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Is it guaranteed that division by power of 10 gives accurate result in Floating-point arithmetic standard)?

I found out that 9 * 0.0001 = 0.0009000000000000001, but 9 / 10_000 = 0.0009

Is it guaranteed that when you divide by power of 10 you will receive exact value (by IEEE 754-2019: IEEE Standard for Floating-Point Arithmetic)

I am expecting someone who knows specific behavior of Floating-point standard to explain if there is such guarantees or not.

Solution

  • 0.009 cannot be represented precisely by IEEE 754 floating point. The binary representation of 0.009 would be:

    0.0000001001001101110100101111000110101001111110111110011101101100100010110100001110010101100000010000011
     0001001001101110100101111000110101001111110111110011101101100100010110100001110010101100000010000011
     ...

    It is a rational number with an infinite number of (binary) digits after the decimal point.

    So this cannot be represented in floating point which only has a finite number of binary digits available in the mantissa part.

    0.009 in floating point is represented as follows:

    sign exponent    mantissa
0    01111000    00100110111010010111100

    But as the binary sequence is finite here, it actually represents this number:

    0.0089999996125698089599609375....

    A similar deviation will occur for other positive powers of 10 in 9/10n

    You can make the conversion using this tool.