Average Error: 23.2 → 11.0
Time: 1.1m
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 4.992243589258637039989530371711132621231 \cdot 10^{155}:\\ \;\;\;\;\frac{\left(\frac{1}{\frac{-\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}{-1}} \cdot \frac{1}{\frac{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}{\beta - \alpha}}\right) \cdot \frac{\beta + \alpha}{\left(\alpha + \beta\right) + 2 \cdot i} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha} + \left(\frac{8}{{\alpha}^{3}} - \frac{4}{\alpha \cdot \alpha}\right)}{2}\\ \end{array}\]
\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} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 4.992243589258637039989530371711132621231 \cdot 10^{155}:\\
\;\;\;\;\frac{\left(\frac{1}{\frac{-\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}{-1}} \cdot \frac{1}{\frac{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}{\beta - \alpha}}\right) \cdot \frac{\beta + \alpha}{\left(\alpha + \beta\right) + 2 \cdot i} + 1}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{2}{\alpha} + \left(\frac{8}{{\alpha}^{3}} - \frac{4}{\alpha \cdot \alpha}\right)}{2}\\

\end{array}
double f(double alpha, double beta, double i) {
        double r171629 = alpha;
        double r171630 = beta;
        double r171631 = r171629 + r171630;
        double r171632 = r171630 - r171629;
        double r171633 = r171631 * r171632;
        double r171634 = 2.0;
        double r171635 = i;
        double r171636 = r171634 * r171635;
        double r171637 = r171631 + r171636;
        double r171638 = r171633 / r171637;
        double r171639 = r171637 + r171634;
        double r171640 = r171638 / r171639;
        double r171641 = 1.0;
        double r171642 = r171640 + r171641;
        double r171643 = r171642 / r171634;
        return r171643;
}
↓
double f(double alpha, double beta, double i) {
        double r171644 = alpha;
        double r171645 = 4.992243589258637e+155;
        bool r171646 = r171644 <= r171645;
        double r171647 = 1.0;
        double r171648 = beta;
        double r171649 = r171644 + r171648;
        double r171650 = 2.0;
        double r171651 = i;
        double r171652 = r171650 * r171651;
        double r171653 = r171649 + r171652;
        double r171654 = r171653 + r171650;
        double r171655 = sqrt(r171654);
        double r171656 = -r171655;
        double r171657 = -1.0;
        double r171658 = r171656 / r171657;
        double r171659 = r171647 / r171658;
        double r171660 = r171648 - r171644;
        double r171661 = r171655 / r171660;
        double r171662 = r171647 / r171661;
        double r171663 = r171659 * r171662;
        double r171664 = r171648 + r171644;
        double r171665 = r171664 / r171653;
        double r171666 = r171663 * r171665;
        double r171667 = 1.0;
        double r171668 = r171666 + r171667;
        double r171669 = r171668 / r171650;
        double r171670 = r171650 / r171644;
        double r171671 = 8.0;
        double r171672 = 3.0;
        double r171673 = pow(r171644, r171672);
        double r171674 = r171671 / r171673;
        double r171675 = 4.0;
        double r171676 = r171644 * r171644;
        double r171677 = r171675 / r171676;
        double r171678 = r171674 - r171677;
        double r171679 = r171670 + r171678;
        double r171680 = r171679 / r171650;
        double r171681 = r171646 ? r171669 : r171680;
        return r171681;
}

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 < 4.992243589258637e+155

    1. Initial program 15.5

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

    3. Applied *-un-lft-identity15.5

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

    4. Applied times-frac5.2

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

    5. Applied associate-/l*5.2

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

    7. Applied associate-/r/5.2

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

    8. Applied add-cube-cbrt5.2

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

    9. Applied *-un-lft-identity5.2

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

    10. Applied times-frac5.2

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

    11. Applied times-frac5.2

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

    13. Applied frac-2neg5.2

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

    15. Applied *-un-lft-identity5.2

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

    16. Applied distribute-lft-neg-in5.2

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

    17. Applied add-sqr-sqrt5.3

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

    18. Applied distribute-lft-neg-in5.3

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

    19. Applied times-frac5.3

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

    20. Applied add-cube-cbrt5.3

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

    21. Applied times-frac5.3

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

    22. Applied times-frac5.3

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

    23. Simplified5.3

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

    if 4.992243589258637e+155 < 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 40.9

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

    3. Simplified40.9

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

  4. Final simplification11.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 4.992243589258637039989530371711132621231 \cdot 10^{155}:\\ \;\;\;\;\frac{\left(\frac{1}{\frac{-\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}{-1}} \cdot \frac{1}{\frac{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}{\beta - \alpha}}\right) \cdot \frac{\beta + \alpha}{\left(\alpha + \beta\right) + 2 \cdot i} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha} + \left(\frac{8}{{\alpha}^{3}} - \frac{4}{\alpha \cdot \alpha}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 
(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))