Comment by mbitsnbites

Given my search criteria, the optimal magic number turns out to be: 0x7ef311c2

  Initial approximation:
    Good bits min: 4
    Good bits avg: 5.242649912834
    Error max: 0.0505102872849 (4.30728 bits)
    Error avg: 0.0327344845327 (4.93304 bits)

  1 NR step:
    Good bits min: 8
    Good bits avg: 10.642581939697
    Error max: 0.00255139507338 (8.61450 bits)
    Error avg: 0.00132373889641 (9.56117 bits)

  2 NR steps:
    Good bits min: 17
    Good bits avg: 19.922843217850
    Error max: 6.62494557693e-06 (17.20366 bits)
    Error avg: 2.62858584054e-06 (18.53728 bits)

  3 NR steps:
    Good bits min: 23
    Good bits avg: 23.674004554749
    Error max: 1.19249960972e-07 (22.99951 bits)
    Error avg: 3.44158509521e-08 (24.79235 bits)

Here, "good bits" is 24 minus the number of trailing non-zero-bits in the integer difference between the approximation and the correct value, looking at the IEEE 754 binary representation (if that makes sense).

Also, for the NR steps I used double precision for the inner (2.0 - x * y) part, then rounded to single precision, to simulate FMA, but single precision for the outer multiplication.