Floating Point

Question 1

Most modern architectures implement ___ standard for floating point representations

Answer 1

IEEE 754

Question 2

True or False: “floating-point units (FPUs) are outdated and included in CPUs anymore”

Answer 2

False. Most CPUs include FPUs now

Question 3

Benefits and purpose of FPUs

Answer 3

Have instructions to do FP arithmetic very quickly. If no FPU, FPU simulation is very slow.

Question 4

Explain the difference between fixed and floating point representations

Answer 4

Fixed: number has a fixed number of decimal digits, more precise, smaller range

Floating: number of decimal digits may vary, less precise, large range

Question 5

How is the position of the binary point established in a fixed-point number?

Answer 5

By convention. It is agreed upon by the programmer and the system so it may not be the same for every program.

Question 6

If the binary point comes after the second bit in a fixed point number, 0111, what decimal number will result?

Answer 6

0x2^1 + 1x2^0 + 1x2^(-1) + 1x2^(-2) = 1.75

Question 6

What is the mantissa (significand) and what is used to represent it in a floating point representation?

Answer 6

4.65 in 465 x10^(-4)

Fixed point is used to represent the mantissa

Question 7

What are the differences between floating-point single format and scientific notation?

Answer 7

• Base 2 instead of 10
• The significand (mantissa) and exponent are in 0b
• The number is normalized so that 1<= significand <2
  
  • Since MSB is always 1, it’s not stored
  • the exponent is biased by adding the constant 127 so it’s stored as a + integer
• IEEE 754 standard

Question 8

What does IEEE 754 standard look like for floating point single-format numbers?

Answer 8

• Uses 4 bytes
• Number = (-1)^s x 1.f x 2^(e-127)

s= signed bit
f = fractional part of significand
e = biased exponent (add 127 to unbiased)

-check notes for diagram

Question 9

Range of biased exponents allowed

Answer 9

1-254

• 255 used to represent not numbers (NaNs)
• 0 used for subnormal numbers (tiny fractional quantities)

Question 10

Range for floating-point single format. Also approximately how many digits of precision?

Answer 10

1.175e-38 to 3.403e+38
1.0 x2^(-126) to (2.0 -ε) x2^(127)

total range:
-3.403e+38 to -1.175e-38
Gap to 0.0 gap to
+1.175e-38 to +3.403

~7 digits of precision

Question 11

What are the s, e and f values for |0|01111101|0000000…| (0x3E800000)

Answer 11

sign = 0
biased exponent = 125
fractional part of significand: 0.0

Question 12

What are the s, e and f values for |0|01111101|0100000…| (0x3E900000)

Answer 12

sign = 0
biased exponent = 125
fractional part of significand: 0.01

Question 13

What does IEEE 754 standard look like for floating point double-format numbers?

Answer 13

• 8 bytes
• Number = (-1)^s x 1.f x 2^(e-1023)
• check diagram

Question 14

Range for floating-point double format. Also approximately how many digits of precision?

Answer 14

~2.2e-308 to ~1.8e+308
~17 digits

Also NaNs are represented with biased exponent 2047

Question 15

+/- infinity, root(-1) NaNs in single format

Answer 15

+: 0x7f800000
-:0xff800000
root(-1): 0x7fffffff

Question 16

+/- infinity, root(-1) NaNs in double format

Answer 16

+: 0x7ff00000 00000000
-:0xfff00000 00000000
root(-1): 0x7fffffff ffffffff

Question 17

How can NaN be used?

Answer 17

• in most instructions causes an exception but can be compared to a number using fcmp
• can be used as an argument to some functions

Question 18

How many floating point registers does ARMv8 have? Also what types of registers are they?

Answer 18

32 128-bit fp registers

s registers use low-order 32 bits for single-precision fp numbers

d registers use low-order 64 bits for double-precision fp numbers

Question 19

opcode for multiply-negate of a fp number

Answer 19

fnmul

Question 20

How is #fpimm encoded? How is it expressed?

Answer 20

With 1 sign bit, 4 bits of fraction and 3 bits for exponent

as +-n/16x2^r
n in range: 16-31
r in range: -3 to +4

Question 21

Instruction for converting s <-> d

Instruction for converting s or d to nearest 32-bit or 64-bit signed int

instruction for converting s or d to unsigned ints

Answer 21

fcvt

fcvtns

fcvtnu

Question 22

Instruction for converting Wn or Xn to Sd or Dd

Instruction for converting unsigned ints to floats

Answer 22

scvtf
ucvtf

Question 23

What registers are used to pass fp arguments into a subroutine?

Answer 23

d0-d7 / s0-s7

Question 24

What are registers d8-d15 called?

Answer 24

Callee-saved
- if used in a subroutine, it must save & restore their values on the stack
- only the bottom 64-bits of the 128 bit register need to be preserved

Question 25

Which registers can be overwritten by a subroutine?

Answer 25

0-7 and 16-31
caller is responsible for saving/restoring if they need to be preserved over a subroutine call

Question 26

What are two things that are stored in .text?

Answer 26

String literals
floating point values

Question 27

What is the bias we add to the exponent for floating point double format?

Answer 27

1023