What is the purpose of max_digits10 and how is it different from digits10?

Question

I am confused about what max_digits10 represents. According to its documentation, it is 0 for all integral types. The formula for floating-point types for max_digits10 looks similar to int's digits10's.

Arne

Related: What is the meaning of numeric_limits<double>::digits10 – Tenrec

Answer 1

To put it simple,

• digits10 is the number of decimal digits guaranteed to survive text → float → text round-trip.
• max_digits10 is the number of decimal digits needed to guarantee correct float → text → float round-trip.

There will be exceptions to both but these values give the minimum guarantee. Read the original proposal on max_digits10 for a clear example, Prof. W. Kahan's words and further details. Most C++ implementations follow IEEE 754 for their floating-point data types. For an IEEE 754 float, digits10 is 6 and max_digits10 is 9; for a double it is 15 and 17. Note that both these numbers should not be confused with the actual decimal precision of floating-point numbers.

Example digits10

char const *s1 = "8.589973e9";
char const *s2 = "0.100000001490116119384765625";
float const f1 = strtof(s1, nullptr);
float const f2 = strtof(s2, nullptr);
std::cout << "'" << s1 << "'" << '\t' << std::scientific << f1 << '\n';
std::cout << "'" << s2 << "'" << '\t' << std::fixed << std::setprecision(27) << f2 << '\n';

Prints

'8.589973e9'      8.589974e+009
'0.100000001490116119384765625'   0.100000001490116119384765625

All digits up to the 6^th significant digit were preserved, while the 7^th digit didn't survive for the first number. However, all 27 digits of the second survived; this is an exception. However, most numbers become different beyond 7 digits and all numbers would be the same within 6 digits.

In summary, digits10 gives the number of significant digits you can count on in a given float as being the same as the original real number in its decimal form from which it was created i.e. the digits that survived after the conversion into a float.

Example max_digits10

void f_s_f(float &f, int p) {
    std::ostringstream oss;
    oss << std::fixed << std::setprecision(p) << f;
    f = strtof(oss.str().c_str(), nullptr);
}

float f3 = 3.145900f;
float f4 = std::nextafter(f3, 3.2f);
std::cout << std::hexfloat << std::showbase << f3 << '\t' << f4 << '\n';
f_s_f(f3, std::numeric_limits<float>::max_digits10);
f_s_f(f4, std::numeric_limits<float>::max_digits10);
std::cout << f3 << '\t' << f4 << '\n';
f_s_f(f3, 6);
f_s_f(f4, 6);
std::cout << f3 << '\t' << f4 << '\n';

Prints

0x1.92acdap+1   0x1.92acdcp+1
0x1.92acdap+1   0x1.92acdcp+1
0x1.92acdap+1   0x1.92acdap+1

Here two different floats, when printed with max_digits10 digits of precision, they give different strings and these strings when read back would give back the original floats they are from. When printed with lesser precision they give the same output due to rounding and hence when read back lead to the same float, when in reality they are from different values.

In summary, max_digits10 are at least required to disambiguate two floats in their decimal form, so that when converted back to a binary float, we get the original bits again and not of the one slightly before or after it due to rounding errors.

Jeffereyjefferies

you use decimal digit as: "base10 digit" or "right of . " – Arne

@Arne base 10 digits in the significand. From the digits10 example: 8.589973e9 has 7 decimal digits and doesn't survive text->float->text roundtrip. – Jeffereyjefferies

btw why is your A mentioning floats when this is also defined for integers/chars seems weird... but explains all the 0 for int/char types in documentetion... why would std do that – Arne

@Arne zeroes are used when the value is not meaningful for the type: max_digits_10 was only defined for floating-point types, just like epsilon, for example. Integer epsilons are also zero. – Jeffereyjefferies

@Tenrec I don't think the edit added anything it didn't already say. I suppose it could be saved with more specific examples? – Jeffereyjefferies

@Tenrec If you want to call somebody's code UB, you'd better understand the answer you link to first. Not all uses of oss.str().c_str() is UB, and the original code is fine as written. – Peacetime

OK, I might be wrong. But I'd like to learn, please tell me how the temporary string returned would live beyond the expression? Wouldn't what c_str be now pointing to a deconstructed string object? – Tenrec

@Tenrec It's solely used inside the full-expression it is created in. The call to strtof is not performed after any temporaries have been destroyed. – Feme

Answer 2

In my opinion, it is explained sufficiently at the linked site (and the site for digits10):

digits10 is the (max.) amount of "decimal" digits where numbers
can be represented by a type in any case, independent of their actual value.
A usual 4-byte unsigned integer as example: As everybody should know, it has exactly 32bit,
that is 32 digits of a binary number.
But in terms of decimal numbers?
Probably 9.
Because, it can store 100000000 as well as 999999999.
But if take numbers with 10 digits: 4000000000 can be stored, but 5000000000 not.
So, if we need a guarantee for minimum decimal digit capacity, it is 9.
And that is the result of digits10.

max_digits10 is only interesting for float/double... and gives the decimal digit count
which we need to output/save/process... to take the whole precision
the floating point type can offer.
Theoretical example: A variable with content 123.112233445566
If you show 123.11223344 to the user, it is not as precise as it can be.
If you show 123.1122334455660000000 to the user, it makes no sense because
you could omit the trailing zeros (because your variable can´t hold that much anyways)
Therefore, max_digits10 says how many digits precision you have available in a type.

Dowd

Answer 3

Lets build some context

After going through lots of answers and reading stuff following is the simplest and layman answer i could reach upto for this.

Floating point numbers in computers (Single precision i.e float type in C/C++ etc. OR double precision i.e double in C/C++ etc.) have to be represented using fixed number of bits.

float is a 32-bit IEEE 754 single precision Floating Point Number – 1 bit for the sign, 8 bits for the exponent, and 23* for the value. float has 7 decimal digits of precision.

And for double type

The C++ double should have a floating-point precision of up to 15 digits as it contains a precision that is twice the precision of the float data type. When you declare a variable as double, you should initialize it with a decimal value

What the heck above means to me?

Its possible that sometimes the floating point number which you have cannot fit into the number of bits available for that type. for eg. float value of 0.1 cannot FIT into available number of BITS in a computer. You may ask why. Try converting this value to binary and you will see that the binary representation is never ending and we have only finite number of bits so we need to stop at one point even though the binary conversion logic says keep going on.

If the given floating point number can be represented by the number of bits available, then we are good. If its not possible to represent the given floating point number in the available number of bits, then the bits are stored a value which is as close as possible to the actual value. This is also known as "Rounding the float value" OR "Rounding error". Now how this value is calculated depends of specific implementation but its safe to assume that given a specific implementation, the most closest value is chosen.

Now lets come to std::numeric_limits<T>::digits10

The value of std::numeric_limits::digits10 is the number of base-10 digits that are necessary to uniquely represent all distinct values of the type T, such as necessary for serialization/deserialization to text. This constant is meaningful for all floating-point types.

What this std::numeric_limits<T>::digits10 is saying is that whenever you fall into a scenario where rounding MUST happen then you can be assured that after given floating point value is rounded to its closest representable value by the computer, then its guarantied that the closest representable value's std::numeric_limits<T>::digits10 number of Decimal digits will be exactly same as your input floating point. For single precision floating point value this number is usually 6 and for double precision float value this number is usually 15.

Now you may ask why i used the word "guarantied". Well i used this because its possible that more number of digits may survive while conversion to float BUT if you ask me give me a guarantee that how many will survive in all the cases, then that number is std::numeric_limits<T>::digits10. Not convinced yet?

OK, consider example of unsigned char which has 8 bits of storage. When you convert a decimal value to unsigned char, then what's the guarantee that how many decimal digits will survive? I will say "2". Then you will say that even 145 will survive, so it should be 3. BUT i will say NO. Because if you take 256, then it won't survive. Of course 255 will survive, but since you are asking for guarantee so i can only guarantee that 2 digits will survive because answer 3 is not true if i am trying to use values higher than 255.

Now use the same analogy for floating number types when someone asks for a guarantee. That guarantee is given by std::numeric_limits<T>::digits10

Now what the heck is std::numeric_limits<T>::max_digits10

Here comes a bit of another level of complexity. BUT I will try to explain as simple as I can

As i mentioned previously that due to limited number of bits available to represent a floating type on a computer, its not possible to represent every float value exactly. Few can be represented exactly BUT not all values. Now lets consider a hypothetical situation. Someone asks you to write down all the possible float values which the computer can represent (ooohhh...i know what you are thinking). Luckily you don't have write all those :) Just imagine that you started and reached the last float value which a computer can represent. The max float value which the computer can represent will have certain number of decimal digits. These are the number of decimal digits which std::numeric_limits<T>::max_digits10 tells us. BUT an actual explanation for std::numeric_limits<T>::max_digits10 is the maximum number of decimal digits you need to represent all possible representable values. Thats why i asked you to write all the value initially and you will see that you need maximum std::numeric_limits<T>::max_digits10 of decimal digits to write all representable values of type T.

Please note that this max float value is also the float value which can survive the text to float to text conversion but its number of decimal digits are NOT the guaranteed number of digits (remember the unsigned char example i gave where 3 digits of 255 doesn't mean all 3 digits values can be stored in unsigned char?)

Hope this attempt of mine gives people some understanding. I know i may have over simplified things BUT I have spent sleepless night thinking and reading stuff and this is the explanation which was able to give me some peace of mind.

Cheers !!!

Furze

Why is std::numeric_limits<T>::max_digits10 defined as 0 for integral types? – Kaden