Octave 3.8, jcobi/3

Average Error: 0.4 → 0.3

Time: 32.1s

Precision: 64

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

Math

\[\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\beta \cdot \alpha\right)}\right)}{\left(1.0\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]

↓

\[\frac{\frac{\frac{1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\frac{\beta + \left(\alpha + 2 \cdot 1\right)}{\left(\mathsf{qma}\left(\left(\mathsf{qma}\left(\left(\left(\alpha + \beta\right)\right), \beta, \alpha\right)\right), 1.0, 1.0\right)\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

C

double f(double alpha, double beta) {
        double r3277690 = alpha;
        double r3277691 = beta;
        double r3277692 = r3277690 + r3277691;
        double r3277693 = r3277691 * r3277690;
        double r3277694 = r3277692 + r3277693;
        double r3277695 = 1.0;
        double r3277696 = /* ERROR: no posit support in C */;
        double r3277697 = r3277694 + r3277696;
        double r3277698 = 2.0;
        double r3277699 = /* ERROR: no posit support in C */;
        double r3277700 = 1.0;
        double r3277701 = /* ERROR: no posit support in C */;
        double r3277702 = r3277699 * r3277701;
        double r3277703 = r3277692 + r3277702;
        double r3277704 = r3277697 / r3277703;
        double r3277705 = r3277704 / r3277703;
        double r3277706 = r3277703 + r3277696;
        double r3277707 = r3277705 / r3277706;
        return r3277707;
}

↓

double f(double alpha, double beta) {
        double r3277708 = 1.0;
        double r3277709 = alpha;
        double r3277710 = beta;
        double r3277711 = r3277709 + r3277710;
        double r3277712 = 2.0;
        double r3277713 = 1.0;
        double r3277714 = r3277712 * r3277713;
        double r3277715 = r3277711 + r3277714;
        double r3277716 = r3277708 / r3277715;
        double r3277717 = r3277709 + r3277714;
        double r3277718 = r3277710 + r3277717;
        double r3277719 = /*Error: no posit support in C */;
        double r3277720 = /*Error: no posit support in C */;
        double r3277721 = /*Error: no posit support in C */;
        double r3277722 = /*Error: no posit support in C */;
        double r3277723 = r3277718 / r3277722;
        double r3277724 = r3277716 / r3277723;
        double r3277725 = r3277715 + r3277708;
        double r3277726 = r3277724 / r3277725;
        return r3277726;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

1. Initial program 0.4

   \[\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\beta \cdot \alpha\right)}\right)}{\left(1.0\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]
2. Using strategy rm

3. Applied introduce-quire 0.4

   \[\leadsto \frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\color{blue}{\left(\left(\left(\frac{\alpha}{\beta}\right)\right)\right)}}{\left(\beta \cdot \alpha\right)}\right)}{\left(1.0\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]

4. Applied insert-quire-fdp-add 0.4

   \[\leadsto \frac{\left(\frac{\left(\frac{\left(\frac{\color{blue}{\left(\left(\mathsf{qma}\left(\left(\left(\frac{\alpha}{\beta}\right)\right), \beta, \alpha\right)\right)\right)}}{\left(1.0\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]

5. Applied insert-quire-add 0.4

   \[\leadsto \frac{\left(\frac{\left(\frac{\color{blue}{\left(\left(\mathsf{qma}\left(\left(\mathsf{qma}\left(\left(\left(\frac{\alpha}{\beta}\right)\right), \beta, \alpha\right)\right), \left(1.0\right), \left(1.0\right)\right)\right)\right)}}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]
6. Using strategy rm

7. Applied *p16-rgt-identity-expand 0.4

   \[\leadsto \frac{\left(\frac{\left(\frac{\left(\left(\mathsf{qma}\left(\left(\mathsf{qma}\left(\left(\left(\frac{\alpha}{\beta}\right)\right), \beta, \alpha\right)\right), \left(1.0\right), \left(1.0\right)\right)\right)\right)}{\color{blue}{\left(\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right) \cdot \left(1.0\right)\right)}}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]

8. Applied p16-*-un-lft-identity 0.4

   \[\leadsto \frac{\left(\frac{\left(\frac{\color{blue}{\left(\left(1.0\right) \cdot \left(\left(\mathsf{qma}\left(\left(\mathsf{qma}\left(\left(\left(\frac{\alpha}{\beta}\right)\right), \beta, \alpha\right)\right), \left(1.0\right), \left(1.0\right)\right)\right)\right)\right)}}{\left(\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right) \cdot \left(1.0\right)\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]

9. Applied p16-times-frac 0.4

   \[\leadsto \frac{\left(\frac{\color{blue}{\left(\left(\frac{\left(1.0\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right) \cdot \left(\frac{\left(\left(\mathsf{qma}\left(\left(\mathsf{qma}\left(\left(\left(\frac{\alpha}{\beta}\right)\right), \beta, \alpha\right)\right), \left(1.0\right), \left(1.0\right)\right)\right)\right)}{\left(1.0\right)}\right)\right)}}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]

10. Applied associate-/l* 0.4

    \[\leadsto \frac{\color{blue}{\left(\frac{\left(\frac{\left(1.0\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(\frac{\left(\left(\mathsf{qma}\left(\left(\mathsf{qma}\left(\left(\left(\frac{\alpha}{\beta}\right)\right), \beta, \alpha\right)\right), \left(1.0\right), \left(1.0\right)\right)\right)\right)}{\left(1.0\right)}\right)}\right)}\right)}}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]

11. Simplified 0.3

    \[\leadsto \frac{\left(\frac{\left(\frac{\left(1.0\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\color{blue}{\left(\frac{\left(\frac{\beta}{\left(\frac{\alpha}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\left(\mathsf{qma}\left(\left(\mathsf{qma}\left(\left(\left(\frac{\alpha}{\beta}\right)\right), \beta, \alpha\right)\right), \left(1.0\right), \left(1.0\right)\right)\right)\right)}\right)}}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]

12. Final simplification 0.3

    \[\leadsto \frac{\frac{\frac{1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\frac{\beta + \left(\alpha + 2 \cdot 1\right)}{\left(\mathsf{qma}\left(\left(\mathsf{qma}\left(\left(\left(\alpha + \beta\right)\right), \beta, \alpha\right)\right), 1.0, 1.0\right)\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :pre (and (>.p16 alpha (real->posit16 -1)) (>.p16 beta (real->posit16 -1)))
  (/.p16 (/.p16 (/.p16 (+.p16 (+.p16 (+.p16 alpha beta) (*.p16 beta alpha)) (real->posit16 1.0)) (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) (real->posit16 1)))) (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) (real->posit16 1)))) (+.p16 (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) (real->posit16 1))) (real->posit16 1.0))))