Octave 3.8, jcobi/2

Average Error: 24.5 → 11.8

Time: 38.2s

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 6.818670779247875958936552940178034387386 \cdot 10^{145}:\\ \;\;\;\;\frac{\sqrt[3]{\left(1 + \left(\beta + \alpha\right) \cdot \left(\frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2} \cdot \sqrt[3]{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2}} \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{2 \cdot i + \left(\beta + \alpha\right)}}{\sqrt[3]{\sqrt[3]{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2} \cdot \left(\sqrt[3]{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2} \cdot \sqrt[3]{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2}\right)}}\right)\right) \cdot \left(\left(1 + \left(\beta + \alpha\right) \cdot \left(\frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2} \cdot \sqrt[3]{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2}} \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{2 \cdot i + \left(\beta + \alpha\right)}}{\sqrt[3]{\sqrt[3]{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2} \cdot \left(\sqrt[3]{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2} \cdot \sqrt[3]{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2}\right)}}\right)\right) \cdot \left(1 + \left(\beta + \alpha\right) \cdot \left(\frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2} \cdot \sqrt[3]{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2}} \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{2 \cdot i + \left(\beta + \alpha\right)}}{\sqrt[3]{\sqrt[3]{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2} \cdot \left(\sqrt[3]{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2} \cdot \sqrt[3]{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2}\right)}}\right)\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{8}{\left(\alpha \cdot \alpha\right) \cdot \alpha} + \left(\frac{2}{\alpha} - \frac{4}{\alpha \cdot \alpha}\right)}{2}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r5326399 = alpha;
        double r5326400 = beta;
        double r5326401 = r5326399 + r5326400;
        double r5326402 = r5326400 - r5326399;
        double r5326403 = r5326401 * r5326402;
        double r5326404 = 2.0;
        double r5326405 = i;
        double r5326406 = r5326404 * r5326405;
        double r5326407 = r5326401 + r5326406;
        double r5326408 = r5326403 / r5326407;
        double r5326409 = r5326407 + r5326404;
        double r5326410 = r5326408 / r5326409;
        double r5326411 = 1.0;
        double r5326412 = r5326410 + r5326411;
        double r5326413 = r5326412 / r5326404;
        return r5326413;
}

↓

double f(double alpha, double beta, double i) {
        double r5326414 = alpha;
        double r5326415 = 6.818670779247876e+145;
        bool r5326416 = r5326414 <= r5326415;
        double r5326417 = 1.0;
        double r5326418 = beta;
        double r5326419 = r5326418 + r5326414;
        double r5326420 = r5326418 - r5326414;
        double r5326421 = cbrt(r5326420);
        double r5326422 = r5326421 * r5326421;
        double r5326423 = 2.0;
        double r5326424 = i;
        double r5326425 = r5326423 * r5326424;
        double r5326426 = r5326425 + r5326419;
        double r5326427 = r5326426 + r5326423;
        double r5326428 = cbrt(r5326427);
        double r5326429 = r5326428 * r5326428;
        double r5326430 = r5326422 / r5326429;
        double r5326431 = r5326421 / r5326426;
        double r5326432 = r5326428 * r5326429;
        double r5326433 = cbrt(r5326432);
        double r5326434 = r5326431 / r5326433;
        double r5326435 = r5326430 * r5326434;
        double r5326436 = r5326419 * r5326435;
        double r5326437 = r5326417 + r5326436;
        double r5326438 = r5326437 * r5326437;
        double r5326439 = r5326437 * r5326438;
        double r5326440 = cbrt(r5326439);
        double r5326441 = r5326440 / r5326423;
        double r5326442 = 8.0;
        double r5326443 = r5326414 * r5326414;
        double r5326444 = r5326443 * r5326414;
        double r5326445 = r5326442 / r5326444;
        double r5326446 = r5326423 / r5326414;
        double r5326447 = 4.0;
        double r5326448 = r5326447 / r5326443;
        double r5326449 = r5326446 - r5326448;
        double r5326450 = r5326445 + r5326449;
        double r5326451 = r5326450 / r5326423;
        double r5326452 = r5326416 ? r5326441 : r5326451;
        return r5326452;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if alpha < 6.818670779247876e+145

   1. Initial program 16.3

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

   3. Applied *-un-lft-identity 16.3

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

   4. Applied *-un-lft-identity 16.3

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

   5. Applied times-frac 5.4

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

   6. Applied times-frac 5.4

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

   7. Simplified 5.4

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

   9. Applied add-cube-cbrt 5.5

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

   10. Applied *-un-lft-identity 5.5

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

   11. Applied add-cube-cbrt 5.5

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

   12. Applied times-frac 5.4

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

   13. Applied times-frac 5.4

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

   15. Applied add-cbrt-cube 5.4

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

   17. Applied add-cbrt-cube 5.4

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

   if 6.818670779247876e+145 < alpha

   1. Initial program 63.3

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

   3. Applied *-un-lft-identity 63.3

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

   4. Applied *-un-lft-identity 63.3

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

   5. Applied times-frac 46.5

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

   6. Applied times-frac 46.6

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

   7. Simplified 46.6

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

   8. Taylor expanded around inf 41.7

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}}{2}\]

   9. Simplified 41.7

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

4. Final simplification 11.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 6.818670779247875958936552940178034387386 \cdot 10^{145}:\\ \;\;\;\;\frac{\sqrt[3]{\left(1 + \left(\beta + \alpha\right) \cdot \left(\frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2} \cdot \sqrt[3]{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2}} \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{2 \cdot i + \left(\beta + \alpha\right)}}{\sqrt[3]{\sqrt[3]{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2} \cdot \left(\sqrt[3]{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2} \cdot \sqrt[3]{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2}\right)}}\right)\right) \cdot \left(\left(1 + \left(\beta + \alpha\right) \cdot \left(\frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2} \cdot \sqrt[3]{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2}} \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{2 \cdot i + \left(\beta + \alpha\right)}}{\sqrt[3]{\sqrt[3]{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2} \cdot \left(\sqrt[3]{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2} \cdot \sqrt[3]{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2}\right)}}\right)\right) \cdot \left(1 + \left(\beta + \alpha\right) \cdot \left(\frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2} \cdot \sqrt[3]{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2}} \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{2 \cdot i + \left(\beta + \alpha\right)}}{\sqrt[3]{\sqrt[3]{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2} \cdot \left(\sqrt[3]{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2} \cdot \sqrt[3]{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2}\right)}}\right)\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{8}{\left(\alpha \cdot \alpha\right) \cdot \alpha} + \left(\frac{2}{\alpha} - \frac{4}{\alpha \cdot \alpha}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (> alpha -1.0) (> beta -1.0) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i))) (+ (+ (+ alpha beta) (* 2.0 i)) 2.0)) 1.0) 2.0))