Lecture 4: Arithmetic for Computers (Part 5)

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Lecture 4 Arithmetic for Computers(Part 5)

• CS 447
• Jason Bakos

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Binary Division

• Like last lecture, well start with some basic
  terminology
• Again, lets assume our numbers are base 10, but
  lets only use 0s and 1s

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Binary Division

• Recall
• DividendQuotientDivisor Remainder
• Lets assume that both the dividend and divisor
  are positive and hence the quotient and the
  remainder are nonnegative
• The division operands and both results are 32-bit
  values and we will ignore the sign for now

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First Hardware Design for Divider
Initialize the Quotient register to 0, initialize
the left-half of the Divisor register with the
divisor, and initialize the Remainder register
with the dividend (right-aligned)
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Second Hardware Design for Divider
Much like with the multiplier, the divisor and
ALU can be reduced to 32-bits if we shift the
remainder right instead of shifting the divisor
to the left
Also, the algorithm must be changed so the
remainder is shifted left before the subtraction
takes place
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Third Hardware Design for Divider
Shift the bits of the quotient into the remainder
register Also, the last step of the algorithm
is to shift the left half of the remainder right
1 bit
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Signed Division

• Simplest solution remember the signs of the
  divisor and the dividend and then negate the
  quotient if the signs disagree
• The dividend and the remainder must have the same
  signs

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Considerations

• The same hardware can be used for both multiply
  and divide
• Requirement 64-bit register that can shift left
  or right and a 32-bit ALU that can add or subtract

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Floating Point

• Floating point (also called real) numbers are
  used to represent values that are fractional or
  that are too big to fit in a 32-bit integer
• Floating point numbers are expressed in
  scientific notation (base 2) and are normalized
  (no leading 0s)
• 1.xxxx2 2yyyy
• In this case, xxxx is the significand and yyyy is
  the exponent

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Floating Point

• In MIPS, a floating point is represented in the
  following manner (IEEE 754 standard)
• bit 31 sign of significand
• bit 30..23 (8) exponent (2s comp)
• bit 22..0 (23) significand
• Note that size of exponent and significand must
  be traded off... accuracy vs. range
• This allows us representation for signed numbers
  as small as 2x10-38 to 2x1038
• Overflow and underflow must be detected
• Double-precision floating point numbers are 2
  words... the significand is extended to 52 bits
  and the exponent to 11 bits
• Also, the first bit of the significand is
  implicit (only the fractional part is specified)
• In order to represent 0 in a float, put 0 in the
  exponent field
• So heres the equation we use (-1)S x
  (1Significand) x 2E
• Or (-1)S X (1 (s1x2-1) (s2x2-2) (s3x2-3)
  (s4x2-4) ...) x 2E

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Considerations

• IEEE 754 sought to make floating-point numbers
  easier to sort
• sign is first bit
• exponent comes first
• But we want an all-0 (1) exponent to represent
  the most-negative exponent and an all-1 exponent
  to be the most positive
• This is called biased-notation, so well use the
  following equation
• (-1)S x (1 Significand) x 2(Exponent-Bias)
• Bias is 127 for single-precision and 1023 for
  double-precision