Average Error: 58.5 → 0.2

Time: 3.3m

Precision: 64

Internal Precision: 1344

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

↓

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

Error

The X axis uses an exponential scale — Bits error versus `eps`

Target

Original	58.5
Target	0.2
Herbie	0.2

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

Derivation

Initial program 58.5
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
Taylor expanded around 0 0.2
\[\leadsto \color{blue}{-\left(\frac{2}{3} \cdot {\varepsilon}^{3} + \left(\frac{2}{5} \cdot {\varepsilon}^{5} + 2 \cdot \varepsilon\right)\right)}\]
Simplified0.2
\[\leadsto \color{blue}{(\left((\frac{2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right) + 2)_*\right) \cdot \left(-\varepsilon\right) + \left(\left(-\frac{2}{5}\right) \cdot {\varepsilon}^{5}\right))_*}\]
Using strategy rm
Applied fma-udef0.2
\[\leadsto \color{blue}{(\frac{2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right) + 2)_* \cdot \left(-\varepsilon\right) + \left(-\frac{2}{5}\right) \cdot {\varepsilon}^{5}}\]
Final simplification0.2
\[\leadsto {\varepsilon}^{5} \cdot \left(-\frac{2}{5}\right) + \left(-\varepsilon\right) \cdot (\frac{2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right) + 2)_*\]

Runtime

herbie shell --seed 2018215 +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))))