Octave 3.8, jcobi/2

Average Error: 0.9 → 0.6

Time: 55.3s

Precision: 64

\[\alpha \gt \left(-1\right) \land \beta \gt \left(-1\right) \land i \gt \left(0\right)\]

Math

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

↓

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

C

double f(double alpha, double beta, double i) {
        double r3898469 = alpha;
        double r3898470 = beta;
        double r3898471 = r3898469 + r3898470;
        double r3898472 = r3898470 - r3898469;
        double r3898473 = r3898471 * r3898472;
        double r3898474 = 2.0;
        double r3898475 = /* ERROR: no posit support in C */;
        double r3898476 = i;
        double r3898477 = r3898475 * r3898476;
        double r3898478 = r3898471 + r3898477;
        double r3898479 = r3898473 / r3898478;
        double r3898480 = 2.0;
        double r3898481 = /* ERROR: no posit support in C */;
        double r3898482 = r3898478 + r3898481;
        double r3898483 = r3898479 / r3898482;
        double r3898484 = 1.0;
        double r3898485 = /* ERROR: no posit support in C */;
        double r3898486 = r3898483 + r3898485;
        double r3898487 = r3898486 / r3898481;
        return r3898487;
}

↓

double f(double alpha, double beta, double i) {
        double r3898488 = alpha;
        double r3898489 = beta;
        double r3898490 = r3898488 + r3898489;
        double r3898491 = i;
        double r3898492 = 2.0;
        double r3898493 = r3898491 * r3898492;
        double r3898494 = r3898488 + r3898493;
        double r3898495 = r3898494 + r3898489;
        double r3898496 = r3898490 / r3898495;
        double r3898497 = r3898489 + r3898488;
        double r3898498 = 2.0;
        double r3898499 = r3898497 + r3898498;
        double r3898500 = r3898493 + r3898499;
        double r3898501 = r3898489 - r3898488;
        double r3898502 = r3898500 / r3898501;
        double r3898503 = r3898496 / r3898502;
        double r3898504 = 1.0;
        double r3898505 = r3898503 + r3898504;
        double r3898506 = r3898505 / r3898498;
        return r3898506;
}

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. Using strategy rm

3. Applied p16-*-un-lft-identity 0.9

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

4. Applied p16-*-un-lft-identity 0.9

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

5. Applied distribute-lft-out 0.9

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

6. Applied *p16-rgt-identity-expand 0.9

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

7. Applied p16-times-frac 0.6

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

8. Applied p16-times-frac 0.6

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

9. Simplified 0.6

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

10. Simplified 0.6

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

12. Applied p16-*-un-lft-identity 0.6

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

13. Applied associate-/l* 0.6

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

15. Applied associate-*r/ 0.6

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

16. Simplified 0.6

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

17. Final simplification 0.6

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

Reproduce

herbie shell --seed 2019163 
(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)))