Toniolo and Linder, Equation (13)

Average Error: 34.7 → 28.1

Time: 48.1s

Precision: binary64

Cost: 27340

Math

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

↓

\[\begin{array}{l} t_1 := \mathsf{fma}\left(\ell, -2, n \cdot \left(\left(U* - U\right) \cdot \frac{\ell}{Om}\right)\right)\\ t_2 := n \cdot \left(2 \cdot U\right)\\ t_3 := 2 \cdot \left(\left(n \cdot U\right) \cdot t\right)\\ \mathbf{if}\;\ell \leq -6 \cdot 10^{+134}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \left(-\sqrt{n \cdot \left(U \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U* - U}{Om}, \frac{-2}{Om}\right)\right)}\right)\right)\\ \mathbf{elif}\;\ell \leq -2.25 \cdot 10^{-154}:\\ \;\;\;\;\sqrt{t_1 \cdot \frac{U \cdot \left(2 \cdot n\right)}{\frac{Om}{\ell}} + t_3}\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-222}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\ell}{Om}, t_1, t\right)} \cdot \sqrt{t_2}\\ \mathbf{elif}\;\ell \leq 5.2 \cdot 10^{-71}:\\ \;\;\;\;\sqrt{t_3 + \mathsf{fma}\left(\ell, -2, n \cdot \frac{U* - U}{\frac{Om}{\ell}}\right) \cdot \left(\frac{\ell}{Om} \cdot t_2\right)}\\ \mathbf{elif}\;\ell \leq 1.05 \cdot 10^{+162}:\\ \;\;\;\;\sqrt{2 \cdot \mathsf{fma}\left(n, U \cdot t, \frac{n}{\frac{\frac{Om}{\ell \cdot U}}{\mathsf{fma}\left(\ell, -2, \frac{n}{Om} \cdot \left(\ell \cdot \left(U* - U\right)\right)\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{n}{Om} \cdot \left(U \cdot \mathsf{fma}\left(\frac{n}{Om}, U* - U, -2\right)\right)}\right)\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (sqrt
  (*
   (* (* 2.0 n) U)
   (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))

↓

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (fma l -2.0 (* n (* (- U* U) (/ l Om)))))
        (t_2 (* n (* 2.0 U)))
        (t_3 (* 2.0 (* (* n U) t))))
   (if (<= l -6e+134)
     (*
      (sqrt 2.0)
      (* l (- (sqrt (* n (* U (fma (/ n Om) (/ (- U* U) Om) (/ -2.0 Om))))))))
     (if (<= l -2.25e-154)
       (sqrt (+ (* t_1 (/ (* U (* 2.0 n)) (/ Om l))) t_3))
       (if (<= l -1e-222)
         (* (sqrt (fma (/ l Om) t_1 t)) (sqrt t_2))
         (if (<= l 5.2e-71)
           (sqrt
            (+
             t_3
             (* (fma l -2.0 (* n (/ (- U* U) (/ Om l)))) (* (/ l Om) t_2))))
           (if (<= l 1.05e+162)
             (sqrt
              (*
               2.0
               (fma
                n
                (* U t)
                (/
                 n
                 (/
                  (/ Om (* l U))
                  (fma l -2.0 (* (/ n Om) (* l (- U* U)))))))))
             (*
              l
              (*
               (sqrt 2.0)
               (sqrt (* (/ n Om) (* U (fma (/ n Om) (- U* U) -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 = fma(l, -2.0, (n * ((U_42_ - U) * (l / Om))));
	double t_2 = n * (2.0 * U);
	double t_3 = 2.0 * ((n * U) * t);
	double tmp;
	if (l <= -6e+134) {
		tmp = sqrt(2.0) * (l * -sqrt((n * (U * fma((n / Om), ((U_42_ - U) / Om), (-2.0 / Om))))));
	} else if (l <= -2.25e-154) {
		tmp = sqrt(((t_1 * ((U * (2.0 * n)) / (Om / l))) + t_3));
	} else if (l <= -1e-222) {
		tmp = sqrt(fma((l / Om), t_1, t)) * sqrt(t_2);
	} else if (l <= 5.2e-71) {
		tmp = sqrt((t_3 + (fma(l, -2.0, (n * ((U_42_ - U) / (Om / l)))) * ((l / Om) * t_2))));
	} else if (l <= 1.05e+162) {
		tmp = sqrt((2.0 * fma(n, (U * t), (n / ((Om / (l * U)) / fma(l, -2.0, ((n / Om) * (l * (U_42_ - U)))))))));
	} else {
		tmp = l * (sqrt(2.0) * sqrt(((n / Om) * (U * fma((n / Om), (U_42_ - U), -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 = fma(l, -2.0, Float64(n * Float64(Float64(U_42_ - U) * Float64(l / Om))))
	t_2 = Float64(n * Float64(2.0 * U))
	t_3 = Float64(2.0 * Float64(Float64(n * U) * t))
	tmp = 0.0
	if (l <= -6e+134)
		tmp = Float64(sqrt(2.0) * Float64(l * Float64(-sqrt(Float64(n * Float64(U * fma(Float64(n / Om), Float64(Float64(U_42_ - U) / Om), Float64(-2.0 / Om))))))));
	elseif (l <= -2.25e-154)
		tmp = sqrt(Float64(Float64(t_1 * Float64(Float64(U * Float64(2.0 * n)) / Float64(Om / l))) + t_3));
	elseif (l <= -1e-222)
		tmp = Float64(sqrt(fma(Float64(l / Om), t_1, t)) * sqrt(t_2));
	elseif (l <= 5.2e-71)
		tmp = sqrt(Float64(t_3 + Float64(fma(l, -2.0, Float64(n * Float64(Float64(U_42_ - U) / Float64(Om / l)))) * Float64(Float64(l / Om) * t_2))));
	elseif (l <= 1.05e+162)
		tmp = sqrt(Float64(2.0 * fma(n, Float64(U * t), Float64(n / Float64(Float64(Om / Float64(l * U)) / fma(l, -2.0, Float64(Float64(n / Om) * Float64(l * Float64(U_42_ - U)))))))));
	else
		tmp = Float64(l * Float64(sqrt(2.0) * sqrt(Float64(Float64(n / Om) * Float64(U * fma(Float64(n / Om), Float64(U_42_ - U), -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[(l * -2.0 + N[(n * N[(N[(U$42$ - U), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(n * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[(N[(n * U), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -6e+134], N[(N[Sqrt[2.0], $MachinePrecision] * N[(l * (-N[Sqrt[N[(n * N[(U * N[(N[(n / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2.25e-154], N[Sqrt[N[(N[(t$95$1 * N[(N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, -1e-222], N[(N[Sqrt[N[(N[(l / Om), $MachinePrecision] * t$95$1 + t), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.2e-71], N[Sqrt[N[(t$95$3 + N[(N[(l * -2.0 + N[(n * N[(N[(U$42$ - U), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / Om), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 1.05e+162], N[Sqrt[N[(2.0 * N[(n * N[(U * t), $MachinePrecision] + N[(n / N[(N[(Om / N[(l * U), $MachinePrecision]), $MachinePrecision] / N[(l * -2.0 + N[(N[(n / Om), $MachinePrecision] * N[(l * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l * N[(N[Sqrt[2.0], $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]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if l < -5.99999999999999993e134

   1. Initial program 59.8

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

   2. Simplified 52.8

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, n \cdot \frac{\ell \cdot \left(U* - U\right)}{Om}\right)\right)}} \]
      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*)))))))))): 0 points increase in error, 5 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, 3 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))))): 3 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))) (*.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*))))))): 1 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 l around inf 60.9

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

   4. Simplified 49.4

      \[\leadsto \sqrt{\color{blue}{n \cdot \left(\left(\mathsf{fma}\left(\frac{n}{Om}, \frac{U* - U}{Om}, \frac{-2}{Om}\right) \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right) \cdot 2\right)}} \]
      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)): 26 points increase in error, 8 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)): 18 points increase in error, 12 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, 17 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 34.2

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

   6. Simplified 29.8

      \[\leadsto \color{blue}{\sqrt{2} \cdot \left(\sqrt{n \cdot \left(U \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U* - U}{Om}, \frac{-2}{Om}\right)\right)} \cdot \left(-\ell\right)\right)} \]
      Proof
      (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (*.f64 n (*.f64 U (fma.f64 (/.f64 n Om) (/.f64 (-.f64 U* U) Om) (/.f64 -2 Om))))) (neg.f64 l))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (*.f64 n (*.f64 U (fma.f64 (/.f64 n Om) (/.f64 (-.f64 U* U) Om) (/.f64 (Rewrite<= metadata-eval (neg.f64 2)) Om))))) (neg.f64 l))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (*.f64 n (*.f64 U (fma.f64 (/.f64 n Om) (/.f64 (-.f64 U* U) Om) (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 2 Om))))))) (neg.f64 l))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (*.f64 n (*.f64 U (fma.f64 (/.f64 n Om) (/.f64 (-.f64 U* U) Om) (neg.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 2 1)) Om)))))) (neg.f64 l))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (*.f64 n (*.f64 U (fma.f64 (/.f64 n Om) (/.f64 (-.f64 U* U) Om) (neg.f64 (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 1 Om)))))))) (neg.f64 l))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (*.f64 n (*.f64 U (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 (/.f64 n Om) (/.f64 (-.f64 U* U) Om)) (*.f64 2 (/.f64 1 Om))))))) (neg.f64 l))): 0 points increase in error, 1 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (*.f64 n (*.f64 U (-.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 n (-.f64 U* U)) (*.f64 Om Om))) (*.f64 2 (/.f64 1 Om)))))) (neg.f64 l))): 23 points increase in error, 7 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (*.f64 n (*.f64 U (-.f64 (/.f64 (*.f64 n (-.f64 U* U)) (Rewrite<= unpow2_binary64 (pow.f64 Om 2))) (*.f64 2 (/.f64 1 Om)))))) (neg.f64 l))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (*.f64 n (Rewrite=> *-commutative_binary64 (*.f64 (-.f64 (/.f64 (*.f64 n (-.f64 U* U)) (pow.f64 Om 2)) (*.f64 2 (/.f64 1 Om))) U)))) (neg.f64 l))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (sqrt.f64 (*.f64 n (*.f64 (-.f64 (/.f64 (*.f64 n (-.f64 U* U)) (pow.f64 Om 2)) (*.f64 2 (/.f64 1 Om))) U))) l)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 l (sqrt.f64 (*.f64 n (*.f64 (-.f64 (/.f64 (*.f64 n (-.f64 U* U)) (pow.f64 Om 2)) (*.f64 2 (/.f64 1 Om))) U))))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (sqrt.f64 2) (*.f64 l (sqrt.f64 (*.f64 n (*.f64 (-.f64 (/.f64 (*.f64 n (-.f64 U* U)) (pow.f64 Om 2)) (*.f64 2 (/.f64 1 Om))) U))))))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (sqrt.f64 2) l) (sqrt.f64 (*.f64 n (*.f64 (-.f64 (/.f64 (*.f64 n (-.f64 U* U)) (pow.f64 Om 2)) (*.f64 2 (/.f64 1 Om))) U)))))): 10 points increase in error, 15 points decrease in error
      (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 (*.f64 (sqrt.f64 2) l) (sqrt.f64 (*.f64 n (*.f64 (-.f64 (/.f64 (*.f64 n (-.f64 U* U)) (pow.f64 Om 2)) (*.f64 2 (/.f64 1 Om))) U)))))): 0 points increase in error, 0 points decrease in error

   if -5.99999999999999993e134 < l < -2.2499999999999999e-154

   1. Initial program 29.1

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

   2. Simplified 29.3

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, n \cdot \frac{\ell \cdot \left(U* - U\right)}{Om}\right)\right)}} \]
      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*)))))))))): 0 points increase in error, 5 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, 3 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))))): 3 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))) (*.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*))))))): 1 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 26.8

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

   4. Applied egg-rr 26.8

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

   if -2.2499999999999999e-154 < l < -1.00000000000000005e-222

   1. Initial program 26.5

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

   2. Simplified 26.2

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, n \cdot \frac{\ell \cdot \left(U* - U\right)}{Om}\right)\right)}} \]
      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*)))))))))): 0 points increase in error, 5 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, 3 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))))): 3 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))) (*.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*))))))): 1 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 41.9

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

   if -1.00000000000000005e-222 < l < 5.1999999999999997e-71

   1. Initial program 26.0

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

   2. Simplified 25.2

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, n \cdot \frac{\ell \cdot \left(U* - U\right)}{Om}\right)\right)}} \]
      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*)))))))))): 0 points increase in error, 5 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, 3 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))))): 3 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))) (*.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*))))))): 1 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 25.0

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

   4. Applied egg-rr 25.0

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

   if 5.1999999999999997e-71 < l < 1.05e162

   1. Initial program 31.6

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

   2. Simplified 32.6

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, n \cdot \frac{\ell \cdot \left(U* - U\right)}{Om}\right)\right)}} \]
      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*)))))))))): 0 points increase in error, 5 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, 3 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))))): 3 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))) (*.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*))))))): 1 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 0 32.2

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

   4. Simplified 29.8

      \[\leadsto \sqrt{\color{blue}{2 \cdot \mathsf{fma}\left(n, U \cdot t, \frac{n}{\frac{\frac{Om}{\ell \cdot U}}{\mathsf{fma}\left(\ell, -2, \frac{n}{Om} \cdot \left(\ell \cdot \left(U* - U\right)\right)\right)}}\right)}} \]
      Proof
      (*.f64 2 (fma.f64 n (*.f64 U t) (/.f64 n (/.f64 (/.f64 Om (*.f64 l U)) (fma.f64 l -2 (*.f64 (/.f64 n Om) (*.f64 l (-.f64 U* U)))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (fma.f64 n (Rewrite<= *-commutative_binary64 (*.f64 t U)) (/.f64 n (/.f64 (/.f64 Om (*.f64 l U)) (fma.f64 l -2 (*.f64 (/.f64 n Om) (*.f64 l (-.f64 U* U)))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (fma.f64 n (*.f64 t U) (/.f64 n (/.f64 (/.f64 Om (*.f64 l U)) (fma.f64 l -2 (Rewrite<= associate-/r/_binary64 (/.f64 n (/.f64 Om (*.f64 l (-.f64 U* U)))))))))): 5 points increase in error, 2 points decrease in error
      (*.f64 2 (fma.f64 n (*.f64 t U) (/.f64 n (/.f64 (/.f64 Om (*.f64 l U)) (fma.f64 l -2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 n (*.f64 l (-.f64 U* U))) Om))))))): 3 points increase in error, 2 points decrease in error
      (*.f64 2 (fma.f64 n (*.f64 t U) (/.f64 n (/.f64 (/.f64 Om (*.f64 l U)) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 l -2) (/.f64 (*.f64 n (*.f64 l (-.f64 U* U))) Om))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (fma.f64 n (*.f64 t U) (/.f64 n (/.f64 (/.f64 Om (*.f64 l U)) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 -2 l)) (/.f64 (*.f64 n (*.f64 l (-.f64 U* U))) Om)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (fma.f64 n (*.f64 t U) (/.f64 n (/.f64 (/.f64 Om (*.f64 l U)) (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 (*.f64 n (*.f64 l (-.f64 U* U))) Om) (*.f64 -2 l))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (fma.f64 n (*.f64 t U) (/.f64 n (Rewrite<= associate-/r*_binary64 (/.f64 Om (*.f64 (*.f64 l U) (+.f64 (/.f64 (*.f64 n (*.f64 l (-.f64 U* U))) Om) (*.f64 -2 l)))))))): 15 points increase in error, 1 points decrease in error
      (*.f64 2 (fma.f64 n (*.f64 t U) (/.f64 n (/.f64 Om (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 (/.f64 (*.f64 n (*.f64 l (-.f64 U* U))) Om) (*.f64 -2 l)) (*.f64 l U))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (fma.f64 n (*.f64 t U) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 n (*.f64 (+.f64 (/.f64 (*.f64 n (*.f64 l (-.f64 U* U))) Om) (*.f64 -2 l)) (*.f64 l U))) Om)))): 7 points increase in error, 14 points decrease in error
      (*.f64 2 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 n (*.f64 t U)) (/.f64 (*.f64 n (*.f64 (+.f64 (/.f64 (*.f64 n (*.f64 l (-.f64 U* U))) Om) (*.f64 -2 l)) (*.f64 l U))) Om)))): 0 points increase in error, 2 points decrease in error
      (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 2 (*.f64 n (*.f64 t U))) (*.f64 2 (/.f64 (*.f64 n (*.f64 (+.f64 (/.f64 (*.f64 n (*.f64 l (-.f64 U* U))) Om) (*.f64 -2 l)) (*.f64 l U))) Om)))): 0 points increase in error, 0 points decrease in error

   if 1.05e162 < l

   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 57.3

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, n \cdot \frac{\ell \cdot \left(U* - U\right)}{Om}\right)\right)}} \]
      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*)))))))))): 0 points increase in error, 5 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, 3 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))))): 3 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))) (*.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*))))))): 1 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 43.4

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

   4. Applied egg-rr 43.4

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

   5. Applied egg-rr 43.4

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

   6. Taylor expanded in l around inf 27.6

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

   7. Simplified 27.0

      \[\leadsto \color{blue}{\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{n}{Om} \cdot \left(U \cdot \mathsf{fma}\left(\frac{n}{Om}, U* - U, -2\right)\right)}\right)} \]
      Proof
      (*.f64 l (*.f64 (sqrt.f64 2) (sqrt.f64 (*.f64 (/.f64 n Om) (*.f64 U (fma.f64 (/.f64 n Om) (-.f64 U* U) -2)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 l (*.f64 (sqrt.f64 2) (sqrt.f64 (*.f64 (/.f64 n Om) (*.f64 U (fma.f64 (/.f64 n Om) (-.f64 U* U) (Rewrite<= metadata-eval (neg.f64 2)))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 l (*.f64 (sqrt.f64 2) (sqrt.f64 (*.f64 (/.f64 n Om) (*.f64 U (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 (/.f64 n Om) (-.f64 U* U)) 2))))))): 0 points increase in error, 1 points decrease in error
      (*.f64 l (*.f64 (sqrt.f64 2) (sqrt.f64 (*.f64 (/.f64 n Om) (*.f64 U (-.f64 (Rewrite<= associate-/r/_binary64 (/.f64 n (/.f64 Om (-.f64 U* U)))) 2)))))): 10 points increase in error, 6 points decrease in error
      (*.f64 l (*.f64 (sqrt.f64 2) (sqrt.f64 (*.f64 (/.f64 n Om) (*.f64 U (-.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 n (-.f64 U* U)) Om)) 2)))))): 8 points increase in error, 8 points decrease in error
      (*.f64 l (*.f64 (sqrt.f64 2) (sqrt.f64 (*.f64 (/.f64 n Om) (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 (/.f64 (*.f64 n (-.f64 U* U)) Om) 2) U)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 l (*.f64 (sqrt.f64 2) (sqrt.f64 (Rewrite<= associate-/r/_binary64 (/.f64 n (/.f64 Om (*.f64 (-.f64 (/.f64 (*.f64 n (-.f64 U* U)) Om) 2) U))))))): 25 points increase in error, 19 points decrease in error
      (*.f64 l (*.f64 (sqrt.f64 2) (sqrt.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 n (*.f64 (-.f64 (/.f64 (*.f64 n (-.f64 U* U)) Om) 2) U)) Om))))): 17 points increase in error, 14 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 l (sqrt.f64 2)) (sqrt.f64 (/.f64 (*.f64 n (*.f64 (-.f64 (/.f64 (*.f64 n (-.f64 U* U)) Om) 2) U)) Om)))): 14 points increase in error, 11 points decrease in error
      (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sqrt.f64 2) l)) (sqrt.f64 (/.f64 (*.f64 n (*.f64 (-.f64 (/.f64 (*.f64 n (-.f64 U* U)) Om) 2) U)) Om))): 0 points increase in error, 0 points decrease in error
3. Recombined 6 regimes into one program.

4. Final simplification 28.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6 \cdot 10^{+134}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \left(-\sqrt{n \cdot \left(U \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U* - U}{Om}, \frac{-2}{Om}\right)\right)}\right)\right)\\ \mathbf{elif}\;\ell \leq -2.25 \cdot 10^{-154}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\ell, -2, n \cdot \left(\left(U* - U\right) \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{U \cdot \left(2 \cdot n\right)}{\frac{Om}{\ell}} + 2 \cdot \left(\left(n \cdot U\right) \cdot t\right)}\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-222}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, n \cdot \left(\left(U* - U\right) \cdot \frac{\ell}{Om}\right)\right), t\right)} \cdot \sqrt{n \cdot \left(2 \cdot U\right)}\\ \mathbf{elif}\;\ell \leq 5.2 \cdot 10^{-71}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot t\right) + \mathsf{fma}\left(\ell, -2, n \cdot \frac{U* - U}{\frac{Om}{\ell}}\right) \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(2 \cdot U\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.05 \cdot 10^{+162}:\\ \;\;\;\;\sqrt{2 \cdot \mathsf{fma}\left(n, U \cdot t, \frac{n}{\frac{\frac{Om}{\ell \cdot U}}{\mathsf{fma}\left(\ell, -2, \frac{n}{Om} \cdot \left(\ell \cdot \left(U* - U\right)\right)\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{n}{Om} \cdot \left(U \cdot \mathsf{fma}\left(\frac{n}{Om}, U* - U, -2\right)\right)}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	25.8
Cost	36552

\[\begin{array}{l} t_1 := U \cdot \left(2 \cdot n\right)\\ t_2 := t_1 \cdot \left(\left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(U* - U\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\\ \mathbf{if}\;t_2 \leq 2 \cdot 10^{-323}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \mathsf{fma}\left(-2, \frac{\ell}{\frac{Om}{\ell}}, t\right)\right)}\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\ell, -2, n \cdot \left(\left(U* - U\right) \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{t_1}{\frac{Om}{\ell}} + 2 \cdot \left(\left(n \cdot U\right) \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{n}{Om} \cdot \left(U \cdot \mathsf{fma}\left(\frac{n}{Om}, U* - U, -2\right)\right)}\right)\\ \end{array} \]

Alternative 2
Error	27.0
Cost	21196

\[\begin{array}{l} \mathbf{if}\;\ell \leq -6 \cdot 10^{+134}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \left(-\sqrt{n \cdot \left(U \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U* - U}{Om}, \frac{-2}{Om}\right)\right)}\right)\right)\\ \mathbf{elif}\;\ell \leq 5.2 \cdot 10^{-71}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\ell, -2, n \cdot \left(\left(U* - U\right) \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{U \cdot \left(2 \cdot n\right)}{\frac{Om}{\ell}} + 2 \cdot \left(\left(n \cdot U\right) \cdot t\right)}\\ \mathbf{elif}\;\ell \leq 1.05 \cdot 10^{+162}:\\ \;\;\;\;\sqrt{2 \cdot \mathsf{fma}\left(n, U \cdot t, \frac{n}{\frac{\frac{Om}{\ell \cdot U}}{\mathsf{fma}\left(\ell, -2, \frac{n}{Om} \cdot \left(\ell \cdot \left(U* - U\right)\right)\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{n}{Om} \cdot \left(U \cdot \mathsf{fma}\left(\frac{n}{Om}, U* - U, -2\right)\right)}\right)\\ \end{array} \]

Alternative 3
Error	27.4
Cost	20684

\[\begin{array}{l} \mathbf{if}\;\ell \leq -6 \cdot 10^{+134}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \left(-\sqrt{n \cdot \left(U \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U* - U}{Om}, \frac{-2}{Om}\right)\right)}\right)\right)\\ \mathbf{elif}\;\ell \leq 5.2 \cdot 10^{-71}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\ell, -2, n \cdot \left(\left(U* - U\right) \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{U \cdot \left(2 \cdot n\right)}{\frac{Om}{\ell}} + 2 \cdot \left(\left(n \cdot U\right) \cdot t\right)}\\ \mathbf{elif}\;\ell \leq 1.05 \cdot 10^{+162}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + -2 \cdot \frac{n \cdot \left(\left(\ell \cdot U\right) \cdot \left(\frac{n \cdot \left(\ell \cdot \left(U - U*\right)\right)}{Om} + \ell \cdot 2\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{n}{Om} \cdot \left(U \cdot \mathsf{fma}\left(\frac{n}{Om}, U* - U, -2\right)\right)}\right)\\ \end{array} \]

Alternative 4
Error	30.9
Cost	15056

\[\begin{array}{l} t_1 := \sqrt{U} \cdot \sqrt{n \cdot \left(2 \cdot t\right)}\\ t_2 := \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot t\right) - \mathsf{fma}\left(\ell, -2, n \cdot \frac{\ell \cdot U*}{Om}\right) \cdot \frac{U \cdot \left(n \cdot -2\right)}{\frac{Om}{\ell}}}\\ \mathbf{if}\;U \leq -4.4 \cdot 10^{-79}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;U \leq 1.32 \cdot 10^{-288}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \frac{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + \ell \cdot -2\right) \cdot \left(n \cdot \left(\ell \cdot U\right)\right)}{Om}}\\ \mathbf{elif}\;U \leq 4.8 \cdot 10^{-112}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;U \leq 6 \cdot 10^{+77}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	30.4
Cost	15056

\[\begin{array}{l} t_1 := \sqrt{U} \cdot \sqrt{n \cdot \left(2 \cdot t\right)}\\ t_2 := 2 \cdot \left(\left(n \cdot U\right) \cdot t\right)\\ \mathbf{if}\;U \leq -4.4 \cdot 10^{-79}:\\ \;\;\;\;\sqrt{t_2 + \frac{U \cdot \left(2 \cdot n\right)}{\frac{Om}{\ell}} \cdot \mathsf{fma}\left(\ell, -2, n \cdot \frac{U* - U}{\frac{Om}{\ell}}\right)}\\ \mathbf{elif}\;U \leq 1.32 \cdot 10^{-288}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \frac{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + \ell \cdot -2\right) \cdot \left(n \cdot \left(\ell \cdot U\right)\right)}{Om}}\\ \mathbf{elif}\;U \leq 4.8 \cdot 10^{-112}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;U \leq 6 \cdot 10^{+77}:\\ \;\;\;\;\sqrt{t_2 - \mathsf{fma}\left(\ell, -2, n \cdot \frac{\ell \cdot U*}{Om}\right) \cdot \frac{U \cdot \left(n \cdot -2\right)}{\frac{Om}{\ell}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	30.4
Cost	15056

\[\begin{array}{l} t_1 := \sqrt{U} \cdot \sqrt{n \cdot \left(2 \cdot t\right)}\\ t_2 := 2 \cdot \left(\left(n \cdot U\right) \cdot t\right)\\ \mathbf{if}\;U \leq -4.4 \cdot 10^{-79}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\ell, -2, n \cdot \left(\left(U* - U\right) \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{U \cdot \left(2 \cdot n\right)}{\frac{Om}{\ell}} + t_2}\\ \mathbf{elif}\;U \leq 1.32 \cdot 10^{-288}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \frac{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + \ell \cdot -2\right) \cdot \left(n \cdot \left(\ell \cdot U\right)\right)}{Om}}\\ \mathbf{elif}\;U \leq 4.8 \cdot 10^{-112}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;U \leq 6 \cdot 10^{+77}:\\ \;\;\;\;\sqrt{t_2 - \mathsf{fma}\left(\ell, -2, n \cdot \frac{\ell \cdot U*}{Om}\right) \cdot \frac{U \cdot \left(n \cdot -2\right)}{\frac{Om}{\ell}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	31.1
Cost	14276

\[\begin{array}{l} t_1 := \sqrt{U} \cdot \sqrt{n \cdot \left(2 \cdot t\right)}\\ t_2 := \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \frac{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + \ell \cdot -2\right) \cdot \left(n \cdot \left(\ell \cdot U\right)\right)}{Om}}\\ \mathbf{if}\;U \leq -10000:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, n \cdot \left(U* \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;U \leq 1.32 \cdot 10^{-288}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;U \leq 4.8 \cdot 10^{-112}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;U \leq 6 \cdot 10^{+77}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	31.6
Cost	13776

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \frac{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + \ell \cdot -2\right) \cdot \left(n \cdot \left(\ell \cdot U\right)\right)}{Om}}\\ t_2 := \sqrt{U} \cdot \sqrt{n \cdot \left(2 \cdot t\right)}\\ \mathbf{if}\;U \leq -4.4 \cdot 10^{-79}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot t\right) - \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(2 \cdot U\right)\right)\right) \cdot \left(U \cdot \left(n \cdot \frac{\ell}{Om}\right) + \ell \cdot 2\right)}\\ \mathbf{elif}\;U \leq 1.32 \cdot 10^{-288}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;U \leq 4.8 \cdot 10^{-112}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;U \leq 6 \cdot 10^{+77}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 9
Error	31.8
Cost	8652

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot t\right) - \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(2 \cdot U\right)\right)\right) \cdot \left(U \cdot \left(n \cdot \frac{\ell}{Om}\right) + \ell \cdot 2\right)}\\ \mathbf{if}\;Om \leq -8.8 \cdot 10^{+93}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;Om \leq 2.6 \cdot 10^{-51}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \frac{\ell \cdot \left(\ell \cdot 2 - \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)\right)}\\ \mathbf{elif}\;Om \leq 2.05 \cdot 10^{+151}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \end{array} \]

Alternative 10
Error	30.8
Cost	8652

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot t\right) - \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(2 \cdot U\right)\right)\right) \cdot \left(U \cdot \left(n \cdot \frac{\ell}{Om}\right) + \ell \cdot 2\right)}\\ \mathbf{if}\;Om \leq -3.7 \cdot 10^{+96}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;Om \leq 2.6 \cdot 10^{-51}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + -2 \cdot \frac{n \cdot \left(\left(\ell \cdot U\right) \cdot \left(\frac{n \cdot \left(\ell \cdot \left(U - U*\right)\right)}{Om} + \ell \cdot 2\right)\right)}{Om}}\\ \mathbf{elif}\;Om \leq 2.05 \cdot 10^{+151}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \end{array} \]

Alternative 11
Error	31.0
Cost	8648

\[\begin{array}{l} \mathbf{if}\;U \leq -4.4 \cdot 10^{-79}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot t\right) - \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(2 \cdot U\right)\right)\right) \cdot \left(U \cdot \left(n \cdot \frac{\ell}{Om}\right) + \ell \cdot 2\right)}\\ \mathbf{elif}\;U \leq 6 \cdot 10^{+77}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \frac{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + \ell \cdot -2\right) \cdot \left(n \cdot \left(\ell \cdot U\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(t - \frac{2}{\frac{Om}{\ell \cdot \ell}}\right)}\\ \end{array} \]

Alternative 12
Error	35.5
Cost	8272

\[\begin{array}{l} \mathbf{if}\;Om \leq -6.5 \cdot 10^{-239}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(\frac{U \cdot \left(n \cdot \frac{\ell}{Om}\right) + \ell \cdot 2}{\frac{Om}{\ell}} - t\right)}\\ \mathbf{elif}\;Om \leq 4.2 \cdot 10^{-277}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\ell \cdot 2 - \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)\right)\right)}{Om}}\\ \mathbf{elif}\;Om \leq 2.4 \cdot 10^{-217}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;Om \leq 5.6 \cdot 10^{-105}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{n \cdot \left(\ell \cdot \ell\right)}{\frac{\frac{Om}{U}}{2 + \frac{n}{Om} \cdot \left(U - U*\right)}}}\\ \mathbf{elif}\;Om \leq 10^{+148}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(t - \frac{2}{\frac{Om}{\ell \cdot \ell}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \end{array} \]

Alternative 13
Error	35.3
Cost	8272

\[\begin{array}{l} \mathbf{if}\;Om \leq -6.5 \cdot 10^{-239}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(\frac{U \cdot \left(n \cdot \frac{\ell}{Om}\right) + \ell \cdot 2}{\frac{Om}{\ell}} - t\right)}\\ \mathbf{elif}\;Om \leq 1.45 \cdot 10^{-273}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{n \cdot \left(\left(\ell \cdot U\right) \cdot \left(\frac{n \cdot \left(\ell \cdot \left(U - U*\right)\right)}{Om} + \ell \cdot 2\right)\right)}{Om}}\\ \mathbf{elif}\;Om \leq 2.4 \cdot 10^{-217}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;Om \leq 5.6 \cdot 10^{-105}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{n \cdot \left(\ell \cdot \ell\right)}{\frac{\frac{Om}{U}}{2 + \frac{n}{Om} \cdot \left(U - U*\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \end{array} \]

Alternative 14
Error	32.3
Cost	8136

\[\begin{array}{l} \mathbf{if}\;Om \leq -2.35 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(\frac{U \cdot \left(n \cdot \frac{\ell}{Om}\right) + \ell \cdot 2}{\frac{Om}{\ell}} - t\right)}\\ \mathbf{elif}\;Om \leq 10^{+161}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \frac{\ell \cdot \left(\ell \cdot 2 - \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \end{array} \]

Alternative 15
Error	34.5
Cost	7880

\[\begin{array}{l} t_1 := t - \frac{2}{\frac{Om}{\ell \cdot \ell}}\\ \mathbf{if}\;U \leq -1.12 \cdot 10^{-38}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot t_1\right)}\\ \mathbf{elif}\;U \leq -7.3 \cdot 10^{-211}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om}\right)\right)\right)}\\ \mathbf{elif}\;U \leq 5 \cdot 10^{+72}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t_1}\\ \end{array} \]

Alternative 16
Error	41.2
Cost	7496

\[\begin{array}{l} \mathbf{if}\;Om \leq 2.4 \cdot 10^{-217}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;Om \leq 2.3 \cdot 10^{-70}:\\ \;\;\;\;\sqrt{n \cdot \left(-2 \cdot \left(\left(\ell \cdot \left(\ell \cdot U\right)\right) \cdot \frac{2}{Om}\right)\right)}\\ \mathbf{elif}\;Om \leq 3.5 \cdot 10^{+145}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(2 \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]

Alternative 17
Error	34.1
Cost	7492

\[\begin{array}{l} \mathbf{if}\;U \leq 5 \cdot 10^{+72}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(t - \frac{2}{\frac{Om}{\ell \cdot \ell}}\right)}\\ \end{array} \]

Alternative 18
Error	41.3
Cost	7368

\[\begin{array}{l} \mathbf{if}\;Om \leq 2.4 \cdot 10^{-217}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;Om \leq 2.3 \cdot 10^{-70}:\\ \;\;\;\;\sqrt{-4 \cdot \left(n \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{U}{Om}\right)\right)}\\ \mathbf{elif}\;Om \leq 1.9 \cdot 10^{+148}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(2 \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]

Alternative 19
Error	35.0
Cost	7360

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

Alternative 20
Error	39.6
Cost	6980

\[\begin{array}{l} \mathbf{if}\;U \leq 1.65 \cdot 10^{+72}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(2 \cdot t\right)}\\ \end{array} \]

Alternative 21
Error	40.2
Cost	6848

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

Reproduce

herbie shell --seed 2022295 
(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*))))))