math - PHP float calculation error when subtracting

Question

I have a very strange issue. If I subtract 2 float vars where one is the result of a mathematical operation I get a wrong value.

Example:

var_dump($remaining);
var_dump($this->hours_sub['personal']);
echo $remaining-$this->hours_sub['personal'];

This it the output:

float 5.4
float 1.4
5.3290705182008E-15

5.4-1.4 should be 4 If I add the two values the result is correct.

Where is my mistake? It can not be a rounding issue.

Answer 1

If still somebody hits this page with similar problems where floating number subtraction causes error or strange values. Below I will explain this problem with a bit more details.

It is not directly related to PHP and it is not a bug. However, every programmer should be aware of this issue.

This problem even took many lives two decades ago.

On 25 February 1991 an incorrect floating-point arithmetic (called rounding error) in a MIM-104 Patriot missile battery prevented it from intercepting an incoming Scud missile in Dhahran, Saudi Arabia, killing 28 soldiers and injuring near 100 servicemen from the U.S. Army's 14th Quartermaster Detachment.

But why it happens?

The reason is that floating point values represent a limited precision. So, a value might not have the same string representation after any processing (chopped off). It also includes writing a floating point value in your script and directly printing it without any mathematical operations.

Just a simple example:

$a = '36';
$b = '-35.99';
echo ($a + $b);

You would expect it to print 0.01, right? But it will print a very strange answer like 0.009999999999998

Like other numbers, floating point numbers double or float is stored in memory as a string of 0's and 1's. How floating point differs from integer is in how we interpret the 0's and 1's when we want to look at them. There are many standards how they are stored.

Floating-point numbers are typically packed into a computer datum as the sign bit, the exponent field, and the significand or mantissa, from left to right....

Decimal numbers are not well represented in binary due to lack of enough space. So, you can't express 1/3 exactly as it's 0.3333333..., right? Why we can't represent 0.01 as a binary float number is for the same reason. 1/100 is 0.00000010100011110101110000..... with a repeating 10100011110101110000.

If 0.01 is kept in simplified and system-truncated form of 01000111101011100001010 in binary, when it is translated back to decimal, it would be read like 0.0099999.... depending on system (64bit computers will give you much better precision than 32-bits). Operating system decides in this case whether to print it as it sees or how to make it in more human-readable way. So, it is machine-dependent how they want to represent it. But it can be protected in language level with different methods.

If you format the result using

echo number_format(0.009999999999998, 2);

it will print 0.01.

It is because in this case you instruct how it should be read and how precision you require.

Note number_format() is not the only function, a few other functions and ways can be used to tell the programming language about the precision expectation.

References:
https://sdqweb.ipd.kit.edu/publications/pdfs/saglam2016a.pdf
https://en.wikipedia.org/wiki/Round-off_error