Average Error: 14.4 → 0.1
Time: 3.2m
Precision: 64
Internal Precision: 832
\[\frac{1}{x + 1} - \frac{1}{x - 1}\]
↓
\[\frac{{\left(1 + 1\right)}^{3} \cdot \frac{-1}{1 + x}}{\left(\left(x - 1\right) + \left(x - 1\right)\right) + \left(\left(x - 1\right) + \left(x - 1\right)\right)}\]

Error

Bits error versus x

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Results

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Derivation

  1. Initial program 14.4

    \[\frac{1}{x + 1} - \frac{1}{x - 1}\]
  2. Using strategy rm

  3. Applied flip--29.4

    \[\leadsto \frac{1}{x + 1} - \frac{1}{\color{blue}{\frac{x \cdot x - 1 \cdot 1}{x + 1}}}\]

  4. Applied associate-/r/29.4

    \[\leadsto \frac{1}{x + 1} - \color{blue}{\frac{1}{x \cdot x - 1 \cdot 1} \cdot \left(x + 1\right)}\]

  5. Applied flip-+14.4

    \[\leadsto \frac{1}{\color{blue}{\frac{x \cdot x - 1 \cdot 1}{x - 1}}} - \frac{1}{x \cdot x - 1 \cdot 1} \cdot \left(x + 1\right)\]

  6. Applied associate-/r/14.4

    \[\leadsto \color{blue}{\frac{1}{x \cdot x - 1 \cdot 1} \cdot \left(x - 1\right)} - \frac{1}{x \cdot x - 1 \cdot 1} \cdot \left(x + 1\right)\]

  7. Applied distribute-lft-out--13.7

    \[\leadsto \color{blue}{\frac{1}{x \cdot x - 1 \cdot 1} \cdot \left(\left(x - 1\right) - \left(x + 1\right)\right)}\]

  8. Applied simplify0.4

    \[\leadsto \frac{1}{x \cdot x - 1 \cdot 1} \cdot \color{blue}{\left(0 - \left(1 + 1\right)\right)}\]
  9. Using strategy rm

  10. Applied difference-of-squares0.4

    \[\leadsto \frac{1}{\color{blue}{\left(x + 1\right) \cdot \left(x - 1\right)}} \cdot \left(0 - \left(1 + 1\right)\right)\]

  11. Applied associate-/r*0.1

    \[\leadsto \color{blue}{\frac{\frac{1}{x + 1}}{x - 1}} \cdot \left(0 - \left(1 + 1\right)\right)\]
  12. Using strategy rm

  13. Applied flip3--0.1

    \[\leadsto \frac{\frac{1}{x + 1}}{x - 1} \cdot \color{blue}{\frac{{0}^{3} - {\left(1 + 1\right)}^{3}}{0 \cdot 0 + \left(\left(1 + 1\right) \cdot \left(1 + 1\right) + 0 \cdot \left(1 + 1\right)\right)}}\]

  14. Applied frac-times0.1

    \[\leadsto \color{blue}{\frac{\frac{1}{x + 1} \cdot \left({0}^{3} - {\left(1 + 1\right)}^{3}\right)}{\left(x - 1\right) \cdot \left(0 \cdot 0 + \left(\left(1 + 1\right) \cdot \left(1 + 1\right) + 0 \cdot \left(1 + 1\right)\right)\right)}}\]

  15. Applied simplify0.1

    \[\leadsto \frac{\color{blue}{{\left(1 + 1\right)}^{3} \cdot \frac{-1}{1 + x}}}{\left(x - 1\right) \cdot \left(0 \cdot 0 + \left(\left(1 + 1\right) \cdot \left(1 + 1\right) + 0 \cdot \left(1 + 1\right)\right)\right)}\]

  16. Applied simplify0.1

    \[\leadsto \frac{{\left(1 + 1\right)}^{3} \cdot \frac{-1}{1 + x}}{\color{blue}{\left(\left(x - 1\right) + \left(x - 1\right)\right) + \left(\left(x - 1\right) + \left(x - 1\right)\right)}}\]

Runtime

Time bar (total: 3.2m)Debug logProfile

herbie shell --seed 2020178 
(FPCore (x)
  :name "Asymptote A"
  (- (/ 1 (+ x 1)) (/ 1 (- x 1))))