Why does the floating point representation of 0.1 end in 1?

Question

I've been writing a decimal to single-precision IEEE754 floating-point converter, and I found a few discrepancies with numbers like 0.1 and 0.2

Let's take 0.1, the first step would be converting it into a simple binary representation (multiplying the fractional part by 2 and taking the integral part) This gives me a well-known recurring binary pattern (001100110011...)

The final binary representation of 0.1 (up to 26-bits) is 0 . 00 0110 0110 0110 0110 0110 0110.

To fit it into a 32-bit floating-point number, the next step is to normalize it by shifting the decimal point 4 times to the right, removing the leading 1., and truncating it to 23 bits. This leaves me with 10011001100110011001100. Most programming languages give me 10011001100110011001101 (with the last bit 1 instead of 0). What am I doing wrong here?

Answer 1

... removing the leading 1., and truncating it to 23 bits.

Wrong operation: round, not truncate. Wrong order of operations - round, then remove the leading 1.

... still have no idea how it gets flipped though

The infinite exact binary answer rounded to the closest float.

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With common float32, the significand is 24-bit^*1. Conversion of values with more than 24-bits is usually a round^*2, not truncate. The remaining bits were 110_0110_...

        123 4567 8901 2345 6789 0123 4
0 . 00 0110 0110 0110 0110 0110 0110 0110 0110 ...
        ^----------------------------^
0 . 00 0110 0110 0110 0110 0110 0110 1 rounded
         10 0110 0110 0110 0110 0110 1 encoded 23-bits

The rounding happens before the leading bit is removed as part of the encoding as the most significant bit place may change.

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Notes:

0.000110011001100110011001100           (binary) = 0.0999999940395355224609375000 (decimal)
0.0001100110011001100110011001100110... (binary) = 0.1                            (decimal)
0.000110011001100110011001101 (closer)  (binary) = 0.1000000014901161193847656250 (decimal)
 
        

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^*1 Most significant 1 bit implied, 23-bits explicitly coded.
^*2 Round to nearest, ties to even.