Average Error: 60.9 → 0.0
Time: 26.1s
Precision: 64
Internal Precision: 128
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\sqrt[3]{\frac{\log_* (1 + \left(-x\right))}{\log_* (1 + x)} \cdot \left(\frac{\log_* (1 + \left(-x\right))}{\log_* (1 + x)} \cdot \frac{\log_* (1 + \left(-x\right))}{\log_* (1 + x)}\right)}\]

Error

Bits error versus x

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Your Program's Arguments

Results

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Target

Original60.9
Target0.3
Herbie0.0
\[-\left(\left(\left(1 + x\right) + \frac{x \cdot x}{2}\right) + \frac{5}{12} \cdot {x}^{3}\right)\]

Derivation

  1. Initial program 60.9

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

  2. Simplified59.9

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

  4. Applied sub-neg59.9

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

  5. Applied log1p-def0.0

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

  7. Applied add-cbrt-cube0.0

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

  8. Final simplification0.0

    \[\leadsto \sqrt[3]{\frac{\log_* (1 + \left(-x\right))}{\log_* (1 + x)} \cdot \left(\frac{\log_* (1 + \left(-x\right))}{\log_* (1 + x)} \cdot \frac{\log_* (1 + \left(-x\right))}{\log_* (1 + x)}\right)}\]

Reproduce

herbie shell --seed 2019026 +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))))