\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]

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

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.248606679275357 \cdot 10^{154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\frac{\alpha + \beta}{1}}{1}, \frac{\sqrt[3]{\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \beta\right) + \alpha}}}{\frac{\left(\mathsf{fma}\left(2, i, \beta\right) + \alpha\right) + 2}{\sqrt[3]{\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \beta\right) + \alpha} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \beta\right) + \alpha}}}}, 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2}\\ \end{array}\]

\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 1.248606679275357 \cdot 10^{154}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{\frac{\alpha + \beta}{1}}{1}, \frac{\sqrt[3]{\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \beta\right) + \alpha}}}{\frac{\left(\mathsf{fma}\left(2, i, \beta\right) + \alpha\right) + 2}{\sqrt[3]{\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \beta\right) + \alpha} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \beta\right) + \alpha}}}}, 1\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2}\\

\end{array}

double f(double alpha, double beta, double i) {
        double r87742 = alpha;
        double r87743 = beta;
        double r87744 = r87742 + r87743;
        double r87745 = r87743 - r87742;
        double r87746 = r87744 * r87745;
        double r87747 = 2.0;
        double r87748 = i;
        double r87749 = r87747 * r87748;
        double r87750 = r87744 + r87749;
        double r87751 = r87746 / r87750;
        double r87752 = r87750 + r87747;
        double r87753 = r87751 / r87752;
        double r87754 = 1.0;
        double r87755 = r87753 + r87754;
        double r87756 = r87755 / r87747;
        return r87756;
}

↓

double f(double alpha, double beta, double i) {
        double r87757 = alpha;
        double r87758 = 1.248606679275357e+154;
        bool r87759 = r87757 <= r87758;
        double r87760 = beta;
        double r87761 = r87757 + r87760;
        double r87762 = 1.0;
        double r87763 = r87761 / r87762;
        double r87764 = r87763 / r87762;
        double r87765 = r87760 - r87757;
        double r87766 = 2.0;
        double r87767 = i;
        double r87768 = fma(r87766, r87767, r87760);
        double r87769 = r87768 + r87757;
        double r87770 = r87765 / r87769;
        double r87771 = cbrt(r87770);
        double r87772 = r87769 + r87766;
        double r87773 = r87770 * r87770;
        double r87774 = cbrt(r87773);
        double r87775 = r87772 / r87774;
        double r87776 = r87771 / r87775;
        double r87777 = 1.0;
        double r87778 = fma(r87764, r87776, r87777);
        double r87779 = r87778 / r87766;
        double r87780 = r87762 / r87757;
        double r87781 = 8.0;
        double r87782 = 3.0;
        double r87783 = pow(r87757, r87782);
        double r87784 = r87762 / r87783;
        double r87785 = r87781 * r87784;
        double r87786 = 4.0;
        double r87787 = 2.0;
        double r87788 = pow(r87757, r87787);
        double r87789 = r87762 / r87788;
        double r87790 = r87786 * r87789;
        double r87791 = r87785 - r87790;
        double r87792 = fma(r87766, r87780, r87791);
        double r87793 = r87792 / r87766;
        double r87794 = r87759 ? r87779 : r87793;
        return r87794;
}

Derivation

Split input into 2 regimes
if alpha < 1.248606679275357e+154
1. Initial program 16.4
  \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
2. Using strategy rm
3. Applied *-un-lft-identity16.4
  \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\color{blue}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)}} + 1}{2}\]
4. Applied *-un-lft-identity16.4
  \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)} + 1}{2}\]
5. Applied times-frac5.5
  \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha + \beta}{1} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)} + 1}{2}\]
6. Applied times-frac5.5
  \[\leadsto \frac{\color{blue}{\frac{\frac{\alpha + \beta}{1}}{1} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} + 1}{2}\]
7. Applied fma-def5.4
  \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{\frac{\alpha + \beta}{1}}{1}, \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}}{2}\]
8. Using strategy rm
9. Applied add-cbrt-cube20.2
  \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{\alpha + \beta}{1}}{1}, \frac{\frac{\beta - \alpha}{\color{blue}{\sqrt[3]{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}{2}\]
10. Applied add-cbrt-cube26.0
  \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{\alpha + \beta}{1}}{1}, \frac{\frac{\color{blue}{\sqrt[3]{\left(\left(\beta - \alpha\right) \cdot \left(\beta - \alpha\right)\right) \cdot \left(\beta - \alpha\right)}}}{\sqrt[3]{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}{2}\]
11. Applied cbrt-undiv26.0
  \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{\alpha + \beta}{1}}{1}, \frac{\color{blue}{\sqrt[3]{\frac{\left(\left(\beta - \alpha\right) \cdot \left(\beta - \alpha\right)\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}{2}\]
12. Simplified5.5
  \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{\alpha + \beta}{1}}{1}, \frac{\sqrt[3]{\color{blue}{{\left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \beta\right) + \alpha}\right)}^{3}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}{2}\]
13. Using strategy rm
14. Applied cube-mult5.5
  \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{\alpha + \beta}{1}}{1}, \frac{\sqrt[3]{\color{blue}{\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \beta\right) + \alpha} \cdot \left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \beta\right) + \alpha} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \beta\right) + \alpha}\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}{2}\]
15. Applied cbrt-prod5.5
  \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{\alpha + \beta}{1}}{1}, \frac{\color{blue}{\sqrt[3]{\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \beta\right) + \alpha}} \cdot \sqrt[3]{\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \beta\right) + \alpha} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \beta\right) + \alpha}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}{2}\]
16. Applied associate-/l*5.5
  \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{\alpha + \beta}{1}}{1}, \color{blue}{\frac{\sqrt[3]{\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \beta\right) + \alpha}}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}{\sqrt[3]{\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \beta\right) + \alpha} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \beta\right) + \alpha}}}}}, 1\right)}{2}\]
17. Simplified5.5
  \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{\alpha + \beta}{1}}{1}, \frac{\sqrt[3]{\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \beta\right) + \alpha}}}{\color{blue}{\frac{\left(\mathsf{fma}\left(2, i, \beta\right) + \alpha\right) + 2}{\sqrt[3]{\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \beta\right) + \alpha} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \beta\right) + \alpha}}}}}, 1\right)}{2}\]
if 1.248606679275357e+154 < 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. Using strategy rm
3. Applied *-un-lft-identity64.0
  \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\color{blue}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)}} + 1}{2}\]
4. Applied *-un-lft-identity64.0
  \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)} + 1}{2}\]
5. Applied times-frac46.7
  \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha + \beta}{1} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)} + 1}{2}\]
6. Applied times-frac46.7
  \[\leadsto \frac{\color{blue}{\frac{\frac{\alpha + \beta}{1}}{1} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} + 1}{2}\]
7. Applied fma-def47.4
  \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{\frac{\alpha + \beta}{1}}{1}, \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}}{2}\]
8. Using strategy rm
9. Applied add-cbrt-cube52.8
  \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{\alpha + \beta}{1}}{1}, \frac{\frac{\beta - \alpha}{\color{blue}{\sqrt[3]{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}{2}\]
10. Applied add-cbrt-cube64.0
  \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{\alpha + \beta}{1}}{1}, \frac{\frac{\color{blue}{\sqrt[3]{\left(\left(\beta - \alpha\right) \cdot \left(\beta - \alpha\right)\right) \cdot \left(\beta - \alpha\right)}}}{\sqrt[3]{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}{2}\]
11. Applied cbrt-undiv64.0
  \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{\alpha + \beta}{1}}{1}, \frac{\color{blue}{\sqrt[3]{\frac{\left(\left(\beta - \alpha\right) \cdot \left(\beta - \alpha\right)\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}{2}\]
12. Simplified47.5
  \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{\alpha + \beta}{1}}{1}, \frac{\sqrt[3]{\color{blue}{{\left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \beta\right) + \alpha}\right)}^{3}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}{2}\]
13. Taylor expanded around inf 41.5
  \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}}{2}\]
14. Simplified41.5
  \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}}{2}\]
Recombined 2 regimes into one program.
Final simplification11.2
\[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.248606679275357 \cdot 10^{154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\frac{\alpha + \beta}{1}}{1}, \frac{\sqrt[3]{\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \beta\right) + \alpha}}}{\frac{\left(\mathsf{fma}\left(2, i, \beta\right) + \alpha\right) + 2}{\sqrt[3]{\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \beta\right) + \alpha} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \beta\right) + \alpha}}}}, 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2}\\ \end{array}\]

Error

Derivation

`if alpha < 1.248606679275357e+154`

`if 1.248606679275357e+154 < alpha`

Reproduce