Toniolo and Linder, Equation (13)

Average Error: 34.6 → 26.1

Time: 36.9s

Precision: binary64

Cost: 49736

Math

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

↓

\[\begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\frac{n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om} + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)\right)}\\ \mathbf{elif}\;t_1 \leq 10^{+297}:\\ \;\;\;\;\sqrt{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\left|\frac{Om}{n \cdot \ell}\right|} \cdot \sqrt{U \cdot \left(U* - U\right)} - \sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{U}{U* - U}}\right)\\ \end{array} \]

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
         (*
          (* (* 2.0 n) U)
          (+
           (+ t (* (/ (* l l) Om) -2.0))
           (* (* n (pow (/ l Om) 2.0)) (- U* U))))))
   (if (<= t_1 0.0)
     (sqrt
      (*
       2.0
       (*
        n
        (*
         U
         (+
          t
          (+ (* (/ n Om) (/ (* l (* l U*)) Om)) (* -2.0 (/ l (/ Om l)))))))))
     (if (<= t_1 1e+297)
       (sqrt t_1)
       (-
        (* (/ (sqrt 2.0) (fabs (/ Om (* n l)))) (sqrt (* U (- U* U))))
        (* (sqrt 2.0) (* l (sqrt (/ U (- 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 t_1 = ((2.0 * n) * U) * ((t + (((l * l) / Om) * -2.0)) + ((n * pow((l / Om), 2.0)) * (U_42_ - U)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = sqrt((2.0 * (n * (U * (t + (((n / Om) * ((l * (l * U_42_)) / Om)) + (-2.0 * (l / (Om / l)))))))));
	} else if (t_1 <= 1e+297) {
		tmp = sqrt(t_1);
	} else {
		tmp = ((sqrt(2.0) / fabs((Om / (n * l)))) * sqrt((U * (U_42_ - U)))) - (sqrt(2.0) * (l * sqrt((U / (U_42_ - U)))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l * l) / om))) - ((n * ((l / om) ** 2.0d0)) * (u - u_42)))))
end function

↓

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((2.0d0 * n) * u) * ((t + (((l * l) / om) * (-2.0d0))) + ((n * ((l / om) ** 2.0d0)) * (u_42 - u)))
    if (t_1 <= 0.0d0) then
        tmp = sqrt((2.0d0 * (n * (u * (t + (((n / om) * ((l * (l * u_42)) / om)) + ((-2.0d0) * (l / (om / l)))))))))
    else if (t_1 <= 1d+297) then
        tmp = sqrt(t_1)
    else
        tmp = ((sqrt(2.0d0) / abs((om / (n * l)))) * sqrt((u * (u_42 - u)))) - (sqrt(2.0d0) * (l * sqrt((u / (u_42 - u)))))
    end if
    code = tmp
end function

Java

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

↓

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = ((2.0 * n) * U) * ((t + (((l * l) / Om) * -2.0)) + ((n * Math.pow((l / Om), 2.0)) * (U_42_ - U)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = Math.sqrt((2.0 * (n * (U * (t + (((n / Om) * ((l * (l * U_42_)) / Om)) + (-2.0 * (l / (Om / l)))))))));
	} else if (t_1 <= 1e+297) {
		tmp = Math.sqrt(t_1);
	} else {
		tmp = ((Math.sqrt(2.0) / Math.abs((Om / (n * l)))) * Math.sqrt((U * (U_42_ - U)))) - (Math.sqrt(2.0) * (l * Math.sqrt((U / (U_42_ - U)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * math.pow((l / Om), 2.0)) * (U - U_42_)))))

↓

def code(n, U, t, l, Om, U_42_):
	t_1 = ((2.0 * n) * U) * ((t + (((l * l) / Om) * -2.0)) + ((n * math.pow((l / Om), 2.0)) * (U_42_ - U)))
	tmp = 0
	if t_1 <= 0.0:
		tmp = math.sqrt((2.0 * (n * (U * (t + (((n / Om) * ((l * (l * U_42_)) / Om)) + (-2.0 * (l / (Om / l)))))))))
	elif t_1 <= 1e+297:
		tmp = math.sqrt(t_1)
	else:
		tmp = ((math.sqrt(2.0) / math.fabs((Om / (n * l)))) * math.sqrt((U * (U_42_ - U)))) - (math.sqrt(2.0) * (l * math.sqrt((U / (U_42_ - U)))))
	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 = 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 <= 0.0)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * Float64(t + Float64(Float64(Float64(n / Om) * Float64(Float64(l * Float64(l * U_42_)) / Om)) + Float64(-2.0 * Float64(l / Float64(Om / l)))))))));
	elseif (t_1 <= 1e+297)
		tmp = sqrt(t_1);
	else
		tmp = Float64(Float64(Float64(sqrt(2.0) / abs(Float64(Om / Float64(n * l)))) * sqrt(Float64(U * Float64(U_42_ - U)))) - Float64(sqrt(2.0) * Float64(l * sqrt(Float64(U / Float64(U_42_ - U))))));
	end
	return tmp
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * ((l / Om) ^ 2.0)) * (U - U_42_)))));
end

↓

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = ((2.0 * n) * U) * ((t + (((l * l) / Om) * -2.0)) + ((n * ((l / Om) ^ 2.0)) * (U_42_ - U)));
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = sqrt((2.0 * (n * (U * (t + (((n / Om) * ((l * (l * U_42_)) / Om)) + (-2.0 * (l / (Om / l)))))))));
	elseif (t_1 <= 1e+297)
		tmp = sqrt(t_1);
	else
		tmp = ((sqrt(2.0) / abs((Om / (n * l)))) * sqrt((U * (U_42_ - U)))) - (sqrt(2.0) * (l * sqrt((U / (U_42_ - U)))));
	end
	tmp_2 = 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[(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]}, If[LessEqual[t$95$1, 0.0], N[Sqrt[N[(2.0 * N[(n * N[(U * N[(t + N[(N[(N[(n / Om), $MachinePrecision] * N[(N[(l * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 1e+297], N[Sqrt[t$95$1], $MachinePrecision], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[Abs[N[(Om / N[(n * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(U * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[2.0], $MachinePrecision] * N[(l * N[Sqrt[N[(U / N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.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 57.2

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

   2. Simplified 41.2

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)}} \]
      Proof
      (sqrt.f64 (*.f64 (*.f64 2 n) (*.f64 U (-.f64 t (+.f64 (*.f64 2 (/.f64 l (/.f64 Om l))) (*.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 2 n) (*.f64 U (-.f64 t (+.f64 (*.f64 2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) Om))) (*.f64 n (*.f64 (pow.f64 (/.f64 l Om) 2) (-.f64 U U*)))))))): 7 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 2 n) (*.f64 U (-.f64 t (+.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, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 2 n) (*.f64 U (Rewrite=> sub-neg_binary64 (+.f64 t (neg.f64 (+.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, 1 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 2 n) (*.f64 U (Rewrite<= sub-neg_binary64 (-.f64 t (+.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 2 n) (*.f64 U (Rewrite<= associate--l-_binary64 (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))))): 1 points increase in error, 1 points decrease in error
      (sqrt.f64 (Rewrite<= associate-*l*_binary64 (*.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 points increase in error, 0 points decrease in error

   3. Taylor expanded in U around 0 45.6

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

   4. Simplified 53.7

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \frac{\left(n \cdot \ell\right) \cdot \ell}{Om} \cdot \frac{U*}{Om}\right)\right) \cdot \left(n \cdot U\right)\right)}} \]
      Proof
      (sqrt.f64 (*.f64 2 (*.f64 (-.f64 t (-.f64 (*.f64 2 (/.f64 l (/.f64 Om l))) (*.f64 (/.f64 (*.f64 (*.f64 n l) l) Om) (/.f64 U* Om)))) (*.f64 n U)))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 2 (*.f64 (-.f64 t (-.f64 (*.f64 2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) Om))) (*.f64 (/.f64 (*.f64 (*.f64 n l) l) Om) (/.f64 U* Om)))) (*.f64 n U)))): 5 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 2 (*.f64 (-.f64 t (-.f64 (*.f64 2 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) Om)) (*.f64 (/.f64 (*.f64 (*.f64 n l) l) Om) (/.f64 U* Om)))) (*.f64 n U)))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 2 (*.f64 (-.f64 t (-.f64 (*.f64 2 (/.f64 (pow.f64 l 2) Om)) (*.f64 (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 n (*.f64 l l))) Om) (/.f64 U* Om)))) (*.f64 n U)))): 0 points increase in error, 1 points decrease in error
      (sqrt.f64 (*.f64 2 (*.f64 (-.f64 t (-.f64 (*.f64 2 (/.f64 (pow.f64 l 2) Om)) (*.f64 (/.f64 (*.f64 n (Rewrite<= unpow2_binary64 (pow.f64 l 2))) Om) (/.f64 U* Om)))) (*.f64 n U)))): 1 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 2 (*.f64 (-.f64 t (-.f64 (*.f64 2 (/.f64 (pow.f64 l 2) Om)) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (*.f64 n (pow.f64 l 2)) U*) (*.f64 Om Om))))) (*.f64 n U)))): 0 points increase in error, 12 points decrease in error
      (sqrt.f64 (*.f64 2 (*.f64 (-.f64 t (-.f64 (*.f64 2 (/.f64 (pow.f64 l 2) Om)) (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 n (*.f64 (pow.f64 l 2) U*))) (*.f64 Om Om)))) (*.f64 n U)))): 12 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 2 (*.f64 (-.f64 t (-.f64 (*.f64 2 (/.f64 (pow.f64 l 2) Om)) (/.f64 (*.f64 n (*.f64 (pow.f64 l 2) U*)) (Rewrite<= unpow2_binary64 (pow.f64 Om 2))))) (*.f64 n U)))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 2 (*.f64 (-.f64 t (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 2 (/.f64 (pow.f64 l 2) Om)) (neg.f64 (/.f64 (*.f64 n (*.f64 (pow.f64 l 2) U*)) (pow.f64 Om 2)))))) (*.f64 n U)))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 2 (*.f64 (-.f64 t (+.f64 (*.f64 2 (/.f64 (pow.f64 l 2) Om)) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (*.f64 n (*.f64 (pow.f64 l 2) U*)) (pow.f64 Om 2)))))) (*.f64 n U)))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 2 (*.f64 (-.f64 t (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (/.f64 (*.f64 n (*.f64 (pow.f64 l 2) U*)) (pow.f64 Om 2))) (*.f64 2 (/.f64 (pow.f64 l 2) Om))))) (*.f64 n U)))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 2 (*.f64 (-.f64 t (+.f64 (*.f64 -1 (/.f64 (*.f64 n (*.f64 (pow.f64 l 2) U*)) (pow.f64 Om 2))) (*.f64 2 (/.f64 (pow.f64 l 2) Om)))) (Rewrite=> *-commutative_binary64 (*.f64 U n))))): 0 points increase in error, 14 points decrease in error
      (sqrt.f64 (*.f64 2 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (-.f64 t (+.f64 (*.f64 -1 (/.f64 (*.f64 n (*.f64 (pow.f64 l 2) U*)) (pow.f64 Om 2))) (*.f64 2 (/.f64 (pow.f64 l 2) Om)))) U) n)))): 14 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 2 (Rewrite<= *-commutative_binary64 (*.f64 n (*.f64 (-.f64 t (+.f64 (*.f64 -1 (/.f64 (*.f64 n (*.f64 (pow.f64 l 2) U*)) (pow.f64 Om 2))) (*.f64 2 (/.f64 (pow.f64 l 2) Om)))) U))))): 0 points increase in error, 14 points decrease in error

   5. Taylor expanded in U around 0 45.6

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

   6. Simplified 40.0

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(n \cdot \left(U \cdot \left(t + \left(\frac{n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om} + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)\right)}} \]
      Proof
      (sqrt.f64 (*.f64 2 (*.f64 n (*.f64 U (+.f64 t (+.f64 (*.f64 (/.f64 n Om) (/.f64 (*.f64 l (*.f64 l U*)) Om)) (*.f64 -2 (/.f64 l (/.f64 Om l))))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 2 (*.f64 n (*.f64 U (+.f64 t (+.f64 (*.f64 (/.f64 n Om) (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 l l) U*)) Om)) (*.f64 -2 (/.f64 l (/.f64 Om l))))))))): 2 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 2 (*.f64 n (*.f64 U (+.f64 t (+.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 n (*.f64 (*.f64 l l) U*)) (*.f64 Om Om))) (*.f64 -2 (/.f64 l (/.f64 Om l))))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 2 (*.f64 n (*.f64 U (+.f64 t (+.f64 (Rewrite=> associate-/l*_binary64 (/.f64 n (/.f64 (*.f64 Om Om) (*.f64 (*.f64 l l) U*)))) (*.f64 -2 (/.f64 l (/.f64 Om l))))))))): 0 points increase in error, 1 points decrease in error
      (sqrt.f64 (*.f64 2 (*.f64 n (*.f64 U (+.f64 t (+.f64 (/.f64 n (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 Om 2)) (*.f64 (*.f64 l l) U*))) (*.f64 -2 (/.f64 l (/.f64 Om l))))))))): 1 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 2 (*.f64 n (*.f64 U (+.f64 t (+.f64 (/.f64 n (/.f64 (pow.f64 Om 2) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) U*))) (*.f64 -2 (/.f64 l (/.f64 Om l))))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 2 (*.f64 n (*.f64 U (+.f64 t (+.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 n (*.f64 (pow.f64 l 2) U*)) (pow.f64 Om 2))) (*.f64 -2 (/.f64 l (/.f64 Om l))))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 2 (*.f64 n (*.f64 U (+.f64 t (+.f64 (/.f64 (*.f64 n (*.f64 (pow.f64 l 2) U*)) (pow.f64 Om 2)) (*.f64 (Rewrite<= metadata-eval (neg.f64 2)) (/.f64 l (/.f64 Om l))))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 2 (*.f64 n (*.f64 U (+.f64 t (+.f64 (/.f64 (*.f64 n (*.f64 (pow.f64 l 2) U*)) (pow.f64 Om 2)) (*.f64 (neg.f64 2) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) Om))))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 2 (*.f64 n (*.f64 U (+.f64 t (+.f64 (/.f64 (*.f64 n (*.f64 (pow.f64 l 2) U*)) (pow.f64 Om 2)) (*.f64 (neg.f64 2) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) Om)))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 2 (*.f64 n (*.f64 U (+.f64 t (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (/.f64 (*.f64 n (*.f64 (pow.f64 l 2) U*)) (pow.f64 Om 2)) (*.f64 2 (/.f64 (pow.f64 l 2) Om))))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 2 (*.f64 n (*.f64 U (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 t (/.f64 (*.f64 n (*.f64 (pow.f64 l 2) U*)) (pow.f64 Om 2))) (*.f64 2 (/.f64 (pow.f64 l 2) Om)))))))): 0 points increase in error, 13 points decrease in error
      (sqrt.f64 (*.f64 2 (*.f64 n (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 (+.f64 t (/.f64 (*.f64 n (*.f64 (pow.f64 l 2) U*)) (pow.f64 Om 2))) (*.f64 2 (/.f64 (pow.f64 l 2) Om))) U))))): 13 points increase in error, 0 points decrease in error

   if 0.0 < (*.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*)))) < 1e297

   1. Initial program 1.8

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

   if 1e297 < (*.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 62.9

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

   2. Simplified 56.0

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\mathsf{fma}\left({\left(\frac{\ell}{Om}\right)}^{2}, n \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right) \cdot \left(n \cdot U\right)\right)}} \]
      Proof
      (sqrt.f64 (*.f64 (*.f64 2 n) (*.f64 U (-.f64 t (+.f64 (*.f64 2 (/.f64 l (/.f64 Om l))) (*.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 2 n) (*.f64 U (-.f64 t (+.f64 (*.f64 2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) Om))) (*.f64 n (*.f64 (pow.f64 (/.f64 l Om) 2) (-.f64 U U*)))))))): 7 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 2 n) (*.f64 U (-.f64 t (+.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, 0 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 2 n) (*.f64 U (Rewrite=> sub-neg_binary64 (+.f64 t (neg.f64 (+.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, 1 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 2 n) (*.f64 U (Rewrite<= sub-neg_binary64 (-.f64 t (+.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 2 n) (*.f64 U (Rewrite<= associate--l-_binary64 (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))))): 1 points increase in error, 1 points decrease in error
      (sqrt.f64 (Rewrite<= associate-*l*_binary64 (*.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 points increase in error, 0 points decrease in error

   3. Taylor expanded in Om around 0 56.1

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

   4. Simplified 56.1

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

   5. Applied egg-rr 55.8

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

   6. Simplified 48.3

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left|\frac{Om}{n \cdot \ell}\right|}} \cdot \sqrt{U \cdot \left(U* - U\right)} - \sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{U}{U* - U}}\right) \]
      Proof
      (-.f64 (*.f64 (/.f64 (sqrt.f64 2) (fabs.f64 (/.f64 Om (*.f64 n l)))) (sqrt.f64 (*.f64 U (-.f64 U* U)))) (*.f64 (sqrt.f64 2) (*.f64 l (sqrt.f64 (/.f64 U (-.f64 U* U)))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 (/.f64 (sqrt.f64 2) (fabs.f64 (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 Om n) l)))) (sqrt.f64 (*.f64 U (-.f64 U* U)))) (*.f64 (sqrt.f64 2) (*.f64 l (sqrt.f64 (/.f64 U (-.f64 U* U)))))): 0 points increase in error, 4 points decrease in error
      (-.f64 (*.f64 (/.f64 (sqrt.f64 2) (Rewrite<= rem-sqrt-square_binary64 (sqrt.f64 (*.f64 (/.f64 (/.f64 Om n) l) (/.f64 (/.f64 Om n) l))))) (sqrt.f64 (*.f64 U (-.f64 U* U)))) (*.f64 (sqrt.f64 2) (*.f64 l (sqrt.f64 (/.f64 U (-.f64 U* U)))))): 4 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 (/.f64 (sqrt.f64 2) (sqrt.f64 (Rewrite<= unpow2_binary64 (pow.f64 (/.f64 (/.f64 Om n) l) 2)))) (sqrt.f64 (*.f64 U (-.f64 U* U)))) (*.f64 (sqrt.f64 2) (*.f64 l (sqrt.f64 (/.f64 U (-.f64 U* U)))))): 2 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 26.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\frac{n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om} + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 10^{+297}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\left|\frac{Om}{n \cdot \ell}\right|} \cdot \sqrt{U \cdot \left(U* - U\right)} - \sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{U}{U* - U}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	28.9
Cost	30728

\[\begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\frac{n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om} + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)\right)}\\ \mathbf{elif}\;t_1 \leq 10^{+297}:\\ \;\;\;\;\sqrt{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{Om}{n \cdot \ell}} \cdot \sqrt{U \cdot U*}\\ \end{array} \]

Alternative 2
Error	30.0
Cost	14992

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := -2 \cdot t_1\\ t_3 := \sqrt{2 \cdot \left(\left(t + \left(\frac{n \cdot \left(\frac{\ell}{Om} \cdot U*\right)}{\frac{Om}{\ell}} + t_2\right)\right) \cdot \left(n \cdot U\right)\right)}\\ \mathbf{if}\;U \leq -1.5 \cdot 10^{-62}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;U \leq -7.5 \cdot 10^{-303}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot t_1 - n \cdot \left(t_1 \cdot \frac{U* - U}{Om}\right)\right) - t\right)\right)}\\ \mathbf{elif}\;U \leq 3.6 \cdot 10^{-289}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;U \leq 5.7 \cdot 10^{+97}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(t_2 - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\frac{U* \cdot \left(n \cdot \ell\right)}{Om \cdot \frac{Om}{\ell}} + t_2\right)\right)\right)}\\ \end{array} \]

Alternative 3
Error	31.6
Cost	14416

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;n \leq -9 \cdot 10^{-72}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot t_1 - n \cdot \left(t_1 \cdot \frac{U* - U}{Om}\right)\right) - t\right)\right)}\\ \mathbf{elif}\;n \leq -7.2 \cdot 10^{-109}:\\ \;\;\;\;\sqrt{U \cdot \left(U* - U\right)} \cdot \left(n \cdot \left(\frac{\ell}{Om} \cdot \sqrt{2}\right)\right)\\ \mathbf{elif}\;n \leq 8.8 \cdot 10^{-290}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq 4.6 \cdot 10^{-180}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \left(\frac{n}{Om} \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(t + \left(\frac{n \cdot \left(\frac{\ell}{Om} \cdot U*\right)}{\frac{Om}{\ell}} + -2 \cdot t_1\right)\right) \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 4
Error	32.0
Cost	13896

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := -2 \cdot t_1\\ \mathbf{if}\;n \leq -9 \cdot 10^{-72}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot t_1 - n \cdot \left(t_1 \cdot \frac{U* - U}{Om}\right)\right) - t\right)\right)}\\ \mathbf{elif}\;n \leq -7.4 \cdot 10^{-109}:\\ \;\;\;\;\sqrt{U \cdot \left(U* - U\right)} \cdot \left(n \cdot \left(\frac{\ell}{Om} \cdot \sqrt{2}\right)\right)\\ \mathbf{elif}\;n \leq -3.3 \cdot 10^{-306}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq 8 \cdot 10^{-106}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\frac{n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om} + t_2\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(t + \left(\frac{n \cdot \left(\frac{\ell}{Om} \cdot U*\right)}{\frac{Om}{\ell}} + t_2\right)\right) \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 5
Error	32.0
Cost	13768

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := -2 \cdot t_1\\ \mathbf{if}\;n \leq -9 \cdot 10^{-72}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot t_1 - n \cdot \left(t_1 \cdot \frac{U* - U}{Om}\right)\right) - t\right)\right)}\\ \mathbf{elif}\;n \leq -7.4 \cdot 10^{-109}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{Om}{n \cdot \ell}} \cdot \sqrt{U \cdot U*}\\ \mathbf{elif}\;n \leq -5.6 \cdot 10^{-306}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq 4 \cdot 10^{-104}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\frac{n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om} + t_2\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(t + \left(\frac{n \cdot \left(\frac{\ell}{Om} \cdot U*\right)}{\frac{Om}{\ell}} + t_2\right)\right) \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 6
Error	30.1
Cost	13512

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;t \leq -2.7 \cdot 10^{+40}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot t_1 - n \cdot \left(t_1 \cdot \frac{U* - U}{Om}\right)\right) - t\right)\right)}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{+62}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(t + \left(\frac{n \cdot \left(\frac{\ell}{Om} \cdot U*\right)}{\frac{Om}{\ell}} + -2 \cdot t_1\right)\right) \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot U} \cdot \sqrt{2 \cdot t}\\ \end{array} \]

Alternative 7
Error	30.1
Cost	13512

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;t \leq -3.1 \cdot 10^{+40}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot t_1 - n \cdot \left(t_1 \cdot \frac{U* - U}{Om}\right)\right) - t\right)\right)}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{+62}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(t + \left(\frac{n \cdot \left(\frac{\ell}{Om} \cdot U*\right)}{\frac{Om}{\ell}} + -2 \cdot t_1\right)\right) \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot U\right)} \cdot \sqrt{t}\\ \end{array} \]

Alternative 8
Error	30.9
Cost	8788

\[\begin{array}{l} t_1 := -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + t_1\right)\right)\right)}\\ t_3 := \sqrt{2 \cdot \left(\left(t + \left(\frac{n \cdot \left(\frac{\ell}{Om} \cdot U*\right)}{\frac{Om}{\ell}} + t_1\right)\right) \cdot \left(n \cdot U\right)\right)}\\ \mathbf{if}\;U \leq -3.1 \cdot 10^{-99}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;U \leq -1.8 \cdot 10^{-244}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;U \leq 3.8 \cdot 10^{-289}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;U \leq 5.8 \cdot 10^{-202}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;U \leq 2.2 \cdot 10^{-198}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(\left(\ell \cdot \left(U \cdot \ell\right)\right) \cdot \left(\frac{2}{Om} - U* \cdot \frac{\frac{n}{Om}}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 9
Error	29.8
Cost	8521

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;U \leq -3.1 \cdot 10^{-61} \lor \neg \left(U \leq 2.2 \cdot 10^{-203}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(\left(t + \left(\frac{n \cdot \left(\frac{\ell}{Om} \cdot U*\right)}{\frac{Om}{\ell}} + -2 \cdot t_1\right)\right) \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\left(2 \cdot t_1 - n \cdot \left(t_1 \cdot \frac{U* - U}{Om}\right)\right) - t\right)\right)}\\ \end{array} \]

Alternative 10
Error	33.0
Cost	8392

\[\begin{array}{l} t_1 := -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;U \leq -2 \cdot 10^{+55}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + t_1\right)\right)}\\ \mathbf{elif}\;U \leq 8.5 \cdot 10^{-195}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\frac{n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om} + t_1\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 11
Error	33.2
Cost	8392

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := -2 \cdot t_1\\ \mathbf{if}\;U \leq -1.2 \cdot 10^{+63}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + t_2\right)\right)}\\ \mathbf{elif}\;U \leq 5.8 \cdot 10^{-204}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\frac{n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om} + t_2\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(n \cdot U\right) \cdot \left(\left(2 \cdot t_1 - \frac{\ell \cdot \left(n \cdot \ell\right)}{Om} \cdot \frac{U*}{Om}\right) - t\right)\right)}\\ \end{array} \]

Alternative 12
Error	35.3
Cost	7890

\[\begin{array}{l} \mathbf{if}\;U \leq -4.2 \cdot 10^{-50} \lor \neg \left(U \leq -8.8 \cdot 10^{-212} \lor \neg \left(U \leq 2.9 \cdot 10^{-280}\right) \land U \leq 1.32 \cdot 10^{-102}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]

Alternative 13
Error	35.3
Cost	7888

\[\begin{array}{l} t_1 := \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\frac{2 \cdot \ell}{\frac{Om}{\ell}} - t\right)\right)\right)}\\ t_2 := \sqrt{n \cdot \left(2 \cdot \left(U \cdot t\right)\right)}\\ \mathbf{if}\;U \leq -2.6 \cdot 10^{-50}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;U \leq -3.8 \cdot 10^{-209}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;U \leq 5.5 \cdot 10^{-281}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;U \leq 1.9 \cdot 10^{-103}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)\right)}\\ \end{array} \]

Alternative 14
Error	31.9
Cost	7625

\[\begin{array}{l} t_1 := t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;U \leq -4.7 \cdot 10^{+29} \lor \neg \left(U \leq 2.2 \cdot 10^{+43}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t_1\right)\right)}\\ \end{array} \]

Alternative 15
Error	32.6
Cost	7624

\[\begin{array}{l} \mathbf{if}\;U \leq -3.9 \cdot 10^{+29}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\frac{2 \cdot \ell}{\frac{Om}{\ell}} - t\right)\right)\right)}\\ \mathbf{elif}\;U \leq 8.1 \cdot 10^{+83}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)\right)}\\ \end{array} \]

Alternative 16
Error	31.8
Cost	7624

\[\begin{array}{l} t_1 := t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;U \leq -4 \cdot 10^{+29}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot t_1\right)}\\ \mathbf{elif}\;U \leq 1:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 17
Error	37.1
Cost	7496

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.6 \cdot 10^{+135}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\ell \cdot \left(2 \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot U\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 240000000:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{-0.5}\right)\right)\right)}\\ \end{array} \]

Alternative 18
Error	38.9
Cost	7364

\[\begin{array}{l} \mathbf{if}\;\ell \leq 240000000:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{-0.5}\right)\right)\right)}\\ \end{array} \]

Alternative 19
Error	39.1
Cost	6980

\[\begin{array}{l} \mathbf{if}\;U \leq -8.6 \cdot 10^{+64}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]

Alternative 20
Error	40.0
Cost	6980

\[\begin{array}{l} \mathbf{if}\;t \leq 3.8 \cdot 10^{-151}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 21
Error	40.1
Cost	6848

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

Reproduce

herbie shell --seed 2022343 
(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*))))))