Error

The X axis uses an exponential scale — Bits error versus `x`

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Target

Original	61.0
Target	0.3
Herbie	0.1

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

Derivation

Initial program 61.0
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
Initial simplification60.1
\[\leadsto \frac{\log \left(1 - x\right)}{\log_* (1 + x)}\]
Using strategy rm
Applied sub-neg60.1
\[\leadsto \frac{\log \color{blue}{\left(1 + \left(-x\right)\right)}}{\log_* (1 + x)}\]
Applied log1p-def0.0
\[\leadsto \frac{\color{blue}{\log_* (1 + \left(-x\right))}}{\log_* (1 + x)}\]
Using strategy rm
Applied *-un-lft-identity0.0
\[\leadsto \frac{\color{blue}{1 \cdot \log_* (1 + \left(-x\right))}}{\log_* (1 + x)}\]
Applied associate-/l*0.0
\[\leadsto \color{blue}{\frac{1}{\frac{\log_* (1 + x)}{\log_* (1 + \left(-x\right))}}}\]
Using strategy rm
Applied div-inv0.2
\[\leadsto \frac{1}{\color{blue}{\log_* (1 + x) \cdot \frac{1}{\log_* (1 + \left(-x\right))}}}\]
Applied associate-/r*0.1
\[\leadsto \color{blue}{\frac{\frac{1}{\log_* (1 + x)}}{\frac{1}{\log_* (1 + \left(-x\right))}}}\]
Final simplification0.1
\[\leadsto \frac{\frac{1}{\log_* (1 + x)}}{\frac{1}{\log_* (1 + \left(-x\right))}}\]

Runtime

herbie shell --seed 2018258 +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))))