Average Error: 58.6 → 0.2
Time: 23.8s
Precision: 64
Internal Precision: 1344
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\left({\varepsilon}^{5} \cdot \left(-\frac{2}{5}\right) + {\varepsilon}^{3} \cdot \left(-\frac{2}{3}\right)\right) + \left(-2 \cdot \varepsilon\right)\]

Error

Bits error versus eps

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.6
Target0.2
Herbie0.2
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.6

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

  2. Initial simplification58.6

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

  3. Taylor expanded around 0 0.2

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

  5. Applied associate-+r+0.2

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

  6. Final simplification0.2

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

Runtime

Time bar (total: 23.8s)Debug logProfile

herbie shell --seed 2018230 
(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))))