$MachineEpsilon¶

Status: Stable

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

Description¶

$MachineEpsilon

gives the difference between 1.0 and the next-nearest number representable as a machine-precision number.

Notes

Equals the platform's DBL\_EPSILON; measures the granularity of machine-precision numbers.

Examples (6)¶

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

Basic examples (4)¶

In[1]:= $MachinePrecision
Out[1]= 15.9546

In[2]:= $MachineEpsilon
Out[2]= 2.22045e-16

In[3]:= {$MinMachineNumber, $MaxMachineNumber}
Out[3]= {2.22507e-308, 1.79769e+308}

MPFR, not machine

In[4]:= MachineNumberQ[$MaxNumber]
Out[4]= False

Applications (2)¶

The smallest e for which 1 + e differs from 1 in machine arithmetic

In[5]:= $MachineEpsilon
Out[5]= 2.22045e-16

Exactly two to the minus fifty-two for an IEEE 754 double

In[6]:= $MachineEpsilon == 2^-52
Out[6]= True

Implementation notes¶

A read-only system constant bound as an OwnValue in system_constants_init (src/core.c) via register_system_constant, then marked ATTR_PROTECTED. Its value is expr_new_real(DBL_EPSILON) — the <float.h> machine epsilon of the local IEEE 754 double.

Attributes: Protected.

References¶

Source: src/core.c
Specification: docs/spec/builtins/expression-information.md

Notes & additional examples¶

Notes¶

$MachineEpsilon is a read-only system constant: the <float.h> DBL_EPSILON of the local IEEE 754 double, bound as a Protected OwnValue. It is the gap between 1.0 and the next representable double, 2^-52 ≈ 2.22045×10⁻¹⁶, and is the natural unit for machine-precision round-off tolerances.

The value is a fixed machine constant on every IEEE 754 platform, so it is safe to quote literally.