Average Error: 23.5 → 11.1
Time: 33.4s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 9.677751062610224 \cdot 10^{+196}:\\ \;\;\;\;\frac{1.0 + \left(\beta + \alpha\right) \cdot \left(\frac{\frac{\sqrt[3]{\beta - \alpha}}{2 \cdot i + \left(\beta + \alpha\right)}}{\sqrt[3]{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2.0}} \cdot \frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2.0} \cdot \sqrt[3]{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2.0}}\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)} + \left(\frac{2.0}{\alpha} - \frac{4.0}{\alpha \cdot \alpha}\right)}{2.0}\\ \end{array}\]
\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 9.677751062610224 \cdot 10^{+196}:\\
\;\;\;\;\frac{1.0 + \left(\beta + \alpha\right) \cdot \left(\frac{\frac{\sqrt[3]{\beta - \alpha}}{2 \cdot i + \left(\beta + \alpha\right)}}{\sqrt[3]{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2.0}} \cdot \frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2.0} \cdot \sqrt[3]{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2.0}}\right)}{2.0}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)} + \left(\frac{2.0}{\alpha} - \frac{4.0}{\alpha \cdot \alpha}\right)}{2.0}\\

\end{array}
double f(double alpha, double beta, double i) {
        double r4271563 = alpha;
        double r4271564 = beta;
        double r4271565 = r4271563 + r4271564;
        double r4271566 = r4271564 - r4271563;
        double r4271567 = r4271565 * r4271566;
        double r4271568 = 2.0;
        double r4271569 = i;
        double r4271570 = r4271568 * r4271569;
        double r4271571 = r4271565 + r4271570;
        double r4271572 = r4271567 / r4271571;
        double r4271573 = 2.0;
        double r4271574 = r4271571 + r4271573;
        double r4271575 = r4271572 / r4271574;
        double r4271576 = 1.0;
        double r4271577 = r4271575 + r4271576;
        double r4271578 = r4271577 / r4271573;
        return r4271578;
}
↓
double f(double alpha, double beta, double i) {
        double r4271579 = alpha;
        double r4271580 = 9.677751062610224e+196;
        bool r4271581 = r4271579 <= r4271580;
        double r4271582 = 1.0;
        double r4271583 = beta;
        double r4271584 = r4271583 + r4271579;
        double r4271585 = r4271583 - r4271579;
        double r4271586 = cbrt(r4271585);
        double r4271587 = 2.0;
        double r4271588 = i;
        double r4271589 = r4271587 * r4271588;
        double r4271590 = r4271589 + r4271584;
        double r4271591 = r4271586 / r4271590;
        double r4271592 = 2.0;
        double r4271593 = r4271590 + r4271592;
        double r4271594 = cbrt(r4271593);
        double r4271595 = r4271591 / r4271594;
        double r4271596 = r4271586 * r4271586;
        double r4271597 = r4271594 * r4271594;
        double r4271598 = r4271596 / r4271597;
        double r4271599 = r4271595 * r4271598;
        double r4271600 = r4271584 * r4271599;
        double r4271601 = r4271582 + r4271600;
        double r4271602 = r4271601 / r4271592;
        double r4271603 = 8.0;
        double r4271604 = r4271579 * r4271579;
        double r4271605 = r4271579 * r4271604;
        double r4271606 = r4271603 / r4271605;
        double r4271607 = r4271592 / r4271579;
        double r4271608 = 4.0;
        double r4271609 = r4271608 / r4271604;
        double r4271610 = r4271607 - r4271609;
        double r4271611 = r4271606 + r4271610;
        double r4271612 = r4271611 / r4271592;
        double r4271613 = r4271581 ? r4271602 : r4271612;
        return r4271613;
}

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 < 9.677751062610224e+196

    1. Initial program 18.2

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

    3. Applied *-un-lft-identity18.2

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

    4. Applied *-un-lft-identity18.2

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

    5. Applied times-frac6.9

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

    6. Applied times-frac6.8

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

    7. Simplified6.8

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

    9. Applied add-cube-cbrt7.0

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

    10. Applied *-un-lft-identity7.0

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

    11. Applied add-cube-cbrt6.9

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

    12. Applied times-frac6.9

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

    13. Applied times-frac6.9

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

    if 9.677751062610224e+196 < alpha

    1. Initial program 63.2

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

    2. Taylor expanded around -inf 42.6

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

    3. Simplified42.6

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

  4. Final simplification11.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 9.677751062610224 \cdot 10^{+196}:\\ \;\;\;\;\frac{1.0 + \left(\beta + \alpha\right) \cdot \left(\frac{\frac{\sqrt[3]{\beta - \alpha}}{2 \cdot i + \left(\beta + \alpha\right)}}{\sqrt[3]{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2.0}} \cdot \frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2.0} \cdot \sqrt[3]{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2.0}}\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)} + \left(\frac{2.0}{\alpha} - \frac{4.0}{\alpha \cdot \alpha}\right)}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019149 
(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))