Average Error: 61.4 → 0.0
Time: 5.9s
Precision: 64
Internal precision: 1408
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
⬇
\[-\left(\frac{1}{2} \cdot {x}^2 + \left(1 + x\right)\right)\]

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 60.4

    \[\leadsto \frac{\log \left(1 - x\right)}{\left(\frac{1}{3} \cdot {x}^{3} + x\right) - \frac{1}{2} \cdot {x}^2}\]

  3. Taylor expanded around 0 60.4

    \[\leadsto \frac{\log \left(1 - x\right)}{\color{blue}{\left(\frac{1}{3} \cdot {x}^{3} + x\right) - \frac{1}{2} \cdot {x}^2}}\]

  4. Applied simplify 60.4

    \[\leadsto \color{blue}{\frac{\log \left(1 - x\right)}{{x}^2 \cdot \left(x \cdot \frac{1}{3} - \frac{1}{2}\right) + x}}\]

  5. Applied taylor 0.4

    \[\leadsto \frac{-\left(\frac{1}{2} \cdot {x}^2 + \left(\frac{1}{3} \cdot {x}^{3} + x\right)\right)}{{x}^2 \cdot \left(x \cdot \frac{1}{3} - \frac{1}{2}\right) + x}\]

  6. Taylor expanded around 0 0.4

    \[\leadsto \frac{\color{blue}{-\left(\frac{1}{2} \cdot {x}^2 + \left(\frac{1}{3} \cdot {x}^{3} + x\right)\right)}}{{x}^2 \cdot \left(x \cdot \frac{1}{3} - \frac{1}{2}\right) + x}\]

  7. Applied taylor 0.0

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

  8. Taylor expanded around 0 0.0

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

Runtime

Time bar (total: 5.9s) Debug logProfile

Please include this information when filing a bug report:

herbie shell --seed '#(1067615470 1817955187 3564058462 2385304812 1026693554 2746013648)'
(FPCore (x)
  :name "qlog (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))))