Average Error: 0.9 → 0.6
Time: 31.3s
Precision: 64
Internal Precision: 320
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\frac{\frac{\beta \cdot \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i} + \left(-\alpha\right) \cdot \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Initial program 0.9

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

  3. Applied associate-/l*0.6

    \[\leadsto \frac{\left(\frac{\left(\frac{\color{blue}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(real->posit(2)\right) \cdot i\right)}\right)}{\left(\beta - \alpha\right)}\right)}\right)}}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(real->posit(2)\right) \cdot i\right)}\right)}{\left(real->posit(2.0)\right)}\right)}\right)}{\left(real->posit(1.0)\right)}\right)}{\left(real->posit(2.0)\right)}\]
  4. Using strategy rm

  5. Applied associate-/r/0.6

    \[\leadsto \frac{\left(\frac{\left(\frac{\color{blue}{\left(\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(real->posit(2)\right) \cdot i\right)}\right)}\right) \cdot \left(\beta - \alpha\right)\right)}}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(real->posit(2)\right) \cdot i\right)}\right)}{\left(real->posit(2.0)\right)}\right)}\right)}{\left(real->posit(1.0)\right)}\right)}{\left(real->posit(2.0)\right)}\]
  6. Using strategy rm

  7. Applied sub-neg0.6

    \[\leadsto \frac{\left(\frac{\left(\frac{\left(\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(real->posit(2)\right) \cdot i\right)}\right)}\right) \cdot \color{blue}{\left(\frac{\beta}{\left(-\alpha\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(real->posit(2)\right) \cdot i\right)}\right)}{\left(real->posit(2.0)\right)}\right)}\right)}{\left(real->posit(1.0)\right)}\right)}{\left(real->posit(2.0)\right)}\]

  8. Applied distribute-rgt-in0.6

    \[\leadsto \frac{\left(\frac{\left(\frac{\color{blue}{\left(\frac{\left(\beta \cdot \left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(real->posit(2)\right) \cdot i\right)}\right)}\right)\right)}{\left(\left(-\alpha\right) \cdot \left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(real->posit(2)\right) \cdot i\right)}\right)}\right)\right)}\right)}}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(real->posit(2)\right) \cdot i\right)}\right)}{\left(real->posit(2.0)\right)}\right)}\right)}{\left(real->posit(1.0)\right)}\right)}{\left(real->posit(2.0)\right)}\]

  9. Final simplification0.6

    \[\leadsto \frac{\frac{\beta \cdot \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i} + \left(-\alpha\right) \cdot \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]

Reproduce

herbie shell --seed 2019094 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (>.p16 alpha (real->posit16 -1)) (>.p16 beta (real->posit16 -1)) (>.p16 i (real->posit16 0)))
  (/.p16 (+.p16 (/.p16 (/.p16 (*.p16 (+.p16 alpha beta) (-.p16 beta alpha)) (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) i))) (+.p16 (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) i)) (real->posit16 2.0))) (real->posit16 1.0)) (real->posit16 2.0)))