Average Error: 3.3 → 1.4
Time: 4.5m
Precision: 64
\[\alpha \gt \left(-1\right) \land \beta \gt \left(-1\right) \land i \gt \left(1\right)\]
\[\frac{\left(\frac{\left(\left(i \cdot \left(\frac{\left(\frac{\alpha}{\beta}\right)}{i}\right)\right) \cdot \left(\frac{\left(\beta \cdot \alpha\right)}{\left(i \cdot \left(\frac{\left(\frac{\alpha}{\beta}\right)}{i}\right)\right)}\right)\right)}{\left(\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right) \cdot \left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)\right)}\right)}{\left(\left(\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right) \cdot \left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)\right) - \left(1.0\right)\right)}\]
↓
\[\frac{\frac{\left(i + \left(\beta + \alpha\right)\right) \cdot i + \beta \cdot \alpha}{\left(i \cdot 2 + \alpha\right) + \left(\beta + 1.0\right)}}{\left(\beta + i \cdot 2\right) + \alpha} \cdot \left(\frac{\left(\alpha + \beta\right) + i}{\beta + \left(\alpha + i \cdot 2\right)} \cdot \frac{i}{\left(i \cdot 2 - 1.0\right) + \left(\alpha + \beta\right)}\right)\]
\frac{\left(\frac{\left(\left(i \cdot \left(\frac{\left(\frac{\alpha}{\beta}\right)}{i}\right)\right) \cdot \left(\frac{\left(\beta \cdot \alpha\right)}{\left(i \cdot \left(\frac{\left(\frac{\alpha}{\beta}\right)}{i}\right)\right)}\right)\right)}{\left(\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right) \cdot \left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)\right)}\right)}{\left(\left(\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right) \cdot \left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)\right) - \left(1.0\right)\right)}
↓
\frac{\frac{\left(i + \left(\beta + \alpha\right)\right) \cdot i + \beta \cdot \alpha}{\left(i \cdot 2 + \alpha\right) + \left(\beta + 1.0\right)}}{\left(\beta + i \cdot 2\right) + \alpha} \cdot \left(\frac{\left(\alpha + \beta\right) + i}{\beta + \left(\alpha + i \cdot 2\right)} \cdot \frac{i}{\left(i \cdot 2 - 1.0\right) + \left(\alpha + \beta\right)}\right)
double f(double alpha, double beta, double i) {
        double r19312640 = i;
        double r19312641 = alpha;
        double r19312642 = beta;
        double r19312643 = r19312641 + r19312642;
        double r19312644 = r19312643 + r19312640;
        double r19312645 = r19312640 * r19312644;
        double r19312646 = r19312642 * r19312641;
        double r19312647 = r19312646 + r19312645;
        double r19312648 = r19312645 * r19312647;
        double r19312649 = 2.0;
        double r19312650 = /* ERROR: no posit support in C */;
        double r19312651 = r19312650 * r19312640;
        double r19312652 = r19312643 + r19312651;
        double r19312653 = r19312652 * r19312652;
        double r19312654 = r19312648 / r19312653;
        double r19312655 = 1.0;
        double r19312656 = /* ERROR: no posit support in C */;
        double r19312657 = r19312653 - r19312656;
        double r19312658 = r19312654 / r19312657;
        return r19312658;
}
↓
double f(double alpha, double beta, double i) {
        double r19312659 = i;
        double r19312660 = beta;
        double r19312661 = alpha;
        double r19312662 = r19312660 + r19312661;
        double r19312663 = r19312659 + r19312662;
        double r19312664 = r19312663 * r19312659;
        double r19312665 = r19312660 * r19312661;
        double r19312666 = r19312664 + r19312665;
        double r19312667 = 2.0;
        double r19312668 = r19312659 * r19312667;
        double r19312669 = r19312668 + r19312661;
        double r19312670 = 1.0;
        double r19312671 = r19312660 + r19312670;
        double r19312672 = r19312669 + r19312671;
        double r19312673 = r19312666 / r19312672;
        double r19312674 = r19312660 + r19312668;
        double r19312675 = r19312674 + r19312661;
        double r19312676 = r19312673 / r19312675;
        double r19312677 = r19312661 + r19312660;
        double r19312678 = r19312677 + r19312659;
        double r19312679 = r19312661 + r19312668;
        double r19312680 = r19312660 + r19312679;
        double r19312681 = r19312678 / r19312680;
        double r19312682 = r19312668 - r19312670;
        double r19312683 = r19312682 + r19312677;
        double r19312684 = r19312659 / r19312683;
        double r19312685 = r19312681 * r19312684;
        double r19312686 = r19312676 * r19312685;
        return r19312686;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Initial program 3.3

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

  3. Applied p16-*-un-lft-identity3.3

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

  4. Applied difference-of-squares3.3

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

  5. Applied *-commutative3.3

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

  6. Applied p16-times-frac1.7

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

  7. Applied p16-times-frac1.7

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

  9. Applied p16-*-un-lft-identity1.7

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

  10. Applied p16-*-un-lft-identity1.7

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

  11. Applied associate-*l*1.7

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

  12. Applied p16-*-un-lft-identity1.7

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

  13. Applied distribute-lft-out1.7

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

  14. Applied distribute-lft-out--1.7

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

  15. Applied *p16-rgt-identity-expand1.7

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

  16. Applied *-commutative1.7

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

  17. Applied p16-times-frac1.5

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

  18. Applied p16-times-frac1.5

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

  19. Simplified1.4

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

  20. Simplified1.4

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

  22. Applied *p16-rgt-identity-expand1.4

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

  23. Applied distribute-lft1-in1.4

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

  24. Applied p16-*-un-lft-identity1.4

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

  25. Applied associate-*l*1.4

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

  26. Applied p16-*-un-lft-identity1.4

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

  27. Applied distribute-lft-out1.4

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

  28. Applied p16-*-un-lft-identity1.4

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

  29. Applied p16-*-un-lft-identity1.4

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

  30. Applied distribute-lft-out1.4

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

  31. Applied p16-times-frac1.4

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

  32. Applied p16-times-frac1.4

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

  33. Simplified1.4

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

  34. Simplified1.4

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

  36. Applied associate-*r/1.4

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

  37. Simplified1.4

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

  38. Final simplification1.4

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

Reproduce

herbie shell --seed 2019158 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :pre (and (>.p16 alpha (real->posit16 -1)) (>.p16 beta (real->posit16 -1)) (>.p16 i (real->posit16 1)))
  (/.p16 (/.p16 (*.p16 (*.p16 i (+.p16 (+.p16 alpha beta) i)) (+.p16 (*.p16 beta alpha) (*.p16 i (+.p16 (+.p16 alpha beta) i)))) (*.p16 (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) i)) (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) i)))) (-.p16 (*.p16 (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) i)) (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) i))) (real->posit16 1.0))))