\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^2}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
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: 6.5 m
Input Error: 15.4
Output Error: 13.5
Log: ⚲
Profile: 🕒
\(\begin{cases} \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)\right)} & \text{when } \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^2}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right) \le 2.87687f-42 \\ \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(t - (\left(\frac{n}{Om}\right) * \left(U - U*\right) + 2)_* \cdot \frac{\ell}{\frac{Om}{\ell}}\right)} & \text{otherwise} \end{cases}\)

    if (* (* (* 2 n) U) (- (- t (* 2 (/ (sqr l) Om))) (* (* n (sqr (/ l Om))) (- U U*)))) < 2.87687f-42

    1. Started with
      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^2}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
      26.3
    2. Using strategy rm
      26.3
    3. Applied square-mult to get
      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\color{red}{{\ell}^2}}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)} \leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
      26.3
    4. Applied associate-/l* to get
      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{red}{\frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)} \leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
      25.3
    5. Using strategy rm
      25.3
    6. Applied associate-*l* to get
      \[\sqrt{\color{red}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}} \leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)\right)}}\]
      19.3

    if 2.87687f-42 < (* (* (* 2 n) U) (- (- t (* 2 (/ (sqr l) Om))) (* (* n (sqr (/ l Om))) (- U U*))))

    1. Started with
      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^2}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
      13.5
    2. Applied taylor to get
      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^2}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)} \leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^2}{Om}\right) - \frac{n \cdot {\ell}^2}{{Om}^2} \cdot \left(U - U*\right)\right)}\]
      15.3
    3. Taylor expanded around 0 to get
      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^2}{Om}\right) - \color{red}{\frac{n \cdot {\ell}^2}{{Om}^2}} \cdot \left(U - U*\right)\right)} \leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^2}{Om}\right) - \color{blue}{\frac{n \cdot {\ell}^2}{{Om}^2}} \cdot \left(U - U*\right)\right)}\]
      15.3
    4. Applied simplify to get
      \[\color{red}{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^2}{Om}\right) - \frac{n \cdot {\ell}^2}{{Om}^2} \cdot \left(U - U*\right)\right)}} \leadsto \color{blue}{\sqrt{\left(t - \frac{{\ell}^2}{Om} \cdot (\left(\frac{n}{Om}\right) * \left(U - U*\right) + 2)_*\right) \cdot \left(2 \cdot \left(U \cdot n\right)\right)}}\]
      14.1
    5. Applied taylor to get
      \[\sqrt{\left(t - \frac{{\ell}^2}{Om} \cdot (\left(\frac{n}{Om}\right) * \left(U - U*\right) + 2)_*\right) \cdot \left(2 \cdot \left(U \cdot n\right)\right)} \leadsto \sqrt{\left(t - \frac{{\ell}^2}{Om} \cdot (\left(\frac{n}{Om}\right) * \left(U - U*\right) + 2)_*\right) \cdot \left(2 \cdot \left(U \cdot n\right)\right)}\]
      14.1
    6. Taylor expanded around 0 to get
      \[\sqrt{\left(t - \color{red}{\frac{{\ell}^2}{Om}} \cdot (\left(\frac{n}{Om}\right) * \left(U - U*\right) + 2)_*\right) \cdot \left(2 \cdot \left(U \cdot n\right)\right)} \leadsto \sqrt{\left(t - \color{blue}{\frac{{\ell}^2}{Om}} \cdot (\left(\frac{n}{Om}\right) * \left(U - U*\right) + 2)_*\right) \cdot \left(2 \cdot \left(U \cdot n\right)\right)}\]
      14.1
    7. Applied simplify to get
      \[\sqrt{\left(t - \frac{{\ell}^2}{Om} \cdot (\left(\frac{n}{Om}\right) * \left(U - U*\right) + 2)_*\right) \cdot \left(2 \cdot \left(U \cdot n\right)\right)} \leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(t - (\left(\frac{n}{Om}\right) * \left(U - U*\right) + 2)_* \cdot \frac{\ell}{\frac{Om}{\ell}}\right)}\]
      12.5
    8. Applied final simplification
  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*))))))