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

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

↓

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

C

double f(double alpha, double beta, double i) {
        double r1549178 = i;
        double r1549179 = alpha;
        double r1549180 = beta;
        double r1549181 = r1549179 + r1549180;
        double r1549182 = r1549181 + r1549178;
        double r1549183 = r1549178 * r1549182;
        double r1549184 = r1549180 * r1549179;
        double r1549185 = r1549184 + r1549183;
        double r1549186 = r1549183 * r1549185;
        double r1549187 = 2.0;
        double r1549188 = /* ERROR: no posit support in C */;
        double r1549189 = r1549188 * r1549178;
        double r1549190 = r1549181 + r1549189;
        double r1549191 = r1549190 * r1549190;
        double r1549192 = r1549186 / r1549191;
        double r1549193 = 1.0;
        double r1549194 = /* ERROR: no posit support in C */;
        double r1549195 = r1549191 - r1549194;
        double r1549196 = r1549192 / r1549195;
        return r1549196;
}

↓

double f(double alpha, double beta, double i) {
        double r1549197 = i;
        double r1549198 = beta;
        double r1549199 = alpha;
        double r1549200 = 2.0;
        double r1549201 = r1549197 * r1549200;
        double r1549202 = r1549199 + r1549201;
        double r1549203 = r1549198 + r1549202;
        double r1549204 = r1549197 / r1549203;
        double r1549205 = r1549199 + r1549197;
        double r1549206 = r1549198 + r1549205;
        double r1549207 = 1.0;
        double r1549208 = r1549207 + r1549201;
        double r1549209 = r1549199 + r1549198;
        double r1549210 = r1549208 + r1549209;
        double r1549211 = r1549207 / r1549210;
        double r1549212 = r1549206 * r1549211;
        double r1549213 = r1549204 * r1549212;
        double r1549214 = r1549200 * r1549197;
        double r1549215 = r1549214 + r1549198;
        double r1549216 = r1549215 - r1549207;
        double r1549217 = r1549199 + r1549216;
        double r1549218 = r1549199 * r1549198;
        double r1549219 = r1549209 + r1549197;
        double r1549220 = r1549197 * r1549219;
        double r1549221 = r1549218 + r1549220;
        double r1549222 = r1549217 / r1549221;
        double r1549223 = r1549214 + r1549209;
        double r1549224 = r1549222 * r1549223;
        double r1549225 = r1549207 / r1549224;
        double r1549226 = r1549213 * r1549225;
        return r1549226;
}

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 1.7

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

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-lft-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-lft-identity-expand 1.4

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

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-lft-identity-expand 1.4

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

23. Applied p16-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 +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))))