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<H3><A NAME="SECTION007651000000000000000">
5.6.5.1 Mitigating round-off error with increased precision</A>
</H3>

<P>
The use of decimal floating point eliminates decimal representation error
(making it possible to represent <tt class="constant">0.1</tt> exactly); however, some
operations can still incur round-off error when non-zero digits exceed the
fixed precision.

<P>
The effects of round-off error can be amplified by the addition or subtraction
of nearly offsetting quantities resulting in loss of significance.  Knuth
provides two instructive examples where rounded floating point arithmetic with
insufficient precision causes the breakdown of the associative and
distributive properties of addition:

<P>
<div class="verbatim"><pre>
# Examples from Seminumerical Algorithms, Section 4.2.2.
&gt;&gt;&gt; from decimal import *
&gt;&gt;&gt; getcontext().prec = 8

&gt;&gt;&gt; u, v, w = Decimal(11111113), Decimal(-11111111), Decimal('7.51111111')
&gt;&gt;&gt; (u + v) + w
Decimal("9.5111111")
&gt;&gt;&gt; u + (v + w)
Decimal("10")

&gt;&gt;&gt; u, v, w = Decimal(20000), Decimal(-6), Decimal('6.0000003')
&gt;&gt;&gt; (u*v) + (u*w)
Decimal("0.01")
&gt;&gt;&gt; u * (v+w)
Decimal("0.0060000")
</pre></div>

<P>
The <tt class="module">decimal</tt> module makes it possible to restore the identities
by expanding the precision sufficiently to avoid loss of significance:

<P>
<div class="verbatim"><pre>
&gt;&gt;&gt; getcontext().prec = 20
&gt;&gt;&gt; u, v, w = Decimal(11111113), Decimal(-11111111), Decimal('7.51111111')
&gt;&gt;&gt; (u + v) + w
Decimal("9.51111111")
&gt;&gt;&gt; u + (v + w)
Decimal("9.51111111")
&gt;&gt;&gt; 
&gt;&gt;&gt; u, v, w = Decimal(20000), Decimal(-6), Decimal('6.0000003')
&gt;&gt;&gt; (u*v) + (u*w)
Decimal("0.0060000")
&gt;&gt;&gt; u * (v+w)
Decimal("0.0060000")
</pre></div>

<P>

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	54
	55	<H3><A NAME="SECTION007651000000000000000">
	56	5.6.5.1 Mitigating round-off error with increased precision</A>
	57	</H3>
	58
	59	<P>
	60	The use of decimal floating point eliminates decimal representation error
	61	(making it possible to represent <tt class="constant">0.1</tt> exactly); however, some
	62	operations can still incur round-off error when non-zero digits exceed the
	63	fixed precision.
	64
65	<P>
66	The effects of round-off error can be amplified by the addition or subtraction
67	of nearly offsetting quantities resulting in loss of significance. Knuth
68	provides two instructive examples where rounded floating point arithmetic with
69	insufficient precision causes the breakdown of the associative and
70	distributive properties of addition:
71
72	<P>
73	<div class="verbatim"><pre>
74	# Examples from Seminumerical Algorithms, Section 4.2.2.
75	>>> from decimal import *
76	>>> getcontext().prec = 8
77
78	>>> u, v, w = Decimal(11111113), Decimal(-11111111), Decimal('7.51111111')
79	>>> (u + v) + w
80	Decimal("9.5111111")
81	>>> u + (v + w)
82	Decimal("10")
83
84	>>> u, v, w = Decimal(20000), Decimal(-6), Decimal('6.0000003')
85	>>> (uv) + (uw)
86	Decimal("0.01")
87	>>> u * (v+w)
88	Decimal("0.0060000")
89	</pre></div>
90
91	<P>
92	The <tt class="module">decimal</tt> module makes it possible to restore the identities
93	by expanding the precision sufficiently to avoid loss of significance:
94
95	<P>
96	<div class="verbatim"><pre>
97	>>> getcontext().prec = 20
98	>>> u, v, w = Decimal(11111113), Decimal(-11111111), Decimal('7.51111111')
99	>>> (u + v) + w
100	Decimal("9.51111111")
101	>>> u + (v + w)
102	Decimal("9.51111111")
103	>>>
104	>>> u, v, w = Decimal(20000), Decimal(-6), Decimal('6.0000003')
105	>>> (uv) + (uw)
106	Decimal("0.0060000")
107	>>> u * (v+w)
108	Decimal("0.0060000")
109	</pre></div>
110
111	<P>
112
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