Integer Internals (GNU MP 6.2.1)
Description: How to install and use the GNU multiple precision arithmetic library, version 6.2.1.

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### 16.1 Integer Internals

`mpz_t` variables represent integers using sign and magnitude, in space dynamically allocated and reallocated. The fields are as follows.

`_mp_size`

  The number of limbs, or the negative of that when representing a negative integer. Zero is represented by `_mp_size` set to zero, in which case the `_mp_d` data is undefined.

`_mp_d`

  A pointer to an array of limbs which is the magnitude. These are stored “little endian” as per the `mpn` functions, so `_mp_d[0]` is the least significant limb and `_mp_d[ABS(_mp_size)-1]` is the most significant. Whenever `_mp_size` is non-zero, the most significant limb is non-zero.

  Currently there’s always at least one readable limb, so for instance `mpz_get_ui` can fetch `_mp_d[0]` unconditionally (though its value is undefined if `_mp_size` is zero).

`_mp_alloc`

  `_mp_alloc` is the number of limbs currently allocated at `_mp_d`, and normally `_mp_alloc >= ABS(_mp_size)`. When an `mpz` routine is about to (or might be about to) increase `_mp_size`, it checks `_mp_alloc` to see whether there’s enough space, and reallocates if not. `MPZ_REALLOC` is generally used for this.

  `mpz_t` variables initialised with the `mpz_roinit_n` function or the `MPZ_ROINIT_N` macro have `_mp_alloc = 0` but can have a non-zero `_mp_size`. They can only be used as read-only constants. See Integer Special Functions for details.

The various bitwise logical functions like `mpz_and` behave as if negative values were twos complement. But sign and magnitude is always used internally, and necessary adjustments are made during the calculations. Sometimes this isn’t pretty, but sign and magnitude are best for other routines.

Some internal temporary variables are setup with `MPZ_TMP_INIT` and these have `_mp_d` space obtained from `TMP_ALLOC` rather than the memory allocation functions. Care is taken to ensure that these are big enough that no reallocation is necessary (since it would have unpredictable consequences).

`_mp_size` and `_mp_alloc` are `int`, although `mp_size_t` is usually a `long`. This is done to make the fields just 32 bits on some 64 bits systems, thereby saving a few bytes of data space but still providing plenty of range.

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