1. 14-1 Bard, Gerstlauer, Valvano, Yerraballi EE 319K Introduction to Embedded Systems Lecture 14: Gaming Engines, Coding Style, Floating Point

2. 14-2 Bard, Gerstlauer, Valvano, Yerraballi Agenda  Recap  Software design  2-D arrays, structs  Bitmaps, sprites  Lab 10  Agenda  Gaming engine design  Coding style  Floating point

3. 14-3 Bard, Gerstlauer, Valvano, Yerraballi 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 oRational numbers, e.g., 5/2 = 2.5 oIrrational numbers, e.g.,  ~ 22/7 = 3.14159265...  Infinity of numbers exist even in the smallest range (Adapted from V. Aagrawal)

4. 14-4 Bard, Gerstlauer, Valvano, Yerraballi Number Representation  Integers  Fixed-width integer number  Reals  Fixed-point number  I  oStore I, but  is fixed oDecimal fixed-point (=10 m ) = I 10 m oBinary fixed-point (=2 m ) = I 2 m  Floating-point number = I B E oStore both I and E (only B is fixed)

5. 14-5 Bard, Gerstlauer, Valvano, Yerraballi Wide Range of Real Numbers  A large number: 976,000,000,000,000 = 9.76 10 14  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)

6. 14-6 Bard, Gerstlauer, Valvano, Yerraballi Floating Point Numbers  Decimal scientific notation  0.513×10 5, 5.13×10 4 and 51.3×10 3  5.13×10 4 is in normalized scientific notation  Binary floating point numbers  Base B = 2  Binary point oMultiplication by 2 moves the point to the left  Normalized scientific notation, e.g., 1.0×2 -1 oKnown as floating point numbers (Adapted from V. Agrawal)

7. 14-7 Bard, Gerstlauer, Valvano, Yerraballi Normalizing Numbers  In scientific notation, we generally choose one digit to the left of the decimal point  13.25 × 10 10 becomes 1.325 × 10 11  Normalizing means  Shifting the decimal point until we have the right number of digits to its left oNormally one  Adding or subtracting from the exponent to reflect the shift (Adapted from V. Agrawal)

8. 14-8 Bard, Gerstlauer, Valvano, Yerraballi Floating Point Numbers  General format ±1.bbbbb two ×2 eeee or(-1) S × (1+F) × 2 E  Where  S = sign, 0 for positive, 1 for negative  F = fraction (or mantissa) as a binary integer, 1+F is called significand  E = exponent as a binary integer, positive or negative (two’s complement) (Adapted from V. Agrawal)

9. 14-9 Bard, Gerstlauer, Valvano, Yerraballi ANSI/IEEE Std 754-1985  Single-precision float format (Adapted from V. Agrawal) Bit 31Mantissa sign, s=0 for positive, s=1 for negative Bits 30:238-bit biased binary exponent 0 ≤ e ≤ 255 Bits 22:024-bit mantissa, m, expressed as a binary fraction, A binary 1 as the most significant bit is implied. m = 1.m 1 m 2 m 3...m 23

10. 14-10 Bard, Gerstlauer, Valvano, Yerraballi 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 127 10 or 01111111 2  255 = special case  First bit of significand is always 1: ± 1.bbbb... b × 2 E  1 before the binary point is implicitly assumed oSo we don’t need to include it – just assume it’s 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)

11. 14-11 Bard, Gerstlauer, Valvano, Yerraballi Negative Overflow Positive Overflow Expressible numbers Numbers in 32-bit Formats  Two’s 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 -2 31 2 31 -10 Expressible negative numbers Expressible positive numbers 0-2 -127 2 -127 Positive underflow Negative underflow (2 – 2 -23 )×2 127 - (2 – 2 -23 )×2 127 (Adapted from V. Agrawal)

12. 14-12 Bard, Gerstlauer, Valvano, Yerraballi Binary to Decimal Conversion Binary(-1) S (1.b 1 b 2 b 3 b 4 ) × 2 E Represents(-1) S × (1 + b 1 ×2 -1 + b 2 ×2 -2 + b 3 ×2 -3 + b 4 ×2 -4 ) × 2 E 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)

13. 14-13 Bard, Gerstlauer, Valvano, Yerraballi Decimal to Binary Conversion  Converting from base 10 to the representation  Single precision example  Covert 100 10  Step 1 – convert to binary - 0110 0100  In a binary representation form of 1.xxx have  0110 0100 = 1.100100 x 2 6

14. 14-14 Bard, Gerstlauer, Valvano, Yerraballi Decimal to Binary Conversion (cont’d)  1.1001 x 2 6 is binary for 100  Thus the exponent is a 6  Biased exponent will be 6+127=133 = 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

15. 14-15 Bard, Gerstlauer, Valvano, Yerraballi Positive Zero in IEEE 754  + 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. 0 00000000 00000000000000000000000 Biased exponent Fraction (Adapted from V. Agrawal)

16. 14-16 Bard, Gerstlauer, Valvano, Yerraballi Negative Zero in IEEE 754  - 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. 1 00000000 00000000000000000000000 Biased exponent Fraction (Adapted from V. Agrawal)

17. 14-17 Bard, Gerstlauer, Valvano, Yerraballi Positive Infinity in IEEE 754  + 1.0 × 2 128  Largest positive number in single-precision IEEE 754 standard.  Interpreted as + ∞  If true exponent = 128 and fraction ≠ 0, then the number is greater than ∞.  It is called “not a number” or NaN and may be interpreted as ∞. 0 11111111 00000000000000000000000 Biased exponent Fraction (Adapted from V. Agrawal)

18. 14-18 Bard, Gerstlauer, Valvano, Yerraballi Negative Infinity in IEEE 754  -1.0 × 2 128  Smallest negative number in single-precision IEEE 754 standard.  Interpreted as - ∞  If true exponent = 128 and fraction ≠ 0, then the number is less than - ∞  It is called “not a number” or NaN and may be interpreted as - ∞. 1 11111111 00000000000000000000000 Biased exponent Fraction (Adapted from V. Agrawal)

19. 14-19 Bard, Gerstlauer, Valvano, Yerraballi IEEE Representation Values  If E=255 and F is nonzero, then V=NaN ("Not a number")  If E=255 and F is zero and S is 1, then V=-Infinity  If E=255 and F is zero and S is 0, then V=+Infinity  If 0<E<255 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 E=0 and F is nonzero, then V=(-1)**S * 2 ** (-126) * (0.F)  These are "unnormalized" values.  If E=0 and F is zero and S is 1, then V=-0  If E=0 and F is zero and S is 0, then V=0

20. 14-20 Bard, Gerstlauer, Valvano, Yerraballi 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)

21. 14-21 Bard, Gerstlauer, Valvano, Yerraballi Rounding  Adjusting significands before addition will produce results that exceed 24 bit  Round toward infinity oselect next largest normalized result  Round toward minus infinity oselect next smallest normalized result  Round toward zero otruncate result  Round to nearest oselect closest normalized result oused by IEEE 754

22. 14-22 Bard, Gerstlauer, Valvano, Yerraballi Example  Subtraction: 0.5 10 - 0.4375 10  Step 0:Floating point numbers to be added 1.000 2 ×2 -1 and -1.110 2 ×2 -2  Step 1: Significand of lesser exponent is shifted right until exponents match -1.110 2 ×2 -2 → - 0.111 2 ×2 -1  Step 2: Add significands, 1.000 2 + (- 0.111 2 ) Result is 0.001 2 ×2 -1  Step 3: Normalize, 1.000 2 × 2 -4 No overflow/underflow since 127 ≥ exponent ≥ -126  Step 4: Rounding, no change since the sum fits in 4 bits. 1.000 2 ×2 -4 = (1+0)/16 = 0.0625 10 (Adapted from V. Agrawal)

23. 14-23 Bard, Gerstlauer, Valvano, Yerraballi 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)

24. 14-24 Bard, Gerstlauer, Valvano, Yerraballi 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)