context
stringlengths 22
3.07k
| embeddings
sequence |
|---|---|
In other words, the output of XOR is
1 if the two sources are dierent. The output is 0 if the two sources are the same.
The truth table for the XOR function is
A
B
XOR
0
0
0
0
1
1
1
0
1
1
1
0 | [
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0.008048162795603275,
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0.002219187328591943,
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0.01447990071028471,
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0.02597145363688469,
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0.006455509457737207,
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-0.02904130145907402,
0.02534661814570427,
0.008075329475104809,
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-0.05150824040174484,
-0.014153899624943733,
0.019587257876992226,
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0.03240998461842537,
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0.03200248256325722,
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0.00875449925661087,
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0.014928152784705162,
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0.02245335467159748,
-0.009678170084953308,
0.000023824004529160447,
0.008109287358820438,
-0.02378452755510807
] |
In the same way that we applied the logical operation AND to two m-bit patterns,
we can apply the XOR operation bit-wise to two m-bit patterns.
Example 2.9
If a and b are 16-bit patterns as before, then c (shown here) is the XOR of a and b.
a: 0011101001101001
b: 0101100100100001
c: 0110001101001000
Note the distinction between the truth table for XOR shown here and the truth table
for OR shown earlier. | [
-0.009749583899974823,
0.007590601220726967,
0.023694153875112534,
-0.013896743766963482,
-0.030280418694019318,
0.025019604712724686,
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0.04410884156823158,
0.0038567904848605394,
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0.019649477675557137,
0.0031069540418684483,
0.03134624660015106,
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0.00681514386087656,
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0.0446280874311924,
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0.010507960803806782,
0.013924073427915573,
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0.0015987404622137547,
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0.012858246453106403,
0.0023263723123818636,
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0.015645792707800865,
0.020551331341266632,
0.0050592622719705105,
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0.0021043249871581793,
0.00538037670776248,
0.030061786994338036,
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0.012988057918846607,
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0.02782081812620163,
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0.02277863584458828,
0.019321531057357788,
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0.033997148275375366,
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0.01501722913235426,
0.006271982099860907,
0.008881891146302223,
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0.0342157781124115,
0.0066443379037082195,
0.009093690663576126,
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0.02381713315844536,
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0.007816064171493053,
0.003949025645852089,
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-0.002780715236440301,
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0.012154527008533478,
0.006989365443587303,
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0.0010384981287643313,
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-0.010501128621399403,
-0.010494296438992023,
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0.0007494096062146127,
0.02560717612504959,
-0.02720591612160206
] |
9
If a and b are 16-bit patterns as before, then c (shown here) is the XOR of a and b.
a: 0011101001101001
b: 0101100100100001
c: 0110001101001000
Note the distinction between the truth table for XOR shown here and the truth table
for OR shown earlier. In the case of exclusive-OR, if both source operands are 1, the
output is 0. That is, the output is 1 if the rst operand is 1 but the second operand is
not 1 or if the second operand is 1 but the rst operand is not 1. | [
-0.0131848668679595,
-0.010834130458533764,
0.0042451536282896996,
0.004290094133466482,
-0.007301111239939928,
0.016220081597566605,
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0.003522648010402918,
0.023410571739077568,
0.004659989383071661,
0.0024596310686320066,
0.02602403610944748,
0.005527688190340996,
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0.04253450408577919,
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0.0029332353733479977,
0.006789480336010456,
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0.020935382694005966,
0.0014743958599865437,
0.019732359796762466,
0.030670199543237686,
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0.0008759086485952139,
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0.011781338602304459,
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] |
In the case of exclusive-OR, if both source operands are 1, the
output is 0. That is, the output is 1 if the rst operand is 1 but the second operand is
not 1 or if the second operand is 1 but the rst operand is not 1. The term exclusive
is used because the output is 1 if only one of the two sources is 1. The OR function,
on the other hand, produces an output 1 if only one of the two sources is 1, or if both
sources are 1. | [
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] |
The term exclusive
is used because the output is 1 if only one of the two sources is 1. The OR function,
on the other hand, produces an output 1 if only one of the two sources is 1, or if both
sources are 1. Ergo, the name inclusive-OR.
Example 2.10
Suppose we wish to know if two patterns are identical. Since the XOR function pro-
duces a 0 only if the corresponding pair of bits is identical, two patterns are identical
if the output of the XOR is all 0s. | [
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] |
10
Suppose we wish to know if two patterns are identical. Since the XOR function pro-
duces a 0 only if the corresponding pair of bits is identical, two patterns are identical
if the output of the XOR is all 0s.
2.6.6 DeMorgans Laws
There are two well-known relationships between AND functions and OR func-
t
1
0
1
0
0
0
0
1
1
1
0
1
Figure 2.2
DeMorgans Law. | [
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the
B =
fun
repr
A AND B = A OR B
We can also state this behavior in English:
It is not the case that both A and B are false is equivalent to saying At least
one of A and B is true.
This equivalence is known as one of two DeMorgans Laws. Question: Is there
a similar result if one inverts both inputs to an OR function, and then inverts the
output?
2.6.7 The Bit Vector
We have discussed the AND, OR, and NOT functions performed on m-bit pat-
terns, where each of the m bits is a logical value (0 or 1) and the operations are per-
formed bit-wise (i. | [
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] |
6.7 The Bit Vector
We have discussed the AND, OR, and NOT functions performed on m-bit pat-
terns, where each of the m bits is a logical value (0 or 1) and the operations are per-
formed bit-wise (i.e., individually and independently). We have also discussed the
use of an m-bit bit mask, where our choice of 0 or 1 for each bit allows us to isolate
the bits we are interested in focusing on and ignore the bits that dont matter. | [
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] |
, individually and independently). We have also discussed the
use of an m-bit bit mask, where our choice of 0 or 1 for each bit allows us to isolate
the bits we are interested in focusing on and ignore the bits that dont matter.
An m-bit pattern where each bit has a logical value (0 or 1) independent of
the other bits is called a bit vector. It is a convenient mechanism for identifying
a property such that some of the bits identify the presence of the property and
other bits identify the absence of the property. | [
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An m-bit pattern where each bit has a logical value (0 or 1) independent of
the other bits is called a bit vector. It is a convenient mechanism for identifying
a property such that some of the bits identify the presence of the property and
other bits identify the absence of the property.
There are many uses for bit vectors. The most common use is a bit mask,
as we saw in Example 2.7. | [
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] |
The most common use is a bit mask,
as we saw in Example 2.7. In that example, we had an eight-bit value, and we
wanted to focus on bit 1 and bit 0 of that value. We did not care about the other
bits. Performing the AND of that value with the bit mask 00000011 caused bit 7
through bit 2 to be ignored, resulting in the AND function producing 00000000,
00000001, 00000010, or 00000011, depending on the values of bit 1 and bit 0. | [
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] |
We did not care about the other
bits. Performing the AND of that value with the bit mask 00000011 caused bit 7
through bit 2 to be ignored, resulting in the AND function producing 00000000,
00000001, 00000010, or 00000011, depending on the values of bit 1 and bit 0.
The bit mask is a bit vector, where the property of each of the bits is whether or
not we care about that bit. In Example 2. | [
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The bit mask is a bit vector, where the property of each of the bits is whether or
not we care about that bit. In Example 2.7, we only cared about bit 1 and bit 0.
Another use of a bit mask could involve a 16-bit 2s complement integer.
Suppose the only thing we cared about was whether the integer was odd or even
and whether it was positive or negative. The bit vector 1000000000000001 has a 1 in
bit 15 that is used to identify a number as positive or negative, and a 1 in bit 0 that is
used to identify if the integer is odd or even. | [
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Suppose the only thing we cared about was whether the integer was odd or even
and whether it was positive or negative. The bit vector 1000000000000001 has a 1 in
bit 15 that is used to identify a number as positive or negative, and a 1 in bit 0 that is
used to identify if the integer is odd or even. If we perform the AND of this bit vector
with a 16-bit 2s complement integer, we would get one of four results, depending
on whether the integer was positive or negative and odd or even:
0000000000000000
0000000000000001
1000000000000000
1000000000000001
Another common use of bit vectors involves managing a complex system
made up of several units, each of which is individually and independently either. | [
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] |
The bit vector 1000000000000001 has a 1 in
bit 15 that is used to identify a number as positive or negative, and a 1 in bit 0 that is
used to identify if the integer is odd or even. If we perform the AND of this bit vector
with a 16-bit 2s complement integer, we would get one of four results, depending
on whether the integer was positive or negative and odd or even:
0000000000000000
0000000000000001
1000000000000000
1000000000000001
Another common use of bit vectors involves managing a complex system
made up of several units, each of which is individually and independently either | [
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] |
busy or available. The system could be a manufacturing plant where each unit is
a particular machine. Or the system could be a taxicab network where each unit
is a particular taxicab. In both cases, it is important to identify which units are
busy and which are available so that work can be properly assigned.
Say we have m such units. We can keep track of these m units with a bit
vector, where a bit is 1 if the unit is free and 0 if the unit is busy. | [
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] |
Say we have m such units. We can keep track of these m units with a bit
vector, where a bit is 1 if the unit is free and 0 if the unit is busy.
Example 2.11
Suppose we have eight machines that we want to monitor with respect to their avail-
ability. We can keep track of them with an eight-bit BUSYNESS bit vector, where a
bit is 1 if the unit is free and 0 if the unit is busy. The bits are labeled, from right to
left, from 0 to 7. | [
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] |
We can keep track of them with an eight-bit BUSYNESS bit vector, where a
bit is 1 if the unit is free and 0 if the unit is busy. The bits are labeled, from right to
left, from 0 to 7.
The BUSYNESS bit vector 11000010 corresponds to the situation where only
units 7, 6, and 1 are free and therefore available for work assignment.
Suppose work is assigned to unit 7. We update our BUSYNESS bit vector by per-
forming the logical AND, where our two sources are the current bit vector 11000010
and the bit mask 01111111. | [
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] |
Suppose work is assigned to unit 7. We update our BUSYNESS bit vector by per-
forming the logical AND, where our two sources are the current bit vector 11000010
and the bit mask 01111111. The purpose of the bit mask is to clear bit 7 of the
BUSYNESS bit vector, while leaving alone the values corresponding to all the other
units. The result is the bit vector 01000010, indicating that unit 7 is now busy. | [
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] |
The purpose of the bit mask is to clear bit 7 of the
BUSYNESS bit vector, while leaving alone the values corresponding to all the other
units. The result is the bit vector 01000010, indicating that unit 7 is now busy.
Suppose unit 5 nishes its task and becomes idle. We can update the
BUSYNESS bit vector by performing the logical OR of it with the bit mask
00100000. The result is 01100010, indicating that unit 5 is now available. | [
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] |
We can update the
BUSYNESS bit vector by performing the logical OR of it with the bit mask
00100000. The result is 01100010, indicating that unit 5 is now available.
2.7 Other Representations
There are many other representations of information that are used in computers.
Two that are among the most useful are the oating point data type and ASCII
codes. We will describe both in this section. | [
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] |
Two that are among the most useful are the oating point data type and ASCII
codes. We will describe both in this section. We will also describe a notation
called hexadecimal that, although not a data type, is convenient for humans to
use when dealing with long strings of 0s and 1s.
2.7.1 Floating Point Data Type (Greater Range, Less Precision)
Most of the arithmetic in this book uses integer values. | [
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] |
7.1 Floating Point Data Type (Greater Range, Less Precision)
Most of the arithmetic in this book uses integer values. The LC-3 computer, which
you will start studying in Chapter 4, uses the 16-bit, 2s complement integer data
type. That data type provides one bit to identify whether the number is positive
or negative and 15 bits to represent the magnitude of the value. With 16 bits used
in this way, we can express integer values between 32,768 and +32,767, that
is, between 215 and +215 1. | [
0.011639759875833988,
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That data type provides one bit to identify whether the number is positive
or negative and 15 bits to represent the magnitude of the value. With 16 bits used
in this way, we can express integer values between 32,768 and +32,767, that
is, between 215 and +215 1. We say the precision of our value is 15 bits, and
the range is 216. As you learned in high school chemistry class, sometimes we
need to express much larger numbers, but we do not require so many digits of
precision. | [
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] |
We say the precision of our value is 15 bits, and
the range is 216. As you learned in high school chemistry class, sometimes we
need to express much larger numbers, but we do not require so many digits of
precision. In fact, recall the value 6.0221023, which you may have been required
to memorize back then. The range needed to express the value 1023 is far greater
than the largest value 2151 that is available with 16-bit 2s complement integers. | [
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] |
0221023, which you may have been required
to memorize back then. The range needed to express the value 1023 is far greater
than the largest value 2151 that is available with 16-bit 2s complement integers.
On the other hand, the 15 bits of precision available with 16-bit 2s complement
integers are overkill | [
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] |
In fact, recall the value 6.0221023, which you may have been required
to memorize back then. The range needed to express the value 1023 is far greater
than the largest value 2151 that is available with 16-bit 2s complement integers.
On the other hand, the 15 bits of precision available with 16-bit 2s complement
integers are overkill. We need only enough bits to express four signicant decimal
digits (6022). | [
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] |
On the other hand, the 15 bits of precision available with 16-bit 2s complement
integers are overkill. We need only enough bits to express four signicant decimal
digits (6022). | [
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] |
So we have a problem. We have more bits than we need for precision. But we
dont have enough bits to represent the range.
The oating point data type solves the problem. Instead of using all the bits
to represent the precision of a value, the oating point data type allocates some of
the bits to the range of values (i.e., how big or how small) that can be expressed.
The rest of the bits (except for the sign bit) are used for precision. | [
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-0.00880445260554552,
-0.0013491528807207942,
0.00962620135396719,
-0.006156211253255606,
0.017194578424096107,
-0.008521328680217266,
0.007236914709210396,
0.02878192812204361,
-0.008093190379440784,
-0.022180315107107162,
-0.014694803394377232,
0.013230848126113415,
-0.0172360111027956,
0.005034075118601322,
-0.021848853677511215,
-0.006684478372335434,
-0.010468666441738605,
0.011856663040816784,
-0.0019576959311962128,
0.02068873681128025,
0.023865245282649994,
0.005607227794826031,
-0.005358631256967783,
-0.015440592542290688,
-0.005917973350733519,
0.0014363342197611928,
0.005003000609576702,
0.007230009417980909,
-0.00648767314851284,
0.03687511757016182,
-0.01006124448031187,
0.005980122368782759,
-0.008134623989462852,
0.0017073731869459152,
0.02226318046450615,
-0.020163921639323235,
0.03731706738471985,
0.019404321908950806,
-0.009494997560977936,
-0.004032784607261419,
0.009833364747464657,
0.022346045821905136,
-0.02908576838672161,
0.034775860607624054,
0.00448163878172636,
0.01041342318058014,
0.0026119875255972147,
0.010026717558503151,
0.015703000128269196,
0.021959340199828148,
0.014252854511141777,
-0.018699966371059418,
0.01898999512195587,
0.02400335483253002,
-0.012264084070920944,
-0.010551531799137592,
0.006698289420455694,
0.005068602506071329,
-0.00020899785158690065,
-0.02439006045460701,
0.01629686914384365,
-0.019597675651311874,
0.024914873763918877,
0.003583929967135191,
0.013147982768714428,
-0.020674925297498703,
0.03919535130262375,
0.007119521964341402,
0.024307195097208023,
0.009646917693316936,
-0.019791027531027794,
0.0014173442032188177,
-0.010247692465782166,
-0.008031041361391544,
0.019404321908950806,
-0.021655499935150146,
-0.014584316872060299,
-0.005672829691320658,
0.03223465383052826,
0.0036046463064849377,
-0.0030798320658504963,
-0.020025813952088356,
0.011780702508985996,
0.0026430620346218348,
0.011304226703941822,
0.0010668925242498517,
-0.014377152547240257,
-0.012063826434314251,
-0.0067880600690841675,
-0.016075894236564636,
0.009895513765513897,
-0.00883897952735424,
-0.012284800410270691,
0.015841109678149223,
0.0014449660666286945,
0.005748789757490158,
-0.04427776485681534,
-0.01470861490815878,
0.017567472532391548,
0.013396578840911388,
-0.03303568810224533,
-0.005192900542169809,
-0.004001710098236799,
0.017581284046173096,
-0.003715133760124445,
0.00004580257518682629,
0.013465633615851402,
0.0007561470847576857,
0.08093190938234329,
0.024749143049120903,
0.028049949556589127,
0.01886569708585739,
-0.013707323931157589,
0.038339074701070786,
0.012229557149112225,
0.01856185682117939,
0.005583058577030897,
-0.04800670966506004,
0.03590835630893707,
-0.00045101239811629057,
-0.0018886413890868425,
-0.004105291794985533,
0.021116875112056732,
0.01705646887421608,
-0.03223465383052826,
0.006242529489099979,
-0.03046685829758644,
-0.005855823867022991,
0.03027350641787052,
0.005258502438664436,
0.02049538493156433,
-0.004063859116286039,
-0.026972699910402298,
-0.004937398713082075,
0.002551564946770668,
0.018147530034184456,
-0.021462148055434227,
-0.021475957706570625,
-0.006042271386831999,
-0.0037531135603785515,
-0.0235199723392725,
-0.00855585653334856,
-0.018506614491343498,
-0.009308550506830215,
-0.016476411372423172,
-0.0163106806576252,
0.02734559401869774,
0.013479444198310375,
-0.0008403073297813535,
0.006011196877807379,
-0.022553209215402603,
-0.023229943588376045,
0.021986961364746094,
0.006926169153302908,
0.00865253247320652,
-0.021337850019335747,
-0.03944394737482071
] |
, how big or how small) that can be expressed.
The rest of the bits (except for the sign bit) are used for precision.
Most ISAs today specify more than one oating point data type. One of them,
usually called oat, consists of 32 bits, allocated as follows:
I
Figure 2.3
The 32-bit oating point data type.
2.7.1.1 Normalized Form
Like Avogadros number that you learned years ago, the oating point data type
represents numbers expressed in scientic notation, and mostly in normalized
form:
N = (1)S 1. | [
0.008420850150287151,
0.0013585570268332958,
0.022333558648824692,
-0.006942794192582369,
-0.018618078902363777,
0.01065149437636137,
-0.03436140716075897,
0.0019119804492220283,
-0.038592174649238586,
-0.030103521421551704,
0.016651857644319534,
0.02985944040119648,
-0.007837763987481594,
-0.005515588913112879,
0.0005720686167478561,
0.01238719280809164,
-0.004620619583874941,
0.004013802856206894,
0.022591201588511467,
0.012115989811718464,
-0.008156427182257175,
0.005868152715265751,
0.0018238394986838102,
-0.0036375089548528194,
0.0029001759830862284,
-0.0038815916050225496,
0.010719294659793377,
-0.006695321761071682,
0.007186877075582743,
-0.033764760941267014,
0.031649377197027206,
-0.005000303499400616,
-0.028096619993448257,
0.004691810347139835,
-0.01924184523522854,
-0.006420728750526905,
-0.022252198308706284,
-0.007315698079764843,
0.017058663070201874,
-0.014183911494910717,
0.023296330124139786,
-0.0015958595322445035,
0.006942794192582369,
-0.008922575041651726,
0.018807921558618546,
0.0061596957966685295,
-0.0047324905171990395,
-0.0037832804955542088,
-0.0030120471492409706,
0.006417338736355305,
0.0018136694561690092,
0.025520192459225655,
-0.014170351438224316,
-0.028964469209313393,
-0.002162843244150281,
-0.002381500555202365,
-0.009587022475898266,
-0.0014136451063677669,
-0.006573280319571495,
-0.0011780376080423594,
0.00270016398280859,
0.015363644808530807,
-0.01924184523522854,
-0.000477571360534057,
-0.016258614137768745,
-0.00461044954136014,
-0.0035188577603548765,
0.010712514631450176,
-0.012393972836434841,
0.006325807888060808,
0.03802264854311943,
0.02077414281666279,
0.007200437132269144,
0.03311387449502945,
0.016258614137768745,
-0.018062112852931023,
-0.0006424118764698505,
0.016950180754065514,
-0.005125734955072403,
0.00005424057962954976,
0.011302380822598934,
-0.02587275765836239,
0.003986682742834091,
-0.004647739697247744,
0.00033179979072883725,
0.003566318191587925,
-0.014278832823038101,
0.013078760355710983,
-0.0008025910938158631,
0.0046002790331840515,
0.03238162770867348,
-0.007512320298701525,
0.013851688243448734,
0.007315698079764843,
-0.005179975647479296,
0.014509355649352074,
0.014183911494910717,
0.017709549516439438,
-0.029588237404823303,
-0.02179115265607834,
-0.02078770287334919,
0.012346511706709862,
-0.02405569702386856,
-0.014360194094479084,
-0.01629929430782795,
0.010244689881801605,
0.0011466797441244125,
0.006288517266511917,
-0.01659761741757393,
-0.006003754213452339,
-0.004342636559158564,
0.05144719034433365,
0.020923303440213203,
-0.00491894269362092,
0.02885598875582218,
-0.0022357290145009756,
0.021248746663331985,
0.0009017496486194432,
-0.021167386323213577,
-0.004332466516643763,
0.020136816427111626,
-0.001152612385340035,
0.027283012866973877,
-0.000614867836702615,
0.009444640949368477,
-0.0026171079371124506,
-0.018672319129109383,
-0.031351055949926376,
-0.020299537107348442,
-0.008603911846876144,
0.011844786815345287,
0.02069278061389923,
0.0015560266328975558,
0.010149768553674221,
-0.007220777217298746,
-0.0038815916050225496,
-0.008278468623757362,
-0.002517102053388953,
-0.030591687187552452,
-0.025194749236106873,
0.01914692483842373,
0.0022764094173908234,
-0.016895940527319908,
-0.0017831590957939625,
0.006719051860272884,
0.012244811281561852,
0.02762201614677906,
0.02375737391412258,
-0.008651372976601124,
-0.034388527274131775,
0.023201407864689827,
0.011004057712852955,
0.010712514631450176,
0.013634725473821163,
-0.001590774510987103,
0.026035478338599205,
-0.01853671856224537,
0.02915431186556816,
-0.014278832823038101,
-0.010231129825115204,
-0.0026730436366051435,
0.02793389931321144,
-0.0218996349722147,
0.01967577077448368,
-0.0020594471134245396,
0.01349912490695715,
-0.0010254859225824475,
-0.007844544015824795,
-0.000878019374795258,
-0.0030222174245864153,
-0.017384106293320656,
0.0336020402610302,
-0.020502939820289612,
0.02160131186246872,
0.0183468759059906,
0.02466590330004692,
0.015350083820521832,
0.0030595078133046627,
0.003532417817041278,
0.013709306716918945,
0.0244489423930645,
-0.016733219847083092,
0.020041894167661667,
0.00682753324508667,
-0.00728857796639204,
0.0029239063151180744,
0.04542648792266846,
0.01268551591783762,
0.00855645164847374,
-0.007959805428981781,
0.022672561928629875,
0.011688845232129097,
-0.011804106645286083,
0.017899392172694206,
-0.6383031606674194,
-0.03137817606329918,
0.006230886559933424,
-0.01784515008330345,
0.01998765394091606,
0.03015776351094246,
0.017763789743185043,
-0.001196682802401483,
0.003449362004175782,
0.004935892764478922,
-0.0269982498139143,
-0.002728979103267193,
0.013363523408770561,
-0.02812374010682106,
-0.01977069117128849,
-0.018116353079676628,
0.0025747325271368027,
-0.0069394041784107685,
0.004396877251565456,
-0.005488468799740076,
-0.029100071638822556,
0.012549914419651031,
-0.023499730974435806,
0.015377204865217209,
0.01456359587609768,
-0.004854531958699226,
-0.00012204130325699225,
-0.012773656286299229,
-0.020624980330467224,
-0.013926269486546516,
-0.024815065786242485,
-0.0030035721138119698,
0.017994312569499016,
0.002347600180655718,
0.032354507595300674,
-0.02312004752457142,
0.011614263989031315,
0.0234319306910038,
0.010481991805136204,
0.018699439242482185,
-0.012970278970897198,
0.0004792663676198572,
0.03295115381479263,
-0.0013153341133147478,
-0.0045697689056396484,
0.0022323390003293753,
0.0029120412655174732,
0.009559902362525463,
-0.007810643408447504,
-0.03658527135848999,
-0.00555626954883337,
0.009505662135779858,
-0.012373632751405239,
-0.01441443432122469,
0.009546342305839062,
-0.03639543056488037,
0.03476821258664131,
-0.00009407350444234908,
0.01395338959991932,
0.000007693793122598436,
0.00603765482082963,
0.005505418870598078,
-0.03894473612308502,
-0.027961019426584244,
0.014455114491283894,
0.002115382580086589,
-0.008380169980227947,
0.004200255032628775,
0.00473927054554224,
-0.0008131849463097751,
-0.002501846756786108,
0.04445015639066696,
0.01986561343073845,
-0.01096337754279375,
-0.0023645502515137196,
0.010102308355271816,
0.0346868522465229,
0.002323869848623872,
0.005434228107333183,
0.013939829543232918,
0.017045103013515472,
-0.016584057360887527,
-0.014550035819411278,
-0.012441433034837246,
0.02589987777173519,
0.020353777334094048,
0.0010687089525163174,
0.024530302733182907,
0.017479026690125465,
-0.001239905715920031,
0.00968872383236885,
0.047541867941617966,
0.023608213290572166,
-0.04084315896034241,
0.005332527216523886,
0.018089232966303825,
-0.017072223126888275,
-0.007322478573769331,
-0.011234580539166927,
-0.006620740983635187,
-0.0168281402438879,
-0.00804794579744339,
0.005583389662206173,
0.012665175832808018,
0.014021189883351326,
0.008454750292003155,
0.003423936665058136,
0.01751970686018467,
0.028557665646076202,
-0.03891761600971222,
0.00491555267944932,
-0.02640160173177719,
-0.0016000971663743258,
0.0025849025696516037,
-0.00009889371722238138,
-0.02477438561618328,
0.027988139539957047,
0.012217691168189049,
-0.005586779676377773,
0.0013543195091187954,
0.031839221715927124,
0.01926896534860134,
0.01492972020059824,
-0.0300492811948061,
-0.01722138375043869,
0.006874993443489075,
0.012115989811718464,
0.006861433386802673,
-0.015526366420090199,
-0.02180471271276474,
0.003051032545045018,
-0.022808164358139038,
-0.004525698255747557,
-0.023459050804376602,
0.02466590330004692,
0.005746111273765564,
0.00024450637283734977,
0.006539380177855492,
0.016055211424827576,
-0.00962092261761427,
-0.014197471551597118,
-0.008536111563444138,
0.01156680379062891,
-0.02385229617357254,
-0.0012560083996504545,
-0.02648296393454075,
-0.003323930548503995,
0.006146135739982128,
0.010454871691763401,
0.023282770067453384,
0.007824203930795193,
-0.009702283889055252,
-0.023309890180826187,
0.015445005148649216,
0.01263127475976944,
-0.011288820765912533,
0.001842484693042934,
-0.05133870989084244,
-0.012909257784485817,
0.00968872383236885,
-0.01442799437791109,
0.014712757430970669,
-0.02599479816854,
-0.011031177826225758,
-0.006763122510164976,
-0.011363402009010315,
-0.010420971550047398,
0.021980995312333107,
-0.010210789740085602,
-0.038076888769865036,
0.01396694965660572,
-0.02455742284655571,
-0.01035317126661539,
-0.004908772651106119,
-0.01917404495179653,
0.021275866776704788,
-0.025954117998480797,
0.0013483869843184948,
0.003925662022083998,
-0.024503182619810104,
0.0020374117884784937,
-0.013424543663859367,
-0.042660217732191086,
0.0023798055481165648,
0.029588237404823303,
0.010536232963204384,
-0.008285248652100563,
-0.012970278970897198,
-0.010061627253890038,
-0.005603729747235775,
0.002884920919314027,
0.024123497307300568,
-0.031106973066926003,
0.03338507562875748,
0.003861251287162304,
-0.00458671897649765,
-0.0008148799533955753,
-0.012448213063180447,
0.02047581970691681,
0.03788704425096512,
0.04287717863917351,
0.0017797690816223621,
-0.0003489618538878858,
-0.010305710136890411,
0.03574454411864281,
-0.022252198308706284,
-0.003432411700487137,
-0.012800777330994606,
0.03273418918251991,
0.009403960779309273,
0.012339731678366661,
-0.005678310990333557,
-0.012000728398561478,
0.004078213591128588,
0.007505540270358324,
0.025303231552243233,
0.04461287707090378,
0.023581093177199364,
0.023987896740436554,
0.014319512993097305,
-0.005607120227068663,
-0.0026967739686369896,
0.00835304893553257,
-0.0008407289860770106,
-0.010251469910144806,
0.01764174923300743,
0.05014541745185852,
0.001590774510987103,
-0.005929173436015844,
-0.017587509006261826,
-0.00952600222080946,
-0.03067304752767086,
-0.0015984021592885256,
0.0005415583145804703,
0.026035478338599205,
0.007485200185328722,
0.022347118705511093,
-0.004359586630016565,
0.022279318422079086,
-0.005542709492146969,
-0.0356631800532341,
0.0008161512087099254,
0.011227800510823727,
-0.025750715285539627,
0.0025374421384185553,
-0.00901749636977911,
0.029940800741314888,
0.022970885038375854,
-0.007010594941675663,
-0.011302380822598934,
-0.005234215874224901,
-0.008827654644846916,
-0.027771176770329475,
0.03240874782204628,
0.023282770067453384,
0.008237788453698158,
0.013797448016703129,
0.0042748358100652695,
0.011688845232129097,
0.025913437828421593,
0.009878565557301044,
0.011485442519187927,
0.008949696086347103,
-0.003179854014888406,
0.016733219847083092,
0.001842484693042934,
-0.004081603605300188,
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0.0074241794645786285,
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] |
1.1 Normalized Form
Like Avogadros number that you learned years ago, the oating point data type
represents numbers expressed in scientic notation, and mostly in normalized
form:
N = (1)S 1. fraction 2exponent127, 1 exponent 254
where S, fraction, and exponent are the binary numbers in the elds of Figure 2.3.
We say mostly in normalized form because (as noted in the equation) the data
type represents a oating point number in normalized form only if the eight-bit
exponent is restricted to the 254 unsigned integer values, 1 (00000001) through
254 (11111110). | [
0.013920610770583153,
0.002875421429052949,
0.014419219456613064,
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0.012599972076714039,
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] |
3.
We say mostly in normalized form because (as noted in the equation) the data
type represents a oating point number in normalized form only if the eight-bit
exponent is restricted to the 254 unsigned integer values, 1 (00000001) through
254 (11111110).
As you know, with eight bits, one can represent 256 values uniquely. For the
other two integer values 0 (00000000) and 255 (11111111), the oating point
data type does not represent normalized numbers. | [
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0.05725857987999916,
0.006525992415845394,
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0.007366640493273735,
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0.011843343265354633,
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0.00397704029455781,
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0.02217419445514679,
-0.012613092549145222,
0.013841991312801838,
0.01789330504834652,
0.005935175810009241,
0.014895332977175713,
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0.00538824824616313,
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0.023551641032099724,
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-0.005584062077105045,
-0.010479399934411049,
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0.009446315467357635,
0.01941930130124092,
0.000586174544878304,
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0.014395670965313911,
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0.017636721953749657,
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0.02654961496591568,
0.009702898561954498,
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0.012349757365882397,
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-0.021255897358059883,
-0.009608367457985878,
0.007332879118621349,
-0.019122203812003136,
-0.013497629202902317,
0.003087439341470599,
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0.021985134109854698,
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0.03265359625220299,
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0.011931121349334717,
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0.00012481003068387508,
0.02776500955224037,
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0.03802834078669548,
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0.03246453404426575,
0.00978392455726862,
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0.016448337584733963,
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0.012707623653113842,
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0.02776500955224037,
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0.005280213430523872,
0.002697500167414546,
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-0.005374744068831205,
0.02623901329934597,
0.002711004577577114,
0.008035107515752316,
-0.004007425159215927,
-0.04831867665052414
] |
As you know, with eight bits, one can represent 256 values uniquely. For the
other two integer values 0 (00000000) and 255 (11111111), the oating point
data type does not represent normalized numbers. We will explain what it does
represent in Section 2.7.1.2 and Section 2.7.1.3.
Recall again Avogadros number: (a) an implied + sign (often left out when
the value is positive), (b) four decimal digits 6. | [
0.0003442821907810867,
0.0019162212265655398,
0.020417610183358192,
-0.03113652393221855,
-0.005882815923541784,
0.014031310565769672,
-0.028910594061017036,
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-0.01338870543986559,
0.007287271786481142,
0.03285897150635719,
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-0.015131025575101376,
0.007989499717950821,
0.0005552404327318072,
0.00036519169225357473,
0.010771912522614002,
0.008850722573697567,
0.002861910266801715,
0.02143782749772072,
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-0.01756894960999489,
-0.016124743968248367,
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0.017144963145256042,
-0.013408579863607883,
0.021225834265351295,
-0.01425655372440815,
0.019370893016457558,
-0.012739475816488266,
-0.04165669530630112,
-0.001411080826073885,
-0.02726764604449272,
0.002860254142433405,
-0.0157802551984787,
-0.011818629689514637,
0.02824811451137066,
-0.005786756053566933,
0.027241146191954613,
0.004614167846739292,
-0.0025969187263399363,
-0.004123933147639036,
0.0071878996677696705,
0.004309427458792925,
0.0047830999828875065,
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0.0024279863573610783,
0.00630349013954401,
0.007366769015789032,
0.02133183181285858,
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0.004865909926593304,
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0.0048559727147221565,
0.0037595697212964296,
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-0.02139808051288128,
-0.017621947452425957,
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0.019993623718619347,
-0.007552263326942921,
0.01367357186973095,
0.02084159664809704,
0.019145650789141655,
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0.010347926057875156,
0.036356862634420395,
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0.009678822010755539,
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0.003299146657809615,
0.005617823917418718,
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0.013183336704969406,
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0.0031567136757075787,
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-0.01338870543986559,
0.006942782551050186,
-0.007340270094573498,
0.0076715098693966866,
0.022073809057474136,
0.005405830685049295,
0.019198648631572723,
-0.012375112622976303,
0.005892752669751644,
0.014163806103169918,
0.008678478188812733,
0.05554226040840149,
-0.01629699021577835,
-0.0123287383466959,
-0.007081903051584959,
-0.0027575697749853134,
-0.010188930667936802,
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-0.004137183073908091,
-0.01412405725568533,
0.006108058616518974,
0.006167681887745857,
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0.05114339664578438,
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0.015541763044893742,
0.011918001808226109,
0.007565512787550688,
-0.01037442497909069,
0.013872315175831318,
0.010844784788787365,
0.0039185648784041405,
-0.03980175405740738,
-0.016111494973301888,
-0.017462952062487602,
0.027479639276862144,
0.03058004193007946,
0.005173963028937578,
0.006356488447636366,
0.002590293763205409,
0.016999216750264168,
-0.006717539392411709,
0.022338800132274628,
-0.0210270918905735,
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0.024789974093437195,
-0.0006310114986263216,
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-0.017926689237356186,
0.007366769015789032,
0.013143587857484818,
0.019119150936603546,
0.03150751441717148,
-0.012229367159307003,
-0.014905783347785473,
-0.005879503209143877,
0.011633135378360748,
0.01689322106540203,
0.007817255333065987,
0.00048319579218514264,
-0.0030639665201306343,
-0.016442734748125076,
0.011931250803172588,
-0.008287615142762661,
-0.007386643439531326,
-0.022431546822190285,
0.03060654178261757,
-0.025982435792684555,
0.025359705090522766,
0.02198106050491333,
0.011606636457145214,
-0.00024594541173428297,
-0.017065465450286865,
-0.0181121826171875,
-0.011328395456075668,
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0.04960644617676735,
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0.01172588299959898,
-0.0015237022889778018,
0.03746982663869858,
0.023690257221460342,
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-0.014574543572962284,
0.01099715568125248,
0.021861815825104713,
-0.0006951891700737178,
0.02557169832289219,
0.01755570061504841,
-0.008247866295278072,
0.013017716817557812,
0.05379331484436989,
0.013925313949584961,
0.021464327350258827,
0.007055404130369425,
0.016257241368293762,
0.005177275277674198,
-0.01023530401289463,
0.006392924580723047,
-0.6381000280380249,
-0.03293846920132637,
0.0022375236731022596,
-0.014375799335539341,
0.022749537602066994,
0.019437141716480255,
0.0169462189078331,
0.008930220268666744,
0.018045933917164803,
0.017992936074733734,
0.005455516744405031,
-0.0066049182787537575,
0.009506577625870705,
-0.044439107179641724,
-0.005872878711670637,
-0.025425953790545464,
0.012547357007861137,
-0.00564432330429554,
0.004740038886666298,
0.0018433484947308898,
-0.017105214297771454,
0.017171461135149002,
-0.01055329479277134,
0.0001247324253199622,
0.01726420968770981,
0.013329082168638706,
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-0.014932282269001007,
-0.014972031116485596,
-0.004504858981817961,
-0.022034060209989548,
0.0010351238306611776,
0.009870940819382668,
0.021517325192689896,
0.07382668554782867,
0.02906958945095539,
-0.0037595697212964296,
0.035747379064559937,
-0.009877565316855907,
0.04250466823577881,
0.027121899649500847,
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0.00861223042011261,
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0.025001967325806618,
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0.008863972499966621,
0.011394643224775791,
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0.025465702638030052,
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0.010500296019017696,
0.014004811644554138,
0.005541638936847448,
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-0.013375455513596535,
0.0035542012192308903,
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-0.012699726969003677,
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0.021769067272543907,
0.009877565316855907,
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0.02641967311501503,
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-0.00026457765488885343,
0.0325939804315567,
0.002454485511407256,
0.0020089682657271624,
-0.01173913199454546,
-0.034846410155296326
] |
3.
Recall again Avogadros number: (a) an implied + sign (often left out when
the value is positive), (b) four decimal digits 6.022 in normalized form (one non-
zero decimal digit 6 before the decimal point) times (c) the radix 10 raised to
the power 23. The computers 32-bit oating point data type, on the other hand,
consists of (a) a sign bit (positive or negative), (b) 24 binary digits in normalized
form (one non-zero binary digit to the left of the binary point) times (c) the radix
2 raised to an exponent expressed in eight bits. | [
0.011247992515563965,
-0.0056538935750722885,
0.015546545386314392,
-0.018576128408312798,
0.0032903533428907394,
0.00746765686199069,
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-0.020183933898806572,
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0.00721519161015749,
0.02963145077228546,
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0.006989301647990942,
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0.006102351471781731,
0.005713687743991613,
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0.020090920850634575,
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0.04084622487425804,
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0.02851528860628605,
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0.010822787880897522,
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0.003438178449869156,
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0.01775229535996914,
0.013181344605982304,
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0.010663336142897606,
0.004321807064116001,
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0.020197222009301186,
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0.023412832990288734,
0.033936649560928345,
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0.0010679945116862655,
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0.010045460425317287,
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0.008935942314565182,
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0.004614135250449181,
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0.028701314702630043,
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0.006168789695948362,
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0.003252151422202587,
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0.016104627400636673,
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0.033830348402261734,
0.02519337646663189,
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0.0035145822912454605,
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0.05676482245326042,
0.005796735640615225,
0.0012590044643729925,
0.027146661654114723,
0.02723967470228672,
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0.020436398684978485,
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0.012151552364230156,
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0.006969370413571596,
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0.0264025516808033,
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0.006165467668324709,
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0.03845444694161415,
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0.03786979243159294,
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0.010856007225811481,
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0.0063182758167386055,
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0.013646412640810013,
0.011726347729563713,
0.027664879336953163,
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0.01578572392463684,
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0.013008605688810349,
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0.00815197080373764,
0.010537102818489075,
0.06792644411325455,
0.039331432431936264,
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0.03755088895559311,
0.005255263764411211,
0.04533744975924492,
0.039995815604925156,
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0.005218722857534885,
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0.03678020462393761,
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0.004597525577992201,
0.0017406815895810723,
0.005225366912782192,
0.014284219592809677,
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0.009606967680156231,
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0.011606758460402489,
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0.04392895847558975,
0.013965316116809845,
0.0014525058213621378,
0.036913082003593445,
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0.005786770023405552,
0.005225366912782192,
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-0.020090920850634575
] |
022 in normalized form (one non-
zero decimal digit 6 before the decimal point) times (c) the radix 10 raised to
the power 23. The computers 32-bit oating point data type, on the other hand,
consists of (a) a sign bit (positive or negative), (b) 24 binary digits in normalized
form (one non-zero binary digit to the left of the binary point) times (c) the radix
2 raised to an exponent expressed in eight bits.
We determine the value of the 32-bit oating point representation shown in
Figure 2. | [
0.009149253368377686,
0.002517388202250004,
0.02048841491341591,
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0.011426489800214767,
0.029852628707885742,
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0.005756909027695656,
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0.01035840529948473,
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0.0231485515832901,
0.008222236298024654,
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0.02588929608464241,
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0.01739835925400257,
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0.027340279892086983,
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0.020031625404953957,
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0.009592609480023384,
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0.00748331006616354,
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0.005575536284595728,
0.0030715828761458397,
0.020380934700369835,
0.011950456537306309,
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0.006737665738910437,
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0.026950662955641747,
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0.01368357427418232,
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0.0004786228237207979,
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0.016699738800525665,
0.004574626684188843,
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0.0006360645638778806,
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0.008732767775654793,
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0.04097682982683182,
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0.011554122902452946,
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0.02297389507293701,
-0.017062483355402946,
0.0015912109520286322,
0.015329365618526936,
0.0021378486417233944,
0.006287592463195324,
-0.013515637256205082,
-0.030390029773116112
] |
The computers 32-bit oating point data type, on the other hand,
consists of (a) a sign bit (positive or negative), (b) 24 binary digits in normalized
form (one non-zero binary digit to the left of the binary point) times (c) the radix
2 raised to an exponent expressed in eight bits.
We determine the value of the 32-bit oating point representation shown in
Figure 2.3 by examining its three parts. | [
-0.0034968992695212364,
0.008655421435832977,
0.023112904280424118,
-0.02822035178542137,
-0.022254852578043938,
0.01124319527298212,
-0.03729118034243584,
-0.0015611766139045358,
-0.0296640582382679,
-0.030372289940714836,
0.011685840785503387,
0.03805389255285263,
-0.03927968069911003,
0.006609037518501282,
-0.0029452950693666935,
0.014300853945314884,
-0.0006443896563723683,
-0.009922068566083908,
0.01720188371837139,
0.007027848158031702,
0.008587322197854519,
0.01826423406600952,
0.0006350260227918625,
-0.00931598525494337,
0.004487744066864252,
-0.004436669871211052,
0.018550250679254532,
-0.011890138499438763,
0.021819017827510834,
-0.02728058211505413,
0.015594741329550743,
-0.01118190586566925,
-0.037917692214250565,
-0.0028720882255584,
-0.01954450085759163,
0.011331724934279919,
-0.02326272241771221,
0.010017408058047295,
0.000833790865726769,
-0.0035275439731776714,
0.04415559023618698,
-0.003365808166563511,
-0.006091482937335968,
-0.0026133109349757433,
0.013395133428275585,
-0.008573702536523342,
0.00568288704380393,
-0.01347685232758522,
-0.018509391695261,
0.0021434256341308355,
-0.016575371846556664,
0.02256811037659645,
-0.015227004885673523,
-0.029309941455721855,
0.01683414913713932,
-0.010786930099129677,
-0.009275125339627266,
-0.006159582175314426,
-0.011958237737417221,
-0.0023579385597258806,
0.013463232666254044,
0.013034206815063953,
-0.013735629618167877,
-0.01390587817877531,
-0.01929934322834015,
-0.005539878737181425,
0.006302590947598219,
0.022990325465798378,
0.001017233356833458,
-0.0015909700887277722,
0.016275733709335327,
0.015948858112096786,
-0.001688862801529467,
0.016983967274427414,
0.0317070372402668,
-0.011903759092092514,
-0.016207635402679443,
0.009288745000958443,
0.003365808166563511,
0.012407693080604076,
0.006108507513999939,
-0.012223825789988041,
0.0019970121793448925,
-0.013647100888192654,
-0.006271946243941784,
-0.025251222774386406,
-0.004964439198374748,
0.0046477774158120155,
-0.02523760311305523,
0.004715877119451761,
0.037999413907527924,
0.006326425354927778,
0.017542380839586258,
0.009431754238903522,
0.005223216954618692,
-0.00538665521889925,
0.016439171507954597,
0.01934020221233368,
-0.030590208247303963,
-0.02941890060901642,
-0.01040557399392128,
0.010718830861151218,
-0.016983967274427414,
-0.012101246975362301,
-0.014654970727860928,
-0.0015867138281464577,
0.0008380470681004226,
-0.002776749199256301,
0.004882720299065113,
-0.0031223532278090715,
-0.01323169469833374,
0.0528450608253479,
0.01720188371837139,
-0.0021195909939706326,
0.014232754707336426,
0.015295104123651981,
0.021968835964798927,
-0.008008478209376335,
-0.010698401369154453,
-0.0054547544568777084,
0.022581730037927628,
0.013960357755422592,
0.021110784262418747,
-0.005226621869951487,
0.01089588925242424,
0.004344735760241747,
-0.011338534764945507,
-0.04576273262500763,
-0.0010470268316566944,
0.001152580720372498,
0.012257874943315983,
0.0025264841970056295,
-0.00496784457936883,
0.004433264955878258,
0.003953164909034967,
-0.005890590138733387,
-0.01063030119985342,
0.005022323690354824,
-0.03674638643860817,
-0.024120774120092392,
0.02715800330042839,
-0.0013543249806389213,
-0.01822337508201599,
-0.011358964256942272,
0.015227004885673523,
0.019081424921751022,
0.02060684934258461,
0.022786026820540428,
-0.007913139648735523,
-0.03503028303384781,
0.015104426071047783,
0.009949308820068836,
0.006353665143251419,
0.018114415928721428,
-0.004474124405533075,
0.00644219433888793,
-0.02121974341571331,
0.02871066704392433,
-0.0023221862502396107,
-0.030535729601979256,
-0.011631361208856106,
0.013742439448833466,
-0.018822647631168365,
0.018332332372665405,
0.03249698877334595,
0.013565381988883018,
-0.00692229438573122,
-0.008798430673778057,
-0.004402620252221823,
0.0007908031693659723,
-0.006071052979677916,
0.025959454476833344,
-0.004327710717916489,
0.011113806627690792,
0.008553273044526577,
0.015867138281464577,
0.002960617421194911,
-0.0014896723441779613,
-0.014818408526480198,
0.0063434503972530365,
0.02322186343371868,
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0.010902699083089828,
0.008546463213860989,
-0.013810539618134499,
-0.0010861839400604367,
0.04192193225026131,
0.0085396533831954,
0.019190384075045586,
-0.001047878060489893,
0.023698559030890465,
-0.005703317001461983,
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0.010916318744421005,
-0.6345765590667725,
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-0.011958237737417221,
-0.019353821873664856,
0.020429791882634163,
0.02879238687455654,
0.020225493237376213,
0.007450064178556204,
-0.0007461130153387785,
0.011570071801543236,
0.00841707456856966,
0.011406634002923965,
0.024447651579976082,
-0.020143775269389153,
-0.005352605599910021,
-0.015867138281464577,
0.00008496452210238203,
-0.006367285270243883,
-0.00839664414525032,
0.002841443521901965,
-0.04570825397968292,
0.01601695641875267,
-0.022636208683252335,
0.012959297746419907,
0.0032040723599493504,
0.007838229648768902,
0.0007571791647933424,
0.009949308820068836,
-0.02019825391471386,
0.004409430082887411,
-0.00156458152923733,
0.00028239929815754294,
0.01069159060716629,
-0.0007716502877883613,
0.029473379254341125,
-0.030263330787420273,
-0.0004494554304983467,
0.010657541453838348,
0.005679482128471136,
0.012203395366668701,
-0.01954450085759163,
-0.019571740180253983,
0.019353821873664856,
0.000821022258605808,
-0.007450064178556204,
-0.010112746618688107,
0.003345378441736102,
0.007627122104167938,
-0.0051346877589821815,
-0.02642253041267395,
0.0037386519834399223,
-0.006571582984179258,
-0.002272814279422164,
-0.02613651379942894,
0.017787538468837738,
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0.034022413194179535,
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0.007055087946355343,
-0.026368051767349243,
-0.0008593281381763518,
0.005703317001461983,
-0.04682508111000061,
-0.038816604763269424,
0.0034100727643817663,
0.0010555392364040017,
-0.029364420101046562,
0.012176156044006348,
0.01622125506401062,
0.0015526642091572285,
0.0014394490281119943,
0.039960671216249466,
0.015717320144176483,
0.0019595574121922255,
-0.0073411050252616405,
0.017882877960801125,
0.04766951501369476,
-0.013102306053042412,
0.015444922260940075,
0.021110784262418747,
0.0036773625761270523,
-0.008124247193336487,
-0.02352149970829487,
-0.013687960803508759,
0.03178875520825386,
0.019639840349555016,
-0.003813561052083969,
0.01415103580802679,
0.007667982019484043,
-0.009452183730900288,
-0.0014130605850368738,
0.022091414779424667,
-0.0005209596711210907,
-0.057911649346351624,
0.00023409136338159442,
0.02109716460108757,
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-0.02974577620625496,
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0.0164664126932621,
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0.02146490104496479,
0.023575980216264725,
-0.03214287385344505,
0.0026780052576214075,
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-0.008648611605167389,
0.004358355421572924,
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0.015349583700299263,
0.009336414746940136,
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-0.0061902268789708614,
0.02605479396879673,
-0.006993798539042473,
-0.004412834998220205,
-0.02835655026137829,
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-0.0032943040132522583,
0.014981847256422043,
0.00870309118181467,
-0.0188907478004694,
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0.002718864707276225,
0.004743116442114115,
-0.0064353845082223415,
-0.010684780776500702,
0.028819626197218895,
-0.0016624743584543467,
-0.010092317126691341,
-0.0031547003891319036,
0.0030542537569999695,
-0.01773305982351303,
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-0.006830360274761915,
0.007790560368448496,
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-0.0015816063387319446,
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0.0046171327121555805,
0.013510902412235737,
0.028901346027851105,
0.006401334889233112,
-0.0055160438641905785,
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0.020375311374664307,
0.005985929165035486,
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0.01740618236362934,
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0.035248201340436935,
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0.00025856454158201814,
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-0.006033598445355892,
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-0.02027997374534607,
-0.02789347618818283
] |
We determine the value of the 32-bit oating point representation shown in
Figure 2.3 by examining its three parts. | [
0.008043266832828522,
0.017567824572324753,
0.012445606291294098,
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0.029293552041053772,
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0.011594210751354694,
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0.0041046966798603535,
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0.00603591138496995,
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0.041393060237169266,
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] |
The sign bit S is just a single binary digit, 0 for positive numbers, 1 for neg-
ative numbers. The formula contains the factor 1S, which evaluates to +1 if
S = 0, and 1 if S = 1.
The 23 fraction bits form the 24-bit quantity 1.fraction, where normalized
form demands exactly one non-zero binary digit to the left of the binary point.
Since there exists only one non-zero binary digit (i. | [
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fraction, where normalized
form demands exactly one non-zero binary digit to the left of the binary point.
Since there exists only one non-zero binary digit (i.e., the value 1), it is unneces-
sary to explicitly store that bit in our 32-bit oating point format. In fact that is
how we get 24 bits of precision, the 1 to the left of the binary point that is always
present in normalized numbers and so is unnecessary to store, and the 23 bits of
fraction that are actually part of the 32-bit data type. | [
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] |
, the value 1), it is unneces-
sary to explicitly store that bit in our 32-bit oating point format. In fact that is
how we get 24 bits of precision, the 1 to the left of the binary point that is always
present in normalized numbers and so is unnecessary to store, and the 23 bits of
fraction that are actually part of the 32-bit data type.
The eight exponent bits are encoded in what we call an excess code, named
for the notion that one can get the *real* exponent by treating the code as
an unsigned integer and subtracting the excess (sometimes called the bias). | [
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] |
In fact that is
how we get 24 bits of precision, the 1 to the left of the binary point that is always
present in normalized numbers and so is unnecessary to store, and the 23 bits of
fraction that are actually part of the 32-bit data type.
The eight exponent bits are encoded in what we call an excess code, named
for the notion that one can get the *real* exponent by treating the code as
an unsigned integer and subtracting the excess (sometimes called the bias). In
the case of the IEEE Floating Point that almost everyone uses, that excess (or
bias) is 127 for 32-bit oating point numbers. | [
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] |
The eight exponent bits are encoded in what we call an excess code, named
for the notion that one can get the *real* exponent by treating the code as
an unsigned integer and subtracting the excess (sometimes called the bias). In
the case of the IEEE Floating Point that almost everyone uses, that excess (or
bias) is 127 for 32-bit oating point numbers. Thus, an exponent eld contain-
ing 10000110 corresponds to the exponent +7 (since 10000110 represents the
unsigned integer 134, from which we subtract 127, yielding +7). | [
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] |
In
the case of the IEEE Floating Point that almost everyone uses, that excess (or
bias) is 127 for 32-bit oating point numbers. Thus, an exponent eld contain-
ing 10000110 corresponds to the exponent +7 (since 10000110 represents the
unsigned integer 134, from which we subtract 127, yielding +7). An exponent
eld containing 00000111 corresponds to the exponent 120 (since 00000111
represents the unsigned integer 7, from which we subtract 127, yielding 120). | [
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] |
Thus, an exponent eld contain-
ing 10000110 corresponds to the exponent +7 (since 10000110 represents the
unsigned integer 134, from which we subtract 127, yielding +7). An exponent
eld containing 00000111 corresponds to the exponent 120 (since 00000111
represents the unsigned integer 7, from which we subtract 127, yielding 120).
The exponent eld gives us numbers as large as 2+127 for an exponent eld con-
taining 254 (11111110) and as small as 2126 for an exponent eld containing
1 (00000001). | [
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-0.02088787965476513,
0.011859145015478134,
-0.014805220067501068,
-0.004500759765505791,
0.005432889796793461,
-0.012825294397771358,
0.025174317881464958,
-0.011137934401631355,
0.028957268223166466,
-0.02155465818941593,
0.016383718699216843,
0.0035278063733130693,
0.00813742820173502,
-0.01602991670370102,
-0.017948606982827187,
-0.03314845263957977,
-0.006120080593973398,
0.017935000360012054,
0.0055961827747523785,
0.016805557534098625,
0.006589547730982304,
0.0839325338602066,
0.03723077476024628,
-0.020670155063271523,
0.04705556109547615,
-0.007817646488547325,
0.028331313282251358,
0.02744681015610695,
0.0004852009587921202,
0.016723912209272385,
-0.07233873754739761,
0.019404636695981026,
-0.0017120236298069358,
0.020153062418103218,
-0.003663883777335286,
0.000700373318977654,
-0.019486282020807266,
0.013138271868228912,
0.03342060744762421,
-0.02032996155321598,
-0.020629333332180977,
0.0356522761285305,
-0.0030515354592353106,
0.045313771814107895,
-0.0020207492634654045,
-0.013539700768887997,
-0.0036366682033985853,
0.007212101481854916,
-0.0022231643088161945,
-0.0017792119178920984,
-0.03279465064406395,
-0.011716263368725777,
-0.009872415103018284,
-0.02584109641611576,
-0.013362799771130085,
-0.004902188200503588,
-0.0016865091165527701,
-0.016478972509503365,
-0.010797740891575813,
0.038918133825063705,
-0.013961540535092354,
-0.019717613235116005,
0.010797740891575813,
0.007416217587888241,
0.010301058180630207,
0.01134205050766468,
-0.006956956814974546,
0.0015087580541148782,
0.016465364024043083,
-0.04452452063560486
] |
An exponent
eld containing 00000111 corresponds to the exponent 120 (since 00000111
represents the unsigned integer 7, from which we subtract 127, yielding 120).
The exponent eld gives us numbers as large as 2+127 for an exponent eld con-
taining 254 (11111110) and as small as 2126 for an exponent eld containing
1 (00000001).
Example 2.12
What does the oating point data type
00111101100000000000000000000000
represent?
The leading bit is a 0. | [
0.005639593582600355,
0.007697940804064274,
0.000404411373892799,
-0.0228782519698143,
-0.031487151980400085,
0.014283262193202972,
-0.029095018282532692,
-0.028816863894462585,
-0.03234943374991417,
-0.0010291737271472812,
0.010987124405801296,
0.04052719101309776,
-0.03310045227408409,
-0.006748736836016178,
-0.002814581384882331,
0.022071603685617447,
-0.005236268974840641,
-0.004147987347096205,
0.003011028515174985,
0.029790405184030533,
-0.010270874947309494,
0.005316238384693861,
-0.0030962133314460516,
-0.0029693052638322115,
0.005663932301104069,
-0.005629162769764662,
0.001396859879605472,
-0.020805997774004936,
-0.006300211884081364,
-0.020388765260577202,
0.020569566637277603,
-0.0027693812735378742,
-0.036076709628105164,
0.0021383168641477823,
-0.013845168054103851,
0.0029936437495052814,
-0.012711686082184315,
-0.02535383217036724,
0.015854837372899055,
0.021876893937587738,
0.04511674866080284,
0.020958982408046722,
-0.016091270372271538,
-0.0005280599580146372,
0.014825664460659027,
0.0019940240308642387,
-0.0008427228895016015,
-0.021445754915475845,
0.0036646928638219833,
0.007558863610029221,
-0.006800890900194645,
0.02446373738348484,
-0.0034786765463650227,
-0.029400989413261414,
0.0040993099100887775,
0.005921225529164076,
0.004335741978138685,
0.007955234497785568,
0.00026098766829818487,
0.011167924851179123,
0.0027641658671200275,
-0.006435812450945377,
-0.025089586153626442,
0.0007484109373763204,
-0.013330580666661263,
-0.007117292378097773,
0.00007393926352960989,
0.024102134630084038,
-0.01716216653585434,
-0.0017315151635557413,
0.00486423633992672,
0.011828543618321419,
-0.005069375969469547,
-0.004127125721424818,
0.01944303885102272,
-0.021557016298174858,
-0.008226435631513596,
0.015242896974086761,
0.010090074501931667,
-0.007117292378097773,
0.01471440214663744,
-0.014909110963344574,
-0.004589558113366365,
-0.003429999342188239,
0.005399684887379408,
-0.023086868226528168,
-0.006780029274523258,
0.0018479926511645317,
-0.014297169633209705,
-0.0029849514830857515,
0.05460183694958687,
0.007614494301378727,
0.04163981229066849,
0.007948280312120914,
0.013184549286961555,
0.006766121368855238,
0.017120443284511566,
0.04556179791688919,
-0.03802379593253136,
-0.02230803482234478,
-0.02453327551484108,
0.01331667322665453,
-0.02727310173213482,
0.0034682457335293293,
-0.014165045693516731,
-0.005156299099326134,
0.002482533920556307,
-0.00029075893689878285,
-0.0037133698351681232,
-0.0019783778116106987,
-0.02173781767487526,
0.05724430829286575,
0.03337860479950905,
-0.02478361502289772,
0.006136795971542597,
-0.0023347639944404364,
0.004544358234852552,
0.004182756412774324,
-0.023921335116028786,
-0.015145542100071907,
0.03866355121135712,
0.009234747849404812,
0.014255446381866932,
-0.016285978257656097,
-0.00886619184166193,
-0.0035256152041256428,
-0.028483077883720398,
-0.037773456424474716,
-0.007012984249740839,
-0.00365426205098629,
0.024616722017526627,
0.014192861504852772,
-0.018093986436724663,
0.008775792084634304,
0.009457271546125412,
0.03126462921500206,
-0.0007214646320790052,
0.003019720781594515,
-0.030430162325501442,
-0.019554300233721733,
0.03896952420473099,
0.00867148395627737,
0.011105339974164963,
0.0012942901812493801,
0.02127886191010475,
0.010090074501931667,
0.010201336815953255,
0.03866355121135712,
-0.013358396477997303,
-0.026341283693909645,
0.011849405243992805,
0.01671711914241314,
0.02827446162700653,
0.022474927827715874,
0.008351605385541916,
0.0213623084127903,
-0.006293258164077997,
0.030012929812073708,
-0.012829901650547981,
-0.030235454440116882,
-0.015117727220058441,
0.0037550930865108967,
-0.002830227604135871,
0.021320585161447525,
0.021723909303545952,
0.027300918474793434,
0.0007944803801365197,
-0.010674200020730495,
-0.017565490677952766,
0.010555984452366829,
-0.013608735986053944,
0.045923396944999695,
-0.0020739936735481024,
0.010514261201024055,
0.006630520801991224,
0.009603302925825119,
0.01564622111618519,
-0.014130276627838612,
-0.006540120579302311,
-0.013608735986053944,
0.021876893937587738,
0.0017975770169869065,
0.031626228243112564,
0.009283424355089664,
-0.016049547120928764,
-0.015785299241542816,
0.03343423828482628,
0.005636116489768028,
-0.012398761697113514,
-0.00038137665251269937,
0.03101428784430027,
0.004391372669488192,
-0.0037064161151647568,
0.008240343071520329,
-0.6003698706626892,
-0.04876058176159859,
-0.016703210771083832,
-0.023434562608599663,
0.015368066728115082,
0.0040680174715816975,
0.028191015124320984,
0.00981887336820364,
-0.007385016418993473,
0.004061063751578331,
0.003030151594430208,
-0.0008861846290528774,
0.022043786942958832,
-0.028316184878349304,
-0.005618731956928968,
-0.02173781767487526,
-0.003197044599801302,
0.001203889842145145,
-0.0016889226390048862,
0.0077535719610750675,
-0.031153365969657898,
0.011432172730565071,
-0.008372467011213303,
0.018733741715550423,
-0.0036646928638219833,
0.0072668008506298065,
-0.0012204053346067667,
-0.00040397673728875816,
-0.009227793663740158,
-0.006936491467058659,
0.011112294159829617,
-0.005632639862596989,
0.006508828140795231,
0.015409789979457855,
0.02827446162700653,
-0.030346715822815895,
-0.006074211094528437,
-0.00627587316557765,
0.002845873823389411,
0.013177595101296902,
-0.006849567871540785,
-0.02040267363190651,
0.02127886191010475,
-0.01887281984090805,
0.004899005871266127,
0.009714565239846706,
-0.00741978595033288,
0.01926223747432232,
0.0024355952627956867,
-0.04917781427502632,
0.023086868226528168,
0.014603139832615852,
-0.0356038473546505,
-0.005451838951557875,
0.008650622330605984,
-0.005667408928275108,
0.014728309586644173,
-0.0008375074830837548,
-0.015423697419464588,
-0.04049937427043915,
-0.0012204053346067667,
0.00619590375572443,
-0.01855294220149517,
-0.006067256908863783,
-0.03540913760662079,
-0.01248220819979906,
-0.03730059042572975,
0.019401315599679947,
-0.0024807953741401434,
-0.0035429999697953463,
0.008622806519269943,
0.04583995044231415,
0.024574998766183853,
0.02027750387787819,
0.005236268974840641,
0.021529201418161392,
0.022919975221157074,
-0.02168218605220318,
0.024755800142884254,
0.03883044421672821,
-0.002628565300256014,
-0.01652240939438343,
-0.05273819714784622,
0.018386049196124077,
0.013984245248138905,
0.016564132645726204,
-0.011084478348493576,
0.013845168054103851,
0.0010126582346856594,
0.00506589887663722,
0.014923018403351307,
0.0010117889614775777,
0.018066169694066048,
-0.05223751813173294,
0.0015420221025124192,
0.013337534852325916,
-0.023434562608599663,
-0.01766284555196762,
-0.0020183625165373087,
-0.017259521409869194,
-0.020555658265948296,
-0.008977454155683517,
-0.004565219860523939,
0.012134513817727566,
0.022224588319659233,
0.02122323028743267,
0.02008279412984848,
0.02725919522345066,
0.02013842575252056,
-0.02560417167842388,
-0.01461704820394516,
-0.01810789294540882,
-0.008268158882856369,
0.018636388704180717,
-0.00486423633992672,
-0.028872493654489517,
0.01385212130844593,
0.0122388219460845,
-0.021264953538775444,
-0.029095018282532692,
0.026104850694537163,
0.008907915093004704,
-0.011800727806985378,
-0.027092302218079567,
0.0023069484159350395,
0.022224588319659233,
0.017009181901812553,
0.030207637697458267,
0.01926223747432232,
-0.013768674805760384,
0.010097028687596321,
-0.0029171511996537447,
0.011522572487592697,
-0.006887814495712519,
0.029067203402519226,
-0.004652142990380526,
-0.0024599337484687567,
-0.01766284555196762,
0.00092747324379161,
-0.008448959328234196,
-0.02072255127131939,
0.005229314789175987,
0.01951257698237896,
-0.028872493654489517,
0.012071928940713406,
-0.015618406236171722,
-0.00362296961247921,
0.004092356190085411,
0.018886728212237358,
0.0066931056790053844,
0.017259521409869194,
-0.0016689302865415812,
-0.019665561616420746,
0.017301244661211967,
0.03279447928071022,
-0.0050172219052910805,
-0.0013864290667697787,
-0.03268321976065636,
0.0034543380606919527,
0.007565817330032587,
-0.003793339477851987,
0.021501384675502777,
-0.01778801530599594,
-0.005608301144093275,
-0.019401315599679947,
-0.005330146290361881,
-0.03310045227408409,
0.025131309404969215,
-0.016285978257656097,
-0.030096376314759254,
-0.011133155785501003,
-0.008977454155683517,
-0.010542076081037521,
0.003228337038308382,
-0.026897592470049858,
0.026800239458680153,
-0.01862248033285141,
0.023281577974557877,
0.01150171086192131,
-0.022739175707101822,
0.008657575584948063,
0.02097289077937603,
-0.00183930026832968,
0.013420981355011463,
-0.005493562202900648,
0.02337893284857273,
0.00748237082734704,
-0.015854837372899055,
0.0022443635389208794,
-0.01185635942965746,
-0.007287662476301193,
0.029456619173288345,
-0.02168218605220318,
0.02859433926641941,
0.015242896974086761,
0.034852828830480576,
0.019178790971636772,
-0.016174716874957085,
0.017370782792568207,
0.042112674564123154,
0.006359319668263197,
-0.03813505545258522,
0.0049163904041051865,
-0.01376172062009573,
0.03421306982636452,
-0.03844102844595909,
-0.01600782386958599,
-0.028761232271790504,
0.03126462921500206,
0.009026131592690945,
0.0014559677802026272,
-0.01956820860505104,
-0.008115173317492008,
0.002652903785929084,
0.007892649620771408,
0.027287010103464127,
0.017746292054653168,
0.033907100558280945,
0.029512250795960426,
0.004533927422016859,
0.0011386972619220614,
0.007489324547350407,
0.019971532747149467,
-0.002951920498162508,
-0.0020931167528033257,
0.01625816337764263,
0.024366382509469986,
-0.0046869125217199326,
-0.026216113939881325,
-0.030291084200143814,
0.006696582771837711,
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-0.002087901346385479,
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-0.016925735399127007,
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0.009645026177167892,
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-0.008692345581948757,
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-0.008045635186135769,
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0.013226272538304329,
0.008789699524641037,
0.02141793817281723,
0.026841962710022926,
0.028705600649118423,
0.013372303918004036,
0.019276145845651627,
0.020611289888620377,
0.005194545723497868,
0.009485087357461452,
-0.01131395623087883,
0.004707774147391319,
-0.007663171738386154,
-0.021835170686244965,
-0.03991524875164032,
0.0012403976870700717,
0.0002596838166937232,
-0.0036959853023290634,
-0.002913674106821418,
-0.004530450329184532,
-0.003998478874564171,
0.0010422122431918979,
-0.001188243622891605,
-0.0035812463611364365,
0.02435247413814068,
0.04962285980582237,
0.006519258953630924,
-0.04144510254263878,
-0.008282066322863102,
0.0275095347315073,
0.03582637012004852,
-0.00243037985637784,
-0.020124517381191254,
-0.00011017547512892634,
0.004669527988880873,
-0.008136034943163395,
-0.012864670716226101,
-0.011195740662515163,
0.01689791865646839,
-0.005476177670061588,
-0.006560982204973698,
-0.0018636388704180717,
-0.02294779196381569,
0.002202640287578106,
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-0.00730852410197258,
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0.0049407291226089,
0.012523931451141834,
-0.010778508149087429,
0.0037307546008378267,
0.05070766434073448,
0.009561579674482346,
0.0014211984816938639,
-0.007033845875412226,
0.00364730809815228,
0.023935241624712944,
0.004137556534260511,
-0.010959308594465256,
-0.023726625367999077,
-0.011654696427285671,
-0.0011004508705809712,
0.0033326451666653156,
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0.012649101205170155,
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-0.013066333718597889,
0.006557505112141371,
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0.015674037858843803,
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-0.0297347754240036,
0.00975628849118948,
0.004996360279619694,
-0.01568794436752796,
0.0060255336575210094,
0.019790731370449066,
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-0.01226663775742054,
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-0.003026674734428525,
0.01471440214663744,
0.01632770150899887,
-0.0036055848468095064,
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] |
Example 2.12
What does the oating point data type
00111101100000000000000000000000
represent?
The leading bit is a 0. This signies a positive number. The next eight bits
represent the unsigned number 123. If we subtract 127, we get the actual expo-
nent 4. The last 23 bits are all 0. Therefore, the number being represented is
+1.000000000000000000000000 24, which is 1
16.
Example 2. | [
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] |
000000000000000000000000 24, which is 1
16.
Example 2.13
How is the number 6 5
8 represented in the oating point data type?
First, we express 6 5
8 as a binary number: 110.101.
(1 22 + 1 21 + 0 20 + 1 21 + 0 22 + 1 23)
Then we normalize the value, yielding 1.10101 22.
The sign bit is 1, reecting the fact that 6 5
8 is a negative number. The exponent
eld contains 10000001, the unsigned number 129, reecting the fact that the real
exponent is +2 (129127 = +2). | [
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] |
The sign bit is 1, reecting the fact that 6 5
8 is a negative number. The exponent
eld contains 10000001, the unsigned number 129, reecting the fact that the real
exponent is +2 (129127 = +2). The fraction is the 23 bits of precision, after remov-
ing the leading 1. That is, the fraction is 10101000000000000000000. The result is
the number 6 5
8, expressed as a oating point number:
1 10000001 10101000000000000000000. | [
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] |
That is, the fraction is 10101000000000000000000. The result is
the number 6 5
8, expressed as a oating point number:
1 10000001 10101000000000000000000 | [
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] |
Example 2.14
The following three examples provide further illustrations of the interpretation of
the 32-bit oating point data type according to the rules of the IEEE standard.
0 10000011 00101000000000000000000 is 1.00101 24 = 18.5
The exponent eld contains the unsigned number 131. Since 131 127 is 4, the
exponent is +4. Combining a 1 to the left of the binary point with the fraction eld
to the right of the binary point yields 1. | [
0.006197545677423477,
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] |
Since 131 127 is 4, the
exponent is +4. Combining a 1 to the left of the binary point with the fraction eld
to the right of the binary point yields 1.00101. If we move the binary point four
positions to the right, we get 10010.1, which is 18.5.
1 10000010 00101000000000000000000 is 1 1.00101 23 = 9.25
The sign bit is 1, signifying a negative number. The exponent is 130, signifying an
exponent of 130 127, or +3. | [
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] |
25
The sign bit is 1, signifying a negative number. The exponent is 130, signifying an
exponent of 130 127, or +3. Combining a 1 to the left of the binary point with
the fraction eld to the right of the binary point yields 1.00101. Moving the binary
point three positions to the right, we get 1001.01, which is 9.25.
0 11111110 11111111111111111111111 is 2128
The sign is +. The exponent is 254 127, or +127. | [
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] |
0 11111110 11111111111111111111111 is 2128
The sign is +. The exponent is 254 127, or +127. Combining a 1 to the left
of the binary point with the fraction eld to the right of the binary point yields
1.11111111 1, which is approximately 2. Therefore, the result is approximately
2128.
2.7.1.2 Innities
We noted above that the oating point data type represented numbers expressed
in scientic notation in normalized form provided the exponent eld does not
contain 00000000 or 11111111. | [
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] |
1.2 Innities
We noted above that the oating point data type represented numbers expressed
in scientic notation in normalized form provided the exponent eld does not
contain 00000000 or 11111111.
If the exponent eld contains 11111111, we use the oating point data type to
represent various things, among them the notion of innity. Innity is represented
by the exponent eld containing all 1s and the fraction eld containing all 0s. | [
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] |
If the exponent eld contains 11111111, we use the oating point data type to
represent various things, among them the notion of innity. Innity is represented
by the exponent eld containing all 1s and the fraction eld containing all 0s. We
represent positive innity if the sign bit is 0 and negative innity if the sign bit is 1.
2.7.1.3 Subnormal Numbers
The smallest number that can be represented in normalized form is
N = 1. | [
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0.02183564007282257,
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0.000002494000000297092,
0.0364154651761055,
0.010004023090004921,
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0.035242531448602676,
0.010910999961197376,
0.028968706727027893,
0.014770766720175743,
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0.008510580286383629,
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0.004995192401111126,
0.017389407381415367,
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-0.016952967271208763,
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0.020089879631996155,
0.003798391902819276,
0.006860291585326195,
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0.016652913764119148,
-0.020048962906003,
0.01200891938060522,
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0.009553944692015648,
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0.022763075307011604,
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0.012711315415799618,
0.0015727183781564236,
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0.007501312531530857,
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0.025422630831599236,
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0.00534298038110137,
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0.009056130424141884,
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0.08232349902391434,
0.04511698707938194,
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0.05207274854183197,
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0.018139537423849106,
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0.005090663675218821,
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0.011640673503279686,
0.019748911261558533,
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0.0049508665688335896,
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0.004739465657621622,
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0.022422105073928833,
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0.002557265805080533,
0.02159014157950878,
0.004555342718958855,
-0.0020764998625963926,
0.004231422208249569,
-0.05973772704601288
] |
1.3 Subnormal Numbers
The smallest number that can be represented in normalized form is
N = 1.00000000000000000000000 2126
What about numbers smaller than 2126 but larger than 0? We call such num-
bers subnormal numbers because they cannot be represented in normalized form.
The largest subnormal number is
N = 0.11111111111111111111111 2126
The smallest subnormal number is
N = 0.00000000000000000000001 2126, i. | [
0.01298023946583271,
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0.03540428727865219,
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0.0006227052072063088,
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0.000008318431355291978,
0.013106778264045715,
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0.03124847821891308,
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0.0022244227584451437,
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0.003459843108430505,
0.0076056611724197865,
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0.008857731707394123,
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] |
11111111111111111111111 2126
The smallest subnormal number is
N = 0.00000000000000000000001 2126, i.e., 223 2126 which is 2149.
Note that the largest subnormal number is 2126 minus 2149. Do you see why
that is the case? | [
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] |
Subnormal numbers are numbers of the form
N = (1)S 0. fraction 2126
We represent them with an exponent eld of 00000000. The fraction eld is
represented in the same way as with normalized numbers. That is, if the expo-
nent eld contains 00000000, the exponent is 126, and the signicant digits are
obtained by starting with a leading 0, followed by a binary point, followed by the
23 bits of the fraction eld. | [
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0.014784904196858406,
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0.010655420832335949,
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0.02965104579925537,
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0.03839742764830589,
0.017777087166905403,
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0.0054089464247226715,
0.011921344324946404,
0.0030480327550321817,
-0.005276938434690237,
-0.020281856879591942,
0.004650746006518602,
0.017601076513528824,
0.03661023825407028,
0.000962982012424618,
-0.015597262419760227,
0.0015671728178858757,
0.019767362624406815,
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-0.020051687955856323,
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-0.001243076054379344,
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0.011196992360055447,
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0.009565507993102074,
0.04189056158065796,
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0.009592587128281593,
0.004390114918351173,
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-0.00917286891490221,
-0.0043833451345562935,
0.017194898799061775,
-0.013579907827079296,
0.016991809010505676,
-0.035852041095495224,
0.015150465071201324,
-0.03688102588057518,
0.05017660930752754,
0.005815125070512295,
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-0.0007408530218526721,
0.03233182430267334,
0.005002767778933048,
0.028188802301883698,
0.0011686099460348487,
0.0002502653223928064,
0.020105846226215363,
0.02954273112118244,
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-0.022001346573233604,
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0.0035845269449055195,
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0.00926764402538538,
0.010188315995037556,
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0.0001249211054528132,
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0.012544152326881886,
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0.01609821617603302,
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0.01259830966591835,
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0.01144069992005825,
0.0052126264199614525,
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0.003279892960563302,
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-0.009315031580626965,
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0.03858697786927223,
0.00699981302022934,
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0.0039230091497302055,
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0.026577625423669815,
0.013776227831840515,
-0.009619665332138538,
-0.008455286733806133,
-0.008888543583452702,
0.012002580799162388,
-0.021744100376963615,
0.007060739677399397,
-0.021662862971425056,
0.008245427161455154,
-0.0411323644220829,
-0.012970639392733574,
-0.00042183350888080895,
-0.015516025945544243,
-0.01819680631160736,
-0.007372143212705851,
-0.03021969459950924,
0.024803979322314262,
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0.026645323261618614,
-0.004342727363109589,
-0.0056661926209926605,
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0.003180040745064616,
0.017695851624011993,
0.030517559498548508,
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0.007047200575470924,
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0.009653513319790363,
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0.01848113164305687,
0.0073789129965007305,
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0.031411152333021164,
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0.002445534337311983,
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0.017357369884848595,
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0.014338107779622078,
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0.026144368574023247,
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0.00042077573016285896,
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0.02538616769015789,
0.01257123053073883,
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0.01585450768470764,
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0.0018904233584180474,
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0.028513744473457336,
0.015055689960718155,
0.005432640202343464,
0.020065227523446083,
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0.009910760447382927,
0.008895313367247581,
0.010912667959928513,
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0.010127388872206211,
0.014974454417824745,
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0.008455286733806133,
0.007087818346917629,
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0.021473314613103867,
0.008888543583452702,
0.040563713759183884,
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0.023910386487841606,
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0.007331525441259146,
0.017628155648708344,
0.0006355004152283072,
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0.02305741049349308,
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0.0020055072382092476,
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-0.010452331975102425,
-0.015651419758796692,
0.0025690803304314613,
0.009897220879793167,
0.03679978847503662,
0.0732746347784996,
0.031546544283628464,
-0.014283950440585613,
0.0416739359498024,
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0.02095882035791874,
0.020051687955856323,
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0.033414967358112335,
-0.001920886803418398,
0.004566125571727753,
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0.004742136225104332,
0.01501507218927145,
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0.0016187913715839386,
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0.024790439754724503,
-0.004034708254039288,
0.035906195640563965,
0.01056064572185278,
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0.025900661945343018,
0.008726072497665882,
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-0.01600344106554985,
0.00012894057726953179,
-0.02078280970454216,
-0.0251966193318367,
0.007541384547948837,
-0.009917530231177807,
0.0057508135214447975,
-0.005266783758997917,
0.02565695345401764,
0.017519840970635414,
-0.012056737206876278,
0.017601076513528824,
-0.008814077824354172,
0.024140553548932076,
0.017221976071596146,
0.003393284510821104,
0.0015223239315673709,
0.008468826301395893,
-0.04213427007198334
] |
The fraction eld is
represented in the same way as with normalized numbers. That is, if the expo-
nent eld contains 00000000, the exponent is 126, and the signicant digits are
obtained by starting with a leading 0, followed by a binary point, followed by the
23 bits of the fraction eld.
Example 2.15
What number corresponds to the following oating point representation?
0 00000000 00001000000000000000000
Answer: The leading 0 means the number is positive. | [
0.01946430653333664,
0.00011728018580470234,
0.007700978312641382,
-0.009126836434006691,
-0.023782234638929367,
-0.002394365379586816,
-0.024078167974948883,
-0.022934790700674057,
-0.020109979435801506,
-0.0046710302121937275,
0.009147013537585735,
0.03755656257271767,
-0.015052217990159988,
-0.010599774308502674,
-0.003182959044352174,
0.01864376664161682,
-0.003305703867226839,
-0.020231042057275772,
-0.0025305617600679398,
0.020325202494859695,
-0.0030686212703585625,
0.015253989957273006,
-0.005081300623714924,
-0.010041537694633007,
-0.007452125661075115,
-0.0010988186113536358,
-0.0008100319537334144,
-0.005787503905594349,
-0.007082209922373295,
-0.008198683150112629,
0.014742833562195301,
0.0069207921624183655,
-0.055581558495759964,
0.01485044602304697,
-0.017809772863984108,
0.022517794743180275,
-0.005861487239599228,
-0.01061322633177042,
0.014231677167117596,
-0.01920872926712036,
0.031019138172268867,
0.006476892624050379,
-0.015146378427743912,
-0.008313021622598171,
0.007082209922373295,
0.007169644348323345,
0.014150967821478844,
-0.0321490615606308,
-0.01541540864855051,
0.008992321789264679,
0.004956874530762434,
0.014218226075172424,
-0.01259059552103281,
-0.03292924910783768,
0.00888470932841301,
-0.01892624795436859,
0.0012501479359343648,
0.017971191555261612,
-0.0026398550253361464,
0.022800276055932045,
-0.01692197471857071,
0.0050611235201358795,
-0.03072320483624935,
-0.017069941386580467,
-0.016545332968235016,
0.003272075206041336,
-0.007647172547876835,
0.010034811682999134,
-0.018388187512755394,
-0.004143059253692627,
0.02487180568277836,
0.009516929276287556,
-0.006268394645303488,
0.008043991401791573,
0.01581895351409912,
-0.014352740719914436,
-0.012913431040942669,
0.01146067027002573,
0.017702162265777588,
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0.00860222801566124,
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-0.02571924962103367,
-0.008413907140493393,
0.011191640049219131,
-0.02168380282819271,
-0.0090932073071599,
0.0072301761247217655,
-0.019679531455039978,
-0.005794229451566935,
0.04775279015302658,
0.0027340154629200697,
0.02518119104206562,
0.015765147283673286,
0.013579280115664005,
-0.01659913919866085,
0.009429494850337505,
0.03925145044922829,
0.0012039084685966372,
-0.022827180102467537,
-0.02518119104206562,
0.012240856885910034,
-0.006584504619240761,
0.000987843843176961,
-0.011931472457945347,
-0.008017088286578655,
0.00967834796756506,
-0.003051806939765811,
-0.014204774051904678,
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-0.017984643578529358,
0.019114568829536438,
0.021589642390608788,
-0.00037117706960998476,
0.011770054697990417,
0.007754784543067217,
0.00722345057874918,
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0.025651993229985237,
0.0132429925724864,
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-0.017460035160183907,
0.03072320483624935,
0.012893253937363625,
0.0013359012082219124,
0.01400300208479166,
0.012846173718571663,
0.01428548339754343,
-0.0015191776910796762,
-0.010276938788592815,
-0.053321707993745804,
-0.018630314618349075,
0.03771797940135002,
0.019840948283672333,
0.009752330370247364,
-0.011911295354366302,
0.019800594076514244,
0.011628814041614532,
0.008709840476512909,
0.021603094413876534,
-0.021939381957054138,
-0.02059423178434372,
0.02394365333020687,
0.018603412434458733,
0.018980054184794426,
0.018267124891281128,
0.0034940247423946857,
0.01462177000939846,
-0.021455127745866776,
0.02016378380358219,
-0.023002048954367638,
-0.014460352249443531,
-0.019262535497546196,
0.007889299653470516,
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0.013189186342060566,
0.017567647621035576,
0.027629362419247627,
0.015603729523718357,
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0.0367225706577301,
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0.04683808982372284,
-0.004267485346645117,
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0.019652627408504486,
-0.003391457023099065,
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0.012287937104701996,
-0.6176924705505371,
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-0.01924908347427845,
-0.013639811426401138,
0.009496752172708511,
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0.024817999452352524,
0.0208363588899374,
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-0.006238128989934921,
-0.012967237271368504,
0.010956238955259323,
0.011191640049219131,
-0.03882100060582161,
-0.0070889354683458805,
-0.009819588623940945,
-0.01879173330962658,
0.0044558062218129635,
-0.0030585327185690403,
-0.0030366741120815277,
-0.026055537164211273,
-0.006688753608614206,
-0.013512022793293,
0.030884621664881706,
0.003480573184788227,
0.0075866407714784145,
0.00807089451700449,
-0.023002048954367638,
-0.019504660740494728,
-0.013801229186356068,
-0.005232629831880331,
-0.00778841320425272,
-0.000987843843176961,
0.024454809725284576,
0.026230406016111374,
-0.02765626460313797,
-0.0015544878551736474,
0.003578096628189087,
0.011164736934006214,
0.02214115299284458,
-0.013296798802912235,
-0.0035478307399898767,
0.02405126579105854,
-0.024844903498888016,
-0.014769736677408218,
0.01592656411230564,
0.011561555787920952,
0.01215342152863741,
-0.013303523883223534,
-0.040246859192848206,
0.013586005195975304,
-0.0034906617365777493,
-0.04083872586488724,
0.0075530121102929115,
0.01666639745235443,
0.003678982611745596,
0.01472938247025013,
-0.003339332528412342,
-0.01170952245593071,
-0.027467943727970123,
-0.014231677167117596,
0.005508385598659515,
-0.01998891495168209,
-0.011783505789935589,
-0.01692197471857071,
0.0037697802763432264,
-0.04541223123669624,
0.011527927592396736,
0.013673440553247929,
0.0029526022262871265,
0.0045634182170033455,
0.06682700663805008,
0.02041936293244362,
0.007008226588368416,
0.02179141528904438,
0.02298859693109989,
0.0300237275660038,
-0.010034811682999134,
0.01618214324116707,
0.04366353899240494,
0.017823224887251854,
-0.018307479098439217,
-0.03882100060582161,
0.0042069535702466965,
0.007647172547876835,
0.01133960671722889,
-0.009012498892843723,
0.023190369829535484,
-0.009342060424387455,
-0.002034537959843874,
-0.004936696961522102,
0.01805189996957779,
0.012341742403805256,
-0.05929417163133621,
-0.015563374385237694,
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] |
Example 2.15
What number corresponds to the following oating point representation?
0 00000000 00001000000000000000000
Answer: The leading 0 means the number is positive. The next eight bits, a zero
exponent, means the exponent is 126, and the bit to the left of the binary point is 0.
The last 23 bits form the number 0.00001000000000000000000, which equals 25.
Thus, the number represented is 25 2126, which is 2131. | [
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] |
00001000000000000000000, which equals 25.
Thus, the number represented is 25 2126, which is 2131.
Including subnormal numbers allows very, very tiny numbers to be repre-
sented.
A detailed understanding of IEEE Floating Point Arithmetic is well beyond
what should be expected in this rst course. Our purpose in including this sec-
tion in the textbook is to at least let you know that there is, in addition to 2s
complement integers, another very important data type available in almost all
ISAs, which is called oating point; it allows very large and very tiny numbers to
be expressed at the cost of reducing the number of binary digits of precision. | [
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] |
A detailed understanding of IEEE Floating Point Arithmetic is well beyond
what should be expected in this rst course. Our purpose in including this sec-
tion in the textbook is to at least let you know that there is, in addition to 2s
complement integers, another very important data type available in almost all
ISAs, which is called oating point; it allows very large and very tiny numbers to
be expressed at the cost of reducing the number of binary digits of precision.
2. | [
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] |
Our purpose in including this sec-
tion in the textbook is to at least let you know that there is, in addition to 2s
complement integers, another very important data type available in almost all
ISAs, which is called oating point; it allows very large and very tiny numbers to
be expressed at the cost of reducing the number of binary digits of precision.
2.7.2 ASCII Codes
Another representation of information is the standard code that almost all com-
puter equipment manufacturers have agreed to use for transferring characters
between the main computer processing unit and the input and output devices. | [
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] |
7.2 ASCII Codes
Another representation of information is the standard code that almost all com-
puter equipment manufacturers have agreed to use for transferring characters
between the main computer processing unit and the input and output devices.
That code is an eight-bit code referred to as ASCII. ASCII stands for Ameri-
can Standard Code for Information Interchange. It (ASCII) greatly simplies the
interface between a keyboard manufactured by one company, a computer made
by another company, and a monitor made by a third company. | [
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ASCII stands for Ameri-
can Standard Code for Information Interchange. It (ASCII) greatly simplies the
interface between a keyboard manufactured by one company, a computer made
by another company, and a monitor made by a third company.
Each key on the keyboard is identied by its unique ASCII code. So, for
example, the digit 3 is represented as 00110011, the digit 2 is 00110010, the
lowercase e is 01100101, and the ENTER key is 00001101. | [
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] |
Each key on the keyboard is identied by its unique ASCII code. So, for
example, the digit 3 is represented as 00110011, the digit 2 is 00110010, the
lowercase e is 01100101, and the ENTER key is 00001101. The entire set of
eight-bit ASCII codes is listed in Figure E.2 of Appendix E. When you type a key
on the keyboard, the corresponding eight-bit code is stored and made available to
the computer. | [
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0.008854980580508709,
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-0.030096545815467834
] |
2 of Appendix E. When you type a key
on the keyboard, the corresponding eight-bit code is stored and made available to
the computer. Where it is stored and how it gets into the computer are discussed
in Chapter 9.
Most keys are associated with more than one code. For example, the ASCII
code for the letter E is 01000101, and the ASCII code for the letter e is 01100101. | [
0.015917792916297913,
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0.017828455194830894,
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] |
Both are associated with the same key, although in one case the Shift key is also
depressed while in the other case, it is not.
In order to display a particular character on the monitor, the computer must
transfer the ASCII code for that character to the electronics associated with the
monitor. That, too, is discussed in Chapter 9.
2.7.3 Hexadecimal Notation
We have seen that information can be represented as 2s complement integers,
as bit vectors, in oating point format, or as an ASCII code. | [
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] |
7.3 Hexadecimal Notation
We have seen that information can be represented as 2s complement integers,
as bit vectors, in oating point format, or as an ASCII code. There are other
representations also, but we will leave them for another book. However, before
we leave this topic, we would like to introduce you to a representation that is used
more as a convenience for humans than as a data type to support operations being
performed by the computer. | [
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] |
There are other
representations also, but we will leave them for another book. However, before
we leave this topic, we would like to introduce you to a representation that is used
more as a convenience for humans than as a data type to support operations being
performed by the computer. This is the hexadecimal notation. As we will see, it
evolves nicely from the positional binary notation and is useful for dealing with
long strings of binary digits without making errors. | [
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0.011166929267346859,
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] |
This is the hexadecimal notation. As we will see, it
evolves nicely from the positional binary notation and is useful for dealing with
long strings of binary digits without making errors.
It will be particularly useful in dealing with the LC-3 where 16-bit binary
strings will be encountered often.
An example of such a binary string is
0011110101101110
Lets try an experiment. Cover the preceding 16-bit binary string of 0s and 1s
with one hand, and try to write it down from memory. | [
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] |
An example of such a binary string is
0011110101101110
Lets try an experiment. Cover the preceding 16-bit binary string of 0s and 1s
with one hand, and try to write it down from memory. How did you do? Hex-
adecimal notation is about being able to do this without making mistakes. We
shall see how.
In general, a 16-bit binary string takes the form
a15 a14 a13 a12 a11 a10 a9 a8 a7 a6 a5 a4 a3 a2 a1 a0
where each of the bits ai is either 0 or 1. | [
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] |
We
shall see how.
In general, a 16-bit binary string takes the form
a15 a14 a13 a12 a11 a10 a9 a8 a7 a6 a5 a4 a3 a2 a1 a0
where each of the bits ai is either 0 or 1.
If we think of this binary string as an unsigned integer, its value can be
computed as
a15 215 + a14 214 + a13 213 + a12 212 + a11 211 + a10 210
+ a9 29 + a8 28 + a7 27 + a6 26 + a5 25 + a4 24 + a3 23
+ a2 22 + a1 21 + a0 20
We can factor 212 from the rst four terms, 28 from the second four terms, 24
from the third set of four terms, and 20 from the last four terms, yielding
212(a15 23 + a14 22 + a13 21 + a12 20)
+ 28(a11 23 + a10 22 + a9 21 + a8 20)
+ 24(a7 23 + a6 22 + a5 21 + a4 20)
+ 20(a3 23 + a2 22 + a1 21 + a0 20)
Note that the largest value inside a set of parentheses is 15, which would be the
case if each of the four bits is 1. | [
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] |
In general, a 16-bit binary string takes the form
a15 a14 a13 a12 a11 a10 a9 a8 a7 a6 a5 a4 a3 a2 a1 a0
where each of the bits ai is either 0 or 1.
If we think of this binary string as an unsigned integer, its value can be
computed as
a15 215 + a14 214 + a13 213 + a12 212 + a11 211 + a10 210
+ a9 29 + a8 28 + a7 27 + a6 26 + a5 25 + a4 24 + a3 23
+ a2 22 + a1 21 + a0 20
We can factor 212 from the rst four terms, 28 from the second four terms, 24
from the third set of four terms, and 20 from the last four terms, yielding
212(a15 23 + a14 22 + a13 21 + a12 20)
+ 28(a11 23 + a10 22 + a9 21 + a8 20)
+ 24(a7 23 + a6 22 + a5 21 + a4 20)
+ 20(a3 23 + a2 22 + a1 21 + a0 20)
Note that the largest value inside a set of parentheses is 15, which would be the
case if each of the four bits is 1. If we replace what is inside each square bracket. | [
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] |
If we think of this binary string as an unsigned integer, its value can be
computed as
a15 215 + a14 214 + a13 213 + a12 212 + a11 211 + a10 210
+ a9 29 + a8 28 + a7 27 + a6 26 + a5 25 + a4 24 + a3 23
+ a2 22 + a1 21 + a0 20
We can factor 212 from the rst four terms, 28 from the second four terms, 24
from the third set of four terms, and 20 from the last four terms, yielding
212(a15 23 + a14 22 + a13 21 + a12 20)
+ 28(a11 23 + a10 22 + a9 21 + a8 20)
+ 24(a7 23 + a6 22 + a5 21 + a4 20)
+ 20(a3 23 + a2 22 + a1 21 + a0 20)
Note that the largest value inside a set of parentheses is 15, which would be the
case if each of the four bits is 1. If we replace what is inside each square bracket | [
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] |
with a symbol representing its value (from 0 to 15), and we replace 212 with its
equivalent 163, 28 with 162, 24 with 161, and 20 with 160, we have
h3 163 + h2 162 + h1 161 + h0 160
where h3, for example, is a symbol representing
a15 23 + a14 22 + a13 21 + a12 20
Since the symbols must represent values from 0 to 15, we assign symbols to these
values as follows: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F. That is, we represent
0000 with the symbol 0, 0001 with the symbol 1, 1001 with 9, 1010 with A,
1011 with B, 1111 with F. The resulting notation is called hexadecimal, or
base 16. | [
0.002466252539306879,
0.016573216766119003,
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] |
That is, we represent
0000 with the symbol 0, 0001 with the symbol 1, 1001 with 9, 1010 with A,
1011 with B, 1111 with F. The resulting notation is called hexadecimal, or
base 16.
So, for example, if the hex digits E92F represent a 16-bit 2s complement
integer, is the value of that integer positive or negative? How do you know?
Now, then, what is this hexadecimal representation good for, anyway? It
seems like just another way to represent a number without adding any benet. | [
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] |
The resulting notation is called hexadecimal, or
base 16.
So, for example, if the hex digits E92F represent a 16-bit 2s complement
integer, is the value of that integer positive or negative? How do you know?
Now, then, what is this hexadecimal representation good for, anyway? It
seems like just another way to represent a number without adding any benet.
Lets return to the exercise where you tried to write from memory the string
0011110101101110
If we had rst broken the string at four-bit boundaries
0011
1101
0110
1110
and then converted each four-bit string to its equivalent hex digit
3
D
6
E
it would have been no problem to jot down (with the string covered) 3D6E. | [
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] |
So, for example, if the hex digits E92F represent a 16-bit 2s complement
integer, is the value of that integer positive or negative? How do you know?
Now, then, what is this hexadecimal representation good for, anyway? It
seems like just another way to represent a number without adding any benet.
Lets return to the exercise where you tried to write from memory the string
0011110101101110
If we had rst broken the string at four-bit boundaries
0011
1101
0110
1110
and then converted each four-bit string to its equivalent hex digit
3
D
6
E
it would have been no problem to jot down (with the string covered) 3D6E.
In summary, although hexadecimal notation can be used to perform base-
16 arithmetic, it is mainly used as a convenience for humans. | [
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Lets return to the exercise where you tried to write from memory the string
0011110101101110
If we had rst broken the string at four-bit boundaries
0011
1101
0110
1110
and then converted each four-bit string to its equivalent hex digit
3
D
6
E
it would have been no problem to jot down (with the string covered) 3D6E.
In summary, although hexadecimal notation can be used to perform base-
16 arithmetic, it is mainly used as a convenience for humans. It can be used to
represent binary strings that are integers or oating point numbers or sequences
of ASCII codes, or bit vectors. | [
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] |
In summary, although hexadecimal notation can be used to perform base-
16 arithmetic, it is mainly used as a convenience for humans. It can be used to
represent binary strings that are integers or oating point numbers or sequences
of ASCII codes, or bit vectors. It simply reduces the number of digits by a factor
of
res
2.1
2.
2.
requiring additional bits for each students unique bit pattern?. | [
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2.
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] |
ital Logic Structures
I
n Chapter 1, we stated that computers were built from very large numbers
of very simple structures. For example, Intels Broadwell-E5 microproces-
sor, introduced in 2016, contained more than seven billion transistors. Similarly,
IBMs Power9 microprocessor, introduced in 2017, contained eight billion tran-
sistors. In this chapter, we will explain how the MOS transistor works (as a logic
element), show how these transistors are connected to form logic gates, and then
show how logic gates are interconnected to form larger units that are needed to
construct a computer. | [
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] |
Similarly,
IBMs Power9 microprocessor, introduced in 2017, contained eight billion tran-
sistors. In this chapter, we will explain how the MOS transistor works (as a logic
element), show how these transistors are connected to form logic gates, and then
show how logic gates are interconnected to form larger units that are needed to
construct a computer. In Chapter 4, we will connect those larger units and form
a computer. | [
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] |
In this chapter, we will explain how the MOS transistor works (as a logic
element), show how these transistors are connected to form logic gates, and then
show how logic gates are interconnected to form larger units that are needed to
construct a computer. In Chapter 4, we will connect those larger units and form
a computer.
But rst, the transistor.
3.1 The Transistor
Most computers today or rather most microprocessors (which form the core of the
computer) are constructed out of MOS transistors. | [
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] |
3.1 The Transistor
Most computers today or rather most microprocessors (which form the core of the
computer) are constructed out of MOS transistors. MOS stands for metal-oxide
semiconductor. The electrical properties of metal-oxide semiconductors are well
beyond the scope of what we want to understand in this course. They are below
our lowest level of abstraction, which means that if somehow transistors start
misbehaving, we are at their mercy. | [
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] |
The electrical properties of metal-oxide semiconductors are well
beyond the scope of what we want to understand in this course. They are below
our lowest level of abstraction, which means that if somehow transistors start
misbehaving, we are at their mercy. However, it is unlikely in this course that we
will have any problems from the transistors.
Still, it is useful to know that there are two types of MOS transistors: P-type
and N-type. | [
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] |
However, it is unlikely in this course that we
will have any problems from the transistors.
Still, it is useful to know that there are two types of MOS transistors: P-type
and N-type. They both operate logically, very similar to the way wall switches
work.
Figure 3.1 shows the most basic of electrical circuits. It consists of (1) a
power supply (in this case, the 120 volts that come into your house if you live in
the United States, or the 220 volts if you live in most of the rest of the world),
(2) a wall switch, and (3) a lamp (plugged into an outlet in the wall). | [
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] |
1 shows the most basic of electrical circuits. It consists of (1) a
power supply (in this case, the 120 volts that come into your house if you live in
the United States, or the 220 volts if you live in most of the rest of the world),
(2) a wall switch, and (3) a lamp (plugged into an outlet in the wall). In order for
the lamp to glow, electrons must ow; in order for electrons to ow, there must be
a closed circuit from the power supply to the lamp and back to the power supply. | [
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] |
It consists of (1) a
power supply (in this case, the 120 volts that come into your house if you live in
the United States, or the 220 volts if you live in most of the rest of the world),
(2) a wall switch, and (3) a lamp (plugged into an outlet in the wall). In order for
the lamp to glow, electrons must ow; in order for electrons to ow, there must be
a closed circuit from the power supply to the lamp and back to the power supply.. | [
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] |
In order for
the lamp to glow, electrons must ow; in order for electrons to ow, there must be
a closed circuit from the power supply to the lamp and back to the power supply. | [
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] |
Figure 3.1
A simple electric circuit showing the use of a wall switch.
The lamp can be turned on and o by simply manipulating the wall switch to
make or break the closed circuit.
Instead of the wall switch, we could use an N-type or a P-type MOS transistor
to make or break the closed circuit. Figure 3.2 shows a schematic rendering of
an N-type transistor (a) by itself, and (b) in a circuit. | [
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] |
Figure 3.2 shows a schematic rendering of
an N-type transistor (a) by itself, and (b) in a circuit. Note (Figure 3.2a) that
the transistor has three terminals. They are called the gate, the source, and the
drain. The reasons for the names source and drain are not of interest to us in this
course. What is of interest is the fact that if the gate of the N-type transistor is
supplied with 1. | [
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] |
The reasons for the names source and drain are not of interest to us in this
course. What is of interest is the fact that if the gate of the N-type transistor is
supplied with 1.2 volts, the connection from source to drain acts like a piece of
wire. We say (in the language of electricity) that we have a short circuit between
the source and drain. If the gate of the N-type transistor is supplied with 0 volts,
the connection between the source and drain is broken. | [
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] |
We say (in the language of electricity) that we have a short circuit between
the source and drain. If the gate of the N-type transistor is supplied with 0 volts,
the connection between the source and drain is broken. We say that between the
source and drain we have an open circuit.
Figure 3.2 shows the N-type transistor in a circuit with a battery and a bulb.
When the gate is supplied with 1. | [
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] |
2 shows the N-type transistor in a circuit with a battery and a bulb.
When the gate is supplied with 1.2 volts, the transistor acts like a piece of wire,
completing the circuit and causing the bulb to glow. When the gate is supplied
Figure 3.2
The N-type MOS transistor. | [
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] |
Figure 3.2c is a shorthand notation for describing the circuit of Figure 3.2b.
Rather than always showing the power supply and the complete circuit, electri-
cal engineers usually show only the terminals of the power supply. The fact that
the power supply itself provides the completion of the completed circuit is well
understood, and so is not usually shown.
The P-type transistor works in exactly the opposite fashion from the N-type
transistor. | [
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] |
The fact that
the power supply itself provides the completion of the completed circuit is well
understood, and so is not usually shown.
The P-type transistor works in exactly the opposite fashion from the N-type
transistor. Figure 3.3 shows the schematic representation of a P-type transistor.
When the gate is supplied with 0 volts, the P-type transistor acts (more or less)
like a piece of wire, closing the circuit. | [
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] |
3 shows the schematic representation of a P-type transistor.
When the gate is supplied with 0 volts, the P-type transistor acts (more or less)
like a piece of wire, closing the circuit. When the gate is supplied with 1.2 volts,
the P-type transistor acts like an open circuit. Because the P-type and N-type
t
s
Drain
Figure 3.3
A P-type MOS transistor.
3.2 Logic Gates
One step up from the transistor is the logic gate. | [
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] |
3.2 Logic Gates
One step up from the transistor is the logic gate. That is, we construct basic logic
structures out of individual MOS transistors. In Chapter 2, we studied the behav-
ior of the AND, the OR, and the NOT functions. In this chapter, we construct
transistor circuits that implement each of these functions. The corresponding
circuits are called AND, OR, and NOT gates.
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