Average Error: 59.4 → 0.1
Time: 40.8s
Precision: 64
Ground Truth: 128
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
⬇
\[-\left(\frac{2}{3} \cdot {\varepsilon}^{3} + \left(\frac{2}{5} \cdot {\varepsilon}^{5} + 2 \cdot \varepsilon\right)\right)\]

Error

Bits error versus eps

Target

Original59.4
Comparison0.1
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

Total time: 40.8s Debug log

Please include this information when filing a bug report:

herbie --seed '#(952131724 1781739391 2778049379 1501882538 2760999206 3961229090)'
(FPCore (eps)
  :name "NMSE problem 3.4.3"

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

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