Error

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

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Enter valid numbers for all inputs

Target

Original	60.9
Target	0.4
Herbie	0.5

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

Derivation

Initial program 60.9
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
Initial simplification60.9
\[\leadsto \frac{\log \left(1 - x\right)}{\log \left(x + 1\right)}\]
Taylor expanded around 0 0.5
\[\leadsto \color{blue}{-\left(x + \left(\frac{1}{2} \cdot {x}^{2} + 1\right)\right)}\]
Simplified0.5
\[\leadsto \color{blue}{\left(-1 - x\right) - x \cdot \left(x \cdot \frac{1}{2}\right)}\]
Using strategy rm
Applied associate--l-0.5
\[\leadsto \color{blue}{-1 - \left(x + x \cdot \left(x \cdot \frac{1}{2}\right)\right)}\]
Final simplification0.5
\[\leadsto -1 - \left(x + \left(\frac{1}{2} \cdot x\right) \cdot x\right)\]

Runtime

	Baseline	Herbie	Oracle	Span	%
Regimes	0.5	0.5	0.0	0.5	0%

herbie shell --seed 2018286 
(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))))