Average Error: 0.7 → 0.8
Time: 26.4s
Precision: 64
Internal Precision: 320
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\frac{\frac{\beta - \alpha}{\beta + \left(\alpha + 2.0\right)} + 1.0}{2.0}\]

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Initial program 0.7

    \[\frac{\left(\frac{\left(\frac{\left(\beta - \alpha\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(real->posit(2.0)\right)}\right)}\right)}{\left(real->posit(1.0)\right)}\right)}{\left(real->posit(2.0)\right)}\]
  2. Using strategy rm

  3. Applied associate-+l+0.7

    \[\leadsto \frac{\left(\frac{\left(\frac{\left(\beta - \alpha\right)}{\color{blue}{\left(\frac{\alpha}{\left(\frac{\beta}{\left(real->posit(2.0)\right)}\right)}\right)}}\right)}{\left(real->posit(1.0)\right)}\right)}{\left(real->posit(2.0)\right)}\]
  4. Using strategy rm

  5. Applied p16-flip--1.3

    \[\leadsto \frac{\left(\frac{\left(\frac{\color{blue}{\left(\frac{\left(\left(\beta \cdot \beta\right) - \left(\alpha \cdot \alpha\right)\right)}{\left(\frac{\beta}{\alpha}\right)}\right)}}{\left(\frac{\alpha}{\left(\frac{\beta}{\left(real->posit(2.0)\right)}\right)}\right)}\right)}{\left(real->posit(1.0)\right)}\right)}{\left(real->posit(2.0)\right)}\]

  6. Applied associate-/l/1.3

    \[\leadsto \frac{\left(\frac{\color{blue}{\left(\frac{\left(\left(\beta \cdot \beta\right) - \left(\alpha \cdot \alpha\right)\right)}{\left(\left(\frac{\alpha}{\left(\frac{\beta}{\left(real->posit(2.0)\right)}\right)}\right) \cdot \left(\frac{\beta}{\alpha}\right)\right)}\right)}}{\left(real->posit(1.0)\right)}\right)}{\left(real->posit(2.0)\right)}\]

  7. Simplified1.2

    \[\leadsto \frac{\left(\frac{\left(\frac{\color{blue}{\left(\left(\beta - \alpha\right) \cdot \left(\frac{\alpha}{\beta}\right)\right)}}{\left(\left(\frac{\alpha}{\left(\frac{\beta}{\left(real->posit(2.0)\right)}\right)}\right) \cdot \left(\frac{\beta}{\alpha}\right)\right)}\right)}{\left(real->posit(1.0)\right)}\right)}{\left(real->posit(2.0)\right)}\]
  8. Using strategy rm

  9. Applied associate-/r*1.2

    \[\leadsto \frac{\left(\frac{\color{blue}{\left(\frac{\left(\frac{\left(\left(\beta - \alpha\right) \cdot \left(\frac{\alpha}{\beta}\right)\right)}{\left(\frac{\alpha}{\left(\frac{\beta}{\left(real->posit(2.0)\right)}\right)}\right)}\right)}{\left(\frac{\beta}{\alpha}\right)}\right)}}{\left(real->posit(1.0)\right)}\right)}{\left(real->posit(2.0)\right)}\]

  10. Simplified1.2

    \[\leadsto \frac{\left(\frac{\left(\frac{\left(\frac{\left(\left(\beta - \alpha\right) \cdot \left(\frac{\alpha}{\beta}\right)\right)}{\left(\frac{\alpha}{\left(\frac{\beta}{\left(real->posit(2.0)\right)}\right)}\right)}\right)}{\color{blue}{\left(\frac{\alpha}{\beta}\right)}}\right)}{\left(real->posit(1.0)\right)}\right)}{\left(real->posit(2.0)\right)}\]

  11. Simplified0.8

    \[\leadsto \color{blue}{\frac{\left(\frac{\left(\frac{\left(\beta - \alpha\right)}{\left(\frac{\beta}{\left(\frac{\alpha}{\left(real->posit(2.0)\right)}\right)}\right)}\right)}{\left(real->posit(1.0)\right)}\right)}{\left(real->posit(2.0)\right)}}\]

  12. Final simplification0.8

    \[\leadsto \frac{\frac{\beta - \alpha}{\beta + \left(\alpha + 2.0\right)} + 1.0}{2.0}\]

Reproduce

herbie shell --seed 2019050 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :pre (and (>.p16 alpha (real->posit16 -1)) (>.p16 beta (real->posit16 -1)))
  (/.p16 (+.p16 (/.p16 (-.p16 beta alpha) (+.p16 (+.p16 alpha beta) (real->posit16 2.0))) (real->posit16 1.0)) (real->posit16 2.0)))