Average Error: 61.1 → 0.4

Time: 51.6s

Precision: 64

Internal Precision: 1344

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

↓

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

Error

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

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Out

Enter valid numbers for all inputs

Target

Original	61.1
Target	0.3
Herbie	0.4

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

Derivation

Initial program 61.1
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
Taylor expanded around 0 60.4
\[\leadsto \frac{\log \left(1 - x\right)}{\color{blue}{\left(x + \frac{1}{3} \cdot {x}^{3}\right) - \frac{1}{2} \cdot {x}^{2}}}\]
Simplified60.4
\[\leadsto \frac{\log \left(1 - x\right)}{\color{blue}{x - \left(x \cdot x\right) \cdot \left(\frac{1}{2} - x \cdot \frac{1}{3}\right)}}\]
Taylor expanded around 0 0.4
\[\leadsto \frac{\color{blue}{-\left(x + \left(\frac{1}{3} \cdot {x}^{3} + \frac{1}{2} \cdot {x}^{2}\right)\right)}}{x - \left(x \cdot x\right) \cdot \left(\frac{1}{2} - x \cdot \frac{1}{3}\right)}\]
Simplified0.4
\[\leadsto \frac{\color{blue}{\left(-x\right) + \left(\left(-x\right) \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \frac{1}{3}\right)}}{x - \left(x \cdot x\right) \cdot \left(\frac{1}{2} - x \cdot \frac{1}{3}\right)}\]
Final simplification0.4
\[\leadsto \frac{\left(\frac{1}{2} + \frac{1}{3} \cdot x\right) \cdot \left(-x \cdot x\right) + \left(-x\right)}{x - \left(x \cdot x\right) \cdot \left(\frac{1}{2} - \frac{1}{3} \cdot x\right)}\]

Runtime

herbie shell --seed 2018216 
(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))))