Skip to content

Round

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

Round[x]

rounds x to the nearest integer, breaking ties to the nearest even integer (banker's rounding).

Round[x, a]

rounds x to the nearest multiple of a.

Notes Round is Listable. Exact inputs return exact integers; Real / MPFR inputs round at the input precision.

Examples (7)

Every input below was run against the current Mathilda build and its output recorded.

Applications (7)

In[1]:= Round[7/2]
Out[1]= 4

In[2]:= Round[5/2]
Out[2]= 2

In[3]:= Round[17, 5]
Out[3]= 15

In[4]:= Round[{1.4, 2.5, 3.6}]
Out[4]= {1, 2, 4}

In[5]:= Round[2.5] + Round[3.5] + Round[4.5]
Out[5]= 10

In[6]:= Round[GoldenRatio, 1/100]
Out[6]= 81/50

In[7]:= Round[N[Pi, 40], 1/10^20]
Out[7]= 157079632679489661923/50000000000000000000

Performance

Against other systems, from the benchmark suite (same input, results cross-checked for agreement):

case Mathilda Wolfram Python
BesselJ[0, .] over 10^6 3.46e+03 s 1.74e+03 s 53 s
Dot 6x6 x 6x6 x 10000 338 s 6.36 s 4.01 s
Eigenvalues 600x600 (general) 270 s 163 s 79.2 s
return {real, int, mask}, then Total 61.2 s 0.344 s 0.983 s
return {real, int}, then Total 42 s 0.219 s 0.41 s
return {real, int, mask}, discarded 41.5 s 0.244 s 0.972 s

Implementation notes

Algorithm. builtin_round is do_piecewise(res, OP_ROUND, "Round", allow_2_args=true), sharing the rounding kernel with Floor/Ceiling/IntegerPart/FractionalPart. Round uses banker's rounding (round-half-to-even) everywhere. For machine double/EXPR_REAL values it calls round_half_even, which floors and inspects the fractional residue, breaking exact .5 ties toward the even integer. For exact rationals it works entirely in GMP: it computes floor((2·num + den) / (2·den)) (round-half-up) and corrects the exact-half tie (rem == 0 and q odd) by decrementing q toward the even neighbour. MPFR values round via mpfr_rint(..., MPFR_RNDN). Complex values round real and imaginary parts independently. An exact non-rational numeric quantity (e.g. 10000000 Pi) is resolved through do_piecewise_numeric_exact, which numericalises at growing precision until the integer result is unambiguous.

The two-argument form Round[x, a] is rewritten as a · Round[x/a] (rounding to the nearest multiple of a). Symbolic arguments return NULL. Attributes include ATTR_LISTABLE | ATTR_NUMERICFUNCTION.

Attributes: Listable, NumericFunction, Protected.

References

Notes & additional examples

Notes

Round breaks ties to the nearest even integer (banker's rounding), so Round[5/2] and Round[2.5] both give 2. Round[x, a] rounds to the nearest multiple of a; Round is Listable.