Average Error: 58.5 → 0.2
Time: 3.2m
Precision: 64
Internal Precision: 1344
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[{\varepsilon}^{3} \cdot \left(-\frac{2}{3}\right) - (\frac{2}{5} \cdot \left({\varepsilon}^{5}\right) + \left(2 \cdot \varepsilon\right))_*\]

Error

Bits error versus eps

Target

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

Derivation

  1. Initial program 58.5

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

  3. Applied add-exp-log58.5

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

  4. Applied add-exp-log58.5

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

  5. Applied div-exp58.5

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

  6. Applied simplify58.5

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

  7. Taylor expanded around 0 58.7

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

  8. Applied simplify0.2

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

Runtime

Time bar (total: 3.2m)Debug logProfile

herbie shell --seed 2018208 +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))))