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Arithmetic for Computers

Arithmetic for Computers. CPSC 252 Computer Organization Ellen Walker, Hiram College. Encoding Numbers. Two symbols (1 vs. 0) Binary logic easiest to implement electronically How to represent arbitrary numbers? Ascii Characters 4 bits per decimal digit (Binary Coded Decimal)

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Arithmetic for Computers

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  1. Arithmetic for Computers CPSC 252 Computer Organization Ellen Walker, Hiram College

  2. Encoding Numbers • Two symbols (1 vs. 0) • Binary logic easiest to implement electronically • How to represent arbitrary numbers? • Ascii Characters • 4 bits per decimal digit (Binary Coded Decimal) • Raw binary (base 2 numbers)

  3. ASCII vs. BCD vs. Binary • 33 in ASCII 00110011 00110011 (8*log10 bits) • 33 in Binary Coded Decimal 0011 0011 (4*log10 bits) • 33 in Binary 100001 log2 bits

  4. Bit Numbering Convention • Bits numbered from right to left, starting with 0 33 = 0 0 1 0 0 0 0 1 Bits: 7 6 5 4 3 2 1 0 • Bit 0 is Least Significant Bit (LSB) • Bit 7 is Most Significant Bit (MSB)

  5. Signed Numbers (Sign / Magnitude) • Sign + Magnitude • One bit used for sign (convention 1 = negative) • N-1 bits for magnitude • Difficulties • 2 representations for 0 • Where does the sign go? (MSB vs. LSB) • Complex addition algorithm

  6. Two’s Complement • MSB (sign bit) is the -2^(N-1) place and all other bits are 2^x (0<=x<N-1) • Advantages: • No changes to addition algorithm • Easy negation (flip the bits and add 1) • Easy test for negative (MSB is 1)

  7. Largest & Smallest Values • N bits can represent 2^N combinations • Unsigned: 0 to (2^N) - 1 • Sign/Magnitude -(2^N-1)+1 to (2^N-1)-1 • 2’s Complement -2^N-1 to (2^N-1)-1 • Arithmetic operations that result in values outside this range cause arithmetic overflow

  8. Arithmetic Overflow • Arithmetic overflow occurs when the sign bit is incorrect after an operation • Example (8 bit 2’s complement) 01111111 127 + 00000011 3 -------------------- 10000010 -126

  9. Unsigned Integers • If negative values aren’t appropriate, use all bits • Values from 0 to (2^N)-1 (all positive) • Example: addresses

  10. Effect on Instruction sets • Instructions for both signed and unsigned values • Loading / storing • Comparisons • Arithmetic / Logic operations

  11. Sign Extension • When copying a shorter signed number into a longer register, don’t change the sign! • 11110000 = 111111110000, not 000011110000 • To avoid this, replicate the sign bit all the way to the MSB • Sign bit replication never changes the value of a number

  12. Shortcuts for Decimal to Binary • Compute minimal bit pattern for absolute value • Repeated divide-by-two, saving remainders • MSB must be 0 (add one if needed) • If negative value is desired, flip the bits and add 1 • Sign-extend to required number of bits

  13. Decimal-to-binary examples • 1037 (16 bit) • -38 (8 bit)

  14. Loading Numbers into Registers • A complete register - lw • No special cases, since all 32 bits specified • A byte • lb (load byte) sign-extends • lbu (load byte unsigned) pads with 0’s • A halfword (2 bytes) • lh (load halfword) sign-extends • Lhu (load halfword unsigned) pads with 0’s

  15. Comparisons • Set on less than (slt, slti) • Treats registers as signed numbers • Set on less than unsigned (sltu, sltui) • Treats registers as unsigned numbers • Example • $s1 holds 00….001, $s2 holds 11..110 • Slt $t0, $s1, $s2 puts 0 in $t0 • Sltu $t0, $s1, $s2 puts 1 in $t0

  16. Shortcut for 0<=x < y • Bit patterns for negative numbers, when treated as unsigned, are larger than all bit patterns for positive numbers. (Why?) • If (unsigned) x < (unsigned) y, and y is known to be positive, we also know that x is positive. • Use sltu to check both negative and too big in one instruction.

  17. Integer Addition & Subtraction • Algorithm in base 2 is same as base 10 • Add up each column of numbers from right to left • A column consists of 2 numbers plus the carry from the previous column (0 if first column) • If the sum is greater than 1, carry a 1 to the next column, e.g. 1+1+1 = 3 (1, carry the 1)

  18. One-Column Addition Table

  19. Addition example (8 bit 2’s complement) • Carry shown above in red 01100000 00110100 52 + 00010010 18 -------------------- 01000110 70

  20. Subtraction • Use the usual subtraction algorithm with borrows • Negate the second operand, then add • Advantage: reuse existing hardware • Negation is easy in hardware • Bit flip • Increment

  21. Overflow (Signed Numbers) • Adding two positive numbers • Overflow if sign bit becomes 1 • Adding two negative numbers • Overflow if sign bit becomes 0 • Adding positive and negative number • Overflow cannot occur (why?)

  22. Signed Overflow Detection Algorithm • IF the sign bits of the operands are equal, • AND the sign bit of the result does not equal the sign bits of the operands • THEN overflow has occurred

  23. Unsigned Integers • Overflow could be detected by having a separate sign bit in addition to the 32 bit register • However, overflow in memory addresses is commonly ignored • “Wrap” from highest to lowest location

  24. Relevant Instructions • Add, addi, sub, subi • Detect overflow and cause an exception when it occurs • Addu, addiu, subu • Never cause an overflow exception

  25. Dealing with Overflow • Exception code (like a procedure call) • Special conditional branch • Many architectures, not MIPS • “Branch on overflow” instruction • Write your own code • Do an unsigned arithmetic operation • Check signs of operands and result • Use xor operation: 1 if different, 0 if equal (see p. 228)

  26. Consequences of Fixed Integer Representations • Limited number of values (both positive and negative) • Moving from shorter to longer format requires sign extension • Unsigned representations allow twice as many values (but no negatives) • Arithmetic overflow must be detected • Correct algorithm on valid inputs yields incorrect result

  27. Multiplication • Multiplication result needs twice as many bits • E.g. 7*7 = 49 (7 is 3 bits, 49 is 6 bits) • Multiplication is more complex than addition/subtraction • Develop algorithm, implement using simpler hardware

  28. Multiplication Algorithms • Repeated addition • Easy to implement in software • Can only multiply positive numbers directly • Time depends on magnitude of operands • Shift-and add • Efficient use of hardware • Time depends only on width of operands • No special case for negative numbers

  29. Shift and Add Algorithm • For each bit in multiplier • If bit = 1, add multiplicand to result • Shift multiplicand left by 1

  30. Shift and Add example 10101 21 X 101 5 --------- 10101 000000 (included for completeness) + 1010100 ------------------ 1101001 105 (1+8+32+64)

  31. Division • Use the long division algorithm (shift and subtract) • Put 1 in the quotient whenever leading bit is 1, else put 0 in the quotient • When all bits from dividend have been used, if difference is not 0, it is the remainder.

  32. Division Example 10101 101 | 1101001 101 0110 101 0101 101 0

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