Arbitrarily Accurate Analytical Approximations for the Error Function

Abstract

A spline-based integral approximation is utilized to define a sequence of approximations to the error function that converge at a significantly faster manner than the default Taylor series. The real case is considered and the approximations can be improved by utilizing the approximation $\mathrm{erf}\left(x\right)\approx 1$ for $\left|x\right|>x_o$ and with $x_o$ optimally chosen. Two generalizations are possible; the first is based on demarcating the integration interval into m equally spaced subintervals. The second, is based on utilizing a larger fixed subinterval, with a known integral, and a smaller subinterval whose integral is to be approximated. Both generalizations lead to significantly improved accuracy. Furthermore, the initial approximations, and those arising from the first generalization, can be utilized as inputs to a custom dynamic system to establish approximations with better convergence properties. Indicative results include those of a fourth-order approximation, based on four subintervals, which leads to a relative error bound of 1.43 × 10⁻⁷ over the interval $\left[0, \infty \right]$. The corresponding sixteenth-order approximation achieves a relative error bound of 2.01 × 10⁻¹⁹. Various approximations that achieve the set relative error bounds of 10⁻⁴, 10⁻⁶, 10⁻¹⁰, and 10⁻¹⁶, over $\left[0, \infty \right]$, are specified. Applications include, first, the definition of functions that are upper and lower bounds, of arbitrary accuracy, for the error function. Second, new series for the error function. Third, new sequences of approximations for $\exp (-x^2)$ that have significantly higher convergence properties than a Taylor series approximation. Fourth, the definition of a complementary demarcation function $e_C(x)$ that satisfies the constraint $e_C^2(x)+{\mathrm{erf}}^2(x)=1$. Fifth, arbitrarily accurate approximations for the power and harmonic distortion for a sinusoidal signal subject to an error function nonlinearity. Sixth, approximate expressions for the linear filtering of a step signal that is modeled by the error function.

1. Introduction

The error function arises in many areas of mathematics, science, and scientific applications including diffusion associated with Brownian motion (Fick’s second law), the heat kernel for the heat equation, e.g., [1], the modeling of magnetization, e.g., [2], the modelling of transitions between two levels, for example, with the modeling of smooth or soft limiters, e.g., [3] and the psychometric function, e.g., [4,5], the modeling of amplifier non-linearities, e.g., [6,7], and the modeling of rubber-like materials and soft tissue, e.g., [8,9]. It is widely used in the modeling of random phenomena as the error function defines the cumulative distribution of the Gaussian probability density function and examples include, the probability of error in signal detection, option pricing via the Black–Scholes formula, etc. Many other applications exist. In general, the error function is associated with a macro description of physical phenomena and the de Moivre–Laplace theorem is illustrative of the link between fundamental outcomes and a higher-level model.

The error function is defined on the complex plane according to

$$\mathrm{erf}(z)=\frac{2}{\sqrt{\pi }}{\displaystyle \underset{\gamma }{\int }{e}^{-{\lambda }^2}\mathrm{d}\lambda }, z\in \mathit{C},$$

(1)

where the path $\gamma$ is between the points zero and $z$ and is arbitrary. Associated functions are the complementary error function, the Faddeyeva function, the Voigt function, and Dawson’s integral, e.g., [10]. The Faddeyeva function and the Voigt function, for example, have applications in spectroscopy, e.g., [11]. The error function can also be defined in terms of the spherical Bessel functions, e.g., Equation (7.6.8) of [10], and the incomplete Gamma function, e.g., Equation (7.11.1) of [10]. Marsaglia [12] provides a brief insight into the history of the error function.

For the real case, which is the case considered in this paper, the error function is defined by the integral

$$\mathrm{erf}(x)=\frac{2}{\sqrt{\pi }}{\displaystyle \underset{0}{\overset{x}{\int }}{e}^{-{\lambda }^2}\mathrm{d}\lambda ,}\hspace{1em} x\in \mathit{R}.$$

(2)

The widely used, and associated, cumulative distribution function for the standard normal distribution is defined according to

$$\Phi (x)=\frac{1}{\sqrt{2\pi }}{\displaystyle \underset{-\infty }{\overset{x}{\int }}\exp \left[\frac{-{\lambda }^2}{2}\right]\mathrm{d}\lambda }=0.5+0.5\mathrm{erf}\left[\frac{x}{\sqrt{2}}\right].$$

(3)

Being defined by an integral, which does not have an explicit analytical form, there is interest in approximations of the error function and, in recent decades, many approximations have been developed. Table 1 details indicative approximations for the real case and their relative errors are shown in Figure 1. Most of the approximations detailed in this table are custom and have a limited relative error bound with bounds in the range of 3.05 × 10⁻⁵ [13] to 7.07 × 10⁻³ [14]. It is preferable to have an approximation form that can be generalized to create approximations that converge to the error function. Examples include the standard Taylor series, the Bürmann series, and the approximation by Abrarov, which are defined in Table 1.

Many of the approximations detailed in Table 1 can be improved upon by approximating the associated residual function, denoted g, via a Padé approximant or a basis set decomposition. Examples of possible approximation forms, and the resulting residual functions, are given in Table 2. One example is that of a 4/2 Padé approximant for the function $g_3$, which leads to the approximation:

$$\mathrm{erf}(x)\approx \sqrt{1-\exp \left[-x^2\cdot \frac{4}{\pi }\left[1+\frac{n_1{x}_1+n_2{x}_1^2+n_3{x}_1^3+n_4{x}_1^4}{1+d_1{x}_1+d_2{x}_1^2}\right]\right]}, x_1=\frac{x}{x+1},x\ge 0,$$

(4)

$$\begin{array}{l}{n}_1=\frac{279}{\mathrm{10,000,000}},n_2=\frac{-\mathrm{303,923}}{\mathrm{10,000,000}},n_3=\frac{\mathrm{34,783}}{\mathrm{5,000,000}},n_4=\frac{\mathrm{40,793}}{\mathrm{10,000,000}}\\ d_1=\frac{-\mathrm{21,941,279}}{\mathrm{10,000,000}},d_2=\frac{\mathrm{3,329,407}}{\mathrm{2,500,000}}.\end{array}$$

(5)

The relative error bound in this approximation is 4.02 × 10⁻⁷. Higher-order Padé approximants can be used to generate approximations with a lower relative error bound. Matic [15] provides a similar approximation with an absolute error of 5.79 × 10⁻⁶.

An approximation of the error function can also be obtained by combining separate approximations, which are accurate, respectively, for $\left|x\right|\ll 1$ and $\left|x\right|\gg 1$, via a demarcation function d

$$\mathrm{erf}(x)=\frac{2x}{\sqrt{\pi }}d(x)+\left[1-\frac{e^{-x^2}}{\sqrt{\pi }x}\right]\cdot [1-d(x)], \hspace{1em} x\ge 0,$$

(6)

where

$$d(x)=\frac{\sqrt{\pi }x\mathrm{erf}(x)-\sqrt{\pi }x+e^{-x^2}}{2x^2-\sqrt{\pi }x+e^{-x^2}}.$$

(7)

Naturally, an approximation of d requires an approximation of the error function. Unsurprisingly, the relative error in the approximation for the error function equals the relative error in the approximation utilized to approximate the error function in d.

Finally, efficient numerical implementation of the error function is of interest and Chevillard [16] and De Schrijver [17] provide results and an overview. Highly accurate piecewise approximations have long since been defined, e.g., [18].

The two-point spline-based approximations for functions and integrals, detailed in [19], have recently been applied to find arbitrarily accurate approximations for the hyperbolic tangent function [20]. In this paper, the general two-point spline approximation form is applied to define a sequence of convergent approximations for the error function. The basic form of the approximation of order n, denoted $f_n$, is

$$\mathrm{erf}(x)\approx f_n(x)=p_{n,0}(x)+p_{n,1}(x)e^{-x^2},$$

(8)

where $p_{n,1}$ is a polynomial of order n and $p_{n,0}$ is a polynomial of order less than n. Convergence of the sequence of approximations $f_1,f_2,\dots$ to $\mathrm{erf}(x)$ is shown and the convergence is significantly better than the default Taylor series. For example, the second-order approximation

$$f_2(x)=\frac{x}{\sqrt{\pi }}\cdot \left[1-\frac{x^2}{30}\right]+\frac{x}{\sqrt{\pi }}\cdot \left[1+\frac{11x^2}{30}+\frac{x^4}{15}\right]e^{-x^2}$$

(9)

yields a relative error bound of 0.056 over the interval [0, 2] which is better than a fifteenth-order Taylor series approximation. The approximations can be improved by utilizing the approximation $\mathrm{erf}(x)\approx 1$ for $\left|x\right|\gg 1$ and thereby establishing approximations with a set relative error bound over the interval $\left[0,\infty \right)$.

Two generalizations are detailed. The first is of the form

$$\mathrm{erf}(x)\approx p_0(x)+p_1(x)e^{-k_1{x}^2}+\dots +p_m(x)e^{-k_m{x}^2}$$

(10)

and is based on utilizing approximations associated with m equally spaced subintervals of the interval [0, x]. The second is based on utilizing a fixed subinterval within [0, x] and then approximating the error function over the remainder of the interval. Both generalizations lead to significantly improved accuracy. For example, a fourth-order approximation based on four variable subintervals, when used with the approximation $\mathrm{erf}\left(x\right)\approx 1$ for $x\gg 1$, has a relative error bound of 1.43 × 10⁻⁷ over the interval $\left[0,\infty \right]$. The corresponding sixteenth-order approximation has a relative error bound of 2.01 × 10⁻¹⁹. Finally, by utilizing the solutions of a custom dynamical system, approximations with better convergence properties can be established.

Applications of the proposed approximations for the error function include, first, the definition of functions that are upper and lower bounds, of arbitrary accuracy, for the error function. Second, new series for the error function. Third, new sequences of approximations for $\exp \left(-x^2\right)$ that have significantly higher convergence properties than a Taylor series approximation. Fourth, the definition of a complementary demarcation function $e_C\left(x\right)$ that satisfies the constraint $e_C^2(x)+er{f}^2(x)=1$. Fifth, arbitrarily accurate approximations for the power and harmonic distortion for a sinusoidal signal subject to an error function nonlinearity. Sixth, approximate expressions for the linear filtering of a step signal modeled by the error function.

Section 2 details the spline-based approximation for the error function and its convergence. Improved approximations, obtained by utilizing the nature of the error function for large arguments, are detailed in Section 3. Two generalizations, with the potential for lower relative error bounds, are detailed in Section 4 and Section 5. Section 6 details how the initial approximations, and the approximations arising from the first generalization, can be utilized as inputs to a custom dynamical system to establish approximations with better convergence properties. Applications are specified in Section 7. Conclusions are stated in Section 8.

1.1. Notes and Notation

As $\mathrm{erf}(-x)=-\mathrm{erf}(x)$, it is sufficient to consider approximations for the interval $\left[0, \infty \right)$.

For a function $f$ defined over the interval $\left[\alpha , \beta \right]$, an approximating function $f_A$ has a relative error, at a point $x_1$, defined according to $\mathrm{re}\left(x_1\right)=1-f_A\left(x_1\right)/f\left(x_1\right)$. The relative error bound for the approximating function over the interval $\left[\alpha , \beta \right]$ is defined according to

$${\mathrm{re}}_{\mathrm{B}}=\max \left\{\right|re(x_1)|:x_1\in [\alpha ,\beta ]\}.$$

(11)

The notation $f^{(k)}(x)=\frac{{\mathrm{d}}^k}{\mathrm{d}{x}^k}f(x)$ is used. The symbol u denotes the unit step function.

Mathematica has been used to facilitate the analysis and to obtain numerical results.

1.2. Background Results

The following result underpins the bounds proposed for the error function:

Lemma 1. 

Upper and Lower Functional Bounds. A positive approximation $f_A$ to a positive function $f$ over the interval $\left[\alpha , \beta \right]$, with a relative error bound

$$-{\mathsf{\epsilon }}_B<1-\frac{f_A(x)}{f(x)}<{\mathsf{\epsilon }}_B,\hspace{1em} x\in [\alpha ,\beta ],{\mathsf{\epsilon }}_B>0,$$

(12)

leads to the following upper and lower bounded functions:

$$\frac{f_A(x)}{1+{\mathsf{\epsilon }}_B}<f(x)<\frac{f_A(x)}{1-{\mathsf{\epsilon }}_B},\hspace{1em} x\in [\alpha ,\beta ].$$

(13)

The relative error bounds, over the interval $\left[\alpha , \beta \right]$, for the upper and lower bounded functions, respectively, are:

$$\frac{2{\mathsf{\epsilon }}_B}{1+{\mathsf{\epsilon }}_B},\frac{2{\mathsf{\epsilon }}_B}{1-{\mathsf{\epsilon }}_B}.$$

(14)

Proof. 

The definition of the relative error bound, as specified by Equation (12), leads to

$$1-{\mathsf{\epsilon }}_B<\frac{f_A(x)}{f(x)}<1+{\mathsf{\epsilon }}_B,$$

(15)

which implies

$$\frac{1-{\mathsf{\epsilon }}_B}{1+{\mathsf{\epsilon }}_B}<\frac{f_A(x)/(1+{\mathsf{\epsilon }}_B)}{f(x)}<1, 1<\frac{f_A(x)/(1-{\mathsf{\epsilon }}_B)}{f(x)}<\frac{1+{\mathsf{\epsilon }}_B}{1-{\mathsf{\epsilon }}_B}$$

(16)

and the relative error bounds:

$$\begin{array}{l}0<1-\frac{f_A(x)/(1+{\mathsf{\epsilon }}_B)}{f(x)}<1-\frac{1-{\mathsf{\epsilon }}_B}{1+{\mathsf{\epsilon }}_B}=\frac{2{\mathsf{\epsilon }}_B}{1+{\mathsf{\epsilon }}_B}\\ 1-\frac{1+{\mathsf{\epsilon }}_B}{1-{\mathsf{\epsilon }}_B}=\frac{-2{\mathsf{\epsilon }}_B}{1-{\mathsf{\epsilon }}_B}<1-\frac{f_A(x)/(1-{\mathsf{\epsilon }}_B)}{f(x)}<0\end{array}$$

(17)

□

Convergent Integral Approximations

One application of the proposed approximations for the error function requires knowledge of when function convergence implies convergence of associated integrals.

Lemma 2. 

Convergent Integral Approximation. If a sequence of functions $f_1,f_2,\dots$ converges, over the interval $\left[0,x\right]$, to a bounded and integrable function $f$, then point-wise convergence is sufficient for the associated integrals to be convergent, i.e., for

$$\underset{n\to \infty }{\lim }{\displaystyle \underset{0}{\overset{x}{\int }}{f}_n(\lambda )d\lambda }={\displaystyle \underset{0}{\overset{x}{\int }}f(\lambda )d\lambda }.$$

(18)

Proof. 

The required result follows if it is possible to interchange the order of limit and integration, i.e.,

$$\underset{n\to \infty }{\lim }{\displaystyle \underset{0}{\overset{x}{\int }}{f}_n(\lambda )d\lambda }={\displaystyle \underset{0}{\overset{x}{\int }}\underset{n\to \infty }{\lim }{f}_n(\lambda )d\lambda }={\displaystyle \underset{0}{\overset{x}{\int }}f(\lambda )d\lambda }.$$

(19)

Standard conditions for when the interchange is valid are specified by the monotone and dominated convergence theorems, e.g., [29], (p. 26). Sufficient conditions for a valid interchange include point-wise function convergence, and for $f$ to be integrable and bounded. □

2. Spline-Based Approximations for Error Function

2.1. Spline Approximation for Error Function

The following nth-order, two-point spline-based approximation for an integral has been detailed in Equation (48) of [19], for a function $f$ that is at least (n + 1)th-order differentiable:

$$\begin{array}{r}{\displaystyle \underset{\alpha }{\overset{x}{\int }}f(\lambda )d\lambda }={\displaystyle \sum _{k=0}^n{c}_{n,k}{(x-\alpha )}^{k+1}\left[f^{(k)}(\alpha )+{(-1)}^k{f}^{(k)}(x)\right]}+R_n(\alpha ,x)\\ n\in \{0,1,2,\dots \}\end{array}$$

(20)

where

$$c_{n,k}=\frac{n!}{(n-k)!(k+1)!}\cdot \frac{(2n+1-k)!}{2(2n+1)!}, k\in \{0,1,\dots ,n\},$$

(21)

$$\begin{array}{l}{R}_n^{(1)}(\alpha ,x)=-{\displaystyle \sum _{k=0}^n{c}_{n,k}(k+1){(x-\alpha )}^k\left[f^{(k)}(\alpha )+{(-1)}^{k+1}\cdot \frac{n+1}{n-k+1}\cdot f^{(k)}(x)\right]}+\\ c_{n,n}{(-1)}^{n+1}{(x-\alpha )}^{n+1}{f}^{(n+1)}(x).\end{array}$$

(22)

Direct application of this result to the integral defining the error function leads to the results stated in Theorem 1:

Theorem 1. 

Spline-Based Integral Approximation for Error Function. The error function can be defined according to

$$\mathrm{erf}(x)=f_n(x)+{\mathsf{\epsilon }}_n(x),$$

(23)

where $f_n$ is the nth-order spline-based integral approximation defined according to

$$f_n(x)=\frac{2}{\sqrt{\pi }}\cdot {\displaystyle \sum _{k=0}^n{c}_{n,k}{x}^{k+1}\left[p(k,0)+{(-1)}^{k}p(k,x)e^{-x^2}\right]}$$

(24)

and ${\mathsf{\epsilon }}_n(x)$ is the associated residual function whose derivative is

$$\begin{array}{ll}{\mathsf{\epsilon }}_n^{(1)}(x)=& \frac{2e^{-x^2}}{\sqrt{\pi }}-\frac{2}{\sqrt{\pi }}\cdot {\displaystyle \sum _{k=0}^n{c}_{n,k}(k+1)x^{k}p(k,0)}-\\ & \frac{2e^{-x^2}}{\sqrt{\pi }}\cdot {\displaystyle \sum _{k=0}^n{c}_{n,k}{x}^k{(-1)}^k\left[(k+1-2x^2)p(k,x)+x{p}^{(1)}(k,x)\right]}\end{array}$$

(25)

In these equations,

$$p(k,x)=p^{(1)}(k-1,x)-2xp(k-1,x),\hspace{1em} p(0,x)=1.$$

(26)

A more general approximation is

$$\begin{array}{ll}\mathrm{erf}(x)-\mathrm{erf}(\alpha )& =\frac{2}{\sqrt{\pi }}\cdot {\displaystyle \sum _{k=0}^n{c}_{n,k}{(x-\alpha )}^{k+1}\left[p(k,\alpha )e^{-{\alpha }^2}+{(-1)}^{k}p(k,x)e^{-x^2}\right]}+{\mathsf{\epsilon }}_n(\alpha ,x)\\ & =f_n(\alpha ,x)+{\mathsf{\epsilon }}_n(\alpha ,x).\end{array}$$

(27)

Proof. 

The proof is detailed in Appendix A. □

2.1.1. Note

The polynomial function $p(k,x)$ is equivalently defined by the kth-order Hermite polynomial function ([21], p. 775, Equation (23.3.10)) and an explicit form is

$$p(k,x)={\displaystyle \sum _{i=0}^{\lfloor k/2\rfloor }\frac{{(-1)}^{i+k}k!}{i!(k-2i)!}\cdot 2^{k-2i}{x}^{k-2i}.}$$

(28)

A change of variable $r=k-2i$, and noting that $i\in \left\{0,1,\dots ,\lfloor k/2\rfloor \right\}$ implies $r\in \left\{k,k-2,\dots ,k-2\lfloor k/2\rfloor \right\}$, leads to the alternative form

$$p(k,x)={\displaystyle \sum _{r=k-2\lfloor k/2\rfloor }^k{d}_{k,r}{x}^r, d_{k,r}=\frac{{(-1)}^{(3k-r)/2}\left[\frac{1+{(-1)}^{r-\left[k-2\lfloor k/2\rfloor \right]}}{2}\right]k!2^r}{\left[(k-r)/2\right]!r!}}.$$

(29)

Substitution of this form into Equation (24) leads to the direct polynomial form for the nth-order approximation to the error function:

$$f_n(x)=\frac{2}{\sqrt{\pi }}\cdot {\displaystyle \sum _{k=0}^n{c}_{n,k}p(k,0)x^{k+1}+\frac{2e^{-x^2}}{\sqrt{\pi }}\cdot {\displaystyle \sum _{k=0}^n\left[{(-1)}^k{c}_{n,k}{\displaystyle \sum _{r=k-2\lfloor k-2\rfloor }^k{d}_{k,r}{x}^{r+k+1}}\right]}}.$$

(30)

2.1.2. Approximations

The polynomial functions p, as defined by Equations (26), (28), and (29), have the explicit forms:

$$\begin{array}{l}p(0,x)=1, p(1,x)=-2x, p(2,x)=-2[1-2x^2],\\ p(3,x)=12x[1-2x^2/3], p(4,x)=12[1-4x^2+4x^4/3], \dots \end{array}$$

(31)

Approximations for the error function, as defined by Equation (24) and for orders zero to five, are:

$$f_0(x)=\frac{x}{\sqrt{\pi }}+\frac{x}{\sqrt{\pi }}\cdot e^{-x^2}$$

(32)

$$f_1(x)=\frac{x}{\sqrt{\pi }}+\frac{x}{\sqrt{\pi }}\left[1+\frac{x^2}{3}\right]e^{-x^2}$$

(33)

$$f_2(x)=\frac{x}{\sqrt{\pi }}\left[1-\frac{x^2}{30}\right]+\frac{x}{\sqrt{\pi }}\left[1+\frac{11x^2}{30}+\frac{x^4}{15}\right]e^{-x^2}$$

(34)

$$f_3(x)=\frac{x}{\sqrt{\pi }}\left[1-\frac{x^2}{21}\right]+\frac{x}{\sqrt{\pi }}\left[1+\frac{8x^2}{21}+\frac{17x^4}{210}+\frac{x^6}{105}\right]e^{-x^2}$$

(35)

$$f_4(x)=\frac{x}{\sqrt{\pi }}\left[1-\frac{x^2}{18}+\frac{x^4}{1260}\right]+\frac{x}{\sqrt{\pi }}\left[1+\frac{7x^2}{18}+\frac{37x^4}{420}+\frac{4x^6}{315}+\frac{x^8}{945}\right]e^{-x^2}$$

(36)

$$f_5(x)=\frac{x}{\sqrt{\pi }}\left[1-\frac{2x^2}{33}+\frac{x^4}{660}\right]+\frac{x}{\sqrt{\pi }}\left[1+\frac{13x^2}{33}+\frac{61x^4}{660}+\frac{67x^6}{4620}+\frac{16x^8}{\mathrm{10,395}}+\frac{x^{10}}{\mathrm{10,395}}\right]e^{-x^2}$$

(37)

2.2. Results

The relative error in the zero- to tenth-order spline-based series approximations, along with the relative error in Taylor series approximations of orders 1–15, are detailed in Figure 2. The clear superiority, in terms of convergence, of the spline-based series relative to the Taylor series is evident. The relative error in the spline approximations, of orders 16, 20, 24, 28, and 32, are shown in Figure 3.

The Mathematica code underpinning the results shown in Figure 2, is detailed in Supplementary Material. Such code is indicative of the code underpinning the results detailed in the paper.

2.3. Approximation for Large Arguments

Zero- and first-order approximations for the error function, and for the case of $x\gg 1$, are

$$\mathrm{erf}(x)\approx 1, \mathrm{erf}(x)\approx 1-\frac{e^{-x^2}}{\sqrt{\pi }x}.$$

(38)

The relative errors in such approximations, respectively, are

$$\mathrm{re}(x)=1-\frac{1}{\mathrm{erf}(x)}, \mathrm{re}(x)\approx 1-\frac{1-e^{-x^2}/\sqrt{\pi }x}{\mathrm{erf}(x)},$$

(39)

and their graphs are shown in Figure 3.

2.4. Convergence

To prove the convergence of the sequence of functions $f_0,f_1,f_2,\dots$, defined by Theorem 1, to the error function, it is sufficient to prove that the corresponding sequence of residual functions ${\mathsf{\epsilon }}_0,{\mathsf{\epsilon }}_1,{\mathsf{\epsilon }}_2,\dots$ converge to zero. This can be shown by considering the derivatives of the residual functions defined by Equation (25). The derivatives of the residual functions of orders zero, one, and two, respectively, are:

$${\mathsf{\epsilon }}_0^{(1)}(x)=\frac{1}{\sqrt{\pi }}\left[1+2x^2\right]e^{-x^2}-\frac{1}{\sqrt{\pi }}$$

(40)

$${\mathsf{\epsilon }}_1^{(1)}(x)=\frac{1}{\sqrt{\pi }}\left[1+x^2+\frac{2x^4}{3}\right]e^{-x^2}-\frac{1}{\sqrt{\pi }}$$

(41)

$${\mathsf{\epsilon }}_2^{(1)}(x)=\frac{1}{\sqrt{\pi }}\left[1+\frac{9x^2}{10}+\frac{2x^4}{5}+\frac{2x^6}{15}\right]e^{-x^2}-\frac{1}{\sqrt{\pi }}\left[1-\frac{x^2}{10}\right].$$

(42)

Theorem 2. 

Convergence of Spline-Based Approximations. For all fixed values of x, the derivatives of the residual functions converge to zero as the order increases, i.e., for all fixed values of x it is the case that

$$\underset{n\to \infty }{\lim }{\mathsf{\epsilon }}_n^{(1)}(x)=0,\hspace{1em} x>0.$$

(43)

This is sufficient for the convergence of the residual functions, i.e.,$\underset{n\to \infty }{\lim }{\mathsf{\epsilon }}_n(x)=0,$ $x>0,$and, hence, for x fixed:

$$\underset{n\to \infty }{\lim }{f}_n(x)=\mathrm{erf}(x),\hspace{1em} x>0.$$

(44)

The convergence is nonuniform.

Proof. 

The proof is detailed in Appendix B. □

2.5. Improved Approximation Based on Iteration

Consider the general result

$${\displaystyle \underset{0}{\overset{x}{\int }}\mathrm{erf}(\lambda )d\lambda }=\frac{1}{\sqrt{\pi }x}\left[1-e^{-x^2}\right]+x\mathrm{erf}(x)$$

(45)

By using the approximations, $f_n$, as defined in Theorem 1, in the integral, improved approximations for the error function can be defined.

Theorem 3. 

Improved Approximation via Iteration. An improved approximation, of order n, for the error function is

$$\begin{array}{ll}{F}_n(x)=& \frac{1}{\sqrt{\pi }x}\left[1-e^{-x^2}\right]+\frac{2}{\sqrt{\pi }}\cdot {\displaystyle \sum _{k=0}^n\frac{c_{n,k}p(k,0)}{k+2}\cdot x^{k+1}}+\\ & \frac{2}{x\sqrt{\pi }}\cdot {\displaystyle \sum _{k=0}^n\left[{(-1)}^k{c}_{n,k}{\displaystyle \sum _{r=k-2\lfloor k/2\rfloor }^k\frac{d_{k,r}}{2}\left[\frac{r+k}{2}\right]!\left[1-e^{-x^2}{\displaystyle \sum _{i=0}^{\frac{r+k}{2}}\frac{x^{2i}}{i!}}\right]}\right]}\end{array}$$

(46)

where $c_{n,k}$ and $d_{k,r}$ are defined, respectively, by Equations (21) and (29).

Proof. 

From Equation (45), it follows that

$$\mathrm{erf}(x)\approx \frac{1}{\sqrt{\pi }x}\left[1-e^{-x^2}\right]+\frac{1}{x}\cdot {\displaystyle \underset{0}{\overset{x}{\int }}{f}_n(\lambda )d\lambda }.$$

(47)

As

$${\displaystyle \underset{0}{\overset{x}{\int }}{x}^u{e}^{-{\lambda }^2}d\lambda }=\frac{1}{2}\left[\frac{u-1}{2}\right]!\left[1-e^{-x^2}{\displaystyle \sum _{i=0}^{(u-1)/2}\frac{x^{2i}}{i!}}\right], u\in \{1,3,5,\dots \},$$

(48)

it follows, from the form for $f_n$ detailed in Equation (30), that

$$\begin{array}{ll}\mathrm{erf}(x)\approx & \frac{1}{\sqrt{\pi }x}\left[1-e^{-x^2}\right]+\frac{2}{\sqrt{\pi }}\cdot {\displaystyle \sum _{k=0}^n\frac{c_{n,k}p(k,0)}{k+2}}\cdot x^{k+1}+\\ & \frac{2}{x\sqrt{\pi }}\cdot {\displaystyle \sum _{k=0}^n\left[{(-1)}^k{c}_{n,k}{\displaystyle \sum _{r=k-2\lfloor k-2\rfloor }^k\frac{d_{k,r}}{2}\left[\frac{r+k}{2}\right]!\left[1-e^{-x^2}{\displaystyle \sum _{i=0}^{\frac{r+k}{2}}\frac{x^{2i}}{i!}}\right]}\right]}\end{array}$$

(49)

which is the required result. □

2.5.1. Explicit Approximations

Approximations to the error function, of orders zero to five, are:

$$f_0(x)=\frac{3}{2\sqrt{\pi }x}\left[1+\frac{x^2}{3}\right]-\frac{3e^{-x^2}}{2\sqrt{\pi }x}$$

(50)

$$f_1(x)=\frac{5}{3\sqrt{\pi }x}\left[1+\frac{3x^2}{10}\right]-\frac{5}{3\sqrt{\pi }x}\left[1+\frac{x^2}{10}\right]e^{-x^2}$$

(51)

$$f_2(x)=\frac{7}{4\sqrt{\pi }x}\left[1+\frac{2x^2}{7}-\frac{x^4}{210}\right]-\frac{7}{4\sqrt{\pi }x}\left[1+\frac{x^2}{7}+\frac{2x^4}{105}\right]e^{-x^2}$$

(52)

$$f_3(x)=\frac{9}{5\sqrt{\pi }x}\left[1+\frac{5x^2}{18}-\frac{5x^4}{756}\right]-\frac{9}{5\sqrt{\pi }x}\left[1+\frac{x^2}{6}+\frac{23x^4}{756}+\frac{x^6}{378}\right]e^{-x^2}$$

(53)

$$\begin{array}{ll}{f}_4(x)=& \frac{11}{6\sqrt{\pi }x}\left[1+\frac{3x^2}{11}-\frac{x^4}{132}+\frac{x^6}{\mathrm{13,860}}\right]-\\ & \frac{11}{6\sqrt{\pi }x}\left[1+\frac{2x^2}{11}+\frac{5x^4}{132}+\frac{16x^6}{3465}+\frac{x^8}{3465}\right]e^{-x^2}\end{array}$$

(54)

$$\begin{array}{ll}{f}_5(x)=& \frac{13}{7\sqrt{\pi }x}\left[1+\frac{7x^2}{26}-\frac{7x^4}{858}+\frac{7x^6}{\mathrm{51,480}}\right]-\\ & \frac{13}{7\sqrt{\pi }x}\left[1+\frac{5x^2}{26}+\frac{37x^4}{858}+\frac{313x^6}{\mathrm{51,480}}+\frac{7x^8}{\mathrm{12,850}}+\frac{x^{10}}{\mathrm{38,610}}\right]e^{-x^2}\end{array}$$

(55)

Note that integration of these expressions leads to functions defined, in part, by the Gamma function that is an integral. This makes further iteration impractical.

2.5.2. Results

The relative errors in even order approximations, of orders 0–10, are shown in Figure 4. A comparison of the results detailed in Figure 2 and Figure 4 show the clear improvement in the approximations specified by Equation (46).

3. Improved Approximations

3.1. Improved Approximation for Error Function

An improved approximation for the error function can be achieved by noting, as illustrated in Figure 3, that the approximation $\mathrm{erf}\left(x\right)\approx 1$ is increasingly accurate for the case of $x\gg 1$ and for $x$ increasing. By switching at a suitable point $x_o$, as illustrated in Figure 5, from a spline-based approximation to the approximation $\mathrm{erf}\left(x\right)\approx 1$, an improved approximation is achieved. Naturally, it is possible to switch to the approximation $\mathrm{erf}\left(x\right)\approx 1-e^{-x^2}/\sqrt{\pi }x$, or higher-order approximations, in a similar manner.

Theorem 4. 

Improved Approximation for Error Function. Improved approximations for the error function, based on the nth-order approximations detailed in Theorems 1 and 3, and consistent with the illustration shown inFigure 5, are

$$\begin{array}{l}\mathrm{erf}(x)\approx f_n(x)u\left[x_o(n)-x\right]+\left[1-u\left[x_o(n)-x\right]\right]\\ \mathrm{erf}(x)\approx F_n(x)u\left[x_o(n)-x\right]+\left[1-u\left[x_o(n)-x\right]\right]\end{array}$$

(56)

where the transition points, respectively, are defined according to

$$\begin{array}{l}{x}_o(n)=x:\left|1-\frac{1}{\mathrm{erf}(x)}\right|=\left|1-\frac{f_n(x)}{\mathrm{erf}(x)}\right|\\ x_o(n)=x:\left|1-\frac{1}{\mathrm{erf}(x)}\right|=\left|1-\frac{F_n(x)}{\mathrm{erf}(x)}\right|\end{array}$$

(57)

Proof. 

The improved approximation results follow from optimally switching, as illustrated in Figure 5 and at the point specified by Equation (57), to the approximation $\mathrm{erf}\left(x\right)\approx 1$, which has a lower relative error magnitude. □

Transition Points and Relative Error Bounds

The transition points, for various orders of spline approximation, are specified in Table 3. The relationship between the transition point and order is shown in Figure 6 for the case of the approximations $f_n$. This relationship can be approximated, with a second-order polynomial, according to

$$x_o(n)=1.3607+0.20511n-0.002932n^2, 0\le n\le 24.$$

(58)

However, as small variations in $x_o\left(n\right)$ can lead to significant changes in the maximum relative error in the approximation for the error function, precise values for $x_o\left(n\right)$ are preferable.

The graphs of the relative errors in the approximations $f_n$ to erf(x), as specified by Equation (56), are shown in Figure 7 for orders 2, 4, 6, …, 20. The relative error bounds that can be achieved over the interval $\left[0,\infty \right]$, using the optimally chosen transition points, are detailed in Table 3.

3.2. Improved Approximation for Taylor Series

The approximation of $\mathrm{erf}\left(x\right)\approx 1$ for $x\gg 1$ can be utilized to improve the relative error bound for a Taylor series approximation according to

$$\mathrm{erf}\left(x\right)\approx T_n\left(x\right)u\left[x_o\left(n\right)-x\right]+\left[1-u\left[x_o\left(n\right)-x\right]\right],$$

(59)

where $T_n$ is a nth-order Taylor series, with n odd, as specified in Table 1. The optimum transition points and relative error bounds, for selected orders, are detailed in Table 4. The variation of the relative errors, with order, are shown in Figure 8. The change in the optimum transition point can be approximated according to

$$x_o\left(n\right)\approx 0.932+0.0560n-0.0003503n^2,\hspace{1em} 3\le n\le 61,$$

(60)

but, again, as small variations in $x_o\left(n\right)$ can lead to significant changes in the maximum relative error in the approximation for the error function, precise values for $x_o\left(n\right)$ are preferable. The clear superiority in the convergence of the spline-based series is evident from a visual comparison of the relative errors shown in Figure 7 and Figure 8.

4. Variable Subinterval Approximations for Error Function

An improved analytic approximation for the error function can be achieved by demarcating the interval $\left[0, x\right]$ into variable subintervals, e.g., the subintervals $\left[0, x/4\right]$, $\left[x/4, x/2\right]$, $\left[x/2, 3x/4\right]$, and $\left[3x/4, x\right]$ for the four-subintervals case, and by utilizing spline-based integral approximations for each subinterval. Chiani [30] utilized subintervals to enhance approximations for the complementary error function.

Theorem 5. 

Variable Subinterval Approximations for Error Function. The nth-order spline-based approximation to the error function, based on m equal-width subintervals, is

$$f_{n,m}(x)=\frac{2}{\sqrt{\pi }}{\displaystyle \sum _{i=0}^{m-1}\left[{\displaystyle \sum _{k=0}^n{c}_{n,k}{\left[\frac{x}{m}\right]}^{k+1}\left[\begin{array}{l}p\left[k,\frac{ix}{m}\right]\exp \left[-\frac{i^2{x}^2}{m^2}\right]+\\ {(-1)}^{k}p\left[k,\frac{(i+1)x}{m}\right]\exp \left[-\frac{{(i+1)}^2{x}^2}{m^2}\right]\end{array}\right]}\right]}$$

(61)

where

$$p(k,x)=p^{(1)}(k-1,x)-2xp(k-1,x),\hspace{1em} p(0,x)=1.$$

(62)

An alternative form is

$$f_{n,m}(x)=\frac{2}{\sqrt{\pi }}\left[p_{n,0}(x)+{\displaystyle \sum _{i=1}^{m-1}{p}_{n,i}(x)\exp \left[-\frac{i^2{x}^2}{m^2}\right]}+p_{n,m}(x)\exp (-x^2)\right]$$

(63)

where

$$\begin{array}{l}{p}_{n,0}(x)={\displaystyle \sum _{k=0}^n\frac{c_{n,k}}{m^{k+1}}\cdot p(k,0)x^{k+1}}\\ p_{n,i}(x)={\displaystyle \sum _{k=0}^n\frac{c_{n,k}}{m^{k+1}}\cdot \left[1+{(-1)}^k\right]p(k,\frac{ix}{m})x^{k+1}}\\ p_{n,m}(x)={\displaystyle \sum _{k=0}^n\frac{c_{n,k}}{m^{k+1}}\cdot {(-1)}^{k}p(k,x)x^{k+1}}.\end{array}$$

(64)

Proof. 

The first result follows by applying Equation (27) in Theorem 1 to the subintervals $\left[0,x/m\right]$, $\left[x/m,2x/m\right]$, $\dots$, $\left[x-\frac{x}{m},x\right]$. The alternative form arises by expanding the outer summation in Equation (61) and collecting terms of similar form. □

4.1. Explicit Expressions

A first-order approximation, based on m subintervals, is

$$f_{1,m}(x)=\frac{x}{m\sqrt{\pi }}\left[1+2{\displaystyle \sum _{i=1}^{m-1}\exp \left[\frac{-i^2{x}^2}{m^2}\right]}+\exp (-x^2)\right]+\frac{x^3}{3m^2\sqrt{\pi }}\cdot \exp (-x^2).$$

(65)

For the four-subintervals case, explicit expressions are

$$f_{0,4}(x)=\frac{x}{4\sqrt{\pi }}\left[1+2\exp \left[\frac{-x^2}{16}\right]+2\exp \left[\frac{-x^2}{4}\right]+2\exp \left[\frac{-9x^2}{16}\right]+\exp (-x^2)\right]$$

(66)

$$f_{1,4}(x)=\frac{x}{4\sqrt{\pi }}\left[1+2\exp \left[\frac{-x^2}{16}\right]+2\exp \left[\frac{-x^2}{4}\right]+2\exp \left[\frac{-9x^2}{16}\right]+\exp (-x^2)\right]+\frac{x^3}{48\sqrt{\pi }}\cdot \exp (-x^2)$$

(67)

Using the alternative form, a fourth-order expression is

$$\begin{array}{ll}{f}_{4,4}(x)=& \frac{x}{4\sqrt{\pi }}\left[1-\frac{x^2}{288}+\frac{x^4}{\mathrm{322,560}}\right]+\\ & \frac{x}{2\sqrt{\pi }}\cdot \exp \left[\frac{-x^2}{16}\right]\left[1-\frac{x^2}{288}+\frac{47x^4}{\mathrm{107,520}}-\frac{x^6}{\mathrm{1,290,040}}+\frac{x^8}{\mathrm{61,931,520}}\right]+\\ & \frac{x}{2\sqrt{\pi }}\cdot \exp \left[\frac{-x^2}{4}\right]\left[1-\frac{x^2}{288}+\frac{187x^4}{\mathrm{107,520}}-\frac{x^6}{\mathrm{322,560}}+\frac{x^8}{\mathrm{3,870,720}}\right]+\\ & \frac{x}{2\sqrt{\pi }}\cdot \exp \left[\frac{-9x^2}{16}\right]\left[1-\frac{x^2}{288}+\frac{1261x^4}{\mathrm{322,560}}-\frac{x^6}{\mathrm{143,360}}+\frac{3x^8}{\mathrm{2,293,760}}\right]+\\ & \frac{x}{4\sqrt{\pi }}\cdot \exp (-x^2)\left[1+\frac{31x^2}{288}+\frac{101x^4}{\mathrm{15,360}}-\frac{19x^6}{\mathrm{80,640}}+\frac{x^8}{\mathrm{241,920}}\right]+\end{array}$$

(68)

A fourth-order spline approximation, which utilizes 16 subintervals, is detailed in Appendix C. This expression, when utilized with the transition point $x_o=7.1544$, yields an approximation with a relative error bound of 4.82 × 10⁻¹⁶.

Results

The relative errors in the spline approximations of orders one to six, and for the case of four equal subintervals $\left[0,x/4\right]$, $\left[x/4,x/2\right]$, $\left[x/2, 3x/4\right]$, and $\left[3x/4, x\right]$, are shown in Figure 9.

4.2. Improved Approximation

The spline approximations utilizing variable subintervals can be improved by using the transition to the approximation $\mathrm{erf}\left(x\right)\approx 1$ at a suitable point, as specified by Equation (56). The relative error in the spline approximations of orders one to seven, and for the case of four equal subintervals $\left[0,x/4\right]$, $\left[x/4,x/2\right]$, $\left[x/2, 3x/4\right]$, and $\left[3x/4, x\right]$, are updated in Figure 10 to show the improvement associated with utilizing the optimum transition point to the approximation $\mathrm{erf}\left(x\right)\approx 1$. The relative error bounds, and transition points, are detailed in Table 5 and Table 6 for the cases of four and 16 subintervals.

Examples

A first-order approximation, based on m subintervals, as specified by Equation (65), yields the relative error bound of 7.21 × 10⁻⁵ for four subintervals and the transition point $x_o=3.2928$; 4.51 × 10⁻⁶ for eight subintervals with a transition point $x_o=4.784$; 2.82 × 10⁻⁷ for 16 subintervals and a transition point of $x_o=6.88$; and 1.10 × 10⁻⁹ for 64 subintervals and a transition point $x_o=15.7888$.

The fourth-order approximation, based on four equal subintervals, as specified by Equation (68), leads to the relative error bound of 1.43 × 10⁻⁷ when used with the transition point $x_o=3.7208$. A sixteenth-order approximation, based on four equal subintervals, leads to an error bound of 2.01 × 10⁻¹⁹ when used with the optimum transition point of $x_o=6.3736$.

5. Dynamic Constant plus Spline Approximation

Consider the demarcation of the areas, as illustrated in Figure 11 and based on a resolution $\Delta$, that define the error function. It follows that

$$\mathrm{erf}(x)={\displaystyle \sum _{k=0}^{\lfloor x/\Delta \rfloor }{c}_k}+\frac{2}{\sqrt{\pi }}{\displaystyle \underset{\Delta \lfloor x/\Delta \rfloor }{\overset{x}{\int }}{e}^{-{\lambda }^2}\mathrm{d}\lambda }\{\begin{array}{c}{c}_0=0\\ c_k=\mathrm{erf}(k\Delta )-\mathrm{erf}\left[(k-1)\Delta \right], k\ge 1.\end{array}$$

(69)

For the general case of nonuniformly spaced intervals, as defined by the set of monotonically increasing points $\left\{x_0,x_1,x_2,\dots ,x_m\right\}$, and where it is not necessarily the case that $x>x_m$, the error function is defined according to

$$\mathrm{erf}(x)={\displaystyle \sum _{k=1}^m{c}_{k}u(x-x_k)}+\frac{2}{\sqrt{\pi }}{\displaystyle \underset{x_S}{\overset{x}{\int }}{e}^{-{\lambda }^2}\mathrm{d}\lambda ,}$$

(70)

where $c_0=0$, $x_0=0$ and

$$c_k=\mathrm{erf}(x_k)-\mathrm{erf}\left[x_{k-1}\right],\hspace{1em} x_S={\displaystyle \sum _{k=1}^m\left[x_k-x_{k-1}\right]u(x-x_k).}$$

(71)

A spline-based approximation, as defined by Equation (27), can be utilized for the unknown integrals in Equations (69) and (70). This leads to the results stated in Theorem 6:

Theorem 6. 

Error Function Approximation: Dynamic Constant Plus a Spline Approximation. The error function, as defined by Equations (69) and (70), can be approximated, respectively, by

$$f_{n,\Delta }(x)={\displaystyle \sum _{k=0}^{\lfloor x/\Delta \rfloor }{c}_k}+\frac{2}{\sqrt{\pi }}\left[{\displaystyle \sum _{k=0}^n{c}_{n,k}{\left[x-\Delta \lfloor \frac{x}{\Delta }\rfloor \right]}^{k+1}\left[\begin{array}{r}p\left[k,\Delta \lfloor \frac{x}{\Delta }\rfloor \right]\exp \left[-{\Delta }^2{\lfloor \frac{x}{\Delta }\rfloor }^2\right]+\\ {(-1)}^{k}p(k,x)exp(-x^2)\end{array}\right]}\right]$$

(72)

$$\mathrm{erf}(x)\approx {\displaystyle \sum _{k=1}^m{c}_{k}u(x-x_k)}+\frac{2}{\sqrt{\pi }}\left[{\displaystyle \sum _{k=0}^n{c}_{n,k}{(x-x_S)}^{k+1}\left[\begin{array}{l}p(k,x_S)exp(-x_S^2)+\\ {(-1)}^{k}p(k,x)exp(-x^2)\end{array}\right]}\right]$$

(73)

Proof. 

These results arise from spline approximation of order n, as defined by Equation (27), for the integrals, respectively, over the intervals $\left[\Delta \lfloor x/\Delta \rfloor ,x\right]$ and $\left[x_S,x\right]$. □

5.1. Approximations of Orders Zeros to Four

Approximations of orders zero to four arising from Theorem 6 are:

$$f_{0,\Delta }(x)={\displaystyle \sum _{k=0}^{\lfloor x/\Delta \rfloor }{c}_k}+\frac{x-\Delta \lfloor x/\Delta \rfloor }{\sqrt{\pi }}\cdot \left[e^{-{\Delta }^2{\lfloor x/\Delta \rfloor }^2}+e^{-x^2}\right]$$

(74)

$$\begin{array}{ll}{f}_{1,\Delta }(x)=& {\displaystyle \sum _{k=0}^{\lfloor x/\Delta \rfloor }{c}_k}+\frac{x-\Delta \lfloor x/\Delta \rfloor }{\sqrt{\pi }}\cdot \left[e^{-{\Delta }^2{\lfloor x/\Delta \rfloor }^2}+e^{-x^2}\right]-\\ & \frac{{\left(x-\Delta \lfloor x/\Delta \rfloor \right)}^2}{3\sqrt{\pi }}\cdot \left[\Delta \lfloor x/\Delta \rfloor e^{-{\Delta }^2{\lfloor x/\Delta \rfloor }^2}-x{e}^{-x^2}\right]\end{array}$$

(75)

$$\begin{array}{ll}{f}_{2,\Delta }(x)=& {\displaystyle \sum _{k=0}^{\lfloor x/\Delta \rfloor }{c}_k}+\frac{x-\Delta \lfloor x/\Delta \rfloor }{\sqrt{\pi }}\cdot \left[e^{-{\Delta }^2{\lfloor x/\Delta \rfloor }^2}+e^{-x^2}\right]-\\ & \frac{2{\left(x-\Delta \lfloor x/\Delta \rfloor \right)}^2}{5\sqrt{\pi }}\cdot \left[\Delta \lfloor x/\Delta \rfloor e^{-{\Delta }^2{\lfloor x/\Delta \rfloor }^2}-x{e}^{-x^2}\right]-\\ & \frac{{\left(x-\Delta \lfloor x/\Delta \rfloor \right)}^3}{30\sqrt{\pi }}\cdot \left[\left[1-2{\Delta }^2{\lfloor x/\Delta \rfloor }^2\right]e^{-{\Delta }^2{\lfloor x/\Delta \rfloor }^2}+\left[1-2x^2\right]e^{-x^2}\right]\end{array}$$

(76)

$$\begin{array}{ll}{f}_{3,\Delta }(x)=& {\displaystyle \sum _{k=0}^{\lfloor x/\Delta \rfloor }{c}_k}+\frac{x-\Delta \lfloor x/\Delta \rfloor }{\sqrt{\pi }}\cdot \left[e^{-{\Delta }^2{\lfloor x/\Delta \rfloor }^2}+e^{-x^2}\right]-\\ & \frac{3{\left(x-\Delta \lfloor x/\Delta \rfloor \right)}^2}{7\sqrt{\pi }}\cdot \left[\Delta \lfloor x/\Delta \rfloor e^{-{\Delta }^2{\lfloor x/\Delta \rfloor }^2}-x{e}^{-x^2}\right]-\\ & \frac{{\left(x-\Delta \lfloor x/\Delta \rfloor \right)}^3}{21\sqrt{\pi }}\cdot \left[\left[1-2{\Delta }^2{\lfloor x/\Delta \rfloor }^2\right]e^{-{\Delta }^2{\lfloor x/\Delta \rfloor }^2}+\left[1-2x^2\right]e^{-x^2}\right]+\\ & \frac{{\left(x-\Delta \lfloor x/\Delta \rfloor \right)}^4}{70\sqrt{\pi }}\cdot \left[\Delta \lfloor x/\Delta \rfloor \left[1-\frac{2{\Delta }^2{\lfloor x/\Delta \rfloor }^2}{3}\right]e^{-{\Delta }^2{\lfloor x/\Delta \rfloor }^2}-x\left[1-\frac{2x^2}{3}\right]e^{-x^2}\right]\end{array}.$$

(77)

$$\begin{array}{ll}{f}_{4,\Delta }(x)=& {\displaystyle \sum _{k=0}^{\lfloor x/\Delta \rfloor }{c}_k}+\frac{x-\Delta \lfloor x/\Delta \rfloor }{\sqrt{\pi }}\cdot \left[e^{-{\Delta }^2{\lfloor x/\Delta \rfloor }^2}+e^{-x^2}\right]-\\ & \frac{4{\left(x-\Delta \lfloor x/\Delta \rfloor \right)}^2}{9\sqrt{\pi }}\cdot \left[\Delta \lfloor x/\Delta \rfloor e^{-{\Delta }^2{\lfloor x/\Delta \rfloor }^2}-x{e}^{-x^2}\right]-\\ & \frac{{\left(x-\Delta \lfloor x/\Delta \rfloor \right)}^3}{18\sqrt{\pi }}\cdot \left[\left[1-2{\Delta }^2{\lfloor x/\Delta \rfloor }^2\right]e^{-{\Delta }^2{\lfloor x/\Delta \rfloor }^2}+\left[1-2x^2\right]e^{-x^2}\right]+\\ & \frac{{\left(x-\Delta \lfloor x/\Delta \rfloor \right)}^4}{42\sqrt{\pi }}\cdot \left[\Delta \lfloor x/\Delta \rfloor \left[1-\frac{2{\Delta }^2{\lfloor x/\Delta \rfloor }^2}{3}\right]e^{-{\Delta }^2{\lfloor x/\Delta \rfloor }^2}-x\left[1-\frac{2x^2}{3}\right]e^{-x^2}\right]+\\ & \frac{{\left(x-\Delta \lfloor x/\Delta \rfloor \right)}^5}{1260\sqrt{\pi }}\cdot \left[\left[1-4{\Delta }^2{\lfloor x/\Delta \rfloor }^2+\frac{4{\Delta }^4{\lfloor x/\Delta \rfloor }^4}{3}\right]e^{-{\Delta }^2{\lfloor x/\Delta \rfloor }^2}+\left[1-4x^2+\frac{4x^4}{3}\right]e^{-x^2}\right]\end{array}$$

(78)

5.2. Results

For a resolution of $\Delta =1/2$, the coefficients are tabulated in Table 7.

A resolution of $\Delta =1/2$ yields a relative error bound of 1.16 × 10⁻⁵ for a second-order approximation, a relative error bound of 1.35 × 10⁻⁹ for a fourth-order approximation, a relative error bound of 7.15 × 10⁻¹⁴ for a sixth-order approximation and a relative error bound of 9.03 × 10⁻³⁷ for a sixteenth-order approximation. These bounds are based on 10,000 equally spaced samples in the interval [0, 8].

The variation of the relative error bound with resolution, and order, is detailed in Figure 12. The nature of the variation of the relative error, for orders two, three, and four, is shown in Figure 13 for a resolution of 0.5. It is possible to obtain better results by using nonuniformly spaced intervals but the improvement, in general, does not warrant the increase in complexity.

6. A Dynamic System to Yield Improved Approximations

It is possible to utilize the approximations detailed in Theorems 1 and 5 as the basis for determining new approximations with a lower relative error. The approach is indirect and based on considering the feedback system illustrated in Figure 14, which has varying feedback. The differential equation characterizing the system is

$$y\prime (t)+f_M(t)y(t)=x(t).$$

(79)

For specific input, x, and modulated feedback, $f_M$, signals the output has a known form. For example, for the case of $x\left(t\right)=f_M\left(t\right)=\mathrm{erf}\left(t\right)u\left(t\right)$, the output signal, assuming zero initial conditions, is

$$y(t)=1-exp\left[\frac{1}{\sqrt{\pi }}\cdot \left[1-e^{-t^2}\right]-t\mathrm{erf}(t)\right],\hspace{1em} t\ge 0.$$

(80)

For the case of

$$x\left(t\right)=f_M\left(t\right)=\frac{4}{\sqrt{\pi }}{e}^{-t^2}\mathrm{erf}\left(t\right)u\left(t\right)$$

(81)

the output signal, assuming zero initial conditions, is

$$y(t)=1-exp\left[-{\mathrm{erf}}^2(t)\right],\hspace{1em} t\ge 0.$$

(82)

This case facilitates approximations for the error function, which can be made arbitrarily accurate and which are valid for the positive real line.

Theorem 7. 

Dynamical System Approximations for Error Function. Based on the differential equation specified by Equation (79), a nth-order approximation to$\mathrm{erf}\left(x\right)$, for the case of$x\ge 0$, can be defined according to

$$f_n(x)=\sqrt{p_{n,0}+p_{n,1}(x)e^{-x^2}+p_{n,2}(x)e^{-2x^2}},$$

(83)

where, for the case of $n\ge 2$:

$$\begin{array}{l}{p}_{n,1}(x)={\alpha }_0+{\alpha }_2{x}^2+\dots +{\alpha }_m{x}^m,\hspace{1em} m=\{\begin{array}{c}n-1 m \mathrm{odd}\\ n\hspace{1em}\hspace{1em}m \mathrm{even}\end{array}\\ p_{n,2}(x)={\beta }_0+{\beta }_2{x}^2+\dots +{\beta }_{2n}{x}^{2n}\\ p_{n,0}(x)=-\left({\alpha }_0+{\beta }_0\right)\end{array}$$

(84)

and with

$$\begin{array}{l}{\alpha }_m=\frac{-4}{\pi }\cdot c_{n,m}{a}_{m,0}\\ {\alpha }_{m-2i}=\frac{(m-2i+2){\alpha }_{m-2i+2}}{2}-\frac{4}{\pi }\cdot c_{n,m-2i}{a}_{m-2i,0},\hspace{1em} i\in \left\{1,\dots ,\frac{m}{2}-1\right\}\\ {\alpha }_0={\alpha }_2-\frac{4}{\pi }\cdot c_{n,0}{a}_{0,0}\end{array}$$

(85)

$$\begin{array}{l}{\beta }_{2n}=\frac{-2}{\pi }\cdot {(-1)}^n{c}_{n,n}{a}_{n,n}\\ {\beta }_{2n-2i}=\frac{(n-i+1){\beta }_{2n-2i+2}}{2}-\frac{2}{\pi }{\displaystyle \sum _{k=n-i}^{\min \{2n-2i,n\}}{(-1)}^k{c}_{n,k}{a}_{k,2(n-i)-k}},\hspace{1em} i\in \left\{1,\dots ,n-1\right\}\\ {\beta }_0=\frac{{\beta }_2}{2}-\frac{2}{\pi }\cdot c_{n,0}{a}_{0,0}\end{array}$$

(86)

Here, the coefficients $a_{i,j}$, $i,j\in \{0,1,\dots ,n\}$ are defined by the expansion

$$p(k,x)=a_{k,0}+a_{k,1}x+a_{k,2}{x}^2+\dots +a_{k,k}{x}^k,\hspace{1em} k\in \{0,1,\dots ,n\},$$

(87)

arising from the polynomials (Equation (26))

$$p(k,x)=p^{(1)}(k-1,x)-2xp[k-1,x],\hspace{1em} p(0,x)=1$$

(88)

Finally, it is the case that

$$\underset{n\to \infty }{\lim }{f}_n(x)=\mathrm{erf}(x),\hspace{1em} x\ge 0,$$

(89)

with the convergence being uniform.

Proof. 

The proof is detailed in Appendix D. □

6.1. Explicit Approximations

Explicit approximations for orders zero to four for $\mathrm{erf}(x)$, $x\ge 0$ are:

$$f_0(x)=\frac{1}{\sqrt{\pi }}\sqrt{3-2e^{-x^2}-e^{-2x^2}}$$

(90)

$$f_1(x)=\frac{1}{\sqrt{\pi }}\sqrt{\frac{19}{6}-2e^{-x^2}-\frac{7e^{-2x^2}}{6}\left[1+\frac{2x^2}{7}\right]}$$

(91)

$$f_2(x)=\frac{1}{\sqrt{\pi }}\sqrt{\frac{63}{20}-\frac{29e^{-x^2}}{15}\left[1-\frac{x^2}{29}\right]-\frac{73e^{-2x^2}}{60}\left[1+\frac{26x^2}{73}+\frac{4x^4}{73}\right]}$$

(92)

$$f_3(x)=\frac{1}{\sqrt{\pi }}\sqrt{\frac{22}{7}-\frac{40e^{-x^2}}{21}\left[1-\frac{x^2}{20}\right]-\frac{26e^{-2x^2}}{21}\left[1+\frac{10x^2}{26}+\frac{x^4}{13}+\frac{x^6}{130}\right]}$$

(93)

$$f_4(x)=\frac{1}{\sqrt{\pi }}\sqrt{\begin{array}{l}\frac{377}{120}-\frac{596e^{-x^2}}{315}\left[1-\frac{17x^2}{298}+\frac{x^4}{1192}\right]-\\ \frac{3149e^{-2x^2}}{2520}\left[1+\frac{1258x^2}{3149}+\frac{278x^4}{3149}+\frac{112x^6}{9447}+\frac{8x^8}{9447}\right]\end{array}}$$

(94)

6.2. Results

The relative error bounds associated with the approximations to $\mathrm{erf}(x)$ are detailed in Table 8. The graphs of the relative errors in the approximations are shown in Figure 15. The clear advantage of the proposed approximations is evident, with the improvement increasing with the order of the initial approximation (i.e., a function with an initial lower relative error bound leads to an increasingly lower relative error bound). The other clear advantage of the approximations, as is evident in Figure 15, is that the relative error is bounded as $x\to \infty$.

6.3. Extension

By utilizing the approximations detailed in Theorem 5, similar approximations can be detailed, with lower relative error. For example, the first-order approximation, $f_{1,4}$, which is based on four equal subintervals and is defined by Equation (67), yields the approximation

$$f_{1,4}(x)=\frac{1}{\sqrt{\pi }}\sqrt{\begin{array}{r}\frac{\mathrm{128,177}}{\mathrm{40,800}}-\frac{e^{-x^2}}{2}-\frac{16e^{-17x^2/16}}{17}-\frac{4e^{-5x^2/4}}{5}-\frac{16e^{-25x^2/16}}{25}\\ -\frac{25e^{-2x^2}}{96}\left[1+\frac{2x^2}{25}\right]\end{array}}$$

(95)

which has a relative error bound of 2.83 × 10⁻⁶. With an optimum transition point of 3.292, the original approximation has a relative error bound of 7.21 × 10⁻⁵ (see Table 5).

6.4. Notes

First, the constants $p_{n,0}$, $n\in \left\{0,1,\dots \right\}$, as defined in Equation (84), form a series that in the limit converges to 1. It then follows that the corresponding series converges to π:

$$3,\frac{19}{6},\frac{63}{20},\frac{22}{7},\frac{377}{120},\frac{\mathrm{174,169}}{\mathrm{55,440}},\frac{\mathrm{4,528,409}}{\mathrm{1,441,440}},\dots .$$

(96)

Second, the square root functional structure has been utilized for approximations to the error function, as is evident from the approximations detailed in Table 1. It is easy to conclude that the form

$$f_n(x)=\sqrt{p_{n,0}-p_{n,1}(x)e^{-k_1{x}^2}-p_{n,2}(x)e^{-k_2{x}^2}-\dots }$$

(97)

is well suited for approximating the error function.

7. Applications

This section details indicative applications for the approximations to the error function that have been detailed above.

The distinct analytical forms specified in Theorems 1, 3, 5 and 7, for approximations to the error function, in general, facilitate analysis for different applications. For example, the form detailed in Theorem 1 underpins approximations to exp(−x²), as detailed in Section 7.2. The form detailed in Theorem 7 underpins analytical approximations for the power associated with the output of a nonlinearity modeled by the error function when subject to a sinusoidal signal. The approximations are detailed in Section 7.6.

For applications, where a set relative error bound over a set interval is required, the suitable approximation will depend, in part, on the domain over which an approximation is required as well as the level of the relative error bound that is acceptable. For example, the approximations detailed in Theorems 1 and 3 lead to simple analytical forms and modest relative error bounds over $\left[0, \infty \right)$ when used with an appropriate transition point for the approximation of $\mathrm{erf}\left(x\right)\approx 1$. Without the use of a transition point, such approximations are likely to be best suited for a restricted domain—for example, the domain $\left[0, \frac{3}{\sqrt{2}}\right]$, which is consistent with the three sigma case arising from a Gaussian distribution. The fourth-order approximations, as specified by Equations (36) and (54), have relative error bounds, respectively, of 1.02 × 10⁻³ and 1.90 × 10⁻⁴ over $\left[0, \frac{3}{\sqrt{2}}\right]$. Section 7.1 provides examples of approximations that are consistent with set relative error bounds, over the interval $\left[0, \infty \right)$, of 10⁻⁴, 10⁻⁶, 10⁻¹⁰, and 10⁻¹⁶.

7.1. Error Function Approximations: Set Relative Error Bounds

Consider the case where an approximation for the error function, with a relative error bound over the positive real line of 10⁻⁴, is required. A 47th-order Taylor series, with a transition point of $x_o=2.752$, yields a relative bound of 1.00 × 10⁴.

An eighth-order spline approximation, with a transition point of $x_o=2.963$, yields a relative error bound of 2.79 × 10⁻⁵. The approximation, according to Equation (56), is

$$\begin{array}{ll}{f}_8(x)=& \frac{x}{\sqrt{\pi }}\cdot u\left[x_o-x\right]\cdot \left[\begin{array}{l}1-\frac{7x^2}{102}+\frac{x^4}{340}-\frac{x^6}{\mathrm{18,564}}+\frac{x^8}{\mathrm{5,250,960}}+\\ \left[\begin{array}{l}1+\frac{41x^2}{102}+\frac{101x^4}{1020}+\frac{1591x^6}{\mathrm{92,820}}+\frac{4793x^8}{\mathrm{2,162,160}}+\\ \frac{2017x^{10}}{\mathrm{9,189,180}}+\frac{38x^{12}}{\mathrm{2,297,295}}+\frac{31x^{14}}{\mathrm{34,459,425}}+\\ \frac{x^{16}}{\mathrm{34,459,425}}\end{array}\right]e^{-x^2}\end{array}\right]+\\ & \left[1-u\left[x_o-u\right]\right]\end{array}$$

(98)

A seventh-order approximation, with a transition point of $x_o=2.65$, yields a relative error bound of 1.79 × 10⁻⁴.

A first-order spline approximation, based on four equal subintervals $\left[0,x/4\right]$, $\left[x/4,x/2\right]$, $\left[x/2, 3x/4\right]$ and $\left[3x/4,x\right]$, is defined according to

$$\begin{array}{ll}{f}_{1,4}(x)=& \left[\begin{array}{r}\frac{x}{4\sqrt{\pi }}\cdot \left[1+2\exp \left[\frac{-x^2}{16}\right]+2\exp \left[\frac{-x^2}{4}\right]+2\exp \left[\frac{-9x^2}{16}\right]+\exp \left[-x^2\right]\right]+\\ \frac{x^3}{48\sqrt{\pi }}\cdot \exp \left[-x^2\right]\end{array}\right]u[x_o-x]+\\ & \left[1-u\left[x_o-u\right]\right]\end{array}$$

(99)

and yields a relative error bound of 7.21 × 10⁻⁵ with the transition point $x_o=3.292$.

A dynamic constant plus a spline approximation of order 2, and based on a resolution of $\Delta =19/20$, achieves a relative error bound of 8.33 × 10⁻⁵ (10,000 points in the interval [0, 5]). The approximation is

$$\begin{array}{ll}{f}_{2,\Delta }\left(x\right)=& {\displaystyle \sum _{k=0}^{\lfloor 20x/19\rfloor }{c}_k}+\frac{1}{\sqrt{\pi }}\left[x-\frac{19}{20}\cdot \lfloor \frac{20x}{19}\rfloor \right]\left[\exp \left[\frac{-361}{400}{\lfloor \frac{20x}{19}\rfloor }^2\right]+e^{-x^2}\right]-\\ & \frac{2}{5\sqrt{\pi }}{\left[x-\frac{19}{20}\cdot \lfloor \frac{20x}{19}\rfloor \right]}^2\left[\frac{19}{20}\cdot \lfloor \frac{20x}{19}\rfloor \exp \left[\frac{-361}{400}{\lfloor \frac{20x}{19}\rfloor }^2\right]-x{e}^{-x^2}\right]-\\ & \frac{1}{30\sqrt{\pi }}{\left[x-\frac{19}{20}\cdot \lfloor \frac{20x}{19}\rfloor \right]}^3\cdot \left[\left[1-\frac{361}{200}\cdot {\lfloor \frac{20x}{19}\rfloor }^2\right]\exp \left[\frac{-361}{400}{\lfloor \frac{20x}{19}\rfloor }^2\right]+(1-2x^2)e^{-x^2}\right]\end{array}$$

(100)

where

$$\begin{array}{l}{c}_0=0,\hspace{1em} c_k=\mathrm{erf}\left[\frac{19k}{20}\right]-\mathrm{erf}\left[\frac{19(k-1)}{20}\right],\hspace{1em} k\in \{1,2,\dots \},\\ c_1=0.82089081,\hspace{1em} c_2=0.17189962,\\ c_3=7.1539145\times {10}^{-3},\hspace{1em} c_4=5.5579\times {10}^{-5},\hspace{1em} c_5=c_6=\dots =0.\end{array}$$

(101)

Here, the approximation of $\mathrm{erf}(x)\approx 1$, for $x\ge 57/20$ (after three intervals) can be utilized without impacting the relative error bound.

Utilizing a fourth-order spline approximation and iteration consistent with Theorem 7, the approximation

$$f_4(x)=\frac{1}{\sqrt{\pi }}\sqrt{\begin{array}{l}\frac{377}{120}-\frac{596e^{-x^2}}{315}\left[1-\frac{17x^2}{298}+\frac{x^4}{1192}\right]-\\ \frac{3149e^{-2x^2}}{2520}\left[1+\frac{1258x^2}{3149}+\frac{278x^4}{3149}+\frac{112x^6}{9447}+\frac{8x^8}{9447}\right]\end{array}}$$

(102)

yields a relative error bound of 1.82 × 10⁻⁵.

Details of approximations that are consistent with higher-order relative error bounds are detailed in Table 9.

7.2. Approximation for Exp(−x²)

A nth-order approximation to the Gaussian function $\exp (-x^2)$ is detailed in the following theorem:

Theorem 8. 

Approximation for Gaussian Function. A nth-order approximation,$g_n,$to the Gaussian function$\exp (-x^2)$is

$$g_n(x)=\frac{{\displaystyle \sum _{k=0}^n{c}_{n,k}(k+1)x^{k}p(k,0)}}{1+{\displaystyle \sum _{k=0}^n{c}_{n,k}{(-1)}^{k+1}{x}^k\left[p(k,x)[k+1-2x^2]+x{p}^{(1)}(k,x)\right]}}$$

(103)

where $c_{n,k}$ is defined by Equation (21) and $p(k,x)$ is defined by Equation (26).

Proof. 

The proof is detailed in Appendix E. □

7.2.1. Approximations

Approximations to $\exp (-x^2)$, of orders zero to five, are:

$$g_0(x)=\frac{1}{1+2x^2}\hspace{1em} g_1(x)=\frac{1}{1+x^2+\frac{2x^4}{3}}\hspace{1em} g_2(x)=\frac{1-x^2/10}{1+\frac{9x^2}{10}+\frac{2x^4}{5}+\frac{2x^6}{15}}$$

(104)

$$\begin{array}{l}{g}_3(x)=\frac{1-x^2/7}{1+\frac{6x^2}{7}+\frac{5x^4}{14}+\frac{2x^6}{21}+\frac{2x^8}{105}}\\ g_4(x)=\frac{1-x^2/6+x^4/252}{1+\frac{5x^2}{6}+\frac{85x^4}{252}+\frac{11x^6}{126}+\frac{x^8}{63}+\frac{2x^{10}}{945}}\end{array}$$

(105)

$$g_5(x)=\frac{1-2x^2/11+x^4/132}{1+\frac{9x^2}{11}+\frac{43x^4}{132}+\frac{x^6}{12}+\frac{x^8}{66}+\frac{x^{10}}{495}+\frac{2x^{12}}{\mathrm{10,395}}}$$

(106)

7.2.2. Results

The relative errors in the above defined approximations to $\exp (-x^2)$ are detailed in Figure 16 for approximations of order 0, 2, 4, 6, 8, 10, and 12, along with the relative error in Taylor series for orders 1, 3, 5, …, 15. The clear superiority of the defined approximations is evident.

7.2.3. Comparison

The following nth-order approximation for exp(−x²) has been proposed in Equation (77) of [19]:

$$h_n(x)=\frac{1-c_{n,0}{x}^2+c_{n,1}{x}^4-c_{n,2}{x}^6+\dots +c_{n,n}{(-1)}^{n+1}{x}^{2n+2}}{1+c_{n,0}{x}^2+c_{n,1}{x}^4+c_{n,2}{x}^6+\dots +c_{n,n}{x}^{2n+2}}$$

(107)

where $c_{n,k}$ is defined by Equation (21). The relative error bounds over the interval $\left[0, 3/\sqrt{2}\right]$ (the three sigma bound case for Gaussian probability distributions) for this approximation, and the approximation defined by Equation (103), are detailed in Table 10. The tabulated results clearly show that this approximation is more accurate than the approximation detailed in Equation (103). The improvement is consistent with the higher-order Padé approximant being used.

The following approximations (seventh- and fifth-order) yield relative error bounds of less than 0.001 over the interval $\left[0, 3/\sqrt{2}\right]$:

$$g_7(x)=\frac{1-\frac{x^2}{5}+\frac{x^4}{78}-\frac{x^6}{4290}}{1+\frac{4x^2}{5}+\frac{61x^4}{195}+\frac{34x^6}{429}+\frac{83x^8}{5270}+\frac{x^{10}}{495}+\frac{7x^{12}}{\mathrm{32,175}}+\frac{4x^{14}}{\mathrm{225,225}}+\frac{2x^{16}}{\mathrm{2,027,025}}}$$

(108)

$$h_5(x)=\frac{1-\frac{x^2}{2}+\frac{5x^4}{44}-\frac{x^6}{66}+\frac{x^8}{792}-\frac{x^{10}}{\mathrm{15,840}}+\frac{x^{12}}{\mathrm{665,280}}}{1+\frac{x^2}{2}+\frac{5x^4}{44}+\frac{x^6}{66}+\frac{x^8}{792}+\frac{x^{10}}{\mathrm{15,840}}+\frac{x^{12}}{\mathrm{665,280}}}$$

(109)

A twenty-seventh-order Taylor series approximation yields a relative bound of 1.03 × 10⁻³.

7.3. Upper and Lower Bounded Approximations to Error Function

Establishing bounds for erf(x) has received modest research interest, e.g., [31], and published bounds for erf(x) for the case of x > 0 include Chu [32]:

$$\sqrt{1-\exp [-p{x}^2]}\le \mathrm{erf}(x)\le \sqrt{1-\exp [-q{x}^2]},\hspace{1em} p\in (0,1],q\in [4/\pi ,\infty ].$$

(110)

Corollary 4.2 of Neuman [33]:

$$\frac{2x}{\sqrt{\pi }}\exp \left[\frac{-x^2}{3}\right]\le \mathrm{erf}(x)\le \frac{4x}{3\sqrt{\pi }}\left[1+\frac{\exp (-x^2)}{2}\right]$$

(111)

and refinements to the form proposed by Chu [32], e.g., Yang [34,35], Corollary 3.4 of [35]:

$$\begin{array}{l}\sqrt{1-\frac{20}{3\pi }\left[1-\frac{\pi }{4}\right]\exp \left[\frac{-8x^2}{5}\right]-\frac{8}{3}\left[1-\frac{5}{2\pi }\right]\exp (-x^2)}\le \mathrm{erf}(x)\le \\ \sqrt{1-\mathsf{\lambda }(p_0)\exp (-p_0{x}^2)-[1-\mathsf{\lambda }(p_0)]\exp [-\mathsf{\mu }(p_0)x^2]}\end{array}$$

(112)

where

$$\begin{array}{l}{p}_0=\frac{21\pi -60+\sqrt{3(147{\pi }^2-920\pi +1440)}}{30(\pi -3)},\\ \mathsf{\lambda }(p)=\frac{4[7\pi -20-5(\pi -3)p]}{\pi (15p^2-40p+28)}, \mathsf{\mu }(p)=\frac{4(5p-7)}{5(3p-4)}.\end{array}$$

(113)

The relative error in these bounds is detailed in Figure 17.

Utilizing the results of Lemma 1, it follows that any of the approximations detailed in Theorems 4, 5, 6, or 7 can be utilized to create upper and lower bounded functions for erf(x), x > 0, of arbitrary accuracy and with an arbitrary relative error bound. For example, the approximation $f_{1,4}$ specified by Equation (99) yields the functional bounds:

$$\frac{f_{1,4}(x)}{1+{\mathsf{\epsilon }}_B}\le \mathrm{erf}(x)\le \frac{f_{1,4}(x)}{1-{\mathsf{\epsilon }}_B}{,\hspace{1em} \mathsf{\epsilon }}_B=7.21\times {10}^{-5},\hspace{1em} x>0,$$

(114)

with a relative error bound of 8.33 × 10⁻⁵ for the lower bounded function and 1.44 × 10⁻⁴ for the upper bounded function. Such accuracy is better than the bounds underpinning the results shown in Figure 17. The sixteenth-order approximation, $f_{4,16}$, based on four equal subintervals, specified in Appendix C and when used with a transition point $x_o=7.1544$, leads to the functional bounds

$$\frac{f_{4,16}(x)}{1+{\mathsf{\epsilon }}_B}\le \mathrm{erf}(x)\le \frac{f_{4,16}(x)}{1-{\mathsf{\epsilon }}_B}{,\hspace{1em} \mathsf{\epsilon }}_B=4.82\times {10}^{-16},\hspace{1em} x>0,$$

(115)

with a relative error bound of less than 9.64 × 10⁻¹⁶ for the lower bounded function and 9.32 × 10⁻¹⁶ for the upper bounded function.

7.4. New Series for Error Function

Consider the exact results

$$\mathrm{erf}(x)=f_n(x)+{\mathsf{\epsilon }}_n(x),\hspace{1em} n\in \{0,1,2,\dots \},$$

(116)

where $f_n$ is specified by Equation (24) and ${\mathsf{\epsilon }}_n^{(1)}(x)$ is specified by Equation (25). By utilizing a Taylor series approximation for $\exp (-x^2)$ in ${\mathsf{\epsilon }}_n^{(1)}(x)$ and then integrating, an approximation for ${\mathsf{\epsilon }}_n(x)$ can be established. This leads to new series for the error function.

Theorem 9. 

New Series for Error Function. Based on zero-, first-, and second-order approximations, the following series for the error function are valid:

$$\begin{array}{ll}\mathrm{erf}(x)=& \frac{x}{\sqrt{\pi }}+\frac{x}{\sqrt{\pi }}\cdot e^{-x^2}+\\ & \frac{x^3}{\sqrt{\pi }}\left[\frac{1}{3}-\frac{3x^2}{2\cdot 5}+\frac{5x^4}{6\cdot 7}-\frac{7x^6}{24\cdot 9}+\frac{9x^8}{120\cdot 11}-\dots +\frac{{(-1)}^k(2k+1)x^{2k}}{(2k+3)\cdot (k+1)!}+\dots \right]\end{array}$$

(117)

$$\begin{array}{ll}\mathrm{erf}(x)=& \frac{x}{\sqrt{\pi }}+\frac{x}{\sqrt{\pi }}\left[1+\frac{x^2}{3}\right]e^{-x^2}+\\ & \frac{x^5}{6\sqrt{\pi }}\left[\frac{1}{5}-\frac{3x^2}{(3/2)\cdot 7}+\frac{5x^4}{4\cdot 9}-\frac{7x^6}{15\cdot 11}+\frac{9x^8}{72\cdot 13}-\dots +\frac{{(-1)}^k(2k+1)(k+1)!x^{2k}}{(2k+5){\displaystyle \prod }_{r=1}^k\left[{\displaystyle \sum _{i=1}^{r}2i+1}\right]}+\dots \right]\end{array}$$

(118)

$$\begin{array}{ll}\mathrm{erf}(x)=& \frac{x}{\sqrt{\pi }}\left[1-\frac{x^2}{30}\right]+\frac{x}{\sqrt{\pi }}\left[1+\frac{11x^2}{30}+\frac{x^4}{15}\right]e^{-x^2}+\\ & \frac{1}{\sqrt{\pi }}\cdot \frac{x^7}{60}\left[\begin{array}{r}\frac{1\cdot 3}{1\cdot 3\cdot 7}-\frac{3\cdot 5x^2}{2\cdot 3\cdot 9}+\frac{5\cdot 7x^4}{4\cdot 5\cdot 11}-\frac{7\cdot 9x^6}{9\cdot 10\cdot 13}+\dots +\\ \frac{{(-1)}^k(2k+1)(2k+3)(k+1)!x^{2k}}{3(2k+7){\displaystyle \prod }_{r=2}^{k+1}\left[2{\displaystyle \sum _{i=2}^{r}i}\right]}+\dots \end{array}\right]\end{array}$$

(119)

Further series, based on higher-order approximations, can also be established.

Proof. 

The proof is detailed in Appendix F. □

Results

The relative errors associated with the zero- and second-order series are shown in Figure 18 and Figure 19. Clearly, the relative error improves as the number of terms used in the series expansion increases. The significant improvement in the relative error, for small values of x, is evident. A comparison with the relative errors associated with Taylor series approximations, as shown in Figure 2, shows the improved performance.

The second-order approximation arising from Equation (118), i.e.,

$$\mathrm{erf}(x)=\frac{x}{\sqrt{\pi }}+\frac{x}{\sqrt{\pi }}\left[1+\frac{x^2}{3}\right]e^{-x^2}+\frac{x^5}{6\sqrt{\pi }}\left[\frac{1}{5}-\frac{2x^2}{7}\right]$$

(120)

yields a relative error bound of less than 0.001 for the interval [0, 0.87] and less than 0.01 for the interval [0, 1.1].

7.5. Complementary Demarcation Functions

Consider a complementary function $e_C$ which is such that

$$e_C^2(x)+{\mathrm{erf}}^2(x)=1,\hspace{1em}x\ge 0.$$

(121)

With the approximation detailed in Theorem 7 (and by noting that $\underset{n\to \infty }{\lim }{p}_{n,0}=1$—see Equation (96)), it is the case that

$$e_C^2(x)=\underset{n\to \infty }{\lim }\left[p_{n,1}(x)e^{-x^2}+p_{n,2}(x)e^{-2x^2}\right]$$

(122)

and, thus, $e_C(x)$ can be defined independently of the error function. This function is shown in Figure 20 along with $\mathrm{erf}{(x)}^2$. These two functions act as complementary demarcation functions for the interval $\left[0,\infty \right)$. The transition point is $x_o=0.74373198514677$ as

$${\mathrm{erf}(x)|}_{x=0.74373198514677 }=\frac{1}{\sqrt{2}}.$$

(123)

7.6. Power and Harmonic Distortion: Erf Modeled Nonlinearity

The error function is often used to model nonlinearities and the harmonic distortion created by such a nonlinearity is of interest. Examples include the harmonic distortion in magnetic recording, e.g., [2,36], and the harmonic distortion arising, in a communication context, by a power amplifier, e.g., [7]. For these cases, the interest was in obtaining, with a sinusoidal input signal defined by $a\sin (2\pi f_{o}t)$, the harmonic distortion created by an error function nonlinearity over the input amplitude range of [−2, 2].

Consider the output signal of a nonlinearity modelled by the error function:

$$y(t)=\mathrm{erf}\left[a\sin (2\pi f_{o}t)\right].$$

(124)

For such a case, the output power is defined according to

$$P=\frac{1}{T}{\displaystyle \underset{0}{\overset{T}{\int }}{y}^2(t)dt}=\frac{1}{T}{\displaystyle \underset{0}{\overset{T}{\int }}{\mathrm{erf}}^2\left[a\sin (2\pi f_{o}t)\right]dt,}\hspace{1em}T=1/f_o,$$

(125)

and the output amplitude associated with the kth harmonic is

$$\frac{\sqrt{2}}{\sqrt{T}}{\displaystyle \underset{0}{\overset{T}{\int }}\mathrm{erf}\left[a\sin (2\pi f_{o}t)\right]\sin (2\pi k{f}_{o}t)dt}.$$

(126)

To determine an analytical approximation to the output power, the approximations stated in Theorem 7 lead to relatively simple expressions. Consider the third-order approximation, as specified by Equation (93), which has a relative error bound of 2.03 × 10⁻⁴ for the positive real line. For such a case, the output signal is approximated according to

$$y_3(t)=\frac{1}{\sqrt{\pi }}{\left[\begin{array}{l}\frac{22}{7}-\frac{40e^{-{[a\sin (2\pi f_{o}t)]}^2}}{21}\left[1-\frac{{[a\sin (2\pi f_{o}t)]}^2}{20}\right]-\\ \frac{26e^{-2{[a\sin (2\pi f_{o}t)]}^2}}{21}\left[\begin{array}{r}1+\frac{10{[a\sin (2\pi f_{o}t)]}^2}{26}+\frac{{[a\sin (2\pi f_{o}t)]}^4}{13}+\\ \frac{{[a\sin (2\pi f_{o}t)]}^6}{130}\end{array}\right]\end{array}\right]}^{1/2}$$

(127)

and is shown in Figure 21.

The power in $y_3$ can be readily determined (e.g., via Mathematica) and it then follows that an approximation to the true power is

$$\begin{array}{ll}P(a)\approx & \frac{22}{7\pi }-\frac{40}{21\pi }{I}_0\left[\frac{a^2}{2}\right]\left[1-\frac{a^2}{40}\right]e^{-a^2/2}-\frac{26}{21\pi }{I}_0\left[a^2\right]\left[1+\frac{5a^2}{26}+\frac{41a^4}{1040}+\frac{a^6}{260}\right]e^{-a^2}-\\ & \frac{a^2}{21\pi }{I}_1\left[\frac{a^2}{2}\right]e^{-a^2/2}+\frac{37a^2}{140\pi }{I}_1\left[a^2\right]\left[1+\frac{43a^2}{222}+\frac{2a^4}{111}\right]e^{-a^2}\end{array}$$

(128)

where $I_0$ and $I_1$, respectively, are the zero- and first-order Bessel functions of the first kind. The variation in output power is shown in Figure 22.

Harmonic Distortion

To establish analytical approximations for the harmonic distortion, the functional forms detailed in Theorem 7 are not suitable. However, the functional forms detailed in Theorem 1 do lead to analytical approximations that are valid over a restricted domain. Consider a fourth-order spline approximation, as specified by Equation (36), which approximates the error function over the range [−2, 2] with a relative error bound that is better than 0.001 and leads to the approximation

$$\begin{array}{ll}{y}_4(t)=& \frac{a\sin (2\pi f_{o}t)}{\sqrt{\pi }}\left[1-\frac{a^2\sin {(2\pi f_{o}t)}^2}{18}+\frac{a^4\sin {(2\pi f_{o}t)}^4}{1260}\right]+\\ & \frac{a\sin (2\pi f_{o}t)}{\sqrt{\pi }}\left[\begin{array}{r}1+\frac{7a^2\sin {(2\pi f_{o}t)}^2}{18}+\frac{37a^4\sin {(2\pi f_{o}t)}^4}{420}+\\ \frac{4a^6\sin {(2\pi f_{o}t)}^6}{315}+\frac{a^8\sin {(2\pi f_{o}t)}^8}{945}\end{array}\right]e^{-a^2\sin {(2\pi f_{o}t)}^2}\end{array}$$

(129)

The amplitude of the kth harmonic in such a signal is given by

$$c_{4,k}=\frac{\sqrt{2}}{\sqrt{T}}{\displaystyle \underset{0}{\overset{T}{\int }}{y}_4\left[a\sin (2\pi f_{o}t)\right]}\sin (2\pi k{f}_{o}t)dt=\sqrt{2T}{\displaystyle \underset{0}{\overset{1}{\int }}{y}_4\left[a\sin (2\pi \lambda )\right]}\sin (2\pi k\lambda )d\lambda$$

(130)

where the change of variable $\lambda =f_{o}t$ has been used. The first, third, fifth, and seventh harmonic levels are:

$$\begin{array}{ll}\frac{c_{4,1}}{\sqrt{T}}=& \frac{\sqrt{2}a}{2\sqrt{\pi }}\left[1-\frac{a^2}{24}+\frac{a^4}{2016}\right]+\frac{\sqrt{2}a}{2\sqrt{\pi }}\cdot I_0\left[\frac{a^2}{2}\right]\left[1+\frac{11a^2}{24}+\frac{11a^4}{105}+\frac{a^6}{70}+\frac{a^8}{945}\right]e^{-a^2/2}-\\ & \frac{5\sqrt{2}a}{6\sqrt{\pi }}\cdot I_1\left[\frac{a^2}{2}\right]\left[1+\frac{1481a^2}{4200}+\frac{38a^4}{525}+\frac{29a^6}{3150}+\frac{a^8}{1575}\right]e^{-a^2/2}\\ \end{array}$$

(131)

$$\begin{array}{ll}\frac{c_{4,3}}{\sqrt{T}}=& \frac{\sqrt{2}{a}^3}{144\sqrt{\pi }}\left[1-\frac{a^2}{56}\right]-\frac{115\sqrt{2}a}{84\sqrt{\pi }}\cdot I_0\left[\frac{a^2}{2}\right]\left[1+\frac{403a^2}{1380}+\frac{6a^4}{115}+\frac{31a^6}{5175}+\frac{2a^8}{5175}\right]e^{-a^2/2}+\\ & \frac{115\sqrt{2}}{21a\sqrt{\pi }}\cdot I_1\left[\frac{a^2}{2}\right]\left[1+\frac{8a^2}{23}+\frac{163a^4}{1840}+\frac{76a^6}{5175}+\frac{11a^8}{6900}+\frac{a^{10}}{\mathrm{10,350}}\right]e^{-a^2/2}\\ \end{array}$$

(132)

$$\begin{array}{ll}\frac{c_{4,5}}{\sqrt{T}}=& \frac{\sqrt{2}{a}^5}{\mathrm{40,320}\sqrt{\pi }}+\frac{262\sqrt{2}}{15a\sqrt{\pi }}\cdot I_0\left[\frac{a^2}{2}\right]\left[1+\frac{1943a^2}{7336}+\frac{1485a^4}{\mathrm{29,344}}+\frac{73a^6}{\mathrm{11,004}}+\frac{13a^8}{\mathrm{22,008}}+\frac{a^{10}}{\mathrm{33,012}}\right]e^{-a^2/2}-\\ & \frac{1048\sqrt{2}}{15a^3\sqrt{\pi }}\cdot I_1\left[\frac{a^2}{2}\right]\left[1+\frac{1943a^2}{7336}+\frac{1201a^4}{\mathrm{14,672}}+\frac{5125a^6}{\mathrm{352,128}}+\frac{5a^8}{2751}+\frac{41a^{10}}{\mathrm{264,096}}+\frac{a^{12}}{\mathrm{132,048}}\right]e^{-a^2/2}\\ \end{array}$$

(133)

$$\begin{array}{ll}\frac{c_{4,7}}{\sqrt{T}}=& \frac{-6784\sqrt{2}}{21a^3\sqrt{\pi }}\cdot I_0\left[\frac{a^2}{2}\right]\left[1+\frac{779a^2}{3392}+\frac{631a^4}{\mathrm{13,568}}+\frac{2047a^6}{\mathrm{325,632}}+\frac{25a^8}{\mathrm{40,704}}+\frac{17a^{10}}{\mathrm{407,040}}+\frac{a^{12}}{\mathrm{610,560}}\right]e^{-a^2/2}-\\ & \frac{\mathrm{27,136}\sqrt{2}}{21a^5\sqrt{\pi }}\cdot I_1\left[\frac{a^2}{2}\right]\left[\begin{array}{r}1+\frac{779a^2}{3392}+\frac{1055a^4}{\mathrm{13,568}}+\frac{137a^6}{\mathrm{10,176}}+\frac{2269a^8}{\mathrm{1,302,528}}+\frac{67a^{10}}{\mathrm{407,040}}+\\ \frac{a^{12}}{\mathrm{92,160}}+\frac{a^{14}}{\mathrm{2,442,240}}\end{array}\right]e^{-a^2/2}\\ \end{array}$$

(134)

The variation, with the input signal amplitude, of the harmonic distortion, as defined by $c_{4,k}^2/c_{4,1}^2$, is shown in Figure 23.

7.7. Linear Filtering of an Error Function Step Signal

Consider the case of a practical step input signal that is modeled by the error function, $\mathrm{erf}\left(t/\gamma \right)$ and the case where such a signal is input to a 2nd-order linear filter with a transfer function defined by

$$H(s)=\frac{1}{{\left[1+\frac{s}{2\pi f_p}\right]}^2} \iff h(t)=\frac{t{e}^{-t/\tau }}{{\tau }^2}\cdot u(t),\hspace{1em} \tau =\frac{1}{2\pi f_p}.$$

(135)

Theorem 10. 

Linear Filtering of an Error Function Signal. The output of a second-order linear filter, defined by Equation (135), to an error function input signal, defined by$\mathrm{erf}\left(t/\gamma \right)$, is

$$\begin{array}{ll}y(t)=& \mathrm{erf}\left[\frac{t}{\gamma }\right]u(t)+\pi \theta \\ & \frac{e^{-t/\tau }}{\tau }\cdot \left[\begin{array}{r}\left[\frac{{\gamma }^2}{2\tau }-(t+\tau )\right]\exp \left[\frac{{\gamma }^2}{4{\tau }^2}\right]\left[\mathrm{erf}\left[\frac{\gamma }{2\tau }\right]-\mathrm{erf}\left[\frac{\gamma }{2\tau }-\frac{t}{\gamma }\right]\right]-\\ \frac{\gamma }{\sqrt{\pi }}\exp \left[\frac{{\gamma }^2}{4{\tau }^2}\right]\exp \left[-{\left[\frac{t}{\gamma }-\frac{\gamma }{2\tau }\right]}^2\right]+\frac{\gamma }{\sqrt{\pi }}\end{array}\right]u(t)\end{array}$$

(136)

and can be approximated by the nth-order signal

$$\begin{array}{ll}{y}_n(t)=& f_n\left[\frac{t}{\gamma }\right]u(t)+\\ & \frac{e^{-t/\tau }}{\tau }\cdot \left[\begin{array}{r}\left[\frac{{\gamma }^2}{2\tau }-(t+\tau )\right]\exp \left[\frac{{\gamma }^2}{4{\tau }^2}\right]\left[f_n\left[\frac{\gamma }{2\tau }\right]-f_n\left[\frac{\gamma }{2\tau }-\frac{t}{\gamma }\right]\right]-\\ \frac{\gamma }{\sqrt{\pi }}\exp \left[\frac{{\gamma }^2}{4{\tau }^2}\right]\exp \left[-{\left[\frac{t}{\gamma }-\frac{\gamma }{2\tau }\right]}^2\right]+\frac{\gamma }{\sqrt{\pi }}\end{array}\right]u(t)\end{array}$$

(137)

where $f_n$ is defined by one of the approximations detailed in Theorems 4, 5, 6, or 7. It is the case that

$$\underset{n\to \infty }{\lim }{y}_n(t)=y(t),\hspace{1em} t\in (0,\infty ).$$

(138)

Proof. 

The proof is detailed in Appendix G. □

Results

For an input signal $\mathrm{erf}\left(t/\gamma \right)u\left(t\right)$, $\gamma =1/2$, input into a second-order linear filter with $f_p=1$, the output signal is shown in Figure 24. The relative errors in the approximations to the output signal are shown in Figure 25 for the case of approximations as specified by Equation (56) and with the use of optimum transition points.

7.8. Extension to Complex Case

By definition, the error function, for the general complex case, is defined according to

$$\mathrm{erf}(z)=\frac{2}{\sqrt{\pi }}{\displaystyle \underset{\gamma }{\int }{e}^{-{\lambda }^2}\mathrm{d}\lambda },\hspace{1em} z\in C,$$

(139)

where the path $\gamma$ is between the points zero and $z$ and is arbitrary. For the case of $z=x+jy$, and a path along the x axis to the point (x, 0) and then to the point $z$, the error function is defined according to Equation (5) of [37]:

$$\begin{array}{ll}\mathrm{erf}(x+jy)=& \frac{2}{\sqrt{\pi }}{\displaystyle \underset{0}{\overset{x}{\int }}{e}^{-{\lambda }^2}\mathrm{d}\lambda }+\frac{2j}{\sqrt{\pi }}{\displaystyle \underset{0}{\overset{y}{\int }}{e}^{-{(x+j\lambda )}^2}\mathrm{d}\lambda }\\ & =\mathrm{erf}(x)+\frac{2e^{-x^2}}{\sqrt{\pi }}\cdot {\displaystyle \underset{0}{\overset{y}{\int }}{e}^{{\lambda }^2}\sin (2x\lambda )\mathrm{d}\lambda }+\frac{2j{e}^{-x^2}}{\sqrt{\pi }}\cdot {\displaystyle \underset{0}{\overset{y}{\int }}{e}^{{\lambda }^2}\cos (2x\lambda )\mathrm{d}\lambda }\end{array}$$

(140)

Explicit approximations for $\mathrm{erf}(x+jy)$ then arise when integrable approximations for the two-dimensional surfaces $\exp (y^2)\sin (2xy)$ and $\exp (y^2)\cos (2xy)$ over $[0,x]\times [0,y]$ are available. Naturally, significant existing research exists, e.g., [28,37].

7.9. Approximation for the Inverse Error Function

There are many applications where the inverse error function is required and accurate approximations for this function are of interest. From the research underpinning this paper, the author’s view is that finding approximations to the inverse error function is best treated directly and as a separate problem, rather than approaching it via finding the inverse of an approximation to the error function.

8. Conclusions

This paper has detailed analytical approximations for the real case of the error function, underpinned by a spline-based integral approximation, which have significantly better convergence than the default Taylor series. The original approximations can be improved by utilizing the approximation $\mathrm{erf}(x)\approx 1$ for $x>x_o$, with $x_o$ being dependent on the order of approximation. The fourth-order approximations arising from Theorems 1 and 3, with respective transition points of $x_o=2.3715$ and $x_o=2.6305$, achieve relative error bounds over the interval $\left[0,\infty \right]$, respectively, of 1.03 × 10⁻³ and 2.28 × 10⁻⁴. The respective sixteenth-order approximations, with $x_o=3.9025$ and $x_o=4.101$, have relative error bounds of 3.44 × 10⁻⁸ and 6.66 × 10⁻⁹.

Further improvements were detailed via two generalizations. The first was based on utilizing integral approximations for each of the m equally spaced subintervals in the required interval of integration. The second was based on utilizing a fixed subinterval within the interval of integration with a known tabulated area, and then utilizing an integral approximation over the remainder of the interval. Both generalizations lead to significantly improved accuracy. For example, a fourth-order approximation based on four subintervals, with $x_o=3.7208$, achieves a relative error bound of 1.43 × 10⁻⁷ over the interval $\left[0,\infty \right]$. A sixteenth-order approximation, with $x_o=6.3726$, has a relative error bound of 2.01 × 10⁻¹⁹.

Finally, it was shown that a custom feedback system, with inputs defined by either the original error function approximations or approximations based on the use of subintervals, leads to analytical approximations with improved accuracy that are valid over the positive real line without utilizing the approximation $\mathrm{erf}(x)\approx 1$ for suitably large values of $x$. The original fourth-order error function approximation yields an approximation with a relative error bound of 1.82 × 10⁻⁵ over the interval $\left[0,\infty \right]$. The original sixteenth-order approximation yields an approximation with a relative error bound of 1.68 × 10⁻¹⁴.

Applications of the approximations include, first, approximations to achieve the specified error bounds of 10⁻⁴, 10⁻⁶, 10⁻¹⁰, and 10⁻¹⁶ over the positive real line. Second, the definitions of functions that are upper and lower bounds, of arbitrary accuracy, for the error function. Third, new series for the error function. Fourth, new sequences of approximations for $\exp (-x^2)$ that have significantly higher convergence properties than a Taylor series approximation. Fifth, a complementary demarcation function satisfying the constraint $e_C^2(x)+er{f}^2(x)=1$ was defined. Sixth, arbitrarily accurate approximations for the power and harmonic distortion for a sinusoidal signal subject to an error function nonlinearity. Seventh, approximate expressions for the linear filtering of a step signal that is modeled by the error function.