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