Input Error: 59.3b
Output Error: 0.1b
Time: 14.3s
Precision: 64b
Ground Truth: 128b
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
⬇
\[-\left(\frac{2}{5} \cdot {\varepsilon}^{5} + \left(\frac{2}{3} \cdot {\varepsilon}^{3} + 2 \cdot \varepsilon\right)\right)\]

Error

Bits error versus eps

Derivation

  1. Initial program 59.3b

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

  2. Applied taylor 0.1b

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

  3. Taylor expanded around 0 0.1b

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

Runtime

Total time: 14.3s Debug log

herbie --seed '#(3107420155 2366827108 293366336 1235941654 3197196612 3811544891)'
(FPCore (eps)
  :name "NMSE problem 3.4.3"
  
  :target
  (* -2 (+ (+ eps (/ (pow eps 3) 3)) (/ (pow eps 5) 5)))(log (/ (- 1 eps) (+ 1 eps))))