ANS FORTH CELEFUNT TEST RESULTS FOR ZSQRT(Z)
zsqrt.fs vs. 0.9.3
dnw 17Jan05

These results were obtained with pfe 0.33.58 compiled with gcc
3.3.3 on my MacOS X 10.3.7, PowerMac dual G4 system.  Leaving
aside the pfe C-primitive tests, Gforth 0.6.2 gives the same
results as pfe on the same system, except for the way NaN and
Inf are treated.

Results from Cody's original Fortran code are included for
reference.  It was compiled with GNU Fortran (GCC) 3.4 20031015
(experimental) under MacOS X.

The "normal" version calls complex.fs vs. 0.8.2, Julian Noble's
original code for ZSQRT.

The "kahan" version calls complex-kahan.fs vs. 0.8.7, which uses
the Kahan algorithm without treatment of IEEE 754
underflow/overflow exceptions.

The "pfe prim" version calls the pfe complex module vs. 0.8.7,
which codes the Kahan algorithm in C with treatment of
underflow/overflow exceptions.  This module is part of the pfe
0.33.xx series distribution.

Each test has three sections corresponding to the magnitude of
the relative complex difference of the compared functions, the
relative error in the real part, and the relative error in the
imaginary part.

N=  = number out of 2000 random trials that are equal
MRE = maximum relative error
RMS = rms relative error
ULP = estimated units in last place
      (loss of base 2 significant digits)

Floating point numbers have 53 significant bits, with at most 7
significant decimal digits printed out below.

The complex numbers following "box" are opposite corners of the
box in the complex plane from which random values of z are
chosen.

We don't know whether the pfe results in the first box below,
compared to the high-level Kahan results, are actually a
degradation, because we don't know that anything is reliable
beyond 0.5 ulp.  The pfe results are clearly better at the
extreme large argument.

z vs. sqrt(z*z), box 0+i0, 10+i10
             N=    MRE       ULP   RMS       ULP
vector
  g77       1921  1.56e-16  0.49  2.64e-17  0.00
  normal    1921  1.56e-16  0.49  2.64e-17  0.00
  kahan     2000  0.00e+00  0.00  0.00e+00  0.00
  pfe prim  1995  1.29E-16  0.22  6.20e-18  0.00
real
  g77       1921  2.17e-16  0.96  2.72e-17  0.00 
  normal    1922  2.17e-16  0.96  2.71e-17  0.00
  kahan     2000  0.00e+00  0.00  0.00e+00  0.00
  pfe prim  1995  1.33e-16  0.26  5.96e-18  0.00
imaginary
  g77       1921  2.01e-16  0.86  2.77e-17  0.00
  normal    1921  6.62e-14  9.22  1.48e-15  3.74
  kahan     2000  0.00e+00  0.00  0.00e+00  0.00
  pfe prim  1995  1.47e-16  0.40  6.64e-18  0.00

z vs. -sqrt(z*z), box 0+i0, -100+i100
             N=    MRE       ULP   RMS       ULP
vector
  g77       1939  1.57e-16  0.50  2.38e-17  0.00
  normal    1936  1.57e-16  0.50  2.43e-17  0.00
  kahan     2000  0.00e+00  0.00  0.00e+00  0.00
  pfe prim  2000  0.00e+00  0.00  0.00e+00  0.00
real
  g77       1939  2.22e-16  1.00  2.44e-17  0.00
  normal    1937  2.22e-16  1.00  2.50e-17  0.00
  kahan     2000  0.00e+00  0.00  0.00e+00  0.00
  pfe prim  2000  0.00e+00  0.00  0.00e+00  0.00
imaginary
  g77       1939  2.10e-16  0.92  2.49e-17  0.00
  normal    1936  4.74e-14  8.74  1.06e-15  3.26
  kahan     2000  0.00e+00  0.00  0.00e+00  0.00
  pfe prim  2000  0.00e+00  0.00  0.00e+00  0.00

SQRT[ 1.0+i0.0 ] =
  g77        1.000000e+00 + i 0.000000e+00
  normal     1.000000e+00 + i 0.000000e+00
  kahan      1.000000e+00 + i 0.000000e+00
  pfe prim   1.000000e+00 + i 0.000000e+00

SQRT[ XMIN + i*XMIN ] =
SQRT[ 2.225074e-308 + i 2.225074e-308 ] =
  g77        1.638872e-154 + i 6.788430e-155
  normal     1.638872e-154 + i 6.788430e-155 
  kahan      1.638872e-154 + i 6.788430e-155 
  pfe prim   1.638872E-154 + i 6.788430E-155 

SQRT[ XMAX + i*XMAX] =
SQRT[ 1.797693e+308 + i 1.797693e+308 ] =
  g77        Inf           + i 0.0000000e+00
  normal     Inf           + i 0.000000e+00 
  kahan      Inf           + i 0.000000e+00 
  pfe prim   1.473095e+154 + i 6.101757e+153