1024 bit Parallel Rational Arithmetic Operators for the GPU

Size: px

Start display at page:

Download "1024 bit Parallel Rational Arithmetic Operators for the GPU"

Brittany Paul
5 years ago
Views:

1 1024 bit Parallel Rational Arithmetic Operators for the GPU Bob Zigon Sr. Staff Research Engineer Beckman Coulter, Inc May 15, 2012

2 Overview What is Rational Arithmetic? What is Particle Characterization? Why Rational Arithmetic for Particle Characterization? Implementation on the GPGPU

3 What is Rational Arithmetic? A number system where values are represented as the ratio of two integers. Example (fraction vs decimal) 1 / 2 = 0.5 (both are exact) 1 / 3 = 0.33 (first is exact, second is approx)

4 Example

5 Example (approximate) (exact)

6 What is Particle Characterization? The study of particles, their attributes and their interaction with surrounding systems.

7 What are Particles? Soil - Cocoa Cement - Salt Printer Toner - Ceramics Drugs - Metals Coffee - Proteins Sludge

8 Why do we care? In any end use application, control over a particle s attributes will have a dramatic impact on the success or failure of the application. For example, Product Quality (coffee, cocoa, cement) Drug Effectiveness

$way of Laser Diffraction Laser$ $diffraction is a method used$ Very popular sizing technique

9 Particle Characterization by way of Laser Diffraction Laser diffraction is a method used to measure particle size. Very popular sizing technique Provides rapid and consistent results

10 Mathematical Model of Diffraction Mie theory is a solution to Maxwell s equations for spherical particles. Mie predicts the amplitude of the scatter for different incident angles. Scattered Rays have different intensities A particle Incident ray

11 2D Scatter Plot Intensity and Angle

12 M = index of refraction of the particle R = Radius of the particle For Angle = 0 to 180 Mu = cos(angle) Intensity = 0 For Lambda = Lo to Hi X = 2 * PI * Radius / Lambda N = X + X^(1/3) Next Lambda Plot (Angle, Intensity) Next Angle

13 Dig a little deeper Easy to compute Legendre Polynomial Easy to compute Hard to compute due to Bessel function oscillation Bessel Function Bessel Function

14 Graph of a Bessel Function

15 Problems Bessel functions are damped and oscillate N is likely to be 1000, or 50000! Numeric instability results from catastrophic floating point cancellation! Solution : Rational arithmetic!

16 Review Rational Arithmetic - Multiply

17 Atomic operators on the GPGPU Add1024 Subtract1024 Multiply512* Divide1024 These 4 manipulate 1024 bit operands GCD used to reduce numerators and denominators Compare used by the GCD Mod used by the GCD

18 Implementation on the GPGPU First build the Adder Subtractor just adds the ones complement plus 1 Multiplier computes partial products then adds Divider computes partial quotients/remainders recursively MOD is a composite of the above operators

19 Ripple Carry Adder (slow) c2 A3 A2 A1 A0 c1 c0 + B3 B2 B1 B0 D3 D2 D1 D0

20 Kogge Stone Adder (fast) Parallel Prefix Carry Look Ahead 3 Stages Step 1. Preprocessing Step 2. Compute carry s Step 3. Compute final sum

21 Kogge Stone Adder details A3 A2 A1 A0 B3 B2 B1 B0 Fully parallel Log 2 N Fully parallel Carry Generate G i = A i and B i Carry Propagate P i = A i + B i Half Sums D i = A i xor B i Carry s C i = G i-1 + P i-1 C i-1 Final Sums S i = D i xor C i-1

22 Performance Function Ripple Carry Adder Kogge Stone Adder # of operand pairs GPU(ms) Host in C (ms) Host in ASM (ms) C / GPU ASM / GPU GPU Mem BW 2,097, GB/sec 2,097, GB/sec

23 The Multiplier A3 A2 A1 A0 * B3 B2 B1 B0 Kogge Stone Adder to sum the partial products Multiplications occur In parallel D15 D11 D14 D7 D10 D13 D3 D6 D9 D12 Kogge Stone adder D2 D5 D8 D1 D4 D0 Result7 Result6 Result5 Result4 Result3 Result2 Result1 Result0

24 The Divider Division is hard! Didn t solve the N bit divided by N bit problem Implemented the N bit divided by 32 bit

25 Divider details q = quotient u = dividend r u vq v = divisor b = radix (2**32) r = remainder q u v n i 1 q b 0 i i

26 Divider recursive equation r i ( br 1 i u i ) mod v q ( br u ) / i i 1 i v r n 0, i n 1..0

27 Euclid s GCD Function GCD( a, b ) while b!= 0 { t = b b = a mod b a = t } Test for equality b = a b*floor(a/b) return a

28 Test for Equality Comparisons occur In parallel == A3 A2 A1 A0 B3 B2 B1 B0 Ballot instruction communicates to all cores in the warp. Warp Level Ballot Instruction Result is a bit field that describes each comparison > All operands are equal > All operands but A0/B0

29 Equality code snippet unsigned int A = gpusrca[gtid]; unsigned int B = gpusrcb[gtid]; // // If gpudst[gtid] returns with 0xFFFFFFFF, then the predicate // is true for all elements of the warp. Therefore, A is equal to B. // unsigned int R = ballot(a == B); if (threadidx.x == 0) gpudst[gtid] = R;

30 Performance Function # of operand pairs GPU(ms) Host in C (ms) Host in ASM (ms) C / GPU ASM / GPU GPU Mem BW Multiply 2,097, , GB/sec Divide 2,097, GB/sec Modulo 2,097, , GB/sec InEquality 2,097, GB/sec

31 References A parallel algorithm for multiple precision division by a single precision integer, Takahashi Efficient parallel scan algorithms for GPUs, Harris A regular layout for parallel adders, Kung Parallel solution of recurrence problems, Kogge

32 Questions? beckman.com Source code

Timing for Ripple Carry Adder

Timing for Ripple Carry Adder 1 2 3 Look Ahead Method 5 6 7 8 9 Look-Ahead, bits wide 10 11 Multiplication Simple Gradeschool Algorithm for 32 Bits (6 Bit Result) Multiplier Multiplicand AND gates 32