Average Error: 58.4 → 0.2
Time: 33.2s
Precision: 64
Internal Precision: 128
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[{\varepsilon}^{5} \cdot \frac{-2}{5} - \left(2 - \left(\varepsilon \cdot \frac{-2}{3}\right) \cdot \varepsilon\right) \cdot \varepsilon\]

Error

Bits error versus eps

Target

Original58.4
Target0.2
Herbie0.2
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.4

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

  2. Taylor expanded around 0 0.2

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

  3. Simplified0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019093 
(FPCore (eps)
  :name "logq (problem 3.4.3)"

  :herbie-target
  (* -2 (+ (+ eps (/ (pow eps 3) 3)) (/ (pow eps 5) 5)))

  (log (/ (- 1 eps) (+ 1 eps))))