Range Reduction Based on Pythagorean Triples for Trigonometric Function Evaluation

Transcription

1 Range Reduction Based on Pythagorean Triples for Trigonometric Function Evaluation Conférence en Parallélisme, Architecture et Systèmes 2015 Hugues de Lassus Saint-Geniès David Defour Guillaume Revy DALI, Université de Perpignan Via Domitia LIRMM, Université de Montpellier 2 CNRS UMR 5506 July 1 st, 2015

2 Context, objectives and achievements Context and objectives Elementary function evaluation IEEE-754 floating-point arithmetic Trigonometric functions Table-based range reduction Remove sources of error Achievements Algorithm to tabulate exact values for the sine and cosine functions For correct-rounding in double precision: Precomputed table size reduced up to 45% Memory accesses and FLOPs reduced up to 45% during the reconstruction H. de Lassus Saint-Geniès Range Reduction Based on Pythagorean Triples for Trigonometric Function Evaluation 2 / 25

3 1 Background on sine and cosine evaluation Additive range reduction Table-lookup range reduction Sources of error 2 Error-free table method for sine and cosine implementations Concentration of the error Pythagorean triples 3 Experimental results Exhaustive search Heuristic search Comparisons with other table-based methods 4 Conclusion and perspectives H. de Lassus Saint-Geniès Range Reduction Based on Pythagorean Triples for Trigonometric Function Evaluation 3 / 25

5 Basic scheme for sine and cosine evaluation (1) Range reduction: use of trigonometric symmetries and periodicities 1 sin(x) π π 0 π 2π H. de Lassus Saint-Geniès Range Reduction Based on Pythagorean Triples for Trigonometric Function Evaluation 4 / 25

6 Basic scheme for sine and cosine evaluation (1) Range reduction: use of trigonometric symmetries and periodicities 1 sin(x) π π 0 π 2π sin(x) = sin(x + 2π) H. de Lassus Saint-Geniès Range Reduction Based on Pythagorean Triples for Trigonometric Function Evaluation 4 / 25

7 Basic scheme for sine and cosine evaluation (1) Range reduction: use of trigonometric symmetries and periodicities 1 sin(x) π π 0 π 2π sin( x) = sin(x) H. de Lassus Saint-Geniès Range Reduction Based on Pythagorean Triples for Trigonometric Function Evaluation 4 / 25

8 Basic scheme for sine and cosine evaluation (1) Range reduction: use of trigonometric symmetries and periodicities 1 sin(x) π π 0 π 2π ( ) sin(x) = ±f k x k π 2 with f k {sin, cos} H. de Lassus Saint-Geniès Range Reduction Based on Pythagorean Triples for Trigonometric Function Evaluation 4 / 25

9 Basic scheme for sine and cosine evaluation (1) Range reduction: use of trigonometric symmetries and periodicities 1 sin(x) π π 0 π 2π ( ) sin(x) = ±f k x k π 2 with f k {sin, cos} H. de Lassus Saint-Geniès Range Reduction Based on Pythagorean Triples for Trigonometric Function Evaluation 4 / 25

10 Basic scheme for sine and cosine evaluation (1) Range reduction: use of trigonometric symmetries and periodicities 1 sin(x) π π 0 π 2π Range reduction F 64 [ 0, π 4 ] H. de Lassus Saint-Geniès Range Reduction Based on Pythagorean Triples for Trigonometric Function Evaluation 4 / 25

11 Basic scheme for sine and cosine evaluation (2) Tang s method H. de Lassus Saint-Geniès Range Reduction Based on Pythagorean Triples for Trigonometric Function Evaluation 5 / 25

14 Basic scheme for sine and cosine evaluation (2) Tang s method x H. de Lassus Saint-Geniès Range Reduction Based on Pythagorean Triples for Trigonometric Function Evaluation 5 / 25

15 Basic scheme for sine and cosine evaluation (2) Tang s method sin h sin(x h ) x x l x h cos h cos(x h ) sin(x) = sin h cos(x l ) cos h sin(x l ) H. de Lassus Saint-Geniès Range Reduction Based on Pythagorean Triples for Trigonometric Function Evaluation 5 / 25

16 Sources of error in the classical method x [0, π 4 ] Splitting x into (x h, x l ) x h x l Table lookup Polynomial approximations cos h sin h cos(x l ) sin(x l ) Reconstruction sin(x) = sin h cos(x l ) cos h sin(x l ) H. de Lassus Saint-Geniès Range Reduction Based on Pythagorean Triples for Trigonometric Function Evaluation 6 / 25

17 Sources of error in the classical method Errors appear in: x [0, π 4 ] Splitting x into (x h, x l ) x, inherently by the range reduction x h x l Table lookup Polynomial approximations cos h sin h cos(x l ) sin(x l ) Reconstruction sin(x) = sin h cos(x l ) cos h sin(x l ) H. de Lassus Saint-Geniès Range Reduction Based on Pythagorean Triples for Trigonometric Function Evaluation 6 / 25

18 Sources of error in the classical method Errors appear in: x [0, π 4 ] Splitting x into (x h, x l ) x, inherently by the range reduction Table lookup x h x l Polynomial approximations sin h and cos h, which are rounded approximations cos h sin h cos(x l ) sin(x l ) Reconstruction sin(x) = sin h cos(x l ) cos h sin(x l ) H. de Lassus Saint-Geniès Range Reduction Based on Pythagorean Triples for Trigonometric Function Evaluation 6 / 25

19 Sources of error in the classical method Errors appear in: Table lookup x [0, π 4 ] Splitting x into (x h, x l ) x h Polynomial approximations cos h sin h cos(x l ) sin(x l ) Reconstruction sin(x) = sin h cos(x l ) cos h sin(x l ) x l x, inherently by the range reduction sin h and cos h, which are rounded approximations sin(x l ) and cos(x l ), which are computed by polynomial evaluation H. de Lassus Saint-Geniès Range Reduction Based on Pythagorean Triples for Trigonometric Function Evaluation 6 / 25

20 Sources of error in the classical method Errors appear in: Table lookup x [0, π 4 ] Splitting x into (x h, x l ) x h Polynomial approximations cos h sin h cos(x l ) sin(x l ) Reconstruction sin(x) = sin h cos(x l ) cos h sin(x l ) x l x, inherently by the range reduction sin h and cos h, which are rounded approximations sin(x l ) and cos(x l ), which are computed by polynomial evaluation in the reconstruction process H. de Lassus Saint-Geniès Range Reduction Based on Pythagorean Triples for Trigonometric Function Evaluation 6 / 25

21 Sources of error in the classical method Errors appear in: Table lookup x [0, π 4 ] Splitting x into (x h, x l ) x h Polynomial approximations cos h sin h cos(x l ) sin(x l ) Reconstruction sin(x) = sin h cos(x l ) cos h sin(x l ) x l x, inherently by the range reduction sin h and cos h, which are rounded approximations sin(x l ) and cos(x l ), which are computed by polynomial evaluation in the reconstruction process Drawback: additional bits in computation steps for accuracy purposes H. de Lassus Saint-Geniès Range Reduction Based on Pythagorean Triples for Trigonometric Function Evaluation 6 / 25

22 Reducing the error on tabulated values x [0, π 4 ] Splitting x into (x h, x l ) Gal s accurate table method: Table lookup x h corr Polynomial approximations cos h sin h cos(x l corr) sin(x l corr) Reconstruction sin(x) = sin h cos(x l corr) cos h sin(x l corr) x l add small perturbations to the table inputs corr make sin h and cos h 10 to 21 bits closer to machine numbers H. de Lassus Saint-Geniès Range Reduction Based on Pythagorean Triples for Trigonometric Function Evaluation 7 / 25

24 Key idea: remove the error on tabulated values x [0, π 4 ] Splitting x into (x h, x l ) Our method: Table lookup x h corr x l Polynomial approximations store exactly sin h and cos h on machine numbers cos h sin h cos(x l corr) sin(x l corr) Reconstruction sin(x) = sin h cos(x l corr) cos h sin(x l corr) concentrate the error on a small corrective term corr H. de Lassus Saint-Geniès Range Reduction Based on Pythagorean Triples for Trigonometric Function Evaluation 9 / 25

25 Pythagorean Triples Let TP and TPP be the sets of pythagorean triples and prime pythagorean triples, respectively. TP = {(a, b, c) N 3 a 2 + b 2 = c 2 } TPP = {(a, b, c) TP a, b, c coprime} c a x b H. de Lassus Saint-Geniès Range Reduction Based on Pythagorean Triples for Trigonometric Function Evaluation 10 / 25

26 Pythagorean Triples Let TP and TPP be the sets of pythagorean triples and prime pythagorean triples, respectively. TP = {(a, b, c) N 3 a 2 + b 2 = c 2 } TPP = {(a, b, c) TP a, b, c coprime} For each pythagorean triple (a, b, c): sin(x) = a c x c b a cos(x) = b c H. de Lassus Saint-Geniès Range Reduction Based on Pythagorean Triples for Trigonometric Function Evaluation 10 / 25

27 How to use PPTs for function evaluation? a sin h x x l x h c cos h b sin(x) = (a cos(x l ) b sin(x l )) c H. de Lassus Saint-Geniès Range Reduction Based on Pythagorean Triples for Trigonometric Function Evaluation 11 / 25

28 Primitive Pythagorean triples generation One easy way: ternary-trees with root (3, 4, 5) 3 linear relationships 3 children/node, e.g: , 2 1 2, H. de Lassus Saint-Geniès Range Reduction Based on Pythagorean Triples for Trigonometric Function Evaluation 12 / 25

29 Primitive Pythagorean triples generation One easy way: ternary-trees with root (3, 4, 5) 3 linear relationships 3 children/node, e.g: , 2 1 2, , 4, 5 5, 12, 13 15, 8, 17 21, 20, 29 H. de Lassus Saint-Geniès Range Reduction Based on Pythagorean Triples for Trigonometric Function Evaluation 12 / 25

30 Primitive Pythagorean triples with c H. de Lassus Saint-Geniès Range Reduction Based on Pythagorean Triples for Trigonometric Function Evaluation 13 / 25

31 Primitive Pythagorean triples with c π/ H. de Lassus Saint-Geniès Range Reduction Based on Pythagorean Triples for Trigonometric Function Evaluation 13 / 25

32 Primitive Pythagorean triples with c π/ H. de Lassus Saint-Geniès Range Reduction Based on Pythagorean Triples for Trigonometric Function Evaluation 13 / 25

33 What are good PPTs? a 2 c 2 a c 1 1 c a 3 3 b 3 b 1 b 2 sin(x) = (a i cos(x l ) b i sin(x l )) c i H. de Lassus Saint-Geniès Range Reduction Based on Pythagorean Triples for Trigonometric Function Evaluation 14 / 25

34 What are good PPTs? A 3 k A 2 k A 1 k B 3 B 2 B 1 sin(x) = A i [cos(x l )/k] B i [sin(x l )/k] H. de Lassus Saint-Geniès Range Reduction Based on Pythagorean Triples for Trigonometric Function Evaluation 14 / 25

35 Actual use of primitive Pythagorean triples For each table entry store exactly A = a c k and B = b c k such that A, B N F 64 H. de Lassus Saint-Geniès Range Reduction Based on Pythagorean Triples for Trigonometric Function Evaluation 15 / 25

36 Actual use of primitive Pythagorean triples For each table entry store exactly A = a c k and B = b c k such that A, B N F 64 Then, incorporate 1 k in polynomial coefficients k should be a small least common multiple (lcm) of any combination of one hypotenuse c per table entry H. de Lassus Saint-Geniès Range Reduction Based on Pythagorean Triples for Trigonometric Function Evaluation 15 / 25

37 Actual use of primitive Pythagorean triples For each table entry store exactly A = a c k and B = b c k such that A, B N F 64 Then, incorporate 1 k in polynomial coefficients k should be a small least common multiple (lcm) of any combination of one hypotenuse c per table entry Reconstruction steps: 1 Table-lookup of A, B, corr 2 Polynomial approximations S l sin(x l corr) k 3 Result reconstruction: and C l cos(x l corr) k sin(x) = A C l B S l H. de Lassus Saint-Geniès Range Reduction Based on Pythagorean Triples for Trigonometric Function Evaluation 15 / 25

39 Table generation algorithm Inputs p: table index size H. de Lassus Saint-Geniès Range Reduction Based on Pythagorean Triples for Trigonometric Function Evaluation 17 / 25

40 Table generation algorithm Inputs p: table index size Algorithm steps 1: n 4 2: repeat 3: Generate all PPTs (a, b, c) such that c 2 n. 4: Search for the LCM k among all generated hypotenuses c. 5: n n + 1 6: until such a k is found 7: Build tabulated values (A, B, corr) for every entry. Outputs table with entries of the form table[i] = {A i, B i, corr i } polynomial approximants of sin(x l corr) k and cos(x l corr) k H. de Lassus Saint-Geniès Range Reduction Based on Pythagorean Triples for Trigonometric Function Evaluation 17 / 25

41 Exhaustive search Intel(R) Xeon(R) CPU E GHz (32 cores) 125 GB RAM p k min n Time (s) , , ,676, ,846, e+11 1e+10 1e+09 1e+08 1e+07 1e Triples stored Hypotenuses stored p H. de Lassus Saint-Geniès Range Reduction Based on Pythagorean Triples for Trigonometric Function Evaluation 18 / 25

42 Exhaustive search Intel(R) Xeon(R) CPU E GHz (32 cores) 125 GB RAM p k min n Time (s) , , ,676, ,846, e+11 1e+10 1e+09 1e+08 1e+07 1e Triples stored Hypotenuses stored Tendency for Triples+Hypotenuses p Bottlenecks = searching the lcm and memory Finding k for a 10-bit addressed table is desperate with this exhaustive technique our estimations show that n = log 2 (k min ) 37 H. de Lassus Saint-Geniès Range Reduction Based on Pythagorean Triples for Trigonometric Function Evaluation 18 / 25

43 Heuristic search p k min Prime Factorization Triples 5 130, , ,676, , ,846, ,365,290 Observation: k is always a product of small Pythagorean primes Pythagorean primes less than 70: p = {5, 13, 17, 29, 37, 41, 53, 61} H. de Lassus Saint-Geniès Range Reduction Based on Pythagorean Triples for Trigonometric Function Evaluation 19 / 25

44 Heuristic search p k min Prime Factorization Triples 5 130, , ,676, , ,846, ,365,290 Observation: k is always a product of small Pythagorean primes Pythagorean primes less than 70: p = {5, 13, 17, 29, 37, 41, 53, 61} During the generation, only keep triples with hypotenuse c = i p[i] r i with { r i = 0 or 1 if p[i] 5 r i N if p[i] = 5 H. de Lassus Saint-Geniès Range Reduction Based on Pythagorean Triples for Trigonometric Function Evaluation 19 / 25

45 Heuristic search results Triples stored Hypotenuses stored p k min n Time (s) ,676, ,846, ,061, ,882,250, ,668,520, p Same results as the exhaustive search for 2 p 7 Exponential speedup Exponential memory reduction H. de Lassus Saint-Geniès Range Reduction Based on Pythagorean Triples for Trigonometric Function Evaluation 20 / 25

46 Heuristic gains 1e+07 1e+06 Speedup Memory gain p H. de Lassus Saint-Geniès Range Reduction Based on Pythagorean Triples for Trigonometric Function Evaluation 21 / 25

48 Comparisons with other table-based methods Context Correct rounding in double precision 2-step Ziv strategy: Quick phase accurate to 2 66 Accurate phase accurate to Table index size p = 10 bits Method Quick phase Accurate phase Table size (B) Tang 4 MA + 64 FLOP 6 MA FLOP 38,640 Gal 3 MA + 53 FLOP 9 MA FLOP 57,960 Proposed 3 MA + 53 FLOP 5 MA FLOP 32,200 Table-size 45% lower than Gal s Accurate phase 45% more efficient than Gal s H. de Lassus Saint-Geniès Range Reduction Based on Pythagorean Triples for Trigonometric Function Evaluation 23 / 25

50 Conclusion and Perspectives In short, we propose: A new algorithm for table-based range reductions: removing the error on two tabulated terms concentrating the error on half the terms in the reconstruction A prototype able to precompute tables up to 10 indexing bits Perspectives: Evaluation of the actual accuracy and performance of sine and cosine implementations (single/double/4096 bit precisions) Apply this technique to other functions Integration in a full code-generation chain (MetaLibm) H. de Lassus Saint-Geniès Range Reduction Based on Pythagorean Triples for Trigonometric Function Evaluation 25 / 25

51 Range Reduction Based on Pythagorean Triples for Trigonometric Function Evaluation Conférence en Parallélisme, Architecture et Systèmes 2015 Hugues de Lassus Saint-Geniès David Defour Guillaume Revy DALI, Université de Perpignan Via Domitia LIRMM, Université de Montpellier 2 CNRS UMR 5506 July 1 st, 2015