Octave 3.8, jcobi/1

Average Error: 0.7 → 0.8

Time: 3.3m

Precision: 64

\[\alpha \gt \left(-1\right) \land \beta \gt \left(-1\right)\]

Math

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

↓

\[\frac{\left(\left(\mathsf{qma}\left(\left(\left(1.0\right)\right), \left(\beta - \alpha\right), \left(\frac{\left(1.0\right)}{\left(\frac{\left(2.0\right)}{\left(\frac{\alpha}{\beta}\right)}\right)}\right)\right)\right)\right)}{\left(2.0\right)}\]

C

double f(double alpha, double beta) {
        double r5686629 = beta;
        double r5686630 = alpha;
        double r5686631 = r5686629 - r5686630;
        double r5686632 = r5686630 + r5686629;
        double r5686633 = 2.0;
        double r5686634 = /* ERROR: no posit support in C */;
        double r5686635 = r5686632 + r5686634;
        double r5686636 = r5686631 / r5686635;
        double r5686637 = 1.0;
        double r5686638 = /* ERROR: no posit support in C */;
        double r5686639 = r5686636 + r5686638;
        double r5686640 = r5686639 / r5686634;
        return r5686640;
}

↓

double f(double alpha, double beta) {
        double r5686641 = 1.0;
        double r5686642 = /* ERROR: no posit support in C */;
        double r5686643 = /*Error: no posit support in C */;
        double r5686644 = beta;
        double r5686645 = alpha;
        double r5686646 = r5686644 - r5686645;
        double r5686647 = 2.0;
        double r5686648 = /* ERROR: no posit support in C */;
        double r5686649 = r5686645 + r5686644;
        double r5686650 = r5686648 + r5686649;
        double r5686651 = r5686642 / r5686650;
        double r5686652 = /*Error: no posit support in C */;
        double r5686653 = /*Error: no posit support in C */;
        double r5686654 = r5686653 / r5686648;
        return r5686654;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

1. Initial program 0.7

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

3. Applied associate-+l+ 0.7

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

5. Applied p16-*-un-lft-identity 0.7

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

6. Applied associate-/r* 0.7

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

7. Simplified 0.8

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

9. Applied p16-*-un-lft-identity 0.8

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

10. Applied *p16-rgt-identity-expand 0.8

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

11. Applied p16-times-frac 0.8

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

12. Applied introduce-quire 0.8

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

13. Applied insert-quire-fdp-add 0.8

    \[\leadsto \frac{\color{blue}{\left(\left(\mathsf{qma}\left(\left(\left(1.0\right)\right), \left(\frac{\left(\beta - \alpha\right)}{\left(1.0\right)}\right), \left(\frac{\left(1.0\right)}{\left(\frac{\left(\frac{\left(2.0\right)}{\alpha}\right)}{\beta}\right)}\right)\right)\right)\right)}}{\left(2.0\right)}\]

14. Simplified 0.8

    \[\leadsto \frac{\left(\color{blue}{\left(\mathsf{qma}\left(\left(\left(1.0\right)\right), \left(\beta - \alpha\right), \left(\frac{\left(1.0\right)}{\left(\frac{\left(2.0\right)}{\left(\frac{\alpha}{\beta}\right)}\right)}\right)\right)\right)}\right)}{\left(2.0\right)}\]

15. Final simplification 0.8

    \[\leadsto \frac{\left(\left(\mathsf{qma}\left(\left(\left(1.0\right)\right), \left(\beta - \alpha\right), \left(\frac{\left(1.0\right)}{\left(\frac{\left(2.0\right)}{\left(\frac{\alpha}{\beta}\right)}\right)}\right)\right)\right)\right)}{\left(2.0\right)}\]

Reproduce

herbie shell --seed 2019168 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :pre (and (>.p16 alpha (real->posit16 -1)) (>.p16 beta (real->posit16 -1)))
  (/.p16 (+.p16 (/.p16 (-.p16 beta alpha) (+.p16 (+.p16 alpha beta) (real->posit16 2.0))) (real->posit16 1.0)) (real->posit16 2.0)))