Average Error: 58.6 → 0.2
Time: 5.3m
Precision: 64
Internal Precision: 1344
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[(\left(\left(-2 \cdot \varepsilon\right) \cdot {\varepsilon}^{4}\right) \cdot \frac{1}{5} + \left((\left(\frac{1}{3} \cdot -2\right) \cdot \left({\varepsilon}^{3}\right) + \left(-2 \cdot \varepsilon\right))_*\right))_*\]

Error

Bits error versus eps

Target

Original58.6
Target0.2
Herbie0.2
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.6

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

  3. Applied add-exp-log59.6

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

  4. Taylor expanded around 0 62.9

    \[\leadsto \color{blue}{\frac{1}{5} \cdot \left({\varepsilon}^{4} \cdot e^{\log \varepsilon + \log -2}\right) + \left(\frac{1}{3} \cdot \left({\varepsilon}^{2} \cdot e^{\log \varepsilon + \log -2}\right) + e^{\log \varepsilon + \log -2}\right)}\]

  5. Applied simplify0.2

    \[\leadsto \color{blue}{(\left(\left(-2 \cdot \varepsilon\right) \cdot {\varepsilon}^{4}\right) \cdot \frac{1}{5} + \left((\left(\frac{1}{3} \cdot -2\right) \cdot \left({\varepsilon}^{3}\right) + \left(-2 \cdot \varepsilon\right))_*\right))_*}\]

Runtime

Time bar (total: 5.3m)Debug logProfile

herbie shell --seed 2018199 +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))))