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12.1 Rounding Modes
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Rounding: the process to obtain the best possible
floating-point representation for a given real value.
ANSI/IEEE standard: round to floating number whose
significand has an LSB of 0 (of two adjacent floatingpoint number, the significand of one must end in 0,
and the other one in 1). This is called round-to-neareven.
For example, 3.5 and 4.5 are both rounded to 4, the
closet even number, based on round-to-near-even.
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• Other rounding methods
– Round inward (toward 0):choose the nearest value
in the same direction as 0.
– Round upward (toward +∞): choose the larger of
the two possible values.
– Round downward (toward -∞): choose the smaller
of the two possible vavlues.
•
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Example 12.1 Rounding to the nearest integer
a. Consider the rounded even integer
corresponding to a real signed-magnitude
number x a rtnei(x). Plot this round-tonearest-even-integer for x in the range [-4,4].
b. Repeat part a for the function rtni(x), that is,
round-to-nearest-integer function, where the
midway values are always rounded up
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Example 12.2 Directed rounding
a. Consider the inward-directed round
corresponding to a real signed-magnitude
number x as a function ritni(x). Plot this
round-inward-to-nearest-integer function for
x in the range [-4,4].
b. Repeat part a for the round-upward-tonearest-integer rutni(x).
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Figure 12.3 Two directed round-to-nearest-integer functions for x in [– 4, 4].
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Figure 12.3 (Continued)
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12.2 Special Values and Execeptions
• Five special values in ANSI/IEEE floating-point
standard
– ±0
Biased exponent=0, significand=0 (no
hidden 1)
–±∞
Biased exponent=255 (short), or 2047
(long), significand=0
– NaN Biased exponent=255 (short), or 2047
(long), significand≠0
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12.3 Floating-Point Addition
Consider the addition of ±2e1s1 and ±2e2s2,
where e1 > e2
(±2e1s1) +(±2e2s2)=±2e1(s1±s2/2e1-e2)
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Figure 12.6 Simplified schematic of a floating-point adder
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12.4 Other Floating-point Operations
Multiplication of ±2e1s1 and ±2e2s2
(±2e1s1)×(±2e2s2)=±2e1+e2(s1×s2/2e1-e2)
Division of ±2e1s1 and ±2e2s2
(±2e1s1)/(±2e2s2)=±2e1-e2(s1/s2)
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Figure 12.6 Simplified schematic of a floating-point multiply/divide unit.
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12.5 Floating-Point Instructions
10 floating-point arithmetic instructions (5 different operations:
add, sub, multiply, divide, negate)
add.s $f0,$f8,$f10
# set $f0 to ($f8)+($f10)
add.d $f0,$f8,$f10
# set $f0 $f1 to ($f8$f9)+($f10$f11)
Single operands can be in any of the floating registers. Double
operands must be in specified to be in even numbered registers
Figure 12.7 The common floating-point instruction format for MiniMIPS and components for arithmetic instructions. The extension (ex) field
distinguishes single (* = s) from double (* = d) operands.
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6 format conversion instructions: integer to single/double,
single to double, double to single, and single/double to integer
cvt.s.w $f0,$f8 # set $f0 to single (integer $f8)
cvt.d.w $f0,$f8 # set $f0 to double (integer $f8)
cvt.d.s $f0,$f8 # set $f0 to double ($f8)
cvt.s.d $f0,$f8 # set $f0 to single ( $f8, $f9,)
cvt.w.s $f0,$f8 # set $f0 to integer ($f8)
cvt.w.d $f0,$f8 # set $f0 to integer ($f8, $f9)
Figure 12.8 Floating-point instructions for format conversion in MiniMIPS.
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6 data transfer instructions: load/store word to/from coprocessor1, move
single/double from one FP register to another, move (copy) between FP
registers and CPU general registers.
lwcl $f8, 40($3) # load mem[40+($s3)] into $f8
swc1 $f8, A($3) # store mem[A+($s3)] into $f8
mv.s $f0,$f8 # load $f0 with ($f8)
mv.d $f0,$f8 # load $f0,$f1 with ( $f8, $f9,)
mfc1 $t0,$f12 # load $t0 with ($f12)
mtc1 $f8,$t4 # load $f8 with ($t4)
Figure 12.9 Instructions for floating-point data movement in MiniMIPS.
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2 branch and 6 comparison instructions. The FP unit has a flag that is set to
T or F based on 6 comparisons (equal, less than, or less or equal for
single/double data type)
bc1t
L
# branch on FP flag true
bc1f
L
# branch on FP flag false
c.eq.* $f0, $f8 # if ($f0)=($f8), set flag to true
c.lt.* $f0, $f8
# if ($f0)<($f8), set flag to true
c.lw.* $f0, $f8 # if ($f0)≤($f8), set flag to true
Figure 12.10 Floating-point branch and comparison instructions in MiniMIPS.
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Table 12.1 The 30 MiniMIPS floating-point instructions:because the op field contains 17 for all but two of the instructions (49 for lwc1 and 50 for
swc1), it is not shown.
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12.6 Result Precision and Errors
• FP arithmetic can be quite dangerous and must be used with
proper care, because results of FP computations are inexact.
• Why?
– Many real numbers do not have exact binary representation within a
finite word format. This is referred as representation error.
– Even for values that are exactly representable, FP arithmetic produces
inexact results. For example, product of 2 short FP numbers will have a
48 bits significant that must be rounded to 23 bits (plus hidden 1) This
is called computation error.
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Example 12. 4
Associate law of addition does not hold in
general in FP arithmetic. For example
a= -25×(1.10101011)
b=25 × (1.10101110)
c=-2-2 × (1.01100101)
(a+b)+c = a+(b+c) ?
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Figure 12.11 Algebraically equivalent computations may yield different results with floating-point arithmetic.
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• Using guard digits to avoid excessive error.
For example, in a 10-digit calculator, 1/3 is
represented as 0.333 333 333 3, multiplying 3
results in 0.999 999 999 9, but not 1.
However, in a calculator with 2 guard bits, 1/3 is
represented as 0.333 333 333 333, but still
displayed as 0.333 333 333 3, multiplying 3
results in 1.
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Figure 12.12 Function evaluation by table lookup and linear interpolation.
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