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This ppt describes one of the interesting algorithms to count the number of bits set in an unsigned integer. x = (x & 0x55555555) + ((x>>1)&0x55555555)); x = (x&0x33333333) + ((x>>2)&0x33333333); ... ....
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Counting the number of Counting the number of bits setbits set
CH Gowri KumarCH Gowri Kumar
[email protected]@gmail.com
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You are given unrestricted freedom to You are given unrestricted freedom to
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Problem StatementProblem Statement
Given an unsigned integer n, count the Given an unsigned integer n, count the number of bits set in it.number of bits set in it.
Ex :Ex : 3 3 11 11 2 2 8 8 1000 1000 1 1 2323101111011144
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 0 10 0 1 10 1 0 00 1 1 0
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 0 10 0 1 10 1 0 00 1 1 0
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 0 10 0 1 10 1 0 00 1 1 0
0 1 1 0
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 0 10 0 1 10 1 0 00 1 1 0
0 1 1 0
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 0 10 0 1 10 1 0 00 1 1 0
0 1 1 0 0 1 0 0
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 0 10 0 1 10 1 0 00 1 1 0
0 1 1 0 0 1 0 0 0 0 1 1 0 0 0 1
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 0 10 0 1 10 1 0 00 1 1 0
0 1 1 0 0 1 0 0 0 0 1 1 0 0 0 1
0 1
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 0 10 0 1 10 1 0 00 1 1 0
0 1 1 0 0 1 0 0 0 0 1 1 0 0 0 1
0 1
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 0 10 0 1 10 1 0 00 1 1 0
0 1 1 0 0 1 0 0 0 0 1 1 0 0 0 1
0 1 1 0 0 1 0 0 0 0 1 1 0 0 0 1
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 0 10 0 1 10 1 0 00 1 1 0
0 1 1 0 0 1 0 0 0 0 1 1 0 0 0 1
0 1 1 0 0 1 0 0 0 0 1 1 0 0 0 1
+
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 0 10 0 1 10 1 0 00 1 1 0
0 1 1 0 0 1 0 0 0 0 1 1 0 0 0 1
0 1 1 0 0 1 0 0 0 0 1 1 0 0 0 1
+
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 0 10 0 1 10 1 0 00 1 1 0
0 1 1 0 0 1 0 0 0 0 1 1 0 0 0 1
1 0 0 1 0 0 0 0 1 1 0 0 0 1
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 0 10 0 1 10 1 0 00 1 1 0
0 1 1 0 0 1 0 0 0 0 1 1 0 0 0 1
1 0 0 1 0 0 0 0 1 1 0 0 0 1
+
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 0 10 0 1 10 1 0 00 1 1 0
0 1 0 1 0 1 0 0 0 0 1 1 0 0 0 1
1 0 0 1 0 0 0 0 1 1 0 0 0 1
+
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 0 10 0 1 10 1 0 00 1 1 0
0 1 0 1 0 1 0 0 0 0 1 1 0 0 0 1
0 1 0 0 0 0 1 1 0 0 0 1
+
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 0 10 0 1 10 1 0 00 1 1 0
0 1 0 1 0 1 0 0 0 0 1 1 0 0 0 1
0 1 0 0 0 0 1 1 0 0 0 1
+
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 0 10 0 1 10 1 0 00 1 1 0
0 1 0 1 0 1 0 0 0 0 1 1 0 0 0 1
0 0 0 0 1 1 0 0 0 1
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 0 10 0 1 10 1 0 00 1 1 0
0 1 0 1 0 1 0 0 0 0 1 1 0 0 0 1
0 0 0 0 1 1 0 0 0 1
+
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 0 10 0 1 10 1 0 00 1 1 0
0 1 0 1 0 1 0 0 0 0 1 1 0 0 0 1
0 0 0 0 1 1 0 0 0 1
+
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 0 10 0 1 10 1 0 00 1 1 0
0 1 0 1 0 1 0 0 0 0 1 1 0 0 0 1
0 0 1 1 0 0 0 1
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 0 10 0 1 10 1 0 00 1 1 0
0 1 0 1 0 1 0 0 0 0 1 1 0 0 0 1
0 0 1 1 0 0 0 1
+
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 0 10 0 1 10 1 0 00 1 1 0
0 1 0 1 0 1 0 0 0 0 1 1 0 0 0 1
0 0 1 1 0 0 0 1
+
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 0 10 0 1 10 1 0 00 1 1 0
0 1 0 1 0 1 0 0 0 0 1 1 0 0 0 1
1 1 0 0 0 1
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 0 10 0 1 10 1 0 00 1 1 0
0 1 0 1 0 1 0 0 0 0 1 1 0 0 0 1
1 1 0 0 0 1
+
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 0 10 0 1 10 1 0 00 1 1 0
0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 1
1 1 0 0 0 1
+
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 0 10 0 1 10 1 0 00 1 1 0
0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 1
0 0 0 1
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 0 10 0 1 10 1 0 00 1 1 0
0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 1
0 0 0 1
+
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 0 10 0 1 10 1 0 00 1 1 0
0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 1
0 0 0 1
+
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 0 10 0 1 10 1 0 00 1 1 0
0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 1
0 1
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 0 10 0 1 10 1 0 00 1 1 0
0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 1
0 1
+
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 0 10 0 1 10 1 0 00 1 1 0
0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 1
0 1
+
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 0 10 0 1 10 1 0 00 1 1 0
0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 1
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 0 10 0 1 10 1 0 00 1 1 0
0 1 1 0 0 1 0 0 0 0 1 1 0 0 0 1
0 1 1 0 0 1 0 0 0 0 1 1 0 0 0 1
+ + + + + + + +
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 0 10 0 1 10 1 0 00 1 1 0
0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 1
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C codeC code
x = (x & 0x5555) + ((x>>1) & 0x5555)x = (x & 0x5555) + ((x>>1) & 0x5555)
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 0 10 0 1 10 1 0 00 1 1 0
0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 1
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 0 10 0 1 10 1 0 00 1 1 0
0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 1+
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 0 10 0 1 10 1 0 00 0 1 0
0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 1+
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 0 10 0 1 10 1 0 00 0 1 0
0 1 0 0 0 0 1 0 0 0 0 1
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 0 10 0 1 10 1 0 00 0 1 0
0 1 0 0 0 0 1 0 0 0 0 1+
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 0 10 0 1 10 0 0 10 0 1 0
0 1 0 0 0 0 1 0 0 0 0 1+
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 0 10 0 1 10 0 0 10 0 1 0
0 0 1 0 0 0 0 1
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 0 10 0 1 10 0 0 10 0 1 0
0 0 1 0 0 0 0 1+
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 0 10 0 1 00 0 0 10 0 1 0
0 0 1 0 0 0 0 1+
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 0 10 0 1 00 0 0 10 0 1 0
0 0 0 1
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 0 10 0 1 00 0 0 10 0 1 0
0 0 0 1+
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 0 10 0 1 00 0 0 10 0 1 0
0 0 0 1+
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 0 10 0 1 00 0 0 10 0 1 0
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 0 10 0 1 10 1 0 00 0 1 0
0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 1+ + + +
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 0 10 0 1 00 0 0 10 0 1 0
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C codeC code
x = (x & 0x5555) + ((x>>1) & 0x5555)x = (x & 0x5555) + ((x>>1) & 0x5555)
x = (x & 0x3333) + ((x>>2) & 0x3333)x = (x & 0x3333) + ((x>>2) & 0x3333)
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 0 10 0 1 00 0 0 10 0 1 0 +
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 1 10 0 0 0 0 0 0 10 0 1 1
0 0 0 10 0 1 00 0 0 10 0 1 0 +
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 1 10 0 0 0 0 0 0 10 0 1 1
0 0 0 10 0 1 0
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 1 10 0 0 0 0 0 0 10 0 1 1
0 0 0 10 0 1 0 +
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 1 10 0 0 0 0 0 1 10 0 0 0
0 0 0 10 0 1 0 +
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 1 10 0 0 0 0 0 1 10 0 0 0
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 1 10 0 0 0 0 0 0 10 0 1 1
0 0 0 10 0 1 00 0 0 10 0 1 0 + +
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 1 10 0 0 0 0 0 1 10 0 0 0
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C codeC code
x = (x & 0x5555) + ((x>>1) & 0x5555)x = (x & 0x5555) + ((x>>1) & 0x5555)
x = (x & 0x3333) + ((x>>2) & 0x3333)x = (x & 0x3333) + ((x>>2) & 0x3333)
x = (x & 0x0F0F) + ((x>>4) & 0x0F0F)x = (x & 0x0F0F) + ((x>>4) & 0x0F0F)
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0 1 0 00 1 1 0 0 0 0 10 0 1 1
0 0 1 10 0 0 0 0 0 1 10 0 0 0+
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0 0 0 00 0 0 0 0 1 1 00 0 0 0
0 0 1 10 0 0 0 0 0 1 10 0 0 0+
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0 0 0 00 0 0 0 0 1 1 00 0 0 0
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C codeC code
x = (x & 0x5555) + ((x>>1) & 0x5555)x = (x & 0x5555) + ((x>>1) & 0x5555)
x = (x & 0x3333) + ((x>>2) & 0x3333)x = (x & 0x3333) + ((x>>2) & 0x3333)
x = (x & 0x0F0F) + ((x>>4) & 0x0F0F)x = (x & 0x0F0F) + ((x>>4) & 0x0F0F)
x = (x & 0x00FF) + (x>>16) & 0x00FF)x = (x & 0x00FF) + (x>>16) & 0x00FF)
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Code for 32-bit integersCode for 32-bit integers
x = (x&0x55555555) + ((x>>1)&0x55555555)x = (x&0x55555555) + ((x>>1)&0x55555555)
x = (x&0x33333333) + ((x>>2)&0x33333333)x = (x&0x33333333) + ((x>>2)&0x33333333)
x = (x&0x0F0F0F0F) + ((x>>4)&0x0F0F0F0F)x = (x&0x0F0F0F0F) + ((x>>4)&0x0F0F0F0F)
x = (x&0x00FF00FF) + ((x>>8)&0x00FF00FF)x = (x&0x00FF00FF) + ((x>>8)&0x00FF00FF)
x = (x&0x0000FFFF) + ((x>>16)&0x0000FFFF)x = (x&0x0000FFFF) + ((x>>16)&0x0000FFFF)
return x;return x;
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ReferenceReference
Hackers DelightHackers Delight
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The EndThe End
