Floating Point

6 October

• Exams

• Division Example

• Floating Point

Exam

Division Hardware

Division Example

Floating Point

• Many numeric applications require numbers over a VERY large range. (e.g. nanoseconds to centuries)

• Most scientific applications require fractions (e.g. )

• But so far we only have integers.

• We *COULD* implement the fractions explicitly (e.g. ½, 1023/102934)

• We *COULD* use bigger integers

• Floating point is a better answer for most applications.

Floating Point

• Just Scientific Notation for computers (e.g. -1.345*10¹²)

• Representation in IEEE 754 floating point standard

• value = (1-2*sign) * significand * 2^exponent

• more bits for significand gives more precision

• more bits for exponent increases range

Normalization

• In Scientific Notation: 1234000 = 1234 x 10³ = 1.234 x 10⁶

• Likewise in Floating Point: 1011000 = 1011 x 2³ = 1.011 x 2⁶

• The standard says we should always keep them normalized so that the first digit is 1 and the binary point comes immediately after.

• But wait! There’s more! If we know the first bit is 1 why store it?

IEEE 754 floating-point standard

• Leading “1” bit of significand is implicit

• We want both positive and negative exponents for big and small numbers.

• Exponent is “biased” to make comparison easier

• all 0s is smallest exponent all 1s is largest

• bias of 127 for single precision and 1023 for double precision

• summary: (1-2*sign) * (1+significand) * 2^{exponent – bias}

Example

• decimal: -.75 = -3/4 = -3/2²

• binary: -.11 = -1.1 x 2^-1

• floating point: exponent = 126 = 01111110

• IEEE single precision: 10111111010000000000000000000000

• What about zero?

Arithmetic in Floating Point

• In Scientific Notation we learned that to add to numbers you must first get a common exponent:

  • 1.23 x 10³ + 4.56 x 10⁶ ==

  • 0.00123 x 10⁶ + 4.56 x 10⁶ ==

  • 4.56123 x 10⁶

• In Scientific Notation, we can multiply numbers by multiplying the significands and adding the exponents

  • 1.23 x 10³ x 4.56 x 10⁶ ==

  • (1.23 x 4.56) x 10³⁺⁶ ==

  • 5.609 x 10⁹

• We use exactly these same rules in Floating point PLUS we add a step at the end to keep the result normalized.

Floating point AIN’T NATURAL

It is CRUCIAL for computer scientists to know that Floating Point arithmetic is NOT the arithmetic you learned since childhood

• 1.0 is NOT EQUAL to 10*0.1 (Why?)

• 1.0 * 10.0 == 10.0

• 0.1 * 10.0 != 1.0

  • 0.1 decimal == 1/16 + 1/32 + 1/256 + 1/512 + 1/4096 + … ==

  • 0.0 0011 0011 0011 0011 0011 …

  • In decimal 1/3 is a repeating fraction 0.333333…

  • If you quit at some fixed number of digits, then 3 * 1/3 != 1

• Floating Point arithmetic IS NOT associative

  • x + (y + z) is not necessarily equal to (x + y) + z

• Addition may not even result in a change

  • (x + 1) MAY == x

Floating Point Complexities

• In addition to overflow we can have “underflow”

• Accuracy can be a big problem

• IEEE 754 keeps two extra bits, guard and round

• four rounding modes

• Non-zero divided by zero yields “infinity” INF

• Zero divided by zero yields “not a number” NAN

• Implementing the standard can be tricky

• Not using the standard can be worse

MIPS floating point

• Floating point “Co-processor”

• 32 Floating point registers separate from 32 general purpose registers

• 32 bits wide each.

• use an even-odd pair for double precision

  • add.d fd, fs, ft # fd = fs + ft in double precision

  • add.s fd, fs, ft # fd = fs + ft in single precision

  • sub.d, sub.s, mul.d, mul.s, div.d, div.s, abs.d, abs.s

  • l.d fd, address # load a double from address

  • l.s, s.d, s.s

• Conversion instructions

• Compare instructions

• Branch (bc1t, bc1f)

Chapter 3 Summary

• Computer arithmetic is constrained by limited precision

• Bit patterns have no inherent meaning but standards do exist

  • two’s complement

  • IEEE 754 floating point

• Computer instructions determine “meaning” of the bit patterns

• Performance and accuracy are important so there are many complexities in real machines (i.e., algorithms and implementation).

• Accurate numerical computing requires methods quite different from those of the math you learned in grade school.

Cultural Highlight

Go to the State Fair!