Floating-Point Numbers

Floating Point Numbers: Real numbers.

• e.g., Fixed point notation: 1.2, rational notation: 1 / 2, floating-point notation 12345 \times 10^{-4}

Floating-Point Notation:

\text{Floating-Point Notation: } \text{Mantissa} \text{Base}^\text{Exponent}

Example:

Number	Scientific Notation	Floating-Point Notation
1,245,000,000,000	1.24510^{12}	0.1245 10^{13}
0.0000001245	1.245 10^{-7}	0.1245 10^{-6}

Representations

Binary Fixed-Point:

We can place an imaginary binary point in a binary pattern to define an integer part and fraction part.

• Math works as normal as long as binary patterns have the same binary point.

Example: Binary pattern with two fractional bits \to decimal

00101011_2 = 001010.11_2 = 10.75_{10}

Note: For more precision you need to use more floating-point bits (you can shift with sra) .

• You also need to be consistent with the binary point for the math to work.

Floating-Point IEEE-754:

• Technical standard for floating-point arithmetic
• Made of: (single precision / double precision)
  1. Sign (1 bit)
  2. Exponent (11 bits / 8 bits)
  3. Mantissa (52 bits / 23 bits)
     • (Always normalized to 0.XXXX)

MIPS Floating Point Coprocessor 1

Instructions are split between single and double precision.

• Don’t mix single and double precision!
  • Only even f-registers are valid operands for double precision operations.
• Single-Precision: 7 fractional bits of precision.
• Double-Precision: 15 fractional bits of precision.

    .data
radius: .float  10.0
area:   .float  0.0
    .text
main:
    l.s $f4, radius
    mul.s   $f5, $f4, $f4   # f4 = radius ^ 2
    li.s    $f6, 3.141592653589793  # f6 = pi
    mul.s   $f12, $f5, $f6  # area = pi * radius^2
    # Save result into area
    s.s $f12, area
    # Print area
    li  $v0, 2
    syscall
    # Exit
    li  $v0, 10
    syscall

Tangent: Swapping Values in Two Registers with Xor

On x86, we have 4 general-purpose registers, A, B, C, and D.

Q: How do we swap A and B without using memory or using another GPR?

A: Use the exclusive or operator

A <- A ^ B
B <- A ^ B
A <- A ^ B

xor   $t0, $t0, $t1
xor   $t1, $t0, $t1
xor   $t0, $t0, $t1