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    <p>(I just wanted to add something to starblue's answer -- it's too long to fit into a comment.)</p> <p>I find it easier to see what's going on -- and I'm hoping you will too -- when I can see the full binary value of a double. I put starblue's examples in a C program and converted the output to binary (using my conversion program at <a href="http://www.exploringbinary.com/converting-floating-point-numbers-to-binary-strings-in-c/" rel="nofollow noreferrer">http://www.exploringbinary.com/converting-floating-point-numbers-to-binary-strings-in-c/</a> ). Here's the output, plus the result of the calculation:</p> <pre> x1 = 101.1100011000011010010000011001001011111001110000111 x2 = 101.11000110000110100100000110010010111110011100001111 c = 0.00101000101100101000111010100110011111010101111100001 r = 100100.01010001101110101101000101101100011111011010101 x1 = 10.111110011111001001111001011010001100001011001111011 x2 = 10.1111100111110010011110010110100011000010110011111 c = 0.0010011101001010001101101010001000011100110010101011 r = 10011.011001001001100010101001001110011100011111011111 x1 = 111.1111110010000001000100001110001111001101010100101 x2 = 111.11111100100000010001000011100011110011010101001011 c = 0.11010100111111110111100101001001011010110100010111011 r = 1001.100110010100010010100101110100000100000110000011 x1 = 0.0001110110111110011011000001011101101100111011010101 x2 = 0.00011101101111100110110000010111011011001110110101010001 c = 0.0001011111010000011100111111110000001001001011101001 r = 1.00111111101111011111001110101010100101010101010101 x1 = 0.1111110010001001000111110100110100001000001101111111 x2 = 0.11111100100010010001111101001101000010000011011111111 c = 0.01100111110011101011111010110111000101001011000000111 r = 10.011011101100011101000000101000110110101011010011111 x1 = 11.1111010101010010010000111001010000100001011000111 x2 = 11.111101010101001001000011100101000010000101100011101 c = 0.110000010101100101010010001010110001110001011111111 r = 101.00111101101010110100110000011111101001010010101111 x1 = 1.1010011101101111101110100000000000011110110111110111 x2 = 1.1010011101101111101110100000000000011110110111111 c = 0.00100101000111101100100101111110100101011010111111001 r = 1011.011010000011101100001011000110000010011111110001 x1 = 101.10111010001001010111100100111110000111100001000011 x2 = 101.101110100010010101111001001111100001111000010001 c = 0.101011101011001100001000111011000001111010111011011 r = 1000.0110010001110100001001010000000101111000011111011 x1 = 11.101101100010000001100111100010010100011000001001111 x2 = 11.10110110001000000110011110001001010001100000101 c = 0.0011011010000011101011110000001111000110010101111111 r = 10001.01101101110011010100011111101110101011001010001 x1 = 101.10101010000101110011111111101001111011111010101111 x2 = 101.1010101000010111001111111110100111101111101011 c = 0.1011000110001100100111111010011000000010100011 r = 1000.0010101011010001010101111000111101110100001000001 </pre> <p>(BTW, the "* const" part of the expression is unnecessary: the division by const alone shows that X1 / const == X2 / const.)</p> <p>You can really see what's going on when you compare the double values with the true, arbitrary precision values. Take the first example, for instance:</p> <pre> x1/c = x2/c (double) = 100100.01010001101110101101000101101100011111011010101 x1/c (true) = 100100.01010001101110101101000101101100011111011010100 1011... x2/c (true) = 100100.01010001101110101101000101101100011111011010101 0111... </pre> <p>I put a space between significant bits 53 and 54, where the rounding occurs in a double. x1/c rounds up, and x2/c rounds down (truncates), becoming the same value.</p>
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    1. POWith IEEE-754, 0 < ABS(const) < 1, is (x / const) * const guaranteed to return distinct results for distinct values of X?
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    1. CO@Rick: regarding "unnesessary expression": since `const` is less than `1`, it may happen so that the quotients are different but the products are not. When I asked this I wasn't sure that the quotients may differ (and was surprised but the fact they can).
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    2. COI was just pointing out that for these ten examples, division alone caused the "problem" (I tried them all on my machine).
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