\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^2}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
Test:
Toniolo and Linder, Equation (13)
Bits:
128 bits
Bits error versus n
Bits error versus U
Bits error versus t
Bits error versus l
Bits error versus Om
Bits error versus U*
Time: 58.2 s
Input Error: 15.3
Output Error: 14.6
Log: ⚲
Profile: 🕒
\(\begin{cases} \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - n \cdot \left({\left(\frac{\ell}{Om}\right)}^2 \cdot \left(U - U*\right)\right)\right)} & \text{when } Om \le -8.772595f-17 \\ \sqrt{\left(t \cdot 2\right) \cdot \left(n \cdot U\right) + \left(\frac{\ell \cdot U}{\frac{Om}{\ell}} \cdot \left(U* - U\right)\right) \cdot \left(\frac{2}{Om} \cdot {n}^2\right)} & \text{when } Om \le 3.7943624f-20 \\ {\left(\sqrt{{\left(\sqrt[3]{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}}\right)}^3}\right)}^2 & \text{otherwise} \end{cases}\)

    if Om < -8.772595f-17

    1. Started with
      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^2}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
      14.1
    2. Using strategy rm
      14.1
    3. Applied square-mult to get
      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\color{red}{{\ell}^2}}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)} \leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
      14.1
    4. Applied associate-/l* to get
      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{red}{\frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)} \leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
      12.6
    5. Using strategy rm
      12.6
    6. Applied associate-*l* to get
      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \color{red}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)}\right)} \leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^2 \cdot \left(U - U*\right)\right)}\right)}\]
      12.9

    if -8.772595f-17 < Om < 3.7943624f-20

    1. Started with
      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^2}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
      18.6
    2. Applied taylor to get
      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^2}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)} \leadsto \sqrt{\left(2 \cdot \left(n \cdot \left(t \cdot U\right)\right) + 2 \cdot \frac{{n}^2 \cdot \left(U* \cdot \left({\ell}^2 \cdot U\right)\right)}{{Om}^2}\right) - 2 \cdot \frac{{n}^2 \cdot \left({\ell}^2 \cdot {U}^2\right)}{{Om}^2}}\]
      26.5
    3. Taylor expanded around inf to get
      \[\sqrt{\color{red}{\left(2 \cdot \left(n \cdot \left(t \cdot U\right)\right) + 2 \cdot \frac{{n}^2 \cdot \left(U* \cdot \left({\ell}^2 \cdot U\right)\right)}{{Om}^2}\right) - 2 \cdot \frac{{n}^2 \cdot \left({\ell}^2 \cdot {U}^2\right)}{{Om}^2}}} \leadsto \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot \left(t \cdot U\right)\right) + 2 \cdot \frac{{n}^2 \cdot \left(U* \cdot \left({\ell}^2 \cdot U\right)\right)}{{Om}^2}\right) - 2 \cdot \frac{{n}^2 \cdot \left({\ell}^2 \cdot {U}^2\right)}{{Om}^2}}}\]
      26.5
    4. Applied simplify to get
      \[\color{red}{\sqrt{\left(2 \cdot \left(n \cdot \left(t \cdot U\right)\right) + 2 \cdot \frac{{n}^2 \cdot \left(U* \cdot \left({\ell}^2 \cdot U\right)\right)}{{Om}^2}\right) - 2 \cdot \frac{{n}^2 \cdot \left({\ell}^2 \cdot {U}^2\right)}{{Om}^2}}} \leadsto \color{blue}{\sqrt{\left(n \cdot 2\right) \cdot \left(t \cdot U\right) + \frac{n \cdot 2}{\frac{Om}{n}} \cdot \left(\frac{\left(\ell \cdot \ell\right) \cdot U*}{\frac{Om}{U}} - \frac{\ell \cdot \ell}{\frac{Om}{U \cdot U}}\right)}}\]
      20.2
    5. Applied simplify to get
      \[\sqrt{\color{red}{\left(n \cdot 2\right) \cdot \left(t \cdot U\right) + \frac{n \cdot 2}{\frac{Om}{n}} \cdot \left(\frac{\left(\ell \cdot \ell\right) \cdot U*}{\frac{Om}{U}} - \frac{\ell \cdot \ell}{\frac{Om}{U \cdot U}}\right)}} \leadsto \sqrt{\color{blue}{\left(t \cdot 2\right) \cdot \left(n \cdot U\right) + \left(\frac{\ell \cdot U}{\frac{Om}{\ell}} \cdot \left(U* - U\right)\right) \cdot \left(\frac{2}{Om} \cdot {n}^2\right)}}\]
      21.1

    if 3.7943624f-20 < Om

    1. Started with
      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^2}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
      15.2
    2. Using strategy rm
      15.2
    3. Applied square-mult to get
      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\color{red}{{\ell}^2}}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)} \leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
      15.2
    4. Applied associate-/l* to get
      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{red}{\frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)} \leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
      13.7
    5. Using strategy rm
      13.7
    6. Applied add-sqr-sqrt to get
      \[\color{red}{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}} \leadsto \color{blue}{{\left(\sqrt{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}}\right)}^2}\]
      13.9
    7. Using strategy rm
      13.9
    8. Applied add-cube-cbrt to get
      \[{\left(\sqrt{\color{red}{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}}}\right)}^2 \leadsto {\left(\sqrt{\color{blue}{{\left(\sqrt[3]{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}}\right)}^3}}\right)}^2\]
      14.0
  1. Removed slow pow expressions

Original test:


(lambda ((n default) (U default) (t default) (l default) (Om default) (U* default))
  #:name "Toniolo and Linder, Equation (13)"
  (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (sqr l) Om))) (* (* n (sqr (/ l Om))) (- U U*))))))