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Introduction to Computer Arithmetic for Efficient Hardware Implementations Arnaud Tisserand CNRS, Lab-STICC CEA-SPEC Seminar, Nov. 2019 -- Babylonian Arithmetic Use of a positional number system with: primary radix 60 auxiliary radix

  1. Fixed-Point Representations Widely used in DSPs and digital integrated circuits for higher speed, lower silicon area and power consumption compared to floating point 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 N16 or Z16 s 2 0 −1 −2 −3 −4 −5 −6 −7 −8 −9 −10 −11 −12 −13 −14 −15 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1Q15 s −1 −2 −3 −4 −5 −6 −7 −8 −9 −10 −11 −12 −13 −14 −15 2 −16 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 Q16 s 8Q16 7 6 5 4 3 2 1 0 −1 −2 −3 −4 −5 −6 −7 −8 −9 −10 −11 −12 −13 −14 −15 −16 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 s 23 16 8 0 MSB ranks LSB Typical fixed-point formats: 16, 24, 32 and 48 bits Arnaud Tisserand. CNRS – Lab-STICC 7/48

  2. Representation(s) of Numbers and Power Consumption Impact of the representation of numbers: • operator speed • circuit area • useful and useless activity cycle value 2’s complement sign/magnitude t c 2 t sm 0 0 0000000000000000 0 0000000000000000 0 1 1 0000000000000001 1 0000000000000001 1 2 -1 1111111111111111 15 1000000000000001 1 3 8 0000000000001000 15 0000000000001000 3 4 -27 1111111111100101 15 1000000000011011 4 5 27 0000000000011011 15 0000000000011011 1 total 61 10 • sign/magnitude (absolute value): n − 2 A = ( s a a n − 2 . . . a 1 a 0 ) = ( − 1) s a × � a i 2 i i =0 • 2’s complement: n − 2 A = ( a n − 1 a n − 2 . . . a 1 a 0 ) = − a n − 1 2 n − 1 + � a i 2 i i =0 Arnaud Tisserand. CNRS – Lab-STICC 8/48

  3. Floating-Point Representation(s) Radix- β floating-point representation of x : • sign s x , 1-bit encoding: 0 ⇒ x > 0 and 1 ⇒ x < 0 • exponent e x ∈ N on k digits and e min ≤ e x ≤ e max • mantissa m x on n + 1 digits • encoding: x = ( − 1) s x × m x × β e x m x = x 0 . x 1 x 2 x 3 · · · x n x i ∈ { 0 , 1 , . . . , β − 1 } For accuracy purpose, the mantissa must be normalized ( x 0 � = 0) Then m x ∈ [1 , β [ and a specific encoding is required for the number 0 Arnaud Tisserand. CNRS – Lab-STICC 9/48

  4. IEEE-754: basic formats Radix β = 2, the first bit of the normalized mantissa is always a “1” (non-stored implicit bit) number of bits format total sign exponent mantissa double precision 64 1 11 52 + 1 simple precision 32 1 8 23 + 1 double precision single precision 63 56 48 40 32 24 16 8 0 MSB ranks LSB Arnaud Tisserand. CNRS – Lab-STICC 10/48

  5. Basic Cells for Addition Useful circuit element in computer arithmetic: counter A ( m , k )-counter is a cell that counts the number of 1 on its m inputs (result expressed as a k -bit integer) a a a a 0 m−1 m−2 1 ... m − 1 k − 1 � � (m,k) s j 2 j a i = i =0 j =0 ... s s k−1 0 Standard counters: • half-adder or HA is a (2,2)-counter • full-adder or FA is a (3,2)-counter Arnaud Tisserand. CNRS – Lab-STICC 11/48

  6. FA Cell a b d c s Arithmetic equation: a b d 0 0 0 0 0 0 0 1 0 1 2 c + s = a + b + d 0 1 0 0 1 FA 0 1 1 1 0 Logic equation: 1 0 0 0 1 1 0 1 1 0 c s s = a ⊕ b ⊕ d 1 1 0 1 0 1 1 1 1 1 = ab + ad + bd c Articles about FA in IEEE Journals 3 2 There many implementations of #articles the FA cell 1 0 1990 1992 1994 1996 1998 2000 2002 2004 Year Arnaud Tisserand. CNRS – Lab-STICC 12/48

  7. Carry Ripple Adder (CRA) Very simple architecture: n FA cells connected in series a b a b a b a b a b a b 5 5 4 4 3 3 2 2 1 1 0 0 r r r r r r 5 4 3 2 1 0 FA FA FA FA FA FA s 6 s s s s s s 5 4 3 2 1 0 complexity delay O ( n ) area O ( n ) Warning: Sometimes a CRA is also called Carry Propagate Adder (CPA), but CPA also means a non-redundant adder (that propagates) Arnaud Tisserand. CNRS – Lab-STICC 13/48

  8. Useless Activity in a Carry Ripple Adder a b a b a b a b a b a b 5 5 4 4 3 3 2 2 1 1 0 0 Very simple architecture: r r r r r r 5 4 3 2 1 0 FA FA FA FA FA FA n FA cells connected in series s 6 s s s s s s 5 4 3 2 1 0 V cycle i 1 1 0 0 1 1 0 0 1 1 0 0 CLK cycle i+1 1 0 1 0 1 0 1 0 1 0 1 1 t 0 FA FA FA FA FA FA V cycle i 0 1 0 1 0 0 CLK cycle i+1 1 0 1 0 1 0 0 1 0 1 0 0 activity 1 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 stable 0 0 0 0 0 0 t Theoretical models (equiprobable and uniform distribution of inputs): • worst case n 2 / 2 transitions • average 3 n / 2 transitions and only n / 2 useful Arnaud Tisserand. CNRS – Lab-STICC 14/48

  9. Carry-Select Adder Idea: computation of the higher half part for the 2 possible input carries (0 and 1) and selection when the output carry from lower half part is known a H b H a L b L 0 s L lower part 1 1 0 0 1 s n higher part s H Recursive version − → O (log n ) delay but there is a fanout problem. . . Arnaud Tisserand. CNRS – Lab-STICC 15/48

  10. Carry Lookahead Adder: 4-Bit Example c 1 = g 0 + p 0 c 0 c 2 = g 1 + p 1 g 0 + p 1 p 0 c 0 c 3 = g 2 + p 2 g 1 + p 2 p 1 g 0 + p 2 p 1 p 0 c 0 c 4 = g 3 + p 3 g 2 + p 3 p 2 g 1 + p 3 p 2 p 1 g 0 + p 3 p 2 p 1 p 0 c 0 p g p g p g p g 3 3 2 2 1 1 0 0 c 0 c c c c 4 3 2 1 Arnaud Tisserand. CNRS – Lab-STICC 16/48

  11. Parallel-Prefix Addition: Standard Architectures 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 0 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 1 0 2 1 3 2 4 3 5 4 6 5 Brent−Kung 6 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 7 0 8 1 9 2 10 11 3 12 13 14 4 15 Kogge−Stone carry ripple 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 0 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 0 1 1 2 2 3 3 4 4 Sklansky 5 Han−Carlson Arnaud Tisserand. CNRS – Lab-STICC 17/48

  12. Redundant or Constant Time Adders To speed-up the addition, one solution consists in “saving” the carries and using them (this makes sense only in case of multiple additions) In 1961, Avizienis suggested to represent numbers in radix β with digits in {− α, − α + 1 , . . . , 0 , . . . , α − 1 , α } instead of { 0 , 1 , 2 , . . . , β − 1 } with α ≤ β − 1 Using this representation, if 2 α + 1 > β some numbers have several possible representation at the bit level. For instance, the value 2345 (in the standard representation) can be represented in radix 10 with digits in {− 5 , − 4 , − 3 , − 2 , − 1 , 0 , 1 , 2 , 3 , 4 , 5 } by the values 2345, 235(-5) or 24(-5)(-5) Such a representation is said redundant In a redundant number system there is constant-time addition algorithm (without carry propagation) where all computations are done in parallel Arnaud Tisserand. CNRS – Lab-STICC 18/48

  13. Addition Q: How can we speed up addition? x 4 y 4 x 3 y 3 x 2 y 2 x 1 y 1 x 0 y 0 r 0 FA FA FA FA FA s 5 s 4 s 3 s 2 s 1 s 0 Arnaud Tisserand. CNRS – Lab-STICC 19/48

  14. Addition Q: How can we speed up addition? R: Save the carries! x 4 y 4 x 3 y 3 x 2 y 2 x 1 y 1 x 0 y 0 z 4 z 3 z 2 z 1 z 0 r 0 0 FA FA FA FA FA s 5 r 5 s 4 r 4 s 3 r 3 s 2 r 2 s 1 r 1 s 0 r 0 Arnaud Tisserand. CNRS – Lab-STICC 19/48

  15. Addition Q: How can we speed up addition? R: Save the carries! x 4 y 4 x 3 y 3 x 2 y 2 x 1 y 1 x 0 y 0 z 4 z 3 z 2 z 1 z 0 r 0 0 FA FA FA FA FA s 5 r 5 s 4 r 4 s 3 r 3 s 2 r 2 s 1 r 1 s 0 r 0 n � ( s i + r i ) 2 i X + Y + Z = S + R = i =0 The computation time does not depend on n T ( n ) = O (1) Arnaud Tisserand. CNRS – Lab-STICC 19/48

  16. Addition using the carry-save representation Q: How can we speed up addition? R: Save the carries! x 4 y 4 x 3 y 3 x 2 y 2 x 1 y 1 x 0 y 0 z 4 z 3 z 2 z 1 z 0 r 0 0 FA FA FA FA FA s 5 r 5 s 4 r 4 s 3 r 3 s 2 r 2 s 1 r 1 s 0 r 0 w 5 w 4 w 3 w 2 w 1 w 0 n � ( s i + r i ) 2 i X + Y + Z = S + R = i =0 n � w i 2 i = W = w i = s i + r i ∈ { 0 , 1 , 2 } avec i =0 The computation time does not depend on n T ( n ) = O (1) Arnaud Tisserand. CNRS – Lab-STICC 19/48

  17. Addition using the carry-save representation Q: How can we speed up addition? R: Save the carries! x 4 y 4 x 3 y 3 x 2 y 2 x 1 y 1 x 0 y 0 z 4 z 3 z 2 z 1 z 0 r 0 0 FA FA FA FA FA s 5 r 5 s 4 r 4 s 3 r 3 s 2 r 2 s 1 r 1 s 0 r 0 w 5 w 4 w 3 w 2 w 1 w 0 n � ( s i + r i ) 2 i X + Y + Z = S + R = i =0 n � w i 2 i = W = w i = s i + r i ∈ { 0 , 1 , 2 } avec i =0 The computation time does not depend on n T ( n ) = O (1) Arnaud Tisserand. CNRS – Lab-STICC 19/48

  18. Addition using the carry-save representation Q: How can we speed up addition? R: Save the carries! x 4 y 4 x 3 y 3 x 2 y 2 x 1 y 1 x 0 y 0 z 4 z 3 z 2 z 1 z 0 r 0 0 FA FA FA FA FA s 5 r 5 s 4 r 4 s 3 r 3 s 2 r 2 s 1 r 1 s 0 r 0 w 5 w 4 w 3 w 2 w 1 w 0 n � ( s i + r i ) 2 i X + Y + Z = S + R = i =0 n � w i 2 i = W = w i = s i + r i ∈ { 0 , 1 , 2 } avec i =0 � s n � � � r n − 1 · · · s 1 s n − 1 s 0 = w n w n − 1 . . . w 1 w 0 = r n r 1 r 0 cs cs The computation time does not depend on n T ( n ) = O (1) Arnaud Tisserand. CNRS – Lab-STICC 19/48

  19. Addition of 2 Carry-Save Numbers x 4 y 4 x 3 y 3 x 2 y 2 x 1 y 1 x 0 y 0 ◦ • ◦ • ◦ • ◦ • ◦ • ◦ • ◦ • ◦ • ◦ • ◦ • FA FA FA FA FA FA FA FA FA FA 0 0 0 ◦ • ◦ • ◦ • ◦ • ◦ • ◦ • w 5 w 4 w 3 w 2 w 1 w 0 n � x i 2 i = x i = x s , i + x r , i = ◦ + • X avec i =0 n � y i 2 i Y = avec y i = y s , i + y r , i = ◦ + • i =0 n � w i 2 i X+Y = W = w i = w s , i + w r , i = ◦ + • avec i =0 Arnaud Tisserand. CNRS – Lab-STICC 20/48

  20. Carry-Save Trees Example with 3 inputs: A , B and C a b c a b c a b c a b c a b c a b c 5 5 5 4 4 4 3 3 3 2 2 2 1 1 1 0 0 0 FA FA FA FA FA FA 6 5 5 4 4 3 3 2 2 1 1 0 0 0 s 6 s 5 s 4 s 3 s 2 s 1 s 0 Carry-save reduction tree: n ( h ) non-redundant inputs can be reduced by a h -level carry-save tree where n ( h ) = ⌊ 3 n ( h − 1) / 2 ⌋ and n (0) = 2 h 1 2 3 4 5 6 7 8 9 10 11 n ( h ) 3 4 6 9 13 19 28 42 63 94 141 Arnaud Tisserand. CNRS – Lab-STICC 21/48

  21. Fast Multipliers B n bits 1. partial products generation a i b j A (with or without recoding) PP generation → delay in O (1) (fanout a i , b j ֒ n bits O (log n )) a b 2 n bits i j 2. sum of the partial products using a carry-save reduction tree reduction ֒ → delay in O (log n ) 4n bits 3. assimilation of the carries using a P (carry−save) fast adder ֒ → delay in O (log n ) 2n bits P Multiplication delay O (log n ), area O ( n 2 ) Arnaud Tisserand. CNRS – Lab-STICC 22/48

  22. Power Consumption in Fast Multipliers 70 70 67% Relative power consumption [%] 60 60 54% Relative delay [%] 50 50 40 40 31% 30 30 20 20 17% 16% 15% 10 10 0 0 PP gen. reduc. assim. PP gen. reduc assim. power delay • 30% to 70% of redundant transitions (useless) • place and route steps based on the internal arrival time • add a pipeline stage Arnaud Tisserand. CNRS – Lab-STICC 23/48

  23. MAC and FMA MAC: multiply and accumulate P ( t ) = A × B + P ( t − 1) A , B are n -bit values and P a m -bit with m >> n (e.g., 16 × 16 + 40 − → 40 in some DSPs) FMA: fused multiply and add P = A × B + C where A , B , C and P can be stored in different registers (recent general purpose processors, e.g., Itanium) C set clk A B reg generation reduction assimilation P Arnaud Tisserand. CNRS – Lab-STICC 24/48

  24. Squarer a 5 a 4 a 3 a 2 a 1 a 0 a 5 a 4 a 3 a 2 a 1 a 0 a 5 a 0 a 4 a 0 a 3 a 0 a 2 a 0 a 1 a 0 a 0 a 0 a 5 a 1 a 4 a 1 a 3 a 1 a 2 a 1 a 1 a 1 a 0 a 1 a i a i = a i a 5 a 2 a 4 a 2 a 3 a 2 a 2 a 2 a 1 a 2 a 0 a 2 a 5 a 3 a 4 a 3 a 3 a 3 a 2 a 3 a 1 a 3 a 0 a 3 a i a j + a j a i = 2 a i a j a 5 a 4 a 4 a 4 a 3 a 4 a 2 a 4 a 1 a 4 a 0 a 4 a 5 a 5 a 4 a 5 a 3 a 5 a 2 a 5 a 1 a 5 a 0 a 5 a 5 a 4 a 5 a 3 a 5 a 2 a 5 a 1 a 5 a 0 a 4 a 0 a 3 a 0 a 2 a 0 a 1 a 0 a 0 a i a j + a i = 2 a i a j + a i − a i a j a 5 a 4 a 3 a 4 a 2 a 4 a 1 a 3 a 1 a 2 a 1 a 1 = 2 a i a j + a i ( 1 − a j ) a 4 a 3 a 2 a 2 = 2 a i a j + a i a j a 3 a 5 a 4 a 5 a 4 a 5 a 3 a 5 a 2 a 5 a 1 a 5 a 0 a 4 a 0 a 3 a 0 a 2 a 0 a 1 a 0 a 0 15 AND + 5 IAND12 a 4 a 3 a 4 a 3 a 4 a 2 a 4 a 1 a 3 a 1 a 2 a 1 a 1 a 0 3 FA + 2 HA a 3 a 2 a 3 a 2 a 2 a 1 a 5 a 4 a 5 a 4 a 3 a 0 a 2 a 0 a 1 a 0 a 0 1 ADD(9 bits) a 2 a 1 a 1 a 0 Arnaud Tisserand. CNRS – Lab-STICC 25/48

  25. Multiplication by Constants (1/2) Problem: substitute a complete multiplier by an optimized sequence of shifts and additions and/or subtractions Example: p = 111463 × x algo. p = 111463 × x = #op. direct ( x ≪ 16)+( x ≪ 15)+( x ≪ 13)+( x ≪ 12)+( x ≪ 9) 10 ± +( x ≪ 8)+( x ≪ 6)+( x ≪ 5)+( x ≪ 2)+( x ≪ 1)+ x CSD ( x ≪ 17) − ( x ≪ 14) − ( x ≪ 12)+( x ≪ 10) 7 ± − ( x ≪ 7) − ( x ≪ 5)+( x ≪ 3) − x Bernstein ((( t 2 ≪ 2)+ x ) ≪ 3) − x 5 ± where t 1 = ((( x ≪ 3) − x ) ≪ 2) − x t 2 = t 1 ≪ 7+ t 1 Our ( t 2 ≪ 12)+( t 2 ≪ 5)+ t 1 4 ± where t 1 = ( x ≪ 3) − x t 2 = ( t 1 ≪ 2) − x CSD: canonical signed digit, 111463 = 11011001101100111 2 = 100101010010101001 2 Arnaud Tisserand. CNRS – Lab-STICC 26/48

  26. Multiplication by Constants (2/2) FIR (1 , 5 , 5 , 1) x[t] Power savings: 30 up to 60% D D D operator init. [1] [2] our 4 DCT 8b 300 94 73 56 4 DCT 12b 368 100 84 70 y[t] A DCT 16b 521 129 114 89 DCT 24b 789 212 — 119 D D D x[t] Power savings: 10% 4 y[t] B operator init. [1] [2] our 8 × 8 Had. 56 24 — 24 D D D y[t] (16 , 11) R.-M. 61 43 31 31 x[t] 4 (15 , 7) BCH 72 48 47 44 C (24 , 12 , 8) Golay 76 — 47 45 x[t] D D D y[t] Power savings: up to 40% z[t] 4 operator init. [22] our D 8 bits 35 32 24 D D z’[t] 16 bits 72 70 46 x[t] y[t] Parks-McClellan filter 4 D E remez (25 , [0 0 . 2 0 . 25 1] , [1 1 0 0]). Arnaud Tisserand. CNRS – Lab-STICC 27/48

  27. Error and Accuracy Question : how many bits are correct ?  = (1 . 000 000 00) 2 x t theoretical value   x c = (0 . 111 111 11) 2 value in the circuit  = (0 . 000 000 01) 2 = 2 − 8 | x t − x c |  Arnaud Tisserand. CNRS – Lab-STICC 28/48

  28. Error and Accuracy Question : how many bits are correct ?  = (1 . 000 000 00) 2 x t theoretical value   x c = (0 . 111 111 11) 2 value in the circuit  = (0 . 000 000 01) 2 = 2 − 8 | x t − x c |  Error, ǫ : distance between 2 objects (e.g. ǫ = || f ( x ) − p ( x ) || ) Accuracy, µ : (fractional) number of bits required to represent values with an error ≤ ǫ µ = − log 2 | ǫ | Arnaud Tisserand. CNRS – Lab-STICC 28/48

  29. Error and Accuracy Question : how many bits are correct ?  = (1 . 000 000 00) 2 x t theoretical value   x c = (0 . 111 111 11) 2 value in the circuit  = (0 . 000 000 01) 2 = 2 − 8 | x t − x c |  Error, ǫ : distance between 2 objects (e.g. ǫ = || f ( x ) − p ( x ) || ) Accuracy, µ : (fractional) number of bits required to represent values with an error ≤ ǫ µ = − log 2 | ǫ | Notation : µ expressed in terms of correct or significant bits ([cb], [sb]) Arnaud Tisserand. CNRS – Lab-STICC 28/48

  30. Error and Accuracy Question : how many bits are correct ?  = (1 . 000 000 00) 2 x t theoretical value   x c = (0 . 111 111 11) 2 value in the circuit  = (0 . 000 000 01) 2 = 2 − 8 | x t − x c |  Error, ǫ : distance between 2 objects (e.g. ǫ = || f ( x ) − p ( x ) || ) Accuracy, µ : (fractional) number of bits required to represent values with an error ≤ ǫ µ = − log 2 | ǫ | Notation : µ expressed in terms of correct or significant bits ([cb], [sb]) Example : error ǫ = 0 . 0000107 is equivalent to accuracy µ = 16 . 5 sb 12 11 10 9 8 7 6 5 4 3 2 1 µ [sb] ǫ 2 − 12 2 − 11 2 − 10 2 − 9 2 − 8 2 − 7 2 − 6 2 − 5 2 − 4 2 − 3 2 − 2 2 − 1 Arnaud Tisserand. CNRS – Lab-STICC 28/48

  31. Polynomial Approximations x x argument f ( x ) [ a , b ] domain b ′ operator f function f f ( x ) a ′ x a b Arnaud Tisserand. CNRS – Lab-STICC 29/48

  32. Polynomial Approximations x x argument f ( x ) [ a , b ] domain b ′ operator f function f p ( x ) ≈ f ( x ) p p polynomial a ′ x a b ǫ ( x ) ǫ approx. error ǫ ( x ) = f ( x ) − p ( x ) x ǫ Arnaud Tisserand. CNRS – Lab-STICC 29/48

  33. Polynomial Approximations x x argument f ( x ) [ a , b ] domain b ′ operator f function f p p ( x ) ≈ f ( x ) p polynomial a ′ x a b ǫ ( x ) ǫ approx. error ǫ ( x ) = f ( x ) − p ( x ) x ǫ Arnaud Tisserand. CNRS – Lab-STICC 29/48

  34. Polynomial Approximations x x argument f ( x ) [ a , b ] domain b ′ operator f function f p p ( x ) ≈ f ( x ) p polynomial a ′ x a b ǫ ( x ) ǫ approx. error ǫ ( x ) = f ( x ) − p ( x ) x ǫ ( x ) ≤ ǫ target ǫ target maximum ǫ allowed error Arnaud Tisserand. CNRS – Lab-STICC 29/48

  35. Polynomial Approximations x x argument f ( x ) [ a , b ] domain b ′ operator f function f p p ( x ) ≈ f ( x ) p polynomial a ′ x a b ǫ ( x ) ǫ approx. error ǫ ( x ) = f ( x ) − p ( x ) x ǫ ( x ) ≤ ǫ target ǫ target maximum ǫ allowed error Arnaud Tisserand. CNRS – Lab-STICC 29/48

  36. Polynomial Approximations x x argument f ( x ) [ a , b ] domain b ′ operator f function f p p ( x ) ≈ f ( x ) p polynomial a ′ Question : what is the best p ? x a b ǫ ( x ) ǫ approx. error ǫ ( x ) = f ( x ) − p ( x ) x ǫ ( x ) ≤ ǫ target ǫ target maximum ǫ allowed error Arnaud Tisserand. CNRS – Lab-STICC 29/48

  37. Accuracy, Degree and Evaluation Cost Degree- d minimax approximation polynomials to sin( x ) with x ∈ [ a , b ]: µ [sb] 24 [ a , b ] 20 16 12 8 4 d 1 2 3 4 5 π π π 0 2 π 4 2 Arnaud Tisserand. CNRS – Lab-STICC 30/48

  38. Accuracy, Degree and Evaluation Cost Degree- d minimax approximation polynomials to sin( x ) with x ∈ [ a , b ]: µ [sb] 24 [ a , b ] 20 16 12 8 4 d 1 2 3 4 5 π π π 0 2 π 4 2 Arnaud Tisserand. CNRS – Lab-STICC 30/48

  39. Accuracy, Degree and Evaluation Cost Degree- d minimax approximation polynomials to sin( x ) with x ∈ [ a , b ]: µ [sb] 24 [ a , b ] 20 16 12 8 4 d 1 2 3 4 5 π π π 0 2 π 4 2 • higher accuracy = ⇒ higher degree • higher degree = ⇒ more costly evaluation Arnaud Tisserand. CNRS – Lab-STICC 30/48

  40. Accuracy, Degree and Evaluation Cost Degree- d minimax approximation polynomials to sin( x ) with x ∈ [ a , b ]: µ [sb] 24 [ a , b ] 20 16 12 8 4 d 1 2 3 4 5 π π π 0 2 π 4 2 • higher accuracy = ⇒ higher degree • higher degree = ⇒ more costly evaluation Arnaud Tisserand. CNRS – Lab-STICC 30/48

  41. Polynomial Evaluation Schemes scheme computations # ± # × p 0 + p 1 x + p 2 x 2 + p 3 x 3 direct 3 5 � � Horner p 0 + p 1 + ( p 2 + p 3 x ) x x 3 3 p 0 + p 1 x + ( p 2 + p 3 x ) x 2 Estrin 3 4 Trade-off: • direct scheme − → high operation cost and smaller accuracy • Horner scheme − → smallest cost but sequential • Estrin scheme − → some internal parallelism Arnaud Tisserand. CNRS – Lab-STICC 31/48

  42. Polynomial Evaluation Schemes scheme computations # ± # × p 0 + p 1 x + p 2 x 2 + p 3 x 3 direct 3 5 � � Horner p 0 + p 1 + ( p 2 + p 3 x ) x x 3 3 p 0 + p 1 x + ( p 2 + p 3 x ) x 2 Estrin 3 4 Trade-off: • direct scheme − → high operation cost and smaller accuracy • Horner scheme − → smallest cost but sequential • Estrin scheme − → some internal parallelism Question : what is the best evaluation scheme? Arnaud Tisserand. CNRS – Lab-STICC 31/48

  43. Round-off Errors Round-off errors occur during most of computations: • due to the finite accuracy during the computations • small for a single operation (fraction of the LSB) • accumulation of such errors may be a problem in long computation sequences • need for a sufficient datapath width in order to limit round-off errors Examples: 1 / 3 = 0 . 33333333 . . . → 0 . 3333 or 0 . 3334 in 1 Q 10 4 format + × Arnaud Tisserand. CNRS – Lab-STICC 32/48

  44. Round-off Errors Round-off errors occur during most of computations: • due to the finite accuracy during the computations • small for a single operation (fraction of the LSB) • accumulation of such errors may be a problem in long computation sequences • need for a sufficient datapath width in order to limit round-off errors Examples: 1 / 3 = 0 . 33333333 . . . → 0 . 3333 or 0 . 3334 in 1 Q 10 4 format + × Arnaud Tisserand. CNRS – Lab-STICC 32/48

  45. Round-off Errors Round-off errors occur during most of computations: • due to the finite accuracy during the computations • small for a single operation (fraction of the LSB) • accumulation of such errors may be a problem in long computation sequences • need for a sufficient datapath width in order to limit round-off errors Examples: 1 / 3 = 0 . 33333333 . . . → 0 . 3333 or 0 . 3334 in 1 Q 10 4 format + × Question : what is the best datapath width? Arnaud Tisserand. CNRS – Lab-STICC 32/48

  46. Rounding Modes and Correct Rounding Notations: • ⊚ is an operation ± , × , ÷ . . . • ⋄ is the active rounding mode (or quantization mode) IEEE-754: △ ( x ) towards + ∞ (up), ∇ ( x ) towards −∞ (down), Z ( x ) towards 0, N ( x ) towards the nearest R representable values midpoints x finite precision values mathematical values r math = a ⊚ math b r finite = a ⊚ finite b Arnaud Tisserand. CNRS – Lab-STICC 33/48

  47. Rounding Modes and Correct Rounding Notations: • ⊚ is an operation ± , × , ÷ . . . • ⋄ is the active rounding mode (or quantization mode) IEEE-754: △ ( x ) towards + ∞ (up), ∇ ( x ) towards −∞ (down), Z ( x ) towards 0, N ( x ) towards the nearest ∇ ( x ) △ ( x ) R representable values midpoints x finite precision values mathematical values r math = a ⊚ math b r finite = a ⊚ finite b Arnaud Tisserand. CNRS – Lab-STICC 33/48

  48. Rounding Modes and Correct Rounding Notations: • ⊚ is an operation ± , × , ÷ . . . • ⋄ is the active rounding mode (or quantization mode) IEEE-754: △ ( x ) towards + ∞ (up), ∇ ( x ) towards −∞ (down), Z ( x ) towards 0, N ( x ) towards the nearest ∇ ( x ) △ ( x ) R representable values midpoints 0 Z ( x ) x finite precision values mathematical values r math = a ⊚ math b r finite = a ⊚ finite b Arnaud Tisserand. CNRS – Lab-STICC 33/48

  49. Rounding Modes and Correct Rounding Notations: • ⊚ is an operation ± , × , ÷ . . . • ⋄ is the active rounding mode (or quantization mode) IEEE-754: △ ( x ) towards + ∞ (up), ∇ ( x ) towards −∞ (down), Z ( x ) towards 0, N ( x ) towards the nearest ∇ ( x ) △ ( x ) R representable values midpoints 0 Z ( x ) x N ( x ) finite precision values mathematical values r math = a ⊚ math b r finite = a ⊚ finite b Arnaud Tisserand. CNRS – Lab-STICC 33/48

  50. Rounding Modes and Correct Rounding Notations: • ⊚ is an operation ± , × , ÷ . . . • ⋄ is the active rounding mode (or quantization mode) IEEE-754: △ ( x ) towards + ∞ (up), ∇ ( x ) towards −∞ (down), Z ( x ) towards 0, N ( x ) towards the nearest ∇ ( x ) △ ( x ) R representable values midpoints 0 Z ( x ) x N ( x ) finite precision values mathematical values r math = a ⊚ math b r finite = a ⊚ finite b a ⊚ math b � � r finite = ⋄ Arnaud Tisserand. CNRS – Lab-STICC 33/48

  51. Bounding Round-off Errors Problem : it is very difficult to get tight bounds Solutions: • worst case: assume 1 / 2 LSB error for each operation � simple but very pessimistic • qualification: exhaustive or selected simulations � simple but only validated bounds for small systems • specific tools: formal accurate analysis (and proof) � we use gappa developed by Guillaume Melquiond Arnaud Tisserand. CNRS – Lab-STICC 34/48

  52. Gappa Overview • developed by Guillaume Melquiond • goal: formal verification of the correctness of numerical programs: ◮ software and hardware ◮ integer, floating-point and fixed-point arithmetic ( ± , × , ÷ , √ ) • uses multiple-precision interval arithmetic, forward error analysis and expression rewriting to bound mathematical expressions (rounded and exact operators) • generates a theorem and its proof which can be automatically checked using a proof assistant (e.g. Coq or HOL Light) • reports tight error bounds for given expressions in a given domain • C++ code and free software licence (CeCILL ≃ GPL) • publication: ACM Transactions on Mathematical Software, n. 1, vol. 37, 2010, pp: 2:1–20, doi: 10.1145/1644001.1644003 • source code and doc: http://gappa.gforge.inria.fr/ Arnaud Tisserand. CNRS – Lab-STICC 35/48

  53. Gappa Example Degree-2 polynomial approximation to e x over [1 / 2 , 1] and format 1Q9: 1 p0 = 571/512; p1 = 275/512; p2 = 545/512; 2 3 x = f i x e d < − 9,dn > (Mx) ; 4 5 y1 f i x e d < − 9,dn > = p2 ∗ x + p1 ; 6 p f i x e d < − 9,dn > = y1 ∗ x + p0 ; 7 8 Mp = ( p2 ∗ Mx + p1 ) ∗ Mx + p0 ; 9 10 { Mx in [ 0 . 5 , 1 ] / \ | Mp − Mf | in [ 0 , 0 . 0 0 1 3 8 5 ] 11 12 − > | p − Mf | in ? 13 14 } x b y = x 2 y ): Gappa-0.14.0 result ([ a , b ], x { ( ≈ x ) 10 , log 2 x } , Results for Mx in [0.5, 1] and |Mp - Mf| in [0, 0.001385]: |p - Mf| in [0, 193518932894171697b-64 {0.0104907, 2^(-6.57475)}] Arnaud Tisserand. CNRS – Lab-STICC 36/48

  54. Still Pending Questions Question : what is the best (or a good) p ? Question : what is the best (or a good) datapath width? Question : what is the best (or a good) evaluation scheme? Arnaud Tisserand. CNRS – Lab-STICC 37/48

  55. Still Pending Questions Question : what is the best (or a good) p ? mathematical p : minimax approximations implemented p : simple selection of representable coefficients links to other methods and tools Question : what is the best (or a good) datapath width? Question : what is the best (or a good) evaluation scheme? Arnaud Tisserand. CNRS – Lab-STICC 37/48

  56. Still Pending Questions Question : what is the best (or a good) p ? mathematical p : minimax approximations implemented p : simple selection of representable coefficients links to other methods and tools Question : what is the best (or a good) datapath width? basic optimization method better heuristics under development. . . Question : what is the best (or a good) evaluation scheme? Arnaud Tisserand. CNRS – Lab-STICC 37/48

  57. Still Pending Questions Question : what is the best (or a good) p ? mathematical p : minimax approximations implemented p : simple selection of representable coefficients links to other methods and tools Question : what is the best (or a good) datapath width? basic optimization method better heuristics under development. . . Question : what is the best (or a good) evaluation scheme? Horner or specific scheme examples. . . work still in progress. . . Arnaud Tisserand. CNRS – Lab-STICC 37/48

  58. Minimax Polynomial Approximations • approximation error ǫ app = || f − p || ∞ = max a ≤ x ≤ b | f ( x ) − p ( x ) | • minimax polynomial approximation to f over [ a , b ] is p ∗ such that: || f − p ∗ || ∞ = min p ∈P d || f − p || ∞ • P d set of polynomials with real coefficients and degree ≤ d • p ∗ computed using an algorithm from Remez (numerically implemented in Maple, Matlab, sollya. . . ) Problems: • p ∗ coefficients in R = ⇒ conversion to finite precision • during p ∗ evaluation, some round-off errors add up to ǫ app Arnaud Tisserand. CNRS – Lab-STICC 38/48

  59. Example f ( x ) = 2 x and x ∈ [0 , 1] f ( x ) 2 x 2 d µ [sb] ǫ app 4 . 31 × 10 − 2 1 4 . 53 2 . 48 × 10 − 3 2 8 . 65 1 . 08 × 10 − 4 3 13 . 18 3 . 71 × 10 − 6 4 18 . 04 1 . 07 × 10 − 7 5 23 . 15 x 1 0 1 p ∗ ? Arnaud Tisserand. CNRS – Lab-STICC 39/48

  60. Example f ( x ) = 2 x and x ∈ [0 , 1] f ( x ) 2 x 2 d µ [sb] ǫ app 4 . 31 × 10 − 2 1 4 . 53 2 . 48 × 10 − 3 2 8 . 65 1 . 08 × 10 − 4 3 13 . 18 3 . 71 × 10 − 6 4 18 . 04 1 . 07 × 10 − 7 5 23 . 15 x 1 0 1 p ∗ = 0 . 956964333 + 1 . 000000000 × Arnaud Tisserand. CNRS – Lab-STICC 39/48

  61. Example f ( x ) = 2 x and x ∈ [0 , 1] f ( x ) 2 x 2 d µ [sb] ǫ app 4 . 31 × 10 − 2 1 4 . 53 2 . 48 × 10 − 3 2 8 . 65 1 . 08 × 10 − 4 3 13 . 18 3 . 71 × 10 − 6 4 18 . 04 1 . 07 × 10 − 7 5 23 . 15 x 1 0 1 p ∗ = 1 . 002476056 + x × (0 . 651046780 + x × 0 . 344001106) Arnaud Tisserand. CNRS – Lab-STICC 39/48

  62. Example f ( x ) = 2 x and x ∈ [0 , 1] f ( x ) 2 x 2 d µ [sb] ǫ app 4 . 31 × 10 − 2 1 4 . 53 2 . 48 × 10 − 3 2 8 . 65 1 . 08 × 10 − 4 3 13 . 18 3 . 71 × 10 − 6 4 18 . 04 1 . 07 × 10 − 7 5 23 . 15 x 1 0 1 p ∗ = 0 . 999892965 + x × (0 . 696457394 + x × (0 . 224338364 + x × 0 . 079204240)) Arnaud Tisserand. CNRS – Lab-STICC 39/48

  63. Example f ( x ) = 2 x and x ∈ [0 , 1] f ( x ) 2 x 2 d µ [sb] ǫ app 4 . 31 × 10 − 2 1 4 . 53 2 . 48 × 10 − 3 2 8 . 65 1 . 08 × 10 − 4 3 13 . 18 3 . 71 × 10 − 6 4 18 . 04 1 . 07 × 10 − 7 5 23 . 15 x 1 0 1 p ∗ = 1 . 000003704 + x × (0 . 692966122 + x × (0 . 241638445 + x × (0 . 051690358 + x × 0 . 013697664))) Arnaud Tisserand. CNRS – Lab-STICC 39/48

  64. Finite Precision Coefficients Selection Problem Example: f ( x ) = e x over [1 / 2 , 1] with d = 2, the remez function from sollya gives: p ∗ = 1 . 116019297 . . . + 0 . 535470348 . . . × x + 1 . 065407185 . . . × x 2 Arnaud Tisserand. CNRS – Lab-STICC 40/48

  65. Finite Precision Coefficients Selection Problem Example: f ( x ) = e x over [1 / 2 , 1] with d = 2, the remez function from sollya gives: p ∗ = 1 . 116019297 . . . + 0 . 535470348 . . . × x + 1 . 065407185 . . . × x 2 Question : what are “good” representable values for p 0 , p 1 and p 2 ? Problem : p ∗ is the best theoretical approximation to f (i.e. p i ∈ R ) Need : find good approximations with “machine-representable” coefficients Arnaud Tisserand. CNRS – Lab-STICC 40/48

  66. Finite Precision Coefficients Selection Problem Example: f ( x ) = e x over [1 / 2 , 1] with d = 2, the remez function from sollya gives: p ∗ = 1 . 116019297 . . . + 0 . 535470348 . . . × x + 1 . 065407185 . . . × x 2 Question : what are “good” representable values for p 0 , p 1 and p 2 ? Problem : p ∗ is the best theoretical approximation to f (i.e. p i ∈ R ) Need : find good approximations with “machine-representable” coefficients Above example with 1Q9 format (all values for domain [1 / 2 , 1]): � • ǫ app = || f − p ∗ || ∞ ≃ 1 . 385 × 10 − 3 ≃ 9 . 4 sb � 571 512 + 137 256 x + 545 512 x 2 • 8 . 1 sb ( ∀ i use N ( p i )) Arnaud Tisserand. CNRS – Lab-STICC 40/48

  67. Finite Precision Coefficients Selection Problem Example: f ( x ) = e x over [1 / 2 , 1] with d = 2, the remez function from sollya gives: p ∗ = 1 . 116019297 . . . + 0 . 535470348 . . . × x + 1 . 065407185 . . . × x 2 Question : what are “good” representable values for p 0 , p 1 and p 2 ? Problem : p ∗ is the best theoretical approximation to f (i.e. p i ∈ R ) Need : find good approximations with “machine-representable” coefficients Above example with 1Q9 format (all values for domain [1 / 2 , 1]): � • ǫ app = || f − p ∗ || ∞ ≃ 1 . 385 × 10 − 3 ≃ 9 . 4 sb � 571 512 + 137 256 x + 545 512 x 2 • 8 . 1 sb ( ∀ i use N ( p i )) � • 571 512 + 275 512 x + 545 512 x 2 9 . 3 sb (best selection) Arnaud Tisserand. CNRS – Lab-STICC 40/48

  68. Basic Coefficient Selection Method Idea: search among all the rounding modes for all the p ∗ i • round up p i = △ ( p ∗ i ), round down p i = ▽ ( p ∗ i ) ⇒ total of 2 d +1 values (but d is small) • 2 values per coeff. = • for each polynomial p evaluate ǫ app = || f − p || ∞ , then select polynomial(s) with the smallest ǫ app height = d + 1 ▽ ( p 0 ) △ ( p 0 ) ▽ ( p 1 ) △ ( p 1 ) ▽ ( p 1 ) △ ( p 1 ) ▽ ( p 2 ) △ ( p 2 ) ▽ ( p 2 ) △ ( p 2 ) ▽ ( p 2 ) △ ( p 2 ) ▽ ( p 2 ) △ ( p 2 ) i =0 p i x i where all p i are representable in target format Result: p ( x ) = � d Arnaud Tisserand. CNRS – Lab-STICC 41/48

  69. Basic Coefficient Selection Method Idea: search among all the rounding modes for all the p ∗ i • round up p i = △ ( p ∗ i ), round down p i = ▽ ( p ∗ i ) ⇒ total of 2 d +1 values (but d is small) • 2 values per coeff. = • for each polynomial p evaluate ǫ app = || f − p || ∞ , then select polynomial(s) with the smallest ǫ app height = d + 1 ▽ ( p 0 ) △ ( p 0 ) ▽ ( p 1 ) △ ( p 1 ) ▽ ( p 1 ) △ ( p 1 ) ▽ ( p 2 ) △ ( p 2 ) ▽ ( p 2 ) △ ( p 2 ) ▽ ( p 2 ) △ ( p 2 ) ▽ ( p 2 ) △ ( p 2 ) ǫ app i =0 p i x i where all p i are representable in target format Result: p ( x ) = � d Arnaud Tisserand. CNRS – Lab-STICC 41/48

  70. Basic Coefficient Selection Method Idea: search among all the rounding modes for all the p ∗ i • round up p i = △ ( p ∗ i ), round down p i = ▽ ( p ∗ i ) ⇒ total of 2 d +1 values (but d is small) • 2 values per coeff. = • for each polynomial p evaluate ǫ app = || f − p || ∞ , then select polynomial(s) with the smallest ǫ app height = d + 1 ▽ ( p 0 ) △ ( p 0 ) ▽ ( p 1 ) △ ( p 1 ) ▽ ( p 1 ) △ ( p 1 ) ▽ ( p 2 ) △ ( p 2 ) ▽ ( p 2 ) △ ( p 2 ) ▽ ( p 2 ) △ ( p 2 ) ▽ ( p 2 ) △ ( p 2 ) ǫ app i =0 p i x i where all p i are representable in target format Result: p ( x ) = � d Arnaud Tisserand. CNRS – Lab-STICC 41/48

  71. Example for f ( x ) = 2 x , x ∈ [0 , 1] and d = 4 � ǫ app ( p ∗ ) 18 . 04 sb ǫ app [sb] ǫ app ( p ) ǫ app ( p ) p p 20 ( ▽ , ▽ , ▽ , ▽ , ▽ ) 12.00 ( ▽ , ▽ , ▽ , ▽ , △ ) 13.00 d = 4 18 ( ▽ , ▽ , ▽ , △ , ▽ ) 13.00 ( ▽ , ▽ , ▽ , △ , △ ) 14.03 ( ▽ , ▽ , △ , ▽ , ▽ ) 13.00 ( ▽ , ▽ , △ , ▽ , △ ) 14.55 16 ( ▽ , ▽ , △ , △ , ▽ ) 14.99 ( ▽ , ▽ , △ , △ , △ ) 13.00 ( ▽ , △ , ▽ , ▽ , ▽ ) 13.00 ( ▽ , △ , ▽ , ▽ , △ ) 16.13 14 ( ▽ , △ , ▽ , △ , ▽ ) 17.12 ( ▽ , △ , ▽ , △ , △ ) 13.00 ( ▽ , △ , △ , ▽ , ▽ ) 15.71 ( ▽ , △ , △ , ▽ , △ ) 13.00 12 ( ▽ , △ , △ , △ , ▽ ) 13.00 ( ▽ , △ , △ , △ , △ ) 12.00 ( △ , ▽ , ▽ , ▽ , ▽ ) 13.00 ( △ , ▽ , ▽ , ▽ , △ ) 13.00 10 ( △ , ▽ , ▽ , △ , ▽ ) 13.00 ( △ , ▽ , ▽ , △ , △ ) 13.00 8 ( △ , ▽ , △ , ▽ , ▽ ) 13.00 ( △ , ▽ , △ , ▽ , △ ) 13.00 ( △ , ▽ , △ , △ , ▽ ) 12.99 ( △ , ▽ , △ , △ , △ ) 12.00 6 ( △ , △ , ▽ , ▽ , ▽ ) 12.99 ( △ , △ , ▽ , ▽ , △ ) 12.98 ( △ , △ , ▽ , △ , ▽ ) 12.91 ( △ , △ , ▽ , △ , △ ) 12.00 4 ( △ , △ , △ , ▽ , ▽ ) 12.79 ( △ , △ , △ , ▽ , △ ) 12.00 ( △ , △ , △ , △ , ▽ ) 12.00 ( △ , △ , △ , △ , △ ) 11.41 2 p represented by ( p 0 , p 1 , p 2 , p 3 , p 4 ) 0 Arnaud Tisserand. CNRS – Lab-STICC 42/48

  72. Example for f ( x ) = 2 x , x ∈ [0 , 1] and d = 4 � ǫ app ( p ∗ ) 18 . 04 sb ǫ app [sb] ǫ app ( p ) ǫ app ( p ) p p 20 ( ▽ , ▽ , ▽ , ▽ , ▽ ) 12.00 ( ▽ , ▽ , ▽ , ▽ , △ ) 13.00 d = 4 18 ( ▽ , ▽ , ▽ , △ , ▽ ) 13.00 ( ▽ , ▽ , ▽ , △ , △ ) 14.03 ( ▽ , ▽ , △ , ▽ , ▽ ) 13.00 ( ▽ , ▽ , △ , ▽ , △ ) 14.55 16 ( ▽ , ▽ , △ , △ , ▽ ) 14.99 ( ▽ , ▽ , △ , △ , △ ) 13.00 ( ▽ , △ , ▽ , ▽ , ▽ ) 13.00 ( ▽ , △ , ▽ , ▽ , △ ) 16.13 14 ( ▽ , △ , ▽ , △ , ▽ ) 17.12 ( ▽ , △ , ▽ , △ , △ ) 13.00 ( ▽ , △ , △ , ▽ , ▽ ) 15.71 ( ▽ , △ , △ , ▽ , △ ) 13.00 12 ( ▽ , △ , △ , △ , ▽ ) 13.00 ( ▽ , △ , △ , △ , △ ) 12.00 ( △ , ▽ , ▽ , ▽ , ▽ ) 13.00 ( △ , ▽ , ▽ , ▽ , △ ) 13.00 10 ( △ , ▽ , ▽ , △ , ▽ ) 13.00 ( △ , ▽ , ▽ , △ , △ ) 13.00 8 ( △ , ▽ , △ , ▽ , ▽ ) 13.00 ( △ , ▽ , △ , ▽ , △ ) 13.00 ( △ , ▽ , △ , △ , ▽ ) 12.99 ( △ , ▽ , △ , △ , △ ) 12.00 6 ( △ , △ , ▽ , ▽ , ▽ ) 12.99 ( △ , △ , ▽ , ▽ , △ ) 12.98 ( △ , △ , ▽ , △ , ▽ ) 12.91 ( △ , △ , ▽ , △ , △ ) 12.00 4 ( △ , △ , △ , ▽ , ▽ ) 12.79 ( △ , △ , △ , ▽ , △ ) 12.00 ( △ , △ , △ , △ , ▽ ) 12.00 ( △ , △ , △ , △ , △ ) 11.41 2 p represented by ( p 0 , p 1 , p 2 , p 3 , p 4 ) 0 Arnaud Tisserand. CNRS – Lab-STICC 42/48

  73. Example for f ( x ) = 2 x , x ∈ [0 , 1] and d = 4 � ǫ app ( p ∗ ) 18 . 04 sb ǫ app [sb] ǫ app ( p ) ǫ app ( p ) p p 20 ( ▽ , ▽ , ▽ , ▽ , ▽ ) 12.00 ( ▽ , ▽ , ▽ , ▽ , △ ) 13.00 d = 4 18 ( ▽ , ▽ , ▽ , △ , ▽ ) 13.00 ( ▽ , ▽ , ▽ , △ , △ ) 14.03 ( ▽ , ▽ , △ , ▽ , ▽ ) 13.00 ( ▽ , ▽ , △ , ▽ , △ ) 14.55 16 ( ▽ , ▽ , △ , △ , ▽ ) 14.99 ( ▽ , ▽ , △ , △ , △ ) 13.00 ( ▽ , △ , ▽ , ▽ , ▽ ) 13.00 ( ▽ , △ , ▽ , ▽ , △ ) 16.13 14 d = 3 ( ▽ , △ , ▽ , △ , ▽ ) 17.12 ( ▽ , △ , ▽ , △ , △ ) 13.00 ( ▽ , △ , △ , ▽ , ▽ ) 15.71 ( ▽ , △ , △ , ▽ , △ ) 13.00 12 ( ▽ , △ , △ , △ , ▽ ) 13.00 ( ▽ , △ , △ , △ , △ ) 12.00 ( △ , ▽ , ▽ , ▽ , ▽ ) 13.00 ( △ , ▽ , ▽ , ▽ , △ ) 13.00 10 d = 2 ( △ , ▽ , ▽ , △ , ▽ ) 13.00 ( △ , ▽ , ▽ , △ , △ ) 13.00 8 ( △ , ▽ , △ , ▽ , ▽ ) 13.00 ( △ , ▽ , △ , ▽ , △ ) 13.00 ( △ , ▽ , △ , △ , ▽ ) 12.99 ( △ , ▽ , △ , △ , △ ) 12.00 6 ( △ , △ , ▽ , ▽ , ▽ ) 12.99 ( △ , △ , ▽ , ▽ , △ ) 12.98 d = 1 ( △ , △ , ▽ , △ , ▽ ) 12.91 ( △ , △ , ▽ , △ , △ ) 12.00 4 ( △ , △ , △ , ▽ , ▽ ) 12.79 ( △ , △ , △ , ▽ , △ ) 12.00 ( △ , △ , △ , △ , ▽ ) 12.00 ( △ , △ , △ , △ , △ ) 11.41 2 p represented by ( p 0 , p 1 , p 2 , p 3 , p 4 ) 0 Arnaud Tisserand. CNRS – Lab-STICC 42/48

  74. Example: 2 x over [0 , 1] and µ ≤ 12 sb (1/2) Let us try with d = 3 (max. theoretical accuracy 13 . 18 sb): p ∗ ( x ) = 0 . 999892965 + 0 . 696457394 x + 0 . 224338364 x 2 + 0 . 079204240 x 3 Coefficients (fractional part) size selection: 12 13 14 15 16 l ǫ app 12 . 38 12 . 45 13 . 00 13 . 00 13 . 02 # polynomials 0 0 2 2 7 Coefficients selection: for n = k + l = 1 + 14 bits, we get: ( ▽ , ▽ , ▽ , ▽ ) 11.41 ( ▽ , ▽ , ▽ , △ ) 12.00 ( ▽ , ▽ , △ , ▽ ) 12.00 ( ▽ , ▽ , △ , △ ) 12.84 ( ▽ , △ , ▽ , ▽ ) 12.00 ( ▽ , △ , ▽ , △ ) 13.00 ( ▽ , △ , △ , ▽ ) 13.00 ( ▽ , △ , △ , △ ) 12.36 ( △ , ▽ , ▽ , ▽ ) 12.00 ( △ , ▽ , ▽ , △ ) 12.25 ( △ , ▽ , △ , ▽ ) 12.23 ( △ , ▽ , △ , △ ) 12.23 ( △ , △ , ▽ , ▽ ) 12.13 ( △ , △ , ▽ , △ ) 12.12 ( △ , △ , △ , ▽ ) 12.05 ( △ , △ , △ , △ ) 11.64 Arnaud Tisserand. CNRS – Lab-STICC 43/48

  75. Example: 2 x over [0 , 1] and µ ≤ 12 sb (2/2) Datapath size selection: n ′ 14 15 16 17 18 19 20 ǫ eval direct 11 . 24 11 . 86 12 . 32 12 . 62 12 . 79 12 . 89 12 . 94 ǫ eval Horner 11 . 32 11 . 93 12 . 36 12 . 65 12 . 81 12 . 90 12 . 95 Solution: d = 3, n = k + l = 1 + 14 and n ′ = 16 Implementation results: solution area period #cycles latency power wo. tools 1 . 00 1 . 00 4 1 . 00 1 . 00 w. tools 0 . 83 0 . 82 3 0 . 61 0 . 68 Arnaud Tisserand. CNRS – Lab-STICC 44/48

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