Average Error: 58.4 → 0.3
Time: 15.1s
Precision: 64
Internal Precision: 1344
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[(\frac{-2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right) + -2)_* \cdot \varepsilon + {\varepsilon}^{5} \cdot \frac{-2}{5}\]

Error

Bits error versus eps

Target

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

Derivation

  1. Initial program 58.4

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

  2. Initial simplification58.4

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

  3. Taylor expanded around 0 0.3

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

  4. Simplified0.3

    \[\leadsto \color{blue}{(\varepsilon \cdot \left((\frac{-2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right) + -2)_*\right) + \left({\varepsilon}^{5} \cdot \frac{-2}{5}\right))_*}\]
  5. Using strategy rm

  6. Applied fma-udef0.3

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

  7. Final simplification0.3

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

Runtime

Time bar (total: 15.1s)Debug logProfile

herbie shell --seed 2018251 +o rules:numerics
(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))))