05 Floating Point Arithmetic

Floating Point¶

S¶

• Sign bit
• 0: non-negative
• 1: negative

Exponent¶

• Excess representation: actual exponent + Bias
• Ensures exponent is unsigned
• Single: Bias = 127; Double: Bias = 1023

Normalized Significand¶

• Significand is Fraction with the 1. restored
• \(1.0 \le |\text{significand}| < 2.0\)
• Always has a leading pre-binary-point 1 bit, so no need to represent it explicitly (hidden bit)

Single Precision (32-bit)¶

Exponents \(00000000\) and \(11111111\) reserved

Double Precision (64-bit)¶

Exponents \(0000…00\) and \(1111…11\) reserved

Floating Point Examples¶

Denormal Numbers¶

Exponent = 000...0 ⇒ hidden bit is 0

• Smaller than normal numbers
• Allow for gradual underflow, with diminishing precision

Denormal with fraction = 000...0 is \(\pm 0.0\) (2 representations of \(0.0\))

Infinities and NaNs¶

Floating-Point Addition¶

1. Align binary points Shift number with smaller exponent
2. Add significands (in binary)
3. Normalize result & check for over/underflow
4. Round and renormalize, if necessary

FP Adder Hardware¶

Much more complex than integer adder

Doing it in one clock cycle would take too long

• Much longer than integer operations
• Slower clock would penalize all instructions

FP adder usually takes several cycles

• Can be pipelined

MIPS Floating Point Instructions¶

FPU co-processor is an adjunct processor that extends the ISA

Separate FP registers

• 32 single-precision: $f0, $f1, … $f31
• Paired for double-precision: $f0/$f1, $f2/$f3,

FP instructions operate only on FP registers

• Programs generally don’t do integer ops on FP data, or vice versa
• More registers with minimal code-size impact

Separate FP instructions for single/double precision

FP Arithmetic Operations¶

FP Load/Store¶

FP Load/Store pseudo Instructions¶

FP Loading immediate value¶

(pseudoinstruction)

li.s $f1, 1.0

li.s $f10, 1.0e-5 ## $f10 = 0.00001

FP Data Movement Instructions¶

Moving data between general purpose and FP registers

FP Convert Instructions¶

m.x.y

Convert

• to destination format \(x\)
• from source format \(y\)

Supported Formats¶

• Single precision float = .s (single precision float in FP register)
• Double precision float = .d (double float in even-odd FP register)
• Signed integer word = .w (signed integer in FP register)

FP Compare and Branch Instructions¶

FP unit (co-processor 1) has a condition flag

• Set to 0 (false) or 1 (true) by any comparison instruction

Three comparisons

• equal, less than, less than or equal

Two branch instructions based on the condition flag

Reading and printing single and double values¶

Floating-Point Data Declarations¶

.data
pi: .float 3.14
tao: .double 6.28

.text
main:

end main

MIPS Floating Point Examples¶

Area of Circle¶

.data
pi: .double 3.14

.text
main:
    #f2,f3 = pi
    ldc1 $f2, pi

    ## read double (radius)
    addi $v0, $zero, 7
    syscall ## f0, f1 = r

    mul.d $f12, $f0, $f0 ## r^2
    mul.d $f12, $f2, $f12 ## area = pi * r^2

  ## print value
  addi $v0, $zero, 3
  syscall

°F \(\to\) °C¶

.data
const5: .float 5.0
const9: .float 9.0
const32: .float 32.0
constf: .float 50.0

.text
main:
l.s $f16, const5
l.s $f18, const9
div.s $f16, $f16, $f18
l.s $f12, constf
l.s $f18, const32
sub.s $f18, $f12, $f18
mul.s $f12, $f16, $f18
li $v0, 2
syscall
li $v0, 10
syscall

\(ax^2 + bx + c\) for user-inputted \(x\)¶

.data
a: .float 1.0
bb: .float 2.0
c: .float 3.0
msg: .asciiz "Enter x: "
blank: .asciiz " "
newl: .asciiz "\n"

.text
main: ## read input
la $a0,msg ## prompt user for x
li $v0,4 ## print string
syscall
li $v0,6 ## read single
syscall ## $f0 <-- x

## evaluate the quadratic
l.s $f2,a ## sum = a
mul.s $f2,$f2,$f0 ## sum = ax
l.s $f4,bb ## get b
add.s $f2,$f2,$f4 ## sum = ax + b
mul.s $f2,$f2,$f0 ## sum = (ax+b)x = ax^2 +bx
l.s $f4,c ## get c
add.s $f2,$f2,$f4 ## sum = ax^2 + bx + c

## print the result
mov.s $f12, $f2 ## $f12 = argument
li $v0,2 ## print single
syscall
la $a0,newl ## new line
li $v0,4 ## print string
syscall
li $v0,10 ## code 10 == exit
syscall ## Return to OS.

Comments