Error

Target

Derivation

1. Initial program 58.5

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

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. Applied simplify0.2

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

5. Applied add-exp-log34.4

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

6. Taylor expanded around 0 34.6

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

7. Applied simplify0.2

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

Runtime

Time bar (total: 16.2s)Debug logProfile

herbie shell --seed '#(1064269945 2896236262 301053905 1701069080 1701464310 1614783279)' +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))))