Bad error computing 1/z

Peter Jeremy peter at
Mon Jul 30 02:51:30 UTC 2012

[Excuse the long lines]

On 2012-Jul-29 19:51:22 -0500, Stephen Montgomery-Smith <stephen at> wrote:
>As I was debugging catanh, I noticed the following oddness.
>If z = cpack(x,y), where
>x = 1
>y = 0x1.25691d4068c910p+512, approximately 1.53672e+154
>then the real part of 1/z is wrong by about 4ULP.  The real part of 1/z 
>is approx 4.2346e-309.  I know the reciprocal of this would cause an 

I think I'm missing a bit of context here.  If I calculate it using
doubles with gcc, I get:
    z:  +1.0000000000000000e+00 0x3ff0000000000000   +1.5367160172612364e+154 0x5ff25691d4068c91
1.0/z: +4.2346036163332497e-309 0x00030b8596824af0   -6.5073832039716640e-155 0x9febeb7fc100edd7

Doing the same thing with 40 digits precision in emacs-calc, I
calculate 1/z as:
(16#C.2E165A092BC21FF1FBEB5F45FAB8DA33*16.^-257, -16#D.F5BFE08076EBB470CE55E8BF55A1B0B5*16.^-129)
(4.234603616333252354574574265966349487777e-309, -6.507383203971664334288137901223264158669e-155)

That give me fractions of:
0x0.30B8596824AF087FC7EFAD7D17EAE368D  0x1.BEB7FC100EDD768E19CABD17EAB43617
Therefore the perfectly rounded double result should be
  0x00030b8596824af1  0x9febeb7fc100edd7
ie, the real part calculates as 1ULP low when using doubles.

Note that since the real part is a denormal, it only has 50 bits of
precision.  Are you sure your ULP calculations are taking into account
the number of fraction bits available when numbers are denormals?

Peter Jeremy
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