\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
Test:
NMSE problem 3.4.3
Bits:
128 bits
Bits error versus eps
Time: 31.1 s
Input Error: 59.4
Output Error: 0.1
Log: ⚲
\(-\left(\left({\varepsilon}^3 \cdot \frac{2}{3} + \frac{2}{5} \cdot {\varepsilon}^{5}\right) + 2 \cdot \varepsilon\right)\)
  1. Started with
    \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
    59.4
  2. Applied taylor to get
    \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \leadsto -\left(\frac{2}{5} \cdot {\varepsilon}^{5} + \left(\frac{2}{3} \cdot {\varepsilon}^{3} + 2 \cdot \varepsilon\right)\right)\]
    0.1
  3. Taylor expanded around 0 to get
    \[\color{red}{-\left(\frac{2}{5} \cdot {\varepsilon}^{5} + \left(\frac{2}{3} \cdot {\varepsilon}^{3} + 2 \cdot \varepsilon\right)\right)} \leadsto \color{blue}{-\left(\frac{2}{5} \cdot {\varepsilon}^{5} + \left(\frac{2}{3} \cdot {\varepsilon}^{3} + 2 \cdot \varepsilon\right)\right)}\]
    0.1
  4. Using strategy rm
    0.1
  5. Applied associate-+r+ to get
    \[-\color{red}{\left(\frac{2}{5} \cdot {\varepsilon}^{5} + \left(\frac{2}{3} \cdot {\varepsilon}^{3} + 2 \cdot \varepsilon\right)\right)} \leadsto -\color{blue}{\left(\left(\frac{2}{5} \cdot {\varepsilon}^{5} + \frac{2}{3} \cdot {\varepsilon}^{3}\right) + 2 \cdot \varepsilon\right)}\]
    0.1
  6. Applied simplify to get
    \[-\left(\color{red}{\left(\frac{2}{5} \cdot {\varepsilon}^{5} + \frac{2}{3} \cdot {\varepsilon}^{3}\right)} + 2 \cdot \varepsilon\right) \leadsto -\left(\color{blue}{\left({\varepsilon}^3 \cdot \frac{2}{3} + \frac{2}{5} \cdot {\varepsilon}^{5}\right)} + 2 \cdot \varepsilon\right)\]
    0.1
  7. Removed slow pow expressions

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))))