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October 24, 2017 21:18
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How to replace multiplication by a constant with shift and add operations.
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// Multiplication by a constant can be replaced by multiple shift and addition operations: | |
// The following table shows that for constants from 1 to 36. | |
// | |
// ============================================================================ | |
// N | Factorized | # Ops | Ops with x as input | |
// ---------------------------------------------------------------------------- | |
// 1 | 1 | 0 | x | |
// 2 | 2 | 1 | x << 1 | |
// 3 | 2 + 1 | 2 | x << 1 + x | |
// 4 | 4 | 1 | x << 2 | |
// 5 | 4 + 1 | 2 | x << 2 + x | |
// 6 | 4 + 2 | 3 | x << 2 + x << 1 | |
// 7 | 4 + 2 + 1 | 4 | x << 2 + x << 1 + x | |
// 8 | 8 | 1 | x << 3 | |
// 9 | 8 + 1 | 2 | x << 3 + x | |
// 10 | 8 + 2 | 3 | x << 3 + x << 1 | |
// 11 | 8 + 2 + 1 | 4 | x << 3 + x << 1 + x | |
// 12 | 8 + 4 | 3 | x << 3 + x << 2 | |
// 13 | 8 + 4 + 1 | 4 | x << 3 + x << 2 + x | |
// 14 | 8 + 4 + 2 | 5 | x << 3 + x << 2 + x << 1 | |
// 15 | 8 + 4 + 2 + 1 | 6 | x << 3 + x << 2 + x << 1 + x | |
// 16 | 16 | 1 | x << 4 | |
// 17 | 16 + 1 | 2 | x << 4 + x | |
// 18 | 16 + 2 | 3 | x << 4 + x << 1 | |
// 19 | 16 + 2 + 1 | 4 | x << 4 + x << 1 + x | |
// 20 | 16 + 4 | 3 | x << 4 + x << 2 | |
// 21 | 16 + 4 + 1 | 4 | x << 4 + x << 2 + x | |
// 22 | 16 + 4 + 2 | 5 | x << 4 + x << 2 + x << 1 | |
// 23 | 16 + 4 + 2 + 1 | 6 | x << 4 + x << 2 + x << 1 + x | |
// 24 | 16 + 8 | 3 | x << 4 + x << 3 | |
// 25 | 16 + 8 + 1 | 4 | x << 4 + x << 3 + x | |
// 26 | 16 + 8 + 2 | 5 | x << 4 + x << 3 + x << 1 | |
// 27 | 16 + 8 + 2 + 1 | 6 | x << 4 + x << 3 + x << 1 + x | |
// 28 | 16 + 8 + 4 | 5 | x << 4 + x << 3 + x << 2 | |
// 29 | 16 + 8 + 4 + 1 | 6 | x << 4 + x << 3 + x << 2 + x | |
// 30 | 16 + 8 + 4 + 2 | 7 | x << 4 + x << 3 + x << 2 + x << 1 | |
// 31 | 16 + 8 + 4 + 2 + 1 | 8 | x << 4 + x << 3 + x << 2 + x << 1 + x | |
// 32 | 32 | 1 | x << 5 | |
// 33 | 32 + 1 | 2 | x << 5 + x | |
// 34 | 32 + 2 | 3 | x << 5 + x << 1 | |
// 35 | 32 + 2 + 1 | 4 | x << 5 + x << 1 + x | |
// 36 | 32 + 4 | 3 | x << 5 + x << 2 |
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