\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
Test:
NMSE example 3.10
Bits:
128 bits
Bits error versus x
Time: 6.3 s
Input Error: 27.2
Output Error: 0.0
Log: ⚲
Profile: 🕒
\(-\left(\left({\left(\frac{1}{2} \cdot {x}^2\right)}^{1} + 1\right) + x\right)\)
  1. Started with
    \[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
    27.2
  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 associate-+r+ to get
    \[-\color{red}{\left(\frac{1}{2} \cdot {x}^2 + \left(1 + x\right)\right)} \leadsto -\color{blue}{\left(\left(\frac{1}{2} \cdot {x}^2 + 1\right) + x\right)}\]
    0.0
  6. Using strategy rm
    0.0
  7. Applied pow1 to get
    \[-\left(\left(\color{red}{\frac{1}{2} \cdot {x}^2} + 1\right) + x\right) \leadsto -\left(\left(\color{blue}{{\left(\frac{1}{2} \cdot {x}^2\right)}^{1}} + 1\right) + x\right)\]
    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)))))