\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
Test:
NMSE problem 3.4.3
Bits:
128 bits
Bits error versus eps
Time: 11.2 s
Input Error: 58.6
Output Error: 0.0
Log: ⚲
Profile: 🕒
\(\log_* (1 + \left(-\varepsilon\right)) + \left(-\log_* (1 + \varepsilon)\right)\)
  1. Started with
    \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
    58.6
  2. Using strategy rm
    58.6
  3. Applied div-inv to get
    \[\log \color{red}{\left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)} \leadsto \log \color{blue}{\left(\left(1 - \varepsilon\right) \cdot \frac{1}{1 + \varepsilon}\right)}\]
    58.6
  4. Applied log-prod to get
    \[\color{red}{\log \left(\left(1 - \varepsilon\right) \cdot \frac{1}{1 + \varepsilon}\right)} \leadsto \color{blue}{\log \left(1 - \varepsilon\right) + \log \left(\frac{1}{1 + \varepsilon}\right)}\]
    58.6
  5. Applied simplify to get
    \[\log \left(1 - \varepsilon\right) + \color{red}{\log \left(\frac{1}{1 + \varepsilon}\right)} \leadsto \log \left(1 - \varepsilon\right) + \color{blue}{\left(-\log_* (1 + \varepsilon)\right)}\]
    50.5
  6. Using strategy rm
    50.5
  7. Applied sub-neg to get
    \[\log \color{red}{\left(1 - \varepsilon\right)} + \left(-\log_* (1 + \varepsilon)\right) \leadsto \log \color{blue}{\left(1 + \left(-\varepsilon\right)\right)} + \left(-\log_* (1 + \varepsilon)\right)\]
    50.5
  8. Applied log1p-def to get
    \[\color{red}{\log \left(1 + \left(-\varepsilon\right)\right)} + \left(-\log_* (1 + \varepsilon)\right) \leadsto \color{blue}{\log_* (1 + \left(-\varepsilon\right))} + \left(-\log_* (1 + \varepsilon)\right)\]
    0.0

Original test:


(lambda ((eps default))
  #:name "NMSE problem 3.4.3"
  (log (/ (- 1 eps) (+ 1 eps)))
  #:target
  (* -2 (+ (+ eps (/ (pow eps 3) 3)) (/ (pow eps 5) 5))))