Can you help me to understand what "significant digits" means in floating point math?

Question

For what I'm learning, once I convert a floating point value to a decimal one, the "significant digits" I need are a fixed number (17 for double, for example). 17 totals: before and after decimal separator.

So for example this code:

typedef std::numeric_limits<double> dbl;

int main()
{
    std::cout.precision(dbl::max_digits10);
    //std::cout << std::fixed;    

    double value1 = 1.2345678912345678912345;
    double value2 = 123.45678912345678912345;
    double value3 = 123456789123.45678912345;

    std::cout << value1 << std::endl;
    std::cout << value2 << std::endl;
    std::cout << value3 << std::endl;
}

will correctly "show me" 17 values:

1.2345678912345679
123.45678912345679
123456789123.45679

But if I increase precision for the cout (i.e. std::cout.precision(100)), I can see there are other numbers after the 17 range:

1.2345678912345678934769921397673897445201873779296875
123.456789123456786683163954876363277435302734375
123456789123.456787109375

Why should ignore them? They are stored within the variables/double as well, so they will affect the whole "math" later (division, multiplication, sum, and so on).

What does it means "significant digits"? There is other...

Answer 1

Can you help me to understand what “significant digits” means in floating point math?

With FP numbers, like mathematical real numbers, significant digits is the leading digits of a value that do not begin with 0 and then, depending on context, to 1) the decimal point, 2) the last non-zero digit, or 3) the last printed digit.

123.         // 3 significant decimal digits
123.125      // 6 significant decimal digits
0.0078125    // 5 significant decimal digits
0x0.00123p45 // 3 significant hexadecimal digits
123000.0     // 3, 6, or 7 significant decimal digits depending on context

When concerned about decimal significant digits and FP types like double. the issue is often "How many decimal significant digits are needed or of concern?"

Nearly all C FP implementations use a binary encoding such that all finite FP are exact sums of power of 2. Each finite FP is exact. Common encoding affords most double to have 53 binary digits is it significand - so 53 significant binary digits. How this appears as a decimal is often the source of confusion.

// Example 0.1 is not an exact sum of powers of 2 so a nearby value is used.
double x = 0.1;
// x takes on the exact value of 
// 0.1000000000000000055511151231257827021181583404541015625
// aka 0x1.999999999999ap-4
// aka base2: 0.000110011001100110011001100110011001100110011001100110011010
// The preceding and subsequent doubles
// 0.09999999999999999167332731531132594682276248931884765625
// 0.10000000000000001942890293094023945741355419158935546875
//   123456789012345678901234567890123456789012345678901234567890

Looking at above, one could say x has over 50 decimal significant digits. Yet the value matches the intended 0.1 to 16 decimal significant digits. Or yet since the preceding and subsequent possible double values differ in the 17 place, one could say x has 17 decimal significant digits.

What does it means "significant digits"?

Various meanings of significant digits exist, but for C, 2 common ones are:

1. The number of decimal significant digits that a textual value to double converts as expected for all double. This is typically 15. C specifies this as DBL_DIG and must be at least 10.

2. The number of decimal significant digits that a textual value of double needs to be printed to distinguish from another double. This is typically 17. C specifies this as DBL_DECIMAL_DIG and must be at least 10.

Why should ignore them?

It depends of coding goals. Rarely are all digits of the exact value needed. (DBL_TRUE_MIN might have 752 od them.) For most applications, DBL_DECIMAL_DIG is enough. In select apps, DBL_DIG will do. So usually, ignoring digits past 17 does not cause problems.

Answer 2

just to add to Pete Becker's answer, I think you're confusing the problem of finding the exact decimal representation of a binary mantissa, with the problem of finding some decimal representation uniquely representing that binary mantissa ( given some fixed rounding scheme ).

Now, regarding the first problem, you always need a finite number of decimal digits to exactly represent a binary mantissa ( because 2 divides 10 ).

For example, you need 18 decimal digits to exactly represent the binary 1.0000000000000001, being 1.00000762939453125 in decimal.

but you need just 17 digits to represent it uniquely as 1.0000076293945312 because no other number having exact value 1.0000076293945312xyz... where 0<=x<5 can exist as a double ( more precisely, the next and prior exactly representable values being 1.0000076293945314720446049250313080847263336181640625 and 1.0000076293945310279553950749686919152736663818359375 ).

Of course, this does not mean that given some decimal number you can ignore all digits past the 17th; it just means that if you apply the same rounding scheme used to produce the decimal at the 17th position and assign it back to a double you'll get the same original double.

Answer 3

Keep in mind that floating-point values are not real numbers. There are gaps between the values, and all those extra digits, while meaningful for real numbers, don’t reflect any difference in the floating-point value. When you convert a floating-point value to text, having std::numeric_limits<...>::max_digits10 digits ensures that you can convert the text string back to floating-point and get the original value. The extra digits don’t affect the result.

The extra digits that you see when you ask for more digits are the result of the conversion algorithm trying to do what you asked. The algorithm typically just keeps extracting digits until it reaches the desired precision; it could be written to start outputting zeros after it’s written max_digits10 digits, but that’s an additional complication that nobody bothers with. It wouldn’t really be helpful.