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Truncated Logarithmic Approximation
Michael Sullivan and Earl E. Swartzlander, Jr.University of Texas
April 10, 2013
Introduction
Approximate arithmetic units have the potential to save power,area, and latency over conventional circuits.
Approximate logarithmic conversion is attractive because it canestimate multiplication, division, rooting, and raising to apower with bounded relative error.
Known application areas include:
I Graphics
I Neural networks
I DSP applicationsI Specialized circuitry
I period meters for nuclear power plants
Example: Approximate division
For example, division of logs can be performed using subtraction.
log2(a/b) = log2(a) − log2(b)
a/b = 2(log2(a)−log2(b))
Approximate logarithms can be used to cheaply approximatefixed-point division.
a/b ≈ 2̃
(l̃og2(a)−l̃og2(b)
)
This WorkMany approximate conversion schemes exist (differ in precision,latency, cost, and flexibility).
The least costly (and least precise) schemes use piece-wise linearinterpolation. All such schemes refine a simple linear interpolationscheme (Mitchell, 1962).
Intuition:The precision of interpolation should be proportional to theprecision of the final result.
The Idea:Rounding off the log and anti-log approximation reduces theircosts, and can actually improve the average precision of the result.
Truncated logarithmic approximation can be used as a drop-inreplacement for Mitchell’s scheme, improving its cost and precision.
Some Background
A fixed-point input, N can be written as N = 2k ∗ (1 + f ).
I k is the characteristic (0≤k< log2(N))
I f is the fractional component (0≤ f < 1)
The binary logarithm of N is log2(N) = k + log2(1 + f ).Approximate logarithmic computations approximate log2(1 + f)with enough fidelity to achieve a set precision.
Mitchell’s Approximation (cont.)
Mitchell estimates log2(1+f ) using a single straight-lineapproximation to the logarithm curve, f ≈ log2(1+f )
26 27 28
6.0
6.5
7.0
7.5
8.0
N
Valu
e
Log2 HNL
Mitchell
Figure: Mitchell’s log approximation.
Mitchell’s Logarithm Generation
LOD Shifter
Encode
X
nlog (n)
n nf
k2
(a) Logarithm Generation
Shifterf'n 2N
z
k'log (n)+12
(b) Anti-log Generation
Figure: Approximate log and anti-log conversion.
Mitchell Hardware Cost
Hardware costs are dominated by two shifters:
X0X1X2X3X4X5X6X7X8X9X10X11X12X13X14
f0f1f2f3f4f5f6f7f8f9f10f11f12f13f14
(a) Logarithm Shifter
1f14f13f12f11f10f9f8f7f6f5f4f3f2f1f0
Z15Z14Z13Z12Z11Z10Z9Z8Z7Z6Z5Z4Z3Z2Z1Z0 Z31Z30Z27Z26 Z29Z28Z19Z18Z17Z16 Z25Z24Z21Z20 Z23Z22
(b) Anti-Log Shifter (after multiplication)
Figure: Shifters for approximate (anti-)logarithmic conversion.
Mitchell’s Approximation
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
f
Valu
e Log2 H1+fLf
(a) Values
0.0 0.2 0.4 0.6 0.8 1.0
-0.08
-0.06
-0.04
-0.02
0.00
f
Absolu
te
Error
f- Log2 H1+fL
(b) Absolute Error
Figure: The error in Mitchell’s log approximation.
Truncated Logarithmic Approximation
Truncated logarithmic approximation replaces Mitchell’s algorithm,retaining only the t most-significant bits of the fractionalcomponent.
log2-T↑(X , t) = k + (f mod 2−t) + 2−t ≈ log2(X ) (1)
LOD Shifter
Encode
X
nlog (n)
tf
k2
n
Shifterf't 2N
z
k'
log (n)+12
Hardware Savings - Log Generation
X0X1X2X3X4X5X6X7X8X9X10X11X12X13X14
f0f1f2f3f4f5f6f7f8f9f10f11f12f13f14
(a) Logarithm Shifter
X0X1X2X3X4X5X6X7X8X9X10X11X12X13X14
f11f12f13f14
tb
The TruncationSpecific Region
(b) Trunc. Log Shift (t = 4)
Figure: The impact of truncation on the log generation shifter.
Hardware Savings - Anti-Log Generation
1f14f13f12f11f10f9f8f7f6f5f4f3f2f1f0
Z15Z14Z13Z12Z11Z10Z9Z8Z7Z6Z5Z4Z3Z2Z1Z0 Z31Z30Z27Z26 Z29Z28Z19Z18Z17Z16 Z25Z24Z21Z20 Z23Z22
(a) Full Anti-Logarithm Shifter
1f14f13f12f11
Z15Z14Z13Z12Z11Z10Z9Z8Z7Z6Z5Z4Z3Z2Z1Z0 Z31Z30Z27Z26 Z29Z28Z19Z18Z17Z16 Z25Z24Z21Z20 Z23Z22
(b) Truncated Anti-Log Shift (t=4)
Truncated Log Error
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
f
Valu
e Exact
Mitchell
Log2-T Ht=4L
Envelope Ht=4L
(c) Up-Rounded Values
0.0 0.2 0.4 0.6 0.8 1.0
-0.08
-0.06
-0.04
-0.02
0.00
0.02
0.04
0.06
f
Error
Mitchell
Log2-T Ht=4L
Log2-T Ht=6L
Envelope Ht=4L
(d) Up-Rounded Error
Figure: The error of upward-rounded truncation.
Cost Savings
0 5 10 15 20 25 30200
400
600
800
1000
t
Cost
HUG
ML
Mitchell
Log2-T�
(a) Cost Across t (n=32)
æ
æ
æ
æ
à
à
à
à
16 32 64 1280
1000
2000
3000
4000
5000
n
Cost
HUG
ML
Mitchell
Log2-T�Ht=4L
(b) Cost Across n (t=4)
Figure: The relative costs of truncated log gen/anti-gen.
Cost Savings (cont.)
10
20
30
4050
16 32 64 128
1
16
32
64
128
n
t
Figure: Exploring the n/t landscape.
Piece-wise Linear Logarithmic Approximation
1. M. Combet, H. Van Zonneveld, and L. Verbeek, “Computation ofthe base two logarithm of binary numbers,” IEEE Transactions onComputers, vol. EC-14, no. 6, pp. 863–867, 1965.
2. E. Hall, D. Lynch, and S. Dwyer, “Generation of products andquotients using approximate binary logarithms for digital filteringapplications,” IEEE Transactions on Computers, vol. C-19, no. 2,pp. 97–105, 1970.
3. S. SanGregory, C. Brothers, D. Gallagher, and R. Siferd, “A fast,low-power logarithm approximation with CMOS VLSIimplementation,” in Midwest Symposium on Circuits and Systems,vol. 1, 1999, pp. 388–391.
4. K. Abed and R. Siferd, “CMOS VLSI implementation of alow-power logarithmic converter,” IEEE Transactions on Computers,vol. 52, no. 11, pp. 1421–1433, 2003.
Using as a Drop-In Replacement for Combet et. al.
0.0 0.2 0.4 0.6 0.8 1.0
-0.05
0.00
0.05
f
Error
Correction
Mitchell
Log2-T Ht=8L
Log2-T+Combet Ht=8L
(a) Truncated Combet et. al.
10 15 20 25 300.000
0.005
0.010
0.015
0.020
0.025
t
Error
Maximum
Average
Log2-T+Combet Ht=8L
(b) Truncated Combet t Exploration
Figure: Truncation applied to Combet et. al.’s correction scheme.
A Novel Error Correction Scheme with Truncation
0.0 0.2 0.4 0.6 0.8 1.0
-0.05
0.
0.05
f
Error
Correction
Mitchell
Log2-T Ht=4L
Log2-T+EC4Ht=4, tec=4L
(a) log2-T↑+EC4 (tec =4)
æ
æ
æ
æ
à
à
à
à
ì
ì
ì
ì
16 32 64 1280
20
40
60
80
n
Cost
for
Correctio
nHU
GM
L
Log2-T+EC4Htec=4L
Log2-T+EC4Htec=3L
Log2-T+EC4Htec=2L
(b) log2-T↑+EC4 (tec =4) cost
Figure: Novel error correction for truncated logarithms.
To Recap (Numerical Results)
Table: Precision and Cost of Truncated Mitchell Analogues (n = 32).
TechniqueMin.Error
Max.Error
Max. Abs.Error
AverageAbs. Error
Cost
Mitchell -0.086 0 0.086 0.057 1.0log2−T↑(t=4) -0.086 0.062 0.086 0.035 0.57
log2−T↑+EC4(tec =2) -0.081 0.078 0.081 0.028 0.60
log2−T↑+EC4(tec =3) -0.066 0.070 0.070 0.024 0.61
log2−T↑+EC4(tec =4) -0.061 0.066 0.061 0.023 0.63
Conclusion
Truncated approximate logarithms improve piecewise-linearapproximate logarithm computations. They are based off of theintuition that the internal precision of a conversion schemeshould be proportional to the precision of the approximation.
Benefits:
I Decrease cost (up to ∼50%)
I May improve the precision of results
I Amenable to existing error reduction techniques
I May allow unique truncation-specific error reduction
Methodology - Unit Gate Model
Delay and cost (energy) estimated using a unit-gate model:
I Simple 2-input gates (AND, OR) [C = 1,T = 1]
I 2-input XOR gates and MUXes [C = 2,T = 2]
I m-input gates composed of a tree of 2-input gates
Advantages of the unit gate model:
I Offers a rough technology-agnostic model for circuit efficiency
I Betters understanding of scaling properties, bottlenecks
I Can be used for rapid design-space exploration
Inverters, buffering, and wiring concerns are ignored, but:
I Limited fan-out components are used
I Wiring stays roughly equivalent after truncation
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