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Introduction. Linus Svensson D4, linsve-7@student.luth.se Åke Östmark D4, akeost-6@student.luth.se. Why we are here. Project called Design and Implementation of a 32-bit FPU in VHDL Complete project presentation fpu.varulv.net. Today's topics. Floating-Point Unit (FPU) Design of our FPU
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Introduction • Linus SvenssonD4, linsve-7@student.luth.se • Åke ÖstmarkD4, akeost-6@student.luth.se
Why we are here • Project called Design and Implementation of a 32-bit FPU in VHDL • Complete project presentation fpu.varulv.net
Today's topics • Floating-Point Unit (FPU) • Design of our FPU • Multiplication, an example of an algorithm • Normalisation and rounding • Implementation in an HDL • Testing of the implementation • Final result • Life after SMD082 • Questions
Floating-Point Unit (FPU) • A unit providing floating-point processing capabilities • Konrad Zuse had a proposal for building a computer in Germany in 1939 • First commercial machine with floating-point hardware was the IBM 704 in 1955 • Professor W. Kahan-Twenty years ago anarchy threatened floating-point arithmetic. • Forming of the IEEE floating-point committee in 1977 • The first chip manufactured in complains with the IEEE standard 754 were the Intel 8087 Math Coprocessor • The Intel486 DX processor for the first time integrated the CPU and the FPU architectures on one chip
Why an FPU LAB3 SMD082 Multiplication … ... bgezal $r0 posbrk #run posbrk on "a" nop beq $v1 $r0 fmulzero #if a==0 or $s0 $r0 $v0 #s0:=a's sign bit or $a0 $r0 $v1 #a0:=a's fraction or $a1 $r0 $a3 #a1:=a's exponent bgezal $r0 prenrm #run prenrm on "a" nop or $s1 $r0 $v0 #s1:=a's fraction or $s2 $r0 $v1 #s2:=a's exponent lw $a0 -8($fp) #fetch "b" bgezal $r0 posbrk #run posbrk on "b" nop … ... LAB3 SMD082 with an FPU Multiplication fmul $v0 $a1 $a2 #Done...
Design of our FPU • We started on a high level of abstraction • Defined I/O
Design of our FPU • We divided the design into smaller blocks …
Design of our FPU • …until we had a low level of abstraction
Design of our FPU • Picked an existing algorithm, or • modified an existing algorithm, or • designed a new algorithm
Start 1 B(i)=1 0 RES:=RES+A 1 RES:=RES>>1 i:=i+1 0 i<3 Done Multiplication • A = 1.11•20 = 1.75 • B = 1.10•2-1 = 0.75 • Iter 1 • B(0) == 0 • RES := RES>>1 • RES = 0000000
Start 1 B(i)=1 0 RES:=RES+A 1 RES:=RES>>1 i:=i+1 0 i<3 Done Multiplication • A = 1.11•20 = 1.75 • B = 1.10•2-1 = 0.75 • Iter 2 • B(1) == 1 • RES := RES + A • RES = 1110000 • RES := RES>>1 • RES = 0111000
Start 1 B(i)=1 0 RES:=RES+A 1 RES:=RES>>1 i:=i+1 0 i<3 Done Multiplication • A = 1.11•20 = 1.75 • B = 1.10•2-1 = 0.75 • Iter 3 • B(2) == 1 • RES := RES + A • RES = 1010100 • RES := RES>>1 • RES = 0101010
Multiplication • signA xor signB = 0 • expA + expB = -1 • man = 10.1010
Normalisation • Express the result in normalised form Binary form Decimal form10.1010•2-1 (2.625•2-1 = 1.3125)1.01010•20 (1.3125•20 = 1.3125)
Rounding • The result has to many digits and must be roundedBinary Decimal1.01010•20 (1.3125) • Identify the nearest numbers that can be representedBinary Decimal1.01•20 (1.25)1.10•20 (1.50)
Rounding • Four different round modesBinary Decimal- Round to nearest 1.01•20 1.25- Round toward + 1.10•20 1.50- Round toward - 1.01•20 1.25- Round toward zero 1.10•20 1.25- Exact 1.01010•20 1.3125
Implementation in an HDL • A software programming language used to model the intended operation of a piece of hardware • Two languages- Verilog- VHDL
What is VHDL • VHSIC (Very High Speed Integrated Circuits) Hardware Description Language • Created by US Department of Defence. Adopted as an IEEE standard in 1987. Latest standard is IEEE 1076 ‘93 • Intended for documenting and modeling digital systems at different abstraction levels ranging from system level down to gate level
Main language concepts • Concurrency • VHDL can describe activities that are happening in parallel • Structure, hierarchy • VHDL allows to structure a design in a hierarchical manner • Sequential statements • VHDL also allows sequential execution of statements. Just like any other programming language
VHDL example if selectX = '1’ then X <= A; else X <= B; end if;
VHDL example 2 if selectX = '1’ then X <= A; else X <= B; end if; if selectY = '1’ then Y <= A; else Y <= B; end if;
Speedup • Sign, exponent and mantissa in parallel
Testing - Verification • Verification • Are we building the product right? • Bottom-up integration • Testing every module on its own before integrating it into higher level modules
Testing - Tools used • ModelSim
Testing - Tools used • Synplify
Testing - Example • Avoid introducing new errors in verified code A := operand_AB := operand_BC := expected_resultif result is equal to C test is okelse raise flag to indicate fault in test
Testing - Validation • Validation • Are we building the right product? • What if we have misinterpretedthe IEEE standard 754?
Testing - Validation • Simulate with a test bench using Modelsim • Compare the simulated result with the computed result from an UltraSparc IIi running Solaris
Final result - ceiling- floor- fraction- float2integer- integer2float • Operations- addition - subtraction- division- multiplication- square root
Final result • Round modes- to nearest- towards +- towards -- towards zero • Exceptions- overflow- underflow- divide by zero- inexact operation- invalid operation
Life after SMD082 • Computer Architecture / DatorarkitekturSMD077, 4p, LP2 • Computation Structures / Beräkningsstrukturer SMD098, 4p, LP2 • Project in Digital Synthesis / Projekt i digital syntesSMD106, 8p, LP3-4 • Computer Science, Advanced course / Datateknik, fördjupningskurs SMD061-3, 2-4p, LP1-4
