Average Error: 61.0 → 0.4
Time: 24.6s
Precision: 64
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[-1 - \left(x - \left(x \cdot x\right) \cdot \frac{-1}{2}\right)\]

Error

Bits error versus x

Target

Original61.0
Target0.3
Herbie0.4
\[-\left(\left(\left(1 + x\right) + \frac{x \cdot x}{2}\right) + \frac{5}{12} \cdot {x}^{3}\right)\]

Derivation

  1. Initial program 61.0

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

  2. Taylor expanded around 0 0.4

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

  3. Simplified0.4

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

  5. Applied associate-+l-0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019100 
(FPCore (x)
  :name "qlog (example 3.10)"
  :pre (and (< -1 x) (< x 1))

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

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