\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
Test:
NMSE problem 3.4.3
Bits:
128 bits
Bits error versus eps
Time: 4.2 s
Input Error: 25.9
Output Error: 0.1
Log: ⚲
Profile: 🕒
\(-\left(2 \cdot \varepsilon + \left(\frac{2}{5} \cdot {\varepsilon}^{5} + \frac{2}{3} \cdot {\varepsilon}^{3}\right)\right)\)
  1. Started with
    \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
    25.9
  2. Applied taylor to get
    \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \leadsto -\left(2 \cdot \varepsilon + \left(\frac{2}{5} \cdot {\varepsilon}^{5} + \frac{2}{3} \cdot {\varepsilon}^{3}\right)\right)\]
    0.1
  3. Taylor expanded around 0 to get
    \[\color{red}{-\left(2 \cdot \varepsilon + \left(\frac{2}{5} \cdot {\varepsilon}^{5} + \frac{2}{3} \cdot {\varepsilon}^{3}\right)\right)} \leadsto \color{blue}{-\left(2 \cdot \varepsilon + \left(\frac{2}{5} \cdot {\varepsilon}^{5} + \frac{2}{3} \cdot {\varepsilon}^{3}\right)\right)}\]
    0.1

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))))