Result when subtrahend calls round function - sql-server

Question

I'm dividing two floats, multiplying it by 100 and then subtracting it by 100. I'm returning a percentage.
My question is: why is the final result a float that isn't rounded when the right part of the subtraction returns a float of 2 digits?
These is one sequence:
/* 1 */
-- Returns 0.956521739130435, which is correct.
select cast(198 as float)/(cast(198 as float) + cast(9 as float)) -- correct
/* 2 */
-- Returns 95.6521739130435, which is correct.
select 100*(cast(198 as float)/(cast(198 as float) + cast(9 as float))) --correct
/* 3 */
-- It's the same as previous one, but with a ROUND
-- Returns 95.65, which is correct.
select round(100*(cast(198 as float)/(cast(198 as float) + cast(9 as float))),2)
/* 4 */
-- Returns 4.34999999999999, should be 100-95.65, but it's not. ROUND is ignored. Why?
select 100-round(100*(cast(198 as float)/(cast(198 as float) + cast(9 as float))),2)
|-------------- This returns 95.65 --------------------------------|
Another sequence:
/* 1 */
-- Returns 0.956521739130435, which is correct.
select cast(198 as float)/(cast(198 as float) + cast(9 as float))
/* 2 */
-- Returns 0.9565, which is correct.
select round(cast(198 as float)/(cast(198 as float) + cast(9 as float)), 4)
/* 3 */
-- Returns 95.65, which is correct.
select 100*round(cast(198 as float)/(cast(198 as float) + cast(9 as float)), 4)
/* 4 */
-- Returns 4.34999999999999, should be 100-95.65, but it's not. ROUND is ignored. Why?
select 100-(100*round(cast(198 as float)/(cast(198 as float) + cast(9 as float)), 4))
|-------------------- This returns 95.65 --------------------------------|
I'm just curious as to why this happens, although it can easily be fixed with one ROUND at the beginning:
select round(100-(100*(cast(198 as float)/(cast(198 as float) + cast(9 as float)))), 2)
The reason I ask is because it's not something that can be easily reproduced. I tried reproducing it, and out of 2,000 times, it only occurred 12 times. That's less than 1%, but with floats with repetitive numbers after the 2nd decimal (ie. 3.47999999999), which makes sense:
declare #rand int = 1
While(#rand <= 2000)
begin
select 100-round(100*(cast(abs(checksum(NewId()) % 1500) as float)/(cast(abs(checksum(NewId()) % 1500) as float) + cast(abs(checksum(NewId()) % 1500) as float))),2)
set #rand = #rand + 1
end
I guess my other question is: what type is the sql editor returning when it returns 95.65 with select round(100*(cast(198 as float)/(cast(198 as float) + cast(9 as float))),2)?

Answer 1

To expand on Jeroen's comment:
SQL Server's FLOAT type is a double-precision floating-point value. As with (most) floating point formats, the value is stored in binary. Just as the number 1/3 cannot be represented with a finite number of digits after the decimal, the number 95.65 cannot be represented with a finite number of bits. The closest value to 95.65 that can be stored in a FLOAT has the exact value:
95.650000000000005684341886080801486968994140625
If you subtract that number from 100, you get exactly:
4.349999999999994315658113919198513031005859375
When displayed, this is rounded to 15 significant digits, and the value printed is:
4.34999999999999
As discussed, you can solve this problem by using DECIMAL type instead of FLOAT.
There are many resources available on StackOverflow and elsewhere if you'd like to learn more about floating-point math.
-- EDIT --
I'm going to use parenthesis notation for repeating decimals. When I write
0.(3)
that means
0.333333333333333333333333333... and so on forever.
Let's start at the beginning. 168 can be stored in a float. 168+9 is 177. That can be stored in a float. If you divide 168 by 177 the mathematically correct answer is:
0.95(6521739130434782608695)
But this value cannot be stored in a float. The closest value that can be stored in a float is:
0.9565217391304348115710354250040836632251739501953125
Take that number and multiply by 100 , the mathematically correct answer is:
95.65217391304348115710354250040836632251739501953125
Since you multiplied a float by 100, you get a float, and that number cannot be stored in a float, so the closest possible value is:
95.6521739130434838216388016007840633392333984375
You ask that this float be rounded to 2 digits after the decimal. The mathematically correct answer is:
95.65
But since you asked to round a float, the answer is also a float, and that value cannot be stored in a float. The closest possible value is:
95.650000000000005684341886080801486968994140625
You asked to subtract that from 100. The mathematically correct value is:
4.349999999999994315658113919198513031005859375
As it happens, that value can be stored in a float. So that's the value that's being selected.
When converting this number to a string, SQL Server rounds the result to 15 significant digits. So that number, when printed, appears as:
4.34999999999999
When you ran the same calculation on your Java console, the exact same calculations were performed, but when the value was printed, Java rounded to 16 significant digits:
4.349999999999994
-- Another EDIT --
Why can't 96.65 be stored exactly in a float? The float type stores numbers in binary format. If you want to express 96.65 in binary, the mathematically exact value is:
1011111.1010011001100110011001100110011001100110011001(1001)
You can see the pattern. Just as 1/3 is represented as an infinite repeating value in decimal, this value has an infinite repeating value in binary. You can see the pattern (1001) being repeated over and over.
A float can only hold 53 significant bits. And so this is rounded to:
1011111.1010011001100110011001100110011001100110011010
If you convert that number back to decimal, you get the exact value:
95.650000000000005684341886080801486968994140625
-- Yet Another Edit --
You ask what happens when you call Round again on the result.
We started with the number:
4.349999999999994315658113919198513031005859375
You ask that this be rounded to 2 places. The mathematically correct answer is:
4.35
Since you are rounding a float, this result must also be a float. Express this value in binary. The mathematically correct answer is:
100.0101100110011001100110011001100110011001100110011001(1001)
Again, this is a repeating binary value. But float can't store an infinite number of bits. The value is rounded to 53 significant bits. The result is:
100.0101100110011001100110011001100110011001100110011
If you convert this to decimal, the exact value is:
4.3499999999999996447286321199499070644378662109375
That is the value you selected. Now SQL Server needs to print that on the screen. As before, it is rounded to 15 significant digits. The result is:
4.35000000000000
It removes the trailing zeros, and the result you see on the screen is:
4.35
The last round did nothing magic. The answer is still stored as a float, and the answer is still not an exact value. As it happens SQL Server chooses to round values to 15 significant digits when printing a float. In this case, that rounded value happened to match the exact value you were expecting.
If values were rounded to 14 places when printing them, the original query would have appeared to have the value you expected.
If values were rounded to 16 places, then the result of the final round would be shown as
4.3499999999999996