Practical Method for log[erfc(a)] Approximation in the Extreme Tail

1. Introduction

Meta-analyses frequently combine results from multiple studies by summing z-scores, which are assumed to be approximately normally distributed. The sum distribution is also normal. Therefore, evaluating the combined p-value requires evaluating the Gaussian tail probabilities of the following form for large a:

$$erfc\left(a\right)={\int }_a^{\infty }\exp \left[-x^2\right]dx.$$

For a ≥ 8, direct computation is numerically unstable: extremely small values underflow to zero in standard double-precision arithmetic.

Existing approximation techniques for the complementary error function $erfc(a)$are almost exclusively based on analytic minimax polynomial approximations, rational-function approximations, Chebyshev-series expansions, or Padé-type expansions, as extensively documented in the NIST Digital Library of Mathematical Functions [1,2,3]. To our knowledge, no prior work has introduced a geometric construction—such as a tangent-based area argument—to derive explicit upper and lower bounds for erfc or $\mathrm{l}\mathrm{o}\mathrm{g}[erfc(a)]$in the extreme tail.

The main purpose of this work was to construct a method that (i) avoids numerical underflow; (ii) provides explicit lower and upper bounds; (iii) admits asymptotic expansions to arbitrary order; and (iv) is easy to implement in practical statistical workflows.

2. Methods

2.1. Geometric Motivation

Consider a positive, decreasing function $h=f(x)$, and point $A=(a, h)$on its graph. Let $s=f\text{'}(a)$ denote the slope, and let l be the horizontal distance from foot $O$ of the vertical through $A$ to x-intercept $P$ of the tangent line at $A$. Then,

$$l={\displaystyle \frac{h}{\left|f^{\text{'}}\left(a\right)\right|}}.$$

The area of triangle AOP is

$$Area\left(AOP\right)={\displaystyle \frac{hl}{2}}={\displaystyle \frac{f{\left(a\right)}^2}{2\left|f^{\text{'}}\left(a\right)\right|}}.$$

If $f$ decays approximately exponentially, then the integral ${\int }_a^{\infty }f\left(x\right)dx$ is comparable to rectangle $h\times l,$whose area is $f{\left(a\right)}^2/\left|f\text{'}(a)\right|$.

At a point of tangency $x=a$ on the exponential curve $y=b\exp [-cx]$, the tangent line and the perpendicular to the x-axis from a right triangle (Figure 1). The area of this triangle is exactly half of tail integral

$${\int }_a^{\infty }b\exp \left[-cx\right]dx$$

2.2. Gaussian Tail and Tangent-Matched Lower Bound

This geometric principle is applied to the Gaussian function. The triangle area provides approximately half of this value, suggesting a natural upper and lower bound pair. For the tangency point chosen, the Gaussian tail $erfc(x)$ (dotted line) lies between the exponential decay curve (solid line) and the area of the right triangle generated by the tangent line (dashed line), demonstrating that the Gaussian tail integral is geometrically sandwiched between two quantities.S

Let

$$f\left(x\right)=\exp \left[-x^2\right],$$

As $f^{\text{'}}\left(a\right)=-2a\exp \left[-a^2\right],$ the rectangle bound gives

$$p_{upper}\left(a\right)={\displaystyle \frac{f{\left(a\right)}^2}{\left|f^{\text{'}}\left(a\right)\right|}}={\displaystyle \frac{\exp \left[-a^2\right]}{2a}}.$$

Let us construct a function

$$g\left(x\right)=A\left(x-u\right)\exp \left[-{\left(x-u\right)}^2\right],$$

that satisfies

$$g\left(a\right)=f \left(a\right), g\text{'}(a)=f(a).$$

Let $t=a-u.$ Solving the resulting system gives

$$t={\displaystyle \frac{a+\sqrt{a^2+2}}{2}},$$

and

$$p_{lower}\left(a\right)={\int }_a^{\infty }g\left(x\right)dx={\displaystyle \frac{\exp \left[-a^2\right]}{2t}}={\displaystyle \frac{\exp \left[-a^2\right]}{a+\sqrt{a^2+2}}}$$

Via construction,

$$p_{lower}\left(a\right)<erfc\left(a\right)<p_{upper}\left(a\right), a>0$$

As $t>a$, the denominators satisfy $a+\sqrt{a^2+2}>2a$; therefore, $p_{\mathrm{lower}}(a)<p_{\mathrm{upper}}(a)$.

Both bounds are explicit and extremely stable in log-space:

$$\log \left[p_{\mathrm{upper}}\left(a\right)\right]=-a^2-\log \left(2a\right),$$

$$\log \left[p_{\mathrm{lower}}(a)\right]=-a^2-\log \left[a+\sqrt{a^2+2}\right].$$

2.3. Asymptotic Expansions Up to $a^{-8}$

Let $a\to \mathrm{\infty }$. In classical form (Mills’ ratio expansion) we get

$$\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{c}(a)\sim {\displaystyle \frac{e^{-a^2}}{2a}}\left(1-{\displaystyle \frac{1}{2a^2}}+{\displaystyle \frac{3}{4a^4}}-{\displaystyle \frac{15}{8a^6}}+{\displaystyle \frac{105}{16a^8}}+O(a^{-10})\right).$$

Our two bounding expressions admit clean asymptotics in the same form.

The upper bound is in the following form:

$$p_{\mathrm{upper}}(a)={\displaystyle \frac{e^{-a^2}}{2a}}.$$

This is exactly the leading term of the Gaussian tail asymptotics.

Using

$$p_{\mathrm{lower}}(a)={\displaystyle \frac{e^{-a^2}}{a+\sqrt{a^2+2}}}={\displaystyle \frac{e^{-a^2}}{2a}}\cdot {\displaystyle \frac{2}{1+\sqrt{1+\frac{2}{a^2}}}},$$

$\sqrt{1+\epsilon }=1+{\displaystyle \frac{\epsilon }{2}}-{\displaystyle \frac{{\epsilon }^2}{8}}+{\displaystyle \frac{{\epsilon }^3}{16}}-{\displaystyle \frac{5{\epsilon }^4}{128}}+O({\epsilon }^5)$ is expanded with $\epsilon =2/a^2$ and inverted to obtain

$${\displaystyle \frac{2}{1+\sqrt{1+\frac{2}{a^2}}}}=1-{\displaystyle \frac{1}{2a^2}}+{\displaystyle \frac{1}{2a^4}}-{\displaystyle \frac{5}{8a^6}}+{\displaystyle \frac{7}{8a^8}}+O(a^{-10}).$$

Therefore,

$$p_{lower}\left(a\right)\sim {\displaystyle \frac{e^{-a^2}}{2a}}\left(1-{\displaystyle \frac{1}{2a^2}}+{\displaystyle \frac{3}{4a^4}}-{\displaystyle \frac{15}{8a^6}}+{\displaystyle \frac{7}{16a^8}}+O(a^{-10})\right).$$

These formulations show that both bounds agree with the exact tail of the first two terms: the lower bound differs starting at $a^{-4}$, whereas the upper bound is the leading term only. In practice, either bound can be used as the starting point for controlled asymptotic refinement.

Table 1 compares $\log \left[erfc(a)\right] ,$ $\log \left[p_{\mathrm{l}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r}}(\mathrm{a})\right]$, and $\log \left[p_{upper}(a)\right]$ for $a = 1, . . . , 20$, with all logarithms being natural. As expected, $\mathrm{l}\mathrm{o}\mathrm{g}{p}_{\mathrm{lower}}(a)<\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{c}(a)<\mathrm{l}\mathrm{o}\mathrm{g}{p}_{\mathrm{upper}}(a)$, and the gap decreases rapidly as $a$ increases. Accurate Gaussian tail values were obtained using the scipy.special.erfc function from the SciPy library (Python).

3. Practical Use and Numerical Stability

Underflow-safe evaluation. The following are always computed in log-space:

$$\log \left[p_{\mathrm{upper}}\left(a\right)\right]=-a^2-\log \left[2a\right],$$

$$\log \left[p_{\mathrm{lower}}(a)\right]=-a^2-\mathrm{l}\mathrm{o}\mathrm{g}[a+\sqrt{a^2+2}].$$

Asymptotic refinement. For very large $a$, we append the known asymptotic correction terms $\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{c}(a)$ to $a^{-8}$ for improved accuracy beyond the single-term upper bound.

Compatibility with the standard normal tail. If the one-sided Gaussian tail

$$Q(z)={\int }_z^{\mathrm{\infty }}{\displaystyle \frac{1}{\sqrt{2\pi }}}{e}^{-x^2/2} dx$$

is needed, then convert via $a=z/\sqrt{2}$ and scale accordingly:

$$Q(z)={\displaystyle \frac{1}{2}}\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{c}({\displaystyle \frac{z}{\sqrt{2}}}).$$

The same bounding strategy applies verbatim after this change of variables.

4. Discussion

The proposed tangent and rectangle bracketing method yields a pair of closed-form expressions, which are

• Simple (two elementary functions of $a$)
• Monotone ($p_{\mathrm{lower}}\le \mathrm{e}\mathrm{r}\mathrm{f}\mathrm{c}\le p_{\mathrm{upper}}$)
• Asymptotically sharp (both match the leading ${\displaystyle \frac{e^{-a^2}}{2a}}$, while the lower bound matches the next term $-{\displaystyle \frac{1}{2a^2}}$)
• Numerically robust in log-space for $a$ in the meta-analytic tail.

In workflows in which a strict bound is acceptable (e.g., to certify that a meta-analytic $p$-value is below a given threshold), the lower bound often suffices. For point estimates, either the midpoint in linear space or the log-mean in log-space provides a “free” estimator with guaranteed bracketing error. If higher accuracy is needed, the classical asymptotic series up to $a^{-8}$ may be added to the leading term; the resulting log can be evaluated via $\log \sum$ with log-sum-exp to avoid catastrophic cancellation.

5. Conclusions

Herein, we introduce a self-contained analytic approximation of $\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{c}(a)$ for large $a$ based on a geometric tangent argument. This method yields closed-form upper and lower bounds with asymptotic expansions up to $a^{-8}$. This is especially useful in meta-analyses in which large $z$-scores must be handled accurately and safely. This approach is straightforward to implement and integrate into statistical codebases.

Figure 2. The Gaussian tail (dotted line) is compared with the exponential function and its tangent-based triangular approximation. For the tangency point chosen, the Gaussian tail lies between the exponential decay curve (solid line) and the area of the right triangle generated by the tangent line (dashed line), demonstrating that the Gaussian tail integral is geometrically sandwiched between these two quantities.

Figure 2. The Gaussian tail (dotted line) is compared with the exponential function and its tangent-based triangular approximation. For the tangency point chosen, the Gaussian tail lies between the exponential decay curve (solid line) and the area of the right triangle generated by the tangent line (dashed line), demonstrating that the Gaussian tail integral is geometrically sandwiched between these two quantities.