Higher radix multiplier encoding #7991
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Codecov ReportAttention:
Additional details and impacted files@@ Coverage Diff @@
## develop #7991 +/- ##
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- Coverage 79.09% 78.63% -0.46%
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Files 1699 1701 +2
Lines 196512 196571 +59
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- Hits 155428 154575 -853
- Misses 41084 41996 +912 ☔ View full report in Codecov by Sentry. |
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We were still repeatedly using `bv[bv.size() - 1]` in place, when using `sign_bit(bv)` adds clarity and avoids having to understand encoding details. Also, use `.back()` to avoid unnecessary repeat arithmetic (which the compiler may or may not optimise away).
We can avoid lowering, and eventually re-use this as part of other algorithms.
1. Duplicate some code to specialise it for signed/unsigned and, thereby, add clarity. 2. De-duplicate code to handle all cases directly in lt_or_le. 3. Make sure we constant-propagate whatever is possible in unsigned comparison to avoid introducing unnecessary fresh literals.
The Radix-4 multiplier pre-computes products for x * 0 (zero), x * 1 (x), x * 2 (left-shift x), x * 3 (x * 2 + x) to reduce the number of sums by a factor of 2. Radix-8 extends this up to x * 7 (reducing sums by a factor of 3), and Radix-16 up to x * 15 (reducing sums by a factor of 4). This modified approach to computing partial products can be freely combined with different (tree) reduction schemes. Benchmarking results can be found at https://tinyurl.com/multiplier-comparison (shortened URL for https://docs.google.com/spreadsheets/d/197uDKVXYRVAQdxB64wZCoWnoQ_TEpyEIw7K5ctJ7taM/). The data suggests that combining Radix-8 partial product pre-computation with Dadda's reduction can yield substantial performance gains while not substantially regressing in other benchmark/solver pairs.
Uses extra sign-bit to keep bit widths small.
Implements the algorithm of Section 4 of "Further Steps Down The Wrong Path : Improving the Bit-Blasting of Multiplication" (see https://ceur-ws.org/Vol-2908/short16.pdf).
Prints a vector of literals in human-readable form, including a decimal representation when all literals are constants.
Implements a symbolic version of the algorithm proposed by Schönhage and Strassen in "Schnelle Multiplikation großer Zahlen", Computing, 7, 1971.
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The Radix-4 multiplier pre-computes products for x * 0 (zero), x * 1 (x), x * 2 (left-shift x), x * 3 (x * 2 + x) to reduce the number of sums by a factor of 2. Radix-8 extends this up to x * 7 (reducing sums by a factor of 3), and Radix-16 up to x * 15 (reducing sums by a factor of 4). This modified approach to computing partial products can be freely combined with different (tree) reduction schemes.
Benchmarking results can be found at
https://tinyurl.com/multiplier-comparison (shortened URL for https://docs.google.com/spreadsheets/d/197uDKVXYRVAQdxB64wZCoWnoQ_TEpyEIw7K5ctJ7taM/). The data suggests that combining Radix-8 partial product pre-computation with Dadda's reduction can yield substantial performance gains while not substantially regressing in other benchmark/solver pairs.