Average Error: 59.4 → 0.1
Time: 5.0s
Precision: 64
Internal precision: 1408
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
⬇
\[-\left(\frac{2}{3} \cdot {\varepsilon}^{3} + \left(\frac{2}{5} \cdot {\varepsilon}^{5} + 2 \cdot \varepsilon\right)\right)\]

Error

Bits error versus eps

Target

Original59.4
Comparison0.0
Herbie0.1
\[ -2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right) \]

Derivation

  1. Initial program 59.4

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

  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. Removed slow pow expressions

Runtime

Time bar (total: 5.0s) Debug logProfile

Please include this information when filing a bug report:

herbie shell --seed '#(1067615470 1817955187 3564058462 2385304812 1026693554 2746013648)'
(FPCore (eps)
  :name "logq (problem 3.4.3)"

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

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