In Gmail, sending emails via Fastmail's SMTP protocol alters the header From:

In the Gmail settings, under "Accounts and Import", I added a new email address under the "Send mail as" setting. This email address using my own domain,, lets say. I've set up Gmail for sending via SMTP servers from Fastmail.

This domain has been configured on Fastmail. The MX records were defined and I added the alias, asking Fastmail to transfer all emails to my @ address.

Receiving mail at works as expected, the mail appears in my Gmail inbox. Sending from the Gmail web interface as also works. However, the recipient receives the email as it comes from my email address @ How can I solve this problem?

differential equations – The increase of MaxExtraPrecision arbitrarily alters the numerical result

I'm trying to confirm that a function $ f $ satisfies a particular differential equation of the type $ D f = 0 $, for a differential operator $ D $. I put $ Df as Diffeq in Mathematica and I tried to check if that gives zero for any numerical value of the variables $ f $ depend on. I wrote

NOT[Difeqx /. x -> 3/4 /. y -> 1/5 , 500]

I had the following:
"N :: meprec: internal precision limit $ MaxExtraPrecision = 50.` reached during the evaluation [the function]
Outside[1]= -1,214057 * 10 ^ -613 –
6.579195 * 10 ^ -614 I ".

I then used

Block[{$MaxExtraPrecision = 550}, N[Difeqx /. x -> 3/4 /. y -> 1/5, 500]]

and I came back
"N :: meprec: internal precision limit $ MaxExtraPrecision = 550.` reached during evaluation [the function]
Outside[2]= 4.131621 * 10 ^ -1089 +
2.022562 * 10 ^ -1089 I "

The numerical result changes (and becomes smaller for this particular case) as I increase MaxExtraPrecision.

The question:
From the different numerical results obtained, I can conclude with some uncertainty that indeed $ Df = $ 0 (or Difeq = 0) is satisfied. But, can I clarify? How can I be safer than it actually gives zero?

P.S. I do not think that Chop, which returns zero, makes it safer that the result is precisely zero.