High Performance Computing - Charles Severance [24]
1.0100 010 1.0100 Rounded down based on guard digits
1.0100 011 1.0100 Rounded down based on guard digits
1.0100 100 1.0100 Rounded down based on sticky bit
1.0100 101 1.0101 Rounded up based on sticky bit
1.0100 110 1.0101 Rounded up based on guard digits
1.0100 111 1.0101 Rounded up based on guard digits
The first priority is to check the guard digits. Never forget that the sticky bit is just a hint, not a real digit. So if we can make a decision without looking at the sticky bit, that is good. The only decision we are making is to round the last storable bit up or down. When that stored value is retrieved for the next computation, its guard digits are set to zeros. It is sometimes helpful to think of the stored value as having the guard digits, but set to zero.
Two guard digits and the sticky bit in the IEEE format insures that operations yield the same rounding as if the intermediate result were computed using unlimited precision and then rounded to fit within the limits of precision of the final computed value.
At this point, you might be asking, “Why do I care about this minutiae?” At some level, unless you are a hardware designer, you don’t care. But when you examine details like this, you can be assured of one thing: when they developed the IEEE floating-point standard, they looked at the details very carefully. The goal was to produce the most accurate possible floating-point standard within the constraints of a fixed-length 32- or 64-bit format. Because they did such a good job, it’s one less thing you have to worry about. Besides, this stuff makes great exam questions.
Special Values*
In addition to specifying the results of operations on numeric data, the IEEE standard also specifies the precise behavior on undefined operations such as dividing by zero. These results are indicated using several special values. These values are bit patterns that are stored in variables that are checked before operations are performed. The IEEE operations are all defined on these special values in addition to the normal numeric values. Table 1.5 summarizes the special values for a 32-bit IEEE floating-point number.
Table 1.5. Special Values for an IEEE 32-Bit Number Special Value Exponent Significand
+ or – 0 00000000 0
Denormalized number 00000000 nonzero
NaN (Not a Number) 11111111 nonzero
+ or – Infinity 11111111 0
The value of the exponent and significand determines which type of special value this particular floating-point number represents. Zero is designed such that integer zero and floating-point zero are the same bit pattern.
Denormalized numbers can occur at some point as a number continues to get smaller, and the exponent has reached the minimum value. We could declare that minimum to be the smallest representable value. However, with denormalized values, we can continue by setting the exponent bits to zero and shifting the significand bits to the right, first adding the leading “1” that was dropped, then continuing to add leading zeros to indicate even smaller values. At some point the last nonzero digit is shifted off to the right, and the value becomes zero. This approach is called gradual underflow where the value keeps approaching zero and then eventually becomes zero. Not all implementations support denormalized numbers in hardware; they might trap to a software routine to handle these numbers at a significant performance cost.
At the top end of the biased exponent value, an exponent of all 1s can represent the Not a Number (NaN) value or infinity. Infinity occurs in computations roughly according to the principles of mathematics. If you continue to increase the magnitude of a number beyond the range of the floating-point format, once the range has been exceeded, the value becomes infinity. Once a value is infinity, further additions won’t increase it, and subtractions won’t decrease it. You can also produce the value infinity by dividing a nonzero value by zero. If you divide a nonzero value by infinity, you get zero as a result.
The NaN value indicates a number that is not mathematically