Average Error: 58.6 → 0.2

Time: 22.5s

Precision: 64

Internal Precision: 128

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

↓

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

Error

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

Target

Original	58.6
Target	0.2
Herbie	0.2

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

Derivation

Initial program 58.6
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
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
\[\leadsto \color{blue}{\frac{-2}{5} \cdot {\varepsilon}^{5} - \left(\left(\frac{2}{3} \cdot \varepsilon\right) \cdot \varepsilon + 2\right) \cdot \varepsilon}\]
Using strategy rm
Applied add-log-exp0.2
\[\leadsto \frac{-2}{5} \cdot {\varepsilon}^{5} - \left(\color{blue}{\log \left(e^{\left(\frac{2}{3} \cdot \varepsilon\right) \cdot \varepsilon}\right)} + 2\right) \cdot \varepsilon\]
Final simplification0.2
\[\leadsto {\varepsilon}^{5} \cdot \frac{-2}{5} - \left(2 + \log \left(e^{\left(\frac{2}{3} \cdot \varepsilon\right) \cdot \varepsilon}\right)\right) \cdot \varepsilon\]

Reproduce

herbie shell --seed 2019090 
(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))))