Why are floating point numbers printed so differently?

问题

It\'s kind of a common knowledge that (most) floating point numbers are not stored precisely (when IEEE-754 format is used). So one shouldn\'t do this:

0.3 - 0.2 === 0.1; // very wrong

... as it will result in false, unless some specific arbitrary-precision type/class was used (BigDecimal in Java/Ruby, BCMath in PHP, Math::BigInt/Math::BigFloat in Perl, to name a few) instead.

Yet I wonder why when one tries to print the result of this expression, 0.3 - 0.2, scripting languages (Perl and PHP) give 0.1, but \"virtual-machine\" ones (Java, JavaScript and Erlang) give something more similar to 0.09999999999999998 instead?

And why is it also inconsistent in Ruby? version 1.8.6 (codepad) gives 0.1, version 1.9.3 (ideone) gives 0.0999...

回答1:

Floating-point numbers are printed differently because printing is done for different purposes, so different choices are made about how to do it.

Printing a floating-point number is a conversion operation: A value encoded in an internal format is converted to a decimal numeral. However, there are choices about the details of the conversion.

(A) If you are doing precise mathematics and want to see the actual value represented by the internal format, then the conversion must be exact: It must produce a decimal numeral that has exactly the same value as the input. (Each floating-point number represents exactly one number. A floating-point number, as defined in the IEEE 754 standard, does not represent an interval.) At times, this may require producing a very large number of digits.

(B) If you do not need the exact value but do need to convert back and forth between the internal format and decimal, then you need to convert it to a decimal numeral precisely (and accurately) enough to distinguish it from any other result. That is, you must produce enough digits that the result is different from what you would get by converting numbers that are adjacent in the internal format. This may require producing a large number of digits, but not so many as to be unmanageable.

(C) If you only want to give the reader a sense of the number, and do not need to produce the exact value in order for your application to function as desired, then you only need to produce as many digits as are needed for your particular application.

Which of these should a conversion do?

Different languages have different defaults because they were developed for different purposes, or because it was not expedient during development to do all the work necessary to produce exact results, or for various other reasons.

(A) requires careful code, and some languages or implementations of them do not provide, or do not guarantee to provide, this behavior.

(B) is required by Java, I believe. However, as we saw in a recent question, it can have some unexpected behavior. (65.12 is printed as “65.12” because the latter has enough digits to distinguish it from nearby values, but 65.12-2 is printed as “63.120000000000005” because there is another floating-point value between it and 63.12, so you need the extra digits to distinguish them.)

(C) is what some languages use by default. It is, in essence, wrong, since no single value for how many digits to print can be suitable for all applications. Indeed, we have seen over decades that it fosters continuing misconceptions about floating-point, largely by concealing the true values involved. It is, however, easy to implement, and hence is attractive to some implementors. Ideally, a language should by default print the correct value of a floating-point number. If fewer digits are to be displayed, the number of digits should be selected only by the application implementor, hopefully including consideration of the appropriate number of digits to produce the desire results.

Worse, some languages, in addition to not displaying the actual value or enough digits to distinguish it, do not even guarantee that the digits produced are correct in some sense (such as being the value you would get by rounding the exact value to the number of digits shown). When programming in an implementation that does not provide a guarantee about this behavior, you are not doing engineering.

回答2:

As for php, output is related to ini settings of precision:

ini_set('precision', 15);
print 0.3 - 0.2; // 0.1

ini_set('precision', 17);
print 0.3 - 0.2; //0.099999999999999978

This may be also cause for other languages

回答3:

PHP automatically rounds the number to an arbitrary precision.

Floating-point numbers in general aren't accurate (as you noted), and you should use the language-specific round() function if you need a comparison with only a few decimal places. Otherwise, take the absolute value of the equation, and test they are within a given range.

PHP Example from php.net:

$a = 1.23456789;
$b = 1.23456780;
$epsilon = 0.00001;
if(abs($a - $b) < $epsilon) {
  echo "true";
}

As for the Ruby issue, they appear to be using different versions. Codepad uses 1.8.6, While Ideaone uses 1.9.3, but it's more likely related to a config somewhere.

回答4:

If we want this property

every two different float has a different printed representation

Or an even stronger one useful for REPL

printed representation shall be re-interpreted unchanged

Then I see 3 solutions for printing a float/double with base 2 internal representation into base 10

print the EXACT representation.
print enough decimal digits (with proper rounding)
print the shortest decimal representation that can be reinterpreted unchanged

Since in base two, the float number is an_integer * 2^an_exponent, its base 10 exact representation has a finite number of digits.
Unfortunately, this can result in very long strings... For example 1.0e-10 is represented exactly as 1.0000000000000000364321973154977415791655470655996396089904010295867919921875e-10

Solution 2 is easy, you use printf with 17 digits for IEEE-754 double...
Drawback: it's not exact, nor the shortest! If you enter 0.1, you get 0.100000000000000006

Solution 3 is the best one for REPL languages, if you enter 0.1, it prints 0.1
Unfortunately it is not found in standard libraries (a shame).
At least, Scheme, Python and recent Squeak/Pharo Smalltalk do it right, I think Java too.