Average Error: 0.9 → 0.6
Time: 52.1s
Precision: 64
\[\alpha \gt \left(-1\right) \land \beta \gt \left(-1\right) \land i \gt \left(0\right)\]
\[\frac{\left(\frac{\left(\frac{\left(\frac{\left(\left(\frac{\alpha}{\beta}\right) \cdot \left(\beta - \alpha\right)\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
↓
\[\frac{\left(0.0 + \frac{\alpha + \beta}{\left(i \cdot 2 + \left(\beta + 2.0\right)\right) + \alpha} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + i \cdot 2}\right) + 1.0}{2.0}\]
\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{\left(0.0 + \frac{\alpha + \beta}{\left(i \cdot 2 + \left(\beta + 2.0\right)\right) + \alpha} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + i \cdot 2}\right) + 1.0}{2.0}
double f(double alpha, double beta, double i) {
        double r4951455 = alpha;
        double r4951456 = beta;
        double r4951457 = r4951455 + r4951456;
        double r4951458 = r4951456 - r4951455;
        double r4951459 = r4951457 * r4951458;
        double r4951460 = 2.0;
        double r4951461 = /* ERROR: no posit support in C */;
        double r4951462 = i;
        double r4951463 = r4951461 * r4951462;
        double r4951464 = r4951457 + r4951463;
        double r4951465 = r4951459 / r4951464;
        double r4951466 = 2.0;
        double r4951467 = /* ERROR: no posit support in C */;
        double r4951468 = r4951464 + r4951467;
        double r4951469 = r4951465 / r4951468;
        double r4951470 = 1.0;
        double r4951471 = /* ERROR: no posit support in C */;
        double r4951472 = r4951469 + r4951471;
        double r4951473 = r4951472 / r4951467;
        return r4951473;
}
↓
double f(double alpha, double beta, double i) {
        double r4951474 = 0.0;
        double r4951475 = alpha;
        double r4951476 = beta;
        double r4951477 = r4951475 + r4951476;
        double r4951478 = i;
        double r4951479 = 2.0;
        double r4951480 = r4951478 * r4951479;
        double r4951481 = 2.0;
        double r4951482 = r4951476 + r4951481;
        double r4951483 = r4951480 + r4951482;
        double r4951484 = r4951483 + r4951475;
        double r4951485 = r4951477 / r4951484;
        double r4951486 = r4951476 - r4951475;
        double r4951487 = r4951477 + r4951480;
        double r4951488 = r4951486 / r4951487;
        double r4951489 = r4951485 * r4951488;
        double r4951490 = r4951474 + r4951489;
        double r4951491 = 1.0;
        double r4951492 = r4951490 + r4951491;
        double r4951493 = r4951492 / r4951481;
        return r4951493;
}

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-expand0.9

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

  4. Applied p16-*-un-lft-identity0.9

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

  5. Applied p16-times-frac0.6

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

  6. Applied p16-times-frac0.6

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

  7. Simplified0.6

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

  8. Simplified0.6

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

  10. Applied p16-*-un-lft-identity0.6

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

  11. Applied associate-*r*0.6

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

  12. Simplified0.6

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

  14. Applied +p16-lft-identity-expand0.6

    \[\leadsto \frac{\left(\frac{\color{blue}{\left(\frac{\left(0.0\right)}{\left(\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\frac{\left(\frac{\left(i \cdot \left(2\right)\right)}{\left(\frac{\beta}{\left(2.0\right)}\right)}\right)}{\alpha}\right)}\right) \cdot \left(\frac{\left(\beta - \alpha\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(i \cdot \left(2\right)\right)}\right)}\right)\right)}\right)}}{\left(1.0\right)}\right)}{\left(2.0\right)}\]

  15. Final simplification0.6

    \[\leadsto \frac{\left(0.0 + \frac{\alpha + \beta}{\left(i \cdot 2 + \left(\beta + 2.0\right)\right) + \alpha} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + i \cdot 2}\right) + 1.0}{2.0}\]

Reproduce

herbie shell --seed 2019162 +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)))