octave-interval 3.1.0-5 / usr / share / octave / packages / interval-3.1.0 / @infsup / fma.m

## Copyright 2014-2016 Oliver Heimlich
##
## This program is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 3 of the License, or
## (at your option) any later version.
##
## This program is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with this program; if not, see <http://www.gnu.org/licenses/>.

## -*- texinfo -*-
## @documentencoding UTF-8
## @defmethod {@@infsup} fma (@var{X}, @var{Y}, @var{Z})
##
## Fused multiply and add @code{@var{X} * @var{Y} + @var{Z}}.
##
## This function is semantically equivalent to evaluating multiplication and
## addition separately, but in addition guarantees a tight enclosure of the
## result.
##
## Accuracy: The result is a tight enclosure.
##
## @example
## @group
## output_precision (16, 'local')
## fma (infsup (1+eps), infsup (7), infsup ("0.1"))
##   @result{} ans ⊂ [7.100000000000001, 7.100000000000003]
## infsup (1+eps) * infsup (7) + infsup ("0.1")
##   @result{} ans ⊂ [7.1, 7.100000000000003]
## @end group
## @end example
## @seealso{@@infsup/plus, @@infsup/times}
## @end defmethod

## Author: Oliver Heimlich
## Keywords: interval
## Created: 2014-10-03

function x = fma (x, y, z)

  if (nargin ~= 3)
    print_usage ();
    return
  endif
  if (not (isa (x, "infsup")))
    x = infsup (x);
  endif
  if (not (isa (y, "infsup")))
    y = infsup (y);
  endif
  if (not (isa (z, "infsup")))
    z = infsup (z);
  endif

  ## Resize, if broadcasting is needed
  if (not (size_equal (x.inf, y.inf)))
    x.inf = ones (size (y.inf)) .* x.inf;
    x.sup = ones (size (y.inf)) .* x.sup;
    y.inf = ones (size (x.inf)) .* y.inf;
    y.sup = ones (size (x.inf)) .* y.sup;
  endif
  if (not (size_equal (y.inf, z.inf)))
    y.inf = ones (size (z.inf)) .* y.inf;
    y.sup = ones (size (z.inf)) .* y.sup;
    z.inf = ones (size (y.inf)) .* z.inf;
    z.sup = ones (size (y.inf)) .* z.sup;
  endif
  if (not (size_equal (x.inf, z.inf)))
    x.inf = ones (size (z.inf)) .* x.inf;
    x.sup = ones (size (z.inf)) .* x.sup;
    z.inf = ones (size (x.inf)) .* z.inf;
    z.sup = ones (size (x.inf)) .* z.sup;
  endif

  ## [Empty] × anything = [Empty]
  ## [0] × anything = [0] × [0]
  ## [Entire] × anything but [0] = [Entire] × [Entire]
  ## This prevents the cases where 0 × inf would produce NaNs.
  entireproduct = isentire (x) | isentire (y);
  zeroproduct = (x.inf == 0 & x.sup == 0) | (y.inf == 0 & y.sup == 0);
  emptyresult = isempty (x) | isempty (y) | isempty (z);
  x.inf(entireproduct) = y.inf(entireproduct) = -inf;
  x.sup(entireproduct) = y.sup(entireproduct) = inf;
  x.inf(zeroproduct) = x.sup(zeroproduct) = ...
  y.inf(zeroproduct) = y.sup(zeroproduct) = 0;

  ## It is hard to determine, which boundaries of x and y take part in the
  ## multiplication of fma.  Therefore, we simply compute the fma for each triple
  ## of boundaries where the min/max could be located.
  ##
  ## How to construct complicated cases: a = rand, b = rand, c = rand,
  ## d = a * b / c (with round towards -infinity for multiplication and towards
  ## +infinity for division).  Then, it is not possible to decide in 50% of all
  ## cases whether a * b would be greater or less than c * d by computing the
  ## products in double-precision.

  l = min (min (min (...
                      mpfr_function_d ('fma', -inf, x.inf, y.inf, z.inf), ...
                      mpfr_function_d ('fma', -inf, x.inf, y.sup, z.inf)), ...
                mpfr_function_d ('fma', -inf, x.sup, y.inf, z.inf)), ...
           mpfr_function_d ('fma', -inf, x.sup, y.sup, z.inf));
  u = max (max (max (...
                      mpfr_function_d ('fma', +inf, x.inf, y.inf, z.sup), ...
                      mpfr_function_d ('fma', +inf, x.inf, y.sup, z.sup)), ...
                mpfr_function_d ('fma', +inf, x.sup, y.inf, z.sup)), ...
           mpfr_function_d ('fma', +inf, x.sup, y.sup, z.sup));

  l(emptyresult) = +inf;
  u(emptyresult) = -inf;

  l(l == 0) = -0;

  x.inf = l;
  x.sup = u;

endfunction

%!# from the documentation string
%!assert (fma (infsup (1+eps), infsup (7), infsup ("0.1")) == "[0x1.C666666666668p2, 0x1.C666666666669p2]");

%!# correct use of signed zeros
%!test
%! x = fma (infsup (0), 0, 0);
%! assert (signbit (inf (x)));
%! assert (not (signbit (sup (x))));
%!test
%! x = fma (infsup (1), 0, 0);
%! assert (signbit (inf (x)));
%! assert (not (signbit (sup (x))));
%!test
%! x = fma (infsup (1), 1, -1);
%! assert (signbit (inf (x)));
%! assert (not (signbit (sup (x))));

%!shared testdata
%! # Load compiled test data (from src/test/*.itl)
%! testdata = load (file_in_loadpath ("test/itl.mat"));

%!test
%! # Scalar evaluation
%! testcases = testdata.NoSignal.infsup.fma;
%! for testcase = [testcases]'
%!   assert (isequaln (...
%!     fma (testcase.in{1}, testcase.in{2}, testcase.in{3}), ...
%!     testcase.out));
%! endfor

%!test
%! # Vector evaluation
%! testcases = testdata.NoSignal.infsup.fma;
%! in1 = vertcat (vertcat (testcases.in){:, 1});
%! in2 = vertcat (vertcat (testcases.in){:, 2});
%! in3 = vertcat (vertcat (testcases.in){:, 3});
%! out = vertcat (testcases.out);
%! assert (isequaln (fma (in1, in2, in3), out));

%!test
%! # N-dimensional array evaluation
%! testcases = testdata.NoSignal.infsup.fma;
%! in1 = vertcat (vertcat (testcases.in){:, 1});
%! in2 = vertcat (vertcat (testcases.in){:, 2});
%! in3 = vertcat (vertcat (testcases.in){:, 3});
%! out = vertcat (testcases.out);
%! # Reshape data
%! i = -1;
%! do
%!   i = i + 1;
%!   testsize = factor (numel (in1) + i);
%! until (numel (testsize) > 2)
%! in1 = reshape ([in1; in1(1:i)], testsize);
%! in2 = reshape ([in2; in2(1:i)], testsize);
%! in3 = reshape ([in3; in3(1:i)], testsize);
%! out = reshape ([out; out(1:i)], testsize);
%! assert (isequaln (fma (in1, in2, in3), out));