1e-58)
		tmp = sqrt(Float64(t_6 + Float64(-2.0 * Float64(Float64(t_1 * Float64(Float64(Float64(n * Float64(l * Float64(U - U_42_))) / Om) + Float64(l * 2.0))) / Om))));
	else
		tmp = t_5;
	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[(U * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(n / N[(Om / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(l / Om), $MachinePrecision] * N[(n * N[(U * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(2.0 * N[(N[(U * n), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(N[(N[(l * -2.0 + N[(N[(t$95$2 * U$42$), $MachinePrecision] - N[(U * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] + t$95$4), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(N[Sqrt[N[(N[(l / Om), $MachinePrecision] * N[(l * -2.0 + N[(n * N[(N[(l / Om), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(U * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[U, -4.5e-267], t$95$5, If[LessEqual[U, 5.2e-301], N[Sqrt[N[(t$95$6 + N[(-2.0 * N[(N[(n * N[(l * N[(U * N[(N[(t$95$1 / Om), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[U, 8.5e-279], t$95$7, If[LessEqual[U, 2.3e-261], N[(N[Sqrt[N[(U * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(l * n), $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[U, 1.5e-215], N[Sqrt[N[(t$95$4 + N[(t$95$3 * N[(l * -2.0 + N[(n * N[(N[(U$42$ - U), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[U, 4e-187], t$95$7, If[LessEqual[U, 1e-58], N[Sqrt[N[(t$95$6 + N[(-2.0 * N[(N[(t$95$1 * N[(N[(N[(n * N[(l * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$5]]]]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if U < -4.4999999999999999e-267 or 1e-58 < U

   1. Initial program 31.7

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

   2. Simplified 30.6

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, n \cdot \frac{\ell \cdot \left(U* - U\right)}{Om}\right)\right)}} \]
      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, 0 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*)))))))))): 1 points increase in error, 8 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)))))): 14 points increase in error, 0 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)))))): 4 points increase in error, 4 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))))): 5 points increase in error, 2 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*))))))): 3 points increase in error, 2 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 26.6

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

   4. Applied egg-rr 25.5

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

   if -4.4999999999999999e-267 < U < 5.1999999999999996e-301

   1. Initial program 43.4

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

   2. Simplified 40.1

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, n \cdot \frac{\ell \cdot \left(U* - U\right)}{Om}\right)\right)}} \]
      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, 0 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*)))))))))): 1 points increase in error, 8 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)))))): 14 points increase in error, 0 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)))))): 4 points increase in error, 4 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))))): 5 points increase in error, 2 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*))))))): 3 points increase in error, 2 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 37.6

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

   4. Applied egg-rr 38.9

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

   5. Taylor expanded in U* around 0 34.0

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

   if 5.1999999999999996e-301 < U < 8.5000000000000002e-279 or 1.50000000000000013e-215 < U < 4.0000000000000001e-187

   1. Initial program 42.4

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

   2. Simplified 39.2

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, n \cdot \frac{\ell \cdot \left(U* - U\right)}{Om}\right)\right)}} \]
      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, 0 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*)))))))))): 1 points increase in error, 8 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)))))): 14 points increase in error, 0 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)))))): 4 points increase in error, 4 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))))): 5 points increase in error, 2 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*))))))): 3 points increase in error, 2 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 49.1

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

   4. Applied egg-rr 41.1

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

   if 8.5000000000000002e-279 < U < 2.3e-261

   1. Initial program 43.2

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

   2. Simplified 40.3

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, n \cdot \frac{\ell \cdot \left(U* - U\right)}{Om}\right)\right)}} \]
      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, 0 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*)))))))))): 1 points increase in error, 8 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)))))): 14 points increase in error, 0 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)))))): 4 points increase in error, 4 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))))): 5 points increase in error, 2 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*))))))): 3 points increase in error, 2 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 36.1

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

   4. Taylor expanded in n around -inf 54.1

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

   5. Simplified 54.1

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(U* - U\right)} \cdot \frac{-\sqrt{2}}{\frac{Om}{n \cdot \ell}}} \]
      Proof
      (*.f64 (sqrt.f64 (*.f64 U (-.f64 U* U))) (/.f64 (neg.f64 (sqrt.f64 2)) (/.f64 Om (*.f64 n l)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 U* U) U))) (/.f64 (neg.f64 (sqrt.f64 2)) (/.f64 Om (*.f64 n l)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (*.f64 (-.f64 U* U) U)) (/.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (sqrt.f64 2))) (/.f64 Om (*.f64 n l)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (*.f64 (-.f64 U* U) U)) (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 (sqrt.f64 2) (/.f64 Om (*.f64 n l)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (*.f64 (-.f64 U* U) U)) (*.f64 -1 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (sqrt.f64 2) (*.f64 n l)) Om)))): 13 points increase in error, 10 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 -1 (/.f64 (*.f64 (sqrt.f64 2) (*.f64 n l)) Om)) (sqrt.f64 (*.f64 (-.f64 U* U) U)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r*_binary64 (*.f64 -1 (*.f64 (/.f64 (*.f64 (sqrt.f64 2) (*.f64 n l)) Om) (sqrt.f64 (*.f64 (-.f64 U* U) U))))): 0 points increase in error, 0 points decrease in error

   6. Taylor expanded in Om around 0 54.1

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

   7. Simplified 54.1

      \[\leadsto \color{blue}{\left(\left(n \cdot \ell\right) \cdot \frac{\sqrt{2}}{Om}\right) \cdot \left(-\sqrt{U \cdot \left(U* - U\right)}\right)} \]
      Proof
      (*.f64 (*.f64 (*.f64 n l) (/.f64 (sqrt.f64 2) Om)) (neg.f64 (sqrt.f64 (*.f64 U (-.f64 U* U))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (sqrt.f64 2) Om) (*.f64 n l))) (neg.f64 (sqrt.f64 (*.f64 U (-.f64 U* U))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (/.f64 (sqrt.f64 2) Om) (*.f64 n l)) (neg.f64 (sqrt.f64 (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 U* U) U))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (sqrt.f64 2) (*.f64 n l)) Om)) (neg.f64 (sqrt.f64 (*.f64 (-.f64 U* U) U)))): 12 points increase in error, 4 points decrease in error
      (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (/.f64 (*.f64 (sqrt.f64 2) (*.f64 n l)) Om) (sqrt.f64 (*.f64 (-.f64 U* U) U))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 (/.f64 (*.f64 (sqrt.f64 2) (*.f64 n l)) Om) (sqrt.f64 (*.f64 (-.f64 U* U) U))))): 0 points increase in error, 0 points decrease in error

   if 2.3e-261 < U < 1.50000000000000013e-215

   1. Initial program 41.5

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

   2. Simplified 39.6

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, n \cdot \frac{\ell \cdot \left(U* - U\right)}{Om}\right)\right)}} \]
      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, 0 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*)))))))))): 1 points increase in error, 8 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)))))): 14 points increase in error, 0 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)))))): 4 points increase in error, 4 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))))): 5 points increase in error, 2 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*))))))): 3 points increase in error, 2 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 34.2

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

   4. Applied egg-rr 34.2

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

   if 4.0000000000000001e-187 < U < 1e-58

   1. Initial program 35.9

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

   2. Simplified 33.0

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, n \cdot \frac{\ell \cdot \left(U* - U\right)}{Om}\right)\right)}} \]
      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, 0 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*)))))))))): 1 points increase in error, 8 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)))))): 14 points increase in error, 0 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)))))): 4 points increase in error, 4 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))))): 5 points increase in error, 2 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*))))))): 3 points increase in error, 2 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 t around inf 30.1

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(t \cdot U\right)\right) + 2 \cdot \frac{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(n \cdot \left(\ell \cdot U\right)\right)}{Om}}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 28.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -4.5 \cdot 10^{-267}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\ell, -2, \frac{n}{\frac{Om}{\ell}} \cdot U* - U \cdot \frac{n}{\frac{Om}{\ell}}\right) \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U \cdot 2\right)\right)\right) + 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)}\\ \mathbf{elif}\;U \leq 5.2 \cdot 10^{-301}:\\ \;\;\;\;\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(U \cdot \ell\right)}{Om} + \ell \cdot 2\right)\right)\right)}{Om}}\\ \mathbf{elif}\;U \leq 8.5 \cdot 10^{-279}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right), t\right)} \cdot \left(\sqrt{U \cdot 2} \cdot \sqrt{n}\right)\\ \mathbf{elif}\;U \leq 2.3 \cdot 10^{-261}:\\ \;\;\;\;\sqrt{U \cdot \left(U* - U\right)} \cdot \left(\left(\ell \cdot n\right) \cdot \frac{-\sqrt{2}}{Om}\right)\\ \mathbf{elif}\;U \leq 1.5 \cdot 10^{-215}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot t\right) + \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U \cdot 2\right)\right)\right) \cdot \mathsf{fma}\left(\ell, -2, n \cdot \frac{U* - U}{\frac{Om}{\ell}}\right)}\\ \mathbf{elif}\;U \leq 4 \cdot 10^{-187}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right), t\right)} \cdot \left(\sqrt{U \cdot 2} \cdot \sqrt{n}\right)\\ \mathbf{elif}\;U \leq 10^{-58}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + -2 \cdot \frac{\left(n \cdot \left(U \cdot \ell\right)\right) \cdot \left(\frac{n \cdot \left(\ell \cdot \left(U - U*\right)\right)}{Om} + \ell \cdot 2\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\ell, -2, \frac{n}{\frac{Om}{\ell}} \cdot U* - U \cdot \frac{n}{\frac{Om}{\ell}}\right) \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U \cdot 2\right)\right)\right) + 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	27.5
Cost	43528

\[\begin{array}{l} t_1 := \sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(U* - U\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot t}\\ \mathbf{elif}\;t_1 \leq 10^{+151}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\frac{n}{\frac{\frac{Om}{\ell}}{U}} \cdot \left(U \cdot \frac{n}{\frac{Om}{\ell}} + \ell \cdot 2\right)\right)}\\ \end{array} \]

Alternative 2
Error	25.5
Cost	41668

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

Alternative 3
Error	26.9
Cost	40964

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

Alternative 4
Error	26.2
Cost	30920

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

Alternative 5
Error	26.2
Cost	29572

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

Alternative 6
Error	30.6
Cost	14548

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.3 \cdot 10^{+139}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\frac{n}{\frac{\frac{Om}{\ell}}{U}} \cdot \left(U \cdot \frac{n}{\frac{Om}{\ell}} + \ell \cdot 2\right)\right)}\\ \mathbf{elif}\;\ell \leq -1.9 \cdot 10^{-91}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + -2 \cdot \frac{\left(n \cdot \left(U \cdot \ell\right)\right) \cdot \left(\frac{n \cdot \left(\ell \cdot \left(U - U*\right)\right)}{Om} + \ell \cdot 2\right)}{Om}}\\ \mathbf{elif}\;\ell \leq 4.8 \cdot 10^{-148}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \left(\frac{2}{\frac{Om}{\ell \cdot \ell}} - t\right)\right)}\\ \mathbf{elif}\;\ell \leq 6.7 \cdot 10^{-63}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\frac{\ell \cdot \left(\ell \cdot 2 - \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om} - t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 8.8 \cdot 10^{+189}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \mathsf{fma}\left(-2, \frac{\ell}{\frac{Om}{\ell}}, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{Om} \cdot \left(U \cdot \left(-2 + \frac{n \cdot U*}{Om}\right)\right)}\right)\\ \end{array} \]

Alternative 7
Error	30.0
Cost	14540

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.3 \cdot 10^{+139}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\frac{n}{\frac{\frac{Om}{\ell}}{U}} \cdot \left(U \cdot \frac{n}{\frac{Om}{\ell}} + \ell \cdot 2\right)\right)}\\ \mathbf{elif}\;\ell \leq -1.45 \cdot 10^{-101}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + -2 \cdot \frac{\left(n \cdot \left(U \cdot \ell\right)\right) \cdot \left(\frac{n \cdot \left(\ell \cdot \left(U - U*\right)\right)}{Om} + \ell \cdot 2\right)}{Om}}\\ \mathbf{elif}\;\ell \leq 7.8 \cdot 10^{+219}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{n}{Om} \cdot \left(\ell \cdot U*\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(-2 - \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)}{Om}}\\ \end{array} \]

Alternative 8
Error	28.4
Cost	14536

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.4 \cdot 10^{+150}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\frac{n}{\frac{\frac{Om}{\ell}}{U}} \cdot \left(U \cdot \frac{n}{\frac{Om}{\ell}} + \ell \cdot 2\right)\right)}\\ \mathbf{elif}\;\ell \leq 7.8 \cdot 10^{+219}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(-2 - \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)}{Om}}\\ \end{array} \]

Alternative 9
Error	28.6
Cost	14408

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.4 \cdot 10^{+150}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\frac{n}{\frac{\frac{Om}{\ell}}{U}} \cdot \left(U \cdot \frac{n}{\frac{Om}{\ell}} + \ell \cdot 2\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.02 \cdot 10^{+226}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, n \cdot \left(U* \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(-2 - \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)}{Om}}\\ \end{array} \]

Alternative 10
Error	31.4
Cost	14292

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.3 \cdot 10^{+139}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\frac{n}{\frac{\frac{Om}{\ell}}{U}} \cdot \left(U \cdot \frac{n}{\frac{Om}{\ell}} + \ell \cdot 2\right)\right)}\\ \mathbf{elif}\;\ell \leq -1.9 \cdot 10^{-91}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + -2 \cdot \frac{\left(n \cdot \left(U \cdot \ell\right)\right) \cdot \left(\frac{n \cdot \left(\ell \cdot \left(U - U*\right)\right)}{Om} + \ell \cdot 2\right)}{Om}}\\ \mathbf{elif}\;\ell \leq 4.8 \cdot 10^{-148}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \left(\frac{2}{\frac{Om}{\ell \cdot \ell}} - t\right)\right)}\\ \mathbf{elif}\;\ell \leq 6.7 \cdot 10^{-63}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\frac{\ell \cdot \left(\ell \cdot 2 - \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om} - t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 8.8 \cdot 10^{+189}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \mathsf{fma}\left(-2, \frac{\ell}{\frac{Om}{\ell}}, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{Om} \cdot \left(U \cdot -2\right)}\right)\\ \end{array} \]

Alternative 11
Error	31.4
Cost	14164

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.3 \cdot 10^{+139}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\frac{n}{\frac{\frac{Om}{\ell}}{U}} \cdot \left(U \cdot \frac{n}{\frac{Om}{\ell}} + \ell \cdot 2\right)\right)}\\ \mathbf{elif}\;\ell \leq -1.9 \cdot 10^{-91}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + -2 \cdot \frac{\left(n \cdot \left(U \cdot \ell\right)\right) \cdot \left(\frac{n \cdot \left(\ell \cdot \left(U - U*\right)\right)}{Om} + \ell \cdot 2\right)}{Om}}\\ \mathbf{elif}\;\ell \leq 4.8 \cdot 10^{-148}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \left(\frac{2}{\frac{Om}{\ell \cdot \ell}} - t\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.6 \cdot 10^{+132}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\frac{\ell \cdot \left(\ell \cdot 2 - \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om} - t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 8.8 \cdot 10^{+189}:\\ \;\;\;\;\sqrt{n \cdot \left(\frac{\ell}{\frac{Om}{\ell}} \cdot \left(U \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{Om} \cdot \left(U \cdot -2\right)}\right)\\ \end{array} \]

Alternative 12
Error	32.1
Cost	9044

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + -2 \cdot \frac{\left(n \cdot \left(U \cdot \ell\right)\right) \cdot \left(\frac{n \cdot \left(\ell \cdot \left(U - U*\right)\right)}{Om} + \ell \cdot 2\right)}{Om}}\\ \mathbf{if}\;\ell \leq -2.3 \cdot 10^{+139}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\frac{n}{\frac{\frac{Om}{\ell}}{U}} \cdot \left(U \cdot \frac{n}{\frac{Om}{\ell}} + \ell \cdot 2\right)\right)}\\ \mathbf{elif}\;\ell \leq -1.9 \cdot 10^{-91}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 4.8 \cdot 10^{-148}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \left(\frac{2}{\frac{Om}{\ell \cdot \ell}} - t\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.6 \cdot 10^{+132}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\frac{\ell \cdot \left(\ell \cdot 2 - \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om} - t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 10^{+185}:\\ \;\;\;\;\sqrt{n \cdot \left(\frac{\ell}{\frac{Om}{\ell}} \cdot \left(U \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	30.3
Cost	8136

\[\begin{array}{l} t_1 := \sqrt{-2 \cdot \left(\frac{n}{\frac{\frac{Om}{\ell}}{U}} \cdot \left(U \cdot \frac{n}{\frac{Om}{\ell}} + \ell \cdot 2\right)\right)}\\ \mathbf{if}\;\ell \leq -1.4 \cdot 10^{+150}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 4.6 \cdot 10^{+132}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\frac{\ell \cdot \left(\ell \cdot 2 - \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om} - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Error	33.3
Cost	8008

\[\begin{array}{l} t_1 := \sqrt{-2 \cdot \left(\frac{U}{\frac{Om}{\ell \cdot n}} \cdot \left(\frac{n \cdot \left(U \cdot \ell\right)}{Om} + \ell \cdot 2\right)\right)}\\ \mathbf{if}\;\ell \leq -2.9 \cdot 10^{+153}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 2.9 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \left(\frac{2}{\frac{Om}{\ell \cdot \ell}} - t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 15
Error	32.6
Cost	8008

\[\begin{array}{l} t_1 := \sqrt{-2 \cdot \left(\left(U \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U \cdot \frac{n}{\frac{Om}{\ell}} + \ell \cdot 2\right)\right)}\\ \mathbf{if}\;\ell \leq -1.4 \cdot 10^{+150}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 6.4 \cdot 10^{+146}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \left(\frac{2}{\frac{Om}{\ell \cdot \ell}} - t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 16
Error	32.7
Cost	8008

\[\begin{array}{l} t_1 := \sqrt{-2 \cdot \left(\frac{n}{\frac{\frac{Om}{\ell}}{U}} \cdot \left(U \cdot \frac{n}{\frac{Om}{\ell}} + \ell \cdot 2\right)\right)}\\ \mathbf{if}\;\ell \leq -1.4 \cdot 10^{+150}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 6.4 \cdot 10^{+146}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \left(\frac{2}{\frac{Om}{\ell \cdot \ell}} - t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 17
Error	35.0
Cost	7624

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.4 \cdot 10^{+150}:\\ \;\;\;\;\sqrt{-2 \cdot \left(2 \cdot \left(\ell \cdot \left(\ell \cdot \left(U \cdot \frac{n}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.6 \cdot 10^{+132}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \left(\frac{2}{\frac{Om}{\ell \cdot \ell}} - t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(\frac{\ell}{\frac{Om}{\ell}} \cdot \left(U \cdot -4\right)\right)}\\ \end{array} \]

Alternative 18
Error	37.5
Cost	7368

\[\begin{array}{l} t_1 := \sqrt{n \cdot \left(\frac{\ell}{\frac{Om}{\ell}} \cdot \left(U \cdot -4\right)\right)}\\ \mathbf{if}\;\ell \leq -4.2 \cdot 10^{+66}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.4 \cdot 10^{+54}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(U \cdot n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 19
Error	37.5
Cost	7368

\[\begin{array}{l} \mathbf{if}\;\ell \leq -4.8 \cdot 10^{+76}:\\ \;\;\;\;\sqrt{-2 \cdot \left(2 \cdot \left(\ell \cdot \left(\ell \cdot \left(U \cdot \frac{n}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.4 \cdot 10^{+54}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(U \cdot n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(\frac{\ell}{\frac{Om}{\ell}} \cdot \left(U \cdot -4\right)\right)}\\ \end{array} \]

Alternative 20
Error	38.7
Cost	7112

\[\begin{array}{l} t_1 := \sqrt{t \cdot \left(2 \cdot \left(U \cdot n\right)\right)}\\ \mathbf{if}\;U \leq -1.86 \cdot 10^{-271}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;U \leq 10^{-58}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 21
Error	39.6
Cost	6980

\[\begin{array}{l} \mathbf{if}\;Om \leq 10^{+115}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(U \cdot n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]

Alternative 22
Error	39.5
Cost	6848

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

Reproduce

herbie shell --seed 2022317 
(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*))))))