\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
Test:
NMSE example 3.10
Bits:
128 bits
Bits error versus x
Time: 6.1 s
Input Error: 26.7
Output Error: 0.1
Log: ⚲
Profile: 🕒
\(\sqrt[3]{{\left(\frac{\log_* (1 + \left(-x\right))}{\log_* (1 + x)}\right)}^3}\)
  1. Started with
    \[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
    26.7
  2. Applied simplify to get
    \[\color{red}{\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}} \leadsto \color{blue}{\frac{\log \left(1 - x\right)}{\log_* (1 + x)}}\]
    26.2
  3. Using strategy rm
    26.2
  4. Applied sub-neg to get
    \[\frac{\log \color{red}{\left(1 - x\right)}}{\log_* (1 + x)} \leadsto \frac{\log \color{blue}{\left(1 + \left(-x\right)\right)}}{\log_* (1 + x)}\]
    26.2
  5. Applied log1p-def to get
    \[\frac{\color{red}{\log \left(1 + \left(-x\right)\right)}}{\log_* (1 + x)} \leadsto \frac{\color{blue}{\log_* (1 + \left(-x\right))}}{\log_* (1 + x)}\]
    0.1
  6. Using strategy rm
    0.1
  7. Applied add-cbrt-cube to get
    \[\frac{\log_* (1 + \left(-x\right))}{\color{red}{\log_* (1 + x)}} \leadsto \frac{\log_* (1 + \left(-x\right))}{\color{blue}{\sqrt[3]{{\left(\log_* (1 + x)\right)}^3}}}\]
    18.7
  8. Applied add-cbrt-cube to get
    \[\frac{\color{red}{\log_* (1 + \left(-x\right))}}{\sqrt[3]{{\left(\log_* (1 + x)\right)}^3}} \leadsto \frac{\color{blue}{\sqrt[3]{{\left(\log_* (1 + \left(-x\right))\right)}^3}}}{\sqrt[3]{{\left(\log_* (1 + x)\right)}^3}}\]
    19.1
  9. Applied cbrt-undiv to get
    \[\color{red}{\frac{\sqrt[3]{{\left(\log_* (1 + \left(-x\right))\right)}^3}}{\sqrt[3]{{\left(\log_* (1 + x)\right)}^3}}} \leadsto \color{blue}{\sqrt[3]{\frac{{\left(\log_* (1 + \left(-x\right))\right)}^3}{{\left(\log_* (1 + x)\right)}^3}}}\]
    0.1
  10. Applied simplify to get
    \[\sqrt[3]{\color{red}{\frac{{\left(\log_* (1 + \left(-x\right))\right)}^3}{{\left(\log_* (1 + x)\right)}^3}}} \leadsto \sqrt[3]{\color{blue}{{\left(\frac{\log_* (1 + \left(-x\right))}{\log_* (1 + x)}\right)}^3}}\]
    0.1

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)))))