Average Error: 23.4 → 12.4
Time: 2.5m
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 2.778527467618903 \cdot 10^{+23}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} + 1.0\right) \cdot \left(\left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} + 1.0\right) \cdot \left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} + 1.0\right)\right)}}{2.0}\\ \mathbf{elif}\;\alpha \le 7.90075049213558 \cdot 10^{+53}:\\ \;\;\;\;\frac{\frac{\frac{\frac{8.0}{\alpha} - 4.0}{\alpha}}{\alpha} + \frac{2.0}{\alpha}}{2.0}\\ \mathbf{elif}\;\alpha \le 4.755359132529547 \cdot 10^{+94}:\\ \;\;\;\;\frac{\left(\beta + \alpha\right) \cdot \frac{\frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{\sqrt{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}}{\sqrt{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}} + 1.0}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{8.0}{\alpha} - 4.0}{\alpha}}{\alpha} + \frac{2.0}{\alpha}}{2.0}\\ \end{array}\]
\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 2.778527467618903 \cdot 10^{+23}:\\
\;\;\;\;\frac{\sqrt[3]{\left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} + 1.0\right) \cdot \left(\left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} + 1.0\right) \cdot \left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} + 1.0\right)\right)}}{2.0}\\

\mathbf{elif}\;\alpha \le 7.90075049213558 \cdot 10^{+53}:\\
\;\;\;\;\frac{\frac{\frac{\frac{8.0}{\alpha} - 4.0}{\alpha}}{\alpha} + \frac{2.0}{\alpha}}{2.0}\\

\mathbf{elif}\;\alpha \le 4.755359132529547 \cdot 10^{+94}:\\
\;\;\;\;\frac{\left(\beta + \alpha\right) \cdot \frac{\frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{\sqrt{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}}{\sqrt{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}} + 1.0}{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 r32217680 = alpha;
        double r32217681 = beta;
        double r32217682 = r32217680 + r32217681;
        double r32217683 = r32217681 - r32217680;
        double r32217684 = r32217682 * r32217683;
        double r32217685 = 2.0;
        double r32217686 = i;
        double r32217687 = r32217685 * r32217686;
        double r32217688 = r32217682 + r32217687;
        double r32217689 = r32217684 / r32217688;
        double r32217690 = 2.0;
        double r32217691 = r32217688 + r32217690;
        double r32217692 = r32217689 / r32217691;
        double r32217693 = 1.0;
        double r32217694 = r32217692 + r32217693;
        double r32217695 = r32217694 / r32217690;
        return r32217695;
}
↓
double f(double alpha, double beta, double i) {
        double r32217696 = alpha;
        double r32217697 = 2.778527467618903e+23;
        bool r32217698 = r32217696 <= r32217697;
        double r32217699 = beta;
        double r32217700 = r32217699 + r32217696;
        double r32217701 = r32217699 - r32217696;
        double r32217702 = i;
        double r32217703 = 2.0;
        double r32217704 = r32217702 * r32217703;
        double r32217705 = r32217704 + r32217700;
        double r32217706 = r32217701 / r32217705;
        double r32217707 = 2.0;
        double r32217708 = r32217707 + r32217705;
        double r32217709 = r32217706 / r32217708;
        double r32217710 = r32217700 * r32217709;
        double r32217711 = 1.0;
        double r32217712 = r32217710 + r32217711;
        double r32217713 = r32217712 * r32217712;
        double r32217714 = r32217712 * r32217713;
        double r32217715 = cbrt(r32217714);
        double r32217716 = r32217715 / r32217707;
        double r32217717 = 7.90075049213558e+53;
        bool r32217718 = r32217696 <= r32217717;
        double r32217719 = 8.0;
        double r32217720 = r32217719 / r32217696;
        double r32217721 = 4.0;
        double r32217722 = r32217720 - r32217721;
        double r32217723 = r32217722 / r32217696;
        double r32217724 = r32217723 / r32217696;
        double r32217725 = r32217707 / r32217696;
        double r32217726 = r32217724 + r32217725;
        double r32217727 = r32217726 / r32217707;
        double r32217728 = 4.755359132529547e+94;
        bool r32217729 = r32217696 <= r32217728;
        double r32217730 = sqrt(r32217708);
        double r32217731 = r32217706 / r32217730;
        double r32217732 = r32217731 / r32217730;
        double r32217733 = r32217700 * r32217732;
        double r32217734 = r32217733 + r32217711;
        double r32217735 = r32217734 / r32217707;
        double r32217736 = r32217729 ? r32217735 : r32217727;
        double r32217737 = r32217718 ? r32217727 : r32217736;
        double r32217738 = r32217698 ? r32217716 : r32217737;
        return r32217738;
}

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 3 regimes

  2. if alpha < 2.778527467618903e+23

    1. Initial program 11.2

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

    3. Applied *-un-lft-identity11.2

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

    4. Applied times-frac0.4

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

    5. Simplified0.4

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

    7. Applied *-un-lft-identity0.4

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

    8. Applied *-un-lft-identity0.4

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

    9. Applied distribute-lft-out0.4

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

    10. Applied times-frac0.4

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

    11. Simplified0.4

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

    13. Applied add-cbrt-cube0.4

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

    if 2.778527467618903e+23 < alpha < 7.90075049213558e+53 or 4.755359132529547e+94 < alpha

    1. Initial program 53.7

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

    3. Applied *-un-lft-identity53.7

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

    4. Applied times-frac39.8

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

    5. Simplified39.8

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

    7. Applied *-un-lft-identity39.8

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

    8. Applied *-un-lft-identity39.8

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

    9. Applied distribute-lft-out39.8

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

    10. Applied times-frac39.8

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

    11. Simplified39.8

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

    12. Taylor expanded around inf 42.0

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

    13. Simplified42.0

      \[\leadsto \frac{\color{blue}{\frac{2.0}{\alpha} + \frac{\frac{\frac{8.0}{\alpha} - 4.0}{\alpha}}{\alpha}}}{2.0}\]

    if 7.90075049213558e+53 < alpha < 4.755359132529547e+94

    1. Initial program 37.6

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

    3. Applied *-un-lft-identity37.6

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

    4. Applied times-frac26.6

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

    5. Simplified26.6

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

    7. Applied *-un-lft-identity26.6

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

    8. Applied *-un-lft-identity26.6

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

    9. Applied distribute-lft-out26.6

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

    10. Applied times-frac26.5

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

    11. Simplified26.5

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

    13. Applied add-sqr-sqrt26.5

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

    14. Applied associate-/r*26.5

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

  4. Final simplification12.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 2.778527467618903 \cdot 10^{+23}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} + 1.0\right) \cdot \left(\left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} + 1.0\right) \cdot \left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} + 1.0\right)\right)}}{2.0}\\ \mathbf{elif}\;\alpha \le 7.90075049213558 \cdot 10^{+53}:\\ \;\;\;\;\frac{\frac{\frac{\frac{8.0}{\alpha} - 4.0}{\alpha}}{\alpha} + \frac{2.0}{\alpha}}{2.0}\\ \mathbf{elif}\;\alpha \le 4.755359132529547 \cdot 10^{+94}:\\ \;\;\;\;\frac{\left(\beta + \alpha\right) \cdot \frac{\frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{\sqrt{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}}{\sqrt{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}} + 1.0}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{8.0}{\alpha} - 4.0}{\alpha}}{\alpha} + \frac{2.0}{\alpha}}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019107 
(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))