自分の備忘用。求め方とかどうでもいいので、一覧が欲しい。
$0\le r \le 1/2$に対する$\sin\pi r, \cos\pi r, \tan\pi r$の値。
分母が51以上については、そのうち追加するかも。
| $r$ | $\sin\pi r$ | $\cos\pi r$ | $\tan\pi r$ |
|---|---|---|---|
| $0$ | $0$ | $1$ | $0$ |
| $\frac{1}{2}$ | $1$ | $0$ | inf |
| $\frac{1}{3}$ | $\frac{\sqrt{3}}{2}$ | $\frac{1}{2}$ | $\sqrt{3}$ |
| $\frac{1}{4}$ | $\frac{\sqrt{2}}{2}$ | $\frac{\sqrt{2}}{2}$ | $1$ |
| $\frac{1}{5}$ | $\frac{\sqrt{10-2\sqrt{5}}}{4}$, $\sqrt{\frac{5-\sqrt{5}}{8}}$ | $\frac{\sqrt{5}+1}{4}$ | $\sqrt{5-2\sqrt{5}}$ |
| $\frac{2}{5}$ | $\frac{\sqrt{10+2\sqrt{5}}}{4}$, $\sqrt{\frac{5+\sqrt{5}}{8}}$ | $\frac{\sqrt{5}-1}{4}$ | $\sqrt{5+2\sqrt{5}}$ |
| $\frac{1}{6}$ | $\frac{1}{2}$ | $\frac{\sqrt{3}}{2}$ | $\frac{\sqrt{3}}{3}$ |
| $\frac{1}{8}$ | $\frac{\sqrt{2-\sqrt{2}}}{2}$, $\sqrt{\frac{2-\sqrt{2}}{4}}$ | $\frac{\sqrt{2+\sqrt{2}}}{2}$, $\sqrt{\frac{2+\sqrt{2}}{4}}$ | $\sqrt{2}-1$ |
| $\frac{3}{8}$ | $\frac{\sqrt{2+\sqrt{2}}}{2}$, $\sqrt{\frac{2+\sqrt{2}}{4}}$ | $\frac{\sqrt{2-\sqrt{2}}}{2}$, $\sqrt{\frac{2-\sqrt{2}}{4}}$ | $\sqrt{2}+1$ |
| $\frac{1}{10}$ | $\frac{\sqrt{5}-1}{4}$ | $\frac{\sqrt{10+2\sqrt{5}}}{4}$, $\sqrt{\frac{5+\sqrt{5}}{8}}$ | $\frac{1}{\sqrt{5+2\sqrt{5}}}$, $\frac{\sqrt{25-10\sqrt{5}}}{5}$, $\sqrt{\frac{5-2\sqrt{5}}{5}}$ |
| $\frac{3}{10}$ | $\frac{\sqrt{5}+1}{4}$ | $\frac{\sqrt{10-2\sqrt{5}}}{4}$, $\sqrt{\frac{5-\sqrt{5}}{8}}$ | $\frac{1}{\sqrt{5-2\sqrt{5}}}$, $\frac{\sqrt{25+10\sqrt{5}}}{5}$, $\sqrt{\frac{5+2\sqrt{5}}{5}}$ |
| $\frac{1}{12}$ | $\frac{\sqrt{6}-\sqrt{2}}{4}$ | $\frac{\sqrt{6}+\sqrt{2}}{4}$ | $2-\sqrt{3}$ |
| $\frac{5}{12}$ | $\frac{\sqrt{6}+\sqrt{2}}{4}$ | $\frac{\sqrt{6}-\sqrt{2}}{4}$ | $2+\sqrt{3}$ |
| $\frac{1}{15}$ | $\frac{\sqrt{3}-\sqrt{15}+\sqrt{2(5+\sqrt{5})}}{8}$ | $\frac{\sqrt{3(10+2\sqrt{5})}+\sqrt{5}-1}{8}$, $\frac{\sqrt{9+\sqrt{5}+\sqrt{3(10-2\sqrt{5})}}}{4}$ | $\sqrt{23-10\sqrt{5}-2\sqrt{3(85-38\sqrt{5})}}$ |
| $\frac{2}{15}$ | $\frac{\sqrt{3}+\sqrt{15}-\sqrt{2(5-\sqrt{5})}}{8}$ | $\frac{\sqrt{3(10-2\sqrt{5})}+\sqrt{5}+1}{8}$, $\frac{\sqrt{9-\sqrt{5}+\sqrt{3(10+2\sqrt{5})}}}{4}$ | $\sqrt{23+10\sqrt{5}-2\sqrt{3(85+38\sqrt{5})}}$ |
| $\frac{4}{15}$ | $\frac{-\sqrt{3}+\sqrt{15}+\sqrt{2(5+\sqrt{5})}}{8}$ | $\frac{\sqrt{3(10+2\sqrt{5})}-\sqrt{5}+1}{8}$, $\frac{\sqrt{9+\sqrt{5}-\sqrt{3(10-2\sqrt{5})}}}{4}$ | $\sqrt{23-10\sqrt{5}+2\sqrt{3(85-38\sqrt{5})}}$ |
| $\frac{7}{15}$ | $\frac{\sqrt{3}+\sqrt{15}+\sqrt{2(5-\sqrt{5})}}{8}$ | $\frac{\sqrt{3(10-2\sqrt{5})}-\sqrt{5}-1}{8}$, $\frac{\sqrt{9-\sqrt{5}-\sqrt{3(10+2\sqrt{5})}}}{4}$ | $\sqrt{23+10\sqrt{5}+2\sqrt{3(85+38\sqrt{5})}}$ |
| $\frac{1}{16}$ | $\frac{\sqrt{2-\sqrt{2+\sqrt{2}}}}{2}$ | $\frac{\sqrt{2+\sqrt{2+\sqrt{2}}}}{2}$ | $\sqrt{2(2+\sqrt{2})}-\sqrt{2}-1$, $\sqrt{\frac{2-\sqrt{2+\sqrt{2}}}{2+\sqrt{2+\sqrt{2}}}}$ |
| $\frac{3}{16}$ | $\frac{\sqrt{2-\sqrt{2-\sqrt{2}}}}{2}$ | $\frac{\sqrt{2+\sqrt{2-\sqrt{2}}}}{2}$ | $\sqrt{2(2-\sqrt{2})}-\sqrt{2}+1$, $\sqrt{\frac{2-\sqrt{2-\sqrt{2}}}{2+\sqrt{2-\sqrt{2}}}}$ |
| $\frac{5}{16}$ | $\frac{\sqrt{2+\sqrt{2-\sqrt{2}}}}{2}$ | $\frac{\sqrt{2-\sqrt{2-\sqrt{2}}}}{2}$ | $\sqrt{2(2-\sqrt{2})}+\sqrt{2}-1$, $\sqrt{\frac{2+\sqrt{2-\sqrt{2}}}{2-\sqrt{2-\sqrt{2}}}}$ |
| $\frac{7}{16}$ | $\frac{\sqrt{2+\sqrt{2+\sqrt{2}}}}{2}$ | $\frac{\sqrt{2-\sqrt{2+\sqrt{2}}}}{2}$ | $\sqrt{2(2+\sqrt{2})}+\sqrt{2}+1$, $\sqrt{\frac{2+\sqrt{2+\sqrt{2}}}{2-\sqrt{2+\sqrt{2}}}}$ |
| $\frac{1}{17}$ | $\sqrt{\frac{17-\sqrt{17}-\sqrt{2(17-\sqrt{17})}-2\sqrt{17+3\sqrt{17}-\sqrt{2(85+19\sqrt{17})}}}{32}}$ | $\sqrt{\frac{15+\sqrt{17}+\sqrt{2(17-\sqrt{17})}+2\sqrt{17+3\sqrt{17}-\sqrt{2(85+19\sqrt{17})}}}{32}}$ | $\sqrt{\frac{17-\sqrt{17}-\sqrt{2(17-\sqrt{17})}-2\sqrt{17+3\sqrt{17}-\sqrt{2(85+19\sqrt{17})}}}{15+\sqrt{17}+\sqrt{2(17-\sqrt{17})}+2\sqrt{17+3\sqrt{17}-\sqrt{2(85+19\sqrt{17})}}}}$ |
| $\frac{2}{17}$ | $\sqrt{\frac{17-\sqrt{17}+\sqrt{2(17-\sqrt{17})}-2\sqrt{17+3\sqrt{17}+\sqrt{2(85+19\sqrt{17})}}}{32}}$ | $\sqrt{\frac{15+\sqrt{17}-\sqrt{2(17-\sqrt{17})}+2\sqrt{17+3\sqrt{17}+\sqrt{2(85+19\sqrt{17})}}}{32}}$ | $\sqrt{\frac{17-\sqrt{17}+\sqrt{2(17-\sqrt{17})}-2\sqrt{17+3\sqrt{17}+\sqrt{2(85+19\sqrt{17})}}}{15+\sqrt{17}-\sqrt{2(17-\sqrt{17})}+2\sqrt{17+3\sqrt{17}+\sqrt{2(85+19\sqrt{17})}}}}$ |
| $\frac{3}{17}$ | $\sqrt{\frac{17+\sqrt{17}-\sqrt{2(17+\sqrt{17})}-2\sqrt{17-3\sqrt{17}+\sqrt{2(85-19\sqrt{17})}}}{32}}$ | $\sqrt{\frac{15-\sqrt{17}+\sqrt{2(17+\sqrt{17})}+2\sqrt{17-3\sqrt{17}+\sqrt{2(85-19\sqrt{17})}}}{32}}$ | $\sqrt{\frac{17+\sqrt{17}-\sqrt{2(17+\sqrt{17})}-2\sqrt{17-3\sqrt{17}+\sqrt{2(85-19\sqrt{17})}}}{15-\sqrt{17}+\sqrt{2(17+\sqrt{17})}+2\sqrt{17-3\sqrt{17}+\sqrt{2(85-19\sqrt{17})}}}}$ |
| $\frac{4}{17}$ | $\sqrt{\frac{17-\sqrt{17}-\sqrt{2(17-\sqrt{17})}+2\sqrt{17+3\sqrt{17}-\sqrt{2(85+19\sqrt{17})}}}{32}}$ | $\sqrt{\frac{15+\sqrt{17}+\sqrt{2(17-\sqrt{17})}-2\sqrt{17+3\sqrt{17}-\sqrt{2(85+19\sqrt{17})}}}{32}}$ | $\sqrt{\frac{17-\sqrt{17}-\sqrt{2(17-\sqrt{17})}+2\sqrt{17+3\sqrt{17}-\sqrt{2(85+19\sqrt{17})}}}{15+\sqrt{17}+\sqrt{2(17-\sqrt{17})}-2\sqrt{17+3\sqrt{17}-\sqrt{2(85+19\sqrt{17})}}}}$ |
| $\frac{5}{17}$ | $\sqrt{\frac{17+\sqrt{17}-\sqrt{2(17+\sqrt{17})}+2\sqrt{17-3\sqrt{17}+\sqrt{2(85-19\sqrt{17})}}}{32}}$ | $\sqrt{\frac{15-\sqrt{17}+\sqrt{2(17+\sqrt{17})}-2\sqrt{17-3\sqrt{17}+\sqrt{2(85-19\sqrt{17})}}}{32}}$ | $\sqrt{\frac{17+\sqrt{17}-\sqrt{2(17+\sqrt{17})}+2\sqrt{17-3\sqrt{17}+\sqrt{2(85-19\sqrt{17})}}}{15-\sqrt{17}+\sqrt{2(17+\sqrt{17})}-2\sqrt{17-3\sqrt{17}+\sqrt{2(85-19\sqrt{17})}}}}$ |
| $\frac{6}{17}$ | $\sqrt{\frac{17+\sqrt{17}+\sqrt{2(17+\sqrt{17})}-2\sqrt{17-3\sqrt{17}-\sqrt{2(85-19\sqrt{17})}}}{32}}$ | $\sqrt{\frac{15-\sqrt{17}-\sqrt{2(17+\sqrt{17})}+2\sqrt{17-3\sqrt{17}-\sqrt{2(85-19\sqrt{17})}}}{32}}$ | $\sqrt{\frac{17+\sqrt{17}+\sqrt{2(17+\sqrt{17})}-2\sqrt{17-3\sqrt{17}-\sqrt{2(85-19\sqrt{17})}}}{15-\sqrt{17}-\sqrt{2(17+\sqrt{17})}+2\sqrt{17-3\sqrt{17}-\sqrt{2(85-19\sqrt{17})}}}}$ |
| $\frac{7}{17}$ | $\sqrt{\frac{17+\sqrt{17}+\sqrt{2(17+\sqrt{17})}+2\sqrt{17-3\sqrt{17}-\sqrt{2(85-19\sqrt{17})}}}{32}}$ | $\sqrt{\frac{15-\sqrt{17}-\sqrt{2(17+\sqrt{17})}-2\sqrt{17-3\sqrt{17}-\sqrt{2(85-19\sqrt{17})}}}{32}}$ | $\sqrt{\frac{17+\sqrt{17}+\sqrt{2(17+\sqrt{17})}+2\sqrt{17-3\sqrt{17}-\sqrt{2(85-19\sqrt{17})}}}{15-\sqrt{17}-\sqrt{2(17+\sqrt{17})}-2\sqrt{17-3\sqrt{17}-\sqrt{2(85-19\sqrt{17})}}}}$ |
| $\frac{8}{17}$ | $\sqrt{\frac{17-\sqrt{17}+\sqrt{2(17-\sqrt{17})}+2\sqrt{17+3\sqrt{17}+\sqrt{2(85+19\sqrt{17})}}}{32}}$ | $\sqrt{\frac{15+\sqrt{17}-\sqrt{2(17-\sqrt{17})}-2\sqrt{17+3\sqrt{17}+\sqrt{2(85+19\sqrt{17})}}}{32}}$ | $\sqrt{\frac{17-\sqrt{17}+\sqrt{2(17-\sqrt{17})}+2\sqrt{17+3\sqrt{17}+\sqrt{2(85+19\sqrt{17})}}}{15+\sqrt{17}-\sqrt{2(17-\sqrt{17})}-2\sqrt{17+3\sqrt{17}+\sqrt{2(85+19\sqrt{17})}}}}$ |
| $\frac{1}{20}$ | $\frac{\sqrt{2}+\sqrt{10}-2\sqrt{5-\sqrt{5}}}{8}$ | $\frac{\sqrt{2}+\sqrt{10}+2\sqrt{5-\sqrt{5}}}{8}$ | $\sqrt{5}+1-\sqrt{5+2\sqrt{5}}$ |
| $\frac{3}{20}$ | $\frac{\sqrt{2}-\sqrt{10}+2\sqrt{5+\sqrt{5}}}{8}$ | $\frac{-\sqrt{2}+\sqrt{10}+2\sqrt{5+\sqrt{5}}}{8}$ | $\sqrt{5}-1-\sqrt{5-2\sqrt{5}}$ |
| $\frac{7}{20}$ | $\frac{-\sqrt{2}+\sqrt{10}+2\sqrt{5+\sqrt{5}}}{8}$ | $\frac{\sqrt{2}-\sqrt{10}+2\sqrt{5+\sqrt{5}}}{8}$ | $\sqrt{5}-1+\sqrt{5-2\sqrt{5}}$ |
| $\frac{9}{20}$ | $\frac{\sqrt{2}+\sqrt{10}+2\sqrt{5-\sqrt{5}}}{8}$ | $\frac{\sqrt{2}+\sqrt{10}-2\sqrt{5-\sqrt{5}}}{8}$ | $\sqrt{5}+1+\sqrt{5+2\sqrt{5}}$ |
| $\frac{1}{24}$ | $\frac{\sqrt{2+\sqrt{2}}-\sqrt{6-3\sqrt{2}}}{4}$ | $\frac{\sqrt{2-\sqrt{2}}+\sqrt{6+3\sqrt{2}}}{4}$ | $\sqrt{6}-\sqrt{3}+\sqrt{2}-2$ |
| $\frac{5}{24}$ | $\frac{-\sqrt{2-\sqrt{2}}+\sqrt{6+3\sqrt{2}}}{4}$ | $\frac{\sqrt{2+\sqrt{2}}+\sqrt{6-3\sqrt{2}}}{4}$ | $\sqrt{6}+\sqrt{3}-\sqrt{2}-2$ |
| $\frac{7}{24}$ | $\frac{\sqrt{2+\sqrt{2}}+\sqrt{6-3\sqrt{2}}}{4}$ | $\frac{-\sqrt{2-\sqrt{2}}+\sqrt{6+3\sqrt{2}}}{4}$ | $\sqrt{6}-\sqrt{3}-\sqrt{2}+2$ |
| $\frac{11}{24}$ | $\frac{\sqrt{2-\sqrt{2}}+\sqrt{6+3\sqrt{2}}}{4}$ | $\frac{\sqrt{2+\sqrt{2}}-\sqrt{6-3\sqrt{2}}}{4}$ | $\sqrt{6}+\sqrt{3}+\sqrt{2}+2$ |
| $\frac{1}{30}$ | $\frac{\sqrt{3(10-2\sqrt{5})}-\sqrt{5}-1}{8}$ | $\frac{\sqrt{3}+\sqrt{15}+\sqrt{2(5-\sqrt{5})}}{8}$ | $\sqrt{7-2\sqrt{5}-2\sqrt{3(5-2\sqrt{5})}}$ |
| $\frac{7}{30}$ | $\frac{\sqrt{3(10+2\sqrt{5})}-\sqrt{5}+1}{8}$ | $\frac{-\sqrt{3}+\sqrt{15}+\sqrt{2(5+\sqrt{5})}}{8}$ | $\sqrt{7+2\sqrt{5}-2\sqrt{3(5+2\sqrt{5})}}$ |
| $\frac{11}{30}$ | $\frac{\sqrt{3(10-2\sqrt{5})}+\sqrt{5}+1}{8}$ | $\frac{\sqrt{3}+\sqrt{15}-\sqrt{2(5-\sqrt{5})}}{8}$ | $\sqrt{7-2\sqrt{5}+2\sqrt{3(5-2\sqrt{5})}}$ |
| $\frac{13}{30}$ | $\frac{\sqrt{3(10+2\sqrt{5})}+\sqrt{5}-1}{8}$ | $\frac{\sqrt{3}-\sqrt{15}+\sqrt{2(5+\sqrt{5})}}{8}$ | $\sqrt{7+2\sqrt{5}+2\sqrt{3(5+2\sqrt{5})}}$ |
| $\frac{1}{32}$ | $\frac{\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2}}}}}{2}$ | $\frac{\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}}{2}$ | $\sqrt{\frac{2-\sqrt{2+\sqrt{2+\sqrt{2}}}}{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}}$ |
| $\frac{3}{32}$ | $\frac{\sqrt{2-\sqrt{2+\sqrt{2-\sqrt{2}}}}}{2}$ | $\frac{\sqrt{2+\sqrt{2+\sqrt{2-\sqrt{2}}}}}{2}$ | $\sqrt{\frac{2-\sqrt{2+\sqrt{2-\sqrt{2}}}}{2+\sqrt{2+\sqrt{2-\sqrt{2}}}}}$ |
| $\frac{5}{32}$ | $\frac{\sqrt{2-\sqrt{2-\sqrt{2-\sqrt{2}}}}}{2}$ | $\frac{\sqrt{2+\sqrt{2-\sqrt{2-\sqrt{2}}}}}{2}$ | $\sqrt{\frac{2-\sqrt{2-\sqrt{2-\sqrt{2}}}}{2+\sqrt{2-\sqrt{2-\sqrt{2}}}}}$ |
| $\frac{7}{32}$ | $\frac{\sqrt{2-\sqrt{2-\sqrt{2+\sqrt{2}}}}}{2}$ | $\frac{\sqrt{2+\sqrt{2-\sqrt{2+\sqrt{2}}}}}{2}$ | $\sqrt{\frac{2-\sqrt{2-\sqrt{2+\sqrt{2}}}}{2+\sqrt{2-\sqrt{2+\sqrt{2}}}}}$ |
| $\frac{9}{32}$ | $\frac{\sqrt{2+\sqrt{2-\sqrt{2+\sqrt{2}}}}}{2}$ | $\frac{\sqrt{2-\sqrt{2-\sqrt{2+\sqrt{2}}}}}{2}$ | $\sqrt{\frac{2+\sqrt{2-\sqrt{2+\sqrt{2}}}}{2-\sqrt{2-\sqrt{2+\sqrt{2}}}}}$ |
| $\frac{11}{32}$ | $\frac{\sqrt{2+\sqrt{2-\sqrt{2-\sqrt{2}}}}}{2}$ | $\frac{\sqrt{2-\sqrt{2-\sqrt{2-\sqrt{2}}}}}{2}$ | $\sqrt{\frac{2+\sqrt{2-\sqrt{2-\sqrt{2}}}}{2-\sqrt{2-\sqrt{2-\sqrt{2}}}}}$ |
| $\frac{13}{32}$ | $\frac{\sqrt{2+\sqrt{2+\sqrt{2-\sqrt{2}}}}}{2}$ | $\frac{\sqrt{2-\sqrt{2+\sqrt{2-\sqrt{2}}}}}{2}$ | $\sqrt{\frac{2+\sqrt{2+\sqrt{2-\sqrt{2}}}}{2-\sqrt{2+\sqrt{2-\sqrt{2}}}}}$ |
| $\frac{15}{32}$ | $\frac{\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}}{2}$ | $\frac{\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2}}}}}{2}$ | $\sqrt{\frac{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}{2-\sqrt{2+\sqrt{2+\sqrt{2}}}}}$ |
| $\frac{1}{34}$ | $\sqrt{\frac{15+\sqrt{17}-\sqrt{2(17-\sqrt{17})}-2\sqrt{17+3\sqrt{17}+\sqrt{2(85+19\sqrt{17})}}}{32}}$ | $\sqrt{\frac{17-\sqrt{17}+\sqrt{2(17-\sqrt{17})}+2\sqrt{17+3\sqrt{17}+\sqrt{2(85+19\sqrt{17})}}}{32}}$ | $\sqrt{\frac{15+\sqrt{17}-\sqrt{2(17-\sqrt{17})}-2\sqrt{17+3\sqrt{17}+\sqrt{2(85+19\sqrt{17})}}}{17-\sqrt{17}+\sqrt{2(17-\sqrt{17})}+2\sqrt{17+3\sqrt{17}+\sqrt{2(85+19\sqrt{17})}}}}$ |
| $\frac{3}{34}$ | $\sqrt{\frac{15-\sqrt{17}-\sqrt{2(17+\sqrt{17})}-2\sqrt{17-3\sqrt{17}-\sqrt{2(85-19\sqrt{17})}}}{32}}$ | $\sqrt{\frac{17+\sqrt{17}+\sqrt{2(17+\sqrt{17})}+2\sqrt{17-3\sqrt{17}-\sqrt{2(85-19\sqrt{17})}}}{32}}$ | $\sqrt{\frac{15-\sqrt{17}-\sqrt{2(17+\sqrt{17})}-2\sqrt{17-3\sqrt{17}-\sqrt{2(85-19\sqrt{17})}}}{17+\sqrt{17}+\sqrt{2(17+\sqrt{17})}+2\sqrt{17-3\sqrt{17}-\sqrt{2(85-19\sqrt{17})}}}}$ |
| $\frac{5}{34}$ | $\sqrt{\frac{15-\sqrt{17}-\sqrt{2(17+\sqrt{17})}+2\sqrt{17-3\sqrt{17}-\sqrt{2(85-19\sqrt{17})}}}{32}}$ | $\sqrt{\frac{17+\sqrt{17}+\sqrt{2(17+\sqrt{17})}-2\sqrt{17-3\sqrt{17}-\sqrt{2(85-19\sqrt{17})}}}{32}}$ | $\sqrt{\frac{15-\sqrt{17}-\sqrt{2(17+\sqrt{17})}+2\sqrt{17-3\sqrt{17}-\sqrt{2(85-19\sqrt{17})}}}{17+\sqrt{17}+\sqrt{2(17+\sqrt{17})}-2\sqrt{17-3\sqrt{17}-\sqrt{2(85-19\sqrt{17})}}}}$ |
| $\frac{7}{34}$ | $\sqrt{\frac{15-\sqrt{17}+\sqrt{2(17+\sqrt{17})}-2\sqrt{17-3\sqrt{17}+\sqrt{2(85-19\sqrt{17})}}}{32}}$ | $\sqrt{\frac{17+\sqrt{17}-\sqrt{2(17+\sqrt{17})}+2\sqrt{17-3\sqrt{17}+\sqrt{2(85-19\sqrt{17})}}}{32}}$ | $\sqrt{\frac{15-\sqrt{17}+\sqrt{2(17+\sqrt{17})}-2\sqrt{17-3\sqrt{17}+\sqrt{2(85-19\sqrt{17})}}}{17+\sqrt{17}-\sqrt{2(17+\sqrt{17})}+2\sqrt{17-3\sqrt{17}+\sqrt{2(85-19\sqrt{17})}}}}$ |
| $\frac{9}{34}$ | $\sqrt{\frac{15+\sqrt{17}+\sqrt{2(17-\sqrt{17})}-2\sqrt{17+3\sqrt{17}-\sqrt{2(85+19\sqrt{17})}}}{32}}$ | $\sqrt{\frac{17-\sqrt{17}-\sqrt{2(17-\sqrt{17})}+2\sqrt{17+3\sqrt{17}-\sqrt{2(85+19\sqrt{17})}}}{32}}$ | $\sqrt{\frac{15+\sqrt{17}+\sqrt{2(17-\sqrt{17})}-2\sqrt{17+3\sqrt{17}-\sqrt{2(85+19\sqrt{17})}}}{17-\sqrt{17}-\sqrt{2(17-\sqrt{17})}+2\sqrt{17+3\sqrt{17}-\sqrt{2(85+19\sqrt{17})}}}}$ |
| $\frac{11}{34}$ | $\sqrt{\frac{15-\sqrt{17}+\sqrt{2(17+\sqrt{17})}+2\sqrt{17-3\sqrt{17}+\sqrt{2(85-19\sqrt{17})}}}{32}}$ | $\sqrt{\frac{17+\sqrt{17}-\sqrt{2(17+\sqrt{17})}-2\sqrt{17-3\sqrt{17}+\sqrt{2(85-19\sqrt{17})}}}{32}}$ | $\sqrt{\frac{15-\sqrt{17}+\sqrt{2(17+\sqrt{17})}+2\sqrt{17-3\sqrt{17}+\sqrt{2(85-19\sqrt{17})}}}{17+\sqrt{17}-\sqrt{2(17+\sqrt{17})}-2\sqrt{17-3\sqrt{17}+\sqrt{2(85-19\sqrt{17})}}}}$ |
| $\frac{13}{34}$ | $\sqrt{\frac{15+\sqrt{17}-\sqrt{2(17-\sqrt{17})}+2\sqrt{17+3\sqrt{17}+\sqrt{2(85+19\sqrt{17})}}}{32}}$ | $\sqrt{\frac{17-\sqrt{17}+\sqrt{2(17-\sqrt{17})}-2\sqrt{17+3\sqrt{17}+\sqrt{2(85+19\sqrt{17})}}}{32}}$ | $\sqrt{\frac{15+\sqrt{17}-\sqrt{2(17-\sqrt{17})}+2\sqrt{17+3\sqrt{17}+\sqrt{2(85+19\sqrt{17})}}}{17-\sqrt{17}+\sqrt{2(17-\sqrt{17})}-2\sqrt{17+3\sqrt{17}+\sqrt{2(85+19\sqrt{17})}}}}$ |
| $\frac{15}{34}$ | $\sqrt{\frac{15+\sqrt{17}+\sqrt{2(17-\sqrt{17})}+2\sqrt{17+3\sqrt{17}-\sqrt{2(85+19\sqrt{17})}}}{32}}$ | $\sqrt{\frac{17-\sqrt{17}-\sqrt{2(17-\sqrt{17})}-2\sqrt{17+3\sqrt{17}-\sqrt{2(85+19\sqrt{17})}}}{32}}$ | $\sqrt{\frac{15+\sqrt{17}+\sqrt{2(17-\sqrt{17})}+2\sqrt{17+3\sqrt{17}-\sqrt{2(85+19\sqrt{17})}}}{17-\sqrt{17}-\sqrt{2(17-\sqrt{17})}-2\sqrt{17+3\sqrt{17}-\sqrt{2(85+19\sqrt{17})}}}}$ |
| $\frac{1}{40}$ | $\frac{\sqrt{2(2-\sqrt{2})(5+\sqrt{5})}-(\sqrt{5}-1)\sqrt{2+\sqrt{2}}}{8}$ | $\frac{\sqrt{2(2+\sqrt{2})(5+\sqrt{5})}+(\sqrt{5}-1)\sqrt{2-\sqrt{2}}}{8}$ | $\frac{\sqrt{2(2-\sqrt{2})(5+\sqrt{5})}-(\sqrt{5}-1)\sqrt{2+\sqrt{2}}}{\sqrt{2(2+\sqrt{2})(5+\sqrt{5})}+(\sqrt{5}-1)\sqrt{2-\sqrt{2}}}$ |
| $\frac{3}{40}$ | $\frac{\sqrt{2(2+\sqrt{2})(5-\sqrt{5})}-(\sqrt{5}+1)\sqrt{2-\sqrt{2}}}{8}$ | $\frac{\sqrt{2(2-\sqrt{2})(5-\sqrt{5})}+(\sqrt{5}+1)\sqrt{2+\sqrt{2}}}{8}$ | $\frac{\sqrt{2(2+\sqrt{2})(5-\sqrt{5})}-(\sqrt{5}+1)\sqrt{2-\sqrt{2}}}{\sqrt{2(2-\sqrt{2})(5-\sqrt{5})}+(\sqrt{5}+1)\sqrt{2+\sqrt{2}}}$ |
| $\frac{7}{40}$ | $\frac{-\sqrt{2(2-\sqrt{2})(5-\sqrt{5})}+(\sqrt{5}+1)\sqrt{2+\sqrt{2}}}{8}$ | $\frac{\sqrt{2(2+\sqrt{2})(5-\sqrt{5})}+(\sqrt{5}+1)\sqrt{2-\sqrt{2}}}{8}$ | $\frac{-\sqrt{2(2-\sqrt{2})(5-\sqrt{5})}+(\sqrt{5}+1)\sqrt{2+\sqrt{2}}}{\sqrt{2(2+\sqrt{2})(5-\sqrt{5})}+(\sqrt{5}+1)\sqrt{2-\sqrt{2}}}$ |
| $\frac{9}{40}$ | $\frac{\sqrt{2(2-\sqrt{2})(5+\sqrt{5})}+(\sqrt{5}-1)\sqrt{2+\sqrt{2}}}{8}$ | $\frac{\sqrt{2(2+\sqrt{2})(5+\sqrt{5})}-(\sqrt{5}-1)\sqrt{2-\sqrt{2}}}{8}$ | $\frac{\sqrt{2(2-\sqrt{2})(5+\sqrt{5})}+(\sqrt{5}-1)\sqrt{2+\sqrt{2}}}{\sqrt{2(2+\sqrt{2})(5+\sqrt{5})}-(\sqrt{5}-1)\sqrt{2-\sqrt{2}}}$ |
| $\frac{11}{40}$ | $\frac{\sqrt{2(2+\sqrt{2})(5+\sqrt{5})}-(\sqrt{5}-1)\sqrt{2-\sqrt{2}}}{8}$ | $\frac{\sqrt{2(2-\sqrt{2})(5+\sqrt{5})}+(\sqrt{5}-1)\sqrt{2+\sqrt{2}}}{8}$ | $\frac{\sqrt{2(2+\sqrt{2})(5+\sqrt{5})}-(\sqrt{5}-1)\sqrt{2-\sqrt{2}}}{\sqrt{2(2-\sqrt{2})(5+\sqrt{5})}+(\sqrt{5}-1)\sqrt{2+\sqrt{2}}}$ |
| $\frac{13}{40}$ | $\frac{\sqrt{2(2+\sqrt{2})(5-\sqrt{5})}+(\sqrt{5}+1)\sqrt{2-\sqrt{2}}}{8}$ | $\frac{-\sqrt{2(2-\sqrt{2})(5-\sqrt{5})}+(\sqrt{5}+1)\sqrt{2+\sqrt{2}}}{8}$ | $\frac{\sqrt{2(2+\sqrt{2})(5-\sqrt{5})}+(\sqrt{5}+1)\sqrt{2-\sqrt{2}}}{-\sqrt{2(2-\sqrt{2})(5-\sqrt{5})}+(\sqrt{5}+1)\sqrt{2+\sqrt{2}}}$ |
| $\frac{17}{40}$ | $\frac{\sqrt{2(2-\sqrt{2})(5-\sqrt{5})}+(\sqrt{5}+1)\sqrt{2+\sqrt{2}}}{8}$ | $\frac{\sqrt{2(2+\sqrt{2})(5-\sqrt{5})}-(\sqrt{5}+1)\sqrt{2-\sqrt{2}}}{8}$ | $\frac{\sqrt{2(2-\sqrt{2})(5-\sqrt{5})}+(\sqrt{5}+1)\sqrt{2+\sqrt{2}}}{\sqrt{2(2+\sqrt{2})(5-\sqrt{5})}-(\sqrt{5}+1)\sqrt{2-\sqrt{2}}}$ |
| $\frac{19}{40}$ | $\frac{\sqrt{2(2+\sqrt{2})(5+\sqrt{5})}+(\sqrt{5}-1)\sqrt{2-\sqrt{2}}}{8}$ | $\frac{\sqrt{2(2-\sqrt{2})(5+\sqrt{5})}-(\sqrt{5}-1)\sqrt{2+\sqrt{2}}}{8}$ | $\frac{\sqrt{2(2+\sqrt{2})(5+\sqrt{5})}+(\sqrt{5}-1)\sqrt{2-\sqrt{2}}}{\sqrt{2(2-\sqrt{2})(5+\sqrt{5})}-(\sqrt{5}-1)\sqrt{2+\sqrt{2}}}$ |
| $\frac{1}{48}$ | $\frac{\sqrt{8-2\sqrt{2-\sqrt{2}}-2\sqrt{3(2+\sqrt{2})}}}{4}$ | $\frac{\sqrt{8+2\sqrt{2-\sqrt{2}}+2\sqrt{3(2+\sqrt{2})}}}{4}$ | $\sqrt{\frac{4-\sqrt{2-\sqrt{2}}-\sqrt{3(2+\sqrt{2})}}{4+\sqrt{2-\sqrt{2}}+\sqrt{3(2+\sqrt{2})}}}$ |
| $\frac{5}{48}$ | $\frac{\sqrt{8-2\sqrt{2+\sqrt{2}}-2\sqrt{3(2-\sqrt{2})}}}{4}$ | $\frac{\sqrt{8+2\sqrt{2+\sqrt{2}}+2\sqrt{3(2-\sqrt{2})}}}{4}$ | $\sqrt{\frac{4-\sqrt{2+\sqrt{2}}-\sqrt{3(2-\sqrt{2})}}{4+\sqrt{2+\sqrt{2}}+\sqrt{3(2-\sqrt{2})}}}$ |
| $\frac{7}{48}$ | $\frac{\sqrt{8+2\sqrt{2-\sqrt{2}}-2\sqrt{3(2+\sqrt{2})}}}{4}$ | $\frac{\sqrt{8-2\sqrt{2-\sqrt{2}}+2\sqrt{3(2+\sqrt{2})}}}{4}$ | $\sqrt{\frac{4+\sqrt{2-\sqrt{2}}-\sqrt{3(2+\sqrt{2})}}{4-\sqrt{2-\sqrt{2}}+\sqrt{3(2+\sqrt{2})}}}$ |
| $\frac{11}{48}$ | $\frac{\sqrt{8-2\sqrt{2+\sqrt{2}}+2\sqrt{3(2-\sqrt{2})}}}{4}$ | $\frac{\sqrt{8+2\sqrt{2+\sqrt{2}}-2\sqrt{3(2-\sqrt{2})}}}{4}$ | $\sqrt{\frac{4-\sqrt{2+\sqrt{2}}+\sqrt{3(2-\sqrt{2})}}{4+\sqrt{2+\sqrt{2}}-\sqrt{3(2-\sqrt{2})}}}$ |
| $\frac{13}{48}$ | $\frac{\sqrt{8+2\sqrt{2+\sqrt{2}}-2\sqrt{3(2-\sqrt{2})}}}{4}$ | $\frac{\sqrt{8-2\sqrt{2+\sqrt{2}}+2\sqrt{3(2-\sqrt{2})}}}{4}$ | $\sqrt{\frac{4+\sqrt{2+\sqrt{2}}-\sqrt{3(2-\sqrt{2})}}{4-\sqrt{2+\sqrt{2}}+\sqrt{3(2-\sqrt{2})}}}$ |
| $\frac{17}{48}$ | $\frac{\sqrt{8-2\sqrt{2-\sqrt{2}}+2\sqrt{3(2+\sqrt{2})}}}{4}$ | $\frac{\sqrt{8+2\sqrt{2-\sqrt{2}}-2\sqrt{3(2+\sqrt{2})}}}{4}$ | $\sqrt{\frac{4-\sqrt{2-\sqrt{2}}+\sqrt{3(2+\sqrt{2})}}{4+\sqrt{2-\sqrt{2}}-\sqrt{3(2+\sqrt{2})}}}$ |
| $\frac{19}{48}$ | $\frac{\sqrt{8+2\sqrt{2+\sqrt{2}}+2\sqrt{3(2-\sqrt{2})}}}{4}$ | $\frac{\sqrt{8-2\sqrt{2+\sqrt{2}}-2\sqrt{3(2-\sqrt{2})}}}{4}$ | $\sqrt{\frac{4+\sqrt{2+\sqrt{2}}+\sqrt{3(2-\sqrt{2})}}{4-\sqrt{2+\sqrt{2}}-\sqrt{3(2-\sqrt{2})}}}$ |
| $\frac{23}{48}$ | $\frac{\sqrt{8+2\sqrt{2-\sqrt{2}}+2\sqrt{3(2+\sqrt{2})}}}{4}$ | $\frac{\sqrt{8-2\sqrt{2-\sqrt{2}}-2\sqrt{3(2+\sqrt{2})}}}{4}$ | $\sqrt{\frac{4+\sqrt{2-\sqrt{2}}+\sqrt{3(2+\sqrt{2})}}{4-\sqrt{2-\sqrt{2}}-\sqrt{3(2+\sqrt{2})}}}$ |
検証コード
import math
from sympy import *
from sympy.parsing.latex import parse_latex
def list_to_markdown(table):
rows = [" | ".join(map(str, row)) for row in table]
separator = " | ".join(["---"] * len(table[0]))
return rows[0] + "\n" + separator + "\n" + "\n".join(rows[1:])
lists = [(S.Zero, [[r'0'], [r'1'], [r'0']]),
(S.Half, [[r'1'], [r'0'], [r'inf']]),
(Rational(1, 3), [[r'\frac{\sqrt{3}}{2}'], [r'\frac{1}{2}'], [r'\sqrt{3}']]),
(Rational(1, 4), [[r'\frac{\sqrt{2}}{2}'], [r'\frac{\sqrt{2}}{2}'], [r'1']]),
(Rational(1, 5), [[r'\frac{\sqrt{10-2\sqrt{5}}}{4}', r'\sqrt{\frac{5-\sqrt{5}}{8}}'], [r'\frac{\sqrt{5}+1}{4}'], [r'\sqrt{5-2\sqrt{5}}']]),
(Rational(2, 5), [[r'\frac{\sqrt{10+2\sqrt{5}}}{4}', r'\sqrt{\frac{5+\sqrt{5}}{8}}'], [r'\frac{\sqrt{5}-1}{4}'], [r'\sqrt{5+2\sqrt{5}}']]),
(Rational(1, 6), [[r'\frac{1}{2}'], [r'\frac{\sqrt{3}}{2}'], [r'\frac{\sqrt{3}}{3}']]),
(Rational(1, 8), [[r'\frac{\sqrt{2-\sqrt{2}}}{2}', r'\sqrt{\frac{2-\sqrt{2}}{4}}'], [r'\frac{\sqrt{2+\sqrt{2}}}{2}', r'\sqrt{\frac{2+\sqrt{2}}{4}}'], [r'\sqrt{2}-1']]),
(Rational(3, 8), [[r'\frac{\sqrt{2+\sqrt{2}}}{2}', r'\sqrt{\frac{2+\sqrt{2}}{4}}'], [r'\frac{\sqrt{2-\sqrt{2}}}{2}', r'\sqrt{\frac{2-\sqrt{2}}{4}}'], [r'\sqrt{2}+1']]),
(Rational(1, 10), [[r'\frac{\sqrt{5}-1}{4}'], [r'\frac{\sqrt{10+2\sqrt{5}}}{4}', r'\sqrt{\frac{5+\sqrt{5}}{8}}'], [r'\frac{1}{\sqrt{5+2\sqrt{5}}}', r'\frac{\sqrt{25-10\sqrt{5}}}{5}', r'\sqrt{\frac{5-2\sqrt{5}}{5}}']]),
(Rational(3, 10), [[r'\frac{\sqrt{5}+1}{4}'], [r'\frac{\sqrt{10-2\sqrt{5}}}{4}', r'\sqrt{\frac{5-\sqrt{5}}{8}}'], [r'\frac{1}{\sqrt{5-2\sqrt{5}}}', r'\frac{\sqrt{25+10\sqrt{5}}}{5}', r'\sqrt{\frac{5+2\sqrt{5}}{5}}']]),
(Rational(1, 12), [[r'\frac{\sqrt{6}-\sqrt{2}}{4}'], [r'\frac{\sqrt{6}+\sqrt{2}}{4}'], [r'2-\sqrt{3}']]),
(Rational(5, 12), [[r'\frac{\sqrt{6}+\sqrt{2}}{4}'], [r'\frac{\sqrt{6}-\sqrt{2}}{4}'], [r'2+\sqrt{3}']]),
(Rational(1, 15), [[r'\frac{\sqrt{3}-\sqrt{15}+\sqrt{2(5+\sqrt{5})}}{8}'], [r'\frac{\sqrt{3(10+2\sqrt{5})}+\sqrt{5}-1}{8}', r'\frac{\sqrt{9+\sqrt{5}+\sqrt{3(10-2\sqrt{5})}}}{4}'], [r'\sqrt{23-10\sqrt{5}-2\sqrt{3(85-38\sqrt{5})}}']]),
(Rational(2, 15), [[r'\frac{\sqrt{3}+\sqrt{15}-\sqrt{2(5-\sqrt{5})}}{8}'], [r'\frac{\sqrt{3(10-2\sqrt{5})}+\sqrt{5}+1}{8}', r'\frac{\sqrt{9-\sqrt{5}+\sqrt{3(10+2\sqrt{5})}}}{4}'], [r'\sqrt{23+10\sqrt{5}-2\sqrt{3(85+38\sqrt{5})}}']]),
(Rational(4, 15), [[r'\frac{-\sqrt{3}+\sqrt{15}+\sqrt{2(5+\sqrt{5})}}{8}'], [r'\frac{\sqrt{3(10+2\sqrt{5})}-\sqrt{5}+1}{8}', r'\frac{\sqrt{9+\sqrt{5}-\sqrt{3(10-2\sqrt{5})}}}{4}'], [r'\sqrt{23-10\sqrt{5}+2\sqrt{3(85-38\sqrt{5})}}']]),
(Rational(7, 15), [[r'\frac{\sqrt{3}+\sqrt{15}+\sqrt{2(5-\sqrt{5})}}{8}'], [r'\frac{\sqrt{3(10-2\sqrt{5})}-\sqrt{5}-1}{8}', r'\frac{\sqrt{9-\sqrt{5}-\sqrt{3(10+2\sqrt{5})}}}{4}'], [r'\sqrt{23+10\sqrt{5}+2\sqrt{3(85+38\sqrt{5})}}']]),
(Rational(1, 16), [[r'\frac{\sqrt{2-\sqrt{2+\sqrt{2}}}}{2}'], [r'\frac{\sqrt{2+\sqrt{2+\sqrt{2}}}}{2}'], [r'\sqrt{2(2+\sqrt{2})}-\sqrt{2}-1', r'\sqrt{\frac{2-\sqrt{2+\sqrt{2}}}{2+\sqrt{2+\sqrt{2}}}}']]),
(Rational(3, 16), [[r'\frac{\sqrt{2-\sqrt{2-\sqrt{2}}}}{2}'], [r'\frac{\sqrt{2+\sqrt{2-\sqrt{2}}}}{2}'], [r'\sqrt{2(2-\sqrt{2})}-\sqrt{2}+1', r'\sqrt{\frac{2-\sqrt{2-\sqrt{2}}}{2+\sqrt{2-\sqrt{2}}}}']]),
(Rational(5, 16), [[r'\frac{\sqrt{2+\sqrt{2-\sqrt{2}}}}{2}'], [r'\frac{\sqrt{2-\sqrt{2-\sqrt{2}}}}{2}'], [r'\sqrt{2(2-\sqrt{2})}+\sqrt{2}-1', r'\sqrt{\frac{2+\sqrt{2-\sqrt{2}}}{2-\sqrt{2-\sqrt{2}}}}']]),
(Rational(7, 16), [[r'\frac{\sqrt{2+\sqrt{2+\sqrt{2}}}}{2}'], [r'\frac{\sqrt{2-\sqrt{2+\sqrt{2}}}}{2}'], [r'\sqrt{2(2+\sqrt{2})}+\sqrt{2}+1', r'\sqrt{\frac{2+\sqrt{2+\sqrt{2}}}{2-\sqrt{2+\sqrt{2}}}}']]),
(Rational(1, 17), [[r'\sqrt{\frac{17-\sqrt{17}-\sqrt{2(17-\sqrt{17})}-2\sqrt{17+3\sqrt{17}-\sqrt{2(85+19\sqrt{17})}}}{32}}'], [r'\sqrt{\frac{15+\sqrt{17}+\sqrt{2(17-\sqrt{17})}+2\sqrt{17+3\sqrt{17}-\sqrt{2(85+19\sqrt{17})}}}{32}}'], [r'\sqrt{\frac{17-\sqrt{17}-\sqrt{2(17-\sqrt{17})}-2\sqrt{17+3\sqrt{17}-\sqrt{2(85+19\sqrt{17})}}}{15+\sqrt{17}+\sqrt{2(17-\sqrt{17})}+2\sqrt{17+3\sqrt{17}-\sqrt{2(85+19\sqrt{17})}}}}']]),
(Rational(2, 17), [[r'\sqrt{\frac{17-\sqrt{17}+\sqrt{2(17-\sqrt{17})}-2\sqrt{17+3\sqrt{17}+\sqrt{2(85+19\sqrt{17})}}}{32}}'], [r'\sqrt{\frac{15+\sqrt{17}-\sqrt{2(17-\sqrt{17})}+2\sqrt{17+3\sqrt{17}+\sqrt{2(85+19\sqrt{17})}}}{32}}'], [r'\sqrt{\frac{17-\sqrt{17}+\sqrt{2(17-\sqrt{17})}-2\sqrt{17+3\sqrt{17}+\sqrt{2(85+19\sqrt{17})}}}{15+\sqrt{17}-\sqrt{2(17-\sqrt{17})}+2\sqrt{17+3\sqrt{17}+\sqrt{2(85+19\sqrt{17})}}}}']]),
(Rational(3, 17), [[r'\sqrt{\frac{17+\sqrt{17}-\sqrt{2(17+\sqrt{17})}-2\sqrt{17-3\sqrt{17}+\sqrt{2(85-19\sqrt{17})}}}{32}}'], [r'\sqrt{\frac{15-\sqrt{17}+\sqrt{2(17+\sqrt{17})}+2\sqrt{17-3\sqrt{17}+\sqrt{2(85-19\sqrt{17})}}}{32}}'], [r'\sqrt{\frac{17+\sqrt{17}-\sqrt{2(17+\sqrt{17})}-2\sqrt{17-3\sqrt{17}+\sqrt{2(85-19\sqrt{17})}}}{15-\sqrt{17}+\sqrt{2(17+\sqrt{17})}+2\sqrt{17-3\sqrt{17}+\sqrt{2(85-19\sqrt{17})}}}}']]),
(Rational(4, 17), [[r'\sqrt{\frac{17-\sqrt{17}-\sqrt{2(17-\sqrt{17})}+2\sqrt{17+3\sqrt{17}-\sqrt{2(85+19\sqrt{17})}}}{32}}'], [r'\sqrt{\frac{15+\sqrt{17}+\sqrt{2(17-\sqrt{17})}-2\sqrt{17+3\sqrt{17}-\sqrt{2(85+19\sqrt{17})}}}{32}}'], [r'\sqrt{\frac{17-\sqrt{17}-\sqrt{2(17-\sqrt{17})}+2\sqrt{17+3\sqrt{17}-\sqrt{2(85+19\sqrt{17})}}}{15+\sqrt{17}+\sqrt{2(17-\sqrt{17})}-2\sqrt{17+3\sqrt{17}-\sqrt{2(85+19\sqrt{17})}}}}']]),
(Rational(5, 17), [[r'\sqrt{\frac{17+\sqrt{17}-\sqrt{2(17+\sqrt{17})}+2\sqrt{17-3\sqrt{17}+\sqrt{2(85-19\sqrt{17})}}}{32}}'], [r'\sqrt{\frac{15-\sqrt{17}+\sqrt{2(17+\sqrt{17})}-2\sqrt{17-3\sqrt{17}+\sqrt{2(85-19\sqrt{17})}}}{32}}'], [r'\sqrt{\frac{17+\sqrt{17}-\sqrt{2(17+\sqrt{17})}+2\sqrt{17-3\sqrt{17}+\sqrt{2(85-19\sqrt{17})}}}{15-\sqrt{17}+\sqrt{2(17+\sqrt{17})}-2\sqrt{17-3\sqrt{17}+\sqrt{2(85-19\sqrt{17})}}}}']]),
(Rational(6, 17), [[r'\sqrt{\frac{17+\sqrt{17}+\sqrt{2(17+\sqrt{17})}-2\sqrt{17-3\sqrt{17}-\sqrt{2(85-19\sqrt{17})}}}{32}}'], [r'\sqrt{\frac{15-\sqrt{17}-\sqrt{2(17+\sqrt{17})}+2\sqrt{17-3\sqrt{17}-\sqrt{2(85-19\sqrt{17})}}}{32}}'], [r'\sqrt{\frac{17+\sqrt{17}+\sqrt{2(17+\sqrt{17})}-2\sqrt{17-3\sqrt{17}-\sqrt{2(85-19\sqrt{17})}}}{15-\sqrt{17}-\sqrt{2(17+\sqrt{17})}+2\sqrt{17-3\sqrt{17}-\sqrt{2(85-19\sqrt{17})}}}}']]),
(Rational(7, 17), [[r'\sqrt{\frac{17+\sqrt{17}+\sqrt{2(17+\sqrt{17})}+2\sqrt{17-3\sqrt{17}-\sqrt{2(85-19\sqrt{17})}}}{32}}'], [r'\sqrt{\frac{15-\sqrt{17}-\sqrt{2(17+\sqrt{17})}-2\sqrt{17-3\sqrt{17}-\sqrt{2(85-19\sqrt{17})}}}{32}}'], [r'\sqrt{\frac{17+\sqrt{17}+\sqrt{2(17+\sqrt{17})}+2\sqrt{17-3\sqrt{17}-\sqrt{2(85-19\sqrt{17})}}}{15-\sqrt{17}-\sqrt{2(17+\sqrt{17})}-2\sqrt{17-3\sqrt{17}-\sqrt{2(85-19\sqrt{17})}}}}']]),
(Rational(8, 17), [[r'\sqrt{\frac{17-\sqrt{17}+\sqrt{2(17-\sqrt{17})}+2\sqrt{17+3\sqrt{17}+\sqrt{2(85+19\sqrt{17})}}}{32}}'], [r'\sqrt{\frac{15+\sqrt{17}-\sqrt{2(17-\sqrt{17})}-2\sqrt{17+3\sqrt{17}+\sqrt{2(85+19\sqrt{17})}}}{32}}'], [r'\sqrt{\frac{17-\sqrt{17}+\sqrt{2(17-\sqrt{17})}+2\sqrt{17+3\sqrt{17}+\sqrt{2(85+19\sqrt{17})}}}{15+\sqrt{17}-\sqrt{2(17-\sqrt{17})}-2\sqrt{17+3\sqrt{17}+\sqrt{2(85+19\sqrt{17})}}}}']]),
(Rational(1, 20), [[r'\frac{\sqrt{2}+\sqrt{10}-2\sqrt{5-\sqrt{5}}}{8}'], [r'\frac{\sqrt{2}+\sqrt{10}+2\sqrt{5-\sqrt{5}}}{8}'], [r'\sqrt{5}+1-\sqrt{5+2\sqrt{5}}']]),
(Rational(3, 20), [[r'\frac{\sqrt{2}-\sqrt{10}+2\sqrt{5+\sqrt{5}}}{8}'], [r'\frac{-\sqrt{2}+\sqrt{10}+2\sqrt{5+\sqrt{5}}}{8}'], [r'\sqrt{5}-1-\sqrt{5-2\sqrt{5}}']]),
(Rational(7, 20), [[r'\frac{-\sqrt{2}+\sqrt{10}+2\sqrt{5+\sqrt{5}}}{8}'], [r'\frac{\sqrt{2}-\sqrt{10}+2\sqrt{5+\sqrt{5}}}{8}'], [r'\sqrt{5}-1+\sqrt{5-2\sqrt{5}}']]),
(Rational(9, 20), [[r'\frac{\sqrt{2}+\sqrt{10}+2\sqrt{5-\sqrt{5}}}{8}'], [r'\frac{\sqrt{2}+\sqrt{10}-2\sqrt{5-\sqrt{5}}}{8}'], [r'\sqrt{5}+1+\sqrt{5+2\sqrt{5}}']]),
(Rational(1, 24), [[r'\frac{\sqrt{2+\sqrt{2}}-\sqrt{6-3\sqrt{2}}}{4}'], [r'\frac{\sqrt{2-\sqrt{2}}+\sqrt{6+3\sqrt{2}}}{4}'], [r'\sqrt{6}-\sqrt{3}+\sqrt{2}-2']]),
(Rational(5, 24), [[r'\frac{-\sqrt{2-\sqrt{2}}+\sqrt{6+3\sqrt{2}}}{4}'], [r'\frac{\sqrt{2+\sqrt{2}}+\sqrt{6-3\sqrt{2}}}{4}'], [r'\sqrt{6}+\sqrt{3}-\sqrt{2}-2']]),
(Rational(7, 24), [[r'\frac{\sqrt{2+\sqrt{2}}+\sqrt{6-3\sqrt{2}}}{4}'], [r'\frac{-\sqrt{2-\sqrt{2}}+\sqrt{6+3\sqrt{2}}}{4}'], [r'\sqrt{6}-\sqrt{3}-\sqrt{2}+2']]),
(Rational(11, 24), [[r'\frac{\sqrt{2-\sqrt{2}}+\sqrt{6+3\sqrt{2}}}{4}'], [r'\frac{\sqrt{2+\sqrt{2}}-\sqrt{6-3\sqrt{2}}}{4}'], [r'\sqrt{6}+\sqrt{3}+\sqrt{2}+2']]),
(Rational(1, 30), [[r'\frac{\sqrt{3(10-2\sqrt{5})}-\sqrt{5}-1}{8}'], [r'\frac{\sqrt{3}+\sqrt{15}+\sqrt{2(5-\sqrt{5})}}{8}'], [r'\sqrt{7-2\sqrt{5}-2\sqrt{3(5-2\sqrt{5})}}']]),
(Rational(7, 30), [[r'\frac{\sqrt{3(10+2\sqrt{5})}-\sqrt{5}+1}{8}'], [r'\frac{-\sqrt{3}+\sqrt{15}+\sqrt{2(5+\sqrt{5})}}{8}'], [r'\sqrt{7+2\sqrt{5}-2\sqrt{3(5+2\sqrt{5})}}']]),
(Rational(11, 30), [[r'\frac{\sqrt{3(10-2\sqrt{5})}+\sqrt{5}+1}{8}'], [r'\frac{\sqrt{3}+\sqrt{15}-\sqrt{2(5-\sqrt{5})}}{8}'], [r'\sqrt{7-2\sqrt{5}+2\sqrt{3(5-2\sqrt{5})}}']]),
(Rational(13, 30), [[r'\frac{\sqrt{3(10+2\sqrt{5})}+\sqrt{5}-1}{8}'], [r'\frac{\sqrt{3}-\sqrt{15}+\sqrt{2(5+\sqrt{5})}}{8}'], [r'\sqrt{7+2\sqrt{5}+2\sqrt{3(5+2\sqrt{5})}}']]),
(Rational(1, 32), [[r'\frac{\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2}}}}}{2}'], [r'\frac{\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}}{2}'], [r'\sqrt{\frac{2-\sqrt{2+\sqrt{2+\sqrt{2}}}}{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}}']]),
(Rational(3, 32), [[r'\frac{\sqrt{2-\sqrt{2+\sqrt{2-\sqrt{2}}}}}{2}'], [r'\frac{\sqrt{2+\sqrt{2+\sqrt{2-\sqrt{2}}}}}{2}'], [r'\sqrt{\frac{2-\sqrt{2+\sqrt{2-\sqrt{2}}}}{2+\sqrt{2+\sqrt{2-\sqrt{2}}}}}']]),
(Rational(5, 32), [[r'\frac{\sqrt{2-\sqrt{2-\sqrt{2-\sqrt{2}}}}}{2}'], [r'\frac{\sqrt{2+\sqrt{2-\sqrt{2-\sqrt{2}}}}}{2}'], [r'\sqrt{\frac{2-\sqrt{2-\sqrt{2-\sqrt{2}}}}{2+\sqrt{2-\sqrt{2-\sqrt{2}}}}}']]),
(Rational(7, 32), [[r'\frac{\sqrt{2-\sqrt{2-\sqrt{2+\sqrt{2}}}}}{2}'], [r'\frac{\sqrt{2+\sqrt{2-\sqrt{2+\sqrt{2}}}}}{2}'], [r'\sqrt{\frac{2-\sqrt{2-\sqrt{2+\sqrt{2}}}}{2+\sqrt{2-\sqrt{2+\sqrt{2}}}}}']]),
(Rational(9, 32), [[r'\frac{\sqrt{2+\sqrt{2-\sqrt{2+\sqrt{2}}}}}{2}'], [r'\frac{\sqrt{2-\sqrt{2-\sqrt{2+\sqrt{2}}}}}{2}'], [r'\sqrt{\frac{2+\sqrt{2-\sqrt{2+\sqrt{2}}}}{2-\sqrt{2-\sqrt{2+\sqrt{2}}}}}']]),
(Rational(11, 32), [[r'\frac{\sqrt{2+\sqrt{2-\sqrt{2-\sqrt{2}}}}}{2}'], [r'\frac{\sqrt{2-\sqrt{2-\sqrt{2-\sqrt{2}}}}}{2}'], [r'\sqrt{\frac{2+\sqrt{2-\sqrt{2-\sqrt{2}}}}{2-\sqrt{2-\sqrt{2-\sqrt{2}}}}}']]),
(Rational(13, 32), [[r'\frac{\sqrt{2+\sqrt{2+\sqrt{2-\sqrt{2}}}}}{2}'], [r'\frac{\sqrt{2-\sqrt{2+\sqrt{2-\sqrt{2}}}}}{2}'], [r'\sqrt{\frac{2+\sqrt{2+\sqrt{2-\sqrt{2}}}}{2-\sqrt{2+\sqrt{2-\sqrt{2}}}}}']]),
(Rational(15, 32), [[r'\frac{\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}}{2}'], [r'\frac{\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2}}}}}{2}'], [r'\sqrt{\frac{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}{2-\sqrt{2+\sqrt{2+\sqrt{2}}}}}']]),
(Rational(1, 34), [[r'\sqrt{\frac{15+\sqrt{17}-\sqrt{2(17-\sqrt{17})}-2\sqrt{17+3\sqrt{17}+\sqrt{2(85+19\sqrt{17})}}}{32}}'], [r'\sqrt{\frac{17-\sqrt{17}+\sqrt{2(17-\sqrt{17})}+2\sqrt{17+3\sqrt{17}+\sqrt{2(85+19\sqrt{17})}}}{32}}'], [r'\sqrt{\frac{15+\sqrt{17}-\sqrt{2(17-\sqrt{17})}-2\sqrt{17+3\sqrt{17}+\sqrt{2(85+19\sqrt{17})}}}{17-\sqrt{17}+\sqrt{2(17-\sqrt{17})}+2\sqrt{17+3\sqrt{17}+\sqrt{2(85+19\sqrt{17})}}}}']]),
(Rational(3, 34), [[r'\sqrt{\frac{15-\sqrt{17}-\sqrt{2(17+\sqrt{17})}-2\sqrt{17-3\sqrt{17}-\sqrt{2(85-19\sqrt{17})}}}{32}}'], [r'\sqrt{\frac{17+\sqrt{17}+\sqrt{2(17+\sqrt{17})}+2\sqrt{17-3\sqrt{17}-\sqrt{2(85-19\sqrt{17})}}}{32}}'], [r'\sqrt{\frac{15-\sqrt{17}-\sqrt{2(17+\sqrt{17})}-2\sqrt{17-3\sqrt{17}-\sqrt{2(85-19\sqrt{17})}}}{17+\sqrt{17}+\sqrt{2(17+\sqrt{17})}+2\sqrt{17-3\sqrt{17}-\sqrt{2(85-19\sqrt{17})}}}}']]),
(Rational(5, 34), [[r'\sqrt{\frac{15-\sqrt{17}-\sqrt{2(17+\sqrt{17})}+2\sqrt{17-3\sqrt{17}-\sqrt{2(85-19\sqrt{17})}}}{32}}'], [r'\sqrt{\frac{17+\sqrt{17}+\sqrt{2(17+\sqrt{17})}-2\sqrt{17-3\sqrt{17}-\sqrt{2(85-19\sqrt{17})}}}{32}}'], [r'\sqrt{\frac{15-\sqrt{17}-\sqrt{2(17+\sqrt{17})}+2\sqrt{17-3\sqrt{17}-\sqrt{2(85-19\sqrt{17})}}}{17+\sqrt{17}+\sqrt{2(17+\sqrt{17})}-2\sqrt{17-3\sqrt{17}-\sqrt{2(85-19\sqrt{17})}}}}']]),
(Rational(7, 34), [[r'\sqrt{\frac{15-\sqrt{17}+\sqrt{2(17+\sqrt{17})}-2\sqrt{17-3\sqrt{17}+\sqrt{2(85-19\sqrt{17})}}}{32}}'], [r'\sqrt{\frac{17+\sqrt{17}-\sqrt{2(17+\sqrt{17})}+2\sqrt{17-3\sqrt{17}+\sqrt{2(85-19\sqrt{17})}}}{32}}'], [r'\sqrt{\frac{15-\sqrt{17}+\sqrt{2(17+\sqrt{17})}-2\sqrt{17-3\sqrt{17}+\sqrt{2(85-19\sqrt{17})}}}{17+\sqrt{17}-\sqrt{2(17+\sqrt{17})}+2\sqrt{17-3\sqrt{17}+\sqrt{2(85-19\sqrt{17})}}}}']]),
(Rational(9, 34), [[r'\sqrt{\frac{15+\sqrt{17}+\sqrt{2(17-\sqrt{17})}-2\sqrt{17+3\sqrt{17}-\sqrt{2(85+19\sqrt{17})}}}{32}}'], [r'\sqrt{\frac{17-\sqrt{17}-\sqrt{2(17-\sqrt{17})}+2\sqrt{17+3\sqrt{17}-\sqrt{2(85+19\sqrt{17})}}}{32}}'], [r'\sqrt{\frac{15+\sqrt{17}+\sqrt{2(17-\sqrt{17})}-2\sqrt{17+3\sqrt{17}-\sqrt{2(85+19\sqrt{17})}}}{17-\sqrt{17}-\sqrt{2(17-\sqrt{17})}+2\sqrt{17+3\sqrt{17}-\sqrt{2(85+19\sqrt{17})}}}}']]),
(Rational(11, 34), [[r'\sqrt{\frac{15-\sqrt{17}+\sqrt{2(17+\sqrt{17})}+2\sqrt{17-3\sqrt{17}+\sqrt{2(85-19\sqrt{17})}}}{32}}'], [r'\sqrt{\frac{17+\sqrt{17}-\sqrt{2(17+\sqrt{17})}-2\sqrt{17-3\sqrt{17}+\sqrt{2(85-19\sqrt{17})}}}{32}}'], [r'\sqrt{\frac{15-\sqrt{17}+\sqrt{2(17+\sqrt{17})}+2\sqrt{17-3\sqrt{17}+\sqrt{2(85-19\sqrt{17})}}}{17+\sqrt{17}-\sqrt{2(17+\sqrt{17})}-2\sqrt{17-3\sqrt{17}+\sqrt{2(85-19\sqrt{17})}}}}']]),
(Rational(13, 34), [[r'\sqrt{\frac{15+\sqrt{17}-\sqrt{2(17-\sqrt{17})}+2\sqrt{17+3\sqrt{17}+\sqrt{2(85+19\sqrt{17})}}}{32}}'], [r'\sqrt{\frac{17-\sqrt{17}+\sqrt{2(17-\sqrt{17})}-2\sqrt{17+3\sqrt{17}+\sqrt{2(85+19\sqrt{17})}}}{32}}'], [r'\sqrt{\frac{15+\sqrt{17}-\sqrt{2(17-\sqrt{17})}+2\sqrt{17+3\sqrt{17}+\sqrt{2(85+19\sqrt{17})}}}{17-\sqrt{17}+\sqrt{2(17-\sqrt{17})}-2\sqrt{17+3\sqrt{17}+\sqrt{2(85+19\sqrt{17})}}}}']]),
(Rational(15, 34), [[r'\sqrt{\frac{15+\sqrt{17}+\sqrt{2(17-\sqrt{17})}+2\sqrt{17+3\sqrt{17}-\sqrt{2(85+19\sqrt{17})}}}{32}}'], [r'\sqrt{\frac{17-\sqrt{17}-\sqrt{2(17-\sqrt{17})}-2\sqrt{17+3\sqrt{17}-\sqrt{2(85+19\sqrt{17})}}}{32}}'], [r'\sqrt{\frac{15+\sqrt{17}+\sqrt{2(17-\sqrt{17})}+2\sqrt{17+3\sqrt{17}-\sqrt{2(85+19\sqrt{17})}}}{17-\sqrt{17}-\sqrt{2(17-\sqrt{17})}-2\sqrt{17+3\sqrt{17}-\sqrt{2(85+19\sqrt{17})}}}}']]),
(Rational(1, 40), [[r'\frac{\sqrt{2(2-\sqrt{2})(5+\sqrt{5})}-(\sqrt{5}-1)\sqrt{2+\sqrt{2}}}{8}'], [r'\frac{\sqrt{2(2+\sqrt{2})(5+\sqrt{5})}+(\sqrt{5}-1)\sqrt{2-\sqrt{2}}}{8}'], [r'\frac{\sqrt{2(2-\sqrt{2})(5+\sqrt{5})}-(\sqrt{5}-1)\sqrt{2+\sqrt{2}}}{\sqrt{2(2+\sqrt{2})(5+\sqrt{5})}+(\sqrt{5}-1)\sqrt{2-\sqrt{2}}}']]),
(Rational(3, 40), [[r'\frac{\sqrt{2(2+\sqrt{2})(5-\sqrt{5})}-(\sqrt{5}+1)\sqrt{2-\sqrt{2}}}{8}'], [r'\frac{\sqrt{2(2-\sqrt{2})(5-\sqrt{5})}+(\sqrt{5}+1)\sqrt{2+\sqrt{2}}}{8}'], [r'\frac{\sqrt{2(2+\sqrt{2})(5-\sqrt{5})}-(\sqrt{5}+1)\sqrt{2-\sqrt{2}}}{\sqrt{2(2-\sqrt{2})(5-\sqrt{5})}+(\sqrt{5}+1)\sqrt{2+\sqrt{2}}}']]),
(Rational(7, 40), [[r'\frac{-\sqrt{2(2-\sqrt{2})(5-\sqrt{5})}+(\sqrt{5}+1)\sqrt{2+\sqrt{2}}}{8}'], [r'\frac{\sqrt{2(2+\sqrt{2})(5-\sqrt{5})}+(\sqrt{5}+1)\sqrt{2-\sqrt{2}}}{8}'], [r'\frac{-\sqrt{2(2-\sqrt{2})(5-\sqrt{5})}+(\sqrt{5}+1)\sqrt{2+\sqrt{2}}}{\sqrt{2(2+\sqrt{2})(5-\sqrt{5})}+(\sqrt{5}+1)\sqrt{2-\sqrt{2}}}']]),
(Rational(9, 40), [[r'\frac{\sqrt{2(2-\sqrt{2})(5+\sqrt{5})}+(\sqrt{5}-1)\sqrt{2+\sqrt{2}}}{8}'], [r'\frac{\sqrt{2(2+\sqrt{2})(5+\sqrt{5})}-(\sqrt{5}-1)\sqrt{2-\sqrt{2}}}{8}'], [r'\frac{\sqrt{2(2-\sqrt{2})(5+\sqrt{5})}+(\sqrt{5}-1)\sqrt{2+\sqrt{2}}}{\sqrt{2(2+\sqrt{2})(5+\sqrt{5})}-(\sqrt{5}-1)\sqrt{2-\sqrt{2}}}']]),
(Rational(11, 40), [[r'\frac{\sqrt{2(2+\sqrt{2})(5+\sqrt{5})}-(\sqrt{5}-1)\sqrt{2-\sqrt{2}}}{8}'], [r'\frac{\sqrt{2(2-\sqrt{2})(5+\sqrt{5})}+(\sqrt{5}-1)\sqrt{2+\sqrt{2}}}{8}'], [r'\frac{\sqrt{2(2+\sqrt{2})(5+\sqrt{5})}-(\sqrt{5}-1)\sqrt{2-\sqrt{2}}}{\sqrt{2(2-\sqrt{2})(5+\sqrt{5})}+(\sqrt{5}-1)\sqrt{2+\sqrt{2}}}']]),
(Rational(13, 40), [[r'\frac{\sqrt{2(2+\sqrt{2})(5-\sqrt{5})}+(\sqrt{5}+1)\sqrt{2-\sqrt{2}}}{8}'], [r'\frac{-\sqrt{2(2-\sqrt{2})(5-\sqrt{5})}+(\sqrt{5}+1)\sqrt{2+\sqrt{2}}}{8}'], [r'\frac{\sqrt{2(2+\sqrt{2})(5-\sqrt{5})}+(\sqrt{5}+1)\sqrt{2-\sqrt{2}}}{-\sqrt{2(2-\sqrt{2})(5-\sqrt{5})}+(\sqrt{5}+1)\sqrt{2+\sqrt{2}}}']]),
(Rational(17, 40), [[r'\frac{\sqrt{2(2-\sqrt{2})(5-\sqrt{5})}+(\sqrt{5}+1)\sqrt{2+\sqrt{2}}}{8}'], [r'\frac{\sqrt{2(2+\sqrt{2})(5-\sqrt{5})}-(\sqrt{5}+1)\sqrt{2-\sqrt{2}}}{8}'], [r'\frac{\sqrt{2(2-\sqrt{2})(5-\sqrt{5})}+(\sqrt{5}+1)\sqrt{2+\sqrt{2}}}{\sqrt{2(2+\sqrt{2})(5-\sqrt{5})}-(\sqrt{5}+1)\sqrt{2-\sqrt{2}}}']]),
(Rational(19, 40), [[r'\frac{\sqrt{2(2+\sqrt{2})(5+\sqrt{5})}+(\sqrt{5}-1)\sqrt{2-\sqrt{2}}}{8}'], [r'\frac{\sqrt{2(2-\sqrt{2})(5+\sqrt{5})}-(\sqrt{5}-1)\sqrt{2+\sqrt{2}}}{8}'], [r'\frac{\sqrt{2(2+\sqrt{2})(5+\sqrt{5})}+(\sqrt{5}-1)\sqrt{2-\sqrt{2}}}{\sqrt{2(2-\sqrt{2})(5+\sqrt{5})}-(\sqrt{5}-1)\sqrt{2+\sqrt{2}}}']]),
(Rational(1, 48), [[r'\frac{\sqrt{8-2\sqrt{2-\sqrt{2}}-2\sqrt{3(2+\sqrt{2})}}}{4}'], [r'\frac{\sqrt{8+2\sqrt{2-\sqrt{2}}+2\sqrt{3(2+\sqrt{2})}}}{4}'], [r'\sqrt{\frac{4-\sqrt{2-\sqrt{2}}-\sqrt{3(2+\sqrt{2})}}{4+\sqrt{2-\sqrt{2}}+\sqrt{3(2+\sqrt{2})}}}']]),
(Rational(5, 48), [[r'\frac{\sqrt{8-2\sqrt{2+\sqrt{2}}-2\sqrt{3(2-\sqrt{2})}}}{4}'], [r'\frac{\sqrt{8+2\sqrt{2+\sqrt{2}}+2\sqrt{3(2-\sqrt{2})}}}{4}'], [r'\sqrt{\frac{4-\sqrt{2+\sqrt{2}}-\sqrt{3(2-\sqrt{2})}}{4+\sqrt{2+\sqrt{2}}+\sqrt{3(2-\sqrt{2})}}}']]),
(Rational(7, 48), [[r'\frac{\sqrt{8+2\sqrt{2-\sqrt{2}}-2\sqrt{3(2+\sqrt{2})}}}{4}'], [r'\frac{\sqrt{8-2\sqrt{2-\sqrt{2}}+2\sqrt{3(2+\sqrt{2})}}}{4}'], [r'\sqrt{\frac{4+\sqrt{2-\sqrt{2}}-\sqrt{3(2+\sqrt{2})}}{4-\sqrt{2-\sqrt{2}}+\sqrt{3(2+\sqrt{2})}}}']]),
(Rational(11, 48), [[r'\frac{\sqrt{8-2\sqrt{2+\sqrt{2}}+2\sqrt{3(2-\sqrt{2})}}}{4}'], [r'\frac{\sqrt{8+2\sqrt{2+\sqrt{2}}-2\sqrt{3(2-\sqrt{2})}}}{4}'], [r'\sqrt{\frac{4-\sqrt{2+\sqrt{2}}+\sqrt{3(2-\sqrt{2})}}{4+\sqrt{2+\sqrt{2}}-\sqrt{3(2-\sqrt{2})}}}']]),
(Rational(13, 48), [[r'\frac{\sqrt{8+2\sqrt{2+\sqrt{2}}-2\sqrt{3(2-\sqrt{2})}}}{4}'], [r'\frac{\sqrt{8-2\sqrt{2+\sqrt{2}}+2\sqrt{3(2-\sqrt{2})}}}{4}'], [r'\sqrt{\frac{4+\sqrt{2+\sqrt{2}}-\sqrt{3(2-\sqrt{2})}}{4-\sqrt{2+\sqrt{2}}+\sqrt{3(2-\sqrt{2})}}}']]),
(Rational(17, 48), [[r'\frac{\sqrt{8-2\sqrt{2-\sqrt{2}}+2\sqrt{3(2+\sqrt{2})}}}{4}'], [r'\frac{\sqrt{8+2\sqrt{2-\sqrt{2}}-2\sqrt{3(2+\sqrt{2})}}}{4}'], [r'\sqrt{\frac{4-\sqrt{2-\sqrt{2}}+\sqrt{3(2+\sqrt{2})}}{4+\sqrt{2-\sqrt{2}}-\sqrt{3(2+\sqrt{2})}}}']]),
(Rational(19, 48), [[r'\frac{\sqrt{8+2\sqrt{2+\sqrt{2}}+2\sqrt{3(2-\sqrt{2})}}}{4}'], [r'\frac{\sqrt{8-2\sqrt{2+\sqrt{2}}-2\sqrt{3(2-\sqrt{2})}}}{4}'], [r'\sqrt{\frac{4+\sqrt{2+\sqrt{2}}+\sqrt{3(2-\sqrt{2})}}{4-\sqrt{2+\sqrt{2}}-\sqrt{3(2-\sqrt{2})}}}']]),
(Rational(23, 48), [[r'\frac{\sqrt{8+2\sqrt{2-\sqrt{2}}+2\sqrt{3(2+\sqrt{2})}}}{4}'], [r'\frac{\sqrt{8-2\sqrt{2-\sqrt{2}}-2\sqrt{3(2+\sqrt{2})}}}{4}'], [r'\sqrt{\frac{4+\sqrt{2-\sqrt{2}}+\sqrt{3(2+\sqrt{2})}}{4-\sqrt{2-\sqrt{2}}-\sqrt{3(2+\sqrt{2})}}}']]),
# (Rational(1, 51), [[r'unknown'], [r'unknown'], [r'unknown']]),
# (Rational(2, 51), [[r'unknown'], [r'unknown'], [r'unknown']]),
# (Rational(4, 51), [[r'unknown'], [r'unknown'], [r'unknown']]),
# (Rational(5, 51), [[r'unknown'], [r'unknown'], [r'unknown']]),
# (Rational(7, 51), [[r'unknown'], [r'unknown'], [r'unknown']]),
# (Rational(8, 51), [[r'unknown'], [r'unknown'], [r'unknown']]),
# (Rational(10, 51), [[r'unknown'], [r'unknown'], [r'unknown']]),
# (Rational(11, 51), [[r'unknown'], [r'unknown'], [r'unknown']]),
# (Rational(13, 51), [[r'unknown'], [r'unknown'], [r'unknown']]),
# (Rational(14, 51), [[r'unknown'], [r'unknown'], [r'unknown']]),
# (Rational(16, 51), [[r'unknown'], [r'unknown'], [r'unknown']]),
# (Rational(19, 51), [[r'unknown'], [r'unknown'], [r'unknown']]),
# (Rational(20, 51), [[r'unknown'], [r'unknown'], [r'unknown']]),
# (Rational(22, 51), [[r'unknown'], [r'unknown'], [r'unknown']]),
# (Rational(23, 51), [[r'unknown'], [r'unknown'], [r'unknown']]),
# (Rational(1, 60), [[r'\frac{\sqrt{5-\sqrt{5}}(1-\sqrt{3}+\sqrt{5}-\sqrt{15})-\sqrt{2}-\sqrt{6}+\sqrt{10}+\sqrt{30}}{16}'], [r'\frac{\sqrt{5-\sqrt{5}}(1+\sqrt{3}+\sqrt{5}+\sqrt{15})+\sqrt{2}-\sqrt{6}-\sqrt{10}+\sqrt{30}}{16}'], [r'unknown']]),
# (Rational(7, 60), [[r'\frac{\sqrt{5+\sqrt{5}}(-1-\sqrt{3}+\sqrt{5}+\sqrt{15})+\sqrt{2}-\sqrt{6}+\sqrt{10}-\sqrt{30}}{16}'], [r'\frac{\sqrt{5+\sqrt{5}}(1-\sqrt{3}-\sqrt{5}+\sqrt{15})+\sqrt{2}+\sqrt{6}+\sqrt{10}+\sqrt{30}}{16}'], [r'unknown']]),
# (Rational(11, 60), [[r'\frac{\sqrt{5-\sqrt{5}}(-1+\sqrt{3}-\sqrt{5}+\sqrt{15})-\sqrt{2}-\sqrt{6}+\sqrt{10}+\sqrt{30}}{16}'], [r'\frac{\sqrt{5-\sqrt{5}}(1+\sqrt{3}+\sqrt{5}+\sqrt{15})-\sqrt{2}+\sqrt{6}+\sqrt{10}-\sqrt{30}}{16}'], [r'unknown']]),
# (Rational(13, 60), [[r'\frac{\sqrt{5+\sqrt{5}}(-1+\sqrt{3}+\sqrt{5}-\sqrt{15})+\sqrt{2}+\sqrt{6}+\sqrt{10}+\sqrt{30}}{16}'], [r'\frac{\sqrt{5+\sqrt{5}}(-1-\sqrt{3}+\sqrt{5}+\sqrt{15})-\sqrt{2}+\sqrt{6}-\sqrt{10}+\sqrt{30}}{16}'], [r'unknown']]),
# (Rational(17, 60), [[r'\frac{\sqrt{5+\sqrt{5}}(-1-\sqrt{3}+\sqrt{5}+\sqrt{15})-\sqrt{2}+\sqrt{6}-\sqrt{10}+\sqrt{30}}{16}'], [r'\frac{\sqrt{5+\sqrt{5}}(-1+\sqrt{3}+\sqrt{5}-\sqrt{15})+\sqrt{2}+\sqrt{6}+\sqrt{10}+\sqrt{30}}{16}'], [r'unknown']]),
# (Rational(19, 60), [[r'\frac{\sqrt{5-\sqrt{5}}(1+\sqrt{3}+\sqrt{5}+\sqrt{15})-\sqrt{2}+\sqrt{6}+\sqrt{10}-\sqrt{30}}{16}'], [r'\frac{\sqrt{5-\sqrt{5}}(-1+\sqrt{3}-\sqrt{5}+\sqrt{15})-\sqrt{2}-\sqrt{6}+\sqrt{10}+\sqrt{30}}{16}'], [r'unknown']]),
# (Rational(23, 60), [[r'\frac{\sqrt{5+\sqrt{5}}(1-\sqrt{3}-\sqrt{5}+\sqrt{15})+\sqrt{2}+\sqrt{6}+\sqrt{10}+\sqrt{30}}{16}'], [r'\frac{\sqrt{5+\sqrt{5}}(-1-\sqrt{3}+\sqrt{5}+\sqrt{15})+\sqrt{2}-\sqrt{6}+\sqrt{10}-\sqrt{30}}{16}'], [r'unknown']]),
# (Rational(29, 60), [[r'\frac{\sqrt{5-\sqrt{5}}(1+\sqrt{3}+\sqrt{5}+\sqrt{15})+\sqrt{2}-\sqrt{6}-\sqrt{10}+\sqrt{30}}{16}'], [r'\frac{\sqrt{5-\sqrt{5}}(1-\sqrt{3}+\sqrt{5}-\sqrt{15})-\sqrt{2}-\sqrt{6}+\sqrt{10}+\sqrt{30}}{16}'], [r'unknown']]),
]
table = [['$r$', '$\\sin\\pi r$', '$\\cos\\pi r$', '$\\tan\\pi r$']]
for r, exprs in lists:
row = []
if r is S.Zero:
row.append('$0$')
else:
row.append(f'${latex(r)}$')
for f, mf, expr in zip([sin, cos, tan], [math.sin, math.cos, math.tan], exprs):
if expr[0] == 'inf' or expr[0] == 'unknown':
row.append(expr[0])
else:
ac = f(r*pi).evalf(n=50)
acm = mf(r*math.pi)
assert abs(float(ac) - acm) < 10e-10, (f, r, ac, acm)
texs = []
for idx, v in enumerate(expr):
assert abs(parse_latex(v).evalf(n=50) - ac) < Float('10e-51'), (f, r, idx)
texs.append(f'${v}$')
row.append(', '.join(texs))
table.append(row)
print(list_to_markdown(table))