Ch 11 computer arithmetic
Steps for addition and subtraction of floating-point numbers
1. Change sign (if subtracting) and check for zero 2. Significand alignment 3. Addition 4. Normalization- make sure result is a normal number
Examples of floating-point formats IEEE 754
Binary 16,32,64,128, 32*(n+4) Decimal 64, 128, 32(n+4) Extended & Extendable precision
Guard bits
Extra zeros at the right-end of the significand in the register. When re-alignment occurs, no bits of significance are lost.
Range extension for twos complement representation
If it is a positive number, simply fill in the spaces with zeros. If it is a negative number, fill in the spaces with ones. Ex: +18=00010010 --> +18= 0000000010010 -18= 10010010 --> -18= 111111111110010010
Binary Multiplication of unsigned numbers by hand
it is carried out like normal multiplication
Base of a floating point number
it is implicit (assumed to be 2) and is thus not stored
round towards zero
simply truncation
Partial remainder
the intermediate remainders during division.
Dividend
what is being divided (first number)
ALU-
"The core or essence of the computer" -The operands for the ALU are stored in registers -The results from the ALU are also stores in registers -The processor provides control signals to the ALU -The ALU can set flags
Arithmetic shift
(part of booth's algorithm). It preserves the sign bit. Everything in A, Q, and Q-1 is shifted one place to the right, except the very first place of A, which remains the same.
Infinity in IEEE 754 is represented by
0 or 1 sign bit indicating +/- inf, all 1s for exponent, 0 for fraction.
Zero in IEEE 754 is represented by
0 or 1 sign bit, 0 exponent, 0 fraction
NaN (not a number) in IEEE 754 is represented by
0 or 1 sign bit, all 1s for exponent, non-zero fraction
Subnormal number in IEEE 754 is represented by
0 or 1 sign bit, exponent of zero, non-zero fraction that isn't a normal number- implicit 1 is no longer true. The true exponent is -126 or -1022.
Three types of floating point formats defined by standard 754
1. Arithmetic Format- All mandatory operations defined by the standard are supported by the format. The format may be used to represent floating-point operands of the results. 2. Basic format- Covers 5 floating point representations 3. Interchange format- Allows for data interchanged storage. Fully specified and fixed length.
Steps for multiplication and division of floating-point numbers
1. check for zero 2. combine exponents (add or subtract, then do the opposite for the bias (ex: multiplication is adding exponents then subtracting bias)) 3. Multiply or divide significands (normally) 4. Normalize & round the significand
Four rounding techniques
1. round to nearest 2. round toward +inf 3. round toward -inf 4. Round toward zero
minuend
A number from which another number is subtracted (first number).
subtrahend
A number that is subtracted (second number)
Machine algorithm for long division
An accumulator register is set to zero while a register Q holds the dividend. In one cycle, the values in Q are shifted toward A while the gaps are filled with zeros. The last digit of Q is set to 1 if the divisor fits into the value in the accumulator. The number of cycles equals the number of digits.
Hardware layout for addition and subtraction
Both operators are stored in registers. A complimenter can reverse (negate) the B register before both are sent to the adder. The adder can set an overflow bit to true.
Possible conditions produced by floating-point operations
Exponent overflow- exponent> exponent max Exponent underflow- exponent< exponent min Significand underflow- digits may be cut off the right end of the significand when allightment of the significands occurs. Significand overflow- two significands added together result in a number that goes too far left. Re-alignment can fix this.
Range extension for sign-magnitude representation
Fill in the newly created positions with zeros
Overflow rule
If two numbers are added, and they are both positive or both negative, then overflow occurs if and only if the result has the opposite sign.
Manual division of unsigned binary integers
It follows the same process as base-10 long division.
Addition of integers in two's complement representation
Simply add the two together, carrying over if a place has a value (in base 10) of two or greater. Ignore the extra bit if it is created (the carry bit)
Round to nearest
The default rounding mode. "The representable value nearest to the infinently precise result shall be delivered." If it is in the middle, the even number is favored.
Sign-magnitude representation
The first bit indicates either zero for positive or one for negative. One problem this raises is that zero could either be 10 or 00. Ex: +18=00010010 or -18=10010010
IEEE Standard 754
The most important floating-point representation. It was created to increase portability of programs.
Binary multiplication implementation
The multiplier and multiplicand are loaded into two registers. If the next digit of the multiplicand is one, the multiplier is added to a result register. This process is repeated, each time shifting by one digit. A carry counter is also used.
Format of a floating point number written out
There is a plus or minus (sign bit), a significand, a base, and and exponent.
Subtraction rule
To subtract one number (subtrahend) from another number (minuend), take the twos complement of the subtrahend and add it to the minuend.
Negation of 2^(n-1)
Where n is the length of the binary number. It cannot occur in twos complement representation. Thus, the max range for a number in twos complement representation is (-2^(n-1), 2^(n-1)-1)
Biased representation for an exponent
a fixed value, called the bias, is subtracted from the field to get the true exponent value. Usually, the bias is half.
Extendable precision format
an example of IEEE 754 format. It has a precision and range that are defined under user control.
Extended precision formats
an example of an IEEE 754 format which extend a supported basic format by providing additional bits in the exponent and significand.
IEEE standard 754 for operations with infinity
follows standard math practices (think indeterminate forms)
positive overflow
greater than (2-2^-23)*2^128
negative underflow
greater than -2^(-127) but less than zero
Fixed-point representation
includes both sign-magnitude and twos complement representations. The radix point is fixed in the rightmost position.
Range Extension
increasing the length (and thus the potential range) of an integer value.
sign of a floating point number
it is the leftmost bit. 0= positive, 1=negative
Twos complement representation
like sign-magnitude, the first bit is a zero for positive of a one for negative. Zero, however, is represented by all zeros. If the number is negative, all places n-2 and earlier add to the negative number. It is the most used representation in computers.
Other names for significand
mantissa, fraction, argument, coefficent
Twos complement multiplication
negative numbers represented in twos complement notation will not multiply properly. If only the multiplicand but not the multiplier is negative, partial products can be filled with ones in front. Otherwise, booth's algorithm must be used.
negative overflow
numbers less than -(2 - 2^-23)*2^(128)
normal number
one in which the most significant digit of the significand is non-zero
positive underflow
positive numbers less than 2^-127
IEEE standard 754 quiet and signaling NaNs-
signaling NaNs report an error immediately, while quiet NaNs may propagate. Any "intermediate forms" in math produce a NaN, like inf/inf
Twos complement operation
taking the twos complement of an integer, which is negating a number in twos complement representation. 1. Take the Boolean complement of every bit of the integer (aka switch 0 and 1) 2. Treating the result as an unsigned binary integer, add one to its value. Ex: +18= 00010010 --> 11101110 = -18
Trailing significand field
the first digit of the significand is implied to be a one and is thus not included. The rest of the significand is referred to as the trailing significand.
multiplicand
the number that is being multiplied (first number)
multiplier
the number that the multiplicand is multiplied by (second number)
Significand
the significant digits that are being stored.
Gradual underflow
the use of subnormal numbers to reduce the gap where underflow occurs.
Format in computer of a floating point number
there is a 1 bit sign bit, a biased exponent, and a significand.
Booth's algorithm
used to multiply two numbers in twos complement representation together, where both are negative.
rounding to plus and minus inf
useful in implementing a technique known as interval arithmetic.
Divisor
what the dividend is divided by by (second number)
Carry-out during twos complement operation
when negating zero, after adding one to the complement, there will be an extra one at the start that has to be removed for zero to negate zero. Ex: 0000 --> 1111 --> 10000 -->0000
rounding
when the result of a floating-point operation is put back into floating-point format.
Trade-off between precision and range
with a given number of bits, only a given number of values can be stored. Thus, if range is increased, precision decreases. There might not be a value nearby to round to.
