\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
Test:
NMSE example 3.10
Bits:
128 bits
Bits error versus x
Time: 7.4 s
Input Error: 61.5
Output Error: 0.0
Log: ⚲
Profile: 🕒
\(-\left(1 + \left(x + {\left(\sqrt{\frac{1}{2}}\right)}^2 \cdot {x}^2\right)\right)\)
  1. Started with
    \[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
    61.5
  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
  4. Using strategy rm
    0.0
  5. Applied add-sqr-sqrt to get
    \[-\color{red}{\left(\frac{1}{2} \cdot {x}^2 + \left(1 + x\right)\right)} \leadsto -\color{blue}{{\left(\sqrt{\frac{1}{2} \cdot {x}^2 + \left(1 + x\right)}\right)}^2}\]
    0.0
  6. Applied taylor to get
    \[-{\left(\sqrt{\frac{1}{2} \cdot {x}^2 + \left(1 + x\right)}\right)}^2 \leadsto -\left(1 + \left(x + {\left(\sqrt{\frac{1}{2}}\right)}^2 \cdot {x}^2\right)\right)\]
    0.0
  7. Taylor expanded around inf to get
    \[-\color{red}{\left(1 + \left(x + {\left(\sqrt{\frac{1}{2}}\right)}^2 \cdot {x}^2\right)\right)} \leadsto -\color{blue}{\left(1 + \left(x + {\left(\sqrt{\frac{1}{2}}\right)}^2 \cdot {x}^2\right)\right)}\]
    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)))))