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

↓

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

Original	61.1
Target	0.3
Herbie	0.4

Derivation

Initial program 61.1
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
Initial simplification61.1
\[\leadsto \frac{\log \left(1 - x\right)}{\log \left(x + 1\right)}\]
Taylor expanded around 0 0.4
\[\leadsto \color{blue}{-\left(x + \left(\frac{1}{2} \cdot {x}^{2} + 1\right)\right)}\]
Simplified0.4
\[\leadsto \color{blue}{\left(-1 - x\right) - x \cdot \left(x \cdot \frac{1}{2}\right)}\]
Using strategy rm
Applied flip3--0.4
\[\leadsto \color{blue}{\frac{{\left(-1 - x\right)}^{3} - {\left(x \cdot \left(x \cdot \frac{1}{2}\right)\right)}^{3}}{\left(-1 - x\right) \cdot \left(-1 - x\right) + \left(\left(x \cdot \left(x \cdot \frac{1}{2}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{2}\right)\right) + \left(-1 - x\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)}}\]
Final simplification0.4
\[\leadsto \frac{{\left(-1 - x\right)}^{3} - {\left(\left(\frac{1}{2} \cdot x\right) \cdot x\right)}^{3}}{\left(\left(\left(\frac{1}{2} \cdot x\right) \cdot x\right) \cdot \left(-1 - x\right) + \left(\left(\frac{1}{2} \cdot x\right) \cdot x\right) \cdot \left(\left(\frac{1}{2} \cdot x\right) \cdot x\right)\right) + \left(-1 - x\right) \cdot \left(-1 - x\right)}\]

Runtime

herbie shell --seed 2018348 
(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))))

Error

Try it out

Target

Derivation

Runtime

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