Computer Architecture & Operations I

Instructor: Yaohang Li Computer Architecture & Operations I

Review • Last Class • Midterm Review • This Class • Division • Floating Point Numbers • Next Class • Floating Point Operations • Final Exam • May. 1, 2014 (3:45PM-6:45PM)

Division • Check for 0 divisor • Long division approach • If divisor ≤ dividend bits • 1 bit in quotient, subtract • Otherwise • 0 bit in quotient, bring down next dividend bit • Restoring division • Do the subtraction, and if remainder goes < 0, add divisor back • Signed division • Divide using absolute values • Adjust sign of quotient and remainder as required quotient dividend 1001 1000 1001010 -1000 1 10 101 1010 -1000 10 divisor remainder n-bit operands yield n-bitquotient and remainder

Division Hardware • First version of Division Hardware • 32-bit quotient • 64-bit ALU • 64-bit remainder Initially divisor in left half Initially dividend

First Version Division Hardware Initially divisor in left half Initially dividend

Example • 7/2

Optimized Divider • One cycle per partial-remainder subtraction • Looks a lot like a multiplier! • Same hardware can be used for both

Faster Division • Can’t use parallel hardware as in multiplier • Subtraction is conditional on sign of remainder • Faster dividers (e.g. SRT devision) generate multiple quotient bits per step • Still require multiple steps

MIPS Division • Use HI/LO registers for result • HI: 32-bit remainder • LO: 32-bit quotient • Instructions • div rs, rt / divu rs, rt • No overflow or divide-by-0 checking • Software must perform checks if required • Use mfhi, mflo to access result

Example Revisit • A better way to do 7/2? • 7>>2 • SRL • Most compiler will replace divide by power of 2 using right shift operations

Scientific Notation • Scientific Notation • Renders numbers with a single digit to the left of the decimal point • Normalization • Scientific notation without leading 0s • Including very small and very large numbers

Floating Point §3.5 Floating Point • Floating Point Numbers • Computer arithmetic that represents numbers in which the binary point is not fixed • Like scientific notation • –2.34 × 1056 • +0.002 × 10–4 • +987.02 × 109 • In binary • ±1.xxxxxxx2 × 2yyyy • Types float and double in C normalized not normalized

Floating-Point Representation • Two components • Fraction • between 0 and 1 • precision of the floating point number • Exponent • numerical value • range of the floating point number • Representation of floating point • Determine the sizes of fraction and exponent • Tradeoff between range and precision S Exponent Fraction

Terms in Floating Point Representation • Overflow • A positive exponent becomes too large to fit in the exponent field • Underflow • A negative exponent becomes too large to fit in the exponent field • Double precision • A floating-point value represented in 64 bits • Single precision • A floating-point value represented in 32 bits

Floating Point Standard • Defined by IEEE Std 754-1985 • Developed in response to divergence of representations • Portability issues for scientific code • Now almost universally adopted • Two representations • Single precision (32-bit) • Double precision (64-bit)

IEEE 754 Encoding

IEEE Floating-Point Format • S: sign bit (0  non-negative, 1  negative) • Normalize significand: 1.0 ≤ |significand| < 2.0 • Always has a leading pre-binary-point 1 bit, so no need to represent it explicitly (hidden bit) • Significand is Fraction with the “1.” restored • Exponent: excess representation: actual exponent + Bias • Ensures exponent is unsigned • Single: Bias = 127; Double: Bias = 1023 single: 8 bitsdouble: 11 bits single: 23 bitsdouble: 52 bits S Exponent Fraction

Floating-Point Example • Represent –0.375 • –0.375 = -3/8=-3/23=(–1)1 × 112 × 2–3 • Normalization=(–1)1 × 1.12 × 2–2 • S = 1 • Fraction = 1000…002 • Exponent = –2 + Bias • Single: –2 + 127 = 125 = 011111012 • Double: –2 + 1023 = 1021 = 011111111012

Floating Point Example • Single Precision • Double Precision

Floating-Point Example • What number is represented by the single-precision float • S = 1 • Fraction = 01000…002 • Exponent = 100000012 = 129 • x = (–1)1 × (1 + 0.012) × 2(129 – 127) = (–1) × 1.25 × 22 = –5.0

Single-Precision Range • Exponents 00000000 and 11111111 reserved • Smallest value • Exponent: 00000001 actual exponent = 1 – 127 = –126 • Fraction: 000…00 significand = 1.0 • ±1.0 × 2–126 ≈ ±1.2 × 10–38 • Largest value • exponent: 11111110 actual exponent = 254 – 127 = +127 • Fraction: 111…11 significand ≈ 2.0 • ±2.0 × 2+127 ≈ ±3.4 × 10+38

Double-Precision Range • Exponents 0000…00 and 1111…11 reserved • Smallest value • Exponent: 00000000001 actual exponent = 1 – 1023 = –1022 • Fraction: 000…00 significand = 1.0 • ±1.0 × 2–1022 ≈ ±2.2 × 10–308 • Largest value • Exponent: 11111111110 actual exponent = 2046 – 1023 = +1023 • Fraction: 111…11 significand ≈ 2.0 • ±2.0 × 2+1023 ≈ ±1.8 × 10+308

Floating-Point Precision • Relative precision • all fraction bits are significant • Single: approx 2–23 • Equivalent to 23 × log102 ≈ 23 × 0.3 ≈ 6 decimal digits of precision • Double: approx 2–52 • Equivalent to 52 × log102 ≈ 52 × 0.3 ≈ 16 decimal digits of precision

Denormal Numbers • Exponent = 000...0  hidden bit is 0 • Smaller than normal numbers • allow for gradual underflow, with diminishing precision • Denormal with fraction = 000...0 Two representations of 0.0!

Infinities and NaNs • Exponent = 111...1, Fraction = 000...0 • ±Infinity • Can be used in subsequent calculations, avoiding need for overflow check • Exponent = 111...1, Fraction ≠ 000...0 • Not-a-Number (NaN) • Indicates illegal or undefined result • e.g., 0.0 / 0.0 • Can be used in subsequent calculations

Summary • Multiplication • First version of multiplication • Optimized multiplication • Division • First version of division • Optimized division • Floating Pointer Number • Exponent • Fraction • IEEE Std 754-1985 • Single Precision • Double Precision

What I want you to do • Review Chapters 3.3 and 3.4

Computer Architecture & Operations I