Average Error: 23.4 → 11.0
Time: 43.2s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.4836991668770156 \cdot 10^{+134}:\\ \;\;\;\;\frac{1.0 + \left(\left(\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + i \cdot 2}\right) \cdot \sqrt{\frac{1}{2.0 + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}\right) \cdot \sqrt{\frac{1}{2.0 + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}}{2.0}\\ \mathbf{elif}\;\alpha \le 7.005757127673538 \cdot 10^{+146}:\\ \;\;\;\;\frac{\left(\frac{2.0}{\alpha} - \frac{4.0}{\alpha \cdot \alpha}\right) + \frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)}}{2.0}\\ \mathbf{elif}\;\alpha \le 1.553504313722894 \cdot 10^{+199}:\\ \;\;\;\;\frac{1.0 + \left(\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + i \cdot 2}\right) \cdot \left(\frac{1}{\sqrt{2.0 + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}} \cdot \frac{1}{\sqrt{2.0 + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{2.0}{\alpha} - \frac{4.0}{\alpha \cdot \alpha}\right) + \frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)}}{2.0}\\ \end{array}\]
\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 1.4836991668770156 \cdot 10^{+134}:\\
\;\;\;\;\frac{1.0 + \left(\left(\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + i \cdot 2}\right) \cdot \sqrt{\frac{1}{2.0 + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}\right) \cdot \sqrt{\frac{1}{2.0 + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}}{2.0}\\

\mathbf{elif}\;\alpha \le 7.005757127673538 \cdot 10^{+146}:\\
\;\;\;\;\frac{\left(\frac{2.0}{\alpha} - \frac{4.0}{\alpha \cdot \alpha}\right) + \frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)}}{2.0}\\

\mathbf{elif}\;\alpha \le 1.553504313722894 \cdot 10^{+199}:\\
\;\;\;\;\frac{1.0 + \left(\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + i \cdot 2}\right) \cdot \left(\frac{1}{\sqrt{2.0 + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}} \cdot \frac{1}{\sqrt{2.0 + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}\right)}{2.0}\\

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

\end{array}
double f(double alpha, double beta, double i) {
        double r6608729 = alpha;
        double r6608730 = beta;
        double r6608731 = r6608729 + r6608730;
        double r6608732 = r6608730 - r6608729;
        double r6608733 = r6608731 * r6608732;
        double r6608734 = 2.0;
        double r6608735 = i;
        double r6608736 = r6608734 * r6608735;
        double r6608737 = r6608731 + r6608736;
        double r6608738 = r6608733 / r6608737;
        double r6608739 = 2.0;
        double r6608740 = r6608737 + r6608739;
        double r6608741 = r6608738 / r6608740;
        double r6608742 = 1.0;
        double r6608743 = r6608741 + r6608742;
        double r6608744 = r6608743 / r6608739;
        return r6608744;
}
↓
double f(double alpha, double beta, double i) {
        double r6608745 = alpha;
        double r6608746 = 1.4836991668770156e+134;
        bool r6608747 = r6608745 <= r6608746;
        double r6608748 = 1.0;
        double r6608749 = beta;
        double r6608750 = r6608745 + r6608749;
        double r6608751 = r6608749 - r6608745;
        double r6608752 = i;
        double r6608753 = 2.0;
        double r6608754 = r6608752 * r6608753;
        double r6608755 = r6608750 + r6608754;
        double r6608756 = r6608751 / r6608755;
        double r6608757 = r6608750 * r6608756;
        double r6608758 = 1.0;
        double r6608759 = 2.0;
        double r6608760 = r6608759 + r6608755;
        double r6608761 = r6608758 / r6608760;
        double r6608762 = sqrt(r6608761);
        double r6608763 = r6608757 * r6608762;
        double r6608764 = r6608763 * r6608762;
        double r6608765 = r6608748 + r6608764;
        double r6608766 = r6608765 / r6608759;
        double r6608767 = 7.005757127673538e+146;
        bool r6608768 = r6608745 <= r6608767;
        double r6608769 = r6608759 / r6608745;
        double r6608770 = 4.0;
        double r6608771 = r6608745 * r6608745;
        double r6608772 = r6608770 / r6608771;
        double r6608773 = r6608769 - r6608772;
        double r6608774 = 8.0;
        double r6608775 = r6608745 * r6608771;
        double r6608776 = r6608774 / r6608775;
        double r6608777 = r6608773 + r6608776;
        double r6608778 = r6608777 / r6608759;
        double r6608779 = 1.553504313722894e+199;
        bool r6608780 = r6608745 <= r6608779;
        double r6608781 = sqrt(r6608760);
        double r6608782 = r6608758 / r6608781;
        double r6608783 = r6608782 * r6608782;
        double r6608784 = r6608757 * r6608783;
        double r6608785 = r6608748 + r6608784;
        double r6608786 = r6608785 / r6608759;
        double r6608787 = r6608780 ? r6608786 : r6608778;
        double r6608788 = r6608768 ? r6608778 : r6608787;
        double r6608789 = r6608747 ? r6608766 : r6608788;
        return r6608789;
}

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 < 1.4836991668770156e+134

    1. Initial program 14.7

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

    3. Applied *-un-lft-identity14.7

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

    4. Applied times-frac4.4

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

    5. Applied associate-/l*4.4

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

    7. Applied div-inv4.4

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

    8. Applied *-un-lft-identity4.4

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

    9. Applied times-frac4.3

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

    10. Simplified4.3

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

    12. Applied add-sqr-sqrt4.4

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

    13. Applied associate-*l*4.3

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

    if 1.4836991668770156e+134 < alpha < 7.005757127673538e+146 or 1.553504313722894e+199 < alpha

    1. Initial program 61.6

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

    2. Taylor expanded around inf 40.0

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

    3. Simplified40.0

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

    if 7.005757127673538e+146 < alpha < 1.553504313722894e+199

    1. Initial program 61.0

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

    3. Applied *-un-lft-identity61.0

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

    4. Applied times-frac40.2

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

    5. Applied associate-/l*40.2

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

    7. Applied div-inv40.2

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

    8. Applied *-un-lft-identity40.2

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

    9. Applied times-frac40.1

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

    10. Simplified40.1

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

    12. Applied add-sqr-sqrt40.3

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

    13. Applied *-un-lft-identity40.3

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

    14. Applied times-frac40.2

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

  4. Final simplification11.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.4836991668770156 \cdot 10^{+134}:\\ \;\;\;\;\frac{1.0 + \left(\left(\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + i \cdot 2}\right) \cdot \sqrt{\frac{1}{2.0 + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}\right) \cdot \sqrt{\frac{1}{2.0 + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}}{2.0}\\ \mathbf{elif}\;\alpha \le 7.005757127673538 \cdot 10^{+146}:\\ \;\;\;\;\frac{\left(\frac{2.0}{\alpha} - \frac{4.0}{\alpha \cdot \alpha}\right) + \frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)}}{2.0}\\ \mathbf{elif}\;\alpha \le 1.553504313722894 \cdot 10^{+199}:\\ \;\;\;\;\frac{1.0 + \left(\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + i \cdot 2}\right) \cdot \left(\frac{1}{\sqrt{2.0 + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}} \cdot \frac{1}{\sqrt{2.0 + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{2.0}{\alpha} - \frac{4.0}{\alpha \cdot \alpha}\right) + \frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)}}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(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))