Average Error: 60.9 → 0.4
Time: 30.0s
Precision: 64
Internal Precision: 1408
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[(\left((\frac{1}{2} \cdot x + 1)_*\right) \cdot \left(-x\right) + \left(-1\right))_*\]

Error

Bits error versus x

Target

Original60.9
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 60.9

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

  2. Applied simplify60.0

    \[\leadsto \color{blue}{\frac{\log \left(1 - x\right)}{\log_* (1 + x)}}\]

  3. Taylor expanded around 0 0.4

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

  4. Applied simplify0.4

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

Runtime

Time bar (total: 30.0s)Debug logProfile

herbie shell --seed '#(1070706311 3771791028 4128836681 4194990999 2341756049 504035650)' +o rules:numerics
(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))))