Average Error: 61.4 → 0.0
Time: 30.7s
Precision: 64
Ground Truth: 128
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
⬇
\[-{\left(\frac{1}{8} \cdot {x}^2 + \left(1 + \frac{1}{2} \cdot x\right)\right)}^2\]

Error

Bits error versus x

Target

Original61.4
Comparison0.4
Herbie0.0
\[ -\left(\left(\left(1 + x\right) + \frac{{x}^2}{2}\right) + \frac{5}{12} \cdot {x}^{3}\right) \]

Derivation

  1. Initial program 61.4

    \[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]

  2. Applied taylor 0.4

    \[\leadsto -\left(\frac{1}{2} \cdot {x}^2 + \left(1 + x\right)\right)\]

  3. Taylor expanded around 0 0.4

    \[\leadsto \color{blue}{-\left(\frac{1}{2} \cdot {x}^2 + \left(1 + x\right)\right)}\]
  4. Using strategy rm

  5. Applied add-sqr-sqrt 0.4

    \[\leadsto -\color{blue}{{\left(\sqrt{\frac{1}{2} \cdot {x}^2 + \left(1 + x\right)}\right)}^2}\]

  6. Applied taylor 0.0

    \[\leadsto -{\left(\frac{1}{8} \cdot {x}^2 + \left(1 + \frac{1}{2} \cdot x\right)\right)}^2\]

  7. Taylor expanded around 0 0.0

    \[\leadsto -{\color{blue}{\left(\frac{1}{8} \cdot {x}^2 + \left(1 + \frac{1}{2} \cdot x\right)\right)}}^2\]
  8. Removed slow pow expressions

Runtime

Total time: 30.7s Debug log

Please include this information when filing a bug report:

herbie --seed '#(3743469905 3666187704 2366022356 303014626 204796633 2150942942)'
(FPCore (x)
  :name "NMSE example 3.10"
  :pre (and (< -1 x) (< x 1))

  :target
  (- (+ (+ (+ 1 x) (/ (sqr x) 2)) (* 5/12 (pow x 3))))

  (/ (log (- 1 x)) (log (+ 1 x))))