Homework 1 graded and returned in class today. Solutions posted online. Request regrades by next class period. Question 10 treated as extra credit

Transcription

1 Announcements Homework 1 graded and returned in class today. Solutions posted online. Request regrades by next class period. Question 10 treated as extra credit Quiz 2 Monday on Number System Conversions Covers lectures on computer number systems Covers all types of problems in Homework 1 Try practice problems (posted to class schedule) CSC Computer Organization 1

2 Representing Real Numbers: Floating Point Finale CSC 242 Computer Organization Dr. Lucas Layman January 31, 2018

3 IEEE 754 Standard for Floating-Point Arithmetic Defines a different format for storing real numbers Specify rules for communicating real numbers between components Rounding rules Arithmetic operations Exception handling (division by zero, overflow, etc) CSC Computer Organization 3

4 IEEE 754 generalized process 1) Convert the integer and significand from decimal to binary 2) Normalize the binary, a.k.a., convert it to scientific notation in binary 3) Store the key values of the normalized binary this is the floating point representation CSC Computer Organization 4

5 1) Convert the integer and significand from decimal to binary Example: convert to 8-bit binary Algorithm: a) Convert the integer part (2) to binary using successive division. b) Convert the significand (.625) to binary using successive multiplication: 1) Multiply only the significand by 2 2) Note the integer value (1 or 0). These are your binary significands starting with the MSB. 3) Repeat steps 1-2 until your significand is 0 or you hit the maximum number of bits c) Write the binary CSC Computer Organization 5

6 2) Normalize the binary Move the radix point so that it is one bit from the left. Adjust the exponent so that the value does not change = = = = = = CSC Computer Organization 6

7 3) Store the key values of the normalized binary 1) the sign 3) the exponent = ± x 2 1 2) the significand CSC Computer Organization 7

8 3) Store the key values of the normalized binary exponent 3 bits An 8-bit float (1 byte): sign bit significand 4 bits Note: 8-bit floats aren t very useful, but they illustrate the process. Modern computers use 32- and 64-bit floats. CSC Computer Organization 8

9 3) Store the key values of the normalized binary (cont.) = x 2 1 exponent 3 bits Sign bit: 1 = negative, 0 = non-negative Exponent bits: sign bit Take the exponent and add the bias to it. The bias is always 2 k-1-1 where k is the number of bits in the exponent field. Convert the exponent (with added bias) to binary. Significand bits: take the first m bits of your normalized binary s significand significand 4 bits CSC Computer Organization 9

10 3) Store the key values of the normalized binary (cont.) = ± x 2 1 exponent 3 bits Sign bit: Exponent bits: We have dropped this 1. It is implied. sign bit mantissa 4 bits Significand bits: CSC Computer Organization 10

11 3) Store the key values of the normalized binary (cont.) = x 2-2 exponent 3 bits Sign bit: Exponent bits: We have dropped this 1. It is implied. sign bit significand 4 bits Significand bits: CSC Computer Organization 11

12 Once more, beginning to end Convert to 8-bit IEEE 754 floating point 1) to binary: 2) Normalize: 3) Store values: 1) Sign bit: 2) Exponent: 3) Significand: Answer: exponent 3 bits sign bit significand 4 bits CSC Computer Organization 12

13 Floats in modern computers A single-precision float is 4 bytes, or 32 bits: exponent 8 bits sign bit significand 23 bits CSC Computer Organization 13

14 Floats in modern computers (cont.) A double-precision float is 8 bytes, or 64 bits. Python uses this by default. 8 bit 32 bit 64 bit exponent 11 bits significand 52 bits CSC Computer Organization 14

15 Converting to IEEE 32-bit floating point format Convert (process is the same as 8-bit) 1) Convert the integer and significand from decimal to binary (example 3) -> ) Normalize: x ) Store the key values of the normalized binary: Sign bit: 1 Exponent: = = Note: The bias for a 32 bit floating point # is always 127 Significand: CSC Computer Organization 15

16 Converting to IEEE 32-bit floating point format (cont.) = Sign bit: 1 Exponent: = = Significand: exponent 8 bits sign bit significand 23 bits CSC Computer Organization 16

17 Another 32-bit example Convert to 32-bit floating point 1) Convert the integer and significand from decimal to binary 2) Normalize the binary 3) Store the key values of the normalized binary 1) sign bit 2) exponent (remember the bias is 127) 3) the significand to 23 bits CSC Computer Organization 17

18 Review on computer representations of numbers Name Base Range Purpose Decimal 10 [0,9] The number system most humans use Binary 2 [0,1] The language of all computers Octal 8 [0,7] Rarely used. Common in early computers Hex 16 [0,F] user-friendly representation of binary data CSC Computer Organization 18

19 Review on computer representations of numbers Name Base Purpose Two s complement 2 A way to represent signed integers Floating point Any A way to represent real numbers CSC Computer Organization 19

20 Storing other types of data We ve discussed storing integers and real numbers But binary bits/bytes can also store: Pixel data for images (e.g., color codes) Digital audio (e.g., compressed or uncompressed waveforms) Program instructions (e.g., from Python or Java) Alphabet characters for display on the monitor Everything is stored in binary, but what the binary represents depends on how the binary is used CSC Computer Organization 20

21 One example ASCII ASCII American Standard Code for Information Interchange 128 characters (including null) represented by 2 7 bits. Characters are usually referred to by their 2-digit hex: 00-7F. Developed in the 60 s when Bell telephone created a digital telegraph CSC Computer Organization 21

22 CSC Computer Organization 22

23 Unicode There are other characters than in ASCII Unicode consortium handles all the characters in the world. Unicode specifies the international standard for representing characters, including emojis CSC Computer Organization 23