1 integer operations. 2 outline arithmetic operations –overflow –unsigned addition,...
TRANSCRIPT
1
Integer Operations
2
Outline
• Arithmetic Operations– overflow– Unsigned addition, multiplication– Signed addition, negation, multiplication– Using Shift to perform power-of-2
multiply/divide
• Suggested reading
– Chap 2.3
Negation ：取反
3
Unsigned Addition
• • •
• • •
u
v+
• • •u + v
• • •
True Sum: w+1 bits
Operands: w bits
Discard Carry: w bits UAddw(u , v)
4
Unsigned Addition
• Standard Addition Function
– Ignores carry output
• Implements Modular Arithmetic
– s = UAddw(u , v) = (u + v) mod 2w
P67 (2.9)
5
Visualizing Unsigned Addition P68 Figure 2.16
• Wraps Around– If true sum ≥ 2w
– At most once
02
46
810
1214
0
2
4
6
8
10
1214
0
2
4
6
8
10
12
14
16
UAdd4(u , v)
u
v
Overflow
0
2w
2w+1
True Sum
Modular Sum
Overflow
Module: 取模
6
Unsigned Addition Forms an Abelian Group P68
• Closed under addition
– 0 UAddw(u , v) 2w –1
• Commutative （交换律）– UAddw(u , v) = UAddw(v , u)
• Associative （结合律）– UAddw (t, UAddw (u,v)) = UAddw (UAddw (t, u ),
v)
7
Unsigned Addition Forms an Abelian Group
• 0 is additive identity
– UAddw (u , 0) = u
• Every element has additive inverse
– Let UCompw (u ) = 2w – u
– UAddw(u , UCompw (u )) = 0
P68 （ 2.10）
8
Signed Addition
• Functionality– True sum requires w+1 bits– Drop off MSB– Treat remaining bits as 2’s comp. integer
)(,2
,
)(,2
),(
NegOverTMinvuvu
TMaxvuTMinvu
PosOvervuTMaxvu
vuTadd
ww
ww
ww
PosOver ： Positive OverflowNegOver ： Negative Overflow
P70 （ 2.12）
9
Signed Addition P70 Figure 2.17
u
v
< 0 > 0
< 0
> 0
NegOver
PosOverTAdd(u , v)
–2w –1
–2w
0
2w –1
2w–1
True Sum
TAdd Result
1 000…0
1 100…0
0 000…0
0 100…0
0 111…1
100…0
000…0
011…1
PosOver
NegOver
10
Visualizing 2’s Comp. Addition
• Values– 4-bit two’s comp.
– Range from -8 to +7
• Wraps Around– If sum 2w-1
• Becomes negative
– If sum < –2w–1
• Becomes positive
11
Visualizing 2’s Comp. Addition P72 Figure 2.19
-8 -6 -4-2 0
24
6
-8
-6
-4
-2
0
2
46
-8
-6
-4
-2
0
2
4
6
8
TAdd4(u , v)
u
v
PosOver
NegOver
12
Detecting Tadd Overflow P71
• Task– Given s = TAddw(u , v)
– Determine if s = Addw(u , v)
• Claim– Overflow iff either:
• u, v < 0, s 0 (NegOver)• u, v 0, s < 0 (PosOver)
– ovf = (u<0 == v<0) && (u<0 != s<0);
0
2w –1
2w–1PosOver
NegOver
13
Mathematical Properties of TAdd
• Two’s Complement Under TAdd Forms a Group– Closed, Commutative, Associative, 0 is
additive identity– Every element has additive inverse
• Let
• TAddw(u , TCompw (u )) = 0
TCompw(u) u u TMinw
TMinw u TMinw
P73 （ 2.13）
14
Mathematical Properties of TAdd
• Isomorphic Algebra to UAdd
– TAddw (u , v) = U2T (UAddw(T2U(u ), T2U(v)))
• Since both have identical bit patterns
– T2U(TAddw (u , v)) = UAddw(T2U(u ), T2U(v))
Isomorphic ：同构
15
Negating with Complement & Increment P73
• In C– ~x + 1 == -x
• Complement– Observation: ~x + x == 1111…111 == -1
1 0 0 1 0 11 1 x
0 1 1 0 1 00 0~x+
1 1 1 1 1 11 1-1
~x ： Complement
16
Signed Addition
• Increment– ~x + 1 = ~x +[x + (-x)] +1– (~x + x) + -x + 1 == -1 + (-x + 1) == -x– So,– ~x + 1 == -x
17
Multiplication P75
• Computing Exact Product of w-bit numbers x, y– Either signed or unsigned
• Ranges– Unsigned: 0 ≤ x * y ≤ (2w – 1) 2 = 22w – 2w+1 + 1
• Up to 2w bits
– Two’s complement min: x *y ≥–2w–1*(2w–1–1) = –22w–2 + 2w–1
• Up to 2w–1 bits
– Two’s complement max: x * y ≤ (–2w–1) 2 = 22w–2
• Up to 2w bits, but only for TMinw2
18
Multiplication
• Maintaining Exact Results– Would need to keep expanding word size with
each product computed
– Done in software by “arbitrary precision” arithmetic packages
19
Power-of-2 Multiply with Shift
• • •
0 0 1 0 0 0•••
u
2k*
u · 2kTrue Product: w+k bits
Operands: w bits
Discard k bits: w bits UMultw(u , 2k)
•••
k
• • • 0 0 0•••
TMultw(u , 2k)
0 0 0••••••
20
Power-of-2 Multiply with Shift
• Operation– u << k gives u * 2k
– Both signed and unsigned
• Examples– u << 3 == u * 8– u << 5 - u << 3 == u * 24– Most machines shift and add much faster
than multiply• Compiler will generate this code automatically
21
Unsigned Power-of-2 Divide with Shift
• Quotient of Unsigned by Power of 2– u >> k gives u / 2k – Uses logical shift
0 0 1 0 0 0•••
u
2k/
u / 2kDivision:
Operands:•••
k••• •••
•••0 0 0••• •••
u / 2k •••0 0•••0Quotient:
.
Binary Point
22
2’s Comp Power-of-2 Divide with Shift P77
• Quotient of Signed by Power of 2– u >> k gives u / 2k – Uses arithmetic shift– Rounds wrong direction when u < 0
0 0 1 0 0 0•••
u
2k/
u / 2kDivision:
Operands:•••
k••• •••
•••0 ••• •••
RoundDown(u / 2k) •••Result:
.
Binary Point
0 •••
23
Correct Power-of-2 Divide
• Quotient of Negative Number by Power of 2– Want u / 2k (Round Toward 0)– Compute as (u+2k-1)/ 2k
• In C: (u + (1<<k)-1) >> k• Biases divided toward 0
Quotient ：商
24
Correct Power-of-2 Divide
Divisor:
Dividend:
Case 1: No rounding
0 0 1 0 0 0•••
u
2k/
u / 2k
•••
k1 ••• 0 0 0•••
1 •••0 1 1••• .
Binary Point
1
0 0 0 1 1 1•••+2k +–1 •••
1 1 1•••
1 ••• 1 1 1•••
Biasing has no effect
25
Correct Power-of-2 Divide
Divisor:
Dividend:
Case 2: Rounding
0 0 1 0 0 0•••
u
2k/
u / 2k
•••
k1 ••• •••
1 •••0 1 1••• .
Binary Point
1
0 0 0 1 1 1•••+2k +–1 •••
1 ••• •••
Biasing adds 1 to final result
•••
Incremented by 1
Incremented by 1
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Floating Point
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Topics
• Fractional Binary Numbers• IEEE 754 Standard• Rounding Mode• FP Operations• Floating Point in C• Suggested Reading: Chap 2.4
28
Encoding Rational Numbers P80
• Form V =• Very useful when >> 0 or <<1• An Approximation to real arithmetic• From programmer’s perspective
– Uninteresting– Arcane and incomprehensive
* Arcane ：神秘的* Incomprehensive: 不可理解的
yx 2V V
29
Encoding Rational Numbers
• Until 1980s– Many idiosyncratic formats, fast speed, easy
implementation, less accuracy
• IEEE 754– Designed by W. Kahan for Intel processors
– Based on a small and consistent set of principles, elegant, understandable, hard to make go fast
Idiosyncratic: 特殊的Elegant ：雅致的
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Fractional Binary Numbers
bm bm–1 b2 b1 b0 b–1 b–2 b–3 b–n• • •• • • .
124
2m–1
2m
• • •
• • •
1/21/4
1/8
2–n
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Fractional Binary Numbers
• Bits to right of “binary point” represent fractional powers of 2
• Represents rational number: 2i
m
niib P81 （ 2.17）
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Fractional Numbers to Binary Bits
unsigned result_bits=0, current_bit=0x80000000
for (i=0;i<32;i++) {
x *= 2
if ( x>= 1 ) {
result_bits |= current_bit ;
if ( x == 1)
break ;
x -= 1 ;
}
current_bit >> 1 ;
}
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Fraction Binary Number Examples
Value Binary Fraction0.2 0.00110011[0011]• Observations:
– The form 0.11111…11 represent numbers just below 1.0 which is noted as 1.0-
– Binary Fractions can only exactly represent x/2k
– Others have repeated bit patterns
34
IEEE Floating-Point Representation P83
• Numeric form– V=(-1)sM 2E
• Sign bit s determines whether number is negative or positive
• Significand M normally a fractional value in range [1.0,2.0).
• Exponent E weights value by power of two
35
IEEE Floating-Point Representation
• Encoding– – s is sign bit– exp field encodes E– frac field encodes M
• Sizes– Single precision (32 bits): 8 exp bits, 23 frac
bits– Double precision (64 bits): 11 exp bits, 52 frac
bits
s exp frac
36
Normalize Values P84
• Condition– exp 000…0 and exp 111…1
• Exponent coded as biased value– E = Exp – Bias
• Exp : unsigned value denoted by exp • Bias : Bias value
– Single precision: 127 (Exp: 1…254, E : -126…127)– Double precision: 1023 (Exp: 1…2046,
E : -1022 …1023)
– In general: Bias = 2m-1 - 1, where m is the number of exponent bits
37
Normalize Values
• Significand coded with implied leading 1
– m = 1.xxx…x2
•xxx…x: bits of frac
• Minimum when 000…0 (M = 1.0)
• Maximum when 111…1 (M = 2.0 – )
• Get extra leading bit for “free”
38
Normalized Encoding Examples
• Value: 12345 (Hex: 0x3039)• Binary bits: 11000000111001• Fraction representation:
1.1000000111001*213
• M: 10000001110010000000000 • E: 10001100 (140)• Binary Encoding
– 0100 0110 0100 0000 1110 0100 0000 0000– 4640E400
39
Denormalized Values P84
• Condition– exp = 000…0
• Values– Exponent Value: E = 1 – Bias
– Significant Value m = 0.xxx…x2
•xxx…x: bits of frac
40
Denormalized Values
• Cases– exp = 000…0, frac = 000…0
• Represents value 0• Note that have distinct values +0 and –0
– exp = 000…0, frac 000…0• Numbers very close to 0.0• Lose precision as get smaller• “Gradual underflow”
41
Special Values P85
• Condition– exp = 111…1
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Special Values
• exp = 111…1, frac = 000…0
– Represents value(infinity)
– Operation that overflows
– Both positive and negative
– E.g., 1.0/0.0 = 1.0/0.0 = +, 1.0/0.0 =
43
Special Values
• exp = 111…1, frac 000…0
– Not-a-Number (NaN)
– Represents case when no numeric value can be determined
– E.g., sqrt(–1),
44
Summary of Real Number Encodings P85 Figure 2.22
NaNNaN
+
0
+Denorm +Normalized-Denorm-Normalized
+0
45
8-bit Floating-Point Representations
s exp frac
02367
46
8-bit Floating-Point Representations
• Exp exp E 2E
• 0 0000 -6 1/64 (denorms)• 1 0001 -6 1/64• 2 0010 -5 1/32• 3 0011 -4 1/16• 4 0100 -3 1/8• 5 0101 -2 1/4• 6 0110 -1 1/2• 7 0111 0 1• 8 1000 +1 2• 9 1001 +2 4• 10 1010 +3 8• 11 1011 +4 16• 12 1100 +5 32• 13 1101 +6 64• 14 1110 +7 128• 15 1111 n/a (inf, NaN)
47
Dynamic Range (Denormalized numbers) P86Figure 2.23
• s exp frac E Value
• 0 0000 000 -6 0• 0 0000 001 -6 1/8*1/64 = 1/512• 0 0000 010 -6 2/8*1/64 = 2/512• …• 0 0000 110 -6 6/8*1/64 = 6/512• 0 0000 111 -6 7/8*1/64 = 7/512
48
Dynamic Range
• s exp frac E Value
• 0 0001 000 -6 8/8*1/64 = 8/512• 0 0001 001 -6 9/8*1/64 = 9/512• …• 0 0110 110 -1 14/8*1/2 = 14/16• 0 0110 111 -1 15/8*1/2 = 15/16• 0 0111 000 0 8/8*1 = 1• 0 0111 001 0 9/8*1 = 9/8
49
Dynamic Range (Denormalized numbers)
• s exp frac E Value
• 0 0111 010 0 10/8*1 = 10/8• …• 0 1110 110 7 14/8*128 = 224• 0 1110 111 7 15/8*128 = 240• 0 1111 000 n/a inf
50
Distribution of Representable Values
• 6-bit IEEE-like format– K = 3 exponent bits
– n = 2 significand bits
– Bias is 3
• Notice how the distribution gets denser toward zero.
51
Distribution of Representable Values
-15 -10 -5 0 5 10 15
Denormalized Normalized Infinity
-1 -0.5 0 0.5 1
Denormalized Normalized Infinity
52
Interesting Numbers P88 Figure 2.24
53
Special Properties of Encoding
• FP Zero Same as Integer Zero– All bits = 0
• Can (Almost) Use Unsigned Integer Comparison– Must first compare sign bits– Must consider -0 = 0– NaNs problematic
• Will be greater than any other values
– Otherwise OK• Denorm vs. normalized• Normalized vs. infinity
54
Round Mode P89
• Round down: – rounded result is close to but no greater than
true result.
• Round up: – rounded result is close to but no less than
true result.
55
Round Mode P90 Figure 2.25
Mode 1.40
1.60
1.50
2.50 -1.50
Round-to-Even 1 2 2 2 -2
Round-toward-zero
1 1 1 2 -1
Round-down 1 1 1 2 -2
Round-up 2 2 2 3 -1
56
Round-to-Even
• Default Rounding Mode– Hard to get any other kind without dropping
into assembly
– All others are statistically biased• Sum of set of positive numbers will consistently be
over- or under- estimated
57
Round-to-Even P89
• Applying to Other Decimal Places
– When exactly halfway between two possible
values
• Round so that least significant digit is even
– E.g., round to nearest hundredth
1.2349999 1.23 (Less than half way)
1.2350001 1.24 (Greater than half way)
1.2350000 1.24 (Half way—round up)
1.2450000 1.24 (Half way—round down)
58
Rounding Binary Number P89
• “ Even” when least significant bit is 0• Half way when bits to right of rounding
position = 100…2
Value Binary Rounded Action Round Decimal
2 3/32 10.00011 10.00 Down 2
2 3/16 10.0011 10.01 Up 2 1/4
2 7/8 10.111 11.00 Up 3
2 5/8 10.101 10.10 Down 2 1/2
59
Floating-Point Operations
• Conceptual View
– First compute exact result
– Make it fit into desired precision
• Possibly overflow if exponent too large
• Possibly round to fit into frac
60
Mathematical Properties of FP Add
• Compare to those of Abelian Group– Closed under addition? YES
• But may generate infinity or NaN
– Commutative? YES– Associative? NO
• Overflow and inexactness of rounding
– 0 is additive identity? YES– Every element has additive inverse ALMOST
• Except for infinities & NaNs
61
Mathematical Properties of FP Add
• Monotonicity– a ≥ b a+c ≥ b+c?
ALMOST• Except for infinities & NaNs
62
Algebraic Properties of FP Mult
• Compare to Commutative Ring– Closed under multiplication? YES
• But may generate infinity or NaN
– Multiplication Commutative? YES– Multiplication is Associative? P92 NO
• Possibility of overflow, inexactness of rounding
– 1 is multiplicative identity?YES
– Multiplication distributes over addition? NO• Possibility of overflow, inexactness of rounding
63
Algebraic Properties of FP Mult P90
• Monotonicity– a ≥ b & c ≥ 0 a *c ≥ b *c? ALMOST
• Except for infinities & NaNs
64
FP Multiplication
• Operands(–1)s1 M1 2E1
(–1)s2 M2 2E2
• Exact Result(–1)s M 2E
– Sign s : s1 ^ s2– Significand M : M1 * M2– Exponent E : E1 + E2
65
FP Multiplication
• Fixing– If M ≥ 2, shift M right, increment E – If E out of range, overflow – Round M to fit frac precision
66
FP Addition
• Operands(–1)s1 M1 2E1
(–1)s2 M2 2E2
– Assume E1 > E2
• Exact Result(–1)s M 2E
– Sign s, significand M: • Result of signed align & add
– Exponent E : E1
67
FP Addition
• Fixing– If M ≥ 2, shift M right, increment E
– if M < 1, shift M left k positions, decrement E by k
– Overflow if E out of range
– Round M to fit frac precision
68
FP Addition
(–1)s1 m1
(–1)s2 m2
E1–E2
+
(–1)s m
69
Answers to Floating Point Puzzles
• int x = …;
• float f = …;
• double d = …;• Assume neither d nor f is NAN or infinity
70
Floating Point in C
• x == (int)(float) xNo: 24 bit significand• x == (int)(double) x Yes: 53 bit significand• f == (float)(double) f Yes: increases precision• d == (float) d No: loses precision• f == -(-f); Yes: Just change sign bit• 2/3 == 2/3.0 No: 2/3 == 0• d < 0.0((d*2) < 0.0) Yes!• d > f -f < -d No• d *d >= 0.0 Yes!• (d+f)-d == f No: Not associative
71
Answers to Floating Point Puzzles
• C Guarantees Two Levels– float single precision– double double precision
72
Answers to Floating Point Puzzles
• Conversions– Casting between int, float, and double changes numeric
values– Double or float to int
• Truncates fractional part• Like rounding toward zero• Not defined when out of range
– Generally saturates to TMin or TMax– int to double
• Exact conversion, as long as int has ≤ 53 bit word size– int to float
• Will round according to rounding mode