Average Error: 59.2 → 0.1
Time: 16.3s
Precision: 64
Internal precision: 1408
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
⬇
\[-\left(\frac{2}{5} \cdot {\varepsilon}^{5} + \left(2 \cdot \varepsilon + \frac{2}{3} \cdot {\varepsilon}^{3}\right)\right)\]

Error

Bits error versus eps

Target

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

Derivation

  1. Initial program 59.2

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

  2. Applied taylor 0.1

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

  3. Taylor expanded around 0 0.1

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

Runtime

Time bar (total: 16.3s) Debug log

Please include this information when filing a bug report:

herbie shell --seed '#(644180380 3784176976 401987740 22459203 1940947670 3323606534)'
(FPCore (eps)
  :name "logq (problem 3.4.3)"

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

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