ساختار و زبان کامپیوتر: نمایش اعداد حقیقی

Computer Structure & Language: Fixed-Point & Floating Point

• This lecture covers fixed-point and floating-point representations of real numbers in computer systems.
• Content is adopted from various sources including "Computer Organization & Design", "Hardware/Software Interface", and other academic materials.
• Topics covered include fixed-point representation, floating-point representation, instructions in MIPS, and FP coprocessors.

Real Numbers

• Real numbers are numbers with fractional parts.
• Examples include 3.14159..., 2.17, 0.0000001, 1.25 * 10^-12, and 1.43 * 10¹².

Real Notations

• Scientific notation represents numbers in the form of a significand (mantissa) multiplied by a power of 10 (exponent).
• Normalized scientific notation specifies that the significand should be between 1.0 and 10.0
• Examples include 0.21 * 10^-12, and 5.33 * 10¹²
• Normalized scientific notation in Binary will be like 1.0_two * 2^-1

Real Numbers in Computers

• Fixed-Point Representation: Represents numbers with a fixed number of bits for the integer and fractional parts.
• Example: d₂d₁d₀.f₁f₀ represents a number with 24 bits for integer and 8 bits for fraction parts.
• Used in CPUs without floating-point units, microcontrollers, and DSP applications.

Fixed-Point Representation (Examples)

• The largest positive number, the smallest positive number, the largest magnitude negative number, and the smallest magnitude negative number in a 5-bit fixed-point representation are discussed.

Fixed-Point Representation (Examples)

• Further examples of simple binary fixed-point operations like addition, subtraction, multiplication, and division are given. Examples include 011.11 + 111.11, 010.10 * 000.10, 011.10 * 010.11 and 000.01 * 000.01

Fixed-Point Representation (Pros & Cons)

• Pros: Simple hardware, fast computation and flexibility for different precisions per application.
• Cons: Low precision and a small range.

Floating-Point Representation

• Floating-point representation represents numbers in the form of 1.XXXXXX_two * 2^yyyy
• A single precision floating-point number consists of a sign bit (S), an exponent (E), and a fraction (F).
  • The fraction has 23 bits.
  • The exponent has 8 bits.
• N = (-1)^S * (1 + F) * 2^E

Floating-Point Representation (Pros & Cons)

• Pros: Wide range and greater precision in comparison to fixed-point
• Cons: Arithmetic operations are more complex, hardware implementation is more complicated.

Floating-Point Representation (Issues)

• Overflow: Occurs if the exponent becomes too large to fit in the allocated bits. In that case, the result may be too large to be represented.
• Underflow: Occurs if the fraction part becomes too small. It may be considered zero in the representation.

Floating-Point Instructions in MIPS

• MIPS architecture supports single and double-precision floating-point instructions.
• Examples include add.s, sub.s, mul.s, div.s, add.d, sub.d, mul.d, div.d, and c.x.s and c.x.d.

FP Coprocessor

• A dedicated coprocessor (often known as the FPU) handles floating-point operations.
• The coprocessor improves performance by offloading floating-point calculations.

FP Registers

• MIPS provides separate floating-point registers (like $f0, $f1, ..., $f31).
• These registers support single and double-precision formats.
• A double precision register uses a pair of even/odd numbers to store values.

FP Instruction Format

• Various floating-point instructions' formats are detailed in a table.

Next Topics

• IEEE 754 standard for floating point definitions, single-precision, and double-precision.