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Floating Point
15-213: Introduction to Computer Systems
4th Lecture, Sep. 10, 2015
Instructors:
Randal E. Bryant and David R. O’Hallaron
Bryant and O’Hallaron, Computer Systems: A Programmer’s Perspective, Third Edition
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Today: Floating Point
Background: Fractional binary numbers
IEEE floating point standard: Definition
Example and properties
Rounding, addition, multiplication
Floating point in C
Summary
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Fractional binary numbers
What is 1011.1012?
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Fractional Binary Numbers
2i
2i-1
4
••• 2
1
bi bi-1 ••• b2 b1 b0 b-1 b-2 b-3 ••• b-j
1/2
1/4 •••
1/8
Representation 2-j
▪ Bits to right of “binary point” represent fractional powers of 2
▪ Represents rational number:
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Fractional Binary Numbers: Examples
Value Representation
5 3/4 101.112
2 7/8 010.1112
1 7/16 001.01112
Observations
▪ Divide by 2 by shifting right (unsigned)
▪ Multiply by 2 by shifting left
▪ Numbers of form 0.111111…2 are just below 1.0
▪ 1/2 + 1/4 + 1/8 + … + 1/2i + … ➙ 1.0
▪ Use notation 1.0 – ε
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Representable Numbers
Limitation #1
▪ Can only exactly represent numbers of the form x/2k
▪ Other rational numbers have repeating bit representations
▪ Value Representation
▪ 1/3 0.0101010101[01]…2
▪ 1/5 0.001100110011[0011]…2
▪ 1/10 0.0001100110011[0011]…2
Limitation #2
▪ Just one setting of binary point within the w bits
▪ Limited range of numbers (very small values? very large?)
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Today: Floating Point
Background: Fractional binary numbers
IEEE floating point standard: Definition
Example and properties
Rounding, addition, multiplication
Floating point in C
Summary
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IEEE Floating Point
IEEE Standard 754
▪ Established in 1985 as uniform standard for floating point arithmetic
Before that, many idiosyncratic formats
▪
▪ Supported by all major CPUs
Driven by numerical concerns
▪ Nice standards for rounding, overflow, underflow
▪ Hard to make fast in hardware
▪ Numerical analysts predominated over hardware designers in defining
standard
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Floating Point Representation
Numerical Form:
(–1)s M 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
Encoding
▪ MSB s is sign bit s
▪ exp field encodes E (but is not equal to E)
▪ frac field encodes M (but is not equal to M)
s exp frac
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Precision options
Single precision: 32 bits
s exp frac
1 8-bits 23-bits
Double precision: 64 bits
s exp frac
1 11-bits 52-bits
Extended precision: 80 bits (Intel only)
s exp frac
1 15-bits 63 or 64-bits
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“Normalized” Values v = (–1)s M 2E
When: exp ≠ 000…0 and exp ≠ 111…1
Exponent coded as a biased value: E = Exp – Bias
▪ Exp: unsigned value of exp field
▪ Bias = 2k-1 - 1, where k is number of exponent bits
▪ Single precision: 127 (Exp: 1…254, E: -126…127)
▪ Double precision: 1023 (Exp: 1…2046, E: -1022…1023)
Significand coded with implied leading 1: M = 1.xxx…x2
▪ xxx…x: bits of frac field
▪ Minimum when frac=000…0 (M = 1.0)
▪ Maximum when frac=111…1 (M = 2.0 – ε)
▪ Get extra leading bit for “free”
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Normalized Encoding Example v = (–1)s M 2E
E = Exp – Bias
Value: float F = 15213.0;
▪ 1521310 = 111011011011012
= 1.11011011011012 x 213
Significand
M = 1.11011011011012
frac = 110110110110100000000002
Exponent
E = 13
Bias = 127
Exp = 140 = 100011002
Result:
0 10001100 11011011011010000000000
s exp frac
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Denormalized Values v = (–1)s M 2E
E = 1 – Bias
Condition: exp = 000…0
Exponent value: E = 1 – Bias (instead of E = 0 – Bias)
Significand coded with implied leading 0: M = 0.xxx…x2
▪ xxx…x: bits of frac
Cases
▪ exp = 000…0, frac = 000…0
Represents zero value
▪
▪ Note distinct values: +0 and –0 (why?)
▪ exp = 000…0, frac ≠ 000…0
▪ Numbers closest to 0.0
▪ Equispaced
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Special Values
Condition: exp = 111…1
Case: 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 = −
Case: exp = 111…1, frac ≠ 000…0
▪ Not-a-Number (NaN)
▪ Represents case when no numeric value can be determined
▪ E.g., sqrt(–1), − , 0
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Visualization: Floating Point Encodings
− +
−Normalized −Denorm +Denorm +Normalized
NaN NaN
−0 +0
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Today: Floating Point
Background: Fractional binary numbers
IEEE floating point standard: Definition
Example and properties
Rounding, addition, multiplication
Floating point in C
Summary
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Tiny Floating Point Example
s exp frac
1 4-bits 3-bits
8-bit Floating Point Representation
▪ the sign bit is in the most significant bit
▪ the next four bits are the exponent, with a bias of 7
▪ the last three bits are the frac
Same general form as IEEE Format
▪ normalized, denormalized
▪ representation of 0, NaN, infinity
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Dynamic Range (Positive Only) v = (–1)s M 2E
s exp frac E Value n: E = Exp – Bias
0 0000 000 -6 0 d: E = 1 – Bias
0 0000 001 -6 1/8*1/64 = 1/512 closest to zero
Denormalized 0 0000 010 -6 2/8*1/64 = 2/512
numbers …
0 0000 110 -6 6/8*1/64 = 6/512
0 0000 111 -6 7/8*1/64 = 7/512 largest denorm
0 0001 000 -6 8/8*1/64 = 8/512
smallest norm
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 closest to 1 below
Normalized 0 0111 000 0 8/8*1 = 1
numbers 0 0111 001 0 9/8*1 = 9/8
closest to 1 above
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 largest norm
0 1111 000 n/a inf
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Distribution of Values
6-bit IEEE-like format
▪ e = 3 exponent bits
▪ f = 2 fraction bits s exp frac
▪ Bias is 23-1-1 = 3 1 3-bits 2-bits
Notice how the distribution gets denser toward zero.
8 values
-15 -10 -5 0 5 10 15
Denormalized Normalized Infinity
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Distribution of Values (close-up view)
6-bit IEEE-like format
▪ e = 3 exponent bits
▪ f = 2 fraction bits s exp frac
▪ Bias is 3 1 3-bits 2-bits
-1 -0.5 0 0.5 1
Denormalized Normalized Infinity
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Special Properties of the IEEE 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
▪
▪ What should comparison yield?
▪ Otherwise OK
▪ Denorm vs. normalized
▪ Normalized vs. infinity
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Today: Floating Point
Background: Fractional binary numbers
IEEE floating point standard: Definition
Example and properties
Rounding, addition, multiplication
Floating point in C
Summary
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Floating Point Operations: Basic Idea
x +f y = Round(x + y)
x f y = Round(x y)
Basic idea
▪ First compute exact result
▪ Make it fit into desired precision
▪ Possibly overflow if exponent too large
▪ Possibly round to fit into frac
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Rounding
Rounding Modes (illustrate with $ rounding)
$1.40 $1.60 $1.50 $2.50 –$1.50
▪ Towards zero $1 $1 $1 $2 –$1
▪ Round down (−) $1 $1 $1 $2 –$2
▪ Round up (+) $2 $2 $2 $3 –$1
▪ Nearest Even (default) $1 $2 $2 $2 –$2
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Closer Look at 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
Applying to Other Decimal Places / Bit Positions
▪ When exactly halfway between two possible values
Round so that least significant digit is even
▪
▪ E.g., round to nearest hundredth
7.8949999 7.89 (Less than half way)
7.8950001 7.90 (Greater than half way)
7.8950000 7.90 (Half way—round up)
7.8850000 7.88 (Half way—round down)
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Rounding Binary Numbers
Binary Fractional Numbers
▪ “Even” when least significant bit is 0
▪ “Half way” when bits to right of rounding position = 100…2
Examples
▪ Round to nearest 1/4 (2 bits right of binary point)
Value Binary Rounded Action Rounded Value
2 3/32 10.000112 10.002 (<1/2—down) 2
2 3/16 10.001102 10.012 (>1/2—up) 2 1/4
2 7/8 10.111002 11.002 ( 1/2—up) 3
2 5/8 10.101002 10.102 ( 1/2—down) 2 1/2
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FP Multiplication
(–1)s1 M1 2E1 x (–1)s2 M2 2E2
s
Exact Result: (–1) M 2
E
▪ Sign s: s1 ^ s2
▪ Significand M: M1 x M2
▪ Exponent E: E1 + E2
Fixing
▪ If M ≥ 2, shift M right, increment E
▪ If E out of range, overflow
▪ Round M to fit frac precision
Implementation
▪ Biggest chore is multiplying significands
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Floating Point Addition
(–1)s1 M1 2E1 + (-1)s2 M2 2E2 Get binary points lined up
▪Assume E1 > E2
E1–E2
(–1)s1 M1
Exact Result: (–1)s M 2E
▪Sign s, significand M: + (–1)s2 M2
Result of signed align & add
▪
▪Exponent E: E1 (–1)s M
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
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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
▪ (3.14+1e10)-1e10 = 0, 3.14+(1e10-1e10) = 3.14
▪ 0 is additive identity?
▪ Every element has additive inverse? Yes
▪ Yes, except for infinities & NaNs Almost
Monotonicity
▪ a ≥ b ⇒ a+c ≥ b+c? Almost
▪ Except for infinities & NaNs
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Mathematical Properties of FP Mult
Compare to Commutative Ring
▪ Closed under multiplication? Yes
▪ But may generate infinity or NaN
▪ Multiplication Commutative? Yes
▪ Multiplication is Associative? No
▪ Possibility of overflow, inexactness of rounding
▪ Ex: (1e20*1e20)*1e-20= inf, 1e20*(1e20*1e-20)= 1e20
▪ 1 is multiplicative identity? Yes
▪ Multiplication distributes over addition? No
▪ Possibility of overflow, inexactness of rounding
▪ 1e20*(1e20-1e20)= 0.0, 1e20*1e20 – 1e20*1e20 = NaN
Monotonicity
▪ a ≥ b & c ≥ 0 ⇒ a * c ≥ b *c? Almost
▪ Except for infinities & NaNs
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Today: Floating Point
Background: Fractional binary numbers
IEEE floating point standard: Definition
Example and properties
Rounding, addition, multiplication
Floating point in C
Summary
Bryant and O’Hallaron, Computer Systems: A Programmer’s Perspective, Third Edition 31
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Floating Point in C
C Guarantees Two Levels
▪float single precision
▪double double precision
Conversions/Casting
▪ Casting between int, float, and double changes bit representation
▪ double/float → int
Truncates fractional part
▪
▪ Like rounding toward zero
▪ Not defined when out of range or NaN: Generally sets to TMin
▪ int → double
▪ Exact conversion, as long as int has ≤ 53 bit word size
▪ int → float
▪ Will round according to rounding mode
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Floating Point Puzzles
For each of the following C expressions, either:
▪ Argue that it is true for all argument values
▪ Explain why not true
• x == (int)(float) x
• x == (int)(double) x
• f == (float)(double) f
int x = …;
float f = …; • d == (double)(float) d
double d = …; • f == -(-f);
• 2/3 == 2/3.0
Assume neither • d < 0.0 ⇒ ((d*2) < 0.0)
d nor f is NaN • d > f ⇒ -f > -d
• d * d >= 0.0
• (d+f)-d == f
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Summary
IEEE Floating Point has clear mathematical properties
Represents numbers of form M x 2
E
One can reason about operations independent of
implementation
▪ As if computed with perfect precision and then rounded
Not the same as real arithmetic
▪ Violates associativity/distributivity
▪ Makes life difficult for compilers & serious numerical applications
programmers
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Additional Slides
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Creating Floating Point Number
Steps s exp frac
▪ Normalize to have leading 1
1 4-bits 3-bits
▪ Round to fit within fraction
▪ Postnormalize to deal with effects of rounding
Case Study
▪ Convert 8-bit unsigned numbers to tiny floating point format
Example Numbers
128 10000000
15 00001101
33 00010001
35 00010011
138 10001010
63 00111111
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Normalize s exp frac
1 4-bits 3-bits
Requirement
▪ Set binary point so that numbers of form 1.xxxxx
▪ Adjust all to have leading one
Decrement exponent as shift left
▪
Value Binary Fraction Exponent
128 10000000 1.0000000 7
15 00001101 1.1010000 3
17 00010001 1.0001000 4
19 00010011 1.0011000 4
138 10001010 1.0001010 7
63 00111111 1.1111100 5
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Rounding 1.BBGRXXX
Guard bit: LSB of result
Sticky bit: OR of remaining bits
Round bit: 1st bit removed
Round up conditions
▪ Round = 1, Sticky = 1 ➙ > 0.5
▪ Guard = 1, Round = 1, Sticky = 0 ➙ Round to even
Value Fraction GRS Incr? Rounded
128 1.0000000 000 N 1.000
15 1.1010000 100 N 1.101
17 1.0001000 010 N 1.000
19 1.0011000 110 Y 1.010
138 1.0001010 011 Y 1.001
63 1.1111100 111 Y 10.000
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Postnormalize
Issue
▪ Rounding may have caused overflow
▪ Handle by shifting right once & incrementing exponent
Value Rounded Exp Adjusted Result
128 1.000 7 128
15 1.101 3 15
17 1.000 4 16
19 1.010 4 20
138 1.001 7 134
63 10.000 5 1.000/6 64
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Interesting Numbers {single,double}
Description exp frac Numeric Value
Zero 00…00 00…00 0.0
Smallest Pos. Denorm. 00…00 00…01 2– {23,52} x 2– {126,1022}
▪ Single ≈ 1.4 x 10–45
▪ Double ≈ 4.9 x 10–324
Largest Denormalized 00…00 11…11 (1.0 – ε) x 2– {126,1022}
▪ Single ≈ 1.18 x 10–38
▪ Double ≈ 2.2 x 10–308
Smallest Pos. Normalized 00…01 00…00 1.0 x 2– {126,1022}
▪ Just larger than largest denormalized
One 01…11 00…00 1.0
Largest Normalized 11…10 11…11 (2.0 – ε) x 2{127,1023}
▪ Single ≈ 3.4 x 1038
▪ Double ≈ 1.8 x 10308
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