Why is there an error in the calculation of 366.0/5.0?
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I input the following code in Matlab and the result obtained has a small error, as shown below:
>> format long
>> 367/5
ans =
73.400000000000006
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Walter Roberson
2024년 5월 2일
By default, MATLAB calculates using IEEE 754 Double Precision representation of numbers. IEEE 754 is a binary floating point representation. One thing about binary floating point representations is that 1/10 is not exactly representable in any finite length binary floating point.
This is not a bug; it is an inherent limitation of binary floating point representation.
The mathematical reasons why 1/10 is not exactly representable in binary is the same reason why 1/3 and 1/6, 1/7, 1/9, and 1/13 are not exactly representable in finite decimal. Every finite base representation has numbers that cannot be exactly finitely represented.
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James Tursa
2024년 5월 4일
편집: James Tursa
2024년 5월 4일
The three closest numbers to 73.4 representable in IEEE double precision are:
format longg
x = [73.4-eps(73.4);73.4;73.4+eps(73.4)];
The binary floating point hex patterns of these three numbers are (they differ by 1 in the trailing bit):
num2hex(x)
The exact decimal conversions of these binary floating point numbers are:
fprintf('%60.50f\n',x)
MATLAB (or more precisely, the CPU floating point processor) picked the closest one for the result. There are no numbers (including 73.4 exactly) inbetween these numbers that are exactly representable in IEEE double precision.
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