Error: 59.4 → 0.1

Time: 16.5s

Precision: 64

Ground Truth: 128

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

⬇

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

Error

The X axis may use a short exponential scale — Bits error versus `eps`

Derivation

Initial program 59.4
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
Applied taylor 0.1
\[\leadsto -\left(\frac{2}{3} \cdot {\varepsilon}^{3} + \left(2 \cdot \varepsilon + \frac{2}{5} \cdot {\varepsilon}^{5}\right)\right)\]
Taylor expanded around 0 0.1

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

Runtime

Please report a bug with the following info:

herbie --seed '#(6413975 3110612524 1166757396 1992150209 139190798 3052506103)'
(FPCore (eps)
  :name "NMSE problem 3.4.3"
  
  :target
  (* -2 (+ (+ eps (/ (pow eps 3) 3)) (/ (pow eps 5) 5)))(log (/ (- 1 eps) (+ 1 eps))))