WONKY calculation bug in Numbers 3.0.1

Question

Hi,
This is wonky. It's a bit hard to explain in words, so best see the attachment to understand. You can download the attachment here: https://app.sugarsync.com/iris/wf/D162135_83773729_7822834
Basically, I have a number in each of A1, B1, and C1. The numbers have up to 2 decimals.
D1 contains the following formula: A1 - SUM( B1:C1 ), giving the result 19782.10.
E1 contains the number 19782.10.
F1 contains the formula D1 - E1.
The answer should be 0.00. Instead, it reads 0.00000000000363797881....
There is no inputted number anywhere with more than 2 decimal places. Clearly there is some kind of bug. The funny thing is if I change the formula in D1 to A1 - B1 - C1, then I get 0.00.
I've also tried changing the numbers in A1, B1, and C1 and then adjusting E1 accordingly. With very few exceptions, I get the correct 0.00. There is something about those particular numbers running through the SUM formula that produces the ghost decimals.
Of course, errors to the 12th decimal aren't going to throw off my spreadsheet too badly. But I have to wonder if these bugs are coming up in such simple math, what else is wonky in Numbers 3.0?
When I get weird results in my math, I like to be able to assume it is because of my error. Knowing that Numbers is wonky is going to add a lot of uncertainty to complex troubleshooting in my formulas.
What's the deal? Any ideas?
A

Answer 1

Hello
Here's a brief anatomy of 1.1 - 1.0 - 0.1 = 8.32667268468867E-17 and 1.1 - 0.1 - 1.0 = 0.0.
legend
    s = sign (0 => +, 1 => -)
    e = exponent (bias = 0x3ff = 1023)
    m = mantissa (hidden bit is explicitly represented; integer bit (1/0) in mantissa is hidden bit)
1.1 = 3ff199999999999a := {s = 0; e = 3ff; m = 1.199999999999a}
1.0 = 3ff0000000000000 := {s = 0; e = 3ff; m = 1.0000000000000}
0.1 = 3fb999999999999a := {s = 0; e = 3fb; m = 1.999999999999a}
1.1 - 1.0 = 3fb99999999999a0 := {s = 0; e = 3fb; m = 1.99999999999a0}
    3ff199999999999a - 3ff0000000000000
    = {s = 0; e = 3ff; m = 1.199999999999a} - {s = 0; e = 3ff; m = 1.0000000000000}
    = {s = 0; e = 3ff; m = 0.199999999999a}    [1]
    = {s = 0; e = 3fb; m = 1.99999999999a0}    [2]    # e -= 4; m << 4
    = 3fb99999999999a0
    = 2 ^ (0x3fb - 0x3ff) * 0x1.99999999999a0
    = 2 ^ (1019 - 1023) * 1.600000000000001421085471520...
    = 2 ^ -4 * 1.600000000000001421085471520...
    = 0.100000000000000088817841970...
1.1 - 1.0 - 0.1 = 3c98000000000000 := {s = 0; e = 3c9; m = 1.000000000000}
    3fb99999999999a0 - 3fb999999999999a
    = {s = 0; e = 3fb; m = 1.99999999999a0} - {s = 0; e = 3fb; m = 1.999999999999a}
    = {s = 0; e = 3fb; m = 0.0000000000006}    [1]
    = {s = 0; e = 3c9; m = 1.8000000000000}    [2]    # e -= 50; m << 50
    = 3c98000000000000
    = 2 ^ (0x3c9 - 0x3ff) * 0x1.8
    = 2 ^ (969 - 1023) * 1.5
    = 2 ^ -54 * 1.5
    = 8.32667268468867E-17
1.1 - 0.1 = 3ff0000000000000 := {s = 0; e = 3ff; m = 1.0000000000000}
    3ff199999999999a - 3fb999999999999a
    = {s = 0; e = 3ff; m = 1.199999999999a} - {s = 0; e = 3fb; m = 1.999999999999a}
    = {s = 0; e = 3ff; m = 1.199999999999a} - {s = 0; e = 3ff; m = 0.199999999999a}    [3]
    = {s = 0; e = 3ff; m = 1.0000000000000}
    = 3ff0000000000000
    = 1.0
1.1 - 0.1 - 1.0 = 0000000000000000
    3ff0000000000000 - 3ff0000000000000
    = {s = 0; e = 3ff; m = 1.0000000000000} - {s = 0; e = 3ff; m = 1.0000000000000}
    = {s = 0; e = 0; m = 0.0000000000000}    [4]
    = 00000000000000
    = 0.0
[1] This is not normalised form with hidden bit = 0
[2] This is normalised form with hidden bit = 1
[3] the 2nd term is not normalised form; LSB of mantissa of the 2nd term is result of rounding.
[4] +0
Cheers,
H