Question

Can somebody explain why multiplying by 100 here gives a less accurate result but multiplying by 10 twice gives a more accurate result?

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>> 129.95 * 100
12994.999999999998
>> 129.95*10
1299.5
>> 129.95*10*10
12995.0

Answer 1

If you do the calculations by hand in double-precision binary, which is limited to 53 significant bits, you’ll see what’s going on:

129.95 = 1.0000001111100110011001100110011001100110011001100110 x 2^7

129.95*100 = 1.1001011000010111111111111111111111111111111111111111011 x 2^13

This is 56 significant bits long, so rounded to 53 bits it’s

1.1001011000010111111111111111111111111111111111111111 x 2^13, which equals

12994.999999999998181010596454143524169921875

Now 129.95*10 = 1.01000100110111111111111111111111111111111111111111111 x 2^10

This is 54 significant bits long, so rounded to 53 bits it’s 1.01000100111 x 2^10 = 1299.5

Now 1299.5 * 10 = 1.1001011000011 x 2^13 = 12995.