why does realmax('single') display 8 digits?
显示 更早的评论
most texts agree that IEEE-754 sinlge persision has 7 signicant digits and double percision has 15
so why does realmax('single') display 8 digits and realmax('double') display 16 digits?
8 个评论
Ameer Hamza
2020-12-17
I guess you are talking about flintmax() instead of realmax(). My answer is based on flintmax().
Bruno Luong
2020-12-17
He talk about realmax
>> format long
>> realmax('single')
ans =
single
3.4028235e+38
^
1 2345678 : 8 digits
>> realmax('double')
ans =
1.797693134862316e+308
^
1 234567890123456 : 16 digits
Walter Roberson
2020-12-17
But that is just the current format output. The format output does not reliably display enough digits to reproduce the input for double.
Bruno Luong
2020-12-17
Exactly he asks about format output.
the question is "realmax('single')" why you guys think he asks about "flintmax()"?
Walter Roberson
2020-12-17
I am too accustomed to floating point, I would not have guessed that the question was about how many digits MATLAB happened to display.
Stephen23
2020-12-18
"most texts agree that IEEE-754 sinlge persision has 7 signicant digits and double percision has 15"
Such a simplistic generalization is not very meaningful, because the behavior of binary floating point numbers is much more nuanced than that. For a start, you need to define what "precision" means to you:
- the precision of decimal values that map to distinct binary values?
- round-trip values binary->decimal->binary giving the same value?
- something else?
For an excellent introduction into this topic:
Sean Doherty
2020-12-18
Sean Doherty
2020-12-18
回答（2 个）
Ameer Hamza
2020-12-17
编辑：Ameer Hamza
2020-12-17
0 个投票
"most texts agree that IEEE-754 sinlge persision has 7 signicant digits and double percision has 15"
But they don't. More precisely, they have 24 and 53 significant binary digits, respectively. Both are capable of representing 2^24 and 2^53 consecutive integers without any loss of precision. MATLAB just returns these values.
Note that it is still possible to represent even bigger integers with full precision, but the only issue is that the number adjacent to them cannot be represented precisely.
Read here: https://en.wikipedia.org/wiki/IEEE_754#Basic_and_interchange_formats. They are capable of representing ~7.22 and ~15.96 decimal digits respectively.
12 个评论
Sean Doherty
2020-12-17
Ameer Hamza
2020-12-17
I guess you meant flintmax() again here.
Yes, the precision for all 8 digits integers is not guaranteed. However, anything less than 16777216 can be represented precisely.
Bruno Luong
2020-12-17
编辑：Bruno Luong
2020-12-17
Why should you care about how many digits realmax('single') displays? Simply use the value it returns.
Beside 24 bits mantissa of single number is more than 7 digits (7.22 as Ameer has mentioned), so the 8-bits display might be not exact but still more accurate than 7-digit, and it might also might be true for 9, 10 digits...
Sean Doherty
2020-12-17
Significant digits are reliable. 7.22 sig digs means digit 8 is not guaranteed significant
Sean Doherty
2020-12-17
The point applies to any SP/DP calculation... why display 8/16 digits when only guaranteed 7/15 significant digits
Bruno Luong
2020-12-17
编辑：Bruno Luong
2020-12-17
"significant" in which sense exactly?
All the 8th digit are still significant in this sense:
Look at this exampe:
1/9 is 0.11111111111... with 1 repeated forever
>> fprintf('%1.7f\n', single(1/9))
0.1111111
>> fprintf('%1.8f\n', single(1/9))
0.11111111
Or look at this exampe:
1/3 is 0.333333333... with 3 repeated forever
>> fprintf('%1.7f\n', single(1/3))
0.3333333
>> fprintf('%1.8f\n', single(1/3))
0.33333334
Granted the 8th digit is "4" and not "3" as expected but:
0.33333334 is 5 times closer to 1/3 than 0.3333333.
So yeah it not correct digit but still quite significant nonetheless.
Again why you care about what is displayed??? If you don't trust the 8 bit just ignore it when look at the screen, or specify the number of digits you think as "significant" with fprintf. It doesn't matter for any calculation or usage of the value.
Ameer Hamza
2020-12-17
"why display 8/16 digits when only guaranteed 7/15 significant digits"
Because that is the purpose of flintmax(), it returns the maximum consecutive integer precisely represented by a floating-point data type. It does not have any mentions anything about the number of digits.
Also, as Bruno mentioned, why do you care how the number is displayed in base10? As long as it is accurately represented in computer memory, there should be no issue.
If integer precision is the primary concern, then why not use an integer data type, uint32 or uint64.
Bruno Luong
2020-12-17
编辑：Bruno Luong
2020-12-17
One more example, not because 8th digit is not warranty that beyond that the digits have no "signification":
>> x = single(2^(-24)); % eps(single(1))/2
>> fprintf('%1.20e\n', x);
5.96046447753906250000e-08 % this is exact result in digital
^
1 2345678901234567 <= obtained only after at least 17 digits, even for single
Feel free to truncate with 7 digits "5.960464e-8". You'll lost the correct 10 digit trailer "...4775390625".
Sean Doherty
2020-12-17
Walter Roberson
2020-12-17
There are some "single" numbers (that do not have trailing 0) where entering only 7 digits decimal is enough to completely reproduce the number, but there are others where 8 digits decimal is required.
Bruno Luong
2020-12-17
编辑：Bruno Luong
2020-12-17
"but there are others where 8 digits decimal is required."
Like
flintmax('single')
Other one need 17 digits in my example of 2^(-24) or even more (up to 20 digits).
Limiting about 7-8 digits is just like see only the emerging side of an ice-berg, and pretend digits after the 8th are not significant is just plain wrong, at least from my point of view.
Given N digits of a single precision number, is entering those N digits enough to exactly reproduce the same bit pattern?
fprintf('%1.20e\n', single(5.960464e-8))
fprintf('%1.20e\n', single(5.9604644e-8))
So 7 input digits is not enough to completely preproduce 2^-24 but 8 input digits is. Therefor, this particular number has 8 significant decimal digits in single precision. But there are other values that use "all" of the digits (no trailing 0 in decimal) that 7 copied digits is enough.
Walter Roberson
2020-12-17
>> flintmax('single')
ans =
single
16777216
You get 7 full digits, up to 9999999 . And there is some spare room beyond that, but not enough for a full digit. It is about 7.22 digits.
类别
在 帮助中心 和 File Exchange 中查找有关 Sparse Matrices 的更多信息
2020-12-17
2020-12-18
Community Treasure Hunt
Find the treasures in MATLAB Central and discover how the community can help you!
Start Hunting!
Translated by 
