EE 319K Introduction to Embedded Systems

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

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

3. 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 Bard, Gerstlauer, Valvano, Yerraballi (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) Bard, Gerstlauer, Valvano, Yerraballi

5. 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 Bard, Gerstlauer, Valvano, Yerraballi (Adapted from V. Aagrawal)

6. Floating Point Numbers • Decimal scientific notation • 0.513×105, 5.13×104 and 51.3×103 • 5.13×104 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., 1.0×2-1 • Known as floating point numbers Bard, Gerstlauer, Valvano, Yerraballi (Adapted from V. Agrawal)

7. 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 Bard, Gerstlauer, Valvano, Yerraballi (Adapted from V. Agrawal)

8. Floating Point Numbers • General format ±1.bbbbb two×2eeee or (-1)S× (1+F) × 2E • 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) Bard, Gerstlauer, Valvano, Yerraballi (Adapted from V. Agrawal)

9. ANSI/IEEE Std 754-1985 • Single-precision float format Bit 31 Mantissa sign, s=0 for positive, s=1 for negative Bits 30:23 8-bit biased binary exponent 0 ≤ e ≤ 255 Bits 22:0 24-bit mantissa, m, expressed as a binary fraction, A binary 1 as the most significant bit is implied. m = 1.m1m2m3...m23 Bard, Gerstlauer, Valvano, Yerraballi (Adapted from V. Agrawal)

10. 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 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) Bard, Gerstlauer, Valvano, Yerraballi (Adapted from V. Agrawal)

11. 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 Expressible numbers -231 0 231-1 Positive underflow Negative underflow Negative Overflow Expressible negative numbers Expressible positive numbers Positive Overflow -2-127 0 2-127 - (2 – 2-23)×2127 (2 – 2-23)×2127 Bard, Gerstlauer, Valvano, Yerraballi (Adapted from V. Agrawal)

12. Binary to Decimal Conversion Binary (-1)S (1.b1b2b3b4) × 2E Represents (-1)S × (1 + b1×2-1 + b2×2-2 + b3×2-3 + b4×2-4) × 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) Bard, Gerstlauer, Valvano, Yerraballi (Adapted from V. Agrawal)

13. 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 Bard, Gerstlauer, Valvano, Yerraballi

14. Decimal to Binary Conversion (cont’d) • 1.1001 x 26 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 Bard, Gerstlauer, Valvano, Yerraballi

15. Positive Zero in IEEE 754 • + 1.0 × 2-127 • Smallest positive number in single-precision IEEE 754 standard. • Interpreted as positive zero. • Exponentless than -127 is positive underflow; can be regarded as zero. 0 0000000000000000000000000000000 Biased exponent Fraction Bard, Gerstlauer, Valvano, Yerraballi (Adapted from V. Agrawal)

16. 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 0000000000000000000000000000000 Biased exponent Fraction Bard, Gerstlauer, Valvano, Yerraballi (Adapted from V. Agrawal)

17. Positive Infinity in IEEE 754 • + 1.0 × 2128 • 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 1111111100000000000000000000000 Biased exponent Fraction Bard, Gerstlauer, Valvano, Yerraballi (Adapted from V. Agrawal)

18. Negative Infinity in IEEE 754 • -1.0 × 2128 • 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 1111111100000000000000000000000 Biased exponent Fraction Bard, Gerstlauer, Valvano, Yerraballi (Adapted from V. Agrawal)

19. 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 Bard, Gerstlauer, Valvano, Yerraballi

20. 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 Bard, Gerstlauer, Valvano, Yerraballi (Adapted from V. Agrawal)

21. 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 Bard, Gerstlauer, Valvano, Yerraballi

22. Example • Subtraction: 0.510- 0.437510 • Step 0: Floating point numbers to be added 1.0002×2-1 and -1.1102×2-2 • Step 1: Significand of lesser exponent is shifted right until exponents match -1.1102×2-2 →- 0.1112×2-1 • 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 = (1+0)/16 = 0.062510 Bard, Gerstlauer, Valvano, Yerraballi (Adapted from V. Agrawal)

23. FP Multiplication: Basic Idea • Separate sign • Add exponents • Multiply significands • Normalize, round, check overflow • Replace sign Bard, Gerstlauer, Valvano, Yerraballi (Adapted from V. Agrawal)

24. FP Division: Basic Idea • Separate sign. • Check for zeros and infinity. • Subtract exponents. • Divide significands. • Normalize/overflow/underflow. • Rounding. • Replace sign. Bard, Gerstlauer, Valvano, Yerraballi (Adapted from V. Agrawal)