Average Error: 61.4 → 0.0
Time: 5.0s
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
Comparison0.5
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 0.0

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

  3. Taylor expanded around 0 0.0

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

Runtime

Time bar (total: 5.0s) Debug logProfile

Please include this information when filing a bug report:

herbie shell --seed '#(1067901057 3396600083 3715501224 3126139233 3908045574 1593683916)'
(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))))