Octave 3.8, jcobi/4

Average Error: 3.3 → 1.4

Time: 1.2m

Precision: 64

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

Math

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

↓

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

C

double f(double alpha, double beta, double i) {
        double r1712038 = i;
        double r1712039 = alpha;
        double r1712040 = beta;
        double r1712041 = r1712039 + r1712040;
        double r1712042 = r1712041 + r1712038;
        double r1712043 = r1712038 * r1712042;
        double r1712044 = r1712040 * r1712039;
        double r1712045 = r1712044 + r1712043;
        double r1712046 = r1712043 * r1712045;
        double r1712047 = 2.0;
        double r1712048 = /* ERROR: no posit support in C */;
        double r1712049 = r1712048 * r1712038;
        double r1712050 = r1712041 + r1712049;
        double r1712051 = r1712050 * r1712050;
        double r1712052 = r1712046 / r1712051;
        double r1712053 = 1.0;
        double r1712054 = /* ERROR: no posit support in C */;
        double r1712055 = r1712051 - r1712054;
        double r1712056 = r1712052 / r1712055;
        return r1712056;
}

↓

double f(double alpha, double beta, double i) {
        double r1712057 = i;
        double r1712058 = beta;
        double r1712059 = alpha;
        double r1712060 = 2.0;
        double r1712061 = r1712057 * r1712060;
        double r1712062 = r1712059 + r1712061;
        double r1712063 = r1712058 + r1712062;
        double r1712064 = r1712057 / r1712063;
        double r1712065 = r1712059 + r1712057;
        double r1712066 = r1712058 + r1712065;
        double r1712067 = 1.0;
        double r1712068 = r1712067 + r1712061;
        double r1712069 = r1712059 + r1712058;
        double r1712070 = r1712068 + r1712069;
        double r1712071 = r1712067 / r1712070;
        double r1712072 = r1712066 * r1712071;
        double r1712073 = r1712064 * r1712072;
        double r1712074 = r1712060 * r1712057;
        double r1712075 = r1712074 + r1712058;
        double r1712076 = r1712075 - r1712067;
        double r1712077 = r1712059 + r1712076;
        double r1712078 = r1712059 * r1712058;
        double r1712079 = r1712069 + r1712057;
        double r1712080 = r1712057 * r1712079;
        double r1712081 = r1712078 + r1712080;
        double r1712082 = r1712077 / r1712081;
        double r1712083 = r1712074 + r1712069;
        double r1712084 = r1712082 * r1712083;
        double r1712085 = r1712067 / r1712084;
        double r1712086 = r1712073 * r1712085;
        return r1712086;
}

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-*-un-lft-identity 1.6

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

9. Applied p16-*-un-lft-identity 1.6

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

10. Applied distribute-lft-out 1.6

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

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

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

12. Applied p16-times-frac 1.5

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

13. Applied p16-times-frac 1.5

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

14. Simplified 1.4

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

15. Simplified 1.4

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

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

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

18. Applied associate-/l* 1.4

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

19. Simplified 1.4

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

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

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

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

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

23. Applied distribute-lft-out 1.4

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

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

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

25. Applied p16-times-frac 1.4

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

26. Simplified 1.4

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

27. Final simplification 1.4

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

Reproduce

herbie shell --seed 2019153 
(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))))