Feat: compute powers of alpha on the fly

[kunxian-xia]

Performance

Summary: we have about 20% improvement. And the performance gain is due to two parts:

1. avoid computing eq(rt, sub_rt) in several places.
2. compute $\alpha$'s powers on the fly.

The cycle count after this PR:

INFO     ┝━ ｉ [info]: ┌╴initial sum
INFO     ┝━ ｉ [info]: └╴162 cycles
INFO     ┝━ ｉ [info]: ┌╴check expected evaluation
INFO     ┝━ ｉ [info]: └╴171 cycles
INFO     ┝━ ｉ [info]: ┌╴derive next layer's expected sum
INFO     ┝━ ｉ [info]: └╴287 cycles
INFO     ┝━ ｉ [info]: ┌╴check expected evaluation
INFO     ┝━ ｉ [info]: └╴186 cycles
INFO     ┝━ ｉ [info]: ┌╴derive next layer's expected sum
INFO     ┝━ ｉ [info]: └╴295 cycles
INFO     ┝━ ｉ [info]: ┌╴check expected evaluation
INFO     ┝━ ｉ [info]: └╴201 cycles
INFO     ┝━ ｉ [info]: ┌╴derive next layer's expected sum
INFO     ┝━ ｉ [info]: └╴303 cycles
INFO     ┝━ ｉ [info]: ┌╴check expected evaluation
INFO     ┝━ ｉ [info]: └╴216 cycles
INFO     ┝━ ｉ [info]: ┌╴derive next layer's expected sum
INFO     ┕━ ｉ [info]: └╴305 cycles

The cycle count before this PR:

INFO     ┝━ ｉ [info]: ┌╴initial sum
INFO     ┝━ ｉ [info]: └╴183 cycles
INFO     ┝━ ｉ [info]: ┌╴check expected evaluation
INFO     ┝━ ｉ [info]: └╴216 cycles
INFO     ┝━ ｉ [info]: ┌╴derive next layer's expected sum
INFO     ┝━ ｉ [info]: └╴360 cycles
INFO     ┝━ ｉ [info]: ┌╴check expected evaluation
INFO     ┝━ ｉ [info]: └╴254 cycles
INFO     ┝━ ｉ [info]: ┌╴derive next layer's expected sum
INFO     ┝━ ｉ [info]: └╴368 cycles
INFO     ┝━ ｉ [info]: ┌╴check expected evaluation
INFO     ┝━ ｉ [info]: └╴292 cycles
INFO     ┝━ ｉ [info]: ┌╴derive next layer's expected sum
INFO     ┝━ ｉ [info]: └╴376 cycles
INFO     ┝━ ｉ [info]: ┌╴check expected evaluation
INFO     ┝━ ｉ [info]: └╴330 cycles
INFO     ┝━ ｉ [info]: ┌╴derive next layer's expected sum
INFO     ┕━ ｉ [info]: └╴378 cycles

[kunxian-xia]

Move this outside the loop.