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Efficient calculation is the primary aim of GMP floats and the use of whole limbs and simple rounding facilitates this.
mpf_t floats have a variable precision mantissa and a single machine
word signed exponent. The mantissa is represented using sign and magnitude.
most least
significant significant
limb limb
_mp_d
|---- _mp_exp ---> |
_____ _____ _____ _____ _____
|_____|_____|_____|_____|_____|
. <------------ radix point
<-------- _mp_size --------->
The fields are as follows.
_mp_size
_mp_size and
_mp_exp both set to zero, and in that case the _mp_d data is
unused. (In the future _mp_exp might be undefined when representing
zero.)
_mp_prec
_mp_prec limbs of result (the most significant being non-zero).
_mp_d
mpn functions, so
_mp_d[0] is the least significant limb and
_mp_d[ABS(_mp_size)-1] the most significant.
The most significant limb is always non-zero, but there are no other restrictions on its value, in particular the highest 1 bit can be anywhere within the limb.
_mp_prec+1 limbs are allocated to _mp_d, the extra limb being
for convenience (see below). There are no reallocations during a calculation,
only in a change of precision with mpf_set_prec.
_mp_exp
Naturally the exponent can be any value, it doesn't have to fall within the
limbs as the diagram shows, it can be a long way above or a long way below.
Limbs other than those included in the {_mp_d,_mp_size} data
are treated as zero.
The following various points should be noted.
_mp_d[0] etc can be zero, though such low
zeros can always be ignored. Routines likely to produce low zeros check and
avoid them to save time in subsequent calculations, but for most routines
they're quite unlikely and aren't checked.
_mp_size count of limbs in use can be less than _mp_prec if
the value can be represented in less. This means low precision values or
small integers stored in a high precision mpf_t can still be operated
on efficiently.
_mp_size can also be greater than _mp_prec. Firstly a value is
allowed to use all of the _mp_prec+1 limbs available at _mp_d,
and secondly when mpf_set_prec_raw lowers _mp_prec it leaves
_mp_size unchanged and so the size can be arbitrarily bigger than
_mp_prec.
_mp_prec limbs
with the high non-zero will ensure the application requested minimum precision
is obtained.
The use of simple "trunc" rounding towards zero is efficient, since there's
no need to examine extra limbs and increment or decrement.
mpf_add and mpf_mul. When differing exponents are
encountered all that's needed is to adjust pointers to line up the relevant
limbs.
Of course mpf_mul_2exp and mpf_div_2exp will require bit shifts,
but the choice is between an exponent in limbs which requires shifts there, or
one in bits which requires them almost everywhere else.
_mp_prec+1 Limbs
_mp_d (_mp_prec+1 rather than just
_mp_prec) helps when an mpf routine might get a carry from its
operation. mpf_add for instance will do an mpn_add of
_mp_prec limbs. If there's no carry then that's the result, but if
there is a carry then it's stored in the extra limb of space and
_mp_size becomes _mp_prec+1.
Whenever _mp_prec+1 limbs are held in a variable, the low limb is not
needed for the intended precision, only the _mp_prec high limbs. But
zeroing it out or moving the rest down is unnecessary. Subsequent routines
reading the value will simply take the high limbs they need, and this will be
_mp_prec if their target has that same precision. This is no more than
a pointer adjustment, and must be checked anyway since the destination
precision can be different from the sources.
Copy functions like mpf_set will retain a full _mp_prec+1 limbs
if available. This ensures that a variable which has _mp_size equal to
_mp_prec+1 will get its full exact value copied. Strictly speaking
this is unnecessary since only _mp_prec limbs are needed for the
application's requested precision, but it's considered that an mpf_set
from one variable into another of the same precision ought to produce an exact
copy.
__GMPF_BITS_TO_PREC converts an application requested precision to an
_mp_prec. The value in bits is rounded up to a whole limb then an
extra limb is added since the most significant limb of _mp_d is only
non-zero and therefore might contain only one bit.
__GMPF_PREC_TO_BITS does the reverse conversion, and removes the extra
limb from _mp_prec before converting to bits. The net effect of
reading back with mpf_get_prec is simply the precision rounded up to a
multiple of mp_bits_per_limb.
Note that the extra limb added here for the high only being non-zero is in
addition to the extra limb allocated to _mp_d. For example with a
32-bit limb, an application request for 250 bits will be rounded up to 8
limbs, then an extra added for the high being only non-zero, giving an
_mp_prec of 9. _mp_d then gets 10 limbs allocated. Reading
back with mpf_get_prec will take _mp_prec subtract 1 limb and
multiply by 32, giving 256 bits.
Strictly speaking, the fact the high limb has at least one bit means that a
float with, say, 3 limbs of 32-bits each will be holding at least 65 bits, but
for the purposes of mpf_t it's considered simply to be 64 bits, a nice
multiple of the limb size.