Toniolo and Linder, Equation (13)

Average Error: 34.2 → 28.1

Time: 12.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := n \cdot \left(U \cdot t\right)\\ t_2 := {\left(2 \cdot \left(t_1 + \frac{n}{Om} \cdot \left(\ell \cdot \left(U \cdot \mathsf{fma}\left(\ell, -2, \frac{n}{Om} \cdot \left(\ell \cdot U*\right)\right)\right)\right)\right)\right)}^{0.5}\\ t_3 := \sqrt{\frac{-2}{\frac{Om}{U \cdot \left(n \cdot \left(2 - \frac{n}{\frac{Om}{U* - U}}\right)\right)}}}\\ \mathbf{if}\;\ell \leq -4.9 \cdot 10^{+120}:\\ \;\;\;\;\ell \cdot \left(-t_3\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{+104}:\\ \;\;\;\;{\left(2 \cdot \left(t_1 + -2 \cdot \frac{n}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq -1.6 \cdot 10^{-39}:\\ \;\;\;\;\sqrt{2 \cdot \mathsf{fma}\left(n, U \cdot t, \frac{\mathsf{fma}\left(\ell, -2, \frac{n}{\frac{\frac{Om}{\ell}}{U* - U}}\right)}{Om} \cdot \left(n \cdot \left(\ell \cdot U\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -5.3 \cdot 10^{-61}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{n}{t_3}, \frac{U}{\frac{\ell}{t}}, \ell \cdot t_3\right)\\ \mathbf{elif}\;\ell \leq -1.75 \cdot 10^{-121}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{-201}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.95 \cdot 10^{+184}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(-2 + \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)}{Om}}\\ \end{array} \]

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 (* n (* U t)))
        (t_2
         (pow
          (*
           2.0
           (+ t_1 (* (/ n Om) (* l (* U (fma l -2.0 (* (/ n Om) (* l U*))))))))
          0.5))
        (t_3 (sqrt (/ -2.0 (/ Om (* U (* n (- 2.0 (/ n (/ Om (- U* U)))))))))))
   (if (<= l -4.9e+120)
     (* l (- t_3))
     (if (<= l -1e+104)
       (pow (* 2.0 (+ t_1 (* -2.0 (/ n (/ (/ Om U) (* l l)))))) 0.5)
       (if (<= l -1.6e-39)
         (sqrt
          (*
           2.0
           (fma
            n
            (* U t)
            (*
             (/ (fma l -2.0 (/ n (/ (/ Om l) (- U* U)))) Om)
             (* n (* l U))))))
         (if (<= l -5.3e-61)
           (- (fma (/ n t_3) (/ U (/ l t)) (* l t_3)))
           (if (<= l -1.75e-121)
             t_2
             (if (<= l 1.1e-201)
               (sqrt
                (*
                 (* 2.0 (* n U))
                 (+ t (* (/ l Om) (fma l -2.0 (* (- U* U) (* n (/ l Om))))))))
               (if (<= l 1.95e+184)
                 t_2
                 (*
                  (* l (sqrt 2.0))
                  (sqrt
                   (/ (* n (* U (+ -2.0 (/ (* n (- U* U)) Om)))) 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 = n * (U * t);
	double t_2 = pow((2.0 * (t_1 + ((n / Om) * (l * (U * fma(l, -2.0, ((n / Om) * (l * U_42_)))))))), 0.5);
	double t_3 = sqrt((-2.0 / (Om / (U * (n * (2.0 - (n / (Om / (U_42_ - U)))))))));
	double tmp;
	if (l <= -4.9e+120) {
		tmp = l * -t_3;
	} else if (l <= -1e+104) {
		tmp = pow((2.0 * (t_1 + (-2.0 * (n / ((Om / U) / (l * l)))))), 0.5);
	} else if (l <= -1.6e-39) {
		tmp = sqrt((2.0 * fma(n, (U * t), ((fma(l, -2.0, (n / ((Om / l) / (U_42_ - U)))) / Om) * (n * (l * U))))));
	} else if (l <= -5.3e-61) {
		tmp = -fma((n / t_3), (U / (l / t)), (l * t_3));
	} else if (l <= -1.75e-121) {
		tmp = t_2;
	} else if (l <= 1.1e-201) {
		tmp = sqrt(((2.0 * (n * U)) * (t + ((l / Om) * fma(l, -2.0, ((U_42_ - U) * (n * (l / Om))))))));
	} else if (l <= 1.95e+184) {
		tmp = t_2;
	} else {
		tmp = (l * sqrt(2.0)) * sqrt(((n * (U * (-2.0 + ((n * (U_42_ - U)) / Om)))) / 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 = Float64(n * Float64(U * t))
	t_2 = Float64(2.0 * Float64(t_1 + Float64(Float64(n / Om) * Float64(l * Float64(U * fma(l, -2.0, Float64(Float64(n / Om) * Float64(l * U_42_)))))))) ^ 0.5
	t_3 = sqrt(Float64(-2.0 / Float64(Om / Float64(U * Float64(n * Float64(2.0 - Float64(n / Float64(Om / Float64(U_42_ - U)))))))))
	tmp = 0.0
	if (l <= -4.9e+120)
		tmp = Float64(l * Float64(-t_3));
	elseif (l <= -1e+104)
		tmp = Float64(2.0 * Float64(t_1 + Float64(-2.0 * Float64(n / Float64(Float64(Om / U) / Float64(l * l)))))) ^ 0.5;
	elseif (l <= -1.6e-39)
		tmp = sqrt(Float64(2.0 * fma(n, Float64(U * t), Float64(Float64(fma(l, -2.0, Float64(n / Float64(Float64(Om / l) / Float64(U_42_ - U)))) / Om) * Float64(n * Float64(l * U))))));
	elseif (l <= -5.3e-61)
		tmp = Float64(-fma(Float64(n / t_3), Float64(U / Float64(l / t)), Float64(l * t_3)));
	elseif (l <= -1.75e-121)
		tmp = t_2;
	elseif (l <= 1.1e-201)
		tmp = 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))))))));
	elseif (l <= 1.95e+184)
		tmp = t_2;
	else
		tmp = Float64(Float64(l * sqrt(2.0)) * sqrt(Float64(Float64(n * Float64(U * Float64(-2.0 + Float64(Float64(n * Float64(U_42_ - U)) / Om)))) / 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[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(2.0 * N[(t$95$1 + N[(N[(n / Om), $MachinePrecision] * N[(l * N[(U * N[(l * -2.0 + N[(N[(n / Om), $MachinePrecision] * N[(l * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(-2.0 / N[(Om / N[(U * N[(n * N[(2.0 - N[(n / N[(Om / N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -4.9e+120], N[(l * (-t$95$3)), $MachinePrecision], If[LessEqual[l, -1e+104], N[Power[N[(2.0 * N[(t$95$1 + N[(-2.0 * N[(n / N[(N[(Om / U), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l, -1.6e-39], N[Sqrt[N[(2.0 * N[(n * N[(U * t), $MachinePrecision] + N[(N[(N[(l * -2.0 + N[(n / N[(N[(Om / l), $MachinePrecision] / N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[(n * N[(l * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, -5.3e-61], (-N[(N[(n / t$95$3), $MachinePrecision] * N[(U / N[(l / t), $MachinePrecision]), $MachinePrecision] + N[(l * t$95$3), $MachinePrecision]), $MachinePrecision]), If[LessEqual[l, -1.75e-121], t$95$2, If[LessEqual[l, 1.1e-201], 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, 1.95e+184], t$95$2, N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(n * N[(U * N[(-2.0 + N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if l < -4.9000000000000001e120

   1. Initial program 56.7

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

   2. Simplified 46.5

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

   3. Taylor expanded in t around inf 44.2

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

   4. Taylor expanded in l around -inf 29.0

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

   5. Simplified 31.9

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

   if -4.9000000000000001e120 < l < -1e104

   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 28.3

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

   3. Taylor expanded in t around inf 34.2

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

   4. Applied egg-rr 32.0

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

   5. Applied egg-rr 31.3

      \[\leadsto \color{blue}{{\left(2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{\left(\frac{n \cdot \ell}{\frac{Om}{U* - U}} + \ell \cdot -2\right) \cdot \left(n \cdot \left(U \cdot \ell\right)\right)}{Om}\right)\right)}^{0.5}} \]

   6. Taylor expanded in n around 0 42.3

      \[\leadsto {\left(2 \cdot \left(n \cdot \left(t \cdot U\right) + \color{blue}{-2 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}}\right)\right)}^{0.5} \]

   7. Simplified 39.2

      \[\leadsto {\left(2 \cdot \left(n \cdot \left(t \cdot U\right) + \color{blue}{-2 \cdot \frac{n}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}}\right)\right)}^{0.5} \]

   if -1e104 < l < -1.5999999999999999e-39

   1. Initial program 32.0

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

   2. Simplified 29.0

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

   3. Taylor expanded in t around 0 30.5

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

   4. Simplified 29.1

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

   if -1.5999999999999999e-39 < l < -5.3e-61

   1. Initial program 26.2

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

   2. Simplified 26.2

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

   3. Taylor expanded in t around inf 26.5

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

   4. Taylor expanded in l around -inf 55.5

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

   5. Simplified 57.5

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

   if -5.3e-61 < l < -1.74999999999999996e-121 or 1.1e-201 < l < 1.94999999999999985e184

   1. Initial program 31.5

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

   2. Simplified 28.7

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

   3. Taylor expanded in t around inf 30.8

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

   4. Applied egg-rr 30.2

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

   5. Applied egg-rr 30.3

      \[\leadsto \color{blue}{{\left(2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{\left(\frac{n \cdot \ell}{\frac{Om}{U* - U}} + \ell \cdot -2\right) \cdot \left(n \cdot \left(U \cdot \ell\right)\right)}{Om}\right)\right)}^{0.5}} \]

   6. Taylor expanded in U around 0 29.9

      \[\leadsto {\left(2 \cdot \left(n \cdot \left(t \cdot U\right) + \color{blue}{\frac{n \cdot \left(\ell \cdot \left(\left(-2 \cdot \ell + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right) \cdot U\right)\right)}{Om}}\right)\right)}^{0.5} \]

   7. Simplified 28.5

      \[\leadsto {\left(2 \cdot \left(n \cdot \left(t \cdot U\right) + \color{blue}{\frac{n}{Om} \cdot \left(\ell \cdot \left(U \cdot \mathsf{fma}\left(\ell, -2, \frac{n}{Om} \cdot \left(\ell \cdot U*\right)\right)\right)\right)}\right)\right)}^{0.5} \]

   if -1.74999999999999996e-121 < l < 1.1e-201

   1. Initial program 24.3

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

   2. Simplified 23.4

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

   if 1.94999999999999985e184 < 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 51.1

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

   3. Taylor expanded in t around inf 50.3

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

   4. Taylor expanded in l around inf 28.2

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

4. Final simplification 28.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.9 \cdot 10^{+120}:\\ \;\;\;\;\ell \cdot \left(-\sqrt{\frac{-2}{\frac{Om}{U \cdot \left(n \cdot \left(2 - \frac{n}{\frac{Om}{U* - U}}\right)\right)}}}\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{+104}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right) + -2 \cdot \frac{n}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq -1.6 \cdot 10^{-39}:\\ \;\;\;\;\sqrt{2 \cdot \mathsf{fma}\left(n, U \cdot t, \frac{\mathsf{fma}\left(\ell, -2, \frac{n}{\frac{\frac{Om}{\ell}}{U* - U}}\right)}{Om} \cdot \left(n \cdot \left(\ell \cdot U\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -5.3 \cdot 10^{-61}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{n}{\sqrt{\frac{-2}{\frac{Om}{U \cdot \left(n \cdot \left(2 - \frac{n}{\frac{Om}{U* - U}}\right)\right)}}}}, \frac{U}{\frac{\ell}{t}}, \ell \cdot \sqrt{\frac{-2}{\frac{Om}{U \cdot \left(n \cdot \left(2 - \frac{n}{\frac{Om}{U* - U}}\right)\right)}}}\right)\\ \mathbf{elif}\;\ell \leq -1.75 \cdot 10^{-121}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{n}{Om} \cdot \left(\ell \cdot \left(U \cdot \mathsf{fma}\left(\ell, -2, \frac{n}{Om} \cdot \left(\ell \cdot U*\right)\right)\right)\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{-201}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.95 \cdot 10^{+184}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{n}{Om} \cdot \left(\ell \cdot \left(U \cdot \mathsf{fma}\left(\ell, -2, \frac{n}{Om} \cdot \left(\ell \cdot U*\right)\right)\right)\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(-2 + \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)}{Om}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022185 
(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*))))))