\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
Test:
NMSE example 3.10
Bits:
128 bits
Bits error versus x
Time: 6.5 s
Input Error: 61.3
Output Error: 0.0
Log: ⚲
Profile: 🕒
\(-\left(\frac{1}{2} \cdot {x}^2 + \left(1 + x\right)\right)\)
  1. Started with
    \[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
    61.3
  2. Applied taylor to get
    \[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)} \leadsto -\left(\frac{1}{2} \cdot {x}^2 + \left(1 + x\right)\right)\]
    0.0
  3. Taylor expanded around 0 to get
    \[\color{red}{-\left(\frac{1}{2} \cdot {x}^2 + \left(1 + x\right)\right)} \leadsto \color{blue}{-\left(\frac{1}{2} \cdot {x}^2 + \left(1 + x\right)\right)}\]
    0.0

Original test:


(lambda ((x default))
  #:name "NMSE example 3.10"
  (/ (log (- 1 x)) (log (+ 1 x)))
  #:target
  (- (+ (+ (+ 1 x) (/ (sqr x) 2)) (* 5/12 (pow x 3)))))