William Stallings Computer Organization and Architecture

1. William Stallings Computer Organization and Architecture Chapter 9 Computer Arithmetic Rev. by Luciano Gualà (2008)

2. Arithmetic & Logic Unit • Does the calculations • Almost everything else in the computer is there to service this unit • Handles integers • May handle floating point (real) numbers Rev. by Luciano Gualà (2008)

3. ALU Inputs and Outputs Rev. by Luciano Gualà (2008)

4. Reminder • Let A=an-1an-2…a1a0 be a (pure) binary number • Its (decimal) value is: Example: 10010 = 24 + 21 = 16 + 2 = 18 Rev. by Luciano Gualà (2008)

5. Integer Representation • Only have 0 & 1 to represent everything • Positive numbers stored in binary • e.g. 41=00101001 • Binary digit = Bit • No minus sign • Negative integer representations: • Sign-Magnitude • Two’s complement Rev. by Luciano Gualà (2008)

6. Sign-Magnitude • Left most bit is sign bit • 0 means positive • 1 means negative +18 = 00010010 -18 = 10010010 • Problems • Need to consider both sign and magnitude in arithmetic • Two representations of zero (+0 and -0) Rev. by Luciano Gualà (2008)

7. A better representation:Two’s Complement • Let A=an-1an-2…a1a0 be a n-length binary string • It’s value is in Two’s Complement representation is: if an-1=0 then A is a non-negative number if an-1=1 then A is a negative number Rev. by Luciano Gualà (2008)

8. Two’s Complement: examples = 70 +64 +4 +2 = -125 -128 +2 +1 = -120 -128 +8 Rev. by Luciano Gualà (2008)

9. Benefits • Only one representation of zero +0 = 0000 -1 = 1111 +1 = 0001 -2 = 1110 +2 = 0010 -3 = 1101 +3 = 0011 -4 = 1100 +4 = 0100 -5 = 1011 +5 = 0101 -6 = 1010 +6 = 0110 -7 = 1001 +7 = 0111 -8 = 1000 • Arithmetic works easily (see later) Rev. by Luciano Gualà (2008)

10. Geometric Depiction of Two’s Complement Integers Rev. by Luciano Gualà (2008)

11. Range of Numbers • 8 bit 2’s complement +127 = 01111111 = 27 -1 -128 = 10000000 = -27 • 16 bit 2’s complement +32767 = 01111111 11111111 = 215 - 1 -32768 = 10000000 00000000 = -215 Rev. by Luciano Gualà (2008)

12. …when you cannot sleep… …don’t count sheep in two’s complement… Rev. by Luciano Gualà (2008)

13. 2’s Complement representation for negative numbers • To represent a negative number using the “two’s complement” technique: • First decide how many bits are used for representation • Then write the modulo of the negative number (in pure binary) • Then, change each 0 in 1, each 1 in 0 (Boolean Complement or “one’s complement”) • Finally, add 1 (as the result of Step 3 was a pure binary number) Rev. by Luciano Gualà (2008)

14. Examples • E.g.: how to represent -3 with 4 bits: • Start from +3 = 0011 • Boolean complement gives 1100 • Add 1 to LSB gives -3 1101 • Represent -20 with 8 bits: • Start from +20 = 00010100 • Bolean complement gives 11101011 • Add 1 11101100 • Negation works in the same way, e.g. negation of -3 is obtained by the “two’s complement” of -3: • Representation of -3 1101 • Boolean complement gives 0010 • Add 1 to LSB gives -(-3)=+3 0011 Rev. by Luciano Gualà (2008)

15. Negation Special Case 1 • 0 = 0000 • Bitwise NOT 1111 (Boolean complement) • Add 1 to LSB +1 • Result 1 0000 • Carry is ignored, so: • - 0 = 0 OK ! Rev. by Luciano Gualà (2008)

16. Negation Special Case 2 • -8 = 1000 • Bitwise NOT 0111 (Boolean complement) • Add 1 to LSB +1 • Result 1000 • So: • -(-8) = -8 WRONG ! • Monitor MSB (sign bit) • If it does not change during negation there is a mistake (but for 0!) Rev. by Luciano Gualà (2008)

17. Conversion Between Lengths • Positive number: pack with leading zeroes +18 = 00010010 +18 = 00000000 00010010 • Negative number: pack with leading ones -18 = 11101110 -18 = 11111111 11101110 • i.e. pack with MSB (sign bit) Rev. by Luciano Gualà (2008)

18. Addition and Subtraction • Addition: standard • Overflow: when the result needs more bits to be represented • Monitor MSB: if it changes there may be an overflow • When Pos + Pos or Neg + Neg the sign bit should not change: if it does there is an overflow • Subtraction: take two’s complement of subtrahend and add to minuend • i.e.: a - b = a + (-b) • So we only need addition and complement circuits Rev. by Luciano Gualà (2008)

19. Addition: examples 1100 + -4 0100 = 4 0000 0 1100 + -4 1111 = -1 1011 -5 • + -7 • 0101 = 5 • 1110 -2 1 1 0011 + 3 0100 = 4 0111 7 0101 + 5 0100 = 4 1001 overflow 1001 + -7 1010 = -6 0011 overflow 1 Rev. by Luciano Gualà (2008)

20. Hardware for Addition and Subtraction Rev. by Luciano Gualà (2008)

21. Real Numbers • …very informally, numbers with the fractional point • Actually, only finite precision real numbers • we need to code the binary point • we need to code the binary point position • Solution: floating point numbers Rev. by Luciano Gualà (2008)

22. Reminder • Let A=an-1an-2…a1a0 ,a-1,a-2,…,a-m be a binary number • Its value is: Example: 1001,1011 = 23 + 20 +2-1 + 2-3 + 2-4 =9,6875 Rev. by Luciano Gualà (2008)

23. Real Numbers • Where is the binary point? • Fixed? • Very limited • Moving? • How do you show where it is? • Example: 976.000.000.000.000 = 9,76 * 1014 = 9,76 * 10-14 0,0000000000000976 Rev. by Luciano Gualà (2008)

24. Floating Point • Represents the value +/- .<significand> * <base><exponent> • “Floating” refers to the fact that the true position of the point “floats” according to the value of the exponent Biased Exponent Significand or Mantissa Sign bit Rev. by Luciano Gualà (2008)

25. Normalization (10,0101)2= 10,0101* 20 = 10010,1 * 2-3 • Floating Point numbers are usually normalized • exponent is adjusted so that there is a single bit equal to 1 before the binary point • Similar to scientific notation for decimal numbers where numbers are normalized to give a single digit before the decimal point 3123 = 3.123 x 103 = 1001010 * 2-5 = 0,00100101 * 24 Rev. by Luciano Gualà (2008)

26. Normalization • A normalized number ( 0) has the following form: • The base (or radix) for the exponent is suppose to be 2 (so it is not represented) • Since MSB of the mantissa is always 1 there is no need to represent it +/- 1,bbb…b * 2 +/- E where b  {0,1} Rev. by Luciano Gualà (2008)

27. Typical Representation of Floating Point with 32 bits • Mantissa uses 23 bits to store a 24 bits pure binary number in the interval [1,2) • Sign is stored in the first bit • Exponent value is represented in excess or biased notation with 8 bits • Excess (bias) 127 means • 8 bit exponent field • Nominal exponent value has range 0-255 • Subtract 127 (= 28-1-1) to get correct exponent value • Real range of exponent value is -127 to +128 Rev. by Luciano Gualà (2008)

28. excess (bias) notation • Two parameters are specified: • the number of bits n • the bias value K (usually, K=2n -1 -1) • the string consisting of all 0s represents the value –K • the string consisting of all 1s represents the value –K + 2n-1 1000 = 1 1001 = 2 1010 = 3 1011 = 4 1100 = 5 1101 = 6 1110 = 7 1111 = 8 n = 4 K=7 0000 = -7 0001 = -6 0010 = -5 0011 = -4 0100 = -3 0101 = -2 0110 = -1 0111 = 0 Rev. by Luciano Gualà (2008)

29. Floating Point Examples 147 = 127+20 107 = 127-20 Rev. by Luciano Gualà (2008)

30. Expressible Numbers Notice: we are representing 232 different numbers in both cases!! 2-127 -2-127 -(2-2-23)*2128 (2-2-23)*2128 Rev. by Luciano Gualà (2008)

31. density • Precision decreases with the increase of modulo (representation is denser for smaller numbers) • How many numbers can we represent in the interval [2,4)? • And in the range [4,8)? • they are the same: 223 Note: Finite precision: (10.0/3.0)*3.0 may be different from 10.0 Rev. by Luciano Gualà (2008)

32. IEEE 754 • Standard for floating point storage • 32 and 64 bit standards • 8 and 11 bit exponent respectively • Extended formats (both mantissa and exponent) for intermediate results Rev. by Luciano Gualà (2008)

33. IEEE 754 Formats Rev. by Luciano Gualà (2008)

34. Special cases (single format) • exponent [-126,127]: normalized numbers • Some strings are interpreted as special strings • Two representations for zero (posit. and negat.) • +/- 0 = 0/1 00000000 00000000000000000000000 • Able to represent infinity (posit. and negat.) • +/-  = 0/1 11111111 00000000000000000000000 • Non-normalized numbers: • 0/1 00000000 (string different to all 0s) • exponent=-126 • form of significand: 0.bbbb.. • NAN = Not A Number • Result of an operation which has no solution • Propagated as NAN through subsequent operations • representation: 0/1 11111111 (string different to all 0s) Rev. by Luciano Gualà (2008)

35. FP Arithmetic: addition and subtraction • Check for zero • Align significands (adjusting exponents) • Add or subtract significands • Normalize result Rev. by Luciano Gualà (2008)

36. FP Arithmetic: multiplication and division • Check for zero • Add/subtract exponents • Multiply/divide significands (watch sign) • Normalize • Round • All intermediate results should be in double length storage Rev. by Luciano Gualà (2008)