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Target

Derivation

1. Initial program 17.5

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

2. Initial simplification17.5

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

4. Applied add-sqr-sqrt17.5

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

5. Applied add-cube-cbrt17.4

   \[\leadsto \log \left(\frac{\color{blue}{\left(\sqrt[3]{1 - \varepsilon} \cdot \sqrt[3]{1 - \varepsilon}\right) \cdot \sqrt[3]{1 - \varepsilon}}}{\sqrt{\varepsilon + 1} \cdot \sqrt{\varepsilon + 1}}\right)\]

6. Applied times-frac17.4

   \[\leadsto \log \color{blue}{\left(\frac{\sqrt[3]{1 - \varepsilon} \cdot \sqrt[3]{1 - \varepsilon}}{\sqrt{\varepsilon + 1}} \cdot \frac{\sqrt[3]{1 - \varepsilon}}{\sqrt{\varepsilon + 1}}\right)}\]

7. Applied log-prod17.4

   \[\leadsto \color{blue}{\log \left(\frac{\sqrt[3]{1 - \varepsilon} \cdot \sqrt[3]{1 - \varepsilon}}{\sqrt{\varepsilon + 1}}\right) + \log \left(\frac{\sqrt[3]{1 - \varepsilon}}{\sqrt{\varepsilon + 1}}\right)}\]
8. Using strategy rm

9. Applied add-sqr-sqrt17.4

   \[\leadsto \log \left(\frac{\sqrt[3]{1 - \varepsilon} \cdot \sqrt[3]{1 - \varepsilon}}{\sqrt{\varepsilon + 1}}\right) + \log \left(\frac{\sqrt[3]{1 - \varepsilon}}{\sqrt{\color{blue}{\sqrt{\varepsilon + 1} \cdot \sqrt{\varepsilon + 1}}}}\right)\]

10. Applied sqrt-prod17.4

    \[\leadsto \log \left(\frac{\sqrt[3]{1 - \varepsilon} \cdot \sqrt[3]{1 - \varepsilon}}{\sqrt{\varepsilon + 1}}\right) + \log \left(\frac{\sqrt[3]{1 - \varepsilon}}{\color{blue}{\sqrt{\sqrt{\varepsilon + 1}} \cdot \sqrt{\sqrt{\varepsilon + 1}}}}\right)\]

11. Applied add-cube-cbrt17.4

    \[\leadsto \log \left(\frac{\sqrt[3]{1 - \varepsilon} \cdot \sqrt[3]{1 - \varepsilon}}{\sqrt{\varepsilon + 1}}\right) + \log \left(\frac{\color{blue}{\left(\sqrt[3]{\sqrt[3]{1 - \varepsilon}} \cdot \sqrt[3]{\sqrt[3]{1 - \varepsilon}}\right) \cdot \sqrt[3]{\sqrt[3]{1 - \varepsilon}}}}{\sqrt{\sqrt{\varepsilon + 1}} \cdot \sqrt{\sqrt{\varepsilon + 1}}}\right)\]

12. Applied times-frac17.4

    \[\leadsto \log \left(\frac{\sqrt[3]{1 - \varepsilon} \cdot \sqrt[3]{1 - \varepsilon}}{\sqrt{\varepsilon + 1}}\right) + \log \color{blue}{\left(\frac{\sqrt[3]{\sqrt[3]{1 - \varepsilon}} \cdot \sqrt[3]{\sqrt[3]{1 - \varepsilon}}}{\sqrt{\sqrt{\varepsilon + 1}}} \cdot \frac{\sqrt[3]{\sqrt[3]{1 - \varepsilon}}}{\sqrt{\sqrt{\varepsilon + 1}}}\right)}\]

13. Applied log-prod17.4

    \[\leadsto \log \left(\frac{\sqrt[3]{1 - \varepsilon} \cdot \sqrt[3]{1 - \varepsilon}}{\sqrt{\varepsilon + 1}}\right) + \color{blue}{\left(\log \left(\frac{\sqrt[3]{\sqrt[3]{1 - \varepsilon}} \cdot \sqrt[3]{\sqrt[3]{1 - \varepsilon}}}{\sqrt{\sqrt{\varepsilon + 1}}}\right) + \log \left(\frac{\sqrt[3]{\sqrt[3]{1 - \varepsilon}}}{\sqrt{\sqrt{\varepsilon + 1}}}\right)\right)}\]

14. Final simplification17.4

    \[\leadsto \log \left(\frac{\sqrt[3]{1 - \varepsilon} \cdot \sqrt[3]{1 - \varepsilon}}{\sqrt{\varepsilon + 1}}\right) + \left(\log \left(\frac{\sqrt[3]{\sqrt[3]{1 - \varepsilon}}}{\sqrt{\sqrt{\varepsilon + 1}}}\right) + \log \left(\frac{\sqrt[3]{\sqrt[3]{1 - \varepsilon}} \cdot \sqrt[3]{\sqrt[3]{1 - \varepsilon}}}{\sqrt{\sqrt{\varepsilon + 1}}}\right)\right)\]

Runtime

Time bar (total: 49.5s)Debug logProfile

herbie shell --seed 2018255 
(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))))