Octave 3.8, jcobi/4

Average Error: 3.3 → 1.4

Time: 2.1m

Precision: 64

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

Math

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

↓

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

C

double f(double alpha, double beta, double i) {
        double r1633221 = i;
        double r1633222 = alpha;
        double r1633223 = beta;
        double r1633224 = r1633222 + r1633223;
        double r1633225 = r1633224 + r1633221;
        double r1633226 = r1633221 * r1633225;
        double r1633227 = r1633223 * r1633222;
        double r1633228 = r1633227 + r1633226;
        double r1633229 = r1633226 * r1633228;
        double r1633230 = 2.0;
        double r1633231 = /* ERROR: no posit support in C */;
        double r1633232 = r1633231 * r1633221;
        double r1633233 = r1633224 + r1633232;
        double r1633234 = r1633233 * r1633233;
        double r1633235 = r1633229 / r1633234;
        double r1633236 = 1.0;
        double r1633237 = /* ERROR: no posit support in C */;
        double r1633238 = r1633234 - r1633237;
        double r1633239 = r1633235 / r1633238;
        return r1633239;
}

↓

double f(double alpha, double beta, double i) {
        double r1633240 = i;
        double r1633241 = beta;
        double r1633242 = alpha;
        double r1633243 = 2.0;
        double r1633244 = r1633240 * r1633243;
        double r1633245 = r1633242 + r1633244;
        double r1633246 = r1633241 + r1633245;
        double r1633247 = r1633240 / r1633246;
        double r1633248 = 1.0;
        double r1633249 = r1633248 + r1633244;
        double r1633250 = r1633242 + r1633241;
        double r1633251 = r1633249 + r1633250;
        double r1633252 = r1633248 / r1633251;
        double r1633253 = r1633242 + r1633240;
        double r1633254 = r1633241 + r1633253;
        double r1633255 = r1633252 * r1633254;
        double r1633256 = r1633247 * r1633255;
        double r1633257 = r1633243 * r1633240;
        double r1633258 = r1633257 + r1633241;
        double r1633259 = r1633258 - r1633248;
        double r1633260 = r1633242 + r1633259;
        double r1633261 = r1633242 * r1633241;
        double r1633262 = r1633250 + r1633240;
        double r1633263 = r1633240 * r1633262;
        double r1633264 = r1633261 + r1633263;
        double r1633265 = r1633260 / r1633264;
        double r1633266 = r1633257 + r1633250;
        double r1633267 = r1633265 * r1633266;
        double r1633268 = r1633248 / r1633267;
        double r1633269 = r1633256 * r1633268;
        return r1633269;
}

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

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

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

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

10. Applied distribute-rgt-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-rgt-identity-expand 1.4

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

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

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

23. Applied p16-times-frac 1.4

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

24. Simplified 1.4

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

25. Final simplification 1.4

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

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))))