GMP ceil/floor/trunc & round-offs

[Hans Aberg]

Here are some inputs on ceil/floor/trunc and round-off functions:

I think that ceil/floor/trunc functions should be viewed as integer
truncation functions
  rational -> integer
  fixed -> integer
  floating -> integer
(I add the "fixed" type here just for mathematical completeness.)

The traditional ceil/floor/trunc should then be replaced by round-off functions
  (fixed, precision) -> fixed
  (floating, precision) -> floating
where the precision argument specoifies the precision of the function
return. Note that there are special round-off techniques favoured by some,
like that the number exactly in the middle in the round-off process is
rounded up/down depending on what the next higher binary digit is. (I am
myself somewhat sceptical about those round-off techniques, and I merely
note that they do exist.)

One should avoid the type-conversions
  rational -> integer
  floating -> integer
  floating -> rational
and similar for the fixed type, and instead make separate functions.

In fact, if I add them to my C++ header, based on mpz_set_q & mpz_set_f, my
C++ compiler becomes confused with the other type conversions. For example,
it cannot add an integer and a rational anymore. It does not make so much
difference under C, but strictly speaking those mpz_set_q & mpz_set_f ought
to be replaced by the ceil/floor/trunc mentioned above.

I added to my rational class the functions corresponding to
  mpz_cdiv_q   mpz_fdiv_q   mpz_tdiv_q
  mpz_cdiv_r   mpz_fdiv_r   mpz_tdiv_r
  mpz_cdiv_qr  mpz_fdiv_qr  mpz_tdiv_r
They the became functions
     int_ceil, int_floor, int_trunc: rational -> integer
  frac_ceil, frac_floor, frac_trunc: rational -> rational
     div_ceil, div_floor, div_trunc: rational -> (integer, rational)
The thing is that the fractional part is only defined as a rational number,
so it would strictly speaking be mathematically ambiguous to define say
  frac_trunc: rational -> integer
as the return integer would depend on the representation of the rational
number.

And then one gets similar functions for the other types, fixed and floating.

  Hans Aberg