nfloats

C++ Arbitrary precision, IEEE-754 floating point numbers, declared as follows:

nfloat<N> f = nfloat<N>("12.34");

where N is the number of bits on which the floating point number will be coded onto (Note that N is such that: 16 <= N <= 2048 and N % 8 == 0, for any nfloat).

Layout

nfloats are stored as std::bitsets of the same size. The first bit always represents the sign of the floating point number, while the exponent and the mantissa follow these rules:

• The exponent of an nfloat is stored in a region of the bitset which occupies round(6.1 * ln(s/16 + 0.7) + 1.8) bits, where s is the size of the nfloat. This funciton was experimentally obtained by interpolating a series of points which had the value s / 16 as x coordinate and the number of bits of the exponent of s-sized IEEE-754 floating point implementation as y coordinate (computed on s values of 16, 32, 64, 128 and 256, standard exponent sizes were obtained from this page).
• The mantissa is stored in the remaining bits.

Initializers

nfloats can be initialized from any already existing floating point supported by C++:

nfloat<32> f1 = nfloat<32>(1.0f);, from floats;

nfloat<64> f2 = nfloat<32>(1.0);, from doubles;

nfloat<128> f3 = nfloat<128>(1.0Q); and from quadmath.h's __float128s.

Note that for any nfloat derived from an already existing floating point number its size will have to be equal to the one of said type. For example, when creating a nfloat from a double, given that their size is 64 bits, then the declaration will have to declare nfloat<64> as type.

nfloats can also be generated from a std::string:

nfloat<80> fstr = nfloat<80>("1.0001");

Manipulation

The only way nfloats can be manipulated is by directly tweaking their sign, exponent or mantissa. As of now, there is no way to perform arithmetic operation between them. However, one operation that can be performed is a cast to another nfloat having a different length. This can be done with the cast<> function:

nfloat<128> F = nfloat<64>("0.3").cast<128>();

Keep in mind that, much like regular floating point casts, the precision of the casted number remains faithful to the size of the nfloat before being casted. In this case, a 128-bit nfloat will be created, but it will hold the value 0.3 with the precision of a 64-bit floating point.

Notes

The aim of nfloats is not efficiency nor any actual application, as some of the algorithms that were created to compute them are very inefficient (yet functiunal). My boredom may have lead to their creation, yet won't lead any further I'm afraid.

However, even if they were somehow efficient, there would be no need to store high precision decimal numbers under the rules of the IEEE-754 format, as many other techniques concerning arbitrary precision numbers do the job much better.

Regardless of that, they can show how the IEEE-754 format can be applied to floating point numbers different from the classic 32-bit floats and the 64-bit doubles found everywhere in programming.