\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^2}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
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: 5.6 m
Input Error: 15.4
Output Error: 14.2
Log: ⚲
Profile: 🕒
\(\sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(t - (\left(\frac{n}{Om}\right) * \left(U - U*\right) + 2)_* \cdot \frac{\ell}{\frac{Om}{\ell}}\right)}\)
  1. Started with
    \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^2}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
    15.4
  2. Applied taylor to get
    \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^2}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)} \leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^2}{Om}\right) - \frac{n \cdot {\ell}^2}{{Om}^2} \cdot \left(U - U*\right)\right)}\]
    16.9
  3. Taylor expanded around 0 to get
    \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^2}{Om}\right) - \color{red}{\frac{n \cdot {\ell}^2}{{Om}^2}} \cdot \left(U - U*\right)\right)} \leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^2}{Om}\right) - \color{blue}{\frac{n \cdot {\ell}^2}{{Om}^2}} \cdot \left(U - U*\right)\right)}\]
    16.9
  4. Applied simplify to get
    \[\color{red}{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^2}{Om}\right) - \frac{n \cdot {\ell}^2}{{Om}^2} \cdot \left(U - U*\right)\right)}} \leadsto \color{blue}{\sqrt{\left(t - \frac{{\ell}^2}{Om} \cdot (\left(\frac{n}{Om}\right) * \left(U - U*\right) + 2)_*\right) \cdot \left(2 \cdot \left(U \cdot n\right)\right)}}\]
    15.7
  5. Applied taylor to get
    \[\sqrt{\left(t - \frac{{\ell}^2}{Om} \cdot (\left(\frac{n}{Om}\right) * \left(U - U*\right) + 2)_*\right) \cdot \left(2 \cdot \left(U \cdot n\right)\right)} \leadsto \sqrt{\left(t - \frac{{\ell}^2}{Om} \cdot (\left(\frac{n}{Om}\right) * \left(U - U*\right) + 2)_*\right) \cdot \left(2 \cdot \left(U \cdot n\right)\right)}\]
    15.7
  6. Taylor expanded around 0 to get
    \[\sqrt{\left(t - \color{red}{\frac{{\ell}^2}{Om}} \cdot (\left(\frac{n}{Om}\right) * \left(U - U*\right) + 2)_*\right) \cdot \left(2 \cdot \left(U \cdot n\right)\right)} \leadsto \sqrt{\left(t - \color{blue}{\frac{{\ell}^2}{Om}} \cdot (\left(\frac{n}{Om}\right) * \left(U - U*\right) + 2)_*\right) \cdot \left(2 \cdot \left(U \cdot n\right)\right)}\]
    15.7
  7. Applied simplify to get
    \[\sqrt{\left(t - \frac{{\ell}^2}{Om} \cdot (\left(\frac{n}{Om}\right) * \left(U - U*\right) + 2)_*\right) \cdot \left(2 \cdot \left(U \cdot n\right)\right)} \leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(t - (\left(\frac{n}{Om}\right) * \left(U - U*\right) + 2)_* \cdot \frac{\ell}{\frac{Om}{\ell}}\right)}\]
    14.2
  8. Applied final simplification

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*))))))