Toniolo and Linder, Equation (13)

Average Error: 34.3 → 25.4

Time: 53.9s

Precision: binary64

Cost: 49352

Math

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

↓

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

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
          (*
           (* U (* n -2.0))
           (+
            (* (* n (pow (/ l Om) 2.0)) (- U U*))
            (- (* 2.0 (/ (* l l) Om)) t)))))
        (t_2 (/ n (/ Om l))))
   (if (<= t_1 0.0)
     (* (sqrt (* 2.0 n)) (sqrt (* U t)))
     (if (<= t_1 INFINITY)
       (sqrt
        (+
         (* (fma l -2.0 (- (* U* t_2) (* U t_2))) (* (/ l Om) (* n (* 2.0 U))))
         (* 2.0 (* t (* n U)))))
       (*
        (sqrt 2.0)
        (* l (sqrt (* (/ n Om) (* U (fma (/ n Om) (- U* U) -2.0))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
}

↓

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt(((U * (n * -2.0)) * (((n * pow((l / Om), 2.0)) * (U - U_42_)) + ((2.0 * ((l * l) / Om)) - t))));
	double t_2 = n / (Om / l);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = sqrt((2.0 * n)) * sqrt((U * t));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = sqrt(((fma(l, -2.0, ((U_42_ * t_2) - (U * t_2))) * ((l / Om) * (n * (2.0 * U)))) + (2.0 * (t * (n * U)))));
	} else {
		tmp = sqrt(2.0) * (l * sqrt(((n / Om) * (U * fma((n / Om), (U_42_ - U), -2.0)))));
	}
	return tmp;
}

Julia

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

↓

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(Float64(U * Float64(n * -2.0)) * Float64(Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_)) + Float64(Float64(2.0 * Float64(Float64(l * l) / Om)) - t))))
	t_2 = Float64(n / Float64(Om / l))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * t)));
	elseif (t_1 <= Inf)
		tmp = sqrt(Float64(Float64(fma(l, -2.0, Float64(Float64(U_42_ * t_2) - Float64(U * t_2))) * Float64(Float64(l / Om) * Float64(n * Float64(2.0 * U)))) + Float64(2.0 * Float64(t * Float64(n * U)))));
	else
		tmp = Float64(sqrt(2.0) * Float64(l * sqrt(Float64(Float64(n / Om) * Float64(U * fma(Float64(n / Om), Float64(U_42_ - U), -2.0))))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

↓

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(U * N[(n * -2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(n / N[(Om / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[Sqrt[N[(N[(N[(l * -2.0 + N[(N[(U$42$ * t$95$2), $MachinePrecision] - N[(U * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / Om), $MachinePrecision] * N[(n * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(l * N[Sqrt[N[(N[(n / Om), $MachinePrecision] * N[(U * N[(N[(n / Om), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 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*))))) < 0.0

   1. Initial program 56.9

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

   2. Taylor expanded in t around inf 42.5

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

   3. Simplified 42.5

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
      Proof
      (*.f64 2 (*.f64 U (*.f64 n t))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 n t) U))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (Rewrite<= associate-*r*_binary64 (*.f64 n (*.f64 t U)))): 50 points increase in error, 46 points decrease in error

   4. Taylor expanded in U around 0 42.5

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

   5. Applied egg-rr 44.3

      \[\leadsto \color{blue}{\sqrt{2 \cdot n} \cdot \sqrt{t \cdot U}} \]

   if 0.0 < (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*))))) < +inf.0

   1. Initial program 23.8

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

   2. Simplified 23.3

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, n \cdot \frac{\ell \cdot \left(U* - U\right)}{Om}\right)\right)}} \]
      Proof
      (sqrt.f64 (*.f64 (*.f64 2 (*.f64 n U)) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l -2 (*.f64 n (/.f64 (*.f64 l (-.f64 U* U)) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 2 n) U)) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l -2 (*.f64 n (/.f64 (*.f64 l (-.f64 U* U)) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (Rewrite<= metadata-eval (neg.f64 2)) (*.f64 n (/.f64 (*.f64 l (-.f64 U* U)) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (Rewrite=> sub-neg_binary64 (+.f64 U* (neg.f64 U)))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (+.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 U*))) (neg.f64 U))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 (neg.f64 U*) U)))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 U (neg.f64 U*))))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 U U*)))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 l Om) (neg.f64 (-.f64 U U*)))))))))): 1 points increase in error, 14 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))))))): 1 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)))))): 15 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (neg.f64 2) l) l)) Om) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (neg.f64 2) (*.f64 l l))) Om) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (Rewrite<= associate-*r/_binary64 (*.f64 (neg.f64 2) (/.f64 (*.f64 l l) Om))) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om)))) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om))) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 l Om) (*.f64 (/.f64 l Om) (-.f64 U U*))) n)))))): 5 points increase in error, 2 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 l Om) (/.f64 l Om)) (-.f64 U U*))) n))))): 5 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*))))))): 3 points increase in error, 7 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 t (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om)))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (Rewrite<= sub-neg_binary64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om)))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 19.2

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

   4. Applied egg-rr 18.1

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

   if +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 57.6

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, n \cdot \frac{\ell \cdot \left(U* - U\right)}{Om}\right)\right)}} \]
      Proof
      (sqrt.f64 (*.f64 (*.f64 2 (*.f64 n U)) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l -2 (*.f64 n (/.f64 (*.f64 l (-.f64 U* U)) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 2 n) U)) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l -2 (*.f64 n (/.f64 (*.f64 l (-.f64 U* U)) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (Rewrite<= metadata-eval (neg.f64 2)) (*.f64 n (/.f64 (*.f64 l (-.f64 U* U)) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (Rewrite=> sub-neg_binary64 (+.f64 U* (neg.f64 U)))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (+.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 U*))) (neg.f64 U))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 (neg.f64 U*) U)))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 U (neg.f64 U*))))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 U U*)))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 l Om) (neg.f64 (-.f64 U U*)))))))))): 1 points increase in error, 14 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))))))): 1 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)))))): 15 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (neg.f64 2) l) l)) Om) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (neg.f64 2) (*.f64 l l))) Om) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (Rewrite<= associate-*r/_binary64 (*.f64 (neg.f64 2) (/.f64 (*.f64 l l) Om))) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om)))) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om))) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 l Om) (*.f64 (/.f64 l Om) (-.f64 U U*))) n)))))): 5 points increase in error, 2 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 l Om) (/.f64 l Om)) (-.f64 U U*))) n))))): 5 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*))))))): 3 points increase in error, 7 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 t (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om)))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (Rewrite<= sub-neg_binary64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om)))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 51.1

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

   4. Taylor expanded in l around inf 47.4

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

   5. Simplified 42.8

      \[\leadsto \color{blue}{\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{Om} \cdot \left(U \cdot \mathsf{fma}\left(\frac{n}{Om}, U* - U, -2\right)\right)}\right)} \]
      Proof
      (*.f64 (sqrt.f64 2) (*.f64 l (sqrt.f64 (*.f64 (/.f64 n Om) (*.f64 U (fma.f64 (/.f64 n Om) (-.f64 U* U) -2)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 l (sqrt.f64 (*.f64 (/.f64 n Om) (*.f64 U (fma.f64 (/.f64 n Om) (-.f64 U* U) (Rewrite<= metadata-eval (neg.f64 2)))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 l (sqrt.f64 (*.f64 (/.f64 n Om) (*.f64 U (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 (/.f64 n Om) (-.f64 U* U)) 2))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 l (sqrt.f64 (*.f64 (/.f64 n Om) (*.f64 U (-.f64 (Rewrite<= associate-/r/_binary64 (/.f64 n (/.f64 Om (-.f64 U* U)))) 2)))))): 12 points increase in error, 1 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 l (sqrt.f64 (*.f64 (/.f64 n Om) (*.f64 U (-.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 n (-.f64 U* U)) Om)) 2)))))): 5 points increase in error, 12 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 l (sqrt.f64 (*.f64 (/.f64 n Om) (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 (/.f64 (*.f64 n (-.f64 U* U)) Om) 2) U)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 l (sqrt.f64 (Rewrite<= associate-/r/_binary64 (/.f64 n (/.f64 Om (*.f64 (-.f64 (/.f64 (*.f64 n (-.f64 U* U)) Om) 2) U))))))): 26 points increase in error, 14 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 l (sqrt.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 n (*.f64 (-.f64 (/.f64 (*.f64 n (-.f64 U* U)) Om) 2) U)) Om))))): 14 points increase in error, 14 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (sqrt.f64 2) l) (sqrt.f64 (/.f64 (*.f64 n (*.f64 (-.f64 (/.f64 (*.f64 n (-.f64 U* U)) Om) 2) U)) Om)))): 8 points increase in error, 17 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 25.4

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

Alternatives

Alternative 1
Error	26.7
Cost	44104

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

Alternative 2
Error	26.5
Cost	44104

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

Alternative 3
Error	27.4
Cost	43720

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

Alternative 4
Error	28.1
Cost	43528

\[\begin{array}{l} t_1 := n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := \sqrt{\left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(t_1 \cdot \left(U - U*\right) + \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right)}\\ \mathbf{if}\;t_2 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t}\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+148}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) + t_1 \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(-2 + \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)}{Om}}\\ \end{array} \]

Alternative 5
Error	28.5
Cost	14808

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + \frac{n}{Om} \cdot \left(\left(\ell \cdot \left(U \cdot \ell\right)\right) \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -9 \cdot 10^{+114}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), \ell \cdot \frac{n \cdot -4}{\frac{Om}{U \cdot \ell}}\right)}\\ \mathbf{elif}\;\ell \leq -9.5 \cdot 10^{-35}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(\frac{n \cdot \left(\ell \cdot \ell\right)}{Om} \cdot \frac{U - U*}{Om} + \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right)}\\ \mathbf{elif}\;\ell \leq -9 \cdot 10^{-172}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{-108}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\frac{\ell \cdot \left(2 \cdot \ell - \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om} - t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+63}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.75 \cdot 10^{+137}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, t \cdot \left(n \cdot U\right), \frac{-4}{\frac{Om}{U \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(-2 + \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)}{Om}}\\ \end{array} \]

Alternative 6
Error	31.2
Cost	14808

\[\begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(U \cdot \frac{n}{\frac{Om}{\ell}} + 2 \cdot \ell\right)\right)}\\ t_2 := \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \frac{\left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om}\right) \cdot \left(n \cdot \left(U \cdot \ell\right)\right)}{Om}}\\ \mathbf{if}\;Om \leq -1.45 \cdot 10^{+76}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;Om \leq -1 \cdot 10^{-160}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;Om \leq -1.1 \cdot 10^{-286}:\\ \;\;\;\;\left(\frac{\ell}{Om} \cdot \sqrt{n \cdot \left(U* - U\right)}\right) \cdot \sqrt{n \cdot \left(2 \cdot U\right)}\\ \mathbf{elif}\;Om \leq 1.65 \cdot 10^{+59}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;Om \leq 7.5 \cdot 10^{+210}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;Om \leq 7.8 \cdot 10^{+295}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{\ell}{\frac{Om}{\ell}} \cdot \mathsf{fma}\left(\frac{n}{Om}, U*, -2\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \frac{2}{\frac{Om}{\ell \cdot \ell}}\right)\right)}\\ \end{array} \]

Alternative 7
Error	29.7
Cost	14680

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + \frac{n}{Om} \cdot \left(\left(\ell \cdot \left(U \cdot \ell\right)\right) \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -9 \cdot 10^{+114}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), \ell \cdot \frac{n \cdot -4}{\frac{Om}{U \cdot \ell}}\right)}\\ \mathbf{elif}\;\ell \leq -9.5 \cdot 10^{-35}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(\frac{n \cdot \left(\ell \cdot \ell\right)}{Om} \cdot \frac{U - U*}{Om} + \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right)}\\ \mathbf{elif}\;\ell \leq -9 \cdot 10^{-172}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{-108}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\frac{\ell \cdot \left(2 \cdot \ell - \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om} - t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+63}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.75 \cdot 10^{+137}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, t \cdot \left(n \cdot U\right), \frac{-4}{\frac{Om}{U \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{n \cdot \left(U \cdot \frac{-2}{Om}\right)}\right)\\ \end{array} \]

Alternative 8
Error	29.7
Cost	14672

\[\begin{array}{l} t_1 := \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, n \cdot \left(\frac{\ell}{Om} \cdot U*\right)\right)\right)}\\ \mathbf{if}\;n \leq -1 \cdot 10^{-61}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq -5 \cdot 10^{-216}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + \frac{n}{Om} \cdot \left(\left(\ell \cdot \left(U \cdot \ell\right)\right) \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)\right)}\\ \mathbf{elif}\;n \leq 4 \cdot 10^{-259}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, t \cdot \left(n \cdot U\right), \frac{-4}{\frac{Om}{U \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)}}\right)}\\ \mathbf{elif}\;n \leq 1.45 \cdot 10^{-139}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	31.1
Cost	14296

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \frac{\left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om}\right) \cdot \left(n \cdot \left(U \cdot \ell\right)\right)}{Om}}\\ t_2 := \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + \frac{n}{Om} \cdot \left(\left(\ell \cdot \left(U \cdot \ell\right)\right) \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -3.3 \cdot 10^{+89}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -2.2 \cdot 10^{-162}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -5.6 \cdot 10^{-294}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \frac{2}{\frac{Om}{\ell \cdot \ell}}\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{-108}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 5.1 \cdot 10^{+63}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 1.75 \cdot 10^{+137}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{n \cdot \left(U \cdot \frac{-2}{Om}\right)}\right)\\ \end{array} \]

Alternative 10
Error	31.2
Cost	14156

\[\begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(U \cdot \frac{n}{\frac{Om}{\ell}} + 2 \cdot \ell\right)\right)}\\ t_2 := \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \frac{\left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om}\right) \cdot \left(n \cdot \left(U \cdot \ell\right)\right)}{Om}}\\ \mathbf{if}\;Om \leq -1.45 \cdot 10^{+76}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;Om \leq -1 \cdot 10^{-160}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;Om \leq -1.1 \cdot 10^{-286}:\\ \;\;\;\;\left(\frac{\ell}{Om} \cdot \sqrt{n \cdot \left(U* - U\right)}\right) \cdot \sqrt{n \cdot \left(2 \cdot U\right)}\\ \mathbf{elif}\;Om \leq 1.65 \cdot 10^{+59}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	31.4
Cost	14032

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \frac{\left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om}\right) \cdot \left(n \cdot \left(U \cdot \ell\right)\right)}{Om}}\\ \mathbf{if}\;\ell \leq -1.9 \cdot 10^{+75}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(-2, \frac{\ell}{\frac{Om}{\ell}}, t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{-108}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 5.1 \cdot 10^{+63}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + \frac{n}{Om} \cdot \left(\left(\ell \cdot \left(U \cdot \ell\right)\right) \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.75 \cdot 10^{+137}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{n \cdot \left(U \cdot \frac{-2}{Om}\right)}\right)\\ \end{array} \]

Alternative 12
Error	30.2
Cost	8648

\[\begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(U \cdot \frac{n}{\frac{Om}{\ell}} + 2 \cdot \ell\right)\right)}\\ \mathbf{if}\;Om \leq -1.45 \cdot 10^{+76}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;Om \leq 1.65 \cdot 10^{+59}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \frac{\left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om}\right) \cdot \left(n \cdot \left(U \cdot \ell\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	35.3
Cost	8268

\[\begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(U \cdot \frac{n}{\frac{Om}{\ell}} + 2 \cdot \ell\right)\right)}\\ \mathbf{if}\;Om \leq -1000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;Om \leq 4.4 \cdot 10^{-186}:\\ \;\;\;\;\sqrt{2 \cdot \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)\right)\right)}{Om}}\\ \mathbf{elif}\;Om \leq 6.5 \cdot 10^{+59}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Error	30.5
Cost	8264

\[\begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(U \cdot \frac{n}{\frac{Om}{\ell}} + 2 \cdot \ell\right)\right)}\\ \mathbf{if}\;Om \leq -1.45 \cdot 10^{+76}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;Om \leq 10^{+44}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + \frac{n}{Om} \cdot \left(\left(\ell \cdot \left(U \cdot \ell\right)\right) \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 15
Error	33.7
Cost	8144

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{-112}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{-30}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.75 \cdot 10^{+137}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \frac{2}{\frac{Om}{\ell \cdot \ell}}\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.2 \cdot 10^{+188}:\\ \;\;\;\;\sqrt{2 \cdot \left(\frac{n}{Om} \cdot \left(\left(\ell \cdot \left(U \cdot \ell\right)\right) \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 16
Error	31.3
Cost	8136

\[\begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(U \cdot \frac{n}{\frac{Om}{\ell}} + 2 \cdot \ell\right)\right)}\\ \mathbf{if}\;Om \leq -1.45 \cdot 10^{+76}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;Om \leq 1.65 \cdot 10^{+59}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\frac{\ell \cdot \left(2 \cdot \ell - \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om} - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 17
Error	36.7
Cost	8012

\[\begin{array}{l} \mathbf{if}\;Om \leq -600000000:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \frac{2}{\frac{Om}{\ell \cdot \ell}}\right)\right)}\\ \mathbf{elif}\;Om \leq 4.4 \cdot 10^{-186}:\\ \;\;\;\;\sqrt{2 \cdot \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)\right)\right)}{Om}}\\ \mathbf{elif}\;Om \leq 1.4 \cdot 10^{-78}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \end{array} \]

Alternative 18
Error	35.3
Cost	7880

\[\begin{array}{l} \mathbf{if}\;Om \leq -2.75 \cdot 10^{+18}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \frac{2}{\frac{Om}{\ell \cdot \ell}}\right)\right)}\\ \mathbf{elif}\;Om \leq 1.4 \cdot 10^{-78}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \end{array} \]

Alternative 19
Error	34.4
Cost	7624

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -2.4 \cdot 10^{-148}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 10^{+130}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \frac{2}{\frac{Om}{\ell \cdot \ell}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 20
Error	34.3
Cost	7492

\[\begin{array}{l} \mathbf{if}\;t \leq 6.5 \cdot 10^{+213}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 21
Error	39.0
Cost	7112

\[\begin{array}{l} t_1 := \sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{if}\;U \leq -1 \cdot 10^{-26}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;U \leq 5.5 \cdot 10^{-286}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 22
Error	39.9
Cost	6848

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

Reproduce

herbie shell --seed 2022309 
(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*))))))