Octave 3.8, jcobi/1

Average Error: 0.8 → 0.8

Time: 2.8m

Precision: 64

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

Math

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

↓

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

C

double f(double alpha, double beta) {
        double r3769737 = beta;
        double r3769738 = alpha;
        double r3769739 = r3769737 - r3769738;
        double r3769740 = r3769738 + r3769737;
        double r3769741 = 2.0;
        double r3769742 = /* ERROR: no posit support in C */;
        double r3769743 = r3769740 + r3769742;
        double r3769744 = r3769739 / r3769743;
        double r3769745 = 1.0;
        double r3769746 = /* ERROR: no posit support in C */;
        double r3769747 = r3769744 + r3769746;
        double r3769748 = r3769747 / r3769742;
        return r3769748;
}

↓

double f(double alpha, double beta) {
        double r3769749 = 1.0;
        double r3769750 = /* ERROR: no posit support in C */;
        double r3769751 = 2.0;
        double r3769752 = /* ERROR: no posit support in C */;
        double r3769753 = /*Error: no posit support in C */;
        double r3769754 = alpha;
        double r3769755 = /*Error: no posit support in C */;
        double r3769756 = beta;
        double r3769757 = /*Error: no posit support in C */;
        double r3769758 = /*Error: no posit support in C */;
        double r3769759 = r3769756 - r3769754;
        double r3769760 = r3769758 / r3769759;
        double r3769761 = r3769750 / r3769760;
        double r3769762 = r3769761 + r3769750;
        double r3769763 = r3769762 / r3769752;
        return r3769763;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

1. Initial program 0.8

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

3. Applied associate-+l+ 0.8

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

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

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

6. Applied associate-/l* 0.8

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

7. Simplified 0.8

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

9. Applied introduce-quire 0.8

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

10. Applied insert-quire-add 0.8

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

11. Applied insert-quire-add 0.7

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

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

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

14. Applied associate-/l* 0.8

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

15. Final simplification 0.8

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

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(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)))