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Miloslav Ciz 2023-04-13 17:43:44 +02:00
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@ -31,7 +31,7 @@ So for example the binary representation `110011` stores mantissa `110` (6) and
Note a few things: firstly our format is [shit](shit.md) because some numbers have multiple representations, e.g. 0 can be represented as `000000`, `000001`, `000010`, `000011` etc., in fact we have 8 zeros! That's unforgivable and formats used in practice address this (usually by prepending an implicit 1 to mantissa).
Secondly notice the non-uniform distribution of our numbers: while we have a nice resolution close to 0 (we can represent 1/16, 2/16, 3/16, ...) but low resolution in higher numbers (the highest number we can represent is 56 but the second highest is 48, we can NOT represent e.g. 50 exactly). Realize that obviously with 6 bits we can still represent only 64 numbers at most! So float is NOT a magical way to get more numbers, with integers on 6 bits we can represent numbers from 0 to 63 spaced exactly by 1 and with our floating point we can represent numbers spaced as close as 1/16th but only in the region near 0, we pay the price of having big gaps in higher numbers.
Secondly notice the non-uniform distribution of our numbers: while we have a nice resolution close to 0 (we can represent 1/16, 2/16, 3/16, ...), our resolution in high numbers is low (the highest number we can represent is 56 but the second highest is 48, we can NOT represent e.g. 50 exactly). Realize that obviously with 6 bits we can still represent only 64 numbers at most! So float is NOT a magical way to get more numbers, with integers on 6 bits we can represent numbers from 0 to 63 spaced exactly by 1 and with our floating point we can represent numbers spaced as close as 1/16th but only in the region near 0, we pay the price of having big gaps in higher numbers.
Also notice that thing like simple addition of numbers become more difficult and time consuming, you have to include conversions and [rounding](rounding.md) -- while with fixed point addition is a single machine instruction, same as integer addition, here with software implementation we might end up with dozens of instructions (specialized hardware can perform addition fast but still, not all computer have that hardware).