# decimal --- Decimal fixed-point and floating-point arithmetic — Mitigating round-off error with increased precision

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

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^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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

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

# Examples from Seminumerical Algorithms, Section 4.2.2. &gt;&gt;&gt; from decimal import Decimal, getcontext &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; (uv) + (uw) Decimal('0.01') &gt;&gt;&gt; u (v+w) Decimal('0.0060000')

The !decimal module makes it possible to restore the identities by expanding the precision sufficiently to avoid loss of significance

&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; (uv) + (uw) Decimal('0.0060000') &gt;&gt;&gt; u (v+w) Decimal('0.0060000')

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