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[CS61C FA20] Lecture 06.3 - Floating Point: Special Numbers — Transcript

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0:00and we're back so now let's play with
0:03some special numbers we can do floating
0:04point numbers and that's
0:05really good we're you know really really
0:07far into this
0:08but let's think about where how we could
0:11use some of the bits in a clever way we
0:13call them special numbers
0:15what's really neat
0:19you now realize that there are numbers
0:22big really big numbers right 10 to the
0:2538ish or so
0:26but what's past that wouldn't it be nice
0:30if there was a representation for
0:31infinity
0:33they said you know we're doing floating
0:35point we're doing you know computational
0:36science
0:37sometimes it's useful to have an
0:39infinity it says that i didn't just
0:41overflow but i overflowed and now
0:42i'm going to store without being an
0:44error the number infinity as part of
0:47this
0:47so why not produce it not an overflow so
0:50and that's actually really useful you
0:51know if i infinity plus some
0:53some other number is still infinity
0:54right um infinity is bigger than every
0:56other number there so you can actually
0:58use infinity
0:58you can by the way if you've ever used
1:01if you've ever done
1:02um an iterative calculation
1:06of how to do um the max of a number
1:10here's you know here's a list of numbers
1:11i'm gonna do the max what's the identity
1:14of max
1:15identity of max says it's the value that
1:18if you maxed
1:19any number with that it wouldn't change
1:20what the max was you know what the
1:22identity of max is negative infinity
1:25whatever number you have in your space
1:27if you max it with negative infinity the
1:30number stays the same
1:31so if you're using a max and you're
1:33starting you say well let's let's
1:34default to the first element of max is
1:36you know how do you initialize max you
1:39know temporary you make some things and
1:40you're
1:41accumulating you're going through a list
1:42and you say what do i initialize max to
1:44you know my max
1:46the max would initialize it to well you
1:48can initialize it to the first element
1:49that's reasonable
1:50you could also initialize it to minus
1:53infinity
1:54and then as you hit the first element
1:55you say well hey my max is
1:57the max of my max the old version the
2:00old number and the first element the
2:02eighth element i'm looking at
2:03well this will always be bigger than
2:06minus infinity bloop
2:07it works so rather than initialize it to
2:08the first element initialize it to
2:10negative infinity which means it helps
2:11to have negative infinity
2:13in your representation pretty cool
2:17so we can store that here's how it's
2:19going to do we're going to represent
2:21the most positive exponent and by the
2:23way here's a cool thing if you look at
2:24the binary bits
2:26you're counting up counting up it
2:27actually counts up in a really beautiful
2:29way we'll talk about how to interpret
2:30this
2:30and then one more from the biggest
2:33number
2:34is infinity it's this beautiful idea
2:38so it's most positive exponent biggest
2:41exponent
2:42all ones eight bits of all ones
2:44significant is all zeros though
2:46okay because the biggest up to that was
2:50all ones minus one so it was 254
2:53and unsigned and so that was the biggest
2:57eight bits seven bits seven bits of one
2:59one bits of zero that was the biggest so
3:01far
3:01all ones in your significant all one one
3:05more
3:05one more let's wrap what drive one more
3:07mile in the binary odometer
3:09this goes up this becomes zero that
3:11becomes all ones
3:12that's your infinity so it is one past
3:14it's like literally getting bigger and
3:16bigger bigger as you're counting your
3:16binodometer bigger and bigger bigger and
3:18then
3:19bloop there is infinity the next one
3:21over really beautiful
3:23zero is before so significant is
3:26all zeros the exponent is all zeros and
3:29the sign we got two zeros we knew this
3:30when we got it
3:31you already knew when i said sign
3:33magnitude you said two zeros immediately
3:35you knew that
3:37so here's what we got here's what we
3:38have if we look at this beautiful table
3:40so what we defined so far is we have all
3:42zeros is zero this is let's assume just
3:43looking at
3:44you know you're ignoring the sign
3:46obviously so far just looking at the
3:47exponent and this significant
3:50we've got if we're going to reserve 255
3:54and none and so for 255 and 0
3:57that's going to be my infinity my normal
4:00number is going to be
4:011 to 254 and anything okay that's the
4:04normal floating point i've seen before
4:06but we haven't told you if we're going
4:07to reserve 0 and 255
4:10when this is non-zero well
4:13that's really interesting let's actually
4:16play with this so far
4:17professor kahan said waste not want not
4:21don't throw bits away if you're saying
4:22ain't no free lunch
4:24and we're having trouble you know
4:25dealing with all the
4:27making use of all the bits and having to
4:28give some to get some well i got these
4:30free bits around if you just hand me
4:31free bits
4:32i want to get some i want something out
4:34of that so what should we use those free
4:36bits for those free bit patterns
4:38here's the really clever idea you ever
4:41have a square root of minus four
4:43if you don't have complex built into
4:44your system or zero over zero
4:48is there error normally errors wouldn't
4:50it be neat if you had a representation
4:52for
4:53not a number it isn't a number it's kind
4:55of some kind of error
4:56so we're going to reserve here's the
4:57cool thing the biggest exponent
5:01non-zero significant for nands
5:04that's awesome now why is that useful
5:07because
5:07here's the cool thing they contaminate
5:11as you're working with something deep
5:13deep down in the bowels of your code
5:15you create a nan well the anything well
5:18this how about this number was the
5:20result of
5:202 plus that and then whatever that is
5:22it's ten times that
5:24and it's six divided by that every time
5:26that nan
5:27has another operation with any other
5:29number it stays a nan nan's like a black
5:31hole
5:31zhu you can't get a nan out of a nan so
5:34that means that nan will percolate its
5:35way
5:36all the way up and be nan at the top
5:38level so again
5:39low low level nan will bubble its way up
5:41because they contaminate other
5:42operations
5:43that's cool so now at the top level you
5:45see nano up some must have been
5:46wrong down and then you trace it down
5:48with your gdp debugger or other debugger
5:51now you're gonna say well dan that's
5:54two to the 23
5:58minus one possible nands
6:03they had they had such brilliance in
6:05their design
6:06they said that's a lot of bits 223 500
6:09folks is 8 million
6:10bits ish they said
6:14well not only could you have a special
6:16code remember
6:17you could do anything with bits right so
6:19you could almost think about those bits
6:20as if there were some agreement
6:23the particular hardware could have some
6:25set of bits say what kind of error it
6:27was
6:27well this was a square root of a
6:29negative number
6:30this was a zero over zero this was a
6:33some
6:34hyperbolic thing whatever errors might
6:36get caused by whatever mathematical
6:37operations you want to do
6:38you'd have those errors and you might
6:40have a code for that
6:42you also might have in some of those
6:44bits
6:45you could actually have
6:48the line number where it happened or any
6:50other debugging information could be the
6:52so you could make use of those eight
6:55million possible bit patterns
6:56to actually encode where things were if
7:00i
7:00down deep know that this happened on
7:01line number 17 why not return a nan
7:04in which this part of my significant was
7:07what kind of nan it is
7:08and this number bit says printf this was
7:11and you could have some kind of encoding
7:12for
7:13what line it was what it was where what
7:15what program it came from what file it
7:17came from you could encode anything
7:19there
7:20so that's really neat and very clever
7:22that they used 8 million bits with the
7:23potential
7:24that people could use that in a clever
7:26standardized way that you could bubble
7:27up in a debugging way what the nan
7:29what caused an end and maybe even how to
7:30resolve it here's some bits for how to
7:32fix it
7:33that's pretty cool so that's the idea
7:35with nands
7:38you remember there also were these bits
7:40where the exponent was zero
7:42but the significant was non-zero and
7:44here is the problem
7:46okay there's a gap
7:49there's a gap around zero by the way
7:53whatever encoding you ever want if
7:55here's a zero
7:57then you have a next number the next
7:59number should be kind of like
8:00equal steps it should be like a you know
8:02like a
8:03a ruler where the the distance between
8:06zero and the first number
8:07and the first or the second number are
8:08the same it should be like that that's
8:10not what happens with floating point the
8:11one i taught you so far
8:13all you know about floating points so
8:14far shows you this
8:16what's the smallest representative
8:18positive number okay
8:20well it's this one point
8:24all zeros right one point all zeros okay
8:27smallest number
8:28times two to them what's what's the
8:30number the exponent was gonna be here's
8:32my field okay let's do this together
8:35exponent is gonna be one remember that's
8:37the smallest one we allowed for
8:39so that 1 minus 127 is 2 to the minus
8:42126.
8:43that's my thing okay so this is 1 times
8:462 minus 126 okay so that's a that's a
8:48pretty small number we're pretty happy
8:49with that
8:51what's the second smallest number it
8:53should be double that right i talked
8:54about before right you want a ruler
8:56where
8:56blink 0 smallest number next smallest
8:59number right equal spaces
9:00so the next one should be 2 times 2 to
9:03the minus 1 26 or you'd want to have
9:052 to minus 1 25 is the next biggest one
9:07right that's what you want okay doubled
9:08minus 126 is 2 to the minus 125. that
9:11should be the next biggest number if
9:12that like that then we're fine or equal
9:13spacing
9:16nope here's how this works the next
9:19smallest number is the one
9:21least number one insignificant goes
9:23bloop and turns on
9:25well that's one plus there's my implicit
9:27one this is two to the minus one twenty
9:29three i told you that before
9:30two to the minus one twenty three well
9:33times two to the minus one twenty six
9:35same thing because my exponent didn't
9:37change
9:38so this is the next so 0 is the smallest
9:40number
9:411 is the next smallest number okay but
9:43that one down there
9:44well that's the truth by 23. this is if
9:47you look at if you
9:48distribute this across this becomes 2 to
9:51the minus 126
9:52which is here remember i wanted the
9:53second nonstop to be double that so it's
9:55equal spacing it's not
9:56the next number is this really small
9:59number
10:00next to it and you're kind of counting
10:02by 2 to the minus one forty nines if you
10:04think about it you're counting by
10:05two to the minus twenty three times two
10:08to the minus one twenty six
10:10so you're counting by two to the minus
10:11one forty nines so all of these guys
10:14all the separations of all the numbers
10:16are two to the minus
10:17149 away from each other but the
10:20smallest number is 2 to the minus 126.
10:22so there's this massive gap
10:268 million times gap it's like an 8
10:28million chasm
10:29before you start counting by a small
10:31thing right eight million is to the
10:32minus twenty
10:33two to the twenty three is eight million
10:35things so this gap is roughly
10:37eight million times bigger than these
10:39gaps can you see that
10:41we don't like that that's crazy and
10:43what's the
10:44who am i to blame i'm blaming
10:46normalization
10:47it's this implicit one that's the issue
10:50that's the problem
10:52well we don't like this gap
10:56what could we do boy i wish we had some
10:58bits we've stored i wish
11:00hey joe do you have any bits left around
11:03i certainly professor cohen you're a bit
11:05smartphone i certainly do have some bits
11:07left around
11:07can i borrow those bits yeah sure take
11:10some bits what are you gonna use
11:11what are you gonna do i forgot what are
11:12you using for i want to
11:15fill in that gap wouldn't it be good if
11:17i did the bits so that
11:19i use all the bits in here
11:22now it'll all be even oh wouldn't that
11:25be amazing
11:26that's exactly what we do these are
11:28called d norms
11:30the solution is the normalization was
11:32bad well how do you
11:34undo a normalization you denormalize it
11:37so that's the idea we don't have an
11:39implicit leading one i have a leading
11:42zero
11:43and the implicit exponent is minus 126.
11:48so now what's my smallest guy my
11:50smallest guy
11:52okay is zero all zero here's my here's
11:54my here's my four lines okay my
11:56three lines all zeros is zero that's
11:58that guy
11:59what's the next one this is still zero
12:02that's a one okay you're telling me the
12:05implicit
12:06exponent is two to the minus one twenty
12:07six two to the minus 1 26 is implicit
12:10okay
12:12times times
12:160 point significand
12:20and that's a 1. well this number is 2 to
12:23the minus
12:2423. 2 to the minus 23
12:28times 2 to the minus 26 is 2 to the
12:30minus 149.
12:33remember we like the 149 as the gap
12:34before so
12:36that's the smallest number what's the
12:39next smallest number
12:41it's double that it's zero zero
12:44and then well okay the smallest number's
12:46a one it's one
12:47zero okay and what's that
12:51that's 2 to the minus 22
12:54times 2 to the minus 126 and that's 2 to
12:58the minus 148.
12:59remember i said i wanted the next number
13:01to be double that so it's like
13:020 and then some number and then double
13:05that number so that spacing is exactly
13:06that
13:07well folks do you realize that 2 to the
13:08minus 48 is double 2 to the minus 149
13:11because essentially i am counting by 2
13:14to the minus 149s
13:16so this is all evenly spaced
13:20i'm counting by the two minus 149 and if
13:22you remember
13:23now so i'm counting so watches i'm
13:25taking a stride zero
13:27take a stride take a baby step because
13:29it's very small and i'm taking a two to
13:30the minus 149 step
13:32bloop and take that step again bloop
13:34bloop now i'm trying to minus 148.
13:36now i'm 3 times 2 minus 4.149 and i keep
13:39counting it's basically
13:41this number encoded times to the minus
13:4420 49.
13:45so when it was when this one when the
13:46significant i'm going a little faster
13:48when the significant was
13:49i don't know 1 100 what's that number
13:53well this is 12 so it'd be 12 times 2
13:55minus 149
13:56and that's 12 in that's that 0 and then
13:5912 more is that number
14:00i'm counting by 2 minus 1 49. amazing
14:04now what happens when i cross over the
14:07threshold
14:08and i cross over into the normalized
14:09numbers what am i counting by then we
14:11saw that the last slide
14:13or a couple slides ago i'm also counting
14:14by 2 to the minus 1 49.
14:16so it turns out i count i take even
14:18steps even steps
14:20until i cross over into the normalized
14:21numbers and now it's still 2 to the
14:23minus 149
14:24even steps what happens when the
14:27exponent changes
14:28get one bigger while that exponent
14:30changes now i'm counting by double that
14:31now i'm counting by 2 to the minus 248
14:33which means my stride
14:34doubles and i do and i have 8 million
14:37steps 2 to the
14:3823 8 million ish steps counting by 2 to
14:42the minus 148.
14:44exponent goes up by one now i'm cutting
14:46by double that
14:47two to the minus 147. so every time i
14:50cross over an exponent changes by one
14:52i double my stride so it's like even set
14:55of numbers
14:55imagine a ruler even set of numbers all
14:58evenly here minus two or minus
14:59one minus one forty nine now i get to
15:02the normalized number
15:03to the minus forty minus because that's
15:05weird that's beautiful that they're
15:06actually the same spacing
15:07across the line the boundary between
15:09denormalized and normalized numbers now
15:12i cross over now i'm still here
15:13now as every time i cross an exponent i
15:16double my stride
15:17so i'm kind of doubling the spacing in
15:19my ruler i go and now it's double size
15:21until it's the next one then double it
15:23again and every time my exponent
15:24increases
15:25i double the size of my stride and
15:27that's how i have a lot of really small
15:29numbers
15:29and i can get to the big numbers as well
15:31like if i kept going by the small count
15:32i would never get to
15:34a big number because i'd be counting by
15:3524.49 but every time my exponent changes
15:38i double my stride i double what that
15:40significant incrementing by one
15:43changes the step count so i'm stepping
15:45by double every time i cross
15:46to a new exponent pretty cool we're
15:49really happy about this so we love
15:51d norms so special number summary we're
15:53doing great love this stuff
15:55all zeros is zero zero non-zero is my d
15:58norm space and now you know those are
15:59counting by two to the minus 49 and they
16:01fit in perfectly
16:02i fit in exactly eight million numbers
16:04in a space so the spacing of them is
16:06exactly the same as the spacing of the
16:07area to the right which is the
16:08normalized node the smallest normalized
16:10numbers
16:10that's awesome my normalized numbers you
16:13saw exponent goes from 1 to 254
16:15in the unsigned space and that goes is 8
16:17million here
16:18then the next 8 million are double
16:19spaced and the next 8 million are double
16:21spaced keep doing that that's that space
16:23until i get to the biggest one i'm
16:25starting by a huge a
16:26huge amount a big giraffe step bloop
16:30bloop so that's almost so it's 254 in
16:33the exponent
16:34all one's insignificant that's my
16:36biggest floating point number
16:37one more ready one more is
16:40infinity isn't that crazy i love that
16:45and now what's one more past infinity
16:48nan oh there's a cnn's
16:51because once my significant gets
16:53non-zero now i'm in my sea of nans
16:55and i'm just a big huge sea of nans
16:57that's it
16:59that's call that's the basics of the
17:01special numbers now before i leave i
17:02want to show you this amazing
17:04demo this is an amazing demo
17:07this is a simulation
17:11of the ieee 754 converter
17:15and i can look at this and i can set the
17:18bits
17:19and watch what happens all the bits are
17:21zero
17:22it's zero
17:25what's the smallest number let's now
17:27count up in our binary odometer what's
17:28the next smallest number it's the
17:30smallest dnorm number
17:32that one is the smallest d-normed number
17:36that's it that's two to the minus 140
17:40nine two to the next the next smallest
17:44is
17:45that that's 2 minus 140 8
17:49or 2 times 2 minus 149
17:52that's 3 times 3 to the minus 149. this
17:55is amazing still d norms okay
17:57until let's do this one let's clear it
17:59let's clear it
18:01let's clear it here until
18:04i get to this number if i go minus one
18:08bloop that is the biggest d norm
18:12that's it that's the biggest team to
18:13look it even says it biggest d norm
18:15okay and i'm representing 1.1 times 10
18:19to the minus 38 but that's you know
18:20that's just some numbers that doesn't
18:21really connect
18:22now watch this this is
18:26exactly 2 to the minus 149 away from the
18:28next number
18:29which is that's the smallest normalized
18:33number that's one point let's look at it
18:35it's 2 to the minus 126. you can see the
18:37exponent says 2 to minus 126
18:39times 1.0 times to the minus one twenty
18:42six is two to minus one twenty six
18:44okay so that's pretty cool
18:47now now we're in normalized space plus
18:50two to the minus one let's get it right
18:54forty nine because i'm still counting by
18:58two that's one forty nine every time
18:59that's right
19:00okay that's that was two to the minus
19:02twenty six plus one two one from two to
19:03minus one forty nine
19:05and this is two to minus one twenty six
19:08minus two to one forty nine because
19:09that's all these things are counted by 2
19:10to the minus 149 that's
19:12that's the spacing that's the inter
19:14ruler line spacing
19:15okay so far this one right smallest
19:19normalized number right there boom okay
19:21which is 2 minus 126.
19:23now stay with me this is going to go
19:27okay counting by this i'm counting by 2
19:30to the minus 1 49 right now
19:32once this gets to be the next number
19:34let's go ahead let's go here
19:35okay the next number bloop there okay
19:37and now go back down by one
19:39that's the biggest i was still counting
19:41by two is my 49s okay
19:42and now i go bloop up
19:46and now i'm counting by two that's my
19:47one now i'm counting by two minus
19:49two to the minus one forty eight so this
19:51number is 2 to the minus 125
19:53plus 2 to the plus and the next one
19:55would be plus 2 to the minus 148
19:59because this is 2 to the minus 23 times
20:022 to the minus 25
20:03125 is 2 to the minus 148 i'm counting
20:05by
20:06get that okay i keep going up
20:09let's go back here this keeps going
20:11bigger double start every time i do this
20:13until i get to
20:15zero oh that was zero sorry
20:18until i didn't i did that wrong until i
20:21get to
20:22all of these what what happens when i'm
20:24counting by ones
20:25where's my one well that's right here
20:27here's my float
20:28watch i just typed this to be a one can
20:31you already figure out what one should
20:32be
20:32what should one be like the number one
20:37well this should be zero and it's one
20:39point zero which that means zero
20:41times something that represents one
20:46well how can i take this set of bits so
20:49it represents one that should be zero
20:51two to the zero is one so
20:54there's a minus 127 here so that means
20:57this has to be
20:581 27. ladies and gentlemen
21:02127 represented here in bits and what
21:04number is it
21:071 because that's 127 minus my thing this
21:10becomes
21:111.0 no significant times 2 to the
21:15that by that unsigned 127 minus my bias
21:18127 is 2 to the 0.
21:20i get 1.0 times 2 to the 0. there's my
21:231.
21:24now what am i counting what's the next
21:26number up
21:27well this is 2 to the 0 this is always 2
21:30to the minus 23 times that thing i'm
21:32counting by
21:33so this the next number bigger than 1 is
21:361 plus 2 to the minus 23.
21:42okay that's the next number above one
21:45how far is one from the next number over
21:47it's two to the minus 23 away
21:50stay with me and now all the numbers
21:53from one to two
21:54how many numbers are there from one to
21:55two it's all these possible bit patterns
21:59eight million bit patterns between one
22:01and two
22:03so what's the next number watch this
22:06what's the next number let's make this
22:08f's make this
22:09f eight eight eight eight let's make
22:11this all fs
22:14okay there that's the biggest number
22:16less than two
22:18how far is it away from two two to the
22:20minus twenty two to the minus twenty
22:21three we just said that we're counting
22:23by the that's the hash marks
22:24what's the next number
22:282 and now what am i counting by
22:322 to the minus 23 times 2 to the 1 or
22:352 to the minus 22 so now the next number
22:39from two
22:39now what happens i have eight million
22:41what between two
22:43and four it's always between
22:46powers of two so between one and two
22:49eight million numbers eight million
22:51stories
22:53eight million between two and four
22:56interesting
22:57eight million between four and eight
23:02counting up by two to minus 22 there
23:05eight million between eight and sixteen
23:078 million between 16 and 32. you kind of
23:09get this idea
23:10it's an equally spaced ruler of 8
23:12million slots
23:13between every power of 2 and the next
23:15power of 2.
23:17isn't that cool so now you're saying
23:19well when is it counting by 1
23:22when is it counting by one can you
23:23figure it out it's when
23:25this number times this number times two
23:28minus 23
23:29is one so what
23:33what over here times two minus twenty is
23:35counting by one
23:36i don't know twenty three what's twenty
23:39plus one twenty seven
23:40let's do it one fifty so what's one
23:44fifty well this is one twenty eight uh
23:47one twenty eight we gotta get to one
23:48fifty what else do i need to do
23:50how about a 32 that's 160.
23:54how about that's right thirteen would
23:56have been what's it okay
23:58how about 150 144 i need six more
24:05ladies and gentlemen two go back here
24:07reset this guy
24:09two to the 23 times mantissa now i'm in
24:12eight what's the number here let's do it
24:14together 8 million
24:17388 608
24:21what's the next number from that how
24:24about 838
24:268609 ladies and gentlemen 838 8609
24:33next one eight six ten i'm counting by
24:36ones now because it's 23 ahead this is
24:38two to twenty three that's through the
24:40minus 23
24:41they cancel now i'm counting by ones
24:44what happens what's the next thing what
24:45am i counting by twos let's keep going
24:48so let's go up let's make this let's go
24:50back to here
24:52let's go back to here one more than that
24:54this is true to 24.
24:55let's go back down ready for ladies and
24:57gentlemen that is
25:00this is two to twenty three oh biggest
25:03one counting by ones right
25:04now still counting by ones two or two
25:06fifteen two
25:08sixteen okay and that number by the way
25:11if you didn't notice is 16 meg what's
25:14the next number above that
25:17i i claim you're counting by twos i
25:19claim that this
25:20single digit is not ones anymore it's
25:22twos
25:23which means this is the first number you
25:26cannot do
25:27this is the smallest energy you cannot
25:29do i claim you cannot do sixteen seven
25:31seven seven
25:32two one seven because when i click plus
25:34one it's gonna say two one eight ladies
25:36and gentlemen
25:37two one eight
25:41so can't do 16 777 217
25:44because i'm counting by twos
25:47awesome i love this simulator so this
25:50simulator is amazing
25:51and i encourage you to play with it oh
25:53by the way i didn't finish i didn't
25:54finish
25:55okay we get really big and we get really
25:58big
25:58this is the biggest number let's get the
26:01biggest number don't don't look at this
26:02thing don't look don't look here
26:03get this here don't look don't look
26:05don't look there
26:06okay now look that's the biggest
26:09possible
26:11floating point number we can store
26:12there's my 3.4
26:14times 10 to the 38. that's a really big
26:17number
26:19what's one more than that i'm still
26:21counting with my this is my odometer
26:23connection let's go by
26:24my odometer by one ready odometer by one
26:28infinity one more the next one over
26:31infinity what's one more past that
26:35nan is that so cool so anyway
26:38that's the end of this lecture i hope
26:40you enjoyed the demo i hope you enjoyed
26:41playing this please do play with the
26:42simulator it's really cool for
26:43understanding how that works
26:44a lot of big ideas we shared with you
26:46now and now we'll have a reflection of
26:48what does that mean for rounding what
26:51does it mean for example let's do some
26:52examples let's have some more fun and
26:54kind of
26:54understand this bit more all right we'll
26:56see the next time

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