Average Error: 58.7 → 0.2
Time: 3.5m
Precision: 64
Internal Precision: 1344
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[(\frac{2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right) + 2)_* \cdot \left(-\varepsilon\right) + \left(-\frac{2}{5}\right) \cdot {\varepsilon}^{5}\]

Error

Bits error versus eps

Target

Original58.7
Target0.2
Herbie0.2
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.7

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

  2. Taylor expanded around 0 0.2

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

  3. Applied simplify0.2

    \[\leadsto \color{blue}{(\left((\frac{2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right) + 2)_*\right) \cdot \left(-\varepsilon\right) + \left(\left(-\frac{2}{5}\right) \cdot {\varepsilon}^{5}\right))_*}\]
  4. Using strategy rm

  5. Applied fma-udef0.2

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

Runtime

Time bar (total: 3.5m)Debug logProfile

herbie shell --seed '#(1072840222 1305617769 1692503039 1353360431 4178980589 1488672652)' +o rules:numerics
(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))))