Octave 3.8, jcobi/2

Average Error: 0.9 → 0.6

Time: 54.7s

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 r3898423 = alpha;
        double r3898424 = beta;
        double r3898425 = r3898423 + r3898424;
        double r3898426 = r3898424 - r3898423;
        double r3898427 = r3898425 * r3898426;
        double r3898428 = 2.0;
        double r3898429 = /* ERROR: no posit support in C */;
        double r3898430 = i;
        double r3898431 = r3898429 * r3898430;
        double r3898432 = r3898425 + r3898431;
        double r3898433 = r3898427 / r3898432;
        double r3898434 = 2.0;
        double r3898435 = /* ERROR: no posit support in C */;
        double r3898436 = r3898432 + r3898435;
        double r3898437 = r3898433 / r3898436;
        double r3898438 = 1.0;
        double r3898439 = /* ERROR: no posit support in C */;
        double r3898440 = r3898437 + r3898439;
        double r3898441 = r3898440 / r3898435;
        return r3898441;
}

↓

double f(double alpha, double beta, double i) {
        double r3898442 = alpha;
        double r3898443 = beta;
        double r3898444 = r3898442 + r3898443;
        double r3898445 = i;
        double r3898446 = 2.0;
        double r3898447 = r3898445 * r3898446;
        double r3898448 = r3898442 + r3898447;
        double r3898449 = r3898448 + r3898443;
        double r3898450 = r3898444 / r3898449;
        double r3898451 = r3898443 + r3898442;
        double r3898452 = 2.0;
        double r3898453 = r3898451 + r3898452;
        double r3898454 = r3898447 + r3898453;
        double r3898455 = r3898443 - r3898442;
        double r3898456 = r3898454 / r3898455;
        double r3898457 = r3898450 / r3898456;
        double r3898458 = 1.0;
        double r3898459 = r3898457 + r3898458;
        double r3898460 = r3898459 / r3898452;
        return r3898460;
}

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 +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)))