Average Error: 61.2 → 0.5
Time: 35.9s
Precision: 64
Internal Precision: 1344
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\frac{1}{x - \left(\frac{1}{2} \cdot {x}^{2} + 1\right)}\]

Error

Bits error versus x

Target

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

Derivation

  1. Initial program 61.2

    \[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
  2. Using strategy rm

  3. Applied clear-num61.2

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

  4. Taylor expanded around 0 0.5

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

Runtime

Time bar (total: 35.9s)Debug logProfile

herbie shell --seed '#(1071948828 1180510430 2986424009 997076509 406109801 420189285)' 
(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))))