Higher radix multiplier encoding #7991
Draft
Conversation
This file contains hidden or bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
tautschnig
requested review from
martin-cs,
kroening and
peterschrammel
as code owners
tautschnig
force-pushed
the
feature/multiplier-encoding
branch
2 times, most recently
from
c5111f1 to
354bb4e
Compare
Codecov ReportAttention:
Additional details and impacted files@@ Coverage Diff @@
## develop #7991 +/- ##
===========================================
- Coverage 79.09% 78.63% -0.46%
===========================================
Files 1699 1701 +2
Lines 196512 196571 +59
===========================================
- Hits 155428 154575 -853
- Misses 41084 41996 +912 ☔ View full report in Codecov by Sentry. |
tautschnig
force-pushed
the
feature/multiplier-encoding
branch
from
354bb4e to
ffe136e
Compare
tautschnig
force-pushed
the
feature/multiplier-encoding
branch
from
ffe136e to
2f37f7b
Compare
tautschnig
force-pushed
the
feature/multiplier-encoding
branch
from
60fcbd3 to
5a5471a
Compare
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.
tautschnig
force-pushed
the
feature/multiplier-encoding
branch
from
ed26c5a to
a08b3f6
Compare
Sign up for free
to join this conversation on GitHub.
Already have an account?
Sign in to comment
Add this suggestion to a batch that can be applied as a single commit.
This suggestion is invalid because no changes were made to the code.
Suggestions cannot be applied while the pull request is closed.
Suggestions cannot be applied while viewing a subset of changes.
Only one suggestion per line can be applied in a batch.
Add this suggestion to a batch that can be applied as a single commit.
Applying suggestions on deleted lines is not supported.
You must change the existing code in this line in order to create a valid suggestion.
Outdated suggestions cannot be applied.
This suggestion has been applied or marked resolved.
Suggestions cannot be applied from pending reviews.
Suggestions cannot be applied on multi-line comments.
Suggestions cannot be applied while the pull request is queued to merge.
Suggestion cannot be applied right now. Please check back later.
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.