Average Error: 61.4 → 0.3
Time: 1.1m
Precision: 64
Internal Precision: 1408
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\frac{\left(\left(-x\right) \cdot x\right) \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot x\right) - x}{\left(x \cdot x\right) \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) + x}\]

Error

Bits error versus x

Target

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

Derivation

  1. Initial program 61.4

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

  2. Taylor expanded around 0 60.6

    \[\leadsto \frac{\log \left(1 - x\right)}{\color{blue}{\left(\frac{1}{3} \cdot {x}^{3} + x\right) - \frac{1}{2} \cdot {x}^{2}}}\]

  3. Taylor expanded around 0 0.3

    \[\leadsto \frac{\color{blue}{-\left(\frac{1}{2} \cdot {x}^{2} + \left(\frac{1}{3} \cdot {x}^{3} + x\right)\right)}}{\left(\frac{1}{3} \cdot {x}^{3} + x\right) - \frac{1}{2} \cdot {x}^{2}}\]

  4. Applied simplify0.3

    \[\leadsto \color{blue}{\frac{\left(\left(-x\right) \cdot x\right) \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot x\right) - x}{\left(x \cdot x\right) \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) + x}}\]

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed '#(1070609872 3456127585 2380521889 2328837196 1765472538 734540918)' 
(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))))