EE 319K Introduction to Embedded Systems

1
EE 319KIntroduction to Embedded Systems

• Lecture 14 Gaming Engines, Coding Style,
  Floating Point

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Agenda

• Recap
• Software design
• 2-D arrays, structs
• Bitmaps, sprites
• Lab 10
• Agenda
• Gaming engine design
• Coding style
• Floating point

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Numbers

• Integers (Z) universe is infinite but discrete
• No fractions
• No numbers between 5 and 6
• A countable (finite) number of items in a finite
  range
• Real numbers (R) universe is infinite
  continuous
• Fractions represented by decimal notation
• Rational numbers, e.g., 5/2 2.5
• Irrational numbers, e.g., p 22/7 3.14159265 .
  . .
• Infinity of numbers exist even in the smallest
  range

(Adapted from V. Aagrawal)
4
Number Representation

• Integers
• Fixed-width integer number
• Reals
• Fixed-point number ? I ?
• Store I, but ? is fixed
• Decimal fixed-point (?10m) I 10m
• Binary fixed-point (?2m) I 2m
• Floating-point number I BE
• Store both I and E (only B is fixed)

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Wide Range of Real Numbers

• A large number
• 976,000,000,000,000 9.76 1014
• A small number
• 0.0000000000000976 9.76 10-14
• No fixed ? that can represent both
• Not representable in single fixed-point format

(Adapted from V. Aagrawal)
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Floating Point Numbers

• Decimal scientific notation
• 0.513105, 5.13104 and 51.3103
• 5.13104 is in normalized scientific notation
• Binary floating point numbers
• Base B 2
• Binary point
• Multiplication by 2 moves the point to the left
• Normalized scientific notation, e.g., 1.02-1
• Known as floating point numbers

(Adapted from V. Agrawal)
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Normalizing Numbers

• In scientific notation, we generally choose one
  digit to the left of the decimal point
• 13.25 1010 becomes 1.325 1011
• Normalizing means
• Shifting the decimal point until we have the
  right number of digits to its left
• Normally one
• Adding or subtracting from the exponent to
  reflect the shift

(Adapted from V. Agrawal)
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Floating Point Numbers

• General format
• 1.bbbbb two2eeee
• or (-1)S (1F) 2E
• Where
• S sign, 0 for positive, 1 for negative
• F fraction (or mantissa) as a binary
  integer, 1F is called significand
• E exponent as a binary integer, positive or
  negative (twos complement)

(Adapted from V. Agrawal)
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ANSI/IEEE Std 754-1985

• Single-precision float format

Bit 31 Mantissa sign, s0 for positive, s1 for
negative Bits 3023 8-bit biased binary exponent
0 e 255 Bits 220 24-bit mantissa, m,
expressed as a binary fraction, A binary 1 as
the most significant bit is implied. m
1.m1m2m3...m23
(Adapted from V. Agrawal)
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IEEE 754 Floating Point Standard

• Biased exponent exponent range -127,127
  changed to 0, 255
• Biased exponent is an 8-bit positive binary
  integer
• True exponent obtained by subtracting 12710 or
  011111112
• 255 special case
• First bit of significand is always 1
• 1.bbbb . . . b 2E
• 1 before the binary point is implicitly assumed
• So we dont need to include it just assume its
  there!
• Significand field is 23 bit fraction after the
  binary point
• Significand range is 1, 2)
• Standard formats
• Single precision 8 (E) 23 (F) 1 (S) 32
  bits (float)
• Double precision 11 (E) 52 (F) 1 (S) 64
  bits (double)

(Adapted from V. Agrawal)
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Numbers in 32-bit Formats

• Twos complement integers
• Floating point numbers
• The range is larger, but the number of numbers
  per unit interval is less than that for a
  comparable fixed point range

Expressible numbers
-231
231-1
0
Positive underflow
Negative underflow
Negative Overflow
Positive Overflow
Expressible negative numbers
Expressible positive numbers
0
-2-127
2-127
(2 2-23)2127
- (2 2-23)2127
(Adapted from V. Agrawal)
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Binary to Decimal Conversion
Binary (-1)S (1.b1b2b3b4) 2E
Represents (-1)S (1 b12-1 b22-2 b32-3
b42-4) 2E
Example -1.1100 2-2 (binary) - (1 2-1
2-2) 2-2 - (1 0.5 0.25)/4 -
1.75/4 - 0.4375 (decimal)
(Adapted from V. Agrawal)
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Decimal to Binary Conversion

• Converting from base 10 to the representation
• Single precision example
• Covert 10010
• Step 1 convert to binary - 0110 0100
• In a binary representation form of 1.xxx have
• 0110 0100 1.100100 x 26

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Decimal to Binary Conversion (contd)

• 1.1001 x 26 is binary for 100
• Thus the exponent is a 6
• Biased exponent will be 6127133 1000 0101
• Sign will be a 0 for positive
• Stored fractional part f will be 1001
• Thus we have
• S E F
• 0 100 0 010 1 1 00 1000.
• 4 2 C 8 0 0 0 0 in
  hexadecimal
• 42C8 0000 is representation for 100

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Positive Zero in IEEE 754
0 00000000 00000000000000000000000
Biased exponent
Fraction

• 1.0 2-127
• Smallest positive number in single-precision IEEE
  754 standard.
• Interpreted as positive zero.
• Exponent less than -127 is positive underflow
  can be regarded as zero.

(Adapted from V. Agrawal)
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Negative Zero in IEEE 754
1 00000000 00000000000000000000000
Biased exponent
Fraction

• - 1.0 2-127
• Smallest negative number in single-precision IEEE
  754 standard.
• Interpreted as negative zero.
• True exponent less than -127 is negative
  underflow may be regarded as 0.

(Adapted from V. Agrawal)
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Positive Infinity in IEEE 754
0 11111111 00000000000000000000000
Biased exponent
Fraction

• 1.0 2128
• Largest positive number in single-precision IEEE
  754 standard.
• Interpreted as 8
• If true exponent 128 and fraction ? 0, then the
  number is greater than 8.
• It is called not a number or NaN and may be
  interpreted as 8.

(Adapted from V. Agrawal)
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Negative Infinity in IEEE 754
1 11111111 00000000000000000000000
Biased exponent
Fraction

• -1.0 2128
• Smallest negative number in single-precision IEEE
  754 standard.
• Interpreted as - 8
• If true exponent 128 and fraction ? 0, then the
  number is less than - 8
• It is called not a number or NaN and may be
  interpreted as - 8.

(Adapted from V. Agrawal)
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IEEE Representation Values

• If E255 and F is nonzero, then VNaN ("Not a
  number")
• If E255 and F is zero and S is 1, then
  V-Infinity
• If E255 and F is zero and S is 0, then
  VInfinity
• If 0ltElt255 then V(-1)S 2 (E-127) (1.F)
  where "1.F" is intended to represent the binary
  number created by prefixing F with an implicit
  leading 1 and a binary point.
• If E0 and F is nonzero, then V(-1)S 2
  (-126) (0.F)
• These are "unnormalized" values.
• If E0 and F is zero and S is 1, then V-0
• If E0 and F is zero and S is 0, then V0

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Addition and Subtraction

• 0. Zero check
• - Change the sign of subtrahend
• - If either operand is 0, the other is the result
• 1. Significand alignment right shift smaller
  significand until two exponents are identical.
• 2. Addition add significands and report
  exception if overflow occurs.
• 3. Normalization
• - Shift significand bits to normalize.
• - report overflow or underflow if exponent goes
  out of range.
• 4. Rounding

(Adapted from V. Agrawal)
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Rounding

• Adjusting significands before addition will
  produce results that exceed 24 bit
• Round toward infinity
• select next largest normalized result
• Round toward minus infinity
• select next smallest normalized result
• Round toward zero
• truncate result
• Round to nearest
• select closest normalized result
• used by IEEE 754

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Example

• Subtraction 0.510- 0.437510
• Step 0 Floating point numbers to be added
• 1.00022-1 and -1.11022-2
• Step 1 Significand of lesser exponent is
  shifted right until exponents match
• -1.11022-2 ? - 0.11122-1
• Step 2 Add significands, 1.0002 (- 0.1112)
• Result is 0.0012 2-1
• Step 3 Normalize, 1.0002 2-4
• No overflow/underflow since
• 127 exponent -126
• Step 4 Rounding, no change since the sum fits
  in 4 bits.
• 1.0002 2-4 (10)/16 0.062510

(Adapted from V. Agrawal)
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FP Multiplication Basic Idea

1. Separate sign
2. Add exponents
3. Multiply significands
4. Normalize, round, check overflow
5. Replace sign

(Adapted from V. Agrawal)
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FP Division Basic Idea

1. Separate sign.
2. Check for zeros and infinity.
3. Subtract exponents.
4. Divide significands.
5. Normalize/overflow/underflow.
6. Rounding.
7. Replace sign.

(Adapted from V. Agrawal)