16.3 Float Internals

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

The number of limbs currently in use, or the negative of that when representing a negative value. Zero is represented by _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

The precision of the mantissa, in limbs. In any calculation the aim is to produce _mp_prec limbs of result (the most significant being non-zero).

_mp_d

A pointer to the array of limbs which is the absolute value of the mantissa. 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] 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

The exponent, in limbs, determining the location of the implied radix point. Zero means the radix point is just above the most significant limb. Positive values mean a radix point offset towards the lower limbs and hence a value >= 1, as for example in the diagram above. Negative exponents mean a radix point further above the highest limb.

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 _mp_size and _mp_prec fields are int, although the mp_size_t type is usually a long. The _mp_exp field is usually long. This is done to make some fields just 32 bits on some 64 bits systems, thereby saving a few bytes of data space but still providing plenty of precision and a very large range.

The following various points should be noted.

Low Zeros

The least significant limbs _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.

Mantissa Size Range

The _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.

Rounding

All rounding is done on limb boundaries. Calculating _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.

Bit Shifts

Since the exponent is in limbs, there are no bit shifts in basic operations like 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.

Use of _mp_prec+1 Limbs

The extra limb on _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.

Application Precisions

__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 that 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.