CS61C Negative Numbers and Logical Operations Lecture 7Review 1/2Review 2/2Numbers: ReviewOverviewWhat if too big?How avoid overflow, allow it sometimes?What if Overflow Detected?How Represent Negative Numbers?Search for Negative Number RepresentationTwo’s ComplementTwo’s Complement Formula, ExampleOverflow for Two’s Complement Numbers?Signed v. Unsigned ComparisonsExample: Signed v. Unsigned ComparisonsAdministrivia“What’s This Stuff Good For?”Two’s complement shortcut: NegationTwo’s complement shortcut: Sign extensionMore Compact Representation v. Binary?Logical OperationsShift InstructionsExtracting a field of bitsAnd instructionOr instructionInserting a field of bits (almost OK)Sign Extension of ImmediatesSummary: 12 new instructions (with formats)Example: show C, assembly, machineSlide 30“And in Conclusion...” 1/1cs 61C L7 Number.1Patterson Spring 99 ©UCBCS61CNegative Numbers and Logical Operations Lecture 7February 10, 1999Dave Patterson (http.cs.berkeley.edu/~patterson)www-inst.eecs.berkeley.edu/~cs61c/schedule.htmlcs 61C L7 Number.2Patterson Spring 99 ©UCBReview 1/2°MIPS assembly language instructions mapped to numbers in 3 formats•Op field determines format °Binary Decimal Assembly Symbolic Assembly C•Reverse Engineering or Disassembly•Its hard to do, therefore people like shipping binary machine language more than assembly or C6 bits 5 bits 5 bits 5 bits 5 bits 6 bitsop rs rt rd functshamtop rs rt immediateop addressRIJcs 61C L7 Number.3Patterson Spring 99 ©UCBReview 2/2°Programming language model of memory allocation and pointers•Allocate in stack vs. heap vs. global areas•Arguments passed call by value vs. call by reference•Pointer in C is HLL version of machine addresscs 61C L7 Number.4Patterson Spring 99 ©UCBNumbers: Review°Number Base B B symbols per digit:•Base 10 (Decimal): 0, 1, 2, 3, 4, 5, 6, 7, 8, 9Base 2 (Binary): 0, 1°Number representation: d31d30 ... d2d1d0•d31 x B31+ d30 x B30 + ... + d2 x B2 + d1 x B1 + d0 x B0•One billion (1,000,000,000ten ) is0011 1011 1001 1010 1100 1010 0000 0000two 228 224 220 216 212 28 24 20= 1x229+1x228+1x227+1x225+1x224+1x223+1x220 +1x219+1x217+1x215 +1x214 +1x211 +1x29 = 536,870,912 + 268,435,456 + 134,217,728+ 33,554,432 + 16,777,216 + 8,388,608 + 1,048,576+ 524,288+ 131,072+ 32,768+ 16,384+ 2,048+ 512cs 61C L7 Number.5Patterson Spring 99 ©UCBOverview°What if Numbers too Big?°How Represent Negative Numbers?°What if Result doesn’t fit in register?°More Compact Notation than Binary?°Administrivia,“What’s this stuff good for”°Shift Instructions°And/Or Instructions°Conclusioncs 61C L7 Number.6Patterson Spring 99 ©UCBWhat if too big?°Binary bit patterns above are simply representatives of numbers °Numbers really have an infinite number of digits•with almost all being zero except for a few of the rightmost digits •Just don’t normally show leading zeros°If result of add (or any other arithmetic operation), cannot be represented by these rightmost hardware bits, overflow is said to have occurred°Up to Compiler and OS what to docs 61C L7 Number.7Patterson Spring 99 ©UCBHow avoid overflow, allow it sometimes?°Some languages detect overflow (Ada), some don’t (C)°MIPS solution is 2 kinds of arithmetic instructions to recognize 2 choices:•add (add), add immediate (addi), and subtract (sub) cause exceptions on overflow•add unsigned (addu), add immediate unsigned (addiu), and subtract unsigned (subu) do not cause exceptions on overflow°Compiler selects appropriate arithmetic•MIPS C compilers produce addu, addiu, subucs 61C L7 Number.8Patterson Spring 99 ©UCBWhat if Overflow Detected?°An “exception” (or “interrupt”) occurs•Address of the instruction that overflowed is saved in a register•Computer jumps to predefined address to invoke appropriate routine for that exception•Like an unplanned hardware function call°Operating system decides what to do•In some situations program continues after corrective code is executed°MIPS support: exception program counter (EPC) contains address of that instruction•move from system control (mfc0) to copy EPCcs 61C L7 Number.9Patterson Spring 99 ©UCBHow Represent Negative Numbers?°Obvious solution: add a separate sign! •sign represented in a single bit! •representation called sign and magnitude°Shortcomings of sign and magnitude•Where to put the sign bit: right? left?•Separate sign bit means it has both a positive and negative zero, lead to problems for inattentive programmers: +0 = -0?•Adder may need extra step size don’t know sign in advance°Thus sign and magnitude was abandonedcs 61C L7 Number.10Patterson Spring 99 ©UCBSearch for Negative Number Representation°Obvious solution didn’t work, find another°What is result for unsigned numbers if tried to subtract large number from a small one?•Would try to borrow from string of leading 0s, so result would have a string of leading 1s•With no obvious better alternative, pick representation that made the hardware simple: leading 0s positive, leading 1s negative•000000...xxx is >=0, 111111...xxx is < 0°This representation called two’s complementcs 61C L7 Number.11Patterson Spring 99 ©UCBTwo’s Complement 0000 ... 0000 0000 0000 0000two = 0ten0000 ... 0000 0000 0000 0001two = 1ten0000 ... 0000 0000 0000 0010two = 2ten. . .0111 ... 1111 1111 1111 1101two = 2,147,483,645ten0111 ... 1111 1111 1111 1110two = 2,147,483,646ten0111 ... 1111 1111 1111 1111two = 2,147,483,647ten1000 ... 0000 0000 0000 0000two = –2,147,483,648ten1000 ... 0000 0000 0000 0001two = –2,147,483,647ten1000 ... 0000 0000 0000 0010two = –2,147,483,646ten. . . 1111 ... 1111 1111 1111 1101two = –3ten1111 ... 1111 1111 1111 1110two = –2ten1111 ... 1111 1111 1111 1111two = –1ten°One zero, 1st bit => >=0 or <0, called sign bit •but one negative with no positive –2,147,483,648tencs 61C L7 Number.12Patterson Spring 99 ©UCBTwo’s Complement Formula, Example°Recognizing role of sign bit, can represent positive and negative numbers in terms of the bit value times a power of 2:•d31 x -231+ d30 x 230 + ... + d2 x 22 + d1 x 21 + d0 x 20°Example1111 1111 1111 1111 1111 1111 1111 1100two= 1x-231 +1x230 +1x229+... +1x22+0x21+0x20= -231 + 230 + 229 + ... + 22 + 0 + 0 = -2,147,483,648ten +
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