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

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

↓

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

C

double f(double alpha, double beta, double i) {
        double r3015207 = i;
        double r3015208 = alpha;
        double r3015209 = beta;
        double r3015210 = r3015208 + r3015209;
        double r3015211 = r3015210 + r3015207;
        double r3015212 = r3015207 * r3015211;
        double r3015213 = r3015209 * r3015208;
        double r3015214 = r3015213 + r3015212;
        double r3015215 = r3015212 * r3015214;
        double r3015216 = 2.0;
        double r3015217 = /* ERROR: no posit support in C */;
        double r3015218 = r3015217 * r3015207;
        double r3015219 = r3015210 + r3015218;
        double r3015220 = r3015219 * r3015219;
        double r3015221 = r3015215 / r3015220;
        double r3015222 = 1.0;
        double r3015223 = /* ERROR: no posit support in C */;
        double r3015224 = r3015220 - r3015223;
        double r3015225 = r3015221 / r3015224;
        return r3015225;
}

↓

double f(double alpha, double beta, double i) {
        double r3015226 = i;
        double r3015227 = alpha;
        double r3015228 = beta;
        double r3015229 = r3015227 + r3015228;
        double r3015230 = 2.0;
        double r3015231 = r3015230 * r3015226;
        double r3015232 = r3015229 + r3015231;
        double r3015233 = r3015226 / r3015232;
        double r3015234 = r3015228 + r3015226;
        double r3015235 = r3015227 + r3015234;
        double r3015236 = r3015233 * r3015235;
        double r3015237 = 1.0;
        double r3015238 = r3015232 + r3015237;
        double r3015239 = r3015236 / r3015238;
        double r3015240 = r3015227 * r3015228;
        double r3015241 = r3015229 + r3015226;
        double r3015242 = r3015226 * r3015241;
        double r3015243 = r3015240 + r3015242;
        double r3015244 = r3015231 + r3015229;
        double r3015245 = r3015243 / r3015244;
        double r3015246 = r3015226 * r3015230;
        double r3015247 = r3015246 + r3015227;
        double r3015248 = r3015228 - r3015237;
        double r3015249 = r3015247 + r3015248;
        double r3015250 = r3015237 / r3015249;
        double r3015251 = r3015245 * r3015250;
        double r3015252 = r3015239 * r3015251;
        return r3015252;
}

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-*-un-lft-identity 1.4

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

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

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

14. Applied distribute-lft-out-- 1.4

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

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

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

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

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

17. Applied p16-times-frac 1.4

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

18. Applied p16-times-frac 1.5

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

19. Simplified 1.5

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

20. Simplified 1.5

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

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

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

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

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

24. Applied p16-times-frac 1.5

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

25. Applied associate-*l* 1.5

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

26. Simplified 1.4

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

27. Final simplification 1.4

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

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(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))))