Some efforts to post-dict particle masses based on Koide's formula and related concepts are discussed here with a link in particular to a November 2011 pre-print. I've been following the online discussions leading up to it, which starts at the open access Physics Forums and had industry engineer Carl Brannen as an important contributor, from its inception.
The desirability of a way to determine quark masses other than via direct experimental measurement is quite real, in part, because it illuminates "within the Standard Model" relationship that shed light on deeper relationships between its parameters, and in part, because the accuracy with which we know the ligher quark masses (up, down, strange) is only about one significant digit, and only about two significant digits for intermediate quark masses (charm, bottom).
The uncertainty in the values of the fundamental quark masses is something that injects meaningful uncertainty into QCD background estimates looking for new physics, and since the same values for quark masses are used by almost everyone doing QCD calculations, the bias is systemmic. Experimentally measured QED and electroweak constants are known to far greater precision.
The paper's post-dictions, which include an interesting 3-1 relationship between quarks and leptons that parallels that 3-1 relationship in the weak force frequencies of the two types of particles attributable to the fact that there are three colors of quarks, while only one lepton color, are as follows:
me = 0.510998910 MeV ± 0.000000013
mμ = 105.6583668 MeV ± 0.0000038
Mq = 3Ml
q = 3 l
m = 1776.96894(7) MeV (Tau)
mt = 173.263947(6) GeV (top)
mb = 4197.57589(15) MeV
mc = 1359.56428(5) MeV (charm)
ms = 92.274758(3) MeV (strange)
md = 5.32 MeV (down)
mu = 0.0356 MeV (up)
The results are within the two sigma error bars of experimental determinations in the case of the tau, top, bottom, charm, strange (80-130 MeV) and down quarks (4.1-5.7 MeV), but has a much lower value than one within the error bars for the up quark mass than the canonical estimate of 1.7-3.1 MeV (mean 2.5 MeV) in a +/- one sigma confidence interval.
The experimental mass values of the heavy quarks cited in the article are:
mt = 172.9 GeV +/- 0.60 +/-.9
mb = 4.19 GeV +0.18−0.06
mc = 1.29 GeV +0.05−0.11 GeV
There have been suggestions by Brannen that one of the neutrinos needs to have a negative sign in the formula to fit the data for that series of masses, and a similar sign adjustment for the up quark (a nice parallel of one reversed sign for the quarks and one for the leptons) would produce a more conventional up quark mass.
Another way to deal with up quark mass is to acknowledge the theoretical issues posed in defining it in terms of observables. There are theoretical virtues to thinking of the fundamental "bare" rest mass of the up quark as zero, which would not be inconsistent with the result achieved from a Koide's mass approximation.
Suffice it to say that I think that this is far more than numerology and that there really is a fundamental physics basis for Koide's formula and its variants that explains the mass matrix.
Taken together with the quark-lepton complementarity principal that relates the parameters of the CKM matrix to those of the PMNS matrix, in a manner that follows a similar ansatz to the extended Koide's formula, there is a mechanism for making very precise phenomenological predictions of Standard Model physical constants that are hard to obtain accurate measurements of experimentally, and for reducing significantly the number of free physical constants in the Standard Model. These precise predictions ought to be possible to confirm continuously over time as experimental data refines the estimated value of these constants.
The predicted low value for the up quark mass, if correct, would also push the total amount of mass in the universe attributable to hadronic glue, as opposed to fundamental particles, to more than 99.6% of the total.
As a footnote, what would be the mass of a b' and t' quarks under the formula used (derived from the top and bottom quark masses and iterated)? It would be approximately 3,555 GeV (i.e. about 3.6 TeV) for the b' and 82,917 GeV (i.e. about 83 TeV) for the t' and 42.96 GeV for the fourth generation charged lepton tau prime (a little less than half of the Z boson mass), barring any sign issues by my back of envelope (literally) calculations. The quark masses implied by this formula for fourth generation quarks would probably be beyond anything that could be discovered at the LHC in the next few years, and the charged lepton mass is experimentally excluded. It would have otherwise been observed in Z boson decays long ago. This, of course, disfavors the notion that there is a fourth generation of fermions at all, something already disfavored by the 45 GeV minimum mass for a fourth generation "fertile" neutrino, which is much, much heavier than the best estimates for the mass of the third generation neutrino without any clear justification or precedent.
The author of some of these papers, Alejandro Rivero, also has the interesting idea, although not stated as clear in his papers as might be hoped, that supersymmetric particles are not fundamental, but are instead composite versions of Standard Model fundamental particles that have the same loop cancellation effects of SUSY particles.