[GNC] Bill total does not match real total

[Michael or Penny Novack]

On 3/6/2024 12:49 PM, Carl Linkletter via gnucash-user wrote:
> OK, Found the bug.  The problem is that the tax is totaled up using sub-penny calculations. That causes a 1.004 + 1.004 to equal 2.008 which is then rounded to 2.01.  Each line should be rounded to the penny first, then rounded up.  So if 1.004 is first rounded to 1.00, then that would cause this calculation to total 2.00, not 2.01.
>
> If that makes any sense :-)  Guess I will enter a bug on this.

Does NOT solve the problem (rounding to the penny first). It is simply 
the case that rounded(A) + rounded(B) /= rounded(A+B).

You have given an example where 1.004 + 1.004 would give 2.00 if rounded 
first and 2.01 if rounded after adding. But how about 1.004 +  1.004 + 
1.004 which would give 3.00 if rounded before adding and 3.01 if rounded 
after.

The problem is actually because of using "reals" to represent decimal 
quantities. Now MY working days were spend with machines whose 
architecture directly supported expressing decimals in BCD (binary coded 
decimal). But the fact that our current CPUs don't directly support BCD 
representation and arithmetic does not mean couldn't emulate that in 
software (in our higher level language). Most of us using c++ are doing 
that so we don't have to "roll our own" << c++ is actually a 
"pre-processor" for c >> Somebody like me COULD come up with a BCD 
arithmetic package and I assume existing developers could also.

The point is that people like the late Admr. Grace Murray Hopper (WWII 
cryptologist, designer of COBOL) understood that using reals to express 
decimals would have this problem and so used BCD for money amounts. 
That's because intended for business and business implies money amounts. 
THEN machine were made with architecture to directly support it.

Michael D Novack

PS --- BCD is essentially "integer" representation and math with the 
"display amount" handling the editing to insert the decimal point. Now 
the only arithmetic operation raising a rounding issue is division.