Is floating point math broken?

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Question:
0.1 + 0.2 == 0.3
-> false
0.1 + 0.2
-> 0.30000000000000004

Why does this happen?


Answer:
Binary floating point math is like this. In most programming languages, it is based on the IEEE 754 standard. JavaScript uses 64-bit floating point representation, which is the same as Java’s double. The crux of the problem is that numbers are represented in this format as a whole number times a power of two; rational numbers (such as 0.1, which is 1/10) whose denominator is not a power of two cannot be exactly represented.

For 0.1 in the standard binary64 format, the representation can be written exactly as
0.1000000000000000055511151231257827021181583404541015625 in decimal, or
0x1.999999999999ap-4 in C99 hexfloat notation.

In contrast, the rational number 0.1, which is 1/10, can be written exactly as
0.1 in decimal, or
0x1.99999999999999…p-4 in an analogue of C99 hexfloat notation, where the … represents an unending sequence of 9’s.

The constants 0.2 and 0.3 in your program will also be approximations to their true values. It happens that the closest double to 0.2 is larger than the rational number 0.2 but that the closest double to 0.3 is smaller than the rational number 0.3. The sum of 0.1 and 0.2 winds up being larger than the rational number 0.3 and hence disagreeing with the constant in your code.

A fairly comprehensive treatment of floating-point arithmetic issues is What Every Computer Scientist Should Know About Floating-Point Arithmetic. For an easier-to-digest explanation, see floating-point-gui.de.
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