\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
Test:
NMSE example 3.10
Bits:
128 bits
Bits error versus x
Time: 5.5 s
Input Error: 61.3
Output Error: 0.0
Log: ⚲
Profile: 🕒
\(\left(-\left(1 + x\right)\right) + \left(x \cdot x\right) \cdot \left(-\frac{1}{2}\right)\)
  1. Started with
    \[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
    61.3
  2. Applied taylor to get
    \[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)} \leadsto \frac{\log \left(1 - x\right)}{\left(\frac{1}{3} \cdot {x}^{3} + x\right) - \frac{1}{2} \cdot {x}^2}\]
    60.3
  3. Taylor expanded around 0 to get
    \[\frac{\log \left(1 - x\right)}{\color{red}{\left(\frac{1}{3} \cdot {x}^{3} + x\right) - \frac{1}{2} \cdot {x}^2}} \leadsto \frac{\log \left(1 - x\right)}{\color{blue}{\left(\frac{1}{3} \cdot {x}^{3} + x\right) - \frac{1}{2} \cdot {x}^2}}\]
    60.3
  4. Applied simplify to get
    \[\color{red}{\frac{\log \left(1 - x\right)}{\left(\frac{1}{3} \cdot {x}^{3} + x\right) - \frac{1}{2} \cdot {x}^2}} \leadsto \color{blue}{\frac{\log \left(1 - x\right)}{x - \left(x \cdot x\right) \cdot \left(\frac{1}{2} - x \cdot \frac{1}{3}\right)}}\]
    60.3
  5. Applied taylor to get
    \[\frac{\log \left(1 - x\right)}{x - \left(x \cdot x\right) \cdot \left(\frac{1}{2} - x \cdot \frac{1}{3}\right)} \leadsto \frac{-\left(\frac{1}{2} \cdot {x}^2 + \left(\frac{1}{3} \cdot {x}^{3} + x\right)\right)}{x - \left(x \cdot x\right) \cdot \left(\frac{1}{2} - x \cdot \frac{1}{3}\right)}\]
    0.4
  6. Taylor expanded around 0 to get
    \[\frac{\color{red}{-\left(\frac{1}{2} \cdot {x}^2 + \left(\frac{1}{3} \cdot {x}^{3} + x\right)\right)}}{x - \left(x \cdot x\right) \cdot \left(\frac{1}{2} - x \cdot \frac{1}{3}\right)} \leadsto \frac{\color{blue}{-\left(\frac{1}{2} \cdot {x}^2 + \left(\frac{1}{3} \cdot {x}^{3} + x\right)\right)}}{x - \left(x \cdot x\right) \cdot \left(\frac{1}{2} - x \cdot \frac{1}{3}\right)}\]
    0.4
  7. Applied taylor to get
    \[\frac{-\left(\frac{1}{2} \cdot {x}^2 + \left(\frac{1}{3} \cdot {x}^{3} + x\right)\right)}{x - \left(x \cdot x\right) \cdot \left(\frac{1}{2} - x \cdot \frac{1}{3}\right)} \leadsto -\left(\frac{1}{2} \cdot {x}^2 + \left(1 + x\right)\right)\]
    0.0
  8. 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
  9. Applied simplify to get
    \[-\left(\frac{1}{2} \cdot {x}^2 + \left(1 + x\right)\right) \leadsto \left(-\left(1 + x\right)\right) + \left(x \cdot x\right) \cdot \left(-\frac{1}{2}\right)\]
    0.0
  10. Applied final simplification
  11. Removed slow pow expressions

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)))))