aproposorg / approx

A library of approximate arithmetic units in Chisel.

approx: A Library of Approximate Arithmetic Units in Chisel

This repository contains a collection of approximate arithmetic units for use in various digital designs. The units are written in Chisel with tests written for the exact units using ChiselTest. Currently only a selection of adders and multipliers and a simple sequential division unit are included - more designs to come!

When you use this library in a research project, please cite it as:

@misc{damsgaard2022approx,
  title={{approx: A Library of Approximate Arithmetic Units in Chisel}},
  author={Damsgaard, Hans Jakob},
  year={2022},
  howpublished={\url{https://github.com/aproposorg/approx}}}

This README only contains a brief overview of the library's current contents. All units are commented to be reasonably understandable for people with prior knowledge of Chisel and Scala. Refer to the Digital Design with Chisel book for more details on these topics.

---

Requirements

Utilizing Chisel and ChiselTest, approx requires a suitable installation of Scala. For this purpose, we use the Scala Build Tool (sbt) for which we provide a suitable build script.

Moreover, some tests run too slow in ChiselTest's built-in Treadle simulator, so we have instead decided to run them using Verilator (see e.g., Radix2MultiplierSpec here). Thus, to run all provided tests, one must have a suitable installation of Verilator available. Note that only specific versions of Verilator are officially supported.

This library is tested in Ubuntu 20.04 with Verilator 4.028.

---

Adders

The approx.addition library contains a vast number of approximate and exact adder designs that are parameterized to be reasonably flexible. The lists below specify which designs are included currently. We gladly accept requests for other designs as issues in this repository.

Exact designs

All exact designs are based on descriptions in Ercegovac and Lang's book on digital arithmetic.

Type	Name	Code location
Half adder	HalfAdder	approx.addition.HalfAdder
Full adder	FullAdder	approx.addition.FullAdder
Ripple-carry adder	RCA	approx.addition.RCA
Adaptive optimized lower-part constant-OR adder	AdaptiveOFLOCA	approx.addition.AdaptiveOFLOCA
Carry-lookahead adder	CLA	approx.addition.CLA
Two-layer carry-lookahead adder	CLA2	approx.addition.CLA2
Carry-select adder	CSA	approx.addition.CSA
Parallel prefix adder	PPA	approx.addition.PPA
Self-timed adder	STA	approx.addition.STA
Parallel carry-completion sensing adder	CCA	approx.addition.CCA

Approximate designs

Type	Name(s)	Code location	Reference
Full adder	AXA1, AXA2, AXA3	approx.addition.AXA	Yang et al.
Full adder	AFA	approx.addition.AFA	Dutt et al.
Full adder	InXA1, InXA2, InXA3	approx.addition.InXA	Almurib et al.
Full adder	SESA1, SESA2, SESA3	approx.addition.SESA	Jha et al.
Full adder	TCAA	approx.addition.TCAA	Yang and Thapliyal
Full adder	TSAA	approx.addition.TSAA	Yang and Thapliyal
Accuracy-configurable adder	ACA	approx.addition.ACA	Kahng and Kang
Approximate parallel prefix adder	AxPPA	approx.addition.AxPPA	da Rosa et al.
Carry cut-back adder	CCBA	approx.addition.CCBA	Camus et al.
Carry estimating simultaneous adder	CESA_PERL	approx.addition.CESA_PERL	Bhattacharjya et al.
Dual-mode ripple-carry adder	DualModeRCA	approx.addition.DualModeRCA	Raha et al.
Dual-mode carry-lookahead adder	DualModeCLA	approx.addition.DualModeCLA	Raha et al.
Error-resilient adder w/o correction	ErrorResilient	approx.addition.ErrorResilient	Dutt et al.
Error-resilient adder w/ correction	ErrorResilientCorrect	approx.addition.ErrorResilientCorrect	Dutt et al.
Error-tolerant adder I	ETAI	approx.addition.ETAI	Zhu et al.
Error-tolerant adder II	ETAII	approx.addition.ETAII	Zhu et al.
Modified error-tolerant adder II	ETAIIM	approx.addition.ETAIIM	Zhu et al.
FAU LUT-based adder	FAU	approx.addition.FAU	Echavarria et al.
Generic accuracy-configurable adder	GeAr	approx.addition.GeAr	Shafique et al.
Hybrid error reduction lower-part OR adder	HERLOA	approx.addition.HERLOA	Seo et al.
Hardware-optimized adder with normal error distribution	HOAANED	approx.addition.HOAANED	Balasubramian et al.
Lower-part OR adder	LOA	approx.addition.LOA
Lower-part constant-OR adder	LOCA	approx.addition.LOCA	Dalloo
Lu's adder	LUA	approx.addition.LUA	Lu
LUT-based adder	LutBased	approx.addition.LUTBased	Becher et al.
Optimized lower-part constant-OR adder	OFLOCA	approx.addition.OFLOCA	Dalloo
Reconfigurable carry-lookahead adder	RAP_CLA	approx.addition.RAP_CLA	Akbari et al.
Speculative carry-skip adder	SCSkip	approx.addition.SCSkip	Kim et al.
Self-adaptive adder	SelfAdaptive	approx.addition.SelfAdaptive	Liu et al.
Static segment adder	SSA	approx.addition.SSA	Jothin and Vasanthanayaki

Beware that some approximate adders (e.g., DualModeRCA, GeAr) do not extend the abstract Adder base class as their IOs do not match those of the basic adder. Almost all designs are purely combinational; only GeAr has an option for adding a register to support flagging errors.

---

Multipliers

Like above, the approx.multiplication library contains several approximate and exact multiplier designs that are also parameterized. The list below specifies which designs are included currently. In addition to these, the tool also includes a generic compressor tree generator for ASIC, Xilinx 7-Series/UltraScale or Versal FPGA (supported by the primitives in approx.util.Xilinx), and Intel FPGA flows. The generator supports fully exact compression as well as approximation by column or row truncation, OR-based column compression, and miscounting (i.e., inexact compression). Multiple of these approximations can also be applied concurrently.

Exact designs

All exact designs are based on descriptions in Ercegovac and Lang's book on digital arithmetic. Note that some exact multiplier implementations, specifically Radix2Multiplier, Radix4Multiplier, RecursiveMultiplier, and AdaptiveRadix2Multiplier, permit approximation through their arguments.

Type	Signed/unsigned	Name	Code location
Compressor 2:2		Compressor2to2	approx.multiplication.Compressor3to2
Compressor 3:2		Compressor3to2	approx.multiplication.Compressor3to2
Compressor 4:2		Compressor4to2, Compressor4to2Opt	approx.multiplication.Compressor4to2
Compressor 5:3		Compressor5to3	approx.multiplication.Compressor5to3
Compressor 7:3		Compressor7to3	approx.multiplication.Compressor7to3
2x2-bit multiplier		TwoxTwo	approx.multiplication.TwoxTwo
Radix-2 array multiplier	Both	Radix2Multiplier	approx.multiplication.Radix2Multiplier
Adaptive radix-2 array multiplier	Both	AdaptiveRadix2Multiplier	approx.multiplication.AdaptiveRadix2Multiplier
Radix-4 array multiplier	Both	Radix4Multiplier	approx.multiplication.Radix4Multiplier
Recursive multiplier	Both	RecursiveMultiplier	approx.multiplication.RecursiveMultiplier
Alphabet-set multiplier	Both	AlphabetSetMultiplier	approx.multiplication.AlphabetSetMultiplier
Radix-2 sequential multiplier	Unsigned	Radix2SeqMultiplier	approx.multiplication.Radix2SeqMultiplier

Approximate designs

Type	Signed/unsigned	Name	Code location	Reference
Reduced compressor 4:2		Compressor4to2D1, Compressor4to2D2	approx.multiplication.Compressor4to2D1	Momeni et al.
Modified compressor 4:2		Compressor4to2CV1, Compressor4to2CV2	approx.multiplication.Compressor4to2CV1	Zanandrea and Meinhardt
Majority-based compressor 4:2		Compressor4to2Maj	approx.multiplication.Compressor4to2Maj	Moaiyeri et al.
Compressor 8:3		Compressor8to3, Compressor8to3SevenSeries, Compressor8to3Versal	approx.multiplication.Compressor8to3	Moaiyeri et al.
Kulkarni-style 2x2-bit multiplier		Kulkarni	approx.multiplication.Kulkarni	Kulkarni et al.
Rehman-style 2x2-bit multiplier		ApproxMul2, ApproxMul3, ApproxMul4, ApproxMul5	approx.multiplication.Rehman	Rehman et al.
Configurable partial error recovery multiplier	Unsigned	CPER	approx.multiplication.CPER	Liu et al.
Dynamic range unbiased multiplier	Both	DRUM	approx.multiplication.DRUM	Hashemi et al.
Error-tolerant multiplier	Both	ETM	approx.multiplication.ETM	Kyaw et al.
Low-power small-area multiplier	Unsigned	LPSA	approx.multiplication.LPSA	Baba et al.
Minimally-biased multiplier	Unsigned	MBM	approx.multiplication.MBM	Saadat et al.
Approximate radix-2 sequential multiplier	Unsigned	ApproxRadix2SeqMultiplier	approx.multiplication.ApproxRadix2SeqMultiplier	Mannepalli et al.

---

Dividers

The approx.division library currently only contains an exact, sequential radix-2 divider. We hope to extend this collection in the future.

Exact designs

Type	Signed/unsigned	Name	Code location
Radix-2 sequential divider	Unsigned	Radix2SeqDivider	approx.division.Radix2SeqDivider

Approximate designs

N/A

---

Accumulators

The approx.accumulation library currently only contains a number of exact, non-pipelined single-lane and parallel accumulators with options for the parallel designs to be approximated with the custom compressor trees from approx.multiplication.comptree. We also hope to extend this collection in the future.

Exact designs

Type	Signed/unsigned	Name	Code location
Simple accumulator	Both	SimpleAccumulator	approx.accumulation.SimpleAccumulator
Multiply accumulator	Both	MultiplyAccumulator	approx.accumulation.MultiplyAccumulator
Bit matrix accumulator		BitMatrixAccumulator	approx.accumulation.BitMatrixAccumulator
Parallel simple accumulator	Both	ParallelSimpleAccumulator	approx.accumulation.ParallelSimpleAccumulator
Parallel multiply accumulator	Both	ParallelMultiplyAccumulator	approx.accumulation.ParallelMultiplyAccumulator

Approximate designs

N/A