Average Error: 58.8 → 0.2
Time: 46.5s
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

Original58.8
Target0.2
Herbie0.2
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.8

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

  2. Taylor expanded around 0 0.2

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

Runtime

Time bar (total: 46.5s)Debug log

herbie shell --seed '#(2094665535 3654809497 1731717781 3082199462 566033875 575777438)' 
(FPCore (eps)
  :name "logq (problem 3.4.3)"

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

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