Average Error: 0.7 → 0.7
Time: 22.2s
Precision: 64
Internal Precision: 320
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\frac{\frac{\beta - \alpha}{\alpha + \left(\beta + 2.0\right)} + 1.0}{2.0}\]

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. Final simplification0.7

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

Reproduce

herbie shell --seed 2019094 +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)))