A CAD Tool for Scalable Floating Point Adder Design and Generation Using C++/VHDL

1. A CAD Tool for Scalable Floating Point Adder Design and GenerationUsing C++/VHDL By Asim J. Al-Khalili AICCSA’06 Sharja

2. Overview • Introduction to Floating point Addition • Architecture of Single Path FADD • Activity Scaling • Triple Data Path Floating Point Adder • VHDL Modeling • Results • Implementation AICCSA’06 Sharja

3. FP Representation FP Representation --1.XXXXX2 * 2YYYY (IEEE 754 floating-point standard, single precision) AICCSA’06 Sharja

4. Start Floating point Addition 1. Compare the exponents of the two numbers. 2. Shift the smaller number to the right until its exponent would match the larger exponent 3. Add the significand 4. Normalize the sum, either shifting right and incrementing the exponent or shifting left and decrementing the exponent Overflow/Underflow Yes Exceptions No 5. Round the significand to the appropriate number No Still Normalized ? Yes Done AICCSA’06 Sharja

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6. What’s the best architecture? Architecture Consideration AICCSA’06 Sharja

7. FP Adder • Function include--- • Sign identification • Exponent comparison • Smaller significand right shift • Significand comparison ( If exp. are equal) • Significand inverter • Addition and Leading Zero anticipation • Normalization shifting left • Rounding • Shift after rounding • Compensation shifting • Exception handler AICCSA’06 Sharja

8. Architecture of TDPFADD AICCSA’06 Sharja

9. Transition activity scaling State Active data path State assertion criterion Activity scaled blocks I Bypass Either exponent is zero or emax +1 or edif > p Entire TDPFADD except Bypass data path and Exponent, Control, and Result Int. Flag units J LZA No Bypass and subtraction and edif 1 (LZsp) Pre-alignment barrel shifter (large) K LZB No Bypass and addition or edif > 1 (LZs 1) LZA logic and normalization barrel shifter (large) •State assertion conditions of TDPFADD AICCSA’06 Sharja

10. Probabilities of the Paths With IEEE single precision floating point data format, the probability that the FADD is in states A, B or C is given by P(A) = 0.8177, P(B) = 0.1765 and P(C) = 0.0058. Here, it is assumed that the exponents are independent, uniformly distributed random variables and the events of addition and subtraction are equally likely. With IEEE double precision floating point format P(A) = 0.9484, P(B) = 0.0509 and P(C) = 7*10-4. The time averaged power consumption (expected value) of a transition activity scaled FADD whose operational states are represented by Fig. 2 is given by Power=P(A)* PA + P(B) PB * + P(C ) * PC where PA, PB and PC represent the time averaged power consumption of the FADD in states A, B and C respectively. AICCSA’06 Sharja

11. Pipelined TDPFADD AICCSA’06 Sharja

12. Architecture Consideration Straightforward IEEE Floating-point addition algorithm 1. Exponent subtraction. 2. Alignment. 3. Significand addition. 4. Conversion. 5. Leading-one detection. 6. Normalization. 7. Rounding. 1 2 3 5 4 6 7 Advantages: 1. Positive result, Eliminate Complement 2. Comparison // Alignment 3. Full Normal // Rounding AICCSA’06 Sharja

13. How can a compound adder compute fastest? Compound Adder AICCSA’06 Sharja

14. Compound Adder The Compound adder computes simultaneously the sum and the sum plus one, and then the correct rounded result is obtained by selecting according to the requirements of the rounding. AICCSA’06 Sharja

15. Architecture Consideration Cont. The latency of the floating-point addition Can be improved if the rounding is combined with the addition/subtraction. (Compare to signal path) Reduce latency FAR data-path: --No Conversion --No Full normalization --No LOP CLOSE data-path: --No Full Alignment Reduce total path delay --eliminate Comparator Increase area --two 2’s COMP ADDER AICCSA’06 Sharja

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17. . CComparison of low latency architectures of TDPFADD and single data path FADD using 0.13 micron CMOS technology Parameters TDPFADD Single data path FADD Maximum Delay, D (ns) 13.62 19.54 Average Power, Pa (mW) at 16.7 MHz 2.95 15.72 Worst case Power, Pw (mW) at 16.7 MHz 4.21 5.13 Power using real data, Preal (mW) at 16.7 MHz 3.41 4.58 Area, A (104 cell-area) 3.62 2.24 Power-Delay Product, PD (ns.mW) 40.18 307.16 Area-Power Product, AP (104cell-area.mW) 10.68 35.21 Area-Delay Product, AT (104cell-area.ns) 49.30 43.76 Area-Delay2 Product, AT2 (104cell-area.ns2) 671.5 855.2 AICCSA’06 Sharja

18. Parameters TDPFADD Single data path FADD Maximum Delay, D (ns) 71.27 109.21 Average Power, Pa (W) at 2.38 MHz 0.113 0.204 Worst case Power, Pw (W) at 2.38 MHz 0.196 0.205 Power using real data, Preal (W) at 2.38 MHz 0.138 0.183 Area, A, Total CLBs (#) 115 73.7 Power-Delay Product, PD (ns.10mW) 8.85 22.27 Area-Power Product, AP (10#.10mW) 12.99 15.03 Area-Delay Product, AT (10#.ns) 8196 8048 Area-Delay2 Product, AT2 (10#.ns2) 58.41 x 104 87.90x 104 •Comparison of low latency architectures of TDPFADD and single data path FADD using FPGA technology AICCSA’06 Sharja

19. •Comparison of pipelined architectures of TDPFADD and single data path FADD using 0.13 micron CMOS technology Parameters TDPFADD Single data path FADD Maximum Delay, D (ns) 5.78 6.35 Average Power, Pa (mW) at 50 MHz 3.87 6.00 Worst case Power, Pw (mW) at 50 MHz 4.51 5.71 Power using real data, Preal (mW) at 50 MHz 3.94 5.50 Area, A (104 cell-area) 5.46 4.44 Power-Delay Product, PD (ns.mW) 22.36 38.1 Area-Power Product, AP (104cell-area.mW) 21.13 26.64 Area-Delay Product, AT (104cell-area.ns) 31.55 28.19 Area-Delay2 Product, AT2 (104cell-area.ns2) 182.40 179.03 AICCSA’06 Sharja

20. •Comparison of pipelined structures of TDPFADD and single data path FADD using FPGA technology Parameters TDPFADD Single data path FADD Maximum Delay, D (ns) 33.70 45.08 Average Power, Pa (W) at 5 MHz 0.089 0.111 Worst case Power, Pw (W) at 5 MHz 0.1130 0.1197 Power using real data, Preal (W) at 5 MHz 0.096 0.1141 Area, A, Total CLBs (#) 147.11 104.66 Power-Delay Product, PD (ns.10mW) 2.999 5.01 11.61 Area-Power Product, AP (10#.10mW) 13.09 Area-Delay Product, AT (10#.ns) 4957.60 4718.07 Area-Delay2 Product, AT2 (10#.ns2) 1.67 x 104 21.26 x 104 AICCSA’06 Sharja

21. VHDL Modeling Design Idea : 1. The length and depth parameters needed by some components are defined in package pkg.vhd 2. The parameters of pkg.vhd are created by C/C++ program with user defined Exponent and Significand length 3. VHDL components and created pkg.vhd together generate FP Adder AICCSA’06 Sharja

22. VHDL Generation Get Parameter Length from user C++ program Calculate needed parameters Structural VHDL code of the floating point adder Package Pkg.vhd VHDL code Synthesize floating point adder hardware AICCSA’06 Sharja

23. Calculating the Parameters Using C/C++ AICCSA’06 Sharja

24. Implementation Example 1 Input: Exponent Length = 8 Significand Length = 23 AICCSA’06 Sharja

25. Generated package pkg.vhd : library ieee; use ieee.std_logic_1164.all; package pkg is constant Exponent_Length : positive :=8; constant Significand_Length : positive :=23; constant HideSig_Length : positive :=27; constant HideSig_Depth : positive :=5; constant LZA_Length : positive :=28; constant LZA_Depth : positive :=5; constant LZA_P2_Length : positive:=32; end pkg; AICCSA’06 Sharja

26. The synthesized FP Adder AICCSA’06 Sharja

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29. Simulation and Test Result AICCSA’06 Sharja

30. Implementation Example 2 Input: Exponent Length = 4 Significand Length = 11 AICCSA’06 Sharja

31. Generated package pkg.vhd : library ieee; use ieee.std_logic_1164.all; package pkg is constant Exponent_Length : positive :=4; constant Significand_Length : positive :=11; constant HideSig_Length : positive :=15; constant HideSig_Depth : positive :=4; constant LZA_Length : positive :=16; constant LZA_Depth : positive :=4; constant LZA_P2_Length : positive:=16; end pkg; AICCSA’06 Sharja

32. The synthesized FP Adder AICCSA’06 Sharja

33. The Synthesized FADD AICCSA’06 Sharja

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35. Conclusion • A scalable-length FP adder is generated • The length of the adder is given by the user through C/C++ • The objective function is also stated • A structural mode FP adder is modeled by VHDL • The adder is Synthesizable • Depending on Power-Area-Delay requirement a • Simple/TDPADD/Pipelined/PTDOADD is generated • The adder can also be pipelined AICCSA’06 Sharja

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37. VHDL Modeling 1. Package for Length and Depth Parameters 2. Components of the FP Adder 3. Top Configuration of the FP Adder AICCSA’06 Sharja

38. 1. Package for Length and Depth Parameters Input parameters : Significand length Exponent length Output parameters: significand length for calculation significand length for shifting significand depth for shifting Exponent length AICCSA’06 Sharja

39. Exponent Difference Calculates the difference of the two exponents. AICCSA’06 Sharja

40. Significand Comparison AICCSA’06 Sharja

41. Equation for Comparison A>B if (an>bn) OR (an=bn) AND an-1>bn-1) OR (an=bn AND an-1=bn-1 AND an-2>bn-2) OR… A>B if an=bn AND an-1=bn-1 AND an-2=bn-2 … A<B if (an<bn) OR (an=bn) AND an-1<bn-1) OR (an=bn AND an-1=bn-1 AND an-2<bn-2) OR… AICCSA’06 Sharja

42. Right Shifter and GRS-bit Generation AICCSA’06 Sharja

43. Right Shifter and GRS-bit Generation Right Shift with variable length AICCSA’06 Sharja

44. Manchester Adder/Subtractor AICCSA’06 Sharja

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46. Leading Zero Anticipation Logic Might one bit anticipate error AICCSA’06 Sharja

47. Leading Zero Counter AICCSA’06 Sharja

48. (left barrel shifter) • Normalization Shifter AICCSA’06 Sharja

49. Rounding Logic =G(M0+R+S) AICCSA’06 Sharja

50. A Half Full Adder AICCSA’06 Sharja