Average Error: 58.4 → 0.3
Time: 14.9s
Precision: 64
Internal Precision: 1344
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[{\varepsilon}^{3} \cdot \frac{-2}{3} + \left(\frac{-2}{5} \cdot {\varepsilon}^{5} + -2 \cdot \varepsilon\right)\]

Error

Bits error versus eps

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Your Program's Arguments

Results

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Target

Original58.4
Target0.3
Herbie0.3
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.4

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

  2. Initial simplification58.4

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

  3. Taylor expanded around 0 0.3

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

  4. Final simplification0.3

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

Runtime

Time bar (total: 14.9s)Debug logProfile

herbie shell --seed 2018251 
(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))))