qlog (example 3.10)

Average Error: 61.3 → 0.4

Time: 7.0s

Precision: binary64

\[-1 \lt x \land x \lt 1\]

Math

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

↓

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

Error

Bits error versus x

Target

Original	61.3
Target	0.3
Herbie	0.4

\[-\left(\left(\left(1 + x\right) + \frac{x \cdot x}{2}\right) + 0.416666666666666685 \cdot {x}^{3}\right)\]

Derivation

1. Initial program 61.3

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

2. Taylor expanded around 0 60.4

   \[\leadsto \frac{\log \left(1 - x\right)}{\color{blue}{\left(1 \cdot x + \log 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}}}\]

3. Simplified 60.4

   \[\leadsto \frac{\log \left(1 - x\right)}{\color{blue}{1 \cdot x + \left(\log 1 + \left(\frac{x}{1} \cdot \frac{x}{1}\right) \cdot \frac{-1}{2}\right)}}\]

4. Taylor expanded around 0 0.4

   \[\leadsto \frac{\color{blue}{\log 1 - \left(1 \cdot x + \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)}}{1 \cdot x + \left(\log 1 + \left(\frac{x}{1} \cdot \frac{x}{1}\right) \cdot \frac{-1}{2}\right)}\]

5. Simplified 0.4

   \[\leadsto \frac{\color{blue}{\left(\log 1 - 1 \cdot x\right) + x \cdot \frac{x \cdot \frac{\frac{-1}{2}}{1}}{1}}}{1 \cdot x + \left(\log 1 + \left(\frac{x}{1} \cdot \frac{x}{1}\right) \cdot \frac{-1}{2}\right)}\]
6. Using strategy rm

7. Applied frac-2neg 0.4

   \[\leadsto \color{blue}{\frac{-\left(\left(\log 1 - 1 \cdot x\right) + x \cdot \frac{x \cdot \frac{\frac{-1}{2}}{1}}{1}\right)}{-\left(1 \cdot x + \left(\log 1 + \left(\frac{x}{1} \cdot \frac{x}{1}\right) \cdot \frac{-1}{2}\right)\right)}}\]

8. Simplified 0.4

   \[\leadsto \frac{\color{blue}{x \cdot \left(1 - \frac{\frac{-1}{2}}{1} \cdot \frac{x}{1}\right) - \log 1}}{-\left(1 \cdot x + \left(\log 1 + \left(\frac{x}{1} \cdot \frac{x}{1}\right) \cdot \frac{-1}{2}\right)\right)}\]

9. Simplified 0.4

   \[\leadsto \frac{x \cdot \left(1 - \frac{\frac{-1}{2}}{1} \cdot \frac{x}{1}\right) - \log 1}{\color{blue}{x \cdot \left(\left(-1\right) - \frac{\frac{-1}{2}}{1} \cdot \frac{x}{1}\right) - \log 1}}\]

10. Final simplification 0.4

    \[\leadsto \frac{x \cdot \left(1 - \frac{\frac{-1}{2}}{1} \cdot \frac{x}{1}\right) - \log 1}{x \cdot \left(\left(-1\right) - \frac{\frac{-1}{2}}{1} \cdot \frac{x}{1}\right) - \log 1}\]

Reproduce

herbie shell --seed 2020179 
(FPCore (x)
  :name "qlog (example 3.10)"
  :precision binary64
  :pre (and (< -1.0 x) (< x 1.0))

  :herbie-target
  (neg (+ (+ (+ 1.0 x) (/ (* x x) 2.0)) (* 0.4166666666666667 (pow x 3.0))))

  (/ (log (- 1.0 x)) (log (+ 1.0 x))))