[CSAPP]Chapter 2 Representing and Manipulating Information

Information storage

Rather than access individuel bits in a memory, many computer use blocks of eight bits, or bytes, as the smallest addressable memory.
A single byte consists of 8 bits. In binary notation, its value ranges from 00000000₂ to 11111111₂. However, this notation seems too verbose to express a number. Thus we use hexadecimal to express a byte in the computer.
To convert a binary number to a hex number: Splite the number into 4-bit groups and convert the groups one by one.
Word Size: Every computer has a word size, which indicates normal size of data and addresses.
exp: 32-bit word size =maximun 4GB virtual address
Addressing and byte ordering: For a object that span multiple bytes, we always focus these problems:

• What will be the address of the object?
  
• How will we order the bytes in the computer memory?
  
  The first problem: We always stored the object in a contiguous sequence. For example, for a integer, the start of its memory is 0x01, then it may be span to 0x02, 0x03, 0x04….
  
  The Second problem: Consider a w-bit question has a bit representation $[x_w-1x_w-2...w<sub>1]$, in which x_w-1 is the most significant bit. The former convention, where the least significant bit comes first is called little endian, while now the most significant bit comes first is called big endian.

• Integer Representation

  Assume we have an integer data type of w bits. We write a bit vector, or as $[x_{w-1}x_{w-2}..x_1]$ to denote its individual bits. When we refer to such a unsigned number, the value of the number is
  

  $$B2U_x = \sum_{i =0}^{w-1}{x_i*2^i}$$
  We should understand that in the computer science, the left side and right side are often defined to be equal. Then in this case, the max number of a w-bit number is [11111..11], while the min number is [000…00].
  However, for many applications, we wish to represent negative values as well. The most common computer representation is called two’s complement form. This defined by interpretation the most significant bit of the word to served as the sign bit.
  The function is here below:
  $$B2T_w(x) =-x_{w-1}2^{w-1}+\sum_{i =0}^{w-2}x_i2^i$$
  We set the sign bit to 1 when the value is negative, while set to 0 when the value is positive.
  Similarly, we have another two methods to represent an integer:
  $$B2O_w(x) =-x_{w-1}(2^{w-1}-1)+\sum_{i =0}^{w-2}x_i2^i$$
  $$B2S_w(x) =(-1)^{x_w-1}+\sum_{i=0}^{w-2}{x_i2^i}$$
  Which is called one’s complement and sign-magnitude.
  Expanding the bit representation of a number:
  In some cases, such as convert a data type to a larger one, we should expand the bit representation of a number!
  
• Unsigned: Zero Extension
  
• Signed: Sign Bit Extension
  Truncating Numbers:
  When truncating a w-bit number to a k-bit one, we just drop the high-order w-k bits.
  Truncating a number can alter its value, which is another type of overflow!

• Integer Arithmetic

  Unsigned arithmetic can be viewed as a form of modular arithmetic!
  

  $$x+y =(x+y)\%2^w$$
  Modular addition forms Abelian group, which has the identity element 0.
  Two’s-Complement Arithmetic as we shown before, the Two’s-Complement Arithmetic may cause positive overflow or negative overflow, which will alter the value of the number!
  Multiply or divide by powers of 2: We can use left or right shift to accomplish this.

  Floating Point

  Floating Point representation encodes rational numbers of the form $V =x*2^y$, which is useful for performing computations involving very largr numbers.
  1985-IEEE754 Standard, under the sponsorship with the design of 8087.

  IEEE Floating-Point Representation

• The sign s determines whether the number is negative or positive.
  
• The significant M is a fractional binary number that ranges ethier between 1 and 2-e or between 0 and 1-e.
  
• The exponent E weights the value by power of 2.
  The value encoded by a given bit representation can be divided into 3 cases, depending on the value of exp.

• Normalized values

• Fraction field: representation $F =0.f_{n-1}f_{n-2}...f_2f_1$, with a implied 1, $M =1+F$.
  
• exponent field: neither all 0 nor all 1

• Denormalized values

• The exponent field is all 0, the exponent value is $M =1-bias$
  
• Significant value is M =f

• Special values

• Exponent field is all 1, representing infinity

• Rounding

  The representation has limited range and precision. So we generally need a systematic method of finding the closest matching $x^{'}$.
  General mode: Round to 0, Round to even, Round up…..
  We finally choose to Round-To-Even because it minimum the plug between round number and actual number!!
  If the number is half-way between 2 possible answers, round to the even one. Otherwise round to the nearest number!!

  Floating Point operations

  The operation is not overflow:
  exp:

  $$(3.14+10e65)-10e65 =0.0(overflow)$$
  $$(10e65-10e65)+3.14 =3.14$$
  The operation is not distribution:
  exp: $$(10e65-10e65)*10e65 =0$$
  $$10e65*10e65-10e65*10e65 =NaN$$

  Transform data type in C

  From int to float: can be overflow, but may be rounded
  From float to double: can be preserved
  From double to float: The value can be overflowed to $+\infty to -\infty$
  Form double or float to int: overflow, but truncated toward 0.