Average Error: 0.4 → 0.4
Time: 27.2s
Precision: 64
Internal Precision: 320
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\frac{\frac{1.0 + \beta}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}{\frac{\alpha + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}}\]

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

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

  7. Applied associate-/l/0.5

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

  8. Simplified0.4

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

  10. Applied p16-times-frac0.3

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

  11. Applied associate-/l*0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2019051 
(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))))