I believe you might be confusing the QED calculations of hydrogen with those of the electron g-factor. Just have a look into the paper I linked (section VII). Most of the QED corrections are given analytically, no computers involved at all. You could in principle calculate this with pen-and-paper (and a good enough table of transcendental functions).
The most accurate hydrogen spectroscopy (of the 1S-2S transition) has reached a relative accuracy of a few parts in 1E15 which is around an order of magnitude above the precision of FP64 numbers.
The "few parts in 1E15" claim is applicable only to the absolute value of the frequency of the 1S-2S transition, which is 1 233 030 706 593 514 Hz.
That absolute frequency is computed from the ratio between an optical frequency and the 9 GHz frequency of a cesium clock, which is affected by large uncertainties due to the need for bridging the gap between optical frequencies and microwave frequencies.
The frequency ratios between distinct lines of the hydrogen atom spectrum or between lines of the hydrogen atom spectrum and lines in the optical spectra of other atoms or ions can be known with uncertainties in parts per 1E18, one thousand times better.
When comparing a simulation with the experiment, the simulation must be able to match those quantities that can be measured with the lowest uncertainty, so the simulated values must also have uncertainties of at most parts per 1E18, or better per 1E19.
This requires more bits than provided by FP64. The extended precision of Intel 8087 would barely be enough to express the final results, but it would not be enough for the intermediate computations, so one really needs quadruple precision computations or double-double-precision computations, which are faster where only FP64 hardware exists.
I have not attempted to compute the QED corrections myself, so I cannot be certain how difficult that really is.
Nevertheless, the section VII from this CODATA paper and also the previous editions of the CODATA publications, some of which had been more detailed, are not consistent with what you say i.e. with them being easy to compute.
For each correction there is a long history of cited research papers that would need to be found and read to determine how exactly they have been computed. For many of them there is a history of refinements in their computations and of discrepancies between the values computed by different teams, discrepancies that have been some times resolved by later more accurate computations, but also some where the right value was not yet known at the date of this publication.
If the computations where so easy that anyone could do them with pen and paper there would have been no need for several years to pass in some cases until the validation of the correct computation and for a very slow improvement in the accuracy of the computed values in other cases.
The accuracy of the hydrogen 1S-2S measurement was mainly limited by the second-order Doppler shift of the moving atoms (and to a lesser degree the AC Stark shift of the excitation laser and the 2S-4P quench light). The comparison between the laser frequency and the Cesium fountain clock was done with an optical frequency comb which introduces a negligible uncertainty (< 1E-19).
Isn't it fun to get your own field of expertise (wrongly) explained to you on the internet?
Edit: I never said that it is easy to derive the corrections listed in the CODATA paper. However, it is relatively easy to calculate them.
The most accurate hydrogen spectroscopy (of the 1S-2S transition) has reached a relative accuracy of a few parts in 1E15 which is around an order of magnitude above the precision of FP64 numbers.