Average Error: 0.9 → 0.6
Time: 54.5s
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 r4951445 = alpha;
        double r4951446 = beta;
        double r4951447 = r4951445 + r4951446;
        double r4951448 = r4951446 - r4951445;
        double r4951449 = r4951447 * r4951448;
        double r4951450 = 2.0;
        double r4951451 = /* ERROR: no posit support in C */;
        double r4951452 = i;
        double r4951453 = r4951451 * r4951452;
        double r4951454 = r4951447 + r4951453;
        double r4951455 = r4951449 / r4951454;
        double r4951456 = 2.0;
        double r4951457 = /* ERROR: no posit support in C */;
        double r4951458 = r4951454 + r4951457;
        double r4951459 = r4951455 / r4951458;
        double r4951460 = 1.0;
        double r4951461 = /* ERROR: no posit support in C */;
        double r4951462 = r4951459 + r4951461;
        double r4951463 = r4951462 / r4951457;
        return r4951463;
}
↓
double f(double alpha, double beta, double i) {
        double r4951464 = 0.0;
        double r4951465 = alpha;
        double r4951466 = beta;
        double r4951467 = r4951465 + r4951466;
        double r4951468 = i;
        double r4951469 = 2.0;
        double r4951470 = r4951468 * r4951469;
        double r4951471 = 2.0;
        double r4951472 = r4951466 + r4951471;
        double r4951473 = r4951470 + r4951472;
        double r4951474 = r4951473 + r4951465;
        double r4951475 = r4951467 / r4951474;
        double r4951476 = r4951466 - r4951465;
        double r4951477 = r4951467 + r4951470;
        double r4951478 = r4951476 / r4951477;
        double r4951479 = r4951475 * r4951478;
        double r4951480 = r4951464 + r4951479;
        double r4951481 = 1.0;
        double r4951482 = r4951480 + r4951481;
        double r4951483 = r4951482 / r4951471;
        return r4951483;
}

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 
(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)))