What is the function of やらで?

Question

What is the meaning of やらで in this tanka and where it comes from?

春のよの月は木のまを出でやらでまづ山のはの花ぞみえ行く

I found that the expression 出でやる could be an equivalent of 出てくる but I don't really understand the equivalence between 出でやる and 出てくる at all... I found that 出で comes from 出づ, and old version of the actual 出る (is this correct?). It can be translated as appear, but i don't really understand the function of やる there.

Also I think the で after やらで could act like "not" here, but i am not really sure about that.

Sources

The original source is this: volume 4 of Shimpen Kokka taikan, ed. Shimpen Kokka Taikan Henshû I'inkai (Tokyo: Kadokawa Shoten, 1986), although I had a hard time finding this book. I also use this English translation, although it's all in rômaji: Just Living, poems & prose by the Japanese monk Tonna, by Steven D. Carter.

Answer 1

You're right about 出{い}で; it is the 連用形 of 出{い}づ, meaning "to emerge, to go out, to come out," etc. Here, the 連用形 is used because 出{い}づ is being combined with the auxiliary verb やる.

As a verb in its own right, やる has a range of meanings, the most basic and common of which is "to send [something or someone somewhere]." By extension, やる can also be used as an auxiliary verb to convey that an action is taking place over a long distance, as in 見{み}遣{や}る, which can mean "to look at [something far away]."

In this poem, though, やる is being used as an auxiliary verb in a different way, a way that conveys the idea of completion. That is, [verb] + やる means "to finish [verb]ing," "to [verb] completely," "to [verb] all the way," etc. However, this auxiliary usage of やる very commonly occurs in a negative form, so it's often used to say "doesn't finish [verb]ing," "didn't [verb] completely," "hasn't [verb]ed all the way," etc. The relevant entry for this usage in Weblio's 古語辞典 is

や・る 【遣る】

[二]補助動詞ラ行四段活用 活用｛ら／り／る／る／れ／れ｝ 〔動詞の連用形に付いて〕 ② 〔多く下に打消の語を伴って〕すっかり…する。最後まで…する。…しきる。▽その>動作が最後まで行われたという意味を表す。

And sure enough, that negative usage is what's happening here, as we have the 未{み}然{ぜん}形{けい} of やる followed by what you have correctly identified as the negative particle で. Here's the Weblio entry for で:

で 接続助詞 《接続》活用語の未然形に付く。〔打消の接続〕…ないで。…ずに。

So putting this all together, 出{い}でやらで tells us that the moon (which seems to be rising from behind a mountain to the east) hasn't yet fully emerged from the trees.