\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^2}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
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: 1.9 m
Input Error: 15.0
Output Error: 10.3
Log: ⚲
Profile: 🕒
\(\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \frac{\ell}{Om} \cdot (\left(\left(-U*\right) \cdot \frac{\ell}{Om}\right) * n + \left((\left(n \cdot U\right) * \left(\frac{\ell}{Om}\right) + \left(2 \cdot \ell\right))_*\right))_*\right)\right)}\)
  1. Started with
    \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^2}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
    15.0
  2. Using strategy rm
    15.0
  3. Applied square-mult to get
    \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\color{red}{{\ell}^2}}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)} \leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
    15.0
  4. Applied associate-/l* to get
    \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{red}{\frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)} \leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
    13.6
  5. Using strategy rm
    13.6
  6. Applied sub-neg to get
    \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \color{red}{\left(U - U*\right)}\right)} \leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \color{blue}{\left(U + \left(-U*\right)\right)}\right)}\]
    13.6
  7. Applied distribute-lft-in to get
    \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \color{red}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U + \left(-U*\right)\right)}\right)} \leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \color{blue}{\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot U + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(-U*\right)\right)}\right)}\]
    13.6
  8. Applied associate--r+ to get
    \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{red}{\left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot U + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(-U*\right)\right)\right)}} \leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot U\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(-U*\right)\right)}}\]
    13.6
  9. Applied simplify to get
    \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{red}{\left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot U\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(-U*\right)\right)} \leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(t - (\left(U \cdot n\right) * \left(\frac{\ell}{Om}\right) + \left(2 \cdot \ell\right))_* \cdot \frac{\ell}{Om}\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(-U*\right)\right)}\]
    13.3
  10. Using strategy rm
    13.3
  11. Applied associate-*l* to get
    \[\sqrt{\color{red}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - (\left(U \cdot n\right) * \left(\frac{\ell}{Om}\right) + \left(2 \cdot \ell\right))_* \cdot \frac{\ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(-U*\right)\right)}} \leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - (\left(U \cdot n\right) * \left(\frac{\ell}{Om}\right) + \left(2 \cdot \ell\right))_* \cdot \frac{\ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(-U*\right)\right)\right)}}\]
    11.8
  12. Applied simplify to get
    \[\sqrt{\left(2 \cdot n\right) \cdot \color{red}{\left(U \cdot \left(\left(t - (\left(U \cdot n\right) * \left(\frac{\ell}{Om}\right) + \left(2 \cdot \ell\right))_* \cdot \frac{\ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(-U*\right)\right)\right)}} \leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(U \cdot \left(t - \frac{\ell}{Om} \cdot (\left(\left(-U*\right) \cdot \frac{\ell}{Om}\right) * n + \left((\left(n \cdot U\right) * \left(\frac{\ell}{Om}\right) + \left(2 \cdot \ell\right))_*\right))_*\right)\right)}}\]
    10.3
  13. 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*))))))