Average Error: 0.9 → 0.6
Time: 51.7s
Precision: 64
\[\alpha \gt \left(-1\right) \land \beta \gt \left(-1\right) \land i \gt \left(0\right)\]
\[\frac{\left(\frac{\left(\frac{\left(\frac{\left(\left(\frac{\alpha}{\beta}\right) \cdot \left(\beta - \alpha\right)\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
↓
\[\frac{\frac{\beta - \alpha}{\alpha + \left(i \cdot 2 + \beta\right)} \cdot \frac{1.0}{\frac{\left(i \cdot 2 + \alpha\right) + \left(2.0 + \beta\right)}{\beta + \alpha}} + 1.0}{2.0}\]
\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(2\right) \cdot i\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}
↓
\frac{\frac{\beta - \alpha}{\alpha + \left(i \cdot 2 + \beta\right)} \cdot \frac{1.0}{\frac{\left(i \cdot 2 + \alpha\right) + \left(2.0 + \beta\right)}{\beta + \alpha}} + 1.0}{2.0}
double f(double alpha, double beta, double i) {
        double r3654602 = alpha;
        double r3654603 = beta;
        double r3654604 = r3654602 + r3654603;
        double r3654605 = r3654603 - r3654602;
        double r3654606 = r3654604 * r3654605;
        double r3654607 = 2.0;
        double r3654608 = /* ERROR: no posit support in C */;
        double r3654609 = i;
        double r3654610 = r3654608 * r3654609;
        double r3654611 = r3654604 + r3654610;
        double r3654612 = r3654606 / r3654611;
        double r3654613 = 2.0;
        double r3654614 = /* ERROR: no posit support in C */;
        double r3654615 = r3654611 + r3654614;
        double r3654616 = r3654612 / r3654615;
        double r3654617 = 1.0;
        double r3654618 = /* ERROR: no posit support in C */;
        double r3654619 = r3654616 + r3654618;
        double r3654620 = r3654619 / r3654614;
        return r3654620;
}
↓
double f(double alpha, double beta, double i) {
        double r3654621 = beta;
        double r3654622 = alpha;
        double r3654623 = r3654621 - r3654622;
        double r3654624 = i;
        double r3654625 = 2.0;
        double r3654626 = r3654624 * r3654625;
        double r3654627 = r3654626 + r3654621;
        double r3654628 = r3654622 + r3654627;
        double r3654629 = r3654623 / r3654628;
        double r3654630 = 1.0;
        double r3654631 = r3654626 + r3654622;
        double r3654632 = 2.0;
        double r3654633 = r3654632 + r3654621;
        double r3654634 = r3654631 + r3654633;
        double r3654635 = r3654621 + r3654622;
        double r3654636 = r3654634 / r3654635;
        double r3654637 = r3654630 / r3654636;
        double r3654638 = r3654629 * r3654637;
        double r3654639 = r3654638 + r3654630;
        double r3654640 = r3654639 / r3654632;
        return r3654640;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Initial program 0.9

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

  2. Simplified1.0

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

  4. Applied p16-times-frac0.6

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

  6. Applied p16-*-un-lft-identity0.6

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

  7. Applied associate-/l*0.6

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

  9. Applied *p16-rgt-identity-expand0.6

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

  10. Applied associate-/l*0.6

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

  11. Simplified0.6

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

  13. Applied associate-+r+0.6

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

  14. Simplified0.6

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

  15. Final simplification0.6

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

Reproduce

herbie shell --seed 2019158 +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)))