Octave 3.8, jcobi/4

Average Error: 3.4 → 1.4

Time: 1.3m

Precision: 64

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

Math

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

↓

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

C

double f(double alpha, double beta, double i) {
        double r1317524 = i;
        double r1317525 = alpha;
        double r1317526 = beta;
        double r1317527 = r1317525 + r1317526;
        double r1317528 = r1317527 + r1317524;
        double r1317529 = r1317524 * r1317528;
        double r1317530 = r1317526 * r1317525;
        double r1317531 = r1317530 + r1317529;
        double r1317532 = r1317529 * r1317531;
        double r1317533 = 2.0;
        double r1317534 = /* ERROR: no posit support in C */;
        double r1317535 = r1317534 * r1317524;
        double r1317536 = r1317527 + r1317535;
        double r1317537 = r1317536 * r1317536;
        double r1317538 = r1317532 / r1317537;
        double r1317539 = 1.0;
        double r1317540 = /* ERROR: no posit support in C */;
        double r1317541 = r1317537 - r1317540;
        double r1317542 = r1317538 / r1317541;
        return r1317542;
}

↓

double f(double alpha, double beta, double i) {
        double r1317543 = i;
        double r1317544 = 1.0;
        double r1317545 = 2.0;
        double r1317546 = r1317543 * r1317545;
        double r1317547 = r1317544 + r1317546;
        double r1317548 = alpha;
        double r1317549 = beta;
        double r1317550 = r1317548 + r1317549;
        double r1317551 = r1317547 + r1317550;
        double r1317552 = r1317543 / r1317551;
        double r1317553 = r1317550 + r1317543;
        double r1317554 = r1317548 + r1317546;
        double r1317555 = r1317549 + r1317554;
        double r1317556 = r1317553 / r1317555;
        double r1317557 = r1317552 * r1317556;
        double r1317558 = r1317548 * r1317549;
        double r1317559 = r1317543 * r1317553;
        double r1317560 = r1317558 + r1317559;
        double r1317561 = r1317545 * r1317543;
        double r1317562 = r1317550 + r1317561;
        double r1317563 = r1317544 / r1317562;
        double r1317564 = r1317560 * r1317563;
        double r1317565 = r1317561 - r1317544;
        double r1317566 = r1317548 + r1317565;
        double r1317567 = r1317549 + r1317566;
        double r1317568 = r1317564 / r1317567;
        double r1317569 = r1317557 * r1317568;
        return r1317569;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Initial program 3.4

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

3. Applied p16-*-un-lft-identity 3.4

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

4. Applied difference-of-squares 3.4

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

5. Applied p16-times-frac 1.8

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

6. Applied p16-times-frac 1.7

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

8. Applied *p16-rgt-identity-expand 1.7

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

9. Applied distribute-lft1-in 1.7

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

10. Applied p16-*-un-lft-identity 1.7

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

11. Applied p16-times-frac 1.5

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

12. Applied p16-times-frac 1.5

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

13. Simplified 1.5

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

14. Simplified 1.4

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

16. Applied p16-*-un-lft-identity 1.4

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

17. Applied *p16-rgt-identity-expand 1.4

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

18. Applied p16-times-frac 1.4

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

19. Simplified 1.4

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

21. Applied /p16-rgt-identity-expand 1.4

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

22. Applied associate-*l/ 1.4

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

23. Applied associate-/l/ 1.4

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

24. Simplified 1.4

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

25. Final simplification 1.4

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

Reproduce

herbie shell --seed 2019155 
(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))))