Average Error: 3.3 → 1.5
Time: 1.2m
Precision: 64
Internal Precision: 320
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\frac{\frac{i}{\left(\alpha + \beta\right) + 2 \cdot i}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 1.0}{\left(\alpha + \beta\right) + i}} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Initial program 3.3

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

  3. Applied difference-of-sqr-13.3

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

  4. Applied p16-times-frac1.7

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

  5. Applied p16-times-frac1.7

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

  7. Applied associate-/l*1.5

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

  9. Applied associate-/r/1.5

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

  10. Applied associate-/l*1.5

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

  11. Final simplification1.5

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

Reproduce

herbie shell --seed 2019091 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :pre (and (>.p16 alpha (real->posit16 -1)) (>.p16 beta (real->posit16 -1)) (>.p16 i (real->posit16 1)))
  (/.p16 (/.p16 (*.p16 (*.p16 i (+.p16 (+.p16 alpha beta) i)) (+.p16 (*.p16 beta alpha) (*.p16 i (+.p16 (+.p16 alpha beta) i)))) (*.p16 (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) i)) (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) i)))) (-.p16 (*.p16 (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) i)) (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) i))) (real->posit16 1.0))))