## 5.6.5 Floating Point Notes

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.
>>> from decimal import *
>>> getcontext().prec = 8

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

>>> u, v, w = Decimal(20000), Decimal(-6), Decimal('6.0000003')
>>> (u*v) + (u*w)
Decimal("0.01")
>>> 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:

```>>> getcontext().prec = 20
>>> u, v, w = Decimal(11111113), Decimal(-11111111), Decimal('7.51111111')
>>> (u + v) + w
Decimal("9.51111111")
>>> u + (v + w)
Decimal("9.51111111")
>>>
>>> u, v, w = Decimal(20000), Decimal(-6), Decimal('6.0000003')
>>> (u*v) + (u*w)
Decimal("0.0060000")
>>> u * (v+w)
Decimal("0.0060000")
```

The number system for the decimal module provides special values including NaN, sNaN, -Infinity, Infinity, and two zeroes, +0 and -0.

Infinities can be constructed directly with: `Decimal('Infinity')`. Also, they can arise from dividing by zero when the DivisionByZero signal is not trapped. Likewise, when the Overflow signal is not trapped, infinity can result from rounding beyond the limits of the largest representable number.

The infinities are signed (affine) and can be used in arithmetic operations where they get treated as very large, indeterminate numbers. For instance, adding a constant to infinity gives another infinite result.

Some operations are indeterminate and return NaN, or if the InvalidOperation signal is trapped, raise an exception. For example, `0/0` returns NaN which means ``not a number''. This variety of NaN is quiet and, once created, will flow through other computations always resulting in another NaN. This behavior can be useful for a series of computations that occasionally have missing inputs -- it allows the calculation to proceed while flagging specific results as invalid.

A variant is sNaN which signals rather than remaining quiet after every operation. This is a useful return value when an invalid result needs to interrupt a calculation for special handling.

The signed zeros can result from calculations that underflow. They keep the sign that would have resulted if the calculation had been carried out to greater precision. Since their magnitude is zero, both positive and negative zeros are treated as equal and their sign is informational.

In addition to the two signed zeros which are distinct yet equal, there are various representations of zero with differing precisions yet equivalent in value. This takes a bit of getting used to. For an eye accustomed to normalized floating point representations, it is not immediately obvious that the following calculation returns a value equal to zero:

```>>> 1 / Decimal('Infinity')
Decimal("0E-1000000026")
```