—   argument ($x\in\mathbb{R}$)

For real $x$, $\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}\,\mathrm{d}t$, $\operatorname{erfc}(x)=1-\operatorname{erf}(x)$, and $\operatorname{erfi}(x)=-i\operatorname{erf}(ix)$ [4].

The functions $\operatorname{erf}$, $\operatorname{erfi}$, $F$, $S$ and $C$ are odd, and $\operatorname{erfc}(-x)=2-\operatorname{erfc}(x)$.

The Fresnel integrals $S$ and $C$, with $S(x)=\int_0^x\sin(\pi t^2/2)\,\mathrm{d}t$ and $C(x)=\int_0^x\cos(\pi t^2/2)\,\mathrm{d}t$, satisfy $C(x)+iS(x)=\frac{1+i}{2}\operatorname{erf}\!\left(\frac{\sqrt{\pi}}{2}(1-i)x\right)$ [3].

$\int_0^1e^{-nt^2}\,dt=\frac{\sqrt\pi}{2\sqrt n}\operatorname{erf}(\sqrt n)$, $n>0$.

The probability integral $\alpha(x)$ is $\operatorname{erf}(x/\sqrt{2})$ [6]. The standard-normal tail $Q(x)$ is $\tfrac12\operatorname{erfc}(x/\sqrt2)$ [1]. The curve $t\mapsto(C(t),S(t))$ is Cornu's spiral, also called the Euler spiral or clothoid, for this Fresnel normalisation [7] [3].

Negative arguments are not listed separately. Values at negative arguments follow from (2).

At $x=0$ the value is exact, $\operatorname{erf}(0)=0$, and is stored as the integer rather than as a hundred places of one.

For a standard normal random variable $Z$, $\Pr(|Z|\leq n)=\operatorname{erf}(n/\sqrt2)$ and $\Pr(|Z|>n)=\operatorname{erfc}(n/\sqrt2)$ [8]. Integer standard-deviation thresholds select these arguments. They are conventional probability references, not arbitrary probability quantiles or mathematically unique cutoffs.

The simple inputs $x=-1,-1/2,1/2,1,3/2,2,3,4$ supplement the conventional standard-deviation references. They are a bounded exploratory Gaussian-integral reference, including negative-side reflection examples and moderate positive tails, not quantile cutoffs [9].

Integer quadratic coefficients in the truncated Gaussian integral $\int_0^1e^{-nt^2}\,dt$ select $x=\sqrt n$. The exploratory range $1\leq n\leq6$ is finite and noncanonical; coincident arguments such as $\sqrt2=2/\sqrt2$ are represented only once [9].

The current generate.py reproduces the complete attached selection, including the retained entries. Their original written values and entry comments are preserved rather than reassigned to a new source. The selection manifest records the predecessor revision and source hashes; earlier generators and checks remain historical provenance, not the enumeration of this extension.

Python

from generate import generator
from numberdb._write import to_text

for params, result in generator.all_entries():
    print(params["x"], to_text(result["number"], generator.digits_for(params)))

Wikipedia: Error function

Wikipedia: Dawson function

Wikipedia: Fresnel integral

DLMF 7.2: Definitions

OEIS A099286: decimal expansion of the error function at 1

MathWorld: Probability Integral

DLMF 7.3: Graphics

DLMF: definitions and natural argument families

DLMF 7.2(i): error function and complement

Values of the complementary error function $\operatorname{erfc}(x)$ —   $\operatorname{erfc}=1-\operatorname{erf}$

Values of the imaginary error function $\operatorname{erfi}(x)$ —   $\operatorname{erfi}(x)=-i\operatorname{erf}(ix)$

Values of Dawson's integral $F(x)$ —   the same Gaussian integrand, with $e^{+t^2}$

Values of the Fresnel sine integral $S(x)$ —   $C(x)+iS(x)$ is an error function of a complex argument (3)

Values of the Fresnel cosine integral $C(x)$ —   $C(x)+iS(x)$ is an error function of a complex argument (3)

Values of the exponential integral —   stores $\operatorname{Ei}(x)$ and $E_1(x)$, with exponential kernels $e^t/t$ and $e^{-t}/t$ rather than the Gaussian kernel $e^{-t^2}$

Values of the sine integral $\operatorname{Si}(x)$ —   stores $\operatorname{Si}(x)=\int_0^x\sin(t)/t\,\mathrm{d}t$, with linear phase in place of the Fresnel quadratic phase

Values of the cosine integral $\operatorname{Ci}(x)$ —   stores $\operatorname{Ci}(x)=\gamma+\log x+\int_0^x(\cos(t)-1)/t\,\mathrm{d}t$, with linear phase in place of the Fresnel quadratic phase

Values of the hyperbolic sine integral $\operatorname{Shi}(x)$ —   stores $\operatorname{Shi}(x)=\int_0^x\sinh(t)/t\,\mathrm{d}t$ at rational arguments

Values of the hyperbolic cosine integral $\operatorname{Chi}(x)$ —   stores $\operatorname{Chi}(x)=\gamma+\log x+\int_0^x(\cosh(t)-1)/t\,\mathrm{d}t$ at rational arguments

Values of the Gamma function at rational numbers —   contains $\Gamma(1/2)=\sqrt{\pi}$, the normalising constant in $\operatorname{erf}$ and $\operatorname{erfc}$

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds the origin and the one- through six-standard-deviation references $x=n/\sqrt2$, $1\leq n\leq6$, under (8); the selection is not exhaustive over the real domain. In addition, it includes the simple inputs $-1,-1/2,1/2,1,3/2,2,3,4$ and $\sqrt n$ for $1\leq n\leq6$, counting equivalent arguments only once. The exploratory families are identification references with practical finite bounds, not claims of documented applications for every input. All prior entries are retained.)

How they were obtained:

All retained values and metadata are preserved, including their precision. New nonexact values have 100 significant digits. balanced_values.py reproduces the complete selection in balanced-selection.json with Arb balls, including uncertainty in exact irrational arguments and certified roots. A 640-bit guard is supplemented by measured accuracy checks and precision escalation.

more

Every written interval contains a freshly recomputed certified enclosure; every value is also checked with independent mpmath computations at 50 and 100 decimal guard digits. Exact origin anchors stay exact. Structural identities, root endpoint signs where applicable, and selected transformed definite integrals supplement these checks. See balanced-checks-evidence.json. Earlier checks-evidence.json and source attachments describe earlier selections and remain for provenance.