Average Error: 29.8 → 0.3
Time: 1.2m
Precision: 64
Internal Precision: 128
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\frac{1}{x + 1} \cdot \frac{(-2 \cdot x + \left(-1 - x\right))_*}{x + -1}\]

Error

Bits error versus x

Derivation

  1. Initial program 29.8

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

  3. Applied div-inv29.9

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

  5. Applied associate-*r/29.8

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

  6. Applied frac-sub30.6

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

  7. Simplified27.0

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

  9. Applied *-un-lft-identity27.0

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

  10. Applied times-frac25.3

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

  11. Simplified0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2019030 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))