Decimal precision of floats

equivalent to log10(2^24) ≈ 7.225开发者_如何学Python decimal digits

Wikipedia

Precision: 7 digits

MSDN

6

std::numeric_limits<float>::digits10

Why numeric_limits return 6 here? Both Wikipedia and MSDN report that floats have 7 decimal digits of precision.

If in doubt, read the spec. The C++ standard says that digits10 is:

Number of base 10 digits that can be represented without change.

That's a little vague; fortunately, there's a footnote:

Equivalent to FLT_DIG, DBL_DIG, LDBL_DIG

Those are defined in the C standard; let's look it up there:

number of decimal digits, q, such that any floating-point number with q decimal digits can be rounded into a floating-point number with p radix b digits and back again without change to the q decimal digits.

So std::numeric_limits<float>::digits10 is the number of decimal digits such that any floating-point number with that many digits is unchanged if you convert it to a float and back to decimal.

As you say, floats have about 7 digits of decimal precision, but the error in representation of both fixed-width decimals and floats is not uniformly logarithmic. The relative error in rounding a number of the form 1.xxx.. to a fixed number of decimal places is nearly ten times larger than the relative error of rounding 9.xxx.. to the same number of decimal places. Similarly, depending on where a value falls in a binade, the relative error in rounding it to 24 binary digits can vary by a factor of nearly two.

The upshot of this is that not all seven-digit decimals survive the round trip to float and back, but all six digit decimals do. Hence, std::numeric_limits<float>::digits10 is 6.

There are not that many six and seven digit decimals with exponents in a valid range for the float type; you can pretty easily write a program to exhaustively test all of them if you're still not convinced.

It's really only 23 bits in the mantissa (there's an implied 1, so it's effectively 24 bits, but the 1 obviously does not vary). This gives 6.923689900271567 decimal digits of precision, which is not quite 7.