3.11 Efficiency

Small Operands

On small operands, the time for function call overheads and memory allocation can be significant in comparison to actual calculation. This is unavoidable in a general purpose variable precision library, although GMP attempts to be as efficient as it can on both large and small operands.

Static Linking

On some CPUs, in particular the x86s, the static libgmp.a should be used for maximum speed, since the PIC code in the shared libgmp.so will have a small overhead on each function call and global data address. For many programs this will be insignificant, but for long calculations there’s a gain to be had.

Initializing and Clearing

Avoid excessive initializing and clearing of variables, since this can be quite time consuming, especially in comparison to otherwise fast operations like addition.

A language interpreter might want to keep a free list or stack of initialized variables ready for use. It should be possible to integrate something like that with a garbage collector too.

Reallocations

An mpz_t or mpq_t variable used to hold successively increasing values will have its memory repeatedly realloced, which could be quite slow or could fragment memory, depending on the C library. If an application can estimate the final size then mpz_init2 or mpz_realloc2 can be called to allocate the necessary space from the beginning (see Initializing Integers).

It doesn’t matter if a size set with mpz_init2 or mpz_realloc2 is too small, since all functions will do a further reallocation if necessary. Badly overestimating memory required will waste space though.

2exp Functions

It’s up to an application to call functions like mpz_mul_2exp when appropriate. General purpose functions like mpz_mul make no attempt to identify powers of two or other special forms, because such inputs will usually be very rare and testing every time would be wasteful.

ui and si Functions

The ui functions and the small number of si functions exist for convenience and should be used where applicable. But if for example an mpz_t contains a value that fits in an unsigned long there’s no need extract it and call a ui function, just use the regular mpz function.

In-Place Operations

mpz_abs, mpq_abs, mpf_abs, mpz_neg, mpq_neg and mpf_neg are fast when used for in-place operations like mpz_abs(x,x), since in the current implementation only a single field of x needs changing. On suitable compilers (GCC for instance) this is inlined too.

mpz_add_ui, mpz_sub_ui, mpf_add_ui and mpf_sub_ui benefit from an in-place operation like mpz_add_ui(x,x,y), since usually only one or two limbs of x will need to be changed. The same applies to the full precision mpz_add etc if y is small. If y is big then cache locality may be helped, but that’s all.

mpz_mul is currently the opposite, a separate destination is slightly better. A call like mpz_mul(x,x,y) will, unless y is only one limb, make a temporary copy of x before forming the result. Normally that copying will only be a tiny fraction of the time for the multiply, so this is not a particularly important consideration.

mpz_set, mpq_set, mpq_set_num, mpf_set, etc, make no attempt to recognise a copy of something to itself, so a call like mpz_set(x,x) will be wasteful. Naturally that would never be written deliberately, but if it might arise from two pointers to the same object then a test to avoid it might be desirable.

if (x != y)
  mpz_set (x, y);

Note that it’s never worth introducing extra mpz_set calls just to get in-place operations. If a result should go to a particular variable then just direct it there and let GMP take care of data movement.

Divisibility Testing (Small Integers)

mpz_divisible_ui_p and mpz_congruent_ui_p are the best functions for testing whether an mpz_t is divisible by an individual small integer. They use an algorithm which is faster than mpz_tdiv_ui, but which gives no useful information about the actual remainder, only whether it’s zero (or a particular value).

However when testing divisibility by several small integers, it’s best to take a remainder modulo their product, to save multi-precision operations. For instance to test whether a number is divisible by any of 23, 29 or 31 take a remainder modulo 23*29*31 = 20677 and then test that.

The division functions like mpz_tdiv_q_ui which give a quotient as well as a remainder are generally a little slower than the remainder-only functions like mpz_tdiv_ui. If the quotient is only rarely wanted then it’s probably best to just take a remainder and then go back and calculate the quotient if and when it’s wanted (mpz_divexact_ui can be used if the remainder is zero).

Rational Arithmetic

The mpq functions operate on mpq_t values with no common factors in the numerator and denominator. Common factors are checked-for and cast out as necessary. In general, cancelling factors every time is the best approach since it minimizes the sizes for subsequent operations.

However, applications that know something about the factorization of the values they’re working with might be able to avoid some of the GCDs used for canonicalization, or swap them for divisions. For example when multiplying by a prime it’s enough to check for factors of it in the denominator instead of doing a full GCD. Or when forming a big product it might be known that very little cancellation will be possible, and so canonicalization can be left to the end.

The mpq_numref and mpq_denref macros give access to the numerator and denominator to do things outside the scope of the supplied mpq functions. See Applying Integer Functions.

The canonical form for rationals allows mixed-type mpq_t and integer additions or subtractions to be done directly with multiples of the denominator. This will be somewhat faster than mpq_add. For example,

/* mpq increment */
mpz_add (mpq_numref(q), mpq_numref(q), mpq_denref(q));

/* mpq += unsigned long */
mpz_addmul_ui (mpq_numref(q), mpq_denref(q), 123UL);

/* mpq -= mpz */
mpz_submul (mpq_numref(q), mpq_denref(q), z);

Number Sequences

Functions like mpz_fac_ui, mpz_fib_ui and mpz_bin_uiui are designed for calculating isolated values. If a range of values is wanted it’s probably best to get a starting point and iterate from there.

Text Input/Output

Hexadecimal or octal are suggested for input or output in text form. Power-of-2 bases like these can be converted much more efficiently than other bases, like decimal. For big numbers there’s usually nothing of particular interest to be seen in the digits, so the base doesn’t matter much.

Maybe we can hope octal will one day become the normal base for everyday use, as proposed by King Charles XII of Sweden and later reformers.