Average Error: 61.0 → 0.1
Time: 16.6s
Precision: 64
Internal Precision: 128
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\frac{\frac{1}{\log_* (1 + x)}}{\frac{1}{\log_* (1 + \left(-x\right))}}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.0
Target0.4
Herbie0.1
\[-\left(\left(\left(1 + x\right) + \frac{x \cdot x}{2}\right) + \frac{5}{12} \cdot {x}^{3}\right)\]

Derivation

  1. Initial program 61.0

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

  2. Simplified60.0

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

  4. Applied sub-neg60.0

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

  5. Applied log1p-def0.0

    \[\leadsto \frac{\color{blue}{\log_* (1 + \left(-x\right))}}{\log_* (1 + x)}\]
  6. Using strategy rm

  7. Applied *-un-lft-identity0.0

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

  8. Applied associate-/l*0.0

    \[\leadsto \color{blue}{\frac{1}{\frac{\log_* (1 + x)}{\log_* (1 + \left(-x\right))}}}\]
  9. Using strategy rm

  10. Applied div-inv0.2

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

  11. Applied associate-/r*0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2019030 +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))))