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DP recurrence for popcount that avoids host bitwise operations: result[i] = result[i / 2] + (i mod 2) Drops the low bit (i / 2 stands in for i lsr 1) and adds it back if it was 1 (i mod 2 stands in for i land 1). sum over 0..100 of popcount(i) = 319 Tests pure-arithmetic popcount, accumulating ref + DP array, classic look-back to half-index pattern. 197 baseline programs total.