Fractions and the Real Numbers

Transcription

1 Fractions and the Real Numbers Many interesting quantities are not normally integer-valued: - the mass of a rocket payload - the batting average of a baseball player - the average score on an assignment 1 Therefore, we have fractions and many fractions can be represented using a positional notation where we allow the powers of the base to be negative as well as non-negative: = Of course, some fractions cannot be represented this way in a finite manner: 2 3 = So, rational numbers are a handy idea, but we'll ignore that for now

2 Fractions: Fixed-Point 2 How can we represent fractions? - Use a binary point to separate positive from negative powers of two just like decimal point. - 2 s comp addition and subtraction still work (if binary points are aligned). But, we cannot represent extremely large or extremely small values, with a reasonable fixed number of bits, in fixed-point notation, if we must specify an alignment for the binary point

3 Floating-point Notation 3 Some numbers are so large or so small that it's inconvenient to write them in the usual manner: Therefore, we have scientific or floating-point notation: = = But, of course: = = So, we frequently adopt a normalized representation requiring the decimal point to be in a specific location, say immediately after the first digit:

4 Anatomy of a Floating-point Value 4 A floating-point value can be viewed as a combination of three distinct components: sign significand exponent the sign of the number the normalized value (to be shifted by the size of the exponent) the amount by which the significand is shifted to obtain the true value (The base of representation is implicit.)

5 Converting base-10 to base-2 At the hardware level, we'll represent values in base-2: = in base-2 5 In general, a base-10 value between 0 and 1 can be converted to base-2 by successively multiplying by 2 and recording whether there was a carry across the decimal point, stopping when you obtain zero (or enough bits to satisfy your needs): fractional value carry-over?

6 Converting base-10 to base-2 6 Here are a couple more examples: fractional value carry-over? So, converts to in base-2. fractional value carry-over? So, is about in base-2.

7 Taking It to Hardware 7 We have to decide how to handle the three components of the floating-point representation. A single bit suffices to represent the sign of the number. We have to allocate bits for the significand and for the exponent. - more bits for the significand provides more accuracy (significant digits) - more bits for the exponent provides a larger range of representation What about normalization? - except for zero, every number will have a left-most (most significant bit) of 1 - why store it?

8 IEEE 754 Floating Point Standard(s) 8 IEEE floating point standard defines fundamental formats: - single precision: 8 bit exponent, 23 bit significand, 1 sign bit - double precision: 11 bit exponent, 52 bit significand, 1 sign bit For both: - the exponent is stored as a non-negative integer, with a bias (127, 1023) - the significand is normalized so that the binary point is to the right of the first nonzero bit (0 being an exception) - the first bit of the significand is not stored (phantom bit) - first bit of significand equals 1 unless biased exponent equals is represented by 32 zeros; also have -0! There are lots of other details, including denormalized numbers; we will ignore them.

9 General Format The general format is: 9 fraction f e significand excluding the high-order bit # of bits in the fraction # of bits in the biased exponent

10 IEEE 754 Floating Point Examples Consider the base-10 value We saw earlier that this converts to in base-2. This normalizes to x 2^1. Sign bit is 0 since number is nonnegative Stored exponent is = 128 Normalized fraction is 11101, padded with 0s to 23 bits

11 Floating Point Addition Addition: - shift so that the exponents are equal - add the mantissas - normalize the result = in IEEE single format = in IEEE single format = The result is already normalized to the IEEE format.

12 Floating Point Multiplication Multiplication: - multiply the mantissas - add the exponents - normalize the result = in IEEE single format = in IEEE single format = Exponent would be 2, so the product equals: = Again, the result is already normalized to the IEEE format.

13 Floating Point Complexities 13 Operations are somewhat more complicated than integer operations. In addition to overflow we can have underflow, Representable values are "relatively" equally spaced. Accuracy can be a big problem - IEEE 754 keeps two extra bits, guard and round - four rounding modes - positive divided by zero yields infinity - zero divide by zero yields not a number (NaN) - other complexities Not following the standard can be even worse - see text for description of 80x86 and Pentium bug!

14 Unrepresentable Numbers Of course we know that many fractions cannot be represented in a finite number of digits. But things may be worse than we would naturally expect: fractional value carry-over? So, 0.1 in base-2 is:

15 Unrepresentable Results Suppose that X = 1.0 and Y = 1.0 x Both are representable as normalized IEEE singles. But: X + Y = = And that value cannot be stored correctly as an IEEE single; in fact, the result will either be truncated to 1.0 or rounded up when it is stored. Either way, an incorrect value will have been stored. Using IEEE doubles merely changes the scale of the problem

16 Some Floating Point Issues 16 Machine epsilon is defined to be ε = 2 -t, where t is the number of bits in the mantissa. This is the smallest distinguishable relative difference between two numbers that have different floating-point representations. Storage errors are inevitable since finitely-many bits are used in the representation. The most we can expect is that: fl(x) = x(1 + δ), where δ <= ε Round-off errors are also inevitable, and may be magnified in interesting ways. Take Numerical Analysis or Numerical Methods to learn more