Average Error: 0.4 → 0.3
Time: 42.2s
Precision: 64
\[\alpha \gt \left(-1\right) \land \beta \gt \left(-1\right)\]
\[\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\beta \cdot \alpha\right)}\right)}{\left(1.0\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]
↓
\[\frac{\left(\left(\frac{\left(\left(\frac{1.0 + \beta}{\frac{\beta + \left(\alpha + 1 \cdot 2\right)}{1.0 + \alpha}}\right)\right)}{\beta + \left(\alpha + 1 \cdot 2\right)}\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\beta \cdot \alpha\right)}\right)}{\left(1.0\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}
↓
\frac{\left(\left(\frac{\left(\left(\frac{1.0 + \beta}{\frac{\beta + \left(\alpha + 1 \cdot 2\right)}{1.0 + \alpha}}\right)\right)}{\beta + \left(\alpha + 1 \cdot 2\right)}\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}
double f(double alpha, double beta) {
        double r3848525 = alpha;
        double r3848526 = beta;
        double r3848527 = r3848525 + r3848526;
        double r3848528 = r3848526 * r3848525;
        double r3848529 = r3848527 + r3848528;
        double r3848530 = 1.0;
        double r3848531 = /* ERROR: no posit support in C */;
        double r3848532 = r3848529 + r3848531;
        double r3848533 = 2.0;
        double r3848534 = /* ERROR: no posit support in C */;
        double r3848535 = 1.0;
        double r3848536 = /* ERROR: no posit support in C */;
        double r3848537 = r3848534 * r3848536;
        double r3848538 = r3848527 + r3848537;
        double r3848539 = r3848532 / r3848538;
        double r3848540 = r3848539 / r3848538;
        double r3848541 = r3848538 + r3848531;
        double r3848542 = r3848540 / r3848541;
        return r3848542;
}
↓
double f(double alpha, double beta) {
        double r3848543 = 1.0;
        double r3848544 = beta;
        double r3848545 = r3848543 + r3848544;
        double r3848546 = alpha;
        double r3848547 = 1.0;
        double r3848548 = 2.0;
        double r3848549 = r3848547 * r3848548;
        double r3848550 = r3848546 + r3848549;
        double r3848551 = r3848544 + r3848550;
        double r3848552 = r3848543 + r3848546;
        double r3848553 = r3848551 / r3848552;
        double r3848554 = r3848545 / r3848553;
        double r3848555 = /*Error: no posit support in C */;
        double r3848556 = /*Error: no posit support in C */;
        double r3848557 = r3848556 / r3848551;
        double r3848558 = /*Error: no posit support in C */;
        double r3848559 = /*Error: no posit support in C */;
        double r3848560 = r3848546 + r3848544;
        double r3848561 = r3848548 * r3848547;
        double r3848562 = r3848560 + r3848561;
        double r3848563 = r3848562 + r3848543;
        double r3848564 = r3848559 / r3848563;
        return r3848564;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Initial program 0.4

    \[\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\beta \cdot \alpha\right)}\right)}{\left(1.0\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]
  2. Using strategy rm

  3. Applied *p16-rgt-identity-expand0.4

    \[\leadsto \frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\beta \cdot \alpha\right)}\right)}{\left(1.0\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\color{blue}{\left(\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right) \cdot \left(1.0\right)\right)}}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]

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

    \[\leadsto \frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\beta \cdot \alpha\right)}\right)}{\left(1.0\right)}\right)}{\color{blue}{\left(\left(1.0\right) \cdot \left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)\right)}}\right)}{\left(\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right) \cdot \left(1.0\right)\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]

  5. Applied *p16-rgt-identity-expand0.4

    \[\leadsto \frac{\left(\frac{\left(\frac{\left(\frac{\color{blue}{\left(\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\beta \cdot \alpha\right)}\right) \cdot \left(1.0\right)\right)}}{\left(1.0\right)}\right)}{\left(\left(1.0\right) \cdot \left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)\right)}\right)}{\left(\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right) \cdot \left(1.0\right)\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]

  6. Applied distribute-lft1-in0.4

    \[\leadsto \frac{\left(\frac{\left(\frac{\color{blue}{\left(\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\beta \cdot \alpha\right)}\right)}{\left(1.0\right)}\right) \cdot \left(1.0\right)\right)}}{\left(\left(1.0\right) \cdot \left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)\right)}\right)}{\left(\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right) \cdot \left(1.0\right)\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]

  7. Applied p16-times-frac0.4

    \[\leadsto \frac{\left(\frac{\color{blue}{\left(\left(\frac{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\beta \cdot \alpha\right)}\right)}{\left(1.0\right)}\right)}{\left(1.0\right)}\right) \cdot \left(\frac{\left(1.0\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)\right)}}{\left(\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right) \cdot \left(1.0\right)\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]

  8. Applied p16-times-frac0.4

    \[\leadsto \frac{\color{blue}{\left(\left(\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\beta \cdot \alpha\right)}\right)}{\left(1.0\right)}\right)}{\left(1.0\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right) \cdot \left(\frac{\left(\frac{\left(1.0\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(1.0\right)}\right)\right)}}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]

  9. Simplified0.4

    \[\leadsto \frac{\left(\left(\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\beta \cdot \alpha\right)}\right)}{\left(1.0\right)}\right)}{\left(1.0\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right) \cdot \color{blue}{\left(\frac{\left(1.0\right)}{\left(\frac{\left(\frac{\beta}{\left(\left(1\right) \cdot \left(2\right)\right)}\right)}{\alpha}\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]
  10. Using strategy rm

  11. Applied introduce-quire0.4

    \[\leadsto \frac{\left(\color{blue}{\left(\left(\left(\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\beta \cdot \alpha\right)}\right)}{\left(1.0\right)}\right)}{\left(1.0\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)\right)\right)} \cdot \left(\frac{\left(1.0\right)}{\left(\frac{\left(\frac{\beta}{\left(\left(1\right) \cdot \left(2\right)\right)}\right)}{\alpha}\right)}\right)\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]

  12. Simplified0.4

    \[\leadsto \frac{\left(\left(\color{blue}{\left(\left(\frac{\left(\frac{\left(\frac{\left(1.0\right)}{\left(\frac{\beta}{\left(\beta \cdot \alpha\right)}\right)}\right)}{\alpha}\right)}{\left(\frac{\left(\frac{\left(\left(2\right) \cdot \left(1\right)\right)}{\beta}\right)}{\alpha}\right)}\right)\right)}\right) \cdot \left(\frac{\left(1.0\right)}{\left(\frac{\left(\frac{\beta}{\left(\left(1\right) \cdot \left(2\right)\right)}\right)}{\alpha}\right)}\right)\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]
  13. Using strategy rm

  14. Applied introduce-quire0.4

    \[\leadsto \frac{\color{blue}{\left(\left(\left(\left(\left(\left(\frac{\left(\frac{\left(\frac{\left(1.0\right)}{\left(\frac{\beta}{\left(\beta \cdot \alpha\right)}\right)}\right)}{\alpha}\right)}{\left(\frac{\left(\frac{\left(\left(2\right) \cdot \left(1\right)\right)}{\beta}\right)}{\alpha}\right)}\right)\right)\right) \cdot \left(\frac{\left(1.0\right)}{\left(\frac{\left(\frac{\beta}{\left(\left(1\right) \cdot \left(2\right)\right)}\right)}{\alpha}\right)}\right)\right)\right)\right)}}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]

  15. Simplified0.3

    \[\leadsto \frac{\left(\color{blue}{\left(\left(\frac{\left(\left(\left(\frac{\left(\left(\frac{\left(1.0\right)}{\beta}\right) \cdot \left(\frac{\left(1.0\right)}{\alpha}\right)\right)}{\left(\frac{\beta}{\left(\frac{\alpha}{\left(\left(1\right) \cdot \left(2\right)\right)}\right)}\right)}\right)\right)\right)}{\left(\frac{\beta}{\left(\frac{\alpha}{\left(\left(1\right) \cdot \left(2\right)\right)}\right)}\right)}\right)\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]
  16. Using strategy rm

  17. Applied associate-/l*0.3

    \[\leadsto \frac{\left(\left(\left(\frac{\left(\left(\color{blue}{\left(\frac{\left(\frac{\left(1.0\right)}{\beta}\right)}{\left(\frac{\left(\frac{\beta}{\left(\frac{\alpha}{\left(\left(1\right) \cdot \left(2\right)\right)}\right)}\right)}{\left(\frac{\left(1.0\right)}{\alpha}\right)}\right)}\right)}\right)\right)}{\left(\frac{\beta}{\left(\frac{\alpha}{\left(\left(1\right) \cdot \left(2\right)\right)}\right)}\right)}\right)\right)\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]

  18. Final simplification0.3

    \[\leadsto \frac{\left(\left(\frac{\left(\left(\frac{1.0 + \beta}{\frac{\beta + \left(\alpha + 1 \cdot 2\right)}{1.0 + \alpha}}\right)\right)}{\beta + \left(\alpha + 1 \cdot 2\right)}\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :pre (and (>.p16 alpha (real->posit16 -1)) (>.p16 beta (real->posit16 -1)))
  (/.p16 (/.p16 (/.p16 (+.p16 (+.p16 (+.p16 alpha beta) (*.p16 beta alpha)) (real->posit16 1.0)) (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) (real->posit16 1)))) (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) (real->posit16 1)))) (+.p16 (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) (real->posit16 1))) (real->posit16 1.0))))