[CS61C FA20] Lecture 06.4 - Floating Point: Examples, Discussion — Transcript

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1. 0:00and welcome back now let's see some
2. 0:02examples and actually make you do some
3. 0:04work
4. 0:04so let's actually think about how to
5. 0:06calculate some floating point
6. 0:08back and forth from floating point to
7. 0:09decimal and back
8. 0:11so here we go first quiz pause the video
9. 0:15what's the decimal equivalent
10. 0:16of one one thousand one and then three
11. 0:19ones and that
12. 0:21go pause the video
13. 0:25and welcome back see i kind of froze
14. 0:27there like i was pretending like i was
15. 0:29first
16. 0:29okay let's do this together all right
17. 0:32well first of all i see a one i think
18. 0:34it's a negative number but i don't know
19. 0:35if the other part
20. 0:36always confuses me let's let's go so i
21. 0:38first label them
22. 0:39sine exponent significant first thing
23. 0:41okay
24. 0:42here's my equation i pull my equation
25. 0:44out minus 1 to the s can be a negative
26. 0:46number
27. 0:471 plus significance so that's 1.111
28. 0:50already times 2 to the exponent minus
29. 0:53127.
30. 0:54okay so first one is
31. 0:58what's the exponent well that's 128 plus
32. 1:00the one is 129.
33. 1:02129 minus 127. we're gonna see that is
34. 1:06minus that's two sorry that's two so one
35. 1:08point one one one times two to the two
36. 1:10means i move the binary point to the
37. 1:13right
38. 1:13two spots so that gives me minus one one
39. 1:17one
40. 1:17point one now you probably hope you can
41. 1:20do this one from here this is like fixed
42. 1:21point which is minus
43. 1:23seven point not bad
44. 1:26okay so that was a great explanation of
45. 1:28how to do that with simple numbers we
46. 1:29don't like to have
47. 1:30a lot of energy in the significant it's
48. 1:32just too much work but that's the basics
49. 1:34of it pretty cool
50. 1:35next one how would you represent
51. 1:37one-third
52. 1:39wait what this is repeating yeah it is a
53. 1:41repeating fraction
54. 1:42so in base 10 it's a repeating fraction
55. 1:440.333
56. 1:46so let's actually start well is there
57. 1:48any halves nope there's no 0.5s in there
58. 1:50is there any quarters yep that's 0.25
59. 1:53okay subtract that off what are you left
60. 1:54with okay keep doing this
61. 1:56and you'll see that it actually works
62. 1:57out to be a it's it's a uh
63. 2:00it's a quarter not an eighth plus a 16th
64. 2:04plus a 64th and so you'll see that it
65. 2:07actually
66. 2:08is a quarter a 16th a 64th 256. it
67. 2:11actually skips one
68. 2:12it starts by skipping one at the half
69. 2:14and then every other one
70. 2:16that's it so now we can do it if we know
71. 2:19that we kind of do a couple of them and
72. 2:20now we can actually do it so 2 to the
73. 2:22minus 2 to the minus 4 to the minus 6 to
74. 2:24the minus 8
75. 2:25or let's write it here 0.010101 etc etc
76. 2:33times 2 to the 0 because i don't need to
77. 2:35move that or shift that at all okay
78. 2:37so that's pretty cool
79. 2:41so now what's that really when i
80. 2:42normalize it remember i have to
81. 2:43normalize it
82. 2:45that becomes 1.01010101 times 2 to the
83. 2:48minus 2
84. 2:49okay because i had to shift the boundary
85. 2:51point to the right so now to the minus
86. 2:532. now you say to yourself
87. 2:54how do i store both that exponent and
88. 2:57the significant in there
89. 2:58sine 0 it's positive number exponent is
90. 3:02i got a minus 2
91. 3:03plus the 127 to get 125. so the bit
92. 3:06pattern has to say 125.
93. 3:09and actually it's kind of interesting if
94. 3:10you look at the bit pattern 125
95. 3:12how many off how many here am i missing
96. 3:14this looks pretty
97. 3:15how am i missing from being all ones
98. 3:17well it's really here i'm missing a two
99. 3:19so you can kind of figure out quickly
100. 3:21minus two that's i know that i'm too off
101. 3:23of that which is 125
102. 3:25because i'm missing the twos bit okay
103. 3:30significand
104. 3:32i just i'll say that another way 127 is
105. 3:350
106. 3:35and all ones minus 2 says
107. 3:39minus 2 which is why there's a hole here
108. 3:43and that 1 minus 1 is 0 and that's what
109. 3:45that hole is just to make sure that it's
110. 3:46significant is zero one zero zero one so
111. 3:49i'm left with
112. 3:50this bit pattern and that is a third
113. 3:52kind of a neat bit pattern
114. 3:55all right so let's see if we can
115. 3:57actually understand the
116. 3:59significant a little bit better and
117. 4:01remember i i mentioned this a couple of
118. 4:02videos but i want to be clear about it
119. 4:060.34 if i give you that some decimal
120. 4:08back in it
121. 4:09you can say you know 0.34 is really like
122. 4:11oh sorry 0.340.
123. 4:13it's like having 340 over a thousand
124. 4:16or it's like 34 over 100. so it's 3400
125. 4:19that's no way to say that
126. 4:21same way 0.110 let's think about that
127. 4:240.110
128. 4:25that's like saying 6 over eight i had
129. 4:27three bits for the eight
130. 4:29six over eight or three
131. 4:32over four same thing point one one over
132. 4:35a hundred so this is nice black people
133. 4:38like this way of thinking about the
134. 4:39significant
135. 4:40there's another way of doing it which is
136. 4:42you just add up the pieces of it we saw
137. 4:45add up the place values
138. 4:46so if i have 1.6732 it's really like
139. 4:49one ones six tenths seven
140. 4:52hundredths etc so 1.100
141. 4:56is another way of saying well i have 1 1
142. 4:591 half and 1 16. so i can either think
143. 5:03of this number as
144. 5:049 16 or a half plus a 16
145. 5:09which is 9 16. so the two ways of
146. 5:11thinking about this
147. 5:12let's uh please choose the one that's
148. 5:14more appropriate for you that kind of
149. 5:15makes
150. 5:16sense for you as you're thinking about
151. 5:17these numbers but that does help to be
152. 5:18able to
153. 5:19to work with binary points on the
154. 5:21fractional side kind of in a fixed point
155. 5:22way
156. 5:23all right see the next video