Assembly Tutorial

# Floating-Point Arithmetic Operations

Floating-point arithmetic is a method of representing real numbers using a format that combines a sign bit, an exponent, and a significand. This representation allows for a wider range of representable numbers than fixed-point arithmetic, but it also introduces some additional complexities in terms of arithmetic operations.

In assembly language, floating-point arithmetic is typically performed using dedicated instructions and hardware units. These instructions handle the complexities of floating-point representation and perform operations such as addition, subtraction, multiplication, and division on floating-point numbers.

## Floating-Point Representation

Floating-point numbers are represented using a three-part format:

1. Sign Bit: The sign bit indicates whether the number is positive or negative. A 0 represents a positive number, while a 1 represents a negative number.
2. Exponent: The exponent represents the power of the base by which the significand is multiplied. It is typically an integer value, either signed or unsigned, depending on the specific floating-point format.
3. Significand: The significand represents the fractional part of the number and is typically normalized to fall within a specific range. It may be stored in binary or hexadecimal form, depending on the floating-point format.

## Floating-Point Arithmetic Instructions

Assembly language provides instructions for performing arithmetic operations on floating-point numbers. These instructions typically take two or three operands, representing the numbers to be operated on and the destination for the result.

For instance, in x86 assembly language, the fadd instruction performs floating-point addition, adding two floating-point values stored in registers ST(1) and ST(0) and storing the result in ST(0).

Similarly, the fmul instruction performs floating-point multiplication, multiplying two floating-point values stored in registers ST(1) and ST(0) and storing the result in ST(0).

fmul st(0), st(1)

## Floating-Point Arithmetic Operations

Floating-point arithmetic operations involve more complex calculations than their fixed-point counterparts due to the representation format. These operations typically involve normalizing the operands, aligning their exponents, and handling special cases such as infinity, not-a-number (NaN), and denormalized numbers.

For example, floating-point addition requires aligning the exponents of the operands before adding their significands. If the exponents differ, the significand of the number with the smaller exponent is shifted to the right until its exponent matches the other operand's exponent.

Load floating-point values into FPU registers or SSE registers.

fld dword [float_variable] ; Load a single-precision float from memory into FPU register movaps xmm0, [float_array] ; Load a float array into SSE register

Add two floating-point values using FPU or SSE instructions.

fadd dword [float_operand] ; Add a float operand to the value in the FPU register addps xmm0, xmm1 ; Add two SSE registers element-wise
Perform Floating-Point Subtraction

Subtract one floating-point value from another.

fsub dword [float_operand] ; Subtract a float operand from the value in the FPU register subps xmm0, xmm1 ; Subtract two SSE registers element-wise
Perform Floating-Point Multiplication

Multiply two floating-point values.

fmul dword [float_operand] ; Multiply the value in the FPU register by a float operand mulps xmm0, xmm1 ; Multiply two SSE registers element-wise
Perform Floating-Point Division

Divide one floating-point value by another.

fdiv dword [float_operand] ; Divide the value in the FPU register by a float operand divps xmm0, xmm1 ; Divide two SSE registers element-wise
Handle Overflow and Underflow

Check for and handle overflow or underflow conditions as needed.

fcomi st0, st1 ; Compare FPU register values fstsw ax ; Store FPU status word sahf ; Set flags based on the status word
Round and Convert

Round floating-point values and convert between different precision formats.

frndint ; Round the value in the FPU register to the nearest integer cvtps2pd xmm0, xmm1 ; Convert single-precision float to double-precision float in SSE
Handle NaN and Infinity

Check for special cases like NaN (Not a Number) and infinity.

fucomip st0, st1 ; Unordered compare and set flags in FPU status word
Store and Retrieve Results

Store the results back to memory or other registers.

fst dword [result_variable] ; Store the result in the FPU register to memory movaps [result_array], xmm0 ; Store the result in SSE register to memory
Handle Exception Conditions

Handle exceptions like division by zero or invalid operation as necessary.

fdiv dword [denominator] ; Handle division by zero exception

### Conclusion

Floating-point arithmetic is an essential tool for numerical computations in assembly language programming. Understanding the representation format, arithmetic operations, and challenges involved in floating-point arithmetic is crucial for writing efficient and accurate code that handles numerical data correctly.