Octave 3.8, jcobi/2

Average Error: 0.9 → 0.6

Time: 51.6s

Precision: 64

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

Math

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

↓

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

C

double f(double alpha, double beta, double i) {
        double r4147429 = alpha;
        double r4147430 = beta;
        double r4147431 = r4147429 + r4147430;
        double r4147432 = r4147430 - r4147429;
        double r4147433 = r4147431 * r4147432;
        double r4147434 = 2.0;
        double r4147435 = /* ERROR: no posit support in C */;
        double r4147436 = i;
        double r4147437 = r4147435 * r4147436;
        double r4147438 = r4147431 + r4147437;
        double r4147439 = r4147433 / r4147438;
        double r4147440 = 2.0;
        double r4147441 = /* ERROR: no posit support in C */;
        double r4147442 = r4147438 + r4147441;
        double r4147443 = r4147439 / r4147442;
        double r4147444 = 1.0;
        double r4147445 = /* ERROR: no posit support in C */;
        double r4147446 = r4147443 + r4147445;
        double r4147447 = r4147446 / r4147441;
        return r4147447;
}

↓

double f(double alpha, double beta, double i) {
        double r4147448 = beta;
        double r4147449 = alpha;
        double r4147450 = r4147448 + r4147449;
        double r4147451 = i;
        double r4147452 = 2.0;
        double r4147453 = r4147451 * r4147452;
        double r4147454 = 2.0;
        double r4147455 = r4147450 + r4147454;
        double r4147456 = r4147453 + r4147455;
        double r4147457 = r4147450 / r4147456;
        double r4147458 = r4147448 - r4147449;
        double r4147459 = r4147452 * r4147451;
        double r4147460 = r4147448 + r4147459;
        double r4147461 = r4147460 + r4147449;
        double r4147462 = r4147458 / r4147461;
        double r4147463 = r4147457 * r4147462;
        double r4147464 = 1.0;
        double r4147465 = r4147463 + r4147464;
        double r4147466 = r4147465 / r4147454;
        return r4147466;
}

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-rgt-identity-expand 0.9

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

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

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

5. Applied p16-times-frac 0.6

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

6. Applied p16-times-frac 0.6

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

7. Simplified 0.6

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

8. Simplified 0.6

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

10. Applied *p16-rgt-identity-expand 0.6

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

11. Applied associate-/l* 0.6

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

12. Simplified 0.6

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

13. Final simplification 0.6

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

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)))