ErfInv

The inverse error function: $\operatorname{erfinv}(x)$ solves $\operatorname{erf}(y) = x$ for $-1 < x < 1$.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaError function↗MathWorldInverseErf↗Erf↗WikidataQ579262↗DLMF7.17.E1↗7.2.E1↗Wolfram LanguageInverseErf!↗

ErfInv(x)the inverse error function, solving $\operatorname{erf}(y) = x$.

SageMatherfinv(x)SymPyerfinv(x)mpmatherfinv(x)Rusterf_inv(x)

Typed signatures

Type	Package	Applies
(complex | infinity) -> number	compute-engine	always
(number, number?) -> number	analyticreplaces compute-engine	always

Domain: Special functions

Details

• Inverse of Erf: $\operatorname{erf}(\operatorname{erfinv}(x)) = x$.
• $\operatorname{erfinv}(0) = 0$, and $\operatorname{erfinv}(x) \to \pm\infty$ as $x \to \pm 1$, the limits where Erf itself saturates.
• Computed by Newton's method on Erf itself, refining an initial approximation to full machine precision.
• Intrinsically ill-conditioned near $x = \pm1$: a tiny change in x there produces a large change in the result.
• compute-engine leaves a call outside $[-1, 1]$ unevaluated rather than erroring or returning NaN.

Plot

Examples

2 more kept as data, checked against the oracles

Basic #

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In$\mathrm{ErfInv}(0)$Out$0$

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In$\mathrm{ErfInv}(1)$Out$∞$

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In$\mathrm{ErfInv}(-1)$Out$-∞$

A floating-point argument evaluates directly to a high-precision decimal

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In$\mathrm{ErfInv}(0.5)$Out$0.476\,936\,276\,204\,469\,873\,381$

A machine-precision argument

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In$\mathrm{ErfInv}(0.6)$Out$0.595\,116\,081\,449\,994\,850\,019$

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In$\operatorname{Series}(\mathrm{ErfInv}(x), x, 0, 3)$Out$\frac{x\sqrt{π}}{2}+\frac{1}{24}x^3\sqrt{π}^{3}+O\left(x^5\right)$

Scope #

Threads element-wise over a list, as Wolfram's Listable heads do

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In$\mathrm{ErfInv}(\bigl\lbrack0, 1\bigr\rbrack)$Out$\bigl\lbrack0, ∞\bigr\rbrack$

Two arguments: $\operatorname{erfinv}(z_0, z)$ solves $z = \operatorname{erf}(z_0, x) = \operatorname{erf}(x) - \operatorname{erf}(z_0)$

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In$\mathrm{ErfInv}(0.4, 0.2)$Out$0.631\,775\,903\,055\,006\,3$

Interval arithmetic: the inverse is increasing, so endpoints map to endpoints

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In$\mathrm{ErfInv}(\mathrm{Interval}(0.5, 0.6))$Out$\mathrm{Interval}(0.476\,936\,276\,204\,469\,9, 0.595\,116\,081\,449\,994\,8)$

Uncertainty propagation

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In$\mathrm{ErfInv}(\mathrm{Around}(0.5, 0.01))$Out$\mathrm{Around}(0.476\,936\,276\,204\,469\,9, 0.011\,125\,848\,189\,719\,498)$

To 30 significant digits, every one correctly rounded

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In$N(\mathrm{ErfInv}(\frac{1}{2}), 30)$Out$0.476\,936\,276\,204\,469\,873\,381\,418\,353\,643$

A 50-digit value of the inverse error function at 0.33

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In$N(\mathrm{ErfInv}(\frac{33}{100}), 50)$Out$0.301\,332\,146\,133\,705\,826\,128\,502\,718\,158\,394\,773\,965\,824\,282\,828\,53$

Matrix function of a 2x2 matrix, expanded into entries involving $\mathrm{erf}^{-1}$ of exact algebraic points

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In$\mathrm{FunctionExpand}(\mathrm{MatrixFunction}(\mathrm{ErfInv}, \bigl\lbrack\bigl\lbrack0, 1/2\bigr\rbrack, \bigl\lbrack1, 0\bigr\rbrack\bigr\rbrack))$Out$\bigl\lbrack\bigl\lbrack\frac{1}{2}\mathrm{ErfInv}(-2^{1/2}/2)+\frac{\mathrm{ErfInv}(2^{1/2}/2)}{2}, \frac{1}{4}\sqrt{2}\mathrm{ErfInv}(2^{1/2}/2)-\frac{1}{4}\sqrt{2}\mathrm{ErfInv}(-2^{1/2}/2)\bigr\rbrack, \bigl\lbrack\frac{1}{2}\sqrt{2}\mathrm{ErfInv}(2^{1/2}/2)-\frac{1}{2}\sqrt{2}\mathrm{ErfInv}(-2^{1/2}/2), \frac{1}{2}\mathrm{ErfInv}(-2^{1/2}/2)+\frac{\mathrm{ErfInv}(2^{1/2}/2)}{2}\bigr\rbrack\bigr\rbrack$

The generalized inverse undoes the generalized erf only up to the plain inverse of erf

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In$\mathrm{ErfInv}(a, \operatorname{erf}(a, z))$Out$\mathrm{ErfInv}(\operatorname{erf}(z))$

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In$\mathrm{FunctionExpand}(\operatorname{erf}(\mathrm{ErfInv}(s)))$Out$s$

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In$\mathrm{ErfInv}(s)=\mathrm{ErfInv}(0, s)$Out$\mathrm{True}$

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In$\operatorname{Series}(\mathrm{ErfInv}(x), x, 0, 8)$Out$\frac{x\sqrt{π}}{2}+\frac{1}{24}x^3\sqrt{π}^{3}+\frac{1}{960}×7x^5\sqrt{π}^{5}+\frac{1}{80\,640}×127x^7\sqrt{π}^{7}+O\left(x^9\right)$

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In$\operatorname{Series}(\mathrm{ErfInv}(\mathrm{z0}, s), \mathrm{z0}, 0, 2)$Out$\mathrm{ErfInv}(s)+\mathrm{z0}e^{\mathrm{ErfInv}(s)^2}+\mathrm{ErfInv}(s)\mathrm{z0}^2e^{2\mathrm{ErfInv}(s)^2}+O\left(\mathrm{z0}^3\right)$

Applications #

The normal quantile $\sqrt2\,\operatorname{erfinv}(2p - 1)$ at $p = 0.975$: the 1.96 of a 95% confidence interval

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In$\sqrt{2}\mathrm{ErfInv}(0.95)$Out$1.959\,963\,984\,540\,054\,235\,527\,460\,400\,177\,851\,935\,713\,577\,92$

Properties #

Inverse of Erf: $\operatorname{erf}(\operatorname{erfinv}(x)) = x$. See Erf

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In$\operatorname{erf}(\mathrm{ErfInv}(0.5))=0.5$Out$\mathrm{True}$

Odd: $\operatorname{erfinv}(-x) = -\operatorname{erfinv}(x)$ applied symbolically

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In$\mathrm{ErfInv}(-x)$Out$-\mathrm{ErfInv}(x)$

$\frac{d}{dx}\operatorname{erfinv}(x) = \frac{\sqrt\pi}{2}e^{\operatorname{erfinv}(x)^2}$

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In$\frac{\mathrm{d}\mathrm{ErfInv}(x)}{\mathrm{d}x}$Out$\frac{1}{2}e^{\mathrm{ErfInv}(x)^2}\sqrt{π}$

A two-argument call with a numeric first argument is numeric

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In$\mathrm{IsNumeric}(\mathrm{ErfInv}(\frac{1}{π}, \frac{1}{2}))$Out$\mathrm{True}$

With both limits 0 the generalized inverse is 0

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In$\mathrm{ErfInv}(0, 0)$Out$0$

Known value · Wolfram MathematicalFunctionData, InverseErf:Generalized

The value 1 is the pole of the inverse: infinity

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In$\mathrm{ErfInv}(0, 1)$Out$∞$

Known value · Wolfram MathematicalFunctionData, InverseErf:Generalized

With lower limit 1 and value 0 it reduces to $\mathrm{erf}^{-1}(\mathrm{erf}(1))$

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In$\mathrm{ErfInv}(1, 0)$Out$\mathrm{ErfInv}(\operatorname{erf}(1))$

Known value · Wolfram MathematicalFunctionData, InverseErf:Generalized

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In$\mathrm{Solve}(\mathrm{ErfInv}(x)^3+2\mathrm{ErfInv}(x)=1, x)$Out$\mathrm{Solve}(\mathrm{ErfInv}(x)^3+2\mathrm{ErfInv}(x)=1, x)$

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In$\mathrm{Attributes}(\mathrm{ErfInv})$Out$\bigl\lbrack\mathrm{Listable}, \mathrm{Protected}\bigr\rbrack$

Possible issues #

Outside its real domain $[-1, 1]$, compute-engine leaves the call unevaluated rather than returning NaN or a complex value

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In$\mathrm{ErfInv}(2)$Out$\mathrm{ErfInv}(2)$

Neat examples #

Blows up near the boundary: $\operatorname{erfinv}(0.9999)$ is already past 2.7

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In$\mathrm{ErfInv}(0.999\,9)$Out$2.751\,063\,905\,712\,060\,796\,15$

About

Examples	32
Checked against	Wolfram Language 28 agree3 differ1 unevaluatedSageMath 13 agree3 differ1 unevaluated1 errorSymPy 14 agree1 differ1 unevaluated3 errormpmath 11 agree3 errorRust 8 agree6 error
Crosswalk	WikipediaMathWorld ×2WikidataDLMF ×2Wolfram LanguageSageMathSymPympmathRust

Implementation

mappedsympypackages/oracle/src/mappings.ts

mappedmpmathpackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

mappedrustpackages/oracle/src/mappings.ts

See also: Erf, Erfc