Average Error: 0.4 → 0.3
Time: 25.3s
Precision: 64
Internal Precision: 320
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\frac{\frac{\frac{\beta + 1.0}{\frac{\left(\alpha + \beta\right) + 2 \cdot 1}{\alpha + 1.0}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Initial program 0.4

    \[\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\beta \cdot \alpha\right)}\right)}{\left(real->posit(1.0)\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(real->posit(2)\right) \cdot \left(real->posit(1)\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(real->posit(2)\right) \cdot \left(real->posit(1)\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(real->posit(2)\right) \cdot \left(real->posit(1)\right)\right)}\right)}{\left(real->posit(1.0)\right)}\right)}\]
  2. Using strategy rm

  3. Applied associate-+l+0.4

    \[\leadsto \frac{\left(\frac{\left(\frac{\left(\frac{\color{blue}{\left(\frac{\alpha}{\left(\frac{\beta}{\left(\beta \cdot \alpha\right)}\right)}\right)}}{\left(real->posit(1.0)\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(real->posit(2)\right) \cdot \left(real->posit(1)\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(real->posit(2)\right) \cdot \left(real->posit(1)\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(real->posit(2)\right) \cdot \left(real->posit(1)\right)\right)}\right)}{\left(real->posit(1.0)\right)}\right)}\]
  4. Using strategy rm

  5. Applied associate-/l/0.4

    \[\leadsto \frac{\color{blue}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\left(\frac{\beta}{\left(\beta \cdot \alpha\right)}\right)}\right)}{\left(real->posit(1.0)\right)}\right)}{\left(\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(real->posit(2)\right) \cdot \left(real->posit(1)\right)\right)}\right) \cdot \left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(real->posit(2)\right) \cdot \left(real->posit(1)\right)\right)}\right)\right)}\right)}}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(real->posit(2)\right) \cdot \left(real->posit(1)\right)\right)}\right)}{\left(real->posit(1.0)\right)}\right)}\]

  6. Simplified0.4

    \[\leadsto \frac{\left(\frac{\color{blue}{\left(\left(\frac{\beta}{\left(real->posit(1.0)\right)}\right) \cdot \left(\frac{\alpha}{\left(real->posit(1.0)\right)}\right)\right)}}{\left(\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(real->posit(2)\right) \cdot \left(real->posit(1)\right)\right)}\right) \cdot \left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(real->posit(2)\right) \cdot \left(real->posit(1)\right)\right)}\right)\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(real->posit(2)\right) \cdot \left(real->posit(1)\right)\right)}\right)}{\left(real->posit(1.0)\right)}\right)}\]
  7. Using strategy rm

  8. Applied associate-/r*0.4

    \[\leadsto \frac{\color{blue}{\left(\frac{\left(\frac{\left(\left(\frac{\beta}{\left(real->posit(1.0)\right)}\right) \cdot \left(\frac{\alpha}{\left(real->posit(1.0)\right)}\right)\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(real->posit(2)\right) \cdot \left(real->posit(1)\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(real->posit(2)\right) \cdot \left(real->posit(1)\right)\right)}\right)}\right)}}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(real->posit(2)\right) \cdot \left(real->posit(1)\right)\right)}\right)}{\left(real->posit(1.0)\right)}\right)}\]
  9. Using strategy rm

  10. Applied associate-/l*0.3

    \[\leadsto \frac{\left(\frac{\color{blue}{\left(\frac{\left(\frac{\beta}{\left(real->posit(1.0)\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(real->posit(2)\right) \cdot \left(real->posit(1)\right)\right)}\right)}{\left(\frac{\alpha}{\left(real->posit(1.0)\right)}\right)}\right)}\right)}}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(real->posit(2)\right) \cdot \left(real->posit(1)\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(real->posit(2)\right) \cdot \left(real->posit(1)\right)\right)}\right)}{\left(real->posit(1.0)\right)}\right)}\]

  11. Final simplification0.3

    \[\leadsto \frac{\frac{\frac{\beta + 1.0}{\frac{\left(\alpha + \beta\right) + 2 \cdot 1}{\alpha + 1.0}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

Reproduce

herbie shell --seed 2019091 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :pre (and (>.p16 alpha (real->posit16 -1)) (>.p16 beta (real->posit16 -1)))
  (/.p16 (/.p16 (/.p16 (+.p16 (+.p16 (+.p16 alpha beta) (*.p16 beta alpha)) (real->posit16 1.0)) (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) (real->posit16 1)))) (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) (real->posit16 1)))) (+.p16 (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) (real->posit16 1))) (real->posit16 1.0))))