Comment 3 for bug 645662

And I unexpectedly had more time to dig.

Findings:

  - asinh() is not in IEEE754, please ignore my comment about
    non-conformity, sorry.
  - The calculation for asinh() is pretty badly conditioned, i.e.
    it blows up errors in the basic calculations.
  - asinh() is implemented in glibc in assembler on the FPU stack.
    This would mean 80 bits float representation.

Error observations:

  Calculations for asinh( -2.1073424255447017e-08), derived from
  the Python 3.1.2 selftest. Formula is asinh(x) = log(x+sqrt(x*x+1))
  Reference is a calculation with long double (128 bit)

  - qemu: 6 correct digits behind the dot
  - C with double: 7 correct digits behind the dot
  - i387: 10 correct digits behind the dot

Possible root cause:

   The observed error may be due to qemu using 64 bit double math in
   the FPU implementation, instead of the 80 bit internal precision
   a real i387 uses.

Comparison with kernel i387 emulation

   As an additional verification, I tried the calculations using the
   Linux kernel coprocessor emulation. This basically failed. With
   kernel 2_6_26 and 2_6_27 and the ''no387'' flag, the kernel locked
   up without output early during boot. With 2.6.34, I could not do
   an ssh login. I may try this again later, and especially try
   to find out what is wrong with qemu and 2.6.34.

Consequences:

  1) qemu is significantly less accurate for some mathematics than a
     genuine i387. This may cause problesm with sensitive mathematics
     developed on a real i387.
  2) qemu can be fingerprinted easily using this problem. This may
     have security implications.

Fix Options:

  1) - Moving the FPU internal representation to long double. It will
       still not be the same results as on an i387, but it will be
       _more_ accurate instead of less, which generally is far less
       of a problem.

       Pro: Higher accuracy
       Cons: Slower

     - Use code derived from the Linux kernel i387 FPU emulator.
       The emulator is not bit-exact, but relatively close and it
       is used at least on some small x86 architectures.

       Pro: Higher accuracy
       Cons: Need to integrate foreign code

     - Leave it as it is but add a clear warning to the "Known qemu
       Problems" Section in the manual and Wiki (I did not find one, I
       think it should be added in an easily findable place), that
       i387 FPU emulation is only 64 bits internally and may be less
       exact for combined calculations done on the FPU stack, such
       as asinh() in glibc(), than a real i387.

       You may recommend to use kernel FPU emulation, but
       this may not work or be a pain to get working.

       Pro: Least work
       Cons: Not really a solution

2) Fixing this is difficult and there are other ways to find
     out that you are running in qemu anyways. I think the security
     implications are insiginficant in most cases.