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Precision

Precision[x]
gives the effective number of digits of precision in the number x.
  • 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].
  • Precision[x] does not normally yield an integer result.
  • Numbers entered in the form 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 , 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.
Machine-precision number:
Arbitrary-precision number:
Exact number:
Machine-precision number:
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Arbitrary-precision number:
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Exact number:
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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 :
The precision of a symbolic expression is the minimum of the precisions of its numbers:
Check the quality of a result:
Track precision loss in a repetitive calculation:
All machine numbers have the same precision, MachinePrecision:
This is 53 bits or about 16 digits:
For non-machine numbers, Precision[x] == RealExponent[x] + Accuracy[x]:
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:
Precision and Accuracy in iterating the tent map:
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