Average Error: 61.3 → 0.0
Time: 21.2s
Precision: 64
Internal precision: 1408
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
⬇
\[-\left(1 + \log \left(e^{\frac{1}{2} \cdot {x}^2 + x}\right)\right)\]

Error

Bits error versus x

Target

Original61.3
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 61.3

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

  2. Applied taylor 0.0

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

  3. Taylor expanded around 0 0.0

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

  5. Applied add-log-exp 0.0

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

Runtime

Time bar (total: 21.2s) Debug log

Please include this information when filing a bug report:

herbie shell --seed '#(2277612311 2645429965 1090895633 2857793080 2144184008 3989768357)'
(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))))