Is there a formula to find the numbers of bits for either exponent or significant in a floating point number?

Question

Recently, I have been interested with using bit shiftings on floating point numbers to do some fast calculations.

To make them work in more generic ways, I would like to make my functions work with different floating point types, probably through templates, that is not limited to float and double, but also "halfwidth" or "quadruple width" floating point numbers and so on.

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Then I noticed:

 - Half   ---  5 exponent bits  ---  10 signicant bits
 - Float  ---  8 exponent bits  ---  23 signicant bits
 - Double --- 11 exponent bits  ---  52 signicant bits

So far I thought exponent bits = logbase2(total byte) * 3 + 2,
which means 128bit float should have 14 exponent bits, and 256bit float should have 17 exponent bits.

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However, then I learned:

 - Quad   --- 15 exponent bits  ---  112 signicant bits
 - Octuple--- 19 exponent bits  ---  237 signicant bits

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So, is there a formula to find it at all? Or, is there a way to call it through some builtin functions?
C or C++ are preferred, but open to other languages.

Thanks.

Answer 1

Characteristics Provided Via Built-In Functions

C++ provides this information via the std::numeric_limits template:

#include <iostream>
#include <limits>
#include <cmath>


template<typename T> void ShowCharacteristics()
{
    int radix = std::numeric_limits<T>::radix;

    std::cout << "The floating-point radix is " << radix << ".\n";

    std::cout << "There are " << std::numeric_limits<T>::digits
        << " base-" << radix << " digits in the significand.\n";

    int min = std::numeric_limits<T>::min_exponent;
    int max = std::numeric_limits<T>::max_exponent;

    std::cout << "Exponents range from " << min << " to " << max << ".\n";
    std::cout << "So there must be " << std::ceil(std::log2(max-min+1))
        << " bits in the exponent field.\n";
}


int main()
{
    ShowCharacteristics<double>();
}

Sample output:

The floating-point radix is 2.
There are 53 base-2 digits in the significand.
Exponents range from -1021 to 1024.
So there must be 11 bits in the exponent field.

C also provides the information, via macro definitions like DBL_MANT_DIG defined in <float.h>, but the standard defines the names only for types float (prefix FLT), double (DBL), and long double (LDBL), so the names in a C implementation that supported additional floating-point types would not be predictable.

Note that the exponent as specified in the C and C++ standards is one off from the usual exponent described in IEEE-754: It is adjusted for a significand scaled to [½, 1) instead of [1, 2), so it is one greater than the usual IEEE-754 exponent. (The example above shows the exponent ranges from −1021 to 1024, but the IEEE-754 exponent range is −1022 to 1023.)

Formulas

IEEE-754 does provide formulas for recommended field widths, but it does not require IEEE-754 implementations to conform to these, and of course the C and C++ standards do not require C and C++ implementations to conform to IEEE-754. The interchange format parameters are specified in IEEE 754-2008 3.6, and the binary parameters are:

• For a floating-point format of 16, 32, 64, or 128 bits, the significand width (including leading bit) should be 11, 24, 53, or 113 bits, and the exponent field width should be 5, 8, 11, or 15 bits.
• Otherwise, for a floating-point format of k bits, k should be a multiple of 32, and the significand width should be k−round(4•log₂k)+13, and the exponent field should be round(4•log₂k)−13.