Toniolo and Linder, Equation (13)

Average Error: 34.3 → 27.1

Time: 32.7s

Precision: binary64

Cost: 27400

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \leq -4 \cdot 10^{+17}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\ell, -2, n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right) \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U \cdot 2\right)\right)\right) + 2 \cdot \left(\left(n \cdot U\right) \cdot t\right)}\\ \mathbf{elif}\;n \leq 3.9 \cdot 10^{-117}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) - -2 \cdot \frac{\left(\frac{\ell \cdot \left(n \cdot U*\right)}{Om} + \ell \cdot -2\right) \cdot \left(n \cdot \left(\ell \cdot U\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n -4e+17)
   (sqrt
    (+
     (* (fma l -2.0 (* n (* (/ l Om) (- U* U)))) (* (/ l Om) (* n (* U 2.0))))
     (* 2.0 (* (* n U) t))))
   (if (<= n 3.9e-117)
     (sqrt
      (-
       (* 2.0 (* n (* U t)))
       (* -2.0 (/ (* (+ (/ (* l (* n U*)) Om) (* l -2.0)) (* n (* l U))) Om))))
     (*
      (sqrt (* n 2.0))
      (sqrt
       (*
        U
        (-
         t
         (fma 2.0 (* l (/ l Om)) (* n (* (pow (/ l Om) 2.0) (- U U*)))))))))))

C

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

↓

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= -4e+17) {
		tmp = sqrt(((fma(l, -2.0, (n * ((l / Om) * (U_42_ - U)))) * ((l / Om) * (n * (U * 2.0)))) + (2.0 * ((n * U) * t))));
	} else if (n <= 3.9e-117) {
		tmp = sqrt(((2.0 * (n * (U * t))) - (-2.0 * (((((l * (n * U_42_)) / Om) + (l * -2.0)) * (n * (l * U))) / Om))));
	} else {
		tmp = sqrt((n * 2.0)) * sqrt((U * (t - fma(2.0, (l * (l / Om)), (n * (pow((l / Om), 2.0) * (U - U_42_)))))));
	}
	return tmp;
}

Julia

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

↓

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= -4e+17)
		tmp = sqrt(Float64(Float64(fma(l, -2.0, Float64(n * Float64(Float64(l / Om) * Float64(U_42_ - U)))) * Float64(Float64(l / Om) * Float64(n * Float64(U * 2.0)))) + Float64(2.0 * Float64(Float64(n * U) * t))));
	elseif (n <= 3.9e-117)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * Float64(U * t))) - Float64(-2.0 * Float64(Float64(Float64(Float64(Float64(l * Float64(n * U_42_)) / Om) + Float64(l * -2.0)) * Float64(n * Float64(l * U))) / Om))));
	else
		tmp = Float64(sqrt(Float64(n * 2.0)) * sqrt(Float64(U * Float64(t - fma(2.0, Float64(l * Float64(l / Om)), Float64(n * Float64((Float64(l / Om) ^ 2.0) * Float64(U - U_42_))))))));
	end
	return tmp
end

Wolfram

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

↓

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, -4e+17], N[Sqrt[N[(N[(N[(l * -2.0 + N[(n * N[(N[(l / Om), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / Om), $MachinePrecision] * N[(n * N[(U * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[(n * U), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 3.9e-117], N[Sqrt[N[(N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-2.0 * N[(N[(N[(N[(N[(l * N[(n * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision] * N[(n * N[(l * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + N[(n * N[(N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -4e17

   1. Initial program 34.7

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

   2. Simplified 31.2

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, n \cdot \frac{\ell \cdot \left(U* - U\right)}{Om}\right)\right)}} \]
      Proof
      (sqrt.f64 (*.f64 (*.f64 2 (*.f64 n U)) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l -2 (*.f64 n (/.f64 (*.f64 l (-.f64 U* U)) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 2 n) U)) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l -2 (*.f64 n (/.f64 (*.f64 l (-.f64 U* U)) Om))))))): 0 points increase in error, 2 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (Rewrite<= metadata-eval (neg.f64 2)) (*.f64 n (/.f64 (*.f64 l (-.f64 U* U)) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (Rewrite=> sub-neg_binary64 (+.f64 U* (neg.f64 U)))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (+.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 U*))) (neg.f64 U))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 (neg.f64 U*) U)))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 U (neg.f64 U*))))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 U U*)))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 l Om) (neg.f64 (-.f64 U U*)))))))))): 0 points increase in error, 8 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (Rewrite=> distribute-rgt-neg-out_binary64 (neg.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)))))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 n (*.f64 (/.f64 l Om) (-.f64 U U*)))))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 l (neg.f64 2)) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 (neg.f64 2) l)) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (Rewrite<= distribute-lft-out--_binary64 (-.f64 (*.f64 (/.f64 l Om) (*.f64 (neg.f64 2) l)) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n))))))): 0 points increase in error, 1 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 l (*.f64 (neg.f64 2) l)) Om)) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 11 points increase in error, 1 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (neg.f64 2) l) l)) Om) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (neg.f64 2) (*.f64 l l))) Om) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (Rewrite<= associate-*r/_binary64 (*.f64 (neg.f64 2) (/.f64 (*.f64 l l) Om))) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om)))) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om))) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 l Om) (*.f64 (/.f64 l Om) (-.f64 U U*))) n)))))): 4 points increase in error, 1 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 l Om) (/.f64 l Om)) (-.f64 U U*))) n))))): 9 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)) n))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om))) (Rewrite<= *-commutative_binary64 (*.f64 n (*.f64 (pow.f64 (/.f64 l Om) 2) (-.f64 U U*)))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om))) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))))): 2 points increase in error, 6 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 t (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om)))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (Rewrite<= sub-neg_binary64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om)))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 29.3

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

   if -4e17 < n < 3.89999999999999992e-117

   1. Initial program 35.3

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

   2. Simplified 34.9

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, n \cdot \frac{\ell \cdot \left(U* - U\right)}{Om}\right)\right)}} \]
      Proof
      (sqrt.f64 (*.f64 (*.f64 2 (*.f64 n U)) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l -2 (*.f64 n (/.f64 (*.f64 l (-.f64 U* U)) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 2 n) U)) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l -2 (*.f64 n (/.f64 (*.f64 l (-.f64 U* U)) Om))))))): 0 points increase in error, 2 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (Rewrite<= metadata-eval (neg.f64 2)) (*.f64 n (/.f64 (*.f64 l (-.f64 U* U)) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (Rewrite=> sub-neg_binary64 (+.f64 U* (neg.f64 U)))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (+.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 U*))) (neg.f64 U))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 (neg.f64 U*) U)))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 U (neg.f64 U*))))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (/.f64 (*.f64 l (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 U U*)))) Om))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 l Om) (neg.f64 (-.f64 U U*)))))))))): 0 points increase in error, 8 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (*.f64 n (Rewrite=> distribute-rgt-neg-out_binary64 (neg.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)))))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 n (*.f64 (/.f64 l Om) (-.f64 U U*)))))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (fma.f64 l (neg.f64 2) (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 l (neg.f64 2)) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (*.f64 (/.f64 l Om) (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 (neg.f64 2) l)) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (Rewrite<= distribute-lft-out--_binary64 (-.f64 (*.f64 (/.f64 l Om) (*.f64 (neg.f64 2) l)) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n))))))): 0 points increase in error, 1 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 l (*.f64 (neg.f64 2) l)) Om)) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 11 points increase in error, 1 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (neg.f64 2) l) l)) Om) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (neg.f64 2) (*.f64 l l))) Om) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (Rewrite<= associate-*r/_binary64 (*.f64 (neg.f64 2) (/.f64 (*.f64 l l) Om))) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om)))) (*.f64 (/.f64 l Om) (*.f64 (*.f64 (/.f64 l Om) (-.f64 U U*)) n)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om))) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 l Om) (*.f64 (/.f64 l Om) (-.f64 U U*))) n)))))): 4 points increase in error, 1 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 l Om) (/.f64 l Om)) (-.f64 U U*))) n))))): 9 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)) n))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om))) (Rewrite<= *-commutative_binary64 (*.f64 n (*.f64 (pow.f64 (/.f64 l Om) 2) (-.f64 U U*)))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (+.f64 t (-.f64 (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om))) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))))): 2 points increase in error, 6 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 t (neg.f64 (*.f64 2 (/.f64 (*.f64 l l) Om)))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (Rewrite<= sub-neg_binary64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om)))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in t around inf 30.5

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

   4. Taylor expanded in U* around inf 30.7

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

   5. Simplified 28.3

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right)\right) + 2 \cdot \frac{\left(\frac{\color{blue}{\ell \cdot \left(n \cdot U*\right)}}{Om} + -2 \cdot \ell\right) \cdot \left(n \cdot \left(\ell \cdot U\right)\right)}{Om}} \]
      Proof
      (*.f64 l (*.f64 n U*)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 l n) U*)): 40 points increase in error, 61 points decrease in error
      (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 n l)) U*): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r*_binary64 (*.f64 n (*.f64 l U*))): 55 points increase in error, 38 points decrease in error

   if 3.89999999999999992e-117 < n

   1. Initial program 32.4

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

   2. Applied egg-rr 23.7

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

4. Final simplification 27.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4 \cdot 10^{+17}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\ell, -2, n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right) \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U \cdot 2\right)\right)\right) + 2 \cdot \left(\left(n \cdot U\right) \cdot t\right)}\\ \mathbf{elif}\;n \leq 3.9 \cdot 10^{-117}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) - -2 \cdot \frac{\left(\frac{\ell \cdot \left(n \cdot U*\right)}{Om} + \ell \cdot -2\right) \cdot \left(n \cdot \left(\ell \cdot U\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	26.1
Cost	30408

\[\begin{array}{l} t_1 := \left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(U* - U\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\\ \mathbf{if}\;t_1 \leq 2 \cdot 10^{-304}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\left(n \cdot \ell\right) \cdot \frac{U*}{Om} + \ell \cdot -2\right)}{Om}\right)\right)\right)}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, n \cdot \left(\frac{\ell}{Om} \cdot U*\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) - -2 \cdot \frac{\left(\frac{\ell \cdot \left(n \cdot U*\right)}{Om} + \ell \cdot -2\right) \cdot \left(n \cdot \left(\ell \cdot U\right)\right)}{Om}}\\ \end{array} \]

Alternative 2
Error	31.4
Cost	14556

\[\begin{array}{l} t_1 := n \cdot \left(\ell \cdot U\right)\\ t_2 := \frac{n \cdot \left(\ell \cdot \left(U - U*\right)\right)}{Om} + \ell \cdot 2\\ t_3 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \mathsf{fma}\left(-2, \frac{\ell}{\frac{Om}{\ell}}, t\right)\right)}\\ t_4 := 2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\\ t_5 := \sqrt{t_4 - -2 \cdot \frac{\left(\frac{\ell \cdot \left(n \cdot U*\right)}{Om} + \ell \cdot -2\right) \cdot t_1}{Om}}\\ \mathbf{if}\;Om \leq -3.4 \cdot 10^{+72}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;Om \leq -1:\\ \;\;\;\;\sqrt{t_4 + -2 \cdot \frac{t_1 \cdot t_2}{Om}}\\ \mathbf{elif}\;Om \leq -2.9 \cdot 10^{-32}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;Om \leq -1.12 \cdot 10^{-182}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\frac{\ell \cdot \left(\ell \cdot 2 - \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om} - t\right)\right)\right)}\\ \mathbf{elif}\;Om \leq 1.5 \cdot 10^{-225}:\\ \;\;\;\;\sqrt{t_4 + -2 \cdot \frac{n \cdot \left(\left(\ell \cdot U\right) \cdot t_2\right)}{Om}}\\ \mathbf{elif}\;Om \leq 10^{-139}:\\ \;\;\;\;\sqrt{U \cdot \left(U* - U\right)} \cdot \frac{-\sqrt{2}}{\frac{\frac{Om}{\ell}}{n}}\\ \mathbf{elif}\;Om \leq 5 \cdot 10^{+138}:\\ \;\;\;\;t_5\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 3
Error	31.4
Cost	14556

\[\begin{array}{l} t_1 := n \cdot \left(\ell \cdot U\right)\\ t_2 := \frac{n \cdot \left(\ell \cdot \left(U - U*\right)\right)}{Om} + \ell \cdot 2\\ t_3 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \mathsf{fma}\left(-2, \frac{\ell}{\frac{Om}{\ell}}, t\right)\right)}\\ t_4 := 2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\\ t_5 := \sqrt{t_4 - -2 \cdot \frac{\left(\frac{\ell \cdot \left(n \cdot U*\right)}{Om} + \ell \cdot -2\right) \cdot t_1}{Om}}\\ \mathbf{if}\;Om \leq -3.4 \cdot 10^{+72}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;Om \leq -1:\\ \;\;\;\;\sqrt{t_4 + -2 \cdot \frac{t_1 \cdot t_2}{Om}}\\ \mathbf{elif}\;Om \leq -2.9 \cdot 10^{-32}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;Om \leq -1.12 \cdot 10^{-182}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\frac{\ell \cdot \left(\ell \cdot 2 - \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om} - t\right)\right)\right)}\\ \mathbf{elif}\;Om \leq 1.5 \cdot 10^{-225}:\\ \;\;\;\;\sqrt{t_4 + -2 \cdot \frac{n \cdot \left(\left(\ell \cdot U\right) \cdot t_2\right)}{Om}}\\ \mathbf{elif}\;Om \leq 10^{-139}:\\ \;\;\;\;\sqrt{U \cdot \left(U* - U\right)} \cdot \frac{-\sqrt{2}}{\frac{Om}{n \cdot \ell}}\\ \mathbf{elif}\;Om \leq 5 \cdot 10^{+138}:\\ \;\;\;\;t_5\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 4
Error	30.3
Cost	14284

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.02 \cdot 10^{+114}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-4, \ell \cdot \frac{\ell \cdot \left(n \cdot U\right)}{Om}, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 10^{-180}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + -2 \cdot \frac{n \cdot \left(\left(\ell \cdot U\right) \cdot \left(\frac{n \cdot \left(\ell \cdot \left(U - U*\right)\right)}{Om} + \ell \cdot 2\right)\right)}{Om}}\\ \mathbf{elif}\;\ell \leq 1.9 \cdot 10^{+142}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\left(n \cdot \ell\right) \cdot \frac{U*}{Om} + \ell \cdot -2\right)}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{Om} \cdot \left(U \cdot \left(-2 + \frac{n \cdot U*}{Om}\right)\right)}\right)\\ \end{array} \]

Alternative 5
Error	30.2
Cost	13900

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.65 \cdot 10^{-58}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) - -2 \cdot \frac{\left(\frac{\ell \cdot \left(n \cdot U*\right)}{Om} + \ell \cdot -2\right) \cdot \left(n \cdot \left(\ell \cdot U\right)\right)}{Om}}\\ \mathbf{elif}\;\ell \leq 10^{-90}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\frac{\ell \cdot \left(\ell \cdot 2 - \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om} - t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.75 \cdot 10^{+144}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{2 - U* \cdot \frac{n}{Om}}{Om} - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{Om} \cdot \left(-2 \cdot U\right)}\right)\\ \end{array} \]

Alternative 6
Error	31.3
Cost	8520

\[\begin{array}{l} \mathbf{if}\;n \leq -5.6 \cdot 10^{+43}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\left(n \cdot \ell\right) \cdot \frac{U*}{Om} + \ell \cdot -2\right)}{Om}\right)\right)\right)}\\ \mathbf{elif}\;n \leq 100:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) - -2 \cdot \frac{\left(\frac{\ell \cdot \left(n \cdot U*\right)}{Om} + \ell \cdot -2\right) \cdot \left(n \cdot \left(\ell \cdot U\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{2 - U* \cdot \frac{n}{Om}}{Om} - t\right)\right)\right)}\\ \end{array} \]

Alternative 7
Error	40.6
Cost	8276

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{n}{Om \cdot Om} \cdot \left(\ell \cdot \left(\ell \cdot U*\right)\right)\right)\right)\right)}\\ \mathbf{if}\;t \leq -1 \cdot 10^{-91}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\frac{\ell \cdot \left(\ell \cdot 2\right)}{Om} - t\right)\right)\right)}\\ \mathbf{elif}\;t \leq -2.3 \cdot 10^{-292}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{n \cdot \left(\ell \cdot \ell\right)}{\frac{\frac{Om}{U}}{2 + \left(U - U*\right) \cdot \frac{n}{Om}}}}\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{-245}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{-76}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{n \cdot \left(\left(\ell \cdot U\right) \cdot \left(\ell \cdot 2 - \frac{\ell \cdot \left(n \cdot U*\right)}{Om}\right)\right)}{Om}}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{+105}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{+227}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(t + t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(t + t\right)\right)}\\ \end{array} \]

Alternative 8
Error	33.6
Cost	8268

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{+165}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{n \cdot \left(\left(\ell \cdot U\right) \cdot \left(\ell \cdot 2 - \frac{\ell \cdot \left(n \cdot U*\right)}{Om}\right)\right)}{Om}}\\ \mathbf{elif}\;\ell \leq -9 \cdot 10^{+31}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(n \cdot U\right) \cdot \left(\frac{2}{\frac{Om}{\ell \cdot \ell}} - t\right)\right)}\\ \mathbf{elif}\;\ell \leq 10^{-90}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\frac{\ell \cdot \left(\ell \cdot 2 - \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om} - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{2 - U* \cdot \frac{n}{Om}}{Om} - t\right)\right)\right)}\\ \end{array} \]

Alternative 9
Error	34.6
Cost	8140

\[\begin{array}{l} t_1 := \sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{2 - U* \cdot \frac{n}{Om}}{Om} - t\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+165}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{n \cdot \left(\left(\ell \cdot U\right) \cdot \left(\ell \cdot 2 - \frac{\ell \cdot \left(n \cdot U*\right)}{Om}\right)\right)}{Om}}\\ \mathbf{elif}\;\ell \leq -8.4 \cdot 10^{-191}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 10^{-176}:\\ \;\;\;\;{\left(\left(n \cdot U\right) \cdot \left(t + t\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	33.3
Cost	8136

\[\begin{array}{l} \mathbf{if}\;\ell \leq -9 \cdot 10^{+31}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(U \cdot \frac{n}{\frac{Om}{\ell}} + \ell \cdot 2\right)\right)}\\ \mathbf{elif}\;\ell \leq 10^{-90}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\frac{\ell \cdot \left(\ell \cdot 2 - \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om} - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{2 - U* \cdot \frac{n}{Om}}{Om} - t\right)\right)\right)}\\ \end{array} \]

Alternative 11
Error	36.4
Cost	8012

\[\begin{array}{l} t_1 := \sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\frac{\ell \cdot \left(\ell \cdot 2\right)}{Om} - t\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -4.4 \cdot 10^{+163}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{\left(n \cdot 2\right) \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)}{Om}}\\ \mathbf{elif}\;\ell \leq 4.5 \cdot 10^{-138}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.72 \cdot 10^{+22}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{n}{Om \cdot Om} \cdot \left(\ell \cdot \left(\ell \cdot U*\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	36.4
Cost	8012

\[\begin{array}{l} t_1 := \sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\frac{\ell \cdot \left(\ell \cdot 2\right)}{Om} - t\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+165}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{n \cdot \left(\left(\ell \cdot U\right) \cdot \left(\ell \cdot 2 - \frac{\ell \cdot \left(n \cdot U*\right)}{Om}\right)\right)}{Om}}\\ \mathbf{elif}\;\ell \leq 4.5 \cdot 10^{-138}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.72 \cdot 10^{+22}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{n}{Om \cdot Om} \cdot \left(\ell \cdot \left(\ell \cdot U*\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	39.9
Cost	8012

\[\begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-91}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\frac{\ell \cdot \left(\ell \cdot 2\right)}{Om} - t\right)\right)\right)}\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{-76}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\ell \cdot 2 - \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)\right)\right)}{Om}}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{+105}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{n}{Om \cdot Om} \cdot \left(\ell \cdot \left(\ell \cdot U*\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{+227}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(t + t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(t + t\right)\right)}\\ \end{array} \]

Alternative 14
Error	37.2
Cost	7492

\[\begin{array}{l} \mathbf{if}\;U \leq -2.45 \cdot 10^{+55}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\frac{\ell \cdot \left(\ell \cdot 2\right)}{Om} - t\right)\right)\right)}\\ \end{array} \]

Alternative 15
Error	39.4
Cost	7044

\[\begin{array}{l} \mathbf{if}\;n \leq 10^{+65}:\\ \;\;\;\;{\left(\left(n \cdot t\right) \cdot \left(U \cdot 2\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(t + t\right)\right)}\\ \end{array} \]

Alternative 16
Error	39.4
Cost	6980

\[\begin{array}{l} \mathbf{if}\;n \leq 10^{+65}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(t + t\right)\right)}\\ \end{array} \]

Alternative 17
Error	61.6
Cost	6848

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

Alternative 18
Error	40.3
Cost	6848

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

Reproduce

herbie shell --seed 2022325 
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  :precision binary64
  (sqrt (* (* (* 2.0 n) U) (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))