Toniolo and Linder, Equation (13)

Average Error: 34.3 → 26.8

Time: 2.3min

Precision: binary64

Cost: 27340

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.6 \cdot 10^{+118}:\\ \;\;\;\;\sqrt{2 \cdot \frac{n \cdot \left(U \cdot \left(-2 + \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)}{Om}} \cdot \left(-\ell\right)\\ \mathbf{elif}\;\ell \leq -4.1 \cdot 10^{-191}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\frac{n \cdot \left(\left(\ell \cdot U\right) \cdot \left(\frac{n}{\frac{Om}{\ell}} \cdot \left(U - U*\right) + \ell \cdot 2\right)\right)}{Om} - n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+71}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, n \cdot \left(\left(U* - U\right) \cdot \frac{\ell}{Om}\right)\right), t\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{2 \cdot \left(\frac{n}{Om} \cdot \left(U \cdot \left(-2 + \left(U* - U\right) \cdot \frac{n}{Om}\right)\right)\right)}\\ \end{array} \]

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
 (if (<= l -2.6e+118)
   (* (sqrt (* 2.0 (/ (* n (* U (+ -2.0 (/ (* n (- U* U)) Om)))) Om))) (- l))
   (if (<= l -4.1e-191)
     (sqrt
      (*
       -2.0
       (-
        (/ (* n (* (* l U) (+ (* (/ n (/ Om l)) (- U U*)) (* l 2.0)))) Om)
        (* n (* U t)))))
     (if (<= l 2.6e+71)
       (*
        (sqrt 2.0)
        (sqrt
         (*
          (* n U)
          (fma (/ l Om) (fma l -2.0 (* n (* (- U* U) (/ l Om)))) t))))
       (*
        l
        (sqrt (* 2.0 (* (/ n Om) (* U (+ -2.0 (* (- U* U) (/ n Om))))))))))))

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 tmp;
	if (l <= -2.6e+118) {
		tmp = sqrt((2.0 * ((n * (U * (-2.0 + ((n * (U_42_ - U)) / Om)))) / Om))) * -l;
	} else if (l <= -4.1e-191) {
		tmp = sqrt((-2.0 * (((n * ((l * U) * (((n / (Om / l)) * (U - U_42_)) + (l * 2.0)))) / Om) - (n * (U * t)))));
	} else if (l <= 2.6e+71) {
		tmp = sqrt(2.0) * sqrt(((n * U) * fma((l / Om), fma(l, -2.0, (n * ((U_42_ - U) * (l / Om)))), t)));
	} else {
		tmp = l * sqrt((2.0 * ((n / Om) * (U * (-2.0 + ((U_42_ - U) * (n / Om)))))));
	}
	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_)
	tmp = 0.0
	if (l <= -2.6e+118)
		tmp = Float64(sqrt(Float64(2.0 * Float64(Float64(n * Float64(U * Float64(-2.0 + Float64(Float64(n * Float64(U_42_ - U)) / Om)))) / Om))) * Float64(-l));
	elseif (l <= -4.1e-191)
		tmp = sqrt(Float64(-2.0 * Float64(Float64(Float64(n * Float64(Float64(l * U) * Float64(Float64(Float64(n / Float64(Om / l)) * Float64(U - U_42_)) + Float64(l * 2.0)))) / Om) - Float64(n * Float64(U * t)))));
	elseif (l <= 2.6e+71)
		tmp = Float64(sqrt(2.0) * sqrt(Float64(Float64(n * U) * fma(Float64(l / Om), fma(l, -2.0, Float64(n * Float64(Float64(U_42_ - U) * Float64(l / Om)))), t))));
	else
		tmp = Float64(l * sqrt(Float64(2.0 * Float64(Float64(n / Om) * Float64(U * Float64(-2.0 + Float64(Float64(U_42_ - U) * Float64(n / Om))))))));
	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$42$_] := If[LessEqual[l, -2.6e+118], N[(N[Sqrt[N[(2.0 * N[(N[(n * N[(U * N[(-2.0 + N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-l)), $MachinePrecision], If[LessEqual[l, -4.1e-191], N[Sqrt[N[(-2.0 * N[(N[(N[(n * N[(N[(l * U), $MachinePrecision] * N[(N[(N[(n / N[(Om / l), $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] - N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 2.6e+71], N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(N[(n * U), $MachinePrecision] * N[(N[(l / Om), $MachinePrecision] * N[(l * -2.0 + N[(n * N[(N[(U$42$ - U), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(l * N[Sqrt[N[(2.0 * N[(N[(n / Om), $MachinePrecision] * N[(U * N[(-2.0 + N[(N[(U$42$ - U), $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.60000000000000016e118

   1. Initial program 58.6

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

   2. Simplified 53.4

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, n \cdot \frac{\ell \cdot \left(U* - U\right)}{Om}\right)\right)}} \]
      Proof
      (sqrt.f64 (*.f64 (*.f64 2 (*.f64 n U)) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l -2 (*.f64 n (/.f64 (*.f64 l (-.f64 U* U)) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 2 n) U)) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l -2 (*.f64 n (/.f64 (*.f64 l (-.f64 U* U)) Om))))))): 0 points increase in error, 2 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (Rewrite<= metadata-eval (neg.f64 2)) (*.f64 n (/.f64 (*.f64 l (-.f64 U* U)) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (Rewrite=> sub-neg_binary64 (+.f64 U* (neg.f64 U)))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (+.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 U*))) (neg.f64 U))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 (neg.f64 U*) U)))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 U (neg.f64 U*))))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 U U*)))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 l Om) (neg.f64 (-.f64 U U*)))))))))): 0 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, 1 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)))))): 11 points increase in error, 1 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (neg.f64 2) l) l)) Om) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (neg.f64 2) (*.f64 l l))) Om) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (Rewrite<= associate-*r/_binary64 (*.f64 (neg.f64 2) (/.f64 (*.f64 l l) Om))) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om)))) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om))) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 l Om) (*.f64 (/.f64 l Om) (-.f64 U U*))) n)))))): 4 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))))): 9 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*))))))): 2 points increase in error, 6 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 28.3

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

   if -2.60000000000000016e118 < l < -4.1000000000000002e-191

   1. Initial program 29.5

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

   2. Simplified 28.9

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, n \cdot \frac{\ell \cdot \left(U* - U\right)}{Om}\right)\right)}} \]
      Proof
      (sqrt.f64 (*.f64 (*.f64 2 (*.f64 n U)) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l -2 (*.f64 n (/.f64 (*.f64 l (-.f64 U* U)) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 2 n) U)) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l -2 (*.f64 n (/.f64 (*.f64 l (-.f64 U* U)) Om))))))): 0 points increase in error, 2 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (Rewrite<= metadata-eval (neg.f64 2)) (*.f64 n (/.f64 (*.f64 l (-.f64 U* U)) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (Rewrite=> sub-neg_binary64 (+.f64 U* (neg.f64 U)))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (+.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 U*))) (neg.f64 U))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 (neg.f64 U*) U)))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 U (neg.f64 U*))))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 U U*)))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 l Om) (neg.f64 (-.f64 U U*)))))))))): 0 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, 1 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)))))): 11 points increase in error, 1 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (neg.f64 2) l) l)) Om) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (neg.f64 2) (*.f64 l l))) Om) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (Rewrite<= associate-*r/_binary64 (*.f64 (neg.f64 2) (/.f64 (*.f64 l l) Om))) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om)))) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om))) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 l Om) (*.f64 (/.f64 l Om) (-.f64 U U*))) n)))))): 4 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))))): 9 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*))))))): 2 points increase in error, 6 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 29.6

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

   4. Simplified 28.4

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{n \cdot \left(\left(\frac{n}{\frac{Om}{\ell}} \cdot \left(U* - U\right) + \ell \cdot -2\right) \cdot \left(\ell \cdot U\right)\right)}{Om}\right)}} \]
      Proof
      (*.f64 2 (+.f64 (*.f64 n (*.f64 U t)) (/.f64 (*.f64 n (*.f64 (+.f64 (*.f64 (/.f64 n (/.f64 Om l)) (-.f64 U* U)) (*.f64 l -2)) (*.f64 l U))) Om))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (+.f64 (*.f64 n (Rewrite<= *-commutative_binary64 (*.f64 t U))) (/.f64 (*.f64 n (*.f64 (+.f64 (*.f64 (/.f64 n (/.f64 Om l)) (-.f64 U* U)) (*.f64 l -2)) (*.f64 l U))) Om))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (+.f64 (*.f64 n (*.f64 t U)) (/.f64 (*.f64 n (*.f64 (+.f64 (*.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 n l) Om)) (-.f64 U* U)) (*.f64 l -2)) (*.f64 l U))) Om))): 3 points increase in error, 1 points decrease in error
      (*.f64 2 (+.f64 (*.f64 n (*.f64 t U)) (/.f64 (*.f64 n (*.f64 (+.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (*.f64 n l) (-.f64 U* U)) Om)) (*.f64 l -2)) (*.f64 l U))) Om))): 8 points increase in error, 1 points decrease in error
      (*.f64 2 (+.f64 (*.f64 n (*.f64 t U)) (/.f64 (*.f64 n (*.f64 (+.f64 (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 n (*.f64 l (-.f64 U* U)))) Om) (*.f64 l -2)) (*.f64 l U))) Om))): 5 points increase in error, 3 points decrease in error
      (*.f64 2 (+.f64 (*.f64 n (*.f64 t U)) (/.f64 (*.f64 n (*.f64 (+.f64 (/.f64 (*.f64 n (*.f64 l (-.f64 U* U))) Om) (Rewrite<= *-commutative_binary64 (*.f64 -2 l))) (*.f64 l U))) Om))): 0 points increase in error, 0 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 -4.1000000000000002e-191 < l < 2.59999999999999991e71

   1. Initial program 25.9

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

   2. Simplified 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, 2 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (Rewrite<= metadata-eval (neg.f64 2)) (*.f64 n (/.f64 (*.f64 l (-.f64 U* U)) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (Rewrite=> sub-neg_binary64 (+.f64 U* (neg.f64 U)))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (+.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 U*))) (neg.f64 U))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 (neg.f64 U*) U)))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 U (neg.f64 U*))))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 U U*)))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 l Om) (neg.f64 (-.f64 U U*)))))))))): 0 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, 1 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)))))): 11 points increase in error, 1 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (neg.f64 2) l) l)) Om) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (neg.f64 2) (*.f64 l l))) Om) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (Rewrite<= associate-*r/_binary64 (*.f64 (neg.f64 2) (/.f64 (*.f64 l l) Om))) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om)))) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om))) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 l Om) (*.f64 (/.f64 l Om) (-.f64 U U*))) n)))))): 4 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))))): 9 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*))))))): 2 points increase in error, 6 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

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

   if 2.59999999999999991e71 < l

   1. Initial program 52.3

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

   2. Simplified 47.7

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, n \cdot \frac{\ell \cdot \left(U* - U\right)}{Om}\right)\right)}} \]
      Proof
      (sqrt.f64 (*.f64 (*.f64 2 (*.f64 n U)) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l -2 (*.f64 n (/.f64 (*.f64 l (-.f64 U* U)) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 2 n) U)) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l -2 (*.f64 n (/.f64 (*.f64 l (-.f64 U* U)) Om))))))): 0 points increase in error, 2 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (Rewrite<= metadata-eval (neg.f64 2)) (*.f64 n (/.f64 (*.f64 l (-.f64 U* U)) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (Rewrite=> sub-neg_binary64 (+.f64 U* (neg.f64 U)))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (+.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 U*))) (neg.f64 U))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 (neg.f64 U*) U)))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 U (neg.f64 U*))))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 U U*)))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 l Om) (neg.f64 (-.f64 U U*)))))))))): 0 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, 1 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)))))): 11 points increase in error, 1 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (neg.f64 2) l) l)) Om) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (neg.f64 2) (*.f64 l l))) Om) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (Rewrite<= associate-*r/_binary64 (*.f64 (neg.f64 2) (/.f64 (*.f64 l l) Om))) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om)))) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om))) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 l Om) (*.f64 (/.f64 l Om) (-.f64 U U*))) n)))))): 4 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))))): 9 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*))))))): 2 points increase in error, 6 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 63.0

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

   4. Simplified 63.0

      \[\leadsto \color{blue}{\sqrt{\frac{-2}{\frac{Om}{\left(2 - \frac{n}{Om} \cdot \left(U* - U\right)\right) \cdot \left(U \cdot n\right)}}} \cdot \left(-\ell\right)} \]
      Proof
      (*.f64 (sqrt.f64 (/.f64 -2 (/.f64 Om (*.f64 (-.f64 2 (*.f64 (/.f64 n Om) (-.f64 U* U))) (*.f64 U n))))) (neg.f64 l)): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (/.f64 -2 (/.f64 Om (*.f64 (-.f64 2 (Rewrite<= associate-/r/_binary64 (/.f64 n (/.f64 Om (-.f64 U* U))))) (*.f64 U n))))) (neg.f64 l)): 11 points increase in error, 3 points decrease in error
      (*.f64 (sqrt.f64 (/.f64 -2 (/.f64 Om (*.f64 (-.f64 2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 n (-.f64 U* U)) Om))) (*.f64 U n))))) (neg.f64 l)): 10 points increase in error, 11 points decrease in error
      (*.f64 (sqrt.f64 (/.f64 -2 (/.f64 Om (*.f64 (Rewrite<= unsub-neg_binary64 (+.f64 2 (neg.f64 (/.f64 (*.f64 n (-.f64 U* U)) Om)))) (*.f64 U n))))) (neg.f64 l)): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (/.f64 -2 (/.f64 Om (*.f64 (+.f64 2 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (*.f64 n (-.f64 U* U)) Om)))) (*.f64 U n))))) (neg.f64 l)): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (/.f64 -2 (/.f64 Om (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (+.f64 2 (*.f64 -1 (/.f64 (*.f64 n (-.f64 U* U)) Om))) U) n))))) (neg.f64 l)): 6 points increase in error, 4 points decrease in error
      (*.f64 (sqrt.f64 (/.f64 -2 (/.f64 Om (Rewrite<= *-commutative_binary64 (*.f64 n (*.f64 (+.f64 2 (*.f64 -1 (/.f64 (*.f64 n (-.f64 U* U)) Om))) U)))))) (neg.f64 l)): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 -2 (*.f64 n (*.f64 (+.f64 2 (*.f64 -1 (/.f64 (*.f64 n (-.f64 U* U)) Om))) U))) Om))) (neg.f64 l)): 4 points increase in error, 7 points decrease in error
      (*.f64 (sqrt.f64 (Rewrite<= associate-*r/_binary64 (*.f64 -2 (/.f64 (*.f64 n (*.f64 (+.f64 2 (*.f64 -1 (/.f64 (*.f64 n (-.f64 U* U)) Om))) U)) Om)))) (neg.f64 l)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (sqrt.f64 (*.f64 -2 (/.f64 (*.f64 n (*.f64 (+.f64 2 (*.f64 -1 (/.f64 (*.f64 n (-.f64 U* U)) Om))) U)) Om))) l))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 l (sqrt.f64 (*.f64 -2 (/.f64 (*.f64 n (*.f64 (+.f64 2 (*.f64 -1 (/.f64 (*.f64 n (-.f64 U* U)) Om))) U)) Om)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 l (sqrt.f64 (*.f64 -2 (/.f64 (*.f64 n (*.f64 (+.f64 2 (*.f64 -1 (/.f64 (*.f64 n (-.f64 U* U)) Om))) U)) Om)))))): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 54.3

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

   6. Simplified 54.3

      \[\leadsto \color{blue}{\sqrt{\frac{n \cdot U}{\frac{Om}{2 - \left(U* - U\right) \cdot \frac{n}{Om}}} \cdot \left(-2 \cdot \left(\ell \cdot \ell\right)\right)}} \]
      Proof
      (sqrt.f64 (*.f64 (/.f64 (*.f64 n U) (/.f64 Om (-.f64 2 (*.f64 (-.f64 U* U) (/.f64 n Om))))) (*.f64 -2 (*.f64 l l)))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (/.f64 (*.f64 n U) (/.f64 Om (-.f64 2 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 n Om) (-.f64 U* U)))))) (*.f64 -2 (*.f64 l l)))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (*.f64 n U) (-.f64 2 (*.f64 (/.f64 n Om) (-.f64 U* U)))) Om)) (*.f64 -2 (*.f64 l l)))): 9 points increase in error, 9 points decrease in error
      (sqrt.f64 (*.f64 (/.f64 (*.f64 (Rewrite=> *-commutative_binary64 (*.f64 U n)) (-.f64 2 (*.f64 (/.f64 n Om) (-.f64 U* U)))) Om) (*.f64 -2 (*.f64 l l)))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 U (*.f64 n (-.f64 2 (*.f64 (/.f64 n Om) (-.f64 U* U)))))) Om) (*.f64 -2 (*.f64 l l)))): 7 points increase in error, 5 points decrease in error
      (sqrt.f64 (*.f64 (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 (*.f64 n (-.f64 2 (*.f64 (/.f64 n Om) (-.f64 U* U)))) U)) Om) (*.f64 -2 (*.f64 l l)))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (/.f64 (*.f64 (Rewrite=> *-commutative_binary64 (*.f64 (-.f64 2 (*.f64 (/.f64 n Om) (-.f64 U* U))) n)) U) Om) (*.f64 -2 (*.f64 l l)))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (/.f64 (Rewrite=> associate-*l*_binary64 (*.f64 (-.f64 2 (*.f64 (/.f64 n Om) (-.f64 U* U))) (*.f64 n U))) Om) (*.f64 -2 (*.f64 l l)))): 5 points increase in error, 7 points decrease in error
      (sqrt.f64 (*.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (-.f64 2 (*.f64 (/.f64 n Om) (-.f64 U* U))) Om) (*.f64 n U))) (*.f64 -2 (*.f64 l l)))): 13 points increase in error, 10 points decrease in error
      (sqrt.f64 (*.f64 (Rewrite<= associate-/r/_binary64 (/.f64 (-.f64 2 (*.f64 (/.f64 n Om) (-.f64 U* U))) (/.f64 Om (*.f64 n U)))) (*.f64 -2 (*.f64 l l)))): 3 points increase in error, 15 points decrease in error
      (sqrt.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 (-.f64 2 (*.f64 (/.f64 n Om) (-.f64 U* U))) (/.f64 Om (*.f64 n U))) -2) (*.f64 l l)))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 -2 (/.f64 (-.f64 2 (*.f64 (/.f64 n Om) (-.f64 U* U))) (/.f64 Om (*.f64 n U))))) (*.f64 l l))): 0 points increase in error, 0 points decrease in error

   7. Taylor expanded in l around 0 30.4

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

   8. Simplified 28.7

      \[\leadsto \color{blue}{\ell \cdot \sqrt{-2 \cdot \left(\frac{n}{Om} \cdot \left(U \cdot \left(2 - \left(U* - U\right) \cdot \frac{n}{Om}\right)\right)\right)}} \]
      Proof
      (*.f64 l (sqrt.f64 (*.f64 -2 (*.f64 (/.f64 n Om) (*.f64 U (-.f64 2 (*.f64 (-.f64 U* U) (/.f64 n Om)))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 l (sqrt.f64 (*.f64 -2 (*.f64 (/.f64 n Om) (*.f64 (Rewrite<= /-rgt-identity_binary64 (/.f64 U 1)) (-.f64 2 (*.f64 (-.f64 U* U) (/.f64 n Om)))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 l (sqrt.f64 (*.f64 -2 (*.f64 (/.f64 n Om) (Rewrite<= associate-/r/_binary64 (/.f64 U (/.f64 1 (-.f64 2 (*.f64 (-.f64 U* U) (/.f64 n Om)))))))))): 1 points increase in error, 2 points decrease in error
      (*.f64 l (sqrt.f64 (*.f64 -2 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 n (/.f64 U (/.f64 1 (-.f64 2 (*.f64 (-.f64 U* U) (/.f64 n Om)))))) Om))))): 24 points increase in error, 6 points decrease in error
      (*.f64 l (sqrt.f64 (*.f64 -2 (/.f64 (*.f64 n (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 U 1) (-.f64 2 (*.f64 (-.f64 U* U) (/.f64 n Om)))))) Om)))): 1 points increase in error, 2 points decrease in error
      (*.f64 l (sqrt.f64 (*.f64 -2 (/.f64 (*.f64 n (*.f64 (Rewrite=> /-rgt-identity_binary64 U) (-.f64 2 (*.f64 (-.f64 U* U) (/.f64 n Om))))) Om)))): 0 points increase in error, 0 points decrease in error
      (*.f64 l (sqrt.f64 (*.f64 -2 (/.f64 (*.f64 n (*.f64 U (-.f64 2 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (-.f64 U* U) n) Om))))) Om)))): 10 points increase in error, 3 points decrease in error
      (*.f64 l (sqrt.f64 (*.f64 -2 (/.f64 (*.f64 n (*.f64 U (-.f64 2 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 n (-.f64 U* U))) Om)))) Om)))): 0 points increase in error, 0 points decrease in error
      (*.f64 l (sqrt.f64 (*.f64 -2 (/.f64 (*.f64 n (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 2 (/.f64 (*.f64 n (-.f64 U* U)) Om)) U))) Om)))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> *-commutative_binary64 (*.f64 (sqrt.f64 (*.f64 -2 (/.f64 (*.f64 n (*.f64 (-.f64 2 (/.f64 (*.f64 n (-.f64 U* U)) Om)) U)) Om))) l)): 0 points increase in error, 0 points decrease in error
3. Recombined 4 regimes into one program.

4. Final simplification 26.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.6 \cdot 10^{+118}:\\ \;\;\;\;\sqrt{2 \cdot \frac{n \cdot \left(U \cdot \left(-2 + \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)}{Om}} \cdot \left(-\ell\right)\\ \mathbf{elif}\;\ell \leq -4.1 \cdot 10^{-191}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\frac{n \cdot \left(\left(\ell \cdot U\right) \cdot \left(\frac{n}{\frac{Om}{\ell}} \cdot \left(U - U*\right) + \ell \cdot 2\right)\right)}{Om} - n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+71}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, n \cdot \left(\left(U* - U\right) \cdot \frac{\ell}{Om}\right)\right), t\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{2 \cdot \left(\frac{n}{Om} \cdot \left(U \cdot \left(-2 + \left(U* - U\right) \cdot \frac{n}{Om}\right)\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	26.8
Cost	21004

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.15 \cdot 10^{+117}:\\ \;\;\;\;\sqrt{2 \cdot \frac{n \cdot \left(U \cdot \left(-2 + \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)}{Om}} \cdot \left(-\ell\right)\\ \mathbf{elif}\;\ell \leq -2.3 \cdot 10^{-192}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\frac{n \cdot \left(\left(\ell \cdot U\right) \cdot \left(\frac{n}{\frac{Om}{\ell}} \cdot \left(U - U*\right) + \ell \cdot 2\right)\right)}{Om} - n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.55 \cdot 10^{+65}:\\ \;\;\;\;{\left(2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, n \cdot \left(\left(U* - U\right) \cdot \frac{\ell}{Om}\right)\right), t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{2 \cdot \left(\frac{n}{Om} \cdot \left(U \cdot \left(-2 + \left(U* - U\right) \cdot \frac{n}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 2
Error	26.7
Cost	15052

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.02 \cdot 10^{+124}:\\ \;\;\;\;\sqrt{2 \cdot \frac{n \cdot \left(U \cdot \left(-2 + \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)}{Om}} \cdot \left(-\ell\right)\\ \mathbf{elif}\;\ell \leq -1.5 \cdot 10^{-192}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\frac{n \cdot \left(\left(\ell \cdot U\right) \cdot \left(\frac{n}{\frac{Om}{\ell}} \cdot \left(U - U*\right) + \ell \cdot 2\right)\right)}{Om} - n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 6.6 \cdot 10^{+69}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right) - \mathsf{fma}\left(\ell, -2, n \cdot \left(\left(U* - U\right) \cdot \frac{\ell}{Om}\right)\right) \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(-2 \cdot U\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{2 \cdot \left(\frac{n}{Om} \cdot \left(U \cdot \left(-2 + \left(U* - U\right) \cdot \frac{n}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 3
Error	27.0
Cost	14668

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.5 \cdot 10^{+118}:\\ \;\;\;\;\sqrt{2 \cdot \frac{n \cdot \left(U \cdot \left(-2 + \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)}{Om}} \cdot \left(-\ell\right)\\ \mathbf{elif}\;\ell \leq -7.5 \cdot 10^{-191}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\frac{n \cdot \left(\left(\ell \cdot U\right) \cdot \left(\frac{n}{\frac{Om}{\ell}} \cdot \left(U - U*\right) + \ell \cdot 2\right)\right)}{Om} - n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.5 \cdot 10^{+50}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, n \cdot \frac{\ell \cdot \left(U* - U\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{2 \cdot \left(\frac{n}{Om} \cdot \left(U \cdot \left(-2 + \left(U* - U\right) \cdot \frac{n}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 4
Error	27.5
Cost	8656

\[\begin{array}{l} t_1 := \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\\ \mathbf{if}\;\ell \leq -1.2 \cdot 10^{+121}:\\ \;\;\;\;\sqrt{2 \cdot \frac{n \cdot \left(U \cdot \left(-2 + \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)}{Om}} \cdot \left(-\ell\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-46}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\frac{n \cdot \left(\left(\ell \cdot U\right) \cdot \left(\ell \cdot 2 - t_1\right)\right)}{Om} - n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{-91}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + t_1\right)}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 6.8 \cdot 10^{+145}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot t + U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{-2 \cdot \frac{2 + \frac{n}{Om} \cdot \left(U - U*\right)}{\frac{\frac{Om}{n}}{U}}}\\ \end{array} \]

Alternative 5
Error	27.3
Cost	8656

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.4 \cdot 10^{+123}:\\ \;\;\;\;\sqrt{2 \cdot \frac{n \cdot \left(U \cdot \left(-2 + \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)}{Om}} \cdot \left(-\ell\right)\\ \mathbf{elif}\;\ell \leq -1.6 \cdot 10^{-110}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\frac{n \cdot \left(\left(\ell \cdot U\right) \cdot \left(\frac{n}{\frac{Om}{\ell}} \cdot \left(U - U*\right) + \ell \cdot 2\right)\right)}{Om} - n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.5 \cdot 10^{-91}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 8.4 \cdot 10^{+144}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot t + U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{-2 \cdot \frac{2 + \frac{n}{Om} \cdot \left(U - U*\right)}{\frac{\frac{Om}{n}}{U}}}\\ \end{array} \]

Alternative 6
Error	28.6
Cost	8400

\[\begin{array}{l} \mathbf{if}\;\ell \leq -8.8 \cdot 10^{+158}:\\ \;\;\;\;\ell \cdot \left(-\sqrt{\frac{-2}{\frac{Om}{n \cdot \left(U \cdot \left(2 - \frac{n \cdot U*}{Om}\right)\right)}}}\right)\\ \mathbf{elif}\;\ell \leq -1.45 \cdot 10^{-47}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{-85}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.7 \cdot 10^{+65}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot t + \left(\ell \cdot \ell\right) \cdot \left(U \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{2 \cdot \left(\frac{n}{Om} \cdot \left(U \cdot \left(-2 + \left(U* - U\right) \cdot \frac{n}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 7
Error	28.5
Cost	8400

\[\begin{array}{l} \mathbf{if}\;\ell \leq -7.2 \cdot 10^{+158}:\\ \;\;\;\;\sqrt{2 \cdot \frac{n \cdot \left(U \cdot \left(-2 + \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)}{Om}} \cdot \left(-\ell\right)\\ \mathbf{elif}\;\ell \leq -6.8 \cdot 10^{-47}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{-86}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.4 \cdot 10^{+64}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot t + \left(\ell \cdot \ell\right) \cdot \left(U \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{2 \cdot \left(\frac{n}{Om} \cdot \left(U \cdot \left(-2 + \left(U* - U\right) \cdot \frac{n}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 8
Error	28.2
Cost	8400

\[\begin{array}{l} t_1 := \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\\ \mathbf{if}\;\ell \leq -3.3 \cdot 10^{+118}:\\ \;\;\;\;\sqrt{2 \cdot \frac{n \cdot \left(U \cdot \left(-2 + \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)}{Om}} \cdot \left(-\ell\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-46}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\frac{n \cdot \left(\left(\ell \cdot U\right) \cdot \left(\ell \cdot 2 - t_1\right)\right)}{Om} - n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 7.2 \cdot 10^{-88}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + t_1\right)}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 9.5 \cdot 10^{+62}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot t + \left(\ell \cdot \ell\right) \cdot \left(U \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{2 \cdot \left(\frac{n}{Om} \cdot \left(U \cdot \left(-2 + \left(U* - U\right) \cdot \frac{n}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 9
Error	28.5
Cost	8268

\[\begin{array}{l} \mathbf{if}\;\ell \leq -9 \cdot 10^{+158}:\\ \;\;\;\;\ell \cdot \left(-\sqrt{\frac{-2}{\frac{Om}{n \cdot \left(U \cdot \left(2 - \frac{n \cdot U*}{Om}\right)\right)}}}\right)\\ \mathbf{elif}\;\ell \leq -1.3 \cdot 10^{-47}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.6:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{2 \cdot \left(\frac{n}{Om} \cdot \left(U \cdot \left(-2 + \left(U* - U\right) \cdot \frac{n}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 10
Error	35.5
Cost	8020

\[\begin{array}{l} \mathbf{if}\;Om \leq -4.1 \cdot 10^{-106}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(\ell \cdot \frac{\ell \cdot 2}{Om} - t\right)\right) \cdot \left(-2 \cdot n\right)}\\ \mathbf{elif}\;Om \leq -6.6 \cdot 10^{-197}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(U \cdot \left(U - U*\right)\right)} \cdot \left(\ell \cdot \left(-n\right)\right)}{Om}\\ \mathbf{elif}\;Om \leq -1.3 \cdot 10^{-282}:\\ \;\;\;\;\sqrt{U \cdot \left(2 \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;Om \leq 5.5 \cdot 10^{-199}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(U* \cdot \left(U \cdot \left(\left(\ell \cdot n\right) \cdot \left(\ell \cdot n\right)\right)\right)\right)}}{Om}\\ \mathbf{elif}\;Om \leq 2.5 \cdot 10^{-140}:\\ \;\;\;\;\frac{n}{Om} \cdot \left(\left(-\ell\right) \cdot \sqrt{\left(U* - U\right) \cdot \left(2 \cdot U\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - t\right)\right)\right)}\\ \end{array} \]

Alternative 11
Error	35.7
Cost	8020

\[\begin{array}{l} \mathbf{if}\;Om \leq -9 \cdot 10^{-106}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(\ell \cdot \frac{\ell \cdot 2}{Om} - t\right)\right) \cdot \left(-2 \cdot n\right)}\\ \mathbf{elif}\;Om \leq -8 \cdot 10^{-197}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(U \cdot \left(U - U*\right)\right)} \cdot \left(\ell \cdot \left(-n\right)\right)}{Om}\\ \mathbf{elif}\;Om \leq -3.4 \cdot 10^{-303}:\\ \;\;\;\;n \cdot \frac{-\sqrt{2 \cdot \left(U \cdot \left(\ell \cdot \left(\ell \cdot \left(U* - U\right)\right)\right)\right)}}{Om}\\ \mathbf{elif}\;Om \leq 5.5 \cdot 10^{-199}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(U* \cdot \left(U \cdot \left(\left(\ell \cdot n\right) \cdot \left(\ell \cdot n\right)\right)\right)\right)}}{Om}\\ \mathbf{elif}\;Om \leq 2.5 \cdot 10^{-140}:\\ \;\;\;\;\frac{n}{Om} \cdot \left(\left(-\ell\right) \cdot \sqrt{\left(U* - U\right) \cdot \left(2 \cdot U\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - t\right)\right)\right)}\\ \end{array} \]

Alternative 12
Error	30.7
Cost	7880

\[\begin{array}{l} \mathbf{if}\;\ell \leq -7.6 \cdot 10^{+158}:\\ \;\;\;\;\ell \cdot \left(-\sqrt{\frac{-2}{\frac{Om}{n \cdot \left(U \cdot \left(2 - \frac{n \cdot U*}{Om}\right)\right)}}}\right)\\ \mathbf{elif}\;\ell \leq 0.19:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{2 \cdot \left(\frac{n}{Om} \cdot \left(U \cdot \left(-2 + \left(U* - U\right) \cdot \frac{n}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 13
Error	33.9
Cost	7756

\[\begin{array}{l} t_1 := \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - t\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -9 \cdot 10^{+158}:\\ \;\;\;\;\ell \cdot \left(-\sqrt{\frac{-2}{0.5 \cdot \frac{\frac{Om}{n}}{U}}}\right)\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{-176}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 7.2 \cdot 10^{+50}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Error	33.7
Cost	7756

\[\begin{array}{l} t_1 := \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - t\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -7.5 \cdot 10^{+158}:\\ \;\;\;\;\ell \cdot \left(-\sqrt{\frac{-2}{0.5 \cdot \frac{\frac{Om}{n}}{U}}}\right)\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{-196}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 3.1 \cdot 10^{+123}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right) \cdot \left(-2 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 15
Error	32.1
Cost	7752

\[\begin{array}{l} \mathbf{if}\;\ell \leq -9 \cdot 10^{+158}:\\ \;\;\;\;\ell \cdot \left(-\sqrt{\frac{-2}{0.5 \cdot \frac{\frac{Om}{n}}{U}}}\right)\\ \mathbf{elif}\;\ell \leq 4.6:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{-2 \cdot \left(\frac{n}{Om} \cdot \left(U \cdot \left(2 - U* \cdot \frac{n}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 16
Error	30.8
Cost	7752

\[\begin{array}{l} \mathbf{if}\;\ell \leq -9.5 \cdot 10^{+158}:\\ \;\;\;\;\ell \cdot \left(-\sqrt{\frac{-2}{\frac{Om}{n \cdot \left(U \cdot \left(2 - \frac{n \cdot U*}{Om}\right)\right)}}}\right)\\ \mathbf{elif}\;\ell \leq 0.93:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{-2 \cdot \left(\frac{n}{Om} \cdot \left(U \cdot \left(2 - U* \cdot \frac{n}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 17
Error	33.8
Cost	7624

\[\begin{array}{l} \mathbf{if}\;\ell \leq -8.8 \cdot 10^{+158}:\\ \;\;\;\;\ell \cdot \left(-\sqrt{\frac{-2}{0.5 \cdot \frac{\frac{Om}{n}}{U}}}\right)\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{-197}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(\ell \cdot \frac{\ell \cdot 2}{Om} - t\right)\right) \cdot \left(-2 \cdot n\right)}\\ \end{array} \]

Alternative 18
Error	38.1
Cost	7300

\[\begin{array}{l} \mathbf{if}\;\ell \leq -3.4 \cdot 10^{+57}:\\ \;\;\;\;\ell \cdot \left(-\sqrt{\frac{-2}{0.5 \cdot \frac{Om}{n \cdot U}}}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(U \cdot t\right) \cdot \left(n \cdot 2\right)\right)}^{0.5}\\ \end{array} \]

Alternative 19
Error	38.4
Cost	7300

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.4 \cdot 10^{+61}:\\ \;\;\;\;\ell \cdot \left(-\sqrt{\frac{-2}{0.5 \cdot \frac{\frac{Om}{n}}{U}}}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(U \cdot t\right) \cdot \left(n \cdot 2\right)\right)}^{0.5}\\ \end{array} \]

Alternative 20
Error	39.4
Cost	7044

\[\begin{array}{l} \mathbf{if}\;n \leq 7.3 \cdot 10^{+62}:\\ \;\;\;\;\sqrt{U \cdot \left(2 \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(U \cdot t\right) \cdot \left(n \cdot 2\right)\right)}^{0.5}\\ \end{array} \]

Alternative 21
Error	39.3
Cost	6980

\[\begin{array}{l} \mathbf{if}\;n \leq 6.6 \cdot 10^{+72}:\\ \;\;\;\;\sqrt{U \cdot \left(2 \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 22
Error	39.4
Cost	6980

\[\begin{array}{l} \mathbf{if}\;n \leq 2 \cdot 10^{+62}:\\ \;\;\;\;\sqrt{U \cdot \left(2 \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot t\right) \cdot \left(n \cdot 2\right)}\\ \end{array} \]

Alternative 23
Error	39.8
Cost	6848

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

Reproduce

herbie shell --seed 2022325 
(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*))))))