Octave 3.8, jcobi/4

Average Error: 3.3 → 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

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

↓

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

C

double f(double alpha, double beta, double i) {
        double r3609987 = i;
        double r3609988 = alpha;
        double r3609989 = beta;
        double r3609990 = r3609988 + r3609989;
        double r3609991 = r3609990 + r3609987;
        double r3609992 = r3609987 * r3609991;
        double r3609993 = r3609989 * r3609988;
        double r3609994 = r3609993 + r3609992;
        double r3609995 = r3609992 * r3609994;
        double r3609996 = 2.0;
        double r3609997 = /* ERROR: no posit support in C */;
        double r3609998 = r3609997 * r3609987;
        double r3609999 = r3609990 + r3609998;
        double r3610000 = r3609999 * r3609999;
        double r3610001 = r3609995 / r3610000;
        double r3610002 = 1.0;
        double r3610003 = /* ERROR: no posit support in C */;
        double r3610004 = r3610000 - r3610003;
        double r3610005 = r3610001 / r3610004;
        return r3610005;
}

↓

double f(double alpha, double beta, double i) {
        double r3610006 = i;
        double r3610007 = alpha;
        double r3610008 = beta;
        double r3610009 = r3610007 + r3610008;
        double r3610010 = 2.0;
        double r3610011 = r3610010 * r3610006;
        double r3610012 = r3610009 + r3610011;
        double r3610013 = r3610006 / r3610012;
        double r3610014 = r3610008 + r3610006;
        double r3610015 = r3610007 + r3610014;
        double r3610016 = r3610013 * r3610015;
        double r3610017 = r3610008 + r3610011;
        double r3610018 = 1.0;
        double r3610019 = r3610018 + r3610007;
        double r3610020 = r3610017 + r3610019;
        double r3610021 = r3610016 / r3610020;
        double r3610022 = r3610011 + r3610009;
        double r3610023 = r3610022 - r3610018;
        double r3610024 = r3610018 / r3610023;
        double r3610025 = r3610007 * r3610008;
        double r3610026 = r3610009 + r3610006;
        double r3610027 = r3610006 * r3610026;
        double r3610028 = r3610025 + r3610027;
        double r3610029 = r3610028 / r3610022;
        double r3610030 = r3610024 * r3610029;
        double r3610031 = r3610021 * r3610030;
        return r3610031;
}

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-identity 3.3

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

4. Applied difference-of-squares 3.3

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

5. Applied p16-times-frac 1.7

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

6. Applied p16-times-frac 1.6

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

8. Applied *p16-rgt-identity-expand 1.6

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

9. Applied p16-times-frac 1.5

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

10. Simplified 1.4

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

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

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

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

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

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

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

15. Applied p16-times-frac 1.4

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

16. Applied p16-times-frac 1.5

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

17. Simplified 1.5

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

18. Simplified 1.5

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

20. Applied /p16-rgt-identity-expand 1.5

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

21. Applied associate-*r/ 1.5

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

22. Applied associate-/l/ 1.5

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

23. Simplified 1.4

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

24. Final simplification 1.4

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

Reproduce

herbie shell --seed 2019162 
(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))))