\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^2}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
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: 6.6 m
Input Error: 14.9
Output Error: 14.0
Log: ⚲
Profile: 🕒
\(\begin{cases} \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)\right)} & \text{when } \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^2}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right) \le 2.422772f+37 \\ \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(\frac{U*}{Om} \cdot n - 2\right) \cdot \frac{\ell}{\frac{Om}{\ell}} + \left(t - {\left(\frac{\ell}{Om}\right)}^2 \cdot \left(n \cdot U\right)\right)\right)} & \text{otherwise} \end{cases}\)

    if (* (* (* 2 n) U) (- (- t (* 2 (/ (sqr l) Om))) (* (* n (sqr (/ l Om))) (- U U*)))) < 2.422772f+37

    1. Started with
      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^2}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
      6.9
    2. Using strategy rm
      6.9
    3. Applied square-mult to get
      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\color{red}{{\ell}^2}}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)} \leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
      6.9
    4. Applied associate-/l* to get
      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{red}{\frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)} \leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
      6.7
    5. Using strategy rm
      6.7
    6. Applied associate-*l* to get
      \[\sqrt{\color{red}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}} \leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)\right)}}\]
      7.3

    if 2.422772f+37 < (* (* (* 2 n) U) (- (- t (* 2 (/ (sqr l) Om))) (* (* n (sqr (/ l Om))) (- U U*))))

    1. Started with
      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^2}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
      28.2
    2. Applied taylor to get
      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^2}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)} \leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^2}{Om}\right) - \left(\frac{n \cdot \left({\ell}^2 \cdot U\right)}{{Om}^2} - \frac{n \cdot \left(U* \cdot {\ell}^2\right)}{{Om}^2}\right)\right)}\]
      28.2
    3. Taylor expanded around 0 to get
      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^2}{Om}\right) - \color{red}{\left(\frac{n \cdot \left({\ell}^2 \cdot U\right)}{{Om}^2} - \frac{n \cdot \left(U* \cdot {\ell}^2\right)}{{Om}^2}\right)}\right)} \leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^2}{Om}\right) - \color{blue}{\left(\frac{n \cdot \left({\ell}^2 \cdot U\right)}{{Om}^2} - \frac{n \cdot \left(U* \cdot {\ell}^2\right)}{{Om}^2}\right)}\right)}\]
      28.2
    4. Applied simplify to get
      \[\color{red}{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^2}{Om}\right) - \left(\frac{n \cdot \left({\ell}^2 \cdot U\right)}{{Om}^2} - \frac{n \cdot \left(U* \cdot {\ell}^2\right)}{{Om}^2}\right)\right)}} \leadsto \color{blue}{\sqrt{\left(\left(U \cdot n\right) \cdot 2\right) \cdot \left(\frac{{\ell}^2}{Om} \cdot \left(\frac{U*}{\frac{Om}{n}} - 2\right) + \left(t - \frac{\ell \cdot n}{\frac{Om}{U}} \cdot \frac{\ell}{Om}\right)\right)}}\]
      28.0
    5. Applied simplify to get
      \[\sqrt{\color{red}{\left(\left(U \cdot n\right) \cdot 2\right) \cdot \left(\frac{{\ell}^2}{Om} \cdot \left(\frac{U*}{\frac{Om}{n}} - 2\right) + \left(t - \frac{\ell \cdot n}{\frac{Om}{U}} \cdot \frac{\ell}{Om}\right)\right)}} \leadsto \sqrt{\color{blue}{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(\frac{U*}{Om} \cdot n - 2\right) \cdot \frac{\ell}{\frac{Om}{\ell}} + \left(t - {\left(\frac{\ell}{Om}\right)}^2 \cdot \left(n \cdot U\right)\right)\right)}}\]
      25.2
  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*))))))