Toniolo and Linder, Equation (13)

Average Error: 34.9 → 26.5

Time: 56.1s

Precision: binary64

Cost: 57868

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t_1 \leq 2 \cdot 10^{-159}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot -2 - \frac{n}{Om} \cdot \left(U \cdot \ell\right), t\right)}\\ \mathbf{elif}\;t_1 \leq 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\sqrt{U \cdot \left(U* - U\right)} \cdot \frac{-\sqrt{2}}{\frac{\frac{Om}{\ell}}{n}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot \mathsf{fma}\left(\ell, -2, \frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om}\right)\right) \cdot \left(\left(U \cdot \ell\right) \cdot \frac{1}{Om}\right)\right)}\\ \end{array} \]

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
         (sqrt
          (*
           (* (* 2.0 n) U)
           (+
            (+ t (* (/ (* l l) Om) -2.0))
            (* (* n (pow (/ l Om) 2.0)) (- U* U)))))))
   (if (<= t_1 2e-159)
     (*
      (sqrt (* 2.0 n))
      (sqrt (* U (fma (/ l Om) (- (* l -2.0) (* (/ n Om) (* U l))) t))))
     (if (<= t_1 1e+154)
       t_1
       (if (<= t_1 INFINITY)
         (* (sqrt (* U (- U* U))) (/ (- (sqrt 2.0)) (/ (/ Om l) n)))
         (sqrt
          (*
           2.0
           (*
            (* n (fma l -2.0 (/ (* (- U* U) (* n l)) Om)))
            (* (* U l) (/ 1.0 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 t_1 = sqrt((((2.0 * n) * U) * ((t + (((l * l) / Om) * -2.0)) + ((n * pow((l / Om), 2.0)) * (U_42_ - U)))));
	double tmp;
	if (t_1 <= 2e-159) {
		tmp = sqrt((2.0 * n)) * sqrt((U * fma((l / Om), ((l * -2.0) - ((n / Om) * (U * l))), t)));
	} else if (t_1 <= 1e+154) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = sqrt((U * (U_42_ - U))) * (-sqrt(2.0) / ((Om / l) / n));
	} else {
		tmp = sqrt((2.0 * ((n * fma(l, -2.0, (((U_42_ - U) * (n * l)) / Om))) * ((U * l) * (1.0 / 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_)
	t_1 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t + Float64(Float64(Float64(l * l) / Om) * -2.0)) + Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U_42_ - U)))))
	tmp = 0.0
	if (t_1 <= 2e-159)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * fma(Float64(l / Om), Float64(Float64(l * -2.0) - Float64(Float64(n / Om) * Float64(U * l))), t))));
	elseif (t_1 <= 1e+154)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = Float64(sqrt(Float64(U * Float64(U_42_ - U))) * Float64(Float64(-sqrt(2.0)) / Float64(Float64(Om / l) / n)));
	else
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * fma(l, -2.0, Float64(Float64(Float64(U_42_ - U) * Float64(n * l)) / Om))) * Float64(Float64(U * l) * Float64(1.0 / 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$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t + N[(N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 2e-159], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(N[(l / Om), $MachinePrecision] * N[(N[(l * -2.0), $MachinePrecision] - N[(N[(n / Om), $MachinePrecision] * N[(U * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+154], t$95$1, If[LessEqual[t$95$1, Infinity], N[(N[Sqrt[N[(U * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / N[(N[(Om / l), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(2.0 * N[(N[(n * N[(l * -2.0 + N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(U * l), $MachinePrecision] * N[(1.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 1.99999999999999998e-159

   1. Initial program 56.5

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

   2. Simplified 56.2

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, n \cdot \frac{\ell \cdot \left(U* - U\right)}{Om}\right)\right)}} \]
      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*)))))))))): 2 points increase in error, 10 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)))))): 10 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)))))): 6 points increase in error, 3 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))))): 6 points increase in error, 1 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)) n))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om))) (Rewrite<= *-commutative_binary64 (*.f64 n (*.f64 (pow.f64 (/.f64 l Om) 2) (-.f64 U U*)))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om))) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))))): 0 points increase in error, 8 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 t (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om)))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (Rewrite<= sub-neg_binary64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om)))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in U* around 0 41.7

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

   4. Simplified 56.5

      \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 - \frac{n}{\frac{Om}{\ell \cdot U}}\right)\right)}} \]
      Proof
      (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (-.f64 (*.f64 l -2) (/.f64 n (/.f64 Om (*.f64 l U))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 -2 l)) (/.f64 n (/.f64 Om (*.f64 l U))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (-.f64 (*.f64 -2 l) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 n (*.f64 l U)) Om)))))): 1 points increase in error, 3 points decrease in error
      (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 -2 l) (neg.f64 (/.f64 (*.f64 n (*.f64 l U)) Om))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (+.f64 (*.f64 -2 l) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (*.f64 n (*.f64 l U)) Om))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (/.f64 (*.f64 n (*.f64 l U)) Om)) (*.f64 -2 l)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (Rewrite<= associate-/r/_binary64 (/.f64 l (/.f64 Om (+.f64 (*.f64 -1 (/.f64 (*.f64 n (*.f64 l U)) Om)) (*.f64 -2 l))))))): 7 points increase in error, 7 points decrease in error
      (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l (+.f64 (*.f64 -1 (/.f64 (*.f64 n (*.f64 l U)) Om)) (*.f64 -2 l))) Om)))): 14 points increase in error, 5 points decrease in error
      (*.f64 (*.f64 (*.f64 2 n) U) (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 (*.f64 l (+.f64 (*.f64 -1 (/.f64 (*.f64 n (*.f64 l U)) Om)) (*.f64 -2 l))) Om) t))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r*_binary64 (*.f64 (*.f64 2 n) (*.f64 U (+.f64 (/.f64 (*.f64 l (+.f64 (*.f64 -1 (/.f64 (*.f64 n (*.f64 l U)) Om)) (*.f64 -2 l))) Om) t)))): 30 points increase in error, 43 points decrease in error
      (*.f64 (*.f64 2 n) (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 (/.f64 (*.f64 l (+.f64 (*.f64 -1 (/.f64 (*.f64 n (*.f64 l U)) Om)) (*.f64 -2 l))) Om) t) U))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r*_binary64 (*.f64 2 (*.f64 n (*.f64 (+.f64 (/.f64 (*.f64 l (+.f64 (*.f64 -1 (/.f64 (*.f64 n (*.f64 l U)) Om)) (*.f64 -2 l))) Om) t) U)))): 1 points increase in error, 1 points decrease in error

   5. Applied egg-rr 40.9

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

   if 1.99999999999999998e-159 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 1.00000000000000004e154

   1. Initial program 1.6

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

   if 1.00000000000000004e154 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < +inf.0

   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 56.4

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, n \cdot \frac{\ell \cdot \left(U* - U\right)}{Om}\right)\right)}} \]
      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, 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*)))))))))): 2 points increase in error, 10 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)))))): 10 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)))))): 6 points increase in error, 3 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))))): 6 points increase in error, 1 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)) n))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om))) (Rewrite<= *-commutative_binary64 (*.f64 n (*.f64 (pow.f64 (/.f64 l Om) 2) (-.f64 U U*)))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om))) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))))): 0 points increase in error, 8 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 t (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om)))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (Rewrite<= sub-neg_binary64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om)))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in t around inf 54.7

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

   4. Taylor expanded in n around -inf 56.0

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

   5. Simplified 55.6

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(U* - U\right)} \cdot \frac{-\sqrt{2}}{\frac{\frac{Om}{\ell}}{n}}} \]
      Proof
      (*.f64 (sqrt.f64 (*.f64 U (-.f64 U* U))) (/.f64 (neg.f64 (sqrt.f64 2)) (/.f64 (/.f64 Om l) n))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 U* U) U))) (/.f64 (neg.f64 (sqrt.f64 2)) (/.f64 (/.f64 Om l) n))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (*.f64 (-.f64 U* U) U)) (/.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (sqrt.f64 2))) (/.f64 (/.f64 Om l) n))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (*.f64 (-.f64 U* U) U)) (/.f64 (*.f64 -1 (sqrt.f64 2)) (Rewrite<= associate-/r*_binary64 (/.f64 Om (*.f64 l n))))): 9 points increase in error, 17 points decrease in error
      (*.f64 (sqrt.f64 (*.f64 (-.f64 U* U) U)) (/.f64 (*.f64 -1 (sqrt.f64 2)) (/.f64 Om (Rewrite<= *-commutative_binary64 (*.f64 n l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (*.f64 (-.f64 U* U) U)) (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 (sqrt.f64 2) (/.f64 Om (*.f64 n l)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (*.f64 (-.f64 U* U) U)) (*.f64 -1 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (sqrt.f64 2) (*.f64 n l)) Om)))): 12 points increase in error, 9 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 -1 (/.f64 (*.f64 (sqrt.f64 2) (*.f64 n l)) Om)) (sqrt.f64 (*.f64 (-.f64 U* U) U)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r*_binary64 (*.f64 -1 (*.f64 (/.f64 (*.f64 (sqrt.f64 2) (*.f64 n l)) Om) (sqrt.f64 (*.f64 (-.f64 U* U) U))))): 0 points increase in error, 0 points decrease in error

   if +inf.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))))

   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 58.1

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, n \cdot \frac{\ell \cdot \left(U* - U\right)}{Om}\right)\right)}} \]
      Proof
      (sqrt.f64 (*.f64 (*.f64 2 (*.f64 n U)) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l -2 (*.f64 n (/.f64 (*.f64 l (-.f64 U* U)) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 2 n) U)) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l -2 (*.f64 n (/.f64 (*.f64 l (-.f64 U* U)) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (Rewrite<= metadata-eval (neg.f64 2)) (*.f64 n (/.f64 (*.f64 l (-.f64 U* U)) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (Rewrite=> sub-neg_binary64 (+.f64 U* (neg.f64 U)))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (+.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 U*))) (neg.f64 U))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 (neg.f64 U*) U)))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 U (neg.f64 U*))))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 U U*)))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 l Om) (neg.f64 (-.f64 U U*)))))))))): 2 points increase in error, 10 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)))))): 10 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)))))): 6 points increase in error, 3 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))))): 6 points increase in error, 1 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)) n))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om))) (Rewrite<= *-commutative_binary64 (*.f64 n (*.f64 (pow.f64 (/.f64 l Om) 2) (-.f64 U U*)))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om))) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))))): 0 points increase in error, 8 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 46.4

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

   4. Applied egg-rr 36.0

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot \mathsf{fma}\left(\ell, -2, \frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om}\right)\right) \cdot \left(\left(\ell \cdot U\right) \cdot \frac{1}{Om}\right)\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 26.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 2 \cdot 10^{-159}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot -2 - \frac{n}{Om} \cdot \left(U \cdot \ell\right), t\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 10^{+154}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{U \cdot \left(U* - U\right)} \cdot \frac{-\sqrt{2}}{\frac{\frac{Om}{\ell}}{n}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot \mathsf{fma}\left(\ell, -2, \frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om}\right)\right) \cdot \left(\left(U \cdot \ell\right) \cdot \frac{1}{Om}\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	27.2
Cost	57868

\[\begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t_1 \leq 2 \cdot 10^{-159}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t}\\ \mathbf{elif}\;t_1 \leq 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\sqrt{U \cdot \left(U* - U\right)} \cdot \frac{-\sqrt{2}}{\frac{\frac{Om}{\ell}}{n}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot \mathsf{fma}\left(\ell, -2, \frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om}\right)\right) \cdot \left(\left(U \cdot \ell\right) \cdot \frac{1}{Om}\right)\right)}\\ \end{array} \]

Alternative 2
Error	32.1
Cost	15448

\[\begin{array}{l} t_1 := \sqrt{\left(2 \cdot n\right) \cdot \left(\mathsf{fma}\left(\frac{n}{Om}, \frac{U* - U}{Om}, \frac{-2}{Om}\right) \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right) + U \cdot t\right)}\\ t_2 := \sqrt{2 \cdot U} \cdot \sqrt{n \cdot t}\\ t_3 := \sqrt{2 \cdot n} \cdot \sqrt{U \cdot t}\\ \mathbf{if}\;U \leq -1 \cdot 10^{-45}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, n \cdot \left(\frac{\ell}{Om} \cdot U*\right)\right)\right)}\\ \mathbf{elif}\;U \leq -1.32 \cdot 10^{-95}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;U \leq -4.4 \cdot 10^{-147}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;U \leq -5.9 \cdot 10^{-164}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;U \leq -9 \cdot 10^{-227}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;U \leq -1 \cdot 10^{-300}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right) - \mathsf{fma}\left(\ell, -2, n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right) \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U \cdot -2\right)\right)\right)}\\ \mathbf{elif}\;U \leq 2.2 \cdot 10^{-300}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;U \leq 10^{-249}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;U \leq 1.1 \cdot 10^{-186}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \frac{\ell \cdot \left(2 \cdot \ell - \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)\right)}\\ \mathbf{elif}\;U \leq 31.5:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) - 2 \cdot \frac{\left(n \cdot \left(U \cdot \ell\right)\right) \cdot \left(\frac{n \cdot \left(\ell \cdot \left(U - U*\right)\right)}{Om} + 2 \cdot \ell\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	32.2
Cost	15064

\[\begin{array}{l} t_1 := n \cdot \left(U \cdot \ell\right)\\ t_2 := n \cdot \left(U \cdot t\right)\\ t_3 := 2 \cdot t_2\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+175}:\\ \;\;\;\;\sqrt{t_3 + -2 \cdot \frac{t_1 \cdot \left(2 \cdot \ell - \frac{n \cdot \ell}{\frac{Om}{U* - U}}\right)}{Om}}\\ \mathbf{elif}\;\ell \leq -3 \cdot 10^{+66}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-38}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, t_2, \frac{n \cdot -4}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)}\\ \mathbf{elif}\;\ell \leq -9.5 \cdot 10^{-129}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t}\\ \mathbf{elif}\;\ell \leq 10^{-300}:\\ \;\;\;\;\sqrt{t_3 + -2 \cdot \frac{t_1 \cdot \left(2 \cdot \ell - \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}}\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{-64}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, n \cdot \frac{\ell \cdot \left(U* - U\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{+142}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + \left(\frac{\left(\ell \cdot \ell\right) \cdot \left(n \cdot U*\right)}{Om \cdot Om} + \frac{-2}{\frac{Om}{\ell \cdot \ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(-2 + \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)}{Om}}\\ \end{array} \]

Alternative 4
Error	33.1
Cost	14940

\[\begin{array}{l} t_1 := n \cdot \left(U \cdot \ell\right)\\ t_2 := n \cdot \left(U \cdot t\right)\\ t_3 := 2 \cdot t_2\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+175}:\\ \;\;\;\;\sqrt{t_3 + -2 \cdot \frac{t_1 \cdot \left(2 \cdot \ell - \frac{n \cdot \ell}{\frac{Om}{U* - U}}\right)}{Om}}\\ \mathbf{elif}\;\ell \leq -3 \cdot 10^{+66}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-38}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, t_2, \frac{n \cdot -4}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)}\\ \mathbf{elif}\;\ell \leq -9.5 \cdot 10^{-129}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t}\\ \mathbf{elif}\;\ell \leq 10^{-255}:\\ \;\;\;\;\sqrt{t_3 + -2 \cdot \frac{t_1 \cdot \left(2 \cdot \ell - \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}}\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{-64}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \frac{U*}{Om} \cdot \frac{n \cdot \left(\ell \cdot \ell\right)}{Om}\right)}\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{+142}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + \left(\frac{\left(\ell \cdot \ell\right) \cdot \left(n \cdot U*\right)}{Om \cdot Om} + \frac{-2}{\frac{Om}{\ell \cdot \ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(-2 + \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)}{Om}}\\ \end{array} \]

Alternative 5
Error	31.5
Cost	14796

\[\begin{array}{l} t_1 := \sqrt{2 \cdot U} \cdot \sqrt{n \cdot t}\\ t_2 := 2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\\ t_3 := n \cdot \left(U \cdot \ell\right)\\ \mathbf{if}\;U \leq -1 \cdot 10^{-45}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, n \cdot \left(\frac{\ell}{Om} \cdot U*\right)\right)\right)}\\ \mathbf{elif}\;U \leq -1.32 \cdot 10^{-95}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t}\\ \mathbf{elif}\;U \leq -1 \cdot 10^{-144}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\mathsf{fma}\left(\frac{n}{Om}, \frac{U* - U}{Om}, \frac{-2}{Om}\right) \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right) + U \cdot t\right)}\\ \mathbf{elif}\;U \leq 2.2 \cdot 10^{-300}:\\ \;\;\;\;\sqrt{t_2 + -2 \cdot \frac{t_3 \cdot \left(2 \cdot \ell - \frac{n \cdot \ell}{\frac{Om}{U* - U}}\right)}{Om}}\\ \mathbf{elif}\;U \leq 10^{-249}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;U \leq 1.1 \cdot 10^{-186}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \frac{\ell \cdot \left(2 \cdot \ell - \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)\right)}\\ \mathbf{elif}\;U \leq 31.5:\\ \;\;\;\;\sqrt{t_2 - 2 \cdot \frac{t_3 \cdot \left(\frac{n \cdot \left(\ell \cdot \left(U - U*\right)\right)}{Om} + 2 \cdot \ell\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	31.3
Cost	14276

\[\begin{array}{l} t_1 := n \cdot \left(U \cdot \ell\right)\\ t_2 := 2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\\ t_3 := \sqrt{2 \cdot U} \cdot \sqrt{n \cdot t}\\ \mathbf{if}\;U \leq -1 \cdot 10^{-45}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, n \cdot \left(\frac{\ell}{Om} \cdot U*\right)\right)\right)}\\ \mathbf{elif}\;U \leq -1.6 \cdot 10^{-92}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t}\\ \mathbf{elif}\;U \leq -3.5 \cdot 10^{-145}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \mathsf{fma}\left(-2, \frac{\ell}{\frac{Om}{\ell}}, t\right)\right)}\\ \mathbf{elif}\;U \leq 2.2 \cdot 10^{-300}:\\ \;\;\;\;\sqrt{t_2 + -2 \cdot \frac{t_1 \cdot \left(2 \cdot \ell - \frac{n \cdot \ell}{\frac{Om}{U* - U}}\right)}{Om}}\\ \mathbf{elif}\;U \leq 3.1 \cdot 10^{-248}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;U \leq 31.5:\\ \;\;\;\;\sqrt{t_2 - 2 \cdot \frac{t_1 \cdot \left(\frac{n \cdot \left(\ell \cdot \left(U - U*\right)\right)}{Om} + 2 \cdot \ell\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 7
Error	32.1
Cost	14040

\[\begin{array}{l} t_1 := n \cdot \left(U \cdot \ell\right)\\ t_2 := 2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\\ t_3 := \sqrt{2 \cdot U} \cdot \sqrt{n \cdot t}\\ \mathbf{if}\;U \leq -1 \cdot 10^{-45}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\ell \cdot \frac{-2}{Om}\right)\right)\right)}\\ \mathbf{elif}\;U \leq -1.32 \cdot 10^{-95}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t}\\ \mathbf{elif}\;U \leq -2.6 \cdot 10^{-136}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right)\right)\right)}\\ \mathbf{elif}\;U \leq 2.2 \cdot 10^{-300}:\\ \;\;\;\;\sqrt{t_2 + -2 \cdot \frac{t_1 \cdot \left(2 \cdot \ell - \frac{n \cdot \ell}{\frac{Om}{U* - U}}\right)}{Om}}\\ \mathbf{elif}\;U \leq 3.1 \cdot 10^{-248}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;U \leq 31.5:\\ \;\;\;\;\sqrt{t_2 - 2 \cdot \frac{t_1 \cdot \left(\frac{n \cdot \left(\ell \cdot \left(U - U*\right)\right)}{Om} + 2 \cdot \ell\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 8
Error	31.9
Cost	14040

\[\begin{array}{l} t_1 := n \cdot \left(U \cdot \ell\right)\\ t_2 := 2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\\ t_3 := \sqrt{2 \cdot U} \cdot \sqrt{n \cdot t}\\ \mathbf{if}\;U \leq -1 \cdot 10^{-45}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\ell \cdot \frac{-2}{Om}\right)\right)\right)}\\ \mathbf{elif}\;U \leq -1.6 \cdot 10^{-92}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t}\\ \mathbf{elif}\;U \leq -3.5 \cdot 10^{-145}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \mathsf{fma}\left(-2, \frac{\ell}{\frac{Om}{\ell}}, t\right)\right)}\\ \mathbf{elif}\;U \leq 2.2 \cdot 10^{-300}:\\ \;\;\;\;\sqrt{t_2 + -2 \cdot \frac{t_1 \cdot \left(2 \cdot \ell - \frac{n \cdot \ell}{\frac{Om}{U* - U}}\right)}{Om}}\\ \mathbf{elif}\;U \leq 3.1 \cdot 10^{-248}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;U \leq 31.5:\\ \;\;\;\;\sqrt{t_2 - 2 \cdot \frac{t_1 \cdot \left(\frac{n \cdot \left(\ell \cdot \left(U - U*\right)\right)}{Om} + 2 \cdot \ell\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 9
Error	31.9
Cost	14040

\[\begin{array}{l} t_1 := \mathsf{fma}\left(-2, \frac{\ell}{\frac{Om}{\ell}}, t\right)\\ t_2 := 2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\\ t_3 := \sqrt{2 \cdot U} \cdot \sqrt{n \cdot t}\\ t_4 := n \cdot \left(U \cdot \ell\right)\\ \mathbf{if}\;U \leq -1 \cdot 10^{-45}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot t_1}\\ \mathbf{elif}\;U \leq -1.6 \cdot 10^{-92}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t}\\ \mathbf{elif}\;U \leq -3.5 \cdot 10^{-145}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t_1\right)}\\ \mathbf{elif}\;U \leq 2.2 \cdot 10^{-300}:\\ \;\;\;\;\sqrt{t_2 + -2 \cdot \frac{t_4 \cdot \left(2 \cdot \ell - \frac{n \cdot \ell}{\frac{Om}{U* - U}}\right)}{Om}}\\ \mathbf{elif}\;U \leq 3.1 \cdot 10^{-248}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;U \leq 31.5:\\ \;\;\;\;\sqrt{t_2 - 2 \cdot \frac{t_4 \cdot \left(\frac{n \cdot \left(\ell \cdot \left(U - U*\right)\right)}{Om} + 2 \cdot \ell\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 10
Error	30.7
Cost	13776

\[\begin{array}{l} t_1 := n \cdot \left(U \cdot \ell\right)\\ t_2 := 2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\\ t_3 := \sqrt{2 \cdot U} \cdot \sqrt{n \cdot t}\\ \mathbf{if}\;U \leq -1.02 \cdot 10^{-45}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\ell \cdot \frac{-2}{Om}\right)\right)\right)}\\ \mathbf{elif}\;U \leq 2.2 \cdot 10^{-300}:\\ \;\;\;\;\sqrt{t_2 + -2 \cdot \frac{t_1 \cdot \left(2 \cdot \ell - \frac{n \cdot \ell}{\frac{Om}{U* - U}}\right)}{Om}}\\ \mathbf{elif}\;U \leq 3.1 \cdot 10^{-248}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;U \leq 31.5:\\ \;\;\;\;\sqrt{t_2 - 2 \cdot \frac{t_1 \cdot \left(\frac{n \cdot \left(\ell \cdot \left(U - U*\right)\right)}{Om} + 2 \cdot \ell\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 11
Error	32.1
Cost	9176

\[\begin{array}{l} t_1 := n \cdot \left(U \cdot \ell\right)\\ t_2 := 2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\\ t_3 := \sqrt{t_2 + -2 \cdot \frac{t_1 \cdot \left(2 \cdot \ell - \frac{n \cdot \ell}{\frac{Om}{U* - U}}\right)}{Om}}\\ t_4 := 2 \cdot \ell - \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+175}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{+66}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -1.8 \cdot 10^{-150}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \frac{\ell \cdot t_4}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -1.1 \cdot 10^{-183}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 10^{-255}:\\ \;\;\;\;\sqrt{t_2 + -2 \cdot \frac{t_1 \cdot t_4}{Om}}\\ \mathbf{elif}\;\ell \leq 9.5 \cdot 10^{-49}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \frac{U*}{Om} \cdot \frac{n \cdot \left(\ell \cdot \ell\right)}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 12
Error	33.4
Cost	8788

\[\begin{array}{l} t_1 := \frac{U \cdot \ell}{\frac{Om}{n}}\\ t_2 := \sqrt{-2 \cdot \left(t_1 \cdot \left(t_1 + 2 \cdot \ell\right)\right)}\\ t_3 := \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \frac{\ell \cdot \left(2 \cdot \ell - \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -1.18 \cdot 10^{+172}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 10^{-300}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\ell \leq 1.42 \cdot 10^{-145}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\ell \cdot \frac{-2}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 5.6 \cdot 10^{-63}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{+142}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + \left(\frac{\left(\ell \cdot \ell\right) \cdot \left(n \cdot U*\right)}{Om \cdot Om} + \frac{-2}{\frac{Om}{\ell \cdot \ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 13
Error	33.6
Cost	8656

\[\begin{array}{l} t_1 := \frac{U \cdot \ell}{\frac{Om}{n}}\\ t_2 := \sqrt{-2 \cdot \left(t_1 \cdot \left(t_1 + 2 \cdot \ell\right)\right)}\\ \mathbf{if}\;\ell \leq -1.18 \cdot 10^{+172}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 10^{-255}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \frac{\ell \cdot \left(2 \cdot \ell - \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{-64}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \frac{U*}{Om} \cdot \frac{n \cdot \left(\ell \cdot \ell\right)}{Om}\right)}\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{+142}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + \left(\frac{\left(\ell \cdot \ell\right) \cdot \left(n \cdot U*\right)}{Om \cdot Om} + \frac{-2}{\frac{Om}{\ell \cdot \ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 14
Error	33.3
Cost	8656

\[\begin{array}{l} t_1 := n \cdot \left(U \cdot \ell\right)\\ t_2 := \frac{U \cdot \ell}{\frac{Om}{n}}\\ \mathbf{if}\;\ell \leq -1.18 \cdot 10^{+172}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) - 2 \cdot \frac{t_1 \cdot \left(\frac{t_1}{Om} + 2 \cdot \ell\right)}{Om}}\\ \mathbf{elif}\;\ell \leq 10^{-255}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \frac{\ell \cdot \left(2 \cdot \ell - \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{-64}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \frac{U*}{Om} \cdot \frac{n \cdot \left(\ell \cdot \ell\right)}{Om}\right)}\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{+142}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + \left(\frac{\left(\ell \cdot \ell\right) \cdot \left(n \cdot U*\right)}{Om \cdot Om} + \frac{-2}{\frac{Om}{\ell \cdot \ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(t_2 \cdot \left(t_2 + 2 \cdot \ell\right)\right)}\\ \end{array} \]

Alternative 15
Error	37.2
Cost	8536

\[\begin{array}{l} t_1 := \frac{U \cdot \ell}{\frac{Om}{n}}\\ t_2 := \sqrt{-2 \cdot \left(t_1 \cdot \left(t_1 + 2 \cdot \ell\right)\right)}\\ \mathbf{if}\;\ell \leq -1.26 \cdot 10^{+213}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -1.18 \cdot 10^{+16}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\ell \cdot \frac{-2}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 10^{-300}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{-73}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;\ell \leq 3 \cdot 10^{+104}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{n \cdot \left(\ell \cdot \ell\right)}{\frac{\frac{Om}{U}}{2 + \left(U - U*\right) \cdot \frac{n}{Om}}}}\\ \mathbf{elif}\;\ell \leq 8.2 \cdot 10^{+155}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 16
Error	34.7
Cost	8400

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \frac{\ell \cdot \left(2 \cdot \ell - \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)\right)}\\ t_2 := \frac{U \cdot \ell}{\frac{Om}{n}}\\ \mathbf{if}\;\ell \leq -1.18 \cdot 10^{+172}:\\ \;\;\;\;\sqrt{-2 \cdot \left(t_2 \cdot \left(t_2 + 2 \cdot \ell\right)\right)}\\ \mathbf{elif}\;\ell \leq 10^{-300}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.42 \cdot 10^{-145}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\ell \cdot \frac{-2}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{-8}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \frac{n \cdot \left(\left(U \cdot \ell\right) \cdot \left(\ell \cdot -2 + \frac{\ell \cdot \left(n \cdot U*\right)}{Om}\right)\right)}{Om}}\\ \end{array} \]

Alternative 17
Error	37.9
Cost	8272

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.18 \cdot 10^{+16}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\ell \cdot \frac{-2}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 10^{-300}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{-73}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;\ell \leq 3 \cdot 10^{+104}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{n \cdot \left(\ell \cdot \ell\right)}{\frac{\frac{Om}{U}}{2 + \left(U - U*\right) \cdot \frac{n}{Om}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \frac{\ell \cdot \left(U \cdot \ell\right)}{\frac{Om}{n}}\right)}\\ \end{array} \]

Alternative 18
Error	36.1
Cost	8272

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\ell \cdot \frac{-2}{Om}\right)\right)\right)}\\ t_2 := \frac{U \cdot \ell}{\frac{Om}{n}}\\ \mathbf{if}\;\ell \leq -1.26 \cdot 10^{+213}:\\ \;\;\;\;\sqrt{-2 \cdot \left(t_2 \cdot \left(t_2 + 2 \cdot \ell\right)\right)}\\ \mathbf{elif}\;\ell \leq -1.18 \cdot 10^{+16}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 10^{-300}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{+75}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \frac{n \cdot \left(\left(U \cdot \ell\right) \cdot \left(\ell \cdot -2 + \frac{\ell \cdot \left(n \cdot U*\right)}{Om}\right)\right)}{Om}}\\ \end{array} \]

Alternative 19
Error	36.9
Cost	7888

\[\begin{array}{l} t_1 := \sqrt{-2 \cdot \left(\left(n \cdot U\right) \cdot \left(\frac{2}{\frac{Om}{\ell \cdot \ell}} - t\right)\right)}\\ t_2 := \sqrt{2 \cdot \left(-2 \cdot \frac{\ell \cdot \left(U \cdot \ell\right)}{\frac{Om}{n}}\right)}\\ \mathbf{if}\;\ell \leq -1.18 \cdot 10^{+172}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -1.18 \cdot 10^{+16}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 10^{-300}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{+142}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 20
Error	40.1
Cost	7760

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(-2 \cdot \frac{\ell \cdot \left(U \cdot \ell\right)}{\frac{Om}{n}}\right)}\\ \mathbf{if}\;\ell \leq -1.9 \cdot 10^{+125}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 10^{-300}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{-73}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;\ell \leq 9.8 \cdot 10^{-49}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 21
Error	35.8
Cost	7624

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\ell \cdot \frac{-2}{Om}\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -1.18 \cdot 10^{+16}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 10^{-300}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 22
Error	40.8
Cost	7112

\[\begin{array}{l} \mathbf{if}\;\ell \leq 10^{-300}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{-73}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]

Alternative 23
Error	40.5
Cost	6980

\[\begin{array}{l} \mathbf{if}\;U \leq 4 \cdot 10^{-119}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]

Alternative 24
Error	40.7
Cost	6848

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

Reproduce

herbie shell --seed 2022306 
(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*))))))