InverseErfc

The inverse complementary error function: solves $\operatorname{erfc}(y) = s$ for $0 < s < 2$ . Not simply ErfInv(1 - s) — that loses precision exactly in the tail ( $s \to 0$ ) InverseErfc exists for.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageInverseErfc!

InverseErfc(s)solves

\operatorname{erfc}(y) = s

for

y

.

SymPyerfcinv(s)Rusterfc_inv(s)

Typed signatures

Type	Package	Applies
`(number) -> number`	`analytic`	always

Domain: Special functions

Details

Built on compute-engine's native ErfInv, but $1 - s$ is computed as a compute-engine expression (exact rational/bignum arithmetic) rather than a plain double, so the final N() sees a fully precise argument even at $s = 10^{-10}$ — where ErfInv of a literal double $1-s$ would already have dropped several digits.
For $s > 1$ , uses $\operatorname{erfc}(-y) = 2 - \operatorname{erfc}(y)$ to reduce to the $s \le 1$ case: $\operatorname{InverseErfc}(s) = -\operatorname{InverseErfc}(2-s)$ .
$\operatorname{Erfc}(\operatorname{InverseErfc}(x)) = x$ holds exactly, symbolic $x$ included — compute-engine has no general inverse-composition simplification (checked: ErfInv(Erf(x)) stays unevaluated too), so Erfc is extended in place with this one structural cancellation.

Plot

Examples

2 more kept as data, checked against the oracles

Basic #

$\operatorname{erfc}^{-1}(1/2) = \operatorname{erf}^{-1}(1/2)$

In

\mathrm{InverseErfc}(0.5)

Out

0.476\,936\,276\,204\,469\,9

Known value · Wolfram 15.0.0 InverseErfc

$\operatorname{erfc}^{-1}(1) = 0$

In

\mathrm{InverseErfc}(1)

Out

0

Scope #

$\operatorname{erfc}^{-1}(0) = \infty$

In

\mathrm{InverseErfc}(0)

Out

∞

$\operatorname{erfc}^{-1}(2) = -\infty$

In

\mathrm{InverseErfc}(2)

Out

-∞

Arguments in $(1, 2)$ give negative values

In

\mathrm{InverseErfc}(1.5)

Out

-0.476\,936\,276\,204\,469\,9

Listable

In

\mathrm{InverseErfc}(\bigl\lbrack0, 1\bigr\rbrack)

Out

\bigl\lbrack∞, 0\bigr\rbrack

A 50-digit value of $\mathrm{erfc}^{-1}(0.33)$

In

N(\mathrm{InverseErfc}(\frac{33}{100}), 50)

Out

0.688\,802\,528\,116\,556\,450\,402\,504\,728\,905\,257\,835\,449\,489\,923\,493\,71

In

\mathrm{FunctionExpand}(\mathrm{MatrixFunction}(\mathrm{InverseErfc}, \bigl\lbrack\bigl\lbrack0, 1/2\bigr\rbrack, \bigl\lbrack1, 0\bigr\rbrack\bigr\rbrack))

Out

\bigl\lbrack\bigl\lbrack\frac{1}{2}\mathrm{InverseErfc}(-2^{1/2}/2)+\frac{\mathrm{InverseErfc}(2^{1/2}/2)}{2}, \frac{1}{4}\sqrt{2}\mathrm{InverseErfc}(2^{1/2}/2)-\frac{1}{4}\sqrt{2}\mathrm{InverseErfc}(-2^{1/2}/2)\bigr\rbrack, \bigl\lbrack\frac{1}{2}\sqrt{2}\mathrm{InverseErfc}(2^{1/2}/2)-\frac{1}{2}\sqrt{2}\mathrm{InverseErfc}(-2^{1/2}/2), \frac{1}{2}\mathrm{InverseErfc}(-2^{1/2}/2)+\frac{\mathrm{InverseErfc}(2^{1/2}/2)}{2}\bigr\rbrack\bigr\rbrack

In

\operatorname{Series}(\mathrm{InverseErfc}(x), x, 1, 8)

Out

(x-1)\mathrm{InverseErfc}^{\prime}(1)+\frac{1}{2}\mathrm{InverseErfc}^{\prime\prime}(1)(x-1)^2+\frac{1}{6}\mathrm{InverseErfc}^{\prime\prime\prime}(1)(x-1)^3+\frac{1}{24}\mathrm{InverseErfc}^{(4)}(1)(x-1)^4+\frac{1}{120}\mathrm{InverseErfc}^{(5)}(1)(x-1)^5+\frac{1}{720}\mathrm{InverseErfc}^{(6)}(1)(x-1)^6+\frac{1}{5\,040}\mathrm{InverseErfc}^{(7)}(1)(x-1)^7+\frac{1}{40\,320}\mathrm{InverseErfc}^{(8)}(1)(x-1)^8+O\left((x-1)^9\right)

Applications #

$\sqrt2$ times this, $2.5758\ldots$ , is the two-sided 99% normal critical value

In

\mathrm{InverseErfc}(0.01)

Out

1.821\,386\,367\,718\,449\,6

Known value · Wolfram 15.0.0 InverseErfc

Properties #

$\operatorname{erfc}(\operatorname{erfc}^{-1}(x)) = x$

In

\operatorname{erfc}(\mathrm{InverseErfc}(x))

Out

x

erfc after its inverse cancels, but not the other way

In

\bigl\lbrack\operatorname{erfc}(\mathrm{InverseErfc}(z)), \mathrm{InverseErfc}(\operatorname{erfc}(z))\bigr\rbrack

Out

\bigl\lbrack z, \mathrm{InverseErfc}(\operatorname{erfc}(z))\bigr\rbrack

The inverse is numeric at an exact rational

In

\mathrm{IsNumeric}(\mathrm{InverseErfc}(\frac{1}{2}))

Out

\mathrm{True}

In

\mathrm{Solve}(\mathrm{InverseErfc}(x)^2+\mathrm{InverseErfc}(x)=1, x)

Out

\bigl\lbrack\operatorname{erfc}(-5^{1/2}/2-\frac{1}{2}), \operatorname{erfc}(\frac{5^{1/2}}{2}-\frac{1}{2})\bigr\rbrack

In

\mathrm{Attributes}(\mathrm{InverseErfc})

Out

\bigl\lbrack\mathrm{Listable}, \mathrm{Protected}\bigr\rbrack

Possible issues #

Deep in the tail it keeps full precision, where $\operatorname{erf}^{-1}(1 - s)$ would lose it

In

\mathrm{InverseErfc}(1.0\cdot10^{-10})

Out

4.572\,824\,967\,389\,485

About

Examples	18
Checked against	Wolfram Language 17 agree1 differSymPy 7 agree7 unevaluated1 errorRust 8 agree3 error
Crosswalk	Wolfram LanguageSymPyRust

Implementation

primitive · kernela kernel — an algorithm over mutable state that would not be readable as a tree

nativeenginetypescriptpackages/symbols/analysis/analytic/src/inverse-erfc.ts

mappedrustpackages/oracle/src/mappings.ts

mappedsympypackages/oracle/src/mappings.ts

See also: Erfc