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[CS61C FA20] Lecture 06.2 - Floating Point: Floating Point — Transcript

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0:00and welcome back when last we left our
0:03hero we saw that
0:04maybe there's an idea of not just
0:07locking in these six bits and saying
0:09that the binary point is
0:10two in from the right but maybe there's
0:12another input on the side that can
0:14actually
0:15change where that binary point is and
0:18that second input
0:20is the aha with floating point
0:23so again before we've locked in fixed
0:26point we fix it there
0:28but what if i were able to also pass in
0:31another input which tells you where the
0:34binary point might be
0:35and that allows the binary point two
0:37drum roll please
0:40float float it's not fixed
0:43it can float and that beautiful idea of
0:46it floating back and forth
0:47is the power of this representation it's
0:50like another level of abstraction rather
0:51than locking it here
0:52i'm now abstracting away and saying what
0:54if it's more general now i can now move
0:56it where it is
0:57so that's just really amazing so that's
0:59a good example
1:01here's 0.1640625 it's in binary
1:05if i represent this in five bits and i
1:07choose where to put the binary point
1:08okay
1:09so here is this i've got point zero it
1:11says here point zero
1:13zero 10101 at 101.01 and then all the
1:17zeros
1:19so all i kind of need to know see the
1:22guys in blue
1:23all the need to know is i need to
1:25somehow send
1:26these five bits if i could somehow send
1:28those five bits i need to send the zeros
1:30before it or zero's after it
1:31or over here i don't need to send those
1:33it's just default it's always zero
1:35i need to i need i almost need to send
1:37you where the energy is where are the
1:38ones and zeros that are interesting
1:40but i kind of need to freeze it on the
1:41left side and the right side with ones
1:42and the zeros outside of it
1:44so if i could just send that set and
1:47with my
1:47other input that's the key idea i tell
1:50you oh
1:50the binary point should be two to the
1:52left of the leftmost
1:54one so i would say like you know somehow
1:56minus two
1:57sitting at minus two that tells you it's
1:59over there
2:00and on this guy i'm sending 10101.
2:04so that's kind of cool so you can
2:05imagine taking two inputs this
2:07is the energy i say what the energy is
2:09and this is
2:10where the the floating point would be
2:13and that's amazing that's amazing
2:17so that is the idea we're gonna have
2:20some fields to our representation one
2:22field tells you kind of where the energy
2:24is
2:24and one field tells you in a way the
2:26exponent tells you where it is it's
2:28it's multiplied times two to the what
2:30right it's times two to the
2:31minus two here's two to the minus two
2:33because it's two over okay it's the
2:35quarter this is a times a quarter
2:36so what it's saying is it's saying
2:40i want to be able to have this exponent
2:42for where that moves every time i move
2:43it to the left it's multiplying by a
2:44factor of two so it's really what two to
2:46the what
2:47two to the zero it says it's exactly
2:48where you think it is to the one says
2:50move it you know move it one to the
2:51to the left and now i have a zero to the
2:53right etcetera
2:54is that cool or one to the right move
2:56one to the right i guess what is but one
2:57to the right okay
2:59so because the binary point
3:02where it is it's different from the
3:03actual stored bits what i call the
3:05energy
3:06now i can do large and small numbers now
3:08i've got the first thing i can do really
3:09small numbers or really big numbers
3:11or big and small and i got it so let's
3:13now review scientific notation
3:15in decimal back in the old days now
3:18we're in i think we're probably in
3:20physics class maybe ap maybe chemistry
3:22back in high school 9th or 10th grade
3:24if you remember here are some words just
3:26to remind you what these words mean
3:28i had a decimal point that's the obvious
3:30thing i've got the radix which we call
3:32the base
3:33i've got the exponent and i've got the
3:35mantissa on the left
3:36okay so if i live in normal form
3:41normal form says that i always have a
3:44single
3:45digit to the left of the decimal point
3:48so i don't have
3:49if i want to store 1 times 10 to the
3:50minus ninth okay so
3:52or a nano something i wouldn't store it
3:55as
3:550.1 times 10 to the 8th and i would not
3:58store it as 10 or 10 to the minus 10
4:00even though they're equivalent
4:00mathematically i would not do that
4:02because they're not in normalized form
4:04so we're going to go with a normalized
4:06form there
4:07so that's pretty cool so let's see what
4:11that looks like when we go to
4:13binary same idea
4:16three out of those four words are
4:17exactly the same mentis is there the
4:20exponents up there the base is base two
4:22not base ten
4:22and now it's the binary point we call
4:26this
4:27floating point numbers officially
4:28floating point numbers and if you've
4:30ever seen the phrase
4:32in c float that's it
4:35this is what happens we're storing this
4:37this story in this way we're storing in
4:39this way
4:40in normalized form where on the left is
4:42always going to be a one
4:44that's the important thing always going
4:45to be one to the left of the binary
4:47point
4:48in normalized form this is it i feel
4:50like i
4:51always it's such a delight to teach
4:53somebody something new for the first
4:54time and this is it this is
4:55the this is the beginning of what we
4:57call the ieee floating point
4:59designation so here's normalized form
5:02you're gonna say but dan
5:04if every number starts at one point
5:06something you might say to me
5:07why do you have to transmit it why would
5:09you store the one on the left if it's
5:11one on the left
5:12always there's always going to be one on
5:13the left unless it's all zeros there's
5:15gonna be a leading one
5:16why store it where it's always gonna be
5:18one and you're like
5:20i'm you're right so we actually don't
5:22store it we only store
5:23the number to the right which is the
5:25significant
5:26so we're one point the one point is
5:29always kind of default there
5:32yyyy is our exponent so this exponent
5:37goes here
5:38here's a significant there and you're
5:40going to say well dan we probably should
5:41have some negative numbers too
5:43and you know to make this easy what
5:45we're going to do is
5:46let's actually use a sine magnitude
5:48model so that left is a sine bit
5:51so there we go s is your sign bit bloop
5:55exponent are these eight bits there that
5:57returns two to the y
5:58and the significant are the x's
6:00represented there with 23 bit that's a
6:02lot
6:02one point 23 more bits that's quite a
6:04bit actually
6:06it's pretty cool and by the way what's
6:08this last bit controlling
6:10what's that last bit if it's always one
6:12point something that last bit is two to
6:14the minus 23.
6:15so it's one plus two to the minus if i
6:18only have a one here and have zeros all
6:20in here
6:20that last guy is two to the minus 23 if
6:22you think about that 23 bits there on
6:24the right it's that it's one point this
6:25to the minus 23. remember there was
6:27there was four bits to the right it was
6:28once one sixteenth it was two minus four
6:30well 23 bits across
6:32two to the minus 23 is the guy all the
6:33way on the right what can we do
6:36well this just this nothing special
6:39nothing more than that no
6:40special cases which we're going to see
6:41in later lectures
6:43we can get to 1.2 times 10 to the minus
6:4638
6:46really small numbers and up to 3.4
6:50times 10 to the 38. that's amazing this
6:53is
6:53really cool now you're going to say well
6:55dan
6:56what if they're too large what if i mean
6:58that's pretty good
6:59that's pretty good and we're going to
7:01see the ain't no free lunch coming up
7:03back again but we'll talk about that in
7:05a second so what if they're bigger than
7:06that what if they're either bigger than
7:08the high side or
7:09smaller than the low side well that's
7:11overflow
7:12you know i want to restore a number
7:14bigger than that i what if i double it
7:15and double that
7:16and i can't always in forever do that
7:18double that double that i got a fixed
7:19bit
7:20bit width at some point it's going to be
7:22bigger than i can store
7:23we call that overflow what if it's even
7:25smaller than it what if it's half of
7:26that half of that on the negative side
7:28half
7:29it should be double a negative number
7:31i'm sorry if you double the negative
7:32number and keep
7:33pushing the left the left side it's also
7:35overflow
7:36some people call it negative overflow
7:37but it's also overflow on the big sides
7:40on the ends towards infinity always
7:42overflow sometimes the left they call it
7:43negative overflow
7:45what if they're too small aha
7:49underflow you thought that the negative
7:51infinity was
7:52under flow it's not if that's overflow
7:54that's not a flow underflow is towards
7:55zero what if i take a number and have it
7:57and have it and have it
7:59and it keeps getting smaller and smaller
8:00at some point i reach the limit
8:02of what i can do and one past
8:05that is underflow i would probably would
8:07say it's zero even though there is some
8:09number nope
8:10the closest number that i can store to
8:12what you're asking me for is zero
8:13so i kind of underflowed to zero and
8:15that's that's a problem because it means
8:17that
8:17oh i made some fractional amount but
8:19then it says zero but it's not really
8:20zero it's close to zero that's an issue
8:22so again underflow is close to zero
8:25overflow
8:26is on the sides we'll just call it
8:27overflow in 621c rather than negative
8:29overflow but that's right what would
8:31help think about what would help reduce
8:33the chances of overflow and underflow
8:36throw more bits at it right throw more
8:38bits at the problem
8:40so here is the ieee 754 floating point
8:44standard used in every computer you have
8:46access to
8:47that's it sign bit
8:50eight exponent bits 23 significant bits
8:54that's great by the way just a refresher
8:56one is going to mean
8:57a negative number and 0 is going to be a
8:59positive number
9:01so that's pretty cool now we mentioned
9:04this before to pack more bits the
9:06leading one is going to be implicit
9:07remember that's normalized form and why
9:09store that one if it's always going to
9:10be a one so
9:11that's pretty cool if i actually want
9:13more
9:14if i want i want to have more resolution
9:16i want to reduce the chance of
9:18overflow or underflow throw more bits of
9:20the problem call it a double
9:22so we'll have actually a double for our
9:24significant for that i have 52 bits for
9:26that
9:28so what's always true is the significand
9:30is between
9:31zero and one for normalized numbers
9:33because if you remember it's one point
9:36all those bits 23 or 52. think about
9:38that one point this
9:40so this the this the significant is
9:42point
9:43something goes from zero if it's all
9:45zeros to all ones but it's still a
9:46fractional number
9:48it's probably with 23 it is one
9:51over two to the 23 away from one so it
9:53is
9:54um i don't know how to say that it's a
9:57it's a
9:57number very close to one how close is it
9:59to one one
10:01over two to the 23 away from 1. that's
10:04how it is
10:05and uh here's the interesting thing 0
10:09has no leading one we said this is weird
10:10case that 0 has no leading 1
10:12so we're going to reserve the exponent
10:14value of all zeros
10:16just for zero which is great so now i
10:19have
10:19exponent of zero so i wanna store zero
10:22there's a zero
10:23there's a zero and ah wait wait sine bit
10:26do you remember our issue with sine bits
10:28with sine magnitude i got two zeros
10:33what do you think we got two zeros here
10:35too
10:37so here's the other part that's
10:39interesting
10:40if zero is the smallest exponent
10:44in terms of the bid pattern and we go up
10:47wouldn't it be nice if all zeros
10:51here's the thing all zeroes were the
10:54smallest one and we kept going up
10:55continually not having to wrap like
10:57remember signed
10:57it starts in the middle goes up and then
10:59it snaps over here and does this thing
11:01i want it to be zero all zeros is the
11:02smallest one and go up
11:04but if i want the small here's the part
11:06that's a little weird i want that
11:07smallest one
11:08to be in the negative because i want to
11:11do fractional numbers right i want that
11:13to be two to the
11:14negative number so if i want all zero
11:16bit pattern just might be a negative
11:17how does this going to work okay
11:22so here's the designer the designers
11:24went back to the drawing board they said
11:26we want to use if i have no floating
11:28port hardware by the way back in the day
11:29they used to have
11:30explicit floating point hardware and we
11:33still have that
11:34but if i didn't in the early days of
11:36machines they didn't have special
11:37purpose floating power they just had a
11:38normal cpu
11:39but they want to be able to use the cpu
11:41to compare and do things to do some
11:43manipulations with
11:44floating point numbers so they came up
11:46with the idea that
11:48bigger integer exponents are bigger
11:49numbers the smaller ones are fractional
11:51numbers
11:52and they thought to themselves they said
11:54think about the odometer the only way to
11:55do that is that bias notation
11:58so therefore kind of a really small bit
12:00patterns very few numbers and low low
12:02bid patterns like you're driving the
12:04binary odometer a little bit
12:05that should be a negative number and a
12:07high thing should be a really big
12:08positive number well that's a bias model
12:11if you think about that that takes this
12:12you know zero and this and here's
12:14all ones and if you shift them down by
12:16some offset
12:17then this is going to be overall a
12:19negative number and this will be a
12:20positive number
12:21if this is from 0 to 55 then roughly
12:23this would be kind of
12:24128 to minus 128 ish somewhere on there
12:27and that's pretty good so we'll shift it
12:29down by some bias and we'll get what we
12:31want
12:32love this so bias notation
12:35and the bias is subtracted and we've got
12:37eight bits you remember how we did eight
12:38bits before
12:39eight bits is two to the n minus one
12:42minus one
12:43so two to the n minus one so eight bits
12:44two to the n minus one eight
12:46n is eight that's 128 minus one is 127.
12:50so we call the bias 127. what that
12:52really means is you subtract 127 from
12:55the value of that so you look at the
12:57unsigned value
12:58subtract 127 and you've got your actual
13:01number
13:02so that's it so there's our equation
13:03let's take a look at our equation now
13:06this is actually a kind of interesting
13:07thing minus 1 to the s when s is 0 minus
13:101 to the 0
13:10is 1. so that's just a normal positive
13:12number if s is 1 it's minus 1 to the 1
13:14which is minus number so actually that's
13:16kind of a cool way to think about the
13:17sign bit it's minus 1 to the sign bit
13:20times 1 plus significant remember this
13:23there's
13:23there's the implicit one times
13:262 to the exponent whatever the raw value
13:29is
13:29minus your 127 bias that's it
13:33folks that is the ieee 754
13:37floating point there's still some more
13:38details there but that's pretty cool and
13:40double is exactly the same
13:41it's just you have more bits for the
13:43significant and more bits for the
13:44exponent but exactly the same equation
13:47that's pretty cool and quad
13:50quad is bigger than that same idea just
13:52more again bits for the exponent
13:54more bits for the significance let me
13:55introduce you to my colleague professor
13:57velvel
13:59kahan he's the father of the floating
14:02point standard uc berkeley
14:05ieee standard 754 for binary floor
14:07arithmetic
14:08earn him the turing award the nobel
14:12prize in computer science
14:14for being the leader of this boy
14:17floating point
14:18space the ecosystem was chaotic before
14:22his team of folks said let's kind of
14:24normalize this so that
14:26if i calculate something on one computer
14:28and bring it to the other computer i've
14:28got the different number
14:29this was happening all around them and
14:31so i would try to compare oh right to
14:32the trajectory and it landed here and
14:34yours that
14:34why because the way the algorithms were
14:37the same but the way the floating point
14:38was done below the line
14:40in the hardware was different on every
14:41machine he said that's crazy let's have
14:43a standard
14:44let's have a way that my calculation can
14:46move to your computer remember porting
14:47was an issue
14:48well it's big indian small indian all
14:50these weird things that are different
14:52about this answer different sizes
14:53it's even worse back in the day in the
14:56floating point space your machine was
14:58completely different from our machine we
14:59couldn't trust our numbers
15:01so the scientific community said we got
15:02to centralize this he led the team to
15:05centralize it came up with ieee standard
15:07754
15:08and won the turing award back in 1994.
15:10amazing right
15:11amazing work so we're going to see now
15:14more details about this we're going to
15:15see
15:16does that cover everything well that's
15:18almost 90
15:19of it in terms of intellectually 90 of
15:20this but there's a lot of interesting
15:22ideas now that you know the basics of
15:24floating point where can we go with that
15:26we're gonna see that in the next set of
15:27videos see you there

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