Octave 3.8, jcobi/2

Average Error: 23.9 → 11.2

Time: 18.5s

Precision: 64

\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]

Math

\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.248606679275357 \cdot 10^{154}:\\ \;\;\;\;\frac{e^{\log \left(\sqrt[3]{{\left(\left(\left(\sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}} \cdot \sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}\right) \cdot \left(\alpha + \beta\right)\right) \cdot \frac{\sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}\right)}^{3}} + 1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}{2}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r422 = alpha;
        double r423 = beta;
        double r424 = r422 + r423;
        double r425 = r423 - r422;
        double r426 = r424 * r425;
        double r427 = 2.0;
        double r428 = i;
        double r429 = r427 * r428;
        double r430 = r424 + r429;
        double r431 = r426 / r430;
        double r432 = r430 + r427;
        double r433 = r431 / r432;
        double r434 = 1.0;
        double r435 = r433 + r434;
        double r436 = r435 / r427;
        return r436;
}

↓

double f(double alpha, double beta, double i) {
        double r437 = alpha;
        double r438 = 1.248606679275357e+154;
        bool r439 = r437 <= r438;
        double r440 = beta;
        double r441 = r440 - r437;
        double r442 = r437 + r440;
        double r443 = 2.0;
        double r444 = i;
        double r445 = r443 * r444;
        double r446 = r442 + r445;
        double r447 = r441 / r446;
        double r448 = cbrt(r447);
        double r449 = r448 * r448;
        double r450 = r449 * r442;
        double r451 = r446 + r443;
        double r452 = r448 / r451;
        double r453 = r450 * r452;
        double r454 = 3.0;
        double r455 = pow(r453, r454);
        double r456 = cbrt(r455);
        double r457 = 1.0;
        double r458 = r456 + r457;
        double r459 = log(r458);
        double r460 = exp(r459);
        double r461 = r460 / r443;
        double r462 = 1.0;
        double r463 = r462 / r437;
        double r464 = r443 * r463;
        double r465 = 8.0;
        double r466 = pow(r437, r454);
        double r467 = r462 / r466;
        double r468 = r465 * r467;
        double r469 = r464 + r468;
        double r470 = 4.0;
        double r471 = 2.0;
        double r472 = pow(r437, r471);
        double r473 = r462 / r472;
        double r474 = r470 * r473;
        double r475 = r469 - r474;
        double r476 = r475 / r443;
        double r477 = r439 ? r461 : r476;
        return r477;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if alpha < 1.248606679275357e+154

   1. Initial program 16.4

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 16.4

      \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\color{blue}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)}} + 1}{2}\]

   4. Applied *-un-lft-identity 16.4

      \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)} + 1}{2}\]

   5. Applied times-frac 5.5

      \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha + \beta}{1} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)} + 1}{2}\]

   6. Applied times-frac 5.5

      \[\leadsto \frac{\color{blue}{\frac{\frac{\alpha + \beta}{1}}{1} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} + 1}{2}\]

   7. Simplified 5.5

      \[\leadsto \frac{\color{blue}{\left(\alpha + \beta\right)} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
   8. Using strategy rm

   9. Applied add-exp-log 5.5

      \[\leadsto \frac{\color{blue}{e^{\log \left(\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\right)}}}{2}\]
   10. Using strategy rm

   11. Applied add-cbrt-cube 15.3

       \[\leadsto \frac{e^{\log \left(\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\color{blue}{\sqrt[3]{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)}}} + 1\right)}}{2}\]

   12. Applied add-cbrt-cube 20.2

       \[\leadsto \frac{e^{\log \left(\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\color{blue}{\sqrt[3]{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}}{\sqrt[3]{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)}} + 1\right)}}{2}\]

   13. Applied add-cbrt-cube 26.0

       \[\leadsto \frac{e^{\log \left(\left(\alpha + \beta\right) \cdot \frac{\frac{\color{blue}{\sqrt[3]{\left(\left(\beta - \alpha\right) \cdot \left(\beta - \alpha\right)\right) \cdot \left(\beta - \alpha\right)}}}{\sqrt[3]{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{\sqrt[3]{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)}} + 1\right)}}{2}\]

   14. Applied cbrt-undiv 26.0

       \[\leadsto \frac{e^{\log \left(\left(\alpha + \beta\right) \cdot \frac{\color{blue}{\sqrt[3]{\frac{\left(\left(\beta - \alpha\right) \cdot \left(\beta - \alpha\right)\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}}{\sqrt[3]{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)}} + 1\right)}}{2}\]

   15. Applied cbrt-undiv 26.0

       \[\leadsto \frac{e^{\log \left(\left(\alpha + \beta\right) \cdot \color{blue}{\sqrt[3]{\frac{\frac{\left(\left(\beta - \alpha\right) \cdot \left(\beta - \alpha\right)\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)}}} + 1\right)}}{2}\]

   16. Applied add-cbrt-cube 26.0

       \[\leadsto \frac{e^{\log \left(\color{blue}{\sqrt[3]{\left(\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right)\right) \cdot \left(\alpha + \beta\right)}} \cdot \sqrt[3]{\frac{\frac{\left(\left(\beta - \alpha\right) \cdot \left(\beta - \alpha\right)\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)}} + 1\right)}}{2}\]

   17. Applied cbrt-unprod 26.0

       \[\leadsto \frac{e^{\log \left(\color{blue}{\sqrt[3]{\left(\left(\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right)\right) \cdot \left(\alpha + \beta\right)\right) \cdot \frac{\frac{\left(\left(\beta - \alpha\right) \cdot \left(\beta - \alpha\right)\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)}}} + 1\right)}}{2}\]

   18. Simplified 5.5

       \[\leadsto \frac{e^{\log \left(\sqrt[3]{\color{blue}{{\left(\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}\right)}^{3}}} + 1\right)}}{2}\]
   19. Using strategy rm

   20. Applied *-un-lft-identity 5.5

       \[\leadsto \frac{e^{\log \left(\sqrt[3]{{\left(\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\color{blue}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)}}\right)}^{3}} + 1\right)}}{2}\]

   21. Applied add-cube-cbrt 5.5

       \[\leadsto \frac{e^{\log \left(\sqrt[3]{{\left(\left(\alpha + \beta\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}} \cdot \sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}\right) \cdot \sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)}\right)}^{3}} + 1\right)}}{2}\]

   22. Applied times-frac 5.5

       \[\leadsto \frac{e^{\log \left(\sqrt[3]{{\left(\left(\alpha + \beta\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}} \cdot \sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{1} \cdot \frac{\sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}\right)}\right)}^{3}} + 1\right)}}{2}\]

   23. Applied associate-*r* 5.5

       \[\leadsto \frac{e^{\log \left(\sqrt[3]{{\color{blue}{\left(\left(\left(\alpha + \beta\right) \cdot \frac{\sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}} \cdot \sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{1}\right) \cdot \frac{\sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}\right)}}^{3}} + 1\right)}}{2}\]

   24. Simplified 5.5

       \[\leadsto \frac{e^{\log \left(\sqrt[3]{{\left(\color{blue}{\left(\left(\sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}} \cdot \sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}\right) \cdot \left(\alpha + \beta\right)\right)} \cdot \frac{\sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}\right)}^{3}} + 1\right)}}{2}\]

   if 1.248606679275357e+154 < alpha

   1. Initial program 64.0

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]

   2. Taylor expanded around inf 41.5

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}}{2}\]
3. Recombined 2 regimes into one program.

4. Final simplification 11.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.248606679275357 \cdot 10^{154}:\\ \;\;\;\;\frac{e^{\log \left(\sqrt[3]{{\left(\left(\left(\sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}} \cdot \sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}\right) \cdot \left(\alpha + \beta\right)\right) \cdot \frac{\sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}\right)}^{3}} + 1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2)) 1) 2))