Average Error: 23.8 → 11.6
Time: 3.1m
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 6.089121605647622 \cdot 10^{+121}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\left(\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)} \cdot \left(\beta + \alpha\right) + 1.0\right) \cdot \left(\frac{1}{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)} \cdot \left(\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i} \cdot \left(\beta + \alpha\right)\right) + 1.0\right)\right) \cdot \left(\frac{1}{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)} \cdot \left(\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i} \cdot \left(\beta + \alpha\right)\right) + 1.0\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{8.0}{\alpha} - 4.0}{\alpha}}{\alpha} + \frac{2.0}{\alpha}}{2.0}\\ \end{array}\]
\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) + 2.0} + 1.0}{2.0}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 6.089121605647622 \cdot 10^{+121}:\\
\;\;\;\;\frac{\sqrt[3]{\left(\left(\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)} \cdot \left(\beta + \alpha\right) + 1.0\right) \cdot \left(\frac{1}{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)} \cdot \left(\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i} \cdot \left(\beta + \alpha\right)\right) + 1.0\right)\right) \cdot \left(\frac{1}{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)} \cdot \left(\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i} \cdot \left(\beta + \alpha\right)\right) + 1.0\right)}}{2.0}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\frac{\frac{8.0}{\alpha} - 4.0}{\alpha}}{\alpha} + \frac{2.0}{\alpha}}{2.0}\\

\end{array}
double f(double alpha, double beta, double i) {
        double r16243439 = alpha;
        double r16243440 = beta;
        double r16243441 = r16243439 + r16243440;
        double r16243442 = r16243440 - r16243439;
        double r16243443 = r16243441 * r16243442;
        double r16243444 = 2.0;
        double r16243445 = i;
        double r16243446 = r16243444 * r16243445;
        double r16243447 = r16243441 + r16243446;
        double r16243448 = r16243443 / r16243447;
        double r16243449 = 2.0;
        double r16243450 = r16243447 + r16243449;
        double r16243451 = r16243448 / r16243450;
        double r16243452 = 1.0;
        double r16243453 = r16243451 + r16243452;
        double r16243454 = r16243453 / r16243449;
        return r16243454;
}
↓
double f(double alpha, double beta, double i) {
        double r16243455 = alpha;
        double r16243456 = 6.089121605647622e+121;
        bool r16243457 = r16243455 <= r16243456;
        double r16243458 = beta;
        double r16243459 = r16243458 - r16243455;
        double r16243460 = r16243458 + r16243455;
        double r16243461 = 2.0;
        double r16243462 = i;
        double r16243463 = r16243461 * r16243462;
        double r16243464 = r16243460 + r16243463;
        double r16243465 = r16243459 / r16243464;
        double r16243466 = 2.0;
        double r16243467 = r16243466 + r16243464;
        double r16243468 = r16243465 / r16243467;
        double r16243469 = r16243468 * r16243460;
        double r16243470 = 1.0;
        double r16243471 = r16243469 + r16243470;
        double r16243472 = 1.0;
        double r16243473 = r16243472 / r16243467;
        double r16243474 = r16243465 * r16243460;
        double r16243475 = r16243473 * r16243474;
        double r16243476 = r16243475 + r16243470;
        double r16243477 = r16243471 * r16243476;
        double r16243478 = r16243477 * r16243476;
        double r16243479 = cbrt(r16243478);
        double r16243480 = r16243479 / r16243466;
        double r16243481 = 8.0;
        double r16243482 = r16243481 / r16243455;
        double r16243483 = 4.0;
        double r16243484 = r16243482 - r16243483;
        double r16243485 = r16243484 / r16243455;
        double r16243486 = r16243485 / r16243455;
        double r16243487 = r16243466 / r16243455;
        double r16243488 = r16243486 + r16243487;
        double r16243489 = r16243488 / r16243466;
        double r16243490 = r16243457 ? r16243480 : r16243489;
        return r16243490;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if alpha < 6.089121605647622e+121

    1. Initial program 14.9

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity14.9

      \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\color{blue}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)}} + 1.0}{2.0}\]

    4. Applied *-un-lft-identity14.9

      \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)} + 1.0}{2.0}\]

    5. Applied times-frac4.0

      \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha + \beta}{1} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)} + 1.0}{2.0}\]

    6. Applied times-frac4.0

      \[\leadsto \frac{\color{blue}{\frac{\frac{\alpha + \beta}{1}}{1} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0}{2.0}\]

    7. Simplified4.0

      \[\leadsto \frac{\color{blue}{\left(\beta + \alpha\right)} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
    8. Using strategy rm

    9. Applied add-cbrt-cube4.0

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0\right) \cdot \left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0\right)\right) \cdot \left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0\right)}}}{2.0}\]
    10. Using strategy rm

    11. Applied div-inv4.0

      \[\leadsto \frac{\sqrt[3]{\left(\left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0\right) \cdot \left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0\right)\right) \cdot \left(\left(\beta + \alpha\right) \cdot \color{blue}{\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}\right)} + 1.0\right)}}{2.0}\]

    12. Applied associate-*r*4.0

      \[\leadsto \frac{\sqrt[3]{\left(\left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0\right) \cdot \left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0\right)\right) \cdot \left(\color{blue}{\left(\left(\beta + \alpha\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}\right) \cdot \frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0\right)}}{2.0}\]
    13. Using strategy rm

    14. Applied div-inv4.0

      \[\leadsto \frac{\sqrt[3]{\left(\left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0\right) \cdot \left(\left(\beta + \alpha\right) \cdot \color{blue}{\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}\right)} + 1.0\right)\right) \cdot \left(\left(\left(\beta + \alpha\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}\right) \cdot \frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0\right)}}{2.0}\]

    15. Applied associate-*r*4.0

      \[\leadsto \frac{\sqrt[3]{\left(\left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0\right) \cdot \left(\color{blue}{\left(\left(\beta + \alpha\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}\right) \cdot \frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0\right)\right) \cdot \left(\left(\left(\beta + \alpha\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}\right) \cdot \frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0\right)}}{2.0}\]

    if 6.089121605647622e+121 < alpha

    1. Initial program 59.8

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]

    2. Taylor expanded around -inf 42.2

      \[\leadsto \frac{\color{blue}{\left(2.0 \cdot \frac{1}{\alpha} + 8.0 \cdot \frac{1}{{\alpha}^{3}}\right) - 4.0 \cdot \frac{1}{{\alpha}^{2}}}}{2.0}\]

    3. Simplified42.2

      \[\leadsto \frac{\color{blue}{\frac{2.0}{\alpha} + \frac{\frac{\frac{8.0}{\alpha} - 4.0}{\alpha}}{\alpha}}}{2.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification11.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 6.089121605647622 \cdot 10^{+121}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\left(\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)} \cdot \left(\beta + \alpha\right) + 1.0\right) \cdot \left(\frac{1}{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)} \cdot \left(\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i} \cdot \left(\beta + \alpha\right)\right) + 1.0\right)\right) \cdot \left(\frac{1}{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)} \cdot \left(\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i} \cdot \left(\beta + \alpha\right)\right) + 1.0\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{8.0}{\alpha} - 4.0}{\alpha}}{\alpha} + \frac{2.0}{\alpha}}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019121 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (> alpha -1) (> beta -1) (> i 0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) 1.0) 2.0))