Average Error: 58.4 → 0.2

Time: 16.1s

Precision: 64

Internal Precision: 1408

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

↓

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

Error

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

Target

Original	58.4
Target	0.2
Herbie	0.2

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

Derivation

Initial program 58.4
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
Taylor expanded around 0 0.2
\[\leadsto \color{blue}{-\left(\frac{2}{3} \cdot {\varepsilon}^{3} + \left(\frac{2}{5} \cdot {\varepsilon}^{5} + 2 \cdot \varepsilon\right)\right)}\]
Applied simplify0.2
\[\leadsto \color{blue}{-(\varepsilon \cdot \left((\frac{2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right) + 2)_*\right) + \left(\frac{2}{5} \cdot {\varepsilon}^{5}\right))_*}\]
Using strategy rm
Applied fma-udef0.2
\[\leadsto -\color{blue}{\left(\varepsilon \cdot (\frac{2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right) + 2)_* + \frac{2}{5} \cdot {\varepsilon}^{5}\right)}\]
Using strategy rm
Applied fma-udef0.2
\[\leadsto -\left(\varepsilon \cdot \color{blue}{\left(\frac{2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right) + 2\right)} + \frac{2}{5} \cdot {\varepsilon}^{5}\right)\]
Applied distribute-lft-in0.2
\[\leadsto -\left(\color{blue}{\left(\varepsilon \cdot \left(\frac{2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + \varepsilon \cdot 2\right)} + \frac{2}{5} \cdot {\varepsilon}^{5}\right)\]

Runtime

herbie shell --seed '#(1070131407 1246090267 3027482374 2150728003 2026520792 2347815650)' +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))))