In "format long" display mode, with output_precision set to 16, odd integers between 2^52 and 2^53 -- all of which are exactly representable in IEEE-754 double precision, and have 16 or fewer decimal digits -- are nevertheless displayed in E format.

Example:

  > format long;
  > output_precision(16);

  > a = 2^52;

  #
  # All values below 2^52 display as expected, i.e. as 16 digit integers:
  #
  > a-3         # ans =  4503599627370493
  > a-2         # ans =  4503599627370494
  > a-1         # ans =  4503599627370495
  > a           # ans =  4503599627370496

  #
  # But odd values between 2^52+1 and 2^53 display alternately in fixed and
  # exponential representation:
  #
  > a+1         # ans =  4.503599627370497e+15
  > a+2         # ans =  4503599627370498
  > a+3         # ans =  4.503599627370499e+15

 
Note that the displayed values for the odd integers above 2^52 are correct (and exact) but it seems inconsistent that they should be displayed in E format since they can be represented exactly as 16 digit integers.

The same behavior is observed for negative integers in the complementary range below zero.

NOTE: This is not a hardware representation issue, at least under the assumption of IEEE-754 compliant hardware, which was the case for the examples shown above: An IEEE-754 double precision float exactly represents all integers on [-2^53, +2^53], all of which fit into 16 digits.  Approximation error begins at 2^53+1. This can be demonstrated by bumping output_precision to 17:

  > output_precision(17)
  >   2^53
  ans =  9007199254740992       # Error = 0
  >   2^53+1
  ans =  9007199254740992       # Error = 1 LSB of significand, as expected
  >   2^53+2
  ans =  9007199254740994       # Error = 0
  >   2^53+3
  ans =  9007199254740996       # Error = 1 LSB of significand, as expected

and so on, all of which is as expected for IEEE-754 hardware.