Built-in Mathematica Symbol

Precision

Precision[x]
gives the effective number of digits of precision in the number x.

MORE INFORMATION

Precision[x] gives a measure of the relative uncertainty in the value of x.

With absolute uncertainty dx, Precision[x] is -Log[10, dx/x].

For exact numbers such as integers, Precision[x] is Infinity.

Precision[x] does not normally yield an integer result.

For any approximate number x, Precision[x] is equal to RealExponent[x]+Accuracy[x].

For machine-precision numbers Precision[x] yields MachinePrecision.

Numbers entered in the form digits`p are taken to have precision p.

Numbers such as 0``a whose overall scale cannot be determined are treated as having zero precision.

Numbers with zero precision are output in StandardForm as 0.10^-a, where a is their accuracy.

If x is not a number, Precision[x] gives the minimum value of Precision for all the numbers that appear in x. MachinePrecision is considered smaller than any explicit precision.

EXAMPLES

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Basic Examples (3)

Machine-precision number:

Arbitrary-precision number:

Exact number:

Machine-precision number:

In[1]:=

Out[1]=

Arbitrary-precision number:

Exact number:

Scope (2)

A zero known to accuracy 20:

The precision is 0.:

The precision of

is the same as the accuracy of

:

N attempts to get a result correct to the given precision:

This cannot always be achieved:

This is because relative error cannot be measured at zero and

:

Generalizations & Extensions (1)

The precision of a symbolic expression is the minimum of the precisions of its numbers:

Applications (2)

Check the quality of a result:

Track precision loss in a repetitive calculation:

Properties & Relations (2)

All machine numbers have the same precision, MachinePrecision:

This is 53 bits or about 16 digits:

For nonmachine numbers, Precision[x]

RealExponent[x] + Accuracy[x]:

Possible Issues (3)

MachinePrecision is always considered effectively smaller than any explicit precision:

Numbers with sufficiently low precision are displayed with zero mantissa:

Since Precision is based on relative error, it is not measurable for zero:

You can measure the absolute size of the error with Accuracy:

If you expect the result to be near zero, you can specify accuracy as a goal for N:

Neat Examples (1)

Precision and Accuracy in iterating the tent map: