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[CS61C FA20] Lecture 06.5 - Floating Point: Floating Point Discussion — Transcript

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0:00and welcome back now let's have a
0:02discussion about some of the issues of
0:04floating point
0:06so first there's some fallacies we want
0:08to
0:09dispel um is add associative you know ad
0:14addition should be associative right
0:15that's you know one plus two plus three
0:17whether you add the one and two first or
0:19the two and three
0:20first it should be the same number it
0:22should be six
0:23well is that true for floating point add
0:26here's an example
0:26so let's make x a really small
0:30uh a really big negative number so minus
0:321.5 times 10 to the 38.
0:34let's make y exactly the opposite but a
0:37positive number
0:38and let z just be one and we even saw
0:40how to do one in a previous video
0:43let's first add y and z together when
0:46you add y and z together get this really
0:48big number
0:49and this really small number and the
0:51number of bits that you can play with is
0:53really 23
0:54bits total if you include the
0:56normalization there's like 24 total bits
0:58from like left most bit to the rightmost
1:00bit that's a one
1:01and the problem is you don't have enough
1:03bits at least in the floating point and
1:05a float
1:06to store both the one and this really
1:08big number over here
1:10so you can't do that um so when that
1:13then when that when this right most let
1:14me get my thing when this when this adds
1:16together that one goes away
1:18i can't store this one and that one the
1:20bigger one wins
1:21we lose the one well that means it's as
1:24if the z wasn't added at all
1:26and i get x plus y which is zero
1:30well now let's try it again seeing if
1:32it's associative or not let's try
1:35the leftmost one
1:38so the leftmost one says
1:42hear the leftmost guy
1:45and x and y and they cancel and then you
1:49have one and so
1:50these two are not equal so therefore in
1:53summary floating point add
1:54is not associative really important that
1:57you saw that
1:59and the reason for that is we you just
2:01have a fixed number of bits you can't do
2:03more than that if you have a really big
2:04number and really small stuff you can't
2:06keep both those guys alive you could
2:08imagine a third
2:09maybe you wanted to invent a third way
2:11to have big numbers and small numbers
2:13and have two pieces of them and
2:14i mean that's kind of like having two
2:16separate numbers anyway but you can't a
2:17single number can't have b both really
2:19big and really small
2:21so that's i mean this bullet says
2:24exactly what i said
2:25the number is so much bigger than one
2:26that it can't the representation cannot
2:28store both of them at once and so it
2:29loses it in one
2:30keeps it in the other because they
2:31cancel each other out and one is not
2:33equal to zero
2:36i wanted to give you a definition just
2:38so that you're using the right words as
2:39you're talking about floating point
2:40numbers
2:41um so precision is if this is
2:44any representation um it's the number of
2:47bits we're throwing at the problem so
2:49precision oh i want more precision
2:50throw more bits of the problem oh okay
2:52float only has this number of bits well
2:54double that or quadruple that
2:55you'll get more precision the more bits
2:57you have the closer you can get to the
2:59actual number you're trying to get to
3:01accuracy is the distance between any
3:04number that you have coded and the
3:06original
3:06actual number so his example high
3:09precision
3:10a lot of bits permits high accuracy but
3:12doesn't guarantee it
3:13here's an example here float pi equals
3:163.14
3:17well there's going to be lots of bits
3:2020 you know 32 bits associated for that
3:23but you're still pretty far in terms of
3:25accuracy from the actual value of pi
3:27now you could have done more i could
3:28have initialized it to a closer number
3:30but that
3:30typical example says i got a lot of bits
3:33but it isn't
3:34close to the actual number so that has
3:36high precision
3:37low accuracy in that case
3:40now let's talk about rounding so
3:42rounding is this complicated thing
3:44uh where you're doing some calculation
3:48and then you figure out okay well i'm
3:49not exactly at
3:50one or the other i'm not exactly at a
3:52floating point number so do i go
3:54the one above or the one below that's
3:56the idea of this
3:57and there's some hardware that usually
4:00carries with
4:00the usually floating point hardware has
4:03a couple of extra bits
4:04on the far right side that's called
4:06rounding bits
4:07and those extra bits can be used to
4:09figure out where do i really want to go
4:11the floating point number above
4:13or the one below in terms of that so
4:14this is the 32 bits and there's extra
4:16bits on the right side of that
4:17that's useful so there are modes when
4:20i'm in the middle let's say i'm some you
4:21know the rounding bits say that there's
4:23something in here that's not a zero down
4:24here my rounding bits means there
4:25actually is something to the right of
4:26that
4:28you could either there's four rounding
4:29modes you could either round
4:31always toward positive infinity always
4:33to the right on the number line
4:35or always to the left of the number line
4:36always towards minus infinity
4:38or just truncating which actually means
4:40always rounding it towards zero
4:42or here's the more clever way you could
4:45have it in an
4:46unbiased way rather than always bigger
4:48or always smaller
4:49or always towards zero you could
4:51sometimes go up and sometimes go down
4:54and that means when you're midway well
4:56this by the way these examples i'm
4:57showing you are all examples
4:59in decimal this is not the same as
5:01binary but you're obviously working with
5:03binary we're working with floating point
5:04calculations so in the first example you
5:07round up that means the 2.001 well you
5:10round toward the next
5:11that's in the middle between two and
5:12three that would round up to three
5:14okay and minus two would round to two we
5:16always round to the right
5:17the same idea always round left one
5:19point nine nine nine rounds to one even
5:20though you're almost at two you go back
5:22to one you lose it
5:24and minus 1.99 also goes down also goes
5:26negative
5:27so here's the case when you're in the
5:29middle so normal rounding
5:31um 2.4 again this is in decimal
5:34is less than halfway so it round down to
5:362. 2.6 we're
5:38more than halfway round up 2.5 is
5:42exactly in the middle
5:43so that was going to round toward the
5:45even number so that rounds to
5:472 but 3.5 would round to 4. 4
5:51nice so think about this i've got some
5:55binary equivalent of this but this is
5:56now in binary
5:57so i've got some binary numbers here
5:59okay 0
6:011 0 1. here are the rounding areas okay
6:05so now
6:07what's halfway between this by the way
6:09if i have many bits here let's say i
6:11have let's say i have three bits let's
6:12just say i have three bits
6:14those bits would be from zero zero zero
6:17to one one
6:18one what's halfway in the middle in
6:20binary
6:21remember what half is this yeah it'd be
6:24a one
6:26and zeros that's halfway in the middle
6:29so when you're that way that's when this
6:31k when when the rounding vowel
6:33bits are one and all zeros that's
6:35halfway in the middle
6:36and then you're making a decision in
6:37this particular mode which way to go
6:40well you round toward the even number so
6:43you ask is that an even number no that's
6:46a one that's an odd number
6:47so therefore i'm going to round not
6:49toward that number but toward the one
6:50above
6:51and that would round up so this would
6:53round to 0
6:541 1 0. okay
6:57if i had 0 1
7:01zero and then here's the rounding guys
7:03and i were one
7:04zero i would round toward the even i
7:06would round down
7:07which means around down toward the even
7:09and i would just cross it off and that
7:11would be the closest even
7:12does that make sense so you round toward
7:14even in binary
7:15which means you have to be a one zero
7:17zero zero zero zero zero in the rounding
7:19bits and you round toward the even
7:20number if it's a one there you round up
7:22you add one to it if it's a zero
7:23you just truncate those rounding bits
7:25okay that way we kind of round
7:27both ways and that's kind of that's
7:29really we really like that as a default
7:30mode so that's the default mode for the
7:31system
7:32because it kind of is fair it doesn't
7:33always go make things bigger or make
7:35things smaller
7:36um however irs if you're calculating my
7:38my return
7:39can you round up always just don't round
7:43the middle so now you know when you see
7:46these errors here's this wonderful comic
7:48strip i found online
7:50uh and here is a robot and it says
7:53welcome to the secret robot internet
7:55prove that you're a human point one plus
7:570.2 0.3004
8:01we can represent powers of two perfectly
8:03fractional powers are two perfectly in
8:05floating point
8:05you know because it's zero in the
8:06significant and some value of the
8:08exponent
8:09but but because we're decimal we can't
8:12do point one or point two
8:13very easily because it's one tenth this
8:15ten thing is different there's ten
8:16fingers with a problem
8:18so this is an example of point three if
8:20you type point three into my simulator
8:22you get 0.300 and it's not exactly
8:26o4 maybe that's a double but in the
8:27float in a single precision or 32 bits
8:30of a float we get 0.3001 so this is
8:32exactly why
8:33you would see this error
8:37addition let's talk about addition now
8:38we're just talking about elements of
8:39this
8:40rounding and addition and truncating and
8:42all those things
8:43so addition uh is a little bit harder
8:46uh because you can't just add the
8:48significance um you have to denormalize
8:51to match the exponent you have to kind
8:52of line them up okay
8:54to match the exponents then you add the
8:57significance to get the resulting guy
8:59and then you have to keep the same
9:00exponent and then normalize to get
9:01the end so you have to kind of shift it
9:03do something and then shift it back
9:04whatever the result is they may cancel
9:06each other out to make something
9:07you may have a big number and a big
9:08number but it turns out that they cancel
9:10most of the big stuff and the small
9:11values of what's left then you have to
9:13renormalize to get that right if the
9:14signs
9:15differ you just do a subtraction instead
9:18this is how you do casting you've seen
9:20casting when you do malik
9:22when you go to the godfather you have to
9:24cast that un
9:25you know uninitialized void star space
9:28into whatever type you want so that you
9:29know
9:30you know pointer plus plus knows how
9:31much to jump that's the reason we do
9:33that
9:34and also to check you know whether
9:35you're whether you're matching these
9:36right and everything ever the pointers
9:38work out
9:40type tracking obviously
9:43here if you've got a floating point
9:45number you want to convert to an integer
9:46just say
9:46int if you remember in python it was it
9:49was in
9:50parentheses there's a function call to
9:51that as a method call to that here you
9:53just type cache with the parenthesis in
9:54so here's 3.14159 times some float
9:58that's obviously a float and now you
9:59cast that to an end to figure what the
10:01closest energy
10:02that might be conversely you do the same
10:04thing on the other side if you have an
10:05integer number you want to float convert
10:07it like that so you say
10:08float of that integer expression and now
10:10you're able to work with that so if i
10:11have f as a float i wanna
10:13f plus equal another float well if i
10:15have i which is int you have to convert
10:17it to a float so
10:18that the compiler will be happy so
10:21now you might ask yourself well if i go
10:23from an into a flow back to an int
10:24do i get the same number does this
10:26always print true
10:30pause the video and think about it talk
10:32to your neighbor and then come on back
10:35and we're back if i equal equal int a
10:39float of i
10:40same thing ain't
10:43no free lunch there are integers
10:48that the float can't handle
10:51so there are certain integers in fact i
10:54even showed you one i think it was
10:55a little bit more than 16 million that
10:58has no
10:59float equivalent you know you can count
11:01up to four billion in in floats and
11:03integers
11:04unsigned in numbers but floats the first
11:06number you can't do we saw that before
11:0816 million plus one i can't do that
11:11number 16 million plus one is
11:13that can't be done so i mean two to the
11:15two to the fourteen two to the 24
11:18plus one can't be done i showed you
11:20before
11:21so we do that it's going to snap it to
11:24the closest float
11:25when you go back to int it's the back
11:27end the same end i can do that it'll
11:28either be
11:29not think about rounding modes that
11:31number is two to the 24
11:33plus one can't be done so does it snap
11:35to the fourth part does it do 22 24
11:38or 2 to the 24 plus 2 well this is an
11:41even number so it'll round to even
11:43so that's an exact case of that rounding
11:45mode there that's going to be
11:462 to the 20. try it 2 to the 24
11:50plus 1. see what it says that'd be
11:52really fun to play with
11:53okay can't be done sorry
11:57double precision so let's talk about
11:59double precision could that do it you
12:00want to ask yourself look at the
12:02double precision could double precision
12:03do that as well
12:06float to int to float does this always
12:09return true
12:10pause the video and think about it and
12:12come on back
12:14and we're back ain't no free lunch
12:19what if the float is 1.5 this is easiest
12:21one 1.5
12:22obviously there's no int that does 1.5
12:24and you go back it's going to be
12:26the integer representation that'd be
12:27either one or two what do you think it
12:29does
12:301.5 round to even
12:33the closest even number is 2. so that
12:36should in theory be
12:37two when it comes back so that should
12:39not be the same that's an obvious one
12:41why that doesn't work we'll see the next
12:42video

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