Average Error: 14.4 → 0.3
Time: 5.8m
Precision: 64
Internal Precision: 128
\[\frac{1}{x + 1} - \frac{1}{x - 1}\]
↓
\[\frac{2}{(\left(1 - x\right) \cdot x + \left(1 - x\right))_*}\]

Error

Bits error versus x

Derivation

  1. Initial program 14.4

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

  3. Applied add-sqr-sqrt14.4

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

  4. Applied associate-/l*14.4

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

  5. Applied frac-2neg14.4

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

  6. Applied frac-sub13.8

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

  7. Simplified0.3

    \[\leadsto \frac{\color{blue}{2}}{\left(-\left(x + 1\right)\right) \cdot \frac{x - 1}{\sqrt{1}}}\]

  8. Simplified0.3

    \[\leadsto \frac{2}{\color{blue}{(\left(1 - x\right) \cdot x + \left(1 - x\right))_*}}\]

  9. Final simplification0.3

    \[\leadsto \frac{2}{(\left(1 - x\right) \cdot x + \left(1 - x\right))_*}\]

Reproduce

herbie shell --seed 2019072 +o rules:numerics
(FPCore (x)
  :name "Asymptote A"
  (- (/ 1 (+ x 1)) (/ 1 (- x 1))))