1.Floating-Point Representation
1.1 Representation of Fractions
Scientific Notation (in Binary):
For single precision, a 32-bit word, can represent number between [2x10e-38, 2x10e38] of [-2x10e38, -2x10e-38]
IEEE 754 single precision Floating-Point Standard:
- 1 bit for sign(s) of floating point number
- 8 bits for exponent(E)
- 23 bits for fraction(F): (get 1 extra bit for precision because leading 1 is implicit)
$$
(-1)^s (1+F) 2^E
$$
Double precision Standard(64 bits)
What if result too large? (> 2x10e38, < -2x10e38):
- Overflow!
What if result too small? (>0 & < 2x10e-38, <0 & > -2x10e-38)
- Underflow!
what could help reduce chances of overflow or underflow?
1.2 two extreme cases: exponents all zeros or all 1s
- What about 0?
- Bit patterns all 0s means 0 (so no implicit leading 1 in this case)
- what if divide 1 by 0?
- can get infinity symbols $+ \infin, - \infin$
- sign bit 0 or 1, largesr exponent(all 1s), 0 in fraction
- what if do something stupid (0/0, $\infin - \infin$)?
- can get special symbols NaN for “Not a Number”
- Sign bit 0 or 1, largest exponent (all 1s), not zero in fraction
1.3 Represent 0:
- exponent all zeros
- significand all zeros
- Sign both + or - is valid
- because it is not really zero!
1.4 Representation $+\infin, -\infin$:
- In FP, divide by 0 should produce $+\infin, -\infin$, not overflow
- because it is for further computations with $\infin$: eg, X/0 > Y may be a valid comparison
- significand all zeros
1.5 special numbers:
1.6 What do I get if calculate sqrt(-4.0) or 0/0:
- thet all get a NaN
- exponent = 255, significand nonzero
- it’s helpful because we can use it debug: op(NaN, x) = NaN
1.7 Representation for denorms
- Problem: there is a gap among representable FP numbers around 0
- smallest positive number: a = 1.0… * $2^{-126}$ = $2^{-126}$
- second smallest: b = 1.000….1 $2^{-126}$ = (1 + $2^{-23}$) $2^{-126}$ = $2^{-126}+2^{-129}$
- Solution:
- Exponont = 0 , significand nonzero
- denormalized number : no (implied) leading 1, implicit exponont = -126
- smallest positive number: a = $2^{-149}(2^{-126} * 2^{-23})$
- second smallest positive number b = $2^{-148}(2^{-126}*2^{-22})$
2. Assembly Language
- Basic job of a CPU: execute lots of instructions
- instructions are the primitive operations that the CPU may execute
- Different CPUs implement different sets of instructions. The set of instructions a particular CPU implements is an instruction Set Architecture(ISA)
2.1 Instruction Set Architecture
- Early trend was to add more and more instructions to new CPUs to do elaborate operations
- VAX architecture had an instruction to multiply polynomials
- RISC (Reduced Instruction Set Computing) philosophy:
- Keep the instruction set small and simple, makes it easier to build fast hardware
- Let software do complicated operations by composing simpler ones
2.2 From RISC-I to RISC-V
RISC-V (pronounced “risk-five”) is a new instruction set architecture (ISA) that was originally designed to support computer architecture research and education.
All RISCs are mostly the same except for one or two silly design decisions
