Toniolo and Linder, Equation (13)

Average Error: 34.8 → 29.4

Time: 19.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -2.0626575918686287 \cdot 10^{+228}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{{Om}^{2}} + \left(\frac{1}{Om} \cdot -2 - \frac{n \cdot U}{{Om}^{2}}\right)\right)\right)} \cdot \left(-\ell\right)\right)\\ \mathbf{elif}\;\ell \leq -5.438743501710158 \cdot 10^{+153}:\\ \;\;\;\;{\left({\left(2 \cdot \left(n \cdot \mathsf{fma}\left(U, t, \left(U \cdot \frac{\ell}{\frac{Om}{\ell}}\right) \cdot -2\right)\right)\right)}^{0.25}\right)}^{2}\\ \mathbf{elif}\;\ell \leq -4.362243388401799 \cdot 10^{+119}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{n \cdot \left(U \cdot t\right) + -2 \cdot \frac{n \cdot \left(U \cdot {\ell}^{2}\right)}{Om}}\\ \mathbf{elif}\;\ell \leq -2.66377957353005 \cdot 10^{-83}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -9.579590953786636 \cdot 10^{-114}:\\ \;\;\;\;{\left({\left(2 \cdot {\left(\sqrt{n \cdot \left(U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right), t\right)\right)}\right)}^{2}\right)}^{0.25}\right)}^{2}\\ \mathbf{elif}\;\ell \leq -4.5557220936498896 \cdot 10^{-266}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{n \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)}{Om} \cdot -4\right)}\\ \mathbf{elif}\;\ell \leq 1.0079913196216918 \cdot 10^{+181}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n}{\frac{Om}{\frac{U*}{Om}}} + \left(\frac{-2}{Om} - \frac{n}{\frac{Om}{\frac{U}{Om}}}\right)\right)\right)}\right)\\ \end{array} \]

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
 (let* ((t_1
         (sqrt
          (*
           (* 2.0 (* n U))
           (+ t (* (/ l Om) (fma l -2.0 (* (- U* U) (* n (/ l Om))))))))))
   (if (<= l -2.0626575918686287e+228)
     (*
      (sqrt 2.0)
      (*
       (sqrt
        (*
         n
         (*
          U
          (+
           (/ (* n U*) (pow Om 2.0))
           (- (* (/ 1.0 Om) -2.0) (/ (* n U) (pow Om 2.0)))))))
       (- l)))
     (if (<= l -5.438743501710158e+153)
       (pow
        (pow (* 2.0 (* n (fma U t (* (* U (/ l (/ Om l))) -2.0)))) 0.25)
        2.0)
       (if (<= l -4.362243388401799e+119)
         (*
          (sqrt 2.0)
          (sqrt (+ (* n (* U t)) (* -2.0 (/ (* n (* U (pow l 2.0))) Om)))))
         (if (<= l -2.66377957353005e-83)
           t_1
           (if (<= l -9.579590953786636e-114)
             (pow
              (pow
               (*
                2.0
                (pow
                 (sqrt
                  (*
                   n
                   (*
                    U
                    (fma
                     (/ l Om)
                     (fma l -2.0 (* n (* (/ l Om) (- U* U))))
                     t))))
                 2.0))
               0.25)
              2.0)
             (if (<= l -4.5557220936498896e-266)
               (sqrt
                (fma 2.0 (* U (* n t)) (* (/ (* n (* U (* l l))) Om) -4.0)))
               (if (<= l 1.0079913196216918e+181)
                 t_1
                 (*
                  (sqrt 2.0)
                  (*
                   l
                   (sqrt
                    (*
                     n
                     (*
                      U
                      (+
                       (/ n (/ Om (/ U* Om)))
                       (- (/ -2.0 Om) (/ n (/ Om (/ U Om)))))))))))))))))))

C

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

↓

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt(((2.0 * (n * U)) * (t + ((l / Om) * fma(l, -2.0, ((U_42_ - U) * (n * (l / Om))))))));
	double tmp;
	if (l <= -2.0626575918686287e+228) {
		tmp = sqrt(2.0) * (sqrt((n * (U * (((n * U_42_) / pow(Om, 2.0)) + (((1.0 / Om) * -2.0) - ((n * U) / pow(Om, 2.0))))))) * -l);
	} else if (l <= -5.438743501710158e+153) {
		tmp = pow(pow((2.0 * (n * fma(U, t, ((U * (l / (Om / l))) * -2.0)))), 0.25), 2.0);
	} else if (l <= -4.362243388401799e+119) {
		tmp = sqrt(2.0) * sqrt(((n * (U * t)) + (-2.0 * ((n * (U * pow(l, 2.0))) / Om))));
	} else if (l <= -2.66377957353005e-83) {
		tmp = t_1;
	} else if (l <= -9.579590953786636e-114) {
		tmp = pow(pow((2.0 * pow(sqrt((n * (U * fma((l / Om), fma(l, -2.0, (n * ((l / Om) * (U_42_ - U)))), t)))), 2.0)), 0.25), 2.0);
	} else if (l <= -4.5557220936498896e-266) {
		tmp = sqrt(fma(2.0, (U * (n * t)), (((n * (U * (l * l))) / Om) * -4.0)));
	} else if (l <= 1.0079913196216918e+181) {
		tmp = t_1;
	} else {
		tmp = sqrt(2.0) * (l * sqrt((n * (U * ((n / (Om / (U_42_ / Om))) + ((-2.0 / Om) - (n / (Om / (U / Om)))))))));
	}
	return tmp;
}

Julia

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

↓

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(Float64(l / Om) * fma(l, -2.0, Float64(Float64(U_42_ - U) * Float64(n * Float64(l / Om))))))))
	tmp = 0.0
	if (l <= -2.0626575918686287e+228)
		tmp = Float64(sqrt(2.0) * Float64(sqrt(Float64(n * Float64(U * Float64(Float64(Float64(n * U_42_) / (Om ^ 2.0)) + Float64(Float64(Float64(1.0 / Om) * -2.0) - Float64(Float64(n * U) / (Om ^ 2.0))))))) * Float64(-l)));
	elseif (l <= -5.438743501710158e+153)
		tmp = (Float64(2.0 * Float64(n * fma(U, t, Float64(Float64(U * Float64(l / Float64(Om / l))) * -2.0)))) ^ 0.25) ^ 2.0;
	elseif (l <= -4.362243388401799e+119)
		tmp = Float64(sqrt(2.0) * sqrt(Float64(Float64(n * Float64(U * t)) + Float64(-2.0 * Float64(Float64(n * Float64(U * (l ^ 2.0))) / Om)))));
	elseif (l <= -2.66377957353005e-83)
		tmp = t_1;
	elseif (l <= -9.579590953786636e-114)
		tmp = (Float64(2.0 * (sqrt(Float64(n * Float64(U * fma(Float64(l / Om), fma(l, -2.0, Float64(n * Float64(Float64(l / Om) * Float64(U_42_ - U)))), t)))) ^ 2.0)) ^ 0.25) ^ 2.0;
	elseif (l <= -4.5557220936498896e-266)
		tmp = sqrt(fma(2.0, Float64(U * Float64(n * t)), Float64(Float64(Float64(n * Float64(U * Float64(l * l))) / Om) * -4.0)));
	elseif (l <= 1.0079913196216918e+181)
		tmp = t_1;
	else
		tmp = Float64(sqrt(2.0) * Float64(l * sqrt(Float64(n * Float64(U * Float64(Float64(n / Float64(Om / Float64(U_42_ / Om))) + Float64(Float64(-2.0 / Om) - Float64(n / Float64(Om / Float64(U / Om))))))))));
	end
	return tmp
end

Wolfram

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

↓

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(N[(l / Om), $MachinePrecision] * N[(l * -2.0 + N[(N[(U$42$ - U), $MachinePrecision] * N[(n * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -2.0626575918686287e+228], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(n * N[(U * N[(N[(N[(n * U$42$), $MachinePrecision] / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 / Om), $MachinePrecision] * -2.0), $MachinePrecision] - N[(N[(n * U), $MachinePrecision] / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-l)), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5.438743501710158e+153], N[Power[N[Power[N[(2.0 * N[(n * N[(U * t + N[(N[(U * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[l, -4.362243388401799e+119], N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[(n * N[(U * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2.66377957353005e-83], t$95$1, If[LessEqual[l, -9.579590953786636e-114], N[Power[N[Power[N[(2.0 * N[Power[N[Sqrt[N[(n * N[(U * N[(N[(l / Om), $MachinePrecision] * N[(l * -2.0 + N[(n * N[(N[(l / Om), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[l, -4.5557220936498896e-266], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(n * N[(U * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 1.0079913196216918e+181], t$95$1, N[(N[Sqrt[2.0], $MachinePrecision] * N[(l * N[Sqrt[N[(n * N[(U * N[(N[(n / N[(Om / N[(U$42$ / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 / Om), $MachinePrecision] - N[(n / N[(Om / N[(U / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Error

Bits error versus n

Bits error versus U

Bits error versus t

Bits error versus l

Bits error versus Om

Bits error versus U*

Derivation

1. Split input into 7 regimes

2. if l < -2.06265759186862865e228

   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 55.4

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

   3. Applied egg-rr 55.4

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

   4. Taylor expanded in l around -inf 31.5

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

   if -2.06265759186862865e228 < l < -5.4387435017101581e153

   1. Initial program 63.8

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

   2. Simplified 38.5

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

   3. Applied egg-rr 38.6

      \[\leadsto \color{blue}{{\left({\left(2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right), t\right)\right)\right)}^{0.25}\right)}^{2}} \]

   4. Taylor expanded in n around 0 63.8

      \[\leadsto {\left({\left(2 \cdot \color{blue}{\left(n \cdot \left(t \cdot U - 2 \cdot \frac{{\ell}^{2} \cdot U}{Om}\right)\right)}\right)}^{0.25}\right)}^{2} \]

   5. Simplified 40.3

      \[\leadsto {\left({\left(2 \cdot \color{blue}{\left(n \cdot \mathsf{fma}\left(U, t, \left(\frac{\ell}{\frac{Om}{\ell}} \cdot U\right) \cdot -2\right)\right)}\right)}^{0.25}\right)}^{2} \]

   if -5.4387435017101581e153 < l < -4.3622433884017993e119

   1. Initial program 36.8

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

   2. Simplified 35.7

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

   3. Applied egg-rr 35.8

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

   4. Taylor expanded in Om around inf 38.5

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

   if -4.3622433884017993e119 < l < -2.6637795735300499e-83 or -4.55572209364988956e-266 < l < 1.0079913196216918e181

   1. Initial program 30.1

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

   2. Simplified 27.8

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

   if -2.6637795735300499e-83 < l < -9.5795909537866356e-114

   1. Initial program 27.9

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

   2. Simplified 26.0

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

   3. Applied egg-rr 26.2

      \[\leadsto \color{blue}{{\left({\left(2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right), t\right)\right)\right)}^{0.25}\right)}^{2}} \]

   4. Applied egg-rr 26.1

      \[\leadsto {\left({\left(2 \cdot \color{blue}{{\left(\sqrt{n \cdot \left(U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, \left(\left(U* - U\right) \cdot \frac{\ell}{Om}\right) \cdot n\right), t\right)\right)}\right)}^{2}}\right)}^{0.25}\right)}^{2} \]

   if -9.5795909537866356e-114 < l < -4.55572209364988956e-266

   1. Initial program 25.5

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

   2. Simplified 24.3

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

   3. Taylor expanded in Om around inf 30.6

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(t \cdot U\right)\right) - 4 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}}} \]

   4. Simplified 29.8

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(2, \left(n \cdot t\right) \cdot U, \frac{n \cdot \left(\left(\ell \cdot \ell\right) \cdot U\right)}{Om} \cdot -4\right)}} \]

   if 1.0079913196216918e181 < l

   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 50.6

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

   3. Applied egg-rr 50.6

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

   4. Taylor expanded in l around inf 35.2

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

   5. Simplified 33.4

      \[\leadsto \sqrt{2} \cdot \color{blue}{\left(\ell \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n}{\frac{Om}{\frac{U*}{Om}}} - \left(\frac{2}{Om} + \frac{n}{\frac{Om}{\frac{U}{Om}}}\right)\right)\right)}\right)} \]
3. Recombined 7 regimes into one program.

4. Final simplification 29.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.0626575918686287 \cdot 10^{+228}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{{Om}^{2}} + \left(\frac{1}{Om} \cdot -2 - \frac{n \cdot U}{{Om}^{2}}\right)\right)\right)} \cdot \left(-\ell\right)\right)\\ \mathbf{elif}\;\ell \leq -5.438743501710158 \cdot 10^{+153}:\\ \;\;\;\;{\left({\left(2 \cdot \left(n \cdot \mathsf{fma}\left(U, t, \left(U \cdot \frac{\ell}{\frac{Om}{\ell}}\right) \cdot -2\right)\right)\right)}^{0.25}\right)}^{2}\\ \mathbf{elif}\;\ell \leq -4.362243388401799 \cdot 10^{+119}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{n \cdot \left(U \cdot t\right) + -2 \cdot \frac{n \cdot \left(U \cdot {\ell}^{2}\right)}{Om}}\\ \mathbf{elif}\;\ell \leq -2.66377957353005 \cdot 10^{-83}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -9.579590953786636 \cdot 10^{-114}:\\ \;\;\;\;{\left({\left(2 \cdot {\left(\sqrt{n \cdot \left(U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right), t\right)\right)}\right)}^{2}\right)}^{0.25}\right)}^{2}\\ \mathbf{elif}\;\ell \leq -4.5557220936498896 \cdot 10^{-266}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{n \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)}{Om} \cdot -4\right)}\\ \mathbf{elif}\;\ell \leq 1.0079913196216918 \cdot 10^{+181}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n}{\frac{Om}{\frac{U*}{Om}}} + \left(\frac{-2}{Om} - \frac{n}{\frac{Om}{\frac{U}{Om}}}\right)\right)\right)}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022146 
(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*))))))