Average Error: 59.4 → 0.1
Time: 16.0s
Precision: 64
Ground Truth: 128
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
⬇
\[-\left(2 \cdot \varepsilon + \left(\frac{2}{3} \cdot {\varepsilon}^{3} + \frac{2}{5} \cdot {\varepsilon}^{5}\right)\right)\]

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

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

  3. Taylor expanded around 0 0.1

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

Runtime

Total time: 16.0s Debug log

Please report a bug with the following info:

herbie --seed '#(2708428187 3839682879 1485032594 271481045 258160024 556816138)'
(FPCore (eps)
  :name "NMSE problem 3.4.3"

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

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