lse-mac-tb

Self-contained Verilator testbench for the log-domain MAC unit (multiply–accumulate in the −log₂(p) probability domain) — the LSE arithmetic block described in Yao et al., IEEE TCAS-I 2025.

Math background

Probabilities are stored as −log₂(p) in 24-bit fixed-point (14 integer + 10 fractional bits). Three identities drive the whole datapath:

Operation	Linear domain	Log domain (this code)
Multiply	p × q	−log₂(p) + (−log₂(q)) — just addition
Add (LSE)	p + q	min(a, b) − log₂(1 + 2^−|a−b|) — log-sum-exp
Zero	p = 0	sentinel 24'h0C3500 (= 800 000 decimal)

lse_mult implements the multiply identity (plain adder with zero-sentinel absorption). lse_add implements the add identity using a 16-entry LUT for the correction term log₂(1 + 2^−|Δ|), with linear interpolation between adjacent entries.

mac_unit chains them: acc_out = lse_add(acc_in, lse_mult(w, x)).

Fixed-point encoding

Bit width : 24 bits (signed 2's complement)
Format    : Q14.10  →  1 ULP = 2^−10 ≈ 0.001 in −log₂
Sentinel  : 24'd800000  →  p = 0  (−log₂(0) = ∞)
Example   : 1024 = 1.0 in −log₂  →  p = 2^−1 = 0.5

Files

File	Description
tb_mac.sv	All modules + testbench in one file (no external dependencies)
Makefile	Compile + run with Verilator

Prerequisites

Verilator ≥ 5.0 is required for --timing support. The system package on Ubuntu 22.04 and earlier is too old; install a current version via one of:

# macOS
brew install verilator

# Linux / macOS — via conda-forge
conda install -c conda-forge verilator

# Linux — from source (https://verilator.org/guide/latest/install.html)

Verify the version:

verilator --version   # must print 5.x or higher

Quick start

git clone https://github.com/lingyunyao/lse-mac-tb.git
cd lse-mac-tb
make

Expected output:

=== tb_mac  (24-bit -log2 fixed-point, tol=2 ULP) ===

-- lse_mult --
  ok    mult(0,0) = 0                     got=0       exp=0
  ok    mult(1k,1k) = 2k                  got=2048    exp=2048
  ok    mult(-inf,x) = -inf               got=800000  exp=800000
  ok    mult(x,-inf) = -inf               got=800000  exp=800000

-- lse_add --
  ok    add(-inf, 2048) = 2048            got=2048    exp=2048
  ok    add(-inf, 0) = 0                  got=0       exp=0
  ok    add(2048,2048) = 1024             got=1024    exp=1024
  ok    add(1024,2048) ~ 426              got=426     exp=426
  ok    add(2048,1024) ~ 426              got=426     exp=426

-- MAC accumulation (3 steps) --
  ok    acc after step 1                  got=1024    exp=1024
  ok    acc after step 2                  got=426     exp=426
  ok    acc after step 3 (~197 ULP)       got=198     exp=198

=== 12 passed, 0 failed ===
RESULT: PASS

If Verilator is not on PATH, pass it explicitly:

make VERILATOR=/path/to/verilator

Without make

verilator --binary --top-module tb_mac --timing -sv \
  -Wno-fatal --Mdir obj_dir tb_mac.sv \
  && ./obj_dir/Vtb_mac

Test descriptions

Group	Test	What it checks
lse_mult	mult(0, 0) = 0	Identity: −log₂(1×1) = 0
lse_mult	mult(1024, 1024) = 2048	−log₂(0.5×0.5) = 2.0
lse_mult	mult(-inf, x) = -inf	Zero-sentinel absorption (left)
lse_mult	mult(x, -inf) = -inf	Zero-sentinel absorption (right)
lse_add	add(-inf, 2048) = 2048	Identity: LSE(0, p) = p
lse_add	add(-inf, 0) = 0	Identity at p=1
lse_add	add(2048, 2048) = 1024	−log₂(0.25+0.25) = 1.0 (exact)
lse_add	add(1024, 2048) ≈ 426	−log₂(0.5+0.25) = −log₂(0.75) ≈ 0.415
lse_add	add(2048, 1024) ≈ 426	Symmetry of log-sum-exp
MAC step 1	acc = lse_add(−∞, 1024) = 1024	Accumulator cold-start
MAC step 2	acc = lse_add(1024, 2048) ≈ 426	Two-term accumulation
MAC step 3	acc = lse_add(426, 3072) ≈ 198	−log₂(0.5+0.25+0.125) = −log₂(0.875) ≈ 0.193

Tolerance is 2 ULP for lse_add results (LUT quantisation error); lse_mult results are exact.

Parameters

Defined as localparam in tb_mac and as module parameters in lse_add:

Parameter	Default	Meaning
INT_BITS	14	Integer bits in Qm.n format
FRAC_BITS	10	Fractional bits
LUT_SIZE	16	Number of LUT entries for LSE correction
LUT_PREC	10	Bit-width of each LUT entry
TOL	2	Acceptance tolerance in ULP

To change tolerance, edit the localparam TOL = 2; line in tb_mac.sv.

Citation

If this testbench or the underlying arithmetic is useful for your work, please cite:

@article{yao2025logsumexp,
  title     = {LogSumExp: Efficient Approximate Logarithm Acceleration for Embedded Tractable Probabilistic Reasoning},
  author    = {Yao, Lingyun and Zhao, Shirui and Trapp, Martin and Leslin, Jelin and Verhelst, Marian and Andraud, Martin},
  journal   = {IEEE Transactions on Circuits and Systems I: Regular Papers},
  year      = {2025},
  publisher = {IEEE},
  url       = {https://ieeexplore.ieee.org/document/11185227}
}

License

MIT