Average Error: 14.3 → 0.1
Time: 2.3s
Precision: binary64
\[\frac{1}{x + 1} - \frac{1}{x - 1}\]
↓
\[\frac{\left(\left(-1\right) - 1\right) \cdot \frac{1}{1 + x}}{x - 1}\]

Error

Bits error versus x

Derivation

  1. Initial program 14.3

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

  3. Applied frac-sub13.7

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

  4. Simplified0.4

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

  5. Simplified0.4

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

  7. Applied difference-of-squares0.4

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

  8. Applied associate-/r*0.1

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

  9. Simplified0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2020179 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))