Convert all numbers in an expression to 20-digit precision:

Convert all numbers to machine precision:

Convert from a machine number to an arbitrary-precision number:

Set precision of a complex number:

Convert approximate numbers to exact rational numbers:

The result has trailing zeros once any hidden digits are exposed:

SetPrecision does not affect exact powers:

This allows you to, for example, change the precision of polynomial coefficients:

Inexact powers are modified:

Special rules may apply to data objects:

For an InterpolatingFunction object, SetPrecision changes the appropriate data only:

It still works as an approximate function, but with the new precision:

Find the roundoff error in evaluating an expression with machine numbers:

This dominates the approximation error since the increment is so small:

Find the representation error of a machine number:

The error is small because this is the closest machine number to :

The distance between nearest machine numbers is a power of two:

Raise the precision of coefficients in a differential equation to check error:

Find the solutions computed at machine precision and precision 32:

Plot the two solutions. They have deviated by , indicating significant error:

Override the default accuracy and precision model:

This loses precision more slowly than the default model that treats operations as independent:

Nonetheless, all digits given are correct:

The bit loss per iteration is justified because the map is effectively a shift map:

SetPrecision just sets the precision of numbers while N works adaptively:

Since N works adaptively, the result has the requested precision of 20:

Use SetPrecision:

The precision is less than 20 because of the conditioning of the sine function at 1000:

SetPrecision effectively evaluates Sin with argument 1000 to precision 20:

N will typically not raise the precision of an expression, while SetPrecision will:

For nonzero numbers , SetPrecision[x,p] is equivalent to SetAccuracy[x,p-e]:

is given by RealExponent:

SetPrecision[x,∞] and Rationalize[x,0] give rational approximations for real x:

Rationalize[x,0] gives a rational that is equivalent to x up to the precision of x:

SetPrecision[x,∞] gets a rational directly from the bitwise representation of x:

SetPrecision[x,∞] and RootApproximant[x] give exact approximations for real x:

RootApproximant[x] gives an algebraic number equivalent to x up to the precision of x:

SetPrecision[x,∞] gets a rational directly from the bitwise representation of x:

SetPrecision[z,prec] will make an exact zero for approximate zeros z:

If you want an approximate zero, use SetAccuracy:

The only exception to this is that when prec is MachinePrecision, you get a machine zero: