Average Error: 59.3 → 0.1
Time: 7.0s
Precision: 64
Internal precision: 1408
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
⬇
\[-\left(\left({\varepsilon}^3 \cdot \frac{2}{3} + \frac{2}{5} \cdot {\varepsilon}^{5}\right) + 2 \cdot \varepsilon\right)\]

Error

Bits error versus eps

Target

Original59.3
Comparison0.1
Herbie0.1
\[ -2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right) \]

Derivation

  1. Initial program 59.3

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

  2. Applied taylor 0.1

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

  3. Taylor expanded around 0 0.1

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

  5. Applied associate-+r+ 0.1

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

  6. Applied simplify 0.1

    \[\leadsto -\left(\color{blue}{\left({\varepsilon}^3 \cdot \frac{2}{3} + \frac{2}{5} \cdot {\varepsilon}^{5}\right)} + 2 \cdot \varepsilon\right)\]
  7. Removed slow pow expressions

Runtime

Time bar (total: 7.0s) Debug logProfile

Please include this information when filing a bug report:

herbie shell --seed '#(1066785882 2324371342 4059510649 1466361199 2701357084 1216585281)'
(FPCore (eps)
  :name "logq (problem 3.4.3)"

  :target
  (* -2 (+ (+ eps (/ (pow eps 3) 3)) (/ (pow eps 5) 5)))

  (log (/ (- 1 eps) (+ 1 eps))))