logq (problem 3.4.3)

Average Error: 58.6 → 0.2

Time: 4.3s

Precision: binary64

Math

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

↓

\[\varepsilon \cdot \left(-2\right) - \left({\varepsilon}^{3} \cdot 0.66666666666666663 + {\varepsilon}^{5} \cdot 0.40000000000000002\right)\]

Error

Bits error versus eps

Target

Original	58.6
Target	0.2
Herbie	0.2

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

Derivation

1. Initial program 58.6

   \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
2. Using strategy rm

3. Applied log-div 58.6

   \[\leadsto \color{blue}{\log \left(1 - \varepsilon\right) - \log \left(1 + \varepsilon\right)}\]

4. Taylor expanded around 0 0.2

   \[\leadsto \color{blue}{-\left(\frac{2}{3} \cdot \frac{{\varepsilon}^{3}}{{1}^{3}} + \left(\frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}} + 2 \cdot \varepsilon\right)\right)}\]

5. Simplified 0.2

   \[\leadsto \color{blue}{{\left(\frac{\varepsilon}{1}\right)}^{3} \cdot \frac{-2}{3} + \left(\frac{{\varepsilon}^{5}}{{1}^{5}} \cdot \frac{-2}{5} - \varepsilon \cdot 2\right)}\]

6. Taylor expanded around 0 0.2

   \[\leadsto \color{blue}{-\left(2 \cdot \varepsilon + \left(0.66666666666666663 \cdot {\varepsilon}^{3} + 0.40000000000000002 \cdot {\varepsilon}^{5}\right)\right)}\]

7. Simplified 0.2

   \[\leadsto \color{blue}{\varepsilon \cdot \left(-2\right) - \left({\varepsilon}^{3} \cdot 0.66666666666666663 + {\varepsilon}^{5} \cdot 0.40000000000000002\right)}\]

8. Final simplification 0.2

   \[\leadsto \varepsilon \cdot \left(-2\right) - \left({\varepsilon}^{3} \cdot 0.66666666666666663 + {\varepsilon}^{5} \cdot 0.40000000000000002\right)\]

Reproduce

herbie shell --seed 2020184 
(FPCore (eps)
  :name "logq (problem 3.4.3)"
  :precision binary64

  :herbie-target
  (* -2.0 (+ (+ eps (/ (pow eps 3.0) 3.0)) (/ (pow eps 5.0) 5.0)))

  (log (/ (- 1.0 eps) (+ 1.0 eps))))