InverseErfc¶

Status: Stable

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

Description¶

InverseErfc[s]
    gives the inverse complementary error function: the z solving s = Erfc[z].
InverseErfc[0] = Infinity, InverseErfc[1] = 0, InverseErfc[2] = -Infinity.
Numerical values are given only for real s in [0, 2], at machine or
arbitrary (MPFR) precision; D[InverseErfc[z], z] =
-(Sqrt[Pi]/2) E^(InverseErfc[z]^2). Listable.

Examples¶

No verified examples yet for this function.

Implementation notes¶

Exact special values: InverseErfc[0] = Infinity, InverseErfc[1] = 0,

Attributes: Listable, NumericFunction, Protected.

Implementation status¶

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

References¶

Source: src/info.c
Specification: docs/spec/builtins/special-functions.md

Notes & additional examples¶

Worked examples¶

In[1]:= InverseErfc[1]
Out[1]= 0

The endpoints of the domain [0, 2] are the infinities, with InverseErfc[2] = -Infinity:

In[1]:= InverseErfc[2]
Out[1]= -Infinity

High-precision evaluation, useful for tail quantiles where Erf underflows:

In[1]:= N[InverseErfc[1/1000000], 40]
Out[1]= 3.4589107372795000221509276359575695199155

The derivative is closed-form, D[InverseErfc[z], z] == -(Sqrt[Pi]/2) E^(InverseErfc[z]^2):

In[1]:= D[InverseErfc[z], z]
Out[1]= -1/2 Sqrt[Pi] E^InverseErfc[z]^2

The reflection identity InverseErf[1 - s] == InverseErfc[s] is recognised symbolically:

In[1]:= InverseErf[1 - 3/10] == InverseErfc[3/10]
Out[1]= True

Notes¶

InverseErfc[s] returns the z solving Erfc[z] == s, with InverseErfc[0] = Infinity, InverseErfc[1] = 0, InverseErfc[2] = -Infinity. Numerical values are produced only for real s in [0, 2], at machine or arbitrary (MPFR) precision. It is the natural function for accurate evaluation of extreme normal-distribution quantiles, where 1 - Erf would lose all precision to cancellation.