Average Error: 0.6 → 0.9
Time: 12.5s
Precision: 64
Internal Precision: 320
\[\frac{1}{x + 1} - \frac{1}{x}\]
↓
\[\frac{\frac{1}{x + 1} + \frac{1}{x}}{\frac{\frac{1}{x + 1} + \frac{1}{x}}{\frac{1}{x + 1} - \frac{1}{x}}}\]

Error

Bits error versus x

Derivation

  1. Initial program 0.6

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

  3. Applied p16-flip--1.3

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

  5. Applied difference-of-squares1.0

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

  6. Applied associate-/l*0.9

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

  7. Final simplification0.9

    \[\leadsto \frac{\frac{1}{x + 1} + \frac{1}{x}}{\frac{\frac{1}{x + 1} + \frac{1}{x}}{\frac{1}{x + 1} - \frac{1}{x}}}\]

Reproduce

herbie shell --seed 2019091 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  (-.p16 (/.p16 (real->posit16 1) (+.p16 x (real->posit16 1))) (/.p16 (real->posit16 1) x)))