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Integer Multiplication/Division CSCi 2021 : Machine Architecture and Organization

Integer Multiplication/Division CSCi 2021 : Machine Architecture and Organization. Chapter 2.3. Today. Representing information as bits Bit-level manipulations Integers Representation: unsigned and signed Conversion, casting Expanding, truncating

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Integer Multiplication/Division CSCi 2021 : Machine Architecture and Organization

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  1. Integer Multiplication/DivisionCSCi 2021: Machine Architecture and Organization Chapter 2.3

  2. Today • Representing information as bits • Bit-level manipulations • Integers • Representation: unsigned and signed • Conversion, casting • Expanding, truncating • Addition, negation, multiplication, division/shifting • Summary Project 1 questions? Homework 1 is posted

  3. Last Time • Addition • What may happen? • How does signed vs. unsigned differ? • Additive inverse • x + (~x + 1) = 0 • also works for two’s complement

  4. Detecting Unsigned Overflow in C • t=a+b • if (t < a) || (t < b)) • Think about: signed • pos overflow, pos underflow

  5. Last Time • Integers: expanding, truncating • Why is truncation important to understand? • truncation => porting • long long => long

  6. Multiplication • Computing Exact Product of w-bit numbers x, y • Either signed or unsigned • Ranges • Unsigned: 0 ≤ x * y ≤ (2w – 1) 2 = 22w – 2w+1 + 1 • Up to 2w bits (1… x 1…) • Two’s complement min (100… x 011…) : • Up to 2w–1 bits • Two’s complement max (100… x 100…) : • Up to 2w bits, but only for (TMinw)2 • Maintaining Exact Results • Would need to keep expanding word size with each product computed • Done in software by “arbitrary precision” arithmetic packages

  7. • • • • • • • • • • • • • • • Unsigned Multiplication in C • Standard Multiplication Function • Ignores high order w bits • Implements Modular Arithmetic UMultw(u , v) = u · v mod 2w u Operands: w bits * v u · v True Product: 2*w bits UMultw(u , v) Discard w bits: w bits

  8. • • • • • • • • • • • • • • • Signed Multiplication in C • Standard Multiplication Function • Ignores high order w bits • Some of which are different for signed vs. unsigned multiplication • Lower bits are the same! • Multiplication can be slow – why? u Operands: w bits * v u · v True Product: 2*w bits TMultw(u , v) Discard w bits: w bits

  9. Security example later if time

  10. Examples/Practice Want full bit patterns long => int * int need a special instruction!

  11. Power-of-2 Multiply with Shift • Operation • u << kgives u * 2k • Both signed and unsigned • Examples • u << 3 == u * 8 • Most machines shift and add faster than multiply • Generalized: • Write k as a sum of powers of 2 (e.g. 14 = 2^3+2^2+2^1) • u*15 => (u << 3) + (u << 2) + (u << 1) + u [3 shifts, 3 adds] • Can you do better? • Good compiler generates this code automatically! k u • • • Operands: w bits * 2k 0 ••• 0 1 0 ••• 0 0

  12. Division • Even slower than multiplication! • Want to optimize this case as well • Focus only on special (but common cases)

  13. ••• ••• Unsigned Power-of-2 Divide with Shift • Quotient of Unsigned by Power of 2 • u >> kgives  u / 2k  • Uses logical shift k u Binary Point ••• Operands: / 2k 0 ••• 0 1 0 ••• 0 0 Division: u / 2k . 0 0 ••• 0 0 ••• Result: u / 2k  0 0 ••• 0 0 ••• Round to zero

  14. Compiled Unsigned Division Code C Function • Uses logical shift for unsigned • For Java Users • Logical shift written as >>> unsigned udiv8(unsigned x) { return x/8; } Compiled Arithmetic Operations Explanation # Logical shift return x >> 3; shrl $3, %eax

  15. ••• ••• Signed Power-of-2 Divide with Shift • Quotient of Signed by Power of 2 • x >> kgives  x / 2k  • Uses arithmetic shift • Rounds wrong dir when a non-zero in purple region Want round toward 0: -x.y => -x k x Binary Point ••• Operands: / 2k 0 ••• 0 1 0 ••• 0 0 Division: x / 2k . 0 ••• ••• RoundDown(x/ 2k) Result: 0 ••• ••• Suppose purple had all 0’s? no rounding (even division)

  16. Correct Power-of-2 Divide • Quotient of Negative Number by Power of 2 • Want  x / 2k  (Round Toward 0) • Compute as  (x+2k-1)/ 2k  • In C: (x + (1<<k)-1) >> k • Biases dividend toward 0 Case 1: No rounding 2k-1/ 2k k Dividend: x 1 ••• 0 ••• 0 0 +2k –1 0 ••• 0 0 1 ••• 1 1 Binary Point 1 ••• 1 ••• 1 1 Divisor: / 2k 0 ••• 0 1 0 ••• 0 0  x / 2k  . 1 0 ••• 1 1 1 ••• 1 ••• 1 1 • Biasing has no effect – shifted off

  17. Correct Power-of-2 Divide (Cont.) a non-zero somewhere Case 2: Rounding k Dividend: x 1 ••• ••• +2k –1 0 ••• 0 0 1 ••• 1 1 1 ••• ••• Incremented by 1 Binary Point Divisor: / 2k 0 ••• 0 1 0 ••• 0 0  x / 2k  . 0 1 ••• 1 1 1 ••• ••• Incremented by 1 • Biasing adds 1 to final result

  18. Examples • u=-8/4, k = 2 • u = -11/8, k =3

  19. Compiled Signed Division Code C Function • Uses arithmetic shift for int • For Java Users • Arith. shift written as >> int idiv8(int x) { return x/8; } Why the if statement? Explanation Compiled Arithmetic Operations testl %eax, %eax js L4 L3: sarl $3, %eax ret L4: addl $7, %eax jmp L3 if x < 0 x += 7; # Arithmetic shift return x >> 3;

  20. Summary: Integer Arithmetic Rules • Addition: • Unsigned/signed: Normal addition followed by truncate,same operation on bit level • Unsigned: addition mod 2w • Mathematical addition + possible subtraction of 2w • Signed: addition mod 2w • Mathematical addition + possible addition or subtraction of 2w • Multiplication: • Unsigned/signed: Normal multiplication followed by truncate, same operation on bit level • Unsigned: multiplication mod 2w • Signed: modified multiplication mod 2w

  21. Arithmetic: Basic Rules • Left shift • Unsigned/signed: multiplication by 2k • Always logical shift • Right shift • Unsigned: logical shift, div (division + round to zero) by 2k • Signed: arithmetic shift • Positive numbers: div (division + round to zero) by 2k • Negative numbers: div (division + round away from zero) by 2kUse biasing to fix

  22. Why Should I Use Unsigned? • Don’t Use Just Because Number Nonnegative • Easy to make mistakes unsigned i; for (i = cnt-2; i >= 0; i--) a[i] += a[i+1]; • Do Use When Greater precision is Needed • Do Use When Using Bits to Represent Sets

  23. Look at security example on your own

  24. Next Time • Floating point representations – Chap 2.4

  25. ele_src malloc(ele_cnt * ele_size) Code Security Example #2 • SUN XDR library • Widely used library for transferring data between machines void* copy_elements(void *ele_src[], int ele_cnt, size_t ele_size);

  26. XDR Code void* copy_elements(void *ele_src[], intele_cnt, size_tele_size) { /* * Allocate buffer for ele_cnt objects, each of ele_size bytes * and copy from locations designated by ele_src */ void *result = malloc(ele_cnt * ele_size); if (result == NULL) /* malloc failed */ return NULL; void *next = result; inti; for (i = 0; i < ele_cnt; i++) { /* Copy object i to destination */ memcpy(next, ele_src[i], ele_size); /* Move pointer to next memory region */ next += ele_size; } return result; } unsigned int

  27. XDR Vulnerability • What if: • ele_cnt = 220 + 1 • ele_size = 4096 = 212 • Allocation = ?? • How can I make this function secure? • P. 146 malloc(ele_cnt * ele_size)

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