Average Error: 1.0 → 1.0
Time: 17.3s
Precision: 64
Internal Precision: 320
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)\]

Error

Bits error versus x

Derivation

  1. Initial program 1.0

    \[\frac{\left(\left(\frac{\left(real->posit(1)\right)}{\left(\frac{x}{\left(real->posit(1)\right)}\right)}\right) - \left(\frac{\left(real->posit(2)\right)}{x}\right)\right)}{\left(\frac{\left(real->posit(1)\right)}{\left(x - \left(real->posit(1)\right)\right)}\right)}\]
  2. Using strategy rm

  3. Applied associate-+l-1.0

    \[\leadsto \color{blue}{\left(\frac{\left(real->posit(1)\right)}{\left(\frac{x}{\left(real->posit(1)\right)}\right)}\right) - \left(\left(\frac{\left(real->posit(2)\right)}{x}\right) - \left(\frac{\left(real->posit(1)\right)}{\left(x - \left(real->posit(1)\right)\right)}\right)\right)}\]

  4. Final simplification1.0

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

Reproduce

herbie shell --seed 2019094 +o rules:numerics
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  (+.p16 (-.p16 (/.p16 (real->posit16 1) (+.p16 x (real->posit16 1))) (/.p16 (real->posit16 2) x)) (/.p16 (real->posit16 1) (-.p16 x (real->posit16 1)))))