Octave 3.8, jcobi/4

Average Error: 3.3 → 1.4

Time: 1.9m

Precision: 64

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

Math

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

↓

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

C

double f(double alpha, double beta, double i) {
        double r1466047 = i;
        double r1466048 = alpha;
        double r1466049 = beta;
        double r1466050 = r1466048 + r1466049;
        double r1466051 = r1466050 + r1466047;
        double r1466052 = r1466047 * r1466051;
        double r1466053 = r1466049 * r1466048;
        double r1466054 = r1466053 + r1466052;
        double r1466055 = r1466052 * r1466054;
        double r1466056 = 2.0;
        double r1466057 = /* ERROR: no posit support in C */;
        double r1466058 = r1466057 * r1466047;
        double r1466059 = r1466050 + r1466058;
        double r1466060 = r1466059 * r1466059;
        double r1466061 = r1466055 / r1466060;
        double r1466062 = 1.0;
        double r1466063 = /* ERROR: no posit support in C */;
        double r1466064 = r1466060 - r1466063;
        double r1466065 = r1466061 / r1466064;
        return r1466065;
}

↓

double f(double alpha, double beta, double i) {
        double r1466066 = i;
        double r1466067 = 1.0;
        double r1466068 = alpha;
        double r1466069 = beta;
        double r1466070 = r1466068 + r1466069;
        double r1466071 = r1466067 + r1466070;
        double r1466072 = 2.0;
        double r1466073 = r1466066 * r1466072;
        double r1466074 = r1466071 + r1466073;
        double r1466075 = r1466066 / r1466074;
        double r1466076 = r1466073 + r1466070;
        double r1466077 = r1466069 + r1466066;
        double r1466078 = r1466077 + r1466068;
        double r1466079 = r1466076 / r1466078;
        double r1466080 = r1466067 / r1466079;
        double r1466081 = r1466075 * r1466080;
        double r1466082 = r1466068 * r1466069;
        double r1466083 = r1466070 + r1466066;
        double r1466084 = r1466083 * r1466066;
        double r1466085 = r1466082 + r1466084;
        double r1466086 = r1466073 + r1466069;
        double r1466087 = r1466068 + r1466086;
        double r1466088 = r1466085 / r1466087;
        double r1466089 = r1466070 - r1466067;
        double r1466090 = r1466073 + r1466089;
        double r1466091 = r1466067 / r1466090;
        double r1466092 = r1466088 * r1466091;
        double r1466093 = r1466081 * r1466092;
        return r1466093;
}

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-lft-identity-expand 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 1.8

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

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-lft-identity-expand 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

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

14. Simplified 1.5

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

16. Applied *p16-rgt-identity-expand 1.5

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

17. Applied *p16-rgt-identity-expand 1.5

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

18. Applied distribute-rgt-out-- 1.5

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

19. Applied *p16-rgt-identity-expand 1.5

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

20. Applied p16-times-frac 1.5

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

21. Simplified 1.5

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

22. Simplified 1.4

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

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

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

25. Applied associate-/l* 1.4

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

26. Final simplification 1.4

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

Reproduce

herbie shell --seed 2019152 
(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))))