Fixed-Point Trigonometric Functions on FPGAs

Transcription

1 Fixed-Point Trigonometric Functions on FPGAs Florent de Dinechin Matei Iştoan Guillaume Sergent LIP, Université de Lyon (CNRS/ENS-Lyon/INRIA/UCBL) 46, allée d Italie, Lyon Cedex 07 June 14th, 2013 de Dinechin, Iştoan, Sergent (ENS Lyon) Fixed-Point Trigonometric Functions on FPGAs June 14th, / 22

2 Outline Outline 1 Introduction 2 Methods for Computing Sine and Cosine 2.1 CORDIC Method 2.2 Table- and DSP-based method 2.3 Polynomial Evaluator-based method 3 Results and Conclusions de Dinechin, Iştoan, Sergent (ENS Lyon) Fixed-Point Trigonometric Functions on FPGAs June 14th, / 22

3 Introduction Outline 1 Introduction 2 Methods for Computing Sine and Cosine 2.1 CORDIC Method 2.2 Table- and DSP-based method 2.3 Polynomial Evaluator-based method 3 Results and Conclusions de Dinechin, Iştoan, Sergent Fixed-Point Trigonometric Functions on FPGAs June 14th, 2013

4 Introduction Introduction Why compute the trigonometric functions sine and cosine? fundamental in signal processing and signal processing applications like FFT, modulation/demodulation, frequency synthesizers,... How to compute them? In this work: 1 the classical CORDIC algorithm, based on additions and shifts 2 a method based on tables and multipliers, suited for modern FPGAs 3 a generic polynomial approximation Which is best on FPGAs? What is the cost of w bits of sine and cosine? de Dinechin, Iştoan, Sergent Fixed-Point Trigonometric Functions on FPGAs June 14th, / 22

5 Introduction Which method is best on FPGAs? A fair comparison of methods computing sine and cosine: same specification (the best possible one) Fixed-point inputs and outputs compute sin(πx) and cos(πx) for x [ 1, 1) Faithful rounding: all the produced bits are useful, no wasted resources same effort (the best possible one) open-source implementations in FloPoCo state-of-the-art? Computing just one, or both? some applications need both sine and cosine (e.g. rotation) some methods compute both de Dinechin, Iştoan, Sergent Fixed-Point Trigonometric Functions on FPGAs June 14th, / 22

6 Introduction Textbook Stuff Decomposition of the exponential in two exponentials e i(a+b) = e ia e ib From complex to real e iϕ = cos(ϕ) + i sin(ϕ) Decompose a rotation in smaller sub-rotations { sin(a + b) = sin(a) cos(b) + cos(a) sin(b) cos(a + b) = cos(a) cos(b) sin(a) sin(b) de Dinechin, Iştoan, Sergent Fixed-Point Trigonometric Functions on FPGAs June 14th, / 22

7 Introduction Argument Reduction based on the 3 MSBs of the input angle x s - sign q - quadrant o - octant remaining argument y [0, 1/4) { 1 y = 4 y if o = 1 y otherwise. compute cos(πy ) and sin(πy ) reconstruction: sqo { Reconstruction sin(πx) = sin(πy ) cos(πx) = cos(πy ) { sin(πx) = cos(πy ) cos(πx) = sin(πy ) { sin(πx) = cos(πy ) cos(πx) = sin(πy ) { sin(πx) = sin(πy ) cos(πx) = cos(πy ) de Dinechin, Iştoan, Sergent Fixed-Point Trigonometric Functions on FPGAs June 14th, / 22

8 Outline 1 Introduction 2 Methods for Computing Sine and Cosine 2.1 CORDIC Method 2.2 Table- and DSP-based method 2.3 Polynomial Evaluator-based method 3 Results and Conclusions de Dinechin, Iştoan, Sergent Fixed-Point Trigonometric Functions on FPGAs June 14th, 2013

9 CORDIC Method CORDIC Method Description compute sine and cosine using a decomposition of the input angle use only additions and shifts as operations The classical CORDIC iteration 1 c 0 = Π n i=1 1+2 i s 0 = 0 α 0 = y (the reduced argument) s i s Pseudo-rotation d i = +1 if α i > 0, otherwise 1 c i+1 = c i 2 i d i s i s i+1 = s i + 2 i d i c i α i+1 = α i d i arctan(2 i ) c n inf = cos(y) s n inf = sin(y) α n inf = 0 s' i+1 s i+1 α i i+1 α c i Rotation c i+1 c' i+1 c de Dinechin, Iştoan, Sergent Fixed-Point Trigonometric Functions on FPGAs June 14th, / 22

10 CORDIC Method CORDIC Method Description compute sine and cosine using a decomposition of the input angle use only additions and shifts as operations The classical CORDIC iteration 1 c 0 = Π n i=1 1+2 i s 0 = 0 α 0 = y (the reduced argument) d i = +1 if α i > 0, otherwise 1 c i+1 = c i 2 i d i s i s i+1 = s i + 2 i d i c i α i+1 = α i d i arctan(2 i ) c n inf = cos(y) s n inf = sin(y) α n inf = 0 de Dinechin, Iştoan, Sergent Fixed-Point Trigonometric Functions on FPGAs June 14th, / 22

11 CORDIC Method CORDIC Method Description compute sine and cosine using a decomposition of the input angle use only additions and shifts as operations The classical CORDIC iteration 1 c 0 = Π n i=1 1+2 i s 0 = 0 α 0 = y (the reduced argument) d i = +1 if α i > 0, otherwise 1 c i+1 = c i 2 i d i s i s i+1 = s i + 2 i d i c i α i+1 = α i d i arctan(2 i ) c n inf = cos(y) s n inf = sin(y) α n inf = 0 de Dinechin, Iştoan, Sergent Fixed-Point Trigonometric Functions on FPGAs June 14th, / 22

12 CORDIC Method CORDIC Method Description compute sine and cosine using a decomposition of the input angle use only additions and shifts as operations The classical CORDIC iteration 1 c 0 = Π n i=1 1+2 i s 0 = 0 α 0 = y (the reduced argument) d i = +1 if α i > 0, otherwise 1 c i+1 = c i 2 i d i s i s i+1 = s i + 2 i d i c i α i+1 = α i d i arctan(2 i ) c n inf = cos(y) s n inf = sin(y) α n inf = 0 de Dinechin, Iştoan, Sergent Fixed-Point Trigonometric Functions on FPGAs June 14th, / 22

13 CORDIC Method CORDIC Method Description compute sine and cosine using a decomposition of the input angle use only additions and shifts as operations The classical CORDIC iteration 1 c 0 = Π n i=1 1+2 i s 0 = 0 α 0 = y (the reduced argument) d i = +1 if α i > 0, otherwise 1 c i+1 = c i 2 i d i s i s i+1 = s i + 2 i d i c i α i+1 = α i d i arctan(2 i ) c n inf = cos(y) s n inf = sin(y) α n inf = 0 de Dinechin, Iştoan, Sergent Fixed-Point Trigonometric Functions on FPGAs June 14th, / 22

14 CORDIC Method CORDIC Method Description compute sine and cosine using a decomposition of the input angle use only additions and shifts as operations The classical CORDIC iteration 1 c 0 = Π n i=1 1+2 i s 0 = 0 α 0 = y (the reduced argument) d i = +1 if α i > 0, otherwise 1 c i+1 = c i 2 i d i s i s i+1 = s i + 2 i d i c i α i+1 = α i d i arctan(2 i ) c n inf = cos(y) s n inf = sin(y) α n inf = 0 de Dinechin, Iştoan, Sergent Fixed-Point Trigonometric Functions on FPGAs June 14th, / 22

15 CORDIC Method CORDIC Method Description compute sine and cosine using a decomposition of the input angle use only additions and shifts as operations The classical CORDIC iteration 1 c 0 = Π n i=1 1+2 i s 0 = 0 α 0 = y (the reduced argument) d i = +1 if α i > 0, otherwise 1 c i+1 = c i 2 i d i s i s i+1 = s i + 2 i d i c i α i+1 = α i d i arctan(2 i ) c n inf = cos(y) s n inf = sin(y) α n inf = 0 de Dinechin, Iştoan, Sergent Fixed-Point Trigonometric Functions on FPGAs June 14th, / 22

16 CORDIC Method CORDIC Method Description compute sine and cosine using a decomposition of the input angle use only additions and shifts as operations The classical CORDIC iteration 1 c 0 = Π n i=1 1+2 i s 0 = 0 α 0 = y (the reduced argument) s i s Pseudo-rotation d i = +1 if α i > 0, otherwise 1 c i+1 = c i 2 i d i s i s i+1 = s i + 2 i d i c i α i+1 = α i d i arctan(2 i ) c n inf = cos(y) s n inf = sin(y) α n inf = 0 s' i+1 s i+1 α i i+1 α c i Rotation c i+1 c' i+1 c de Dinechin, Iştoan, Sergent Fixed-Point Trigonometric Functions on FPGAs June 14th, / 22

17 CORDIC Method CORDIC Architecture c 0 s 0 >>0 >>0 d 0 c 0 = 1 s 0 = Π n 1+2 i=1 0 α 0 = y (the reduced argument) d i = +1 if α i > 0, otherwise 1 c i+1 = c i 2 i d i s i s i+1 = s i + 2 i d i c i α i+1 = α i d i arctan(2 i ) c n inf = cos(y) s n inf = sin(y) α n inf = 0 c 1 s 1 >>1 >>1 c 2 s 2 >>2 >>2 c n 1 s n 1 >>n 1 >>n 1 d 1 d 2 d n 1 x n y n z n de Dinechin, Iştoan, Sergent Fixed-Point Trigonometric Functions on FPGAs June 14th, / 22

18 CORDIC Method CORDIC Improvements c 0 s 0 z 0 d 0 >>0 >>0 α 0 Reduced α-datapath α i < 2 i decrement the α-datapath by 1 bit per iteration benefits saves space saves latency c 1 s 1 z 1 d 1 >>1 >>1 α 1 c 2 s 2 z 2 d 2 >>2 >>2 α 2 c n 1 s n 1 z n 1 >>n 1 >>n 1 d n 1 α n 1 x n y n z n de Dinechin, Iştoan, Sergent Fixed-Point Trigonometric Functions on FPGAs June 14th, / 22

19 CORDIC Method CORDIC Improvements Reduced Iterations stop iterations when they can be replaced by a single rotation, with enough accuracy { sin(α) α cos(α) 1 half the iterations replaced by { xi+1 = x i + α y i y i+1 = y i α x i c 0 s 0 >>0 >>0 c 1 s 1 >>1 >>1 c 2 s 2 >>2 >>2 d 1 d 2 d 0 only 2 multiplications are needed 2 DSPs for input up to 32 bits truncated multiplications for larger sizes c n 1 >>n 1 >>n 1 s n 1 d n 1 x n y n z n de Dinechin, Iştoan, Sergent Fixed-Point Trigonometric Functions on FPGAs June 14th, / 22

20 CORDIC Method CORDIC Improvements Reduced Iterations stop iterations when they can be replaced by a single rotation, with enough accuracy { sin(α) α cos(α) 1 half the iterations replaced by { xi+1 = x i + α y i y i+1 = y i α x i c 0 s 0 >>0 >>0 c 1 s 1 >>1 >>1 c 2 s 2 >>2 >>2 d 0 d 1 d 2 only 2 multiplications are needed 2 DSPs for input up to 32 bits truncated multiplications for larger sizes c n/2+1 s n/2+1 x n y n z n de Dinechin, Iştoan, Sergent Fixed-Point Trigonometric Functions on FPGAs June 14th, / 22

21 CORDIC Method Error Analysis Goal: last-bit accuracy of the result the result is within 1ulp of the mathematical result ulp = weight of least significant bit Intermediate precision approximations and roundings computations on w+g bits internally guard bits g Error budget: total of 1ulp ulp for the final rounding error ulp for the method error 1 4ulp for the rounding errors c 0 s 0 c n 1 >>0 >>0 c 1 s 1 >>1 >>1 c 2 s 2 >>2 >>2 >>n 1 >>n 1 s n 1 d 0 d 1 d 2 d n 1 x n y n z n de Dinechin, Iştoan, Sergent Fixed-Point Trigonometric Functions on FPGAs June 14th, / 22

22 CORDIC Method CORDIC Error Analysis (1) c 0 s 0 >>0 >>0 d 0 Analysis: method error (ε method ) ε method of the order of the value of α final α final can be bounded numerically number of iterations: smallest number for which ε method < 2 w 2 c 1 s 1 >>1 >>1 c 2 s 2 >>2 >>2 d 1 d 2 c n 1 >>n 1 >>n 1 s n 1 d n 1 x n y n z n de Dinechin, Iştoan, Sergent Fixed-Point Trigonometric Functions on FPGAs June 14th, / 22

23 CORDIC Method CORDIC Error Analysis (2) Analysis: rounding errors (ε round ) on the α datapath correct rounding of arctan(2 i ) error bounded by 2 w g 1 total error on the α-datapath: nb iter 2 w g 1 on the sin() and cos() datapath for each shift operation, error bounded by 2 w g total error larger than on the α-datapath must be smaller than 2 w 2 : ε 2 w g < 2 w 2 this gives g ε method + ε round < 2 w 1 Final rounding of sin and cos has error of the order 2 w 1 c 0 s 0 c n 1 >>0 >>0 c 1 s 1 >>1 >>1 c 2 s 2 >>2 >>2 >>n 1 >>n 1 s n 1 x n y n z n de Dinechin, Iştoan, Sergent Fixed-Point Trigonometric Functions on FPGAs June 14th, / 22 d 1 d 2 d 0 d n 1

24 CORDIC Method CORDIC Error Analysis (2) Analysis: rounding errors (ε round ) on the α datapath correct rounding of arctan(2 i ) error bounded by 2 w g 1 total error on the α-datapath: nb iter 2 w g 1 on the sin() and cos() datapath for each shift operation, error bounded by 2 w g total error larger than on the α-datapath must be smaller than 2 w 2 : ε 2 w g < 2 w 2 this gives g ε method + ε round < 2 w 1 Final rounding of sin and cos has error of the order 2 w 1 c 0 s 0 c n 1 >>0 >>0 c 1 s 1 >>1 >>1 c 2 s 2 >>2 >>2 >>n 1 >>n 1 s n 1 x n y n z n de Dinechin, Iştoan, Sergent Fixed-Point Trigonometric Functions on FPGAs June 14th, / 22 d 1 d 2 d 0 d n 1

25 CORDIC Method CORDIC Error Analysis (2) Analysis: rounding errors (ε round ) on the α datapath correct rounding of arctan(2 i ) error bounded by 2 w g 1 total error on the α-datapath: nb iter 2 w g 1 on the sin() and cos() datapath for each shift operation, error bounded by 2 w g total error larger than on the α-datapath must be smaller than 2 w 2 : ε 2 w g < 2 w 2 this gives g ε method + ε round < 2 w 1 Final rounding of sin and cos has error of the order 2 w 1 c 0 s 0 c n 1 >>0 >>0 c 1 s 1 >>1 >>1 c 2 s 2 >>2 >>2 >>n 1 >>n 1 s n 1 x n y n z n de Dinechin, Iştoan, Sergent Fixed-Point Trigonometric Functions on FPGAs June 14th, / 22 d 1 d 2 d 0 d n 1

26 Table- and DSP-based method Table- and DSP-based method Algorithm angle split: y (the reduced angle) = t + y red t on a bits y red such that y red < 2 (a+2) store sin(πt) and cos(πt) in tables evaluate sin(πy red ) and cos(πy red ) using a Taylor polynomial approximation need to compute first z = y red π sin(z) z z 3 /6 cos(z) 1 z 2 /2 reconstruct the values of sin(πy) and cos(πy) using { sin(π(t + yred )) = sin(πt) cos(πy red ) + cos(πt) sin(πy red ) cos(π(t + y red )) = cos(πt) cos(πy red ) sin(πt) sin(πy red ) de Dinechin, Iştoan, Sergent Fixed-Point Trigonometric Functions on FPGAs June 14th, / 22

27 Table- and DSP-based method Table- and DSP-based method Algorithm angle split: y (the reduced angle) = t + y red t on a bits y red such that y red < 2 (a+2) store sin(πt) and cos(πt) in tables evaluate sin(πy red ) and cos(πy red ) using a Taylor polynomial approximation need to compute first z = y red π sin(z) z z 3 /6 cos(z) 1 z 2 /2 reconstruct the values of sin(πy) and cos(πy) using { sin(π(t + yred )) = sin(πt) cos(πy red ) + cos(πt) sin(πy red ) cos(π(t + y red )) = cos(πt) cos(πy red ) sin(πt) sin(πy red ) de Dinechin, Iştoan, Sergent Fixed-Point Trigonometric Functions on FPGAs June 14th, / 22

28 Table- and DSP-based method Table- and DSP-based method Algorithm angle split: y (the reduced angle) = t + y red t on a bits y red such that y red < 2 (a+2) store sin(πt) and cos(πt) in tables evaluate sin(πy red ) and cos(πy red ) using a Taylor polynomial approximation need to compute first z = y red π sin(z) z z 3 /6 cos(z) 1 z 2 /2 reconstruct the values of sin(πy) and cos(πy) using { sin(π(t + yred )) = sin(πt) cos(πy red ) + cos(πt) sin(πy red ) cos(π(t + y red )) = cos(πt) cos(πy red ) sin(πt) sin(πy red ) de Dinechin, Iştoan, Sergent Fixed-Point Trigonometric Functions on FPGAs June 14th, / 22

29 Table- and DSP-based method Table- and DSP-based method Algorithm angle split: y (the reduced angle) = t + y red t on a bits y red such that y red < 2 (a+2) store sin(πt) and cos(πt) in tables evaluate sin(πy red ) and cos(πy red ) using a Taylor polynomial approximation need to compute first z = y red π sin(z) z z 3 /6 cos(z) 1 z 2 /2 reconstruct the values of sin(πy) and cos(πy) using { sin(π(t + yred )) = sin(πt) cos(πy red ) + cos(πt) sin(πy red ) cos(π(t + y red )) = cos(πt) cos(πy red ) sin(πt) sin(πy red ) de Dinechin, Iştoan, Sergent Fixed-Point Trigonometric Functions on FPGAs June 14th, / 22

30 Table- and DSP-based method Table- and DSP-based method Algorithm angle split: y (the reduced angle) = t + y red t on a bits y red such that y red < 2 (a+2) store sin(πt) and cos(πt) in tables evaluate sin(πy red ) and cos(πy red ) using a Taylor polynomial approximation need to compute first z = y red π sin(z) z z 3 /6 cos(z) 1 z 2 /2 reconstruct the values of sin(πy) and cos(πy) using { sin(π(t + yred )) = sin(πt) cos(πy red ) + cos(πt) sin(πy red ) cos(π(t + y red )) = cos(πt) cos(πy red ) sin(πt) sin(πy red ) de Dinechin, Iştoan, Sergent Fixed-Point Trigonometric Functions on FPGAs June 14th, / 22

31 Table- and DSP-based method Table- and DSP-based method Algorithm angle split: y (the reduced angle) = t + y red t on a bits y red such that y red < 2 (a+2) store sin(πt) and cos(πt) in tables evaluate sin(πy red ) and cos(πy red ) using a Taylor polynomial approximation need to compute first z = y red π sin(z) z z 3 /6 cos(z) 1 z 2 /2 reconstruct the values of sin(πy) and cos(πy) using { sin(π(t + yred )) = sin(πt) cos(πy red ) + cos(πt) sin(πy red ) cos(π(t + y red )) = cos(πt) cos(πy red ) sin(πt) sin(πy red ) de Dinechin, Iştoan, Sergent Fixed-Point Trigonometric Functions on FPGAs June 14th, / 22

32 Table- and DSP-based method Table- and DSP-based method: Details approximating y = 1 4 y red as y red choose a such that z w g so that a degree-3 Taylor polynomial may be used means that 4(a + 2) 2 w + g truncated multiplications T Z T constant multiplication by π z 2 /2 computed using a squarer z 3 /6 read from a table for small precisions computed with a dedicated architecture for larger precisions (based on a bit heap and divider by 3, see paper) T T T T T T de Dinechin, Iştoan, Sergent Fixed-Point Trigonometric Functions on FPGAs June 14th, / 22

33 Table- and DSP-based method Table- and DSP-based method: Error Analysis Error Analysis 1 2ulp lost per table 1ulp per truncation and truncated multiplier/squarer T Z T 1ulp for computing 1 4 y red (as y red ) T T T T T T total of 15ulp, independent of the input width gives g=4 de Dinechin, Iştoan, Sergent Fixed-Point Trigonometric Functions on FPGAs June 14th, / 22

34 Polynomial Evaluator-based method Polynomial-based method using existing software (more details in the reference) based on polynomial approximation computes only one of the functions, depending on an input i A D Coef. ROM a n a n 1 a 0 y trunc. mult. + trunc. mult. + round de Dinechin, Iştoan, Sergent Fixed-Point Trigonometric Functions on FPGAs June 14th, / 22 R ỹ 1 ỹ 2 T T y

35 Results and Conclusions Outline 1 Introduction 2 Methods for Computing Sine and Cosine 2.1 CORDIC Method 2.2 Table- and DSP-based method 2.3 Polynomial Evaluator-based method 3 Results and Conclusions de Dinechin, Iştoan, Sergent Fixed-Point Trigonometric Functions on FPGAs June 14th, 2013

36 Results and Conclusions Results 16 bit Precision Approach latency frequency Reg. + LUTs BRAM DSP CORDIC CORDIC CORDIC CORDIC Red. CORDIC Red. CORDIC Red. CORDIC Red. CORDIC SinAndCos SinAndCos SinOrCos (d=2) SinOrCos (d=2) Synthesis Results on Virtex5 FPGA, Using ISE 12.1 de Dinechin, Iştoan, Sergent Fixed-Point Trigonometric Functions on FPGAs June 14th, / 22

37 Results and Conclusions Results 16 bit Precision Approach latency frequency Reg. + LUTs BRAM DSP CORDIC CORDIC CORDIC CORDIC Red. CORDIC Red. CORDIC Red. CORDIC Red. CORDIC SinAndCos SinAndCos SinOrCos (d=2) SinOrCos (d=2) Synthesis Results on Virtex5 FPGA, Using ISE 12.1 de Dinechin, Iştoan, Sergent Fixed-Point Trigonometric Functions on FPGAs June 14th, / 22

38 Results and Conclusions Results 16 bit Precision Approach latency frequency Reg. + LUTs BRAM DSP CORDIC CORDIC CORDIC CORDIC Red. CORDIC Red. CORDIC Red. CORDIC Red. CORDIC SinAndCos SinAndCos SinOrCos (d=2) SinOrCos (d=2) Synthesis Results on Virtex5 FPGA, Using ISE 12.1 de Dinechin, Iştoan, Sergent Fixed-Point Trigonometric Functions on FPGAs June 14th, / 22

39 Results and Conclusions Results 16 bit Precision Approach latency frequency Reg. + LUTs BRAM DSP CORDIC CORDIC CORDIC CORDIC Red. CORDIC Red. CORDIC Red. CORDIC Red. CORDIC SinAndCos SinAndCos SinOrCos (d=2) SinOrCos (d=2) Synthesis Results on Virtex5 FPGA, Using ISE 12.1 de Dinechin, Iştoan, Sergent Fixed-Point Trigonometric Functions on FPGAs June 14th, / 22

40 Results and Conclusions Results 16 bit Precision Approach latency frequency Reg. + LUTs BRAM DSP CORDIC CORDIC CORDIC CORDIC Red. CORDIC Red. CORDIC Red. CORDIC Red. CORDIC SinAndCos SinAndCos SinOrCos (d=2) SinOrCos (d=2) Synthesis Results on Virtex5 FPGA, Using ISE 12.1 de Dinechin, Iştoan, Sergent Fixed-Point Trigonometric Functions on FPGAs June 14th, / 22

41 Results and Conclusions Results Highest Frequency Approach latency frequency Reg. + LUTs BRAM DSP precision = 16 bits CORDIC Red. CORDIC SinAndCos SinOrCos (d=2) precision = 24 bits CORDIC Red. CORDIC SinAndCos SinOrCos (d=2) precision = 32 bits CORDIC Red. CORDIC SinAndCos SinOrCos (d=3) precision = 40 bits CORDIC Red. CORDIC SinAndCos (bit heap) SinAndCos (table z 3 /6) SinOrCos (d=3) precision = 48 bits SinAndCos (bit heap) SinOrCos de Dinechin, Iştoan, Sergent Fixed-Point Trigonometric Functions on FPGAs June 14th, / 22

42 Results and Conclusions Results Highest Frequency Approach latency frequency Reg. + LUTs BRAM DSP precision = 16 bits CORDIC Red. CORDIC SinAndCos SinOrCos (d=2) precision = 24 bits CORDIC Red. CORDIC SinAndCos SinOrCos (d=2) precision = 32 bits CORDIC Red. CORDIC SinAndCos SinOrCos (d=3) precision = 40 bits CORDIC Red. CORDIC SinAndCos (bit heap) SinAndCos (table z 3 /6) SinOrCos (d=3) precision = 48 bits SinAndCos (bit heap) SinOrCos de Dinechin, Iştoan, Sergent Fixed-Point Trigonometric Functions on FPGAs June 14th, / 22

43 Results and Conclusions Results Highest Frequency Approach latency frequency Reg. + LUTs BRAM DSP precision = 16 bits CORDIC Red. CORDIC SinAndCos SinOrCos (d=2) precision = 24 bits CORDIC Red. CORDIC SinAndCos SinOrCos (d=2) precision = 32 bits CORDIC Red. CORDIC SinAndCos SinOrCos (d=3) precision = 40 bits CORDIC Red. CORDIC SinAndCos (bit heap) SinAndCos (table z 3 /6) SinOrCos (d=3) precision = 48 bits SinAndCos (bit heap) SinOrCos de Dinechin, Iştoan, Sergent Fixed-Point Trigonometric Functions on FPGAs June 14th, / 22

44 Results and Conclusions Results Options for Z 3 6 Approach latency frequency Reg. + LUTs BRAM DSP precision = 40 bits CORDIC CORDIC Red. CORDIC Red. CORDIC SinAndCos (bit heap) SinAndCos (table z 3 /6) SinAndCos (bit heap) SinAndCos (table z 3 /6) SinOrCos (d=3) SinOrCos (d=3) precision = 48 bits SinAndCos (bit heap) SinAndCos (bit heap) SinOrCos SinOrCos de Dinechin, Iştoan, Sergent Fixed-Point Trigonometric Functions on FPGAs June 14th, / 22

45 Results and Conclusions Results Options for Z 3 6 Approach latency frequency Reg. + LUTs BRAM DSP precision = 40 bits CORDIC CORDIC Red. CORDIC Red. CORDIC SinAndCos (bit heap) SinAndCos (table z 3 /6) SinAndCos (bit heap) SinAndCos (table z 3 /6) SinOrCos (d=3) SinOrCos (d=3) precision = 48 bits SinAndCos (bit heap) SinAndCos (bit heap) SinOrCos SinOrCos de Dinechin, Iştoan, Sergent Fixed-Point Trigonometric Functions on FPGAs June 14th, / 22

46 Results and Conclusions Results Options for Z 3 6 Approach latency frequency Reg. + LUTs BRAM DSP precision = 40 bits CORDIC CORDIC Red. CORDIC Red. CORDIC SinAndCos (bit heap) SinAndCos (table z 3 /6) SinAndCos (bit heap) SinAndCos (table z 3 /6) SinOrCos (d=3) SinOrCos (d=3) precision = 48 bits SinAndCos (bit heap) SinAndCos (bit heap) SinOrCos SinOrCos de Dinechin, Iştoan, Sergent Fixed-Point Trigonometric Functions on FPGAs June 14th, / 22

47 Results and Conclusions Conclusions A wide range of open-source accurate implementations In terms of performance, CORDIC implementation on par with vendor-provided solutions some tuning still needed in FloPoCo to get better performance on DSP-based methods SinAndCos method slightly better than polynomial evaluator-based method Little point in using CORDIC for FPGAs? Approach latency area CORDIC 16 bits 30.3 ns 1034 LUTs SinAndCos 16 bits 15.0 ns 1211 LUTs CORDIC 24 bits 44.6 ns 2079 LUTs SinAndCos 24 bits 17.0 ns 2183 LUTs CORDIC 32 bits 62.1 ns 3513 LUTs SinAndCos 32 bits 19.4 ns 3539 LUTs Synthesis results for logic-only implementations What is the cost of computing w bits of sine/cosine? de Dinechin, Iştoan, Sergent Fixed-Point Trigonometric Functions on FPGAs June 14th, / 22

48 Results and Conclusions Questions? FloPoCo Thank you for your attention! Questions? de Dinechin, Iştoan, Sergent Fixed-Point Trigonometric Functions on FPGAs June 14th, / 22

49 Results and Conclusions Hidden Frame Hidden Frame de Dinechin, Iştoan, Sergent Fixed-Point Trigonometric Functions on FPGAs June 14th, 2013

50 Results and Conclusions Format Input Fixed-Point modulo 2π periodicity of the trigonometric functions periodicity of the fixed point representation Input in x [ 1, 1) compute sin(πx) and cos(πx) de Dinechin, Iştoan, Sergent Fixed-Point Trigonometric Functions on FPGAs June 14th, 2013

51 Results and Conclusions Format Output Fixed-Point Scaled output fixed-point range not symmetric: -1 representable but not 1 largest value representable 1 2 w, smallest value 1 scale factor 1 2 w compute sin(πx) (1 2 w ) and cos(πx) (1 2 w ) no penalty paid for the (1 2 w ) scaling factor Faithful rounding the returned result produces an error that is less than 1ulp all the produced bits are useful, no wasted resources de Dinechin, Iştoan, Sergent Fixed-Point Trigonometric Functions on FPGAs June 14th, 2013

52 Results and Conclusions Method Description analyzing the terms: Z 3 Z 3 = w 1 2 3i zi 3 + i=0 sin(z) Z Z 3 /6 0 i<j w 1 taking advantage of Boolean properties zi k = z i 2 2 w a = 2 w+1 a after division by i+2j z i z 2 j + 0 i<j<k w i+j+k z i z j z k Z Z 3 /6 = w 1 i=0 2i z i 1/3 i max i=i min 2 3i 1 z i i min i<j i max 2 i+2j 1 z i z j i min i<j<k i max 2 i+j+k z i z j z k w=16 bits de Dinechin, Iştoan, Sergent Fixed-Point Trigonometric Functions on FPGAs June 14th, 2013

53 Results and Conclusions Format Input Fixed-Point Input, Fixed-Point Output modulo 2π periodicity of the trigonometric functions periodicity of the fixed point representation Input in x [ 1, 1) compute sin(πx) and cos(πx) Scaled output fixed-point range not symmetric: -1 representable but not 1 compute sin(πx) (1 2 w ) and cos(πx) (1 2 w ) de Dinechin, Iştoan, Sergent Fixed-Point Trigonometric Functions on FPGAs June 14th, 2013