Arthimetic for Computer

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Intrduction

• Computer words are composed of bits
  • thus words can be represented as binary numbers
  • there are 32bit/word or 64bits/word in RISC-V
  • Contains four bytes or eight bytes
• Generic Implementation
  • use the PC to supply instruction address
  • get the insturction from memory
  • read registers
  • use the instruction to decide exactly what to do

Arithmetic

Multiplication

对于 \(n\) bit 乘法，首先将乘数放到积寄存器的低位，在每轮迭代中检测积的第 0 位是否为 1，如果是则将积的高位加上被乘数后右移，否则仅做右移操作，迭代 \(n\) 轮。

Example

Booth's Algorithm

注意移位的时候要保证符号位不变(算术右移)

Example

Division

Example

Floating point numbers

IEEE 754 standard

• Leading 1 bit of significand is implicit(saves one bit)
• Exponent is biased: (127 for single precision, 1023 for double precision)

\[ (-1)^{sign}\cdot (1+significand) \cdot 2^{exponent - bias} \]

Single-Precision Range

Double-Precision Range

Infinities and NaNs

Floating point addition

• Alignment
• The proper digits have to be added
• Addition of siginificands
• Normalisation of the result
• Rounding

Example

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最后更新: 2024年4月20日 22:02:37
创建日期: 2024年3月18日 19:42:20