ERF in Excel: Calculate Error-Function Intervals with Explicit Bounds

• Introduced: documented by the official Excel 2010 change record, which lists improvements to this existing identity; its first release is not established.
• Availability: official help lists Excel for Microsoft 365, Excel for Microsoft 365 for Mac, Excel 2024, Excel 2024 for Mac, Excel 2021, Excel 2021 for Mac, Excel 2019, Excel 2016.
• Checked: October 14, 2026; official ERF reference. Documentation and independent models; no native Excel execution or wider platform certification.

=ERF(lower_limit,[upper_limit])

Lower_limit is required. Upper_limit is optional. Microsoft explicitly says that omitting the second argument makes the first argument the endpoint of an integral beginning at zero.

Consequently, =ERF(0.5) and =ERF(0.5,1) ask different questions. The first covers zero to 0.5; the second covers 0.5 to one. Do not assume that omitting an upper argument means infinity.

This is the mathematical error function, unrelated to catching spreadsheet errors with IFERROR.

Enter 0, 0.25, 0.5, 1, 1.5 and 2 in A2:A7. Put =ERF(A2) in B2 and fill down.

The table is rounded to nine decimals. Preserve the calculated B values for subsequent subtraction or totals.

These values describe the standard normalized error function. Its mathematical definition is two divided by SQRT(PI()) times the integral of EXP(−t²) from zero to the endpoint. The scale factor matters: it is not simply the unnormalized area under EXP(−t²).

In D2 enter 0.5 and in E2 enter one. Compare:

=ERF(D2,E2)
=ERF(E2)-ERF(D2)

The independent target for both is 0.322200915, rounded. This follows from subtracting two integrals that share the zero origin.

For a second ordered interval, D3=0.25 and E3=1.25, the independent target is approximately 0.646573738. Each row has its own lower and upper bound; do not mix one row’s endpoint with another’s start.

The worked intervals use nonnegative ordered bounds. Reversed limits, negative-input legacy behavior and competing-error precedence are not certified by this guide.

Enter these lower/upper pairs in D5:E8 and fill =ERF(D5,E5) down in F:

Summing the unrounded F results has independent target 0.995322265, equal to ERF(2) at this display precision. The subintervals touch without overlap and cover zero through two.

The intervals all have width 0.5, but their contributions differ. Multiplying the first contribution by four would not represent the full interval because the integrand changes with t.

Use =ABS(SUM(F5:F8)-ERF(2))<1E-9 as an explicitly chosen reconciliation target for this moderate fixture. This is an independent mathematical check, not a universal native precision guarantee.

ERFC describes the complementary error-function integral from a given endpoint to infinity. Mathematically, ERF(x)+ERFC(x)=1.

At x=1, ERF is approximately 0.842700793 and ERFC approximately 0.157299207. Their sum is one in the mathematical model.

For a very small complement, subtracting a displayed ERF value rounded to one would destroy the tail. Call the complementary function directly when the requested quantity is the tail, and retain scientific notation where useful. This advice does not assert which internal algorithm Excel uses.

ERF.PRECISE is a separately documented one-argument identity introduced in the Excel 2010 catalogue. It does not provide ERF’s optional second endpoint.

The official Excel 2010 change record lists algorithm improvements to ERF and lists ERF.PRECISE among new functions. That is documented history, not proof that every present ERF result is less accurate or that ERF has retired.

Both current worksheet references remain available. VBA-method signatures, historical add-in instructions and their negative-input notes are not substituted for this worksheet contract.

Microsoft assigns #VALUE! when either supplied bound is nonnumeric. The dedicated page does not separately resolve every numeric-looking text, Boolean or blank-reference conversion.

Its ERF(0.745) and ERF(1) examples print 0.70792892 and 0.84270079. Independent numerical integration agrees at that printed precision. All new values here are independent mathematical expectations; native Excel execution, extreme tails and cross-client numerical equivalence remain unverified.

• Microsoft ERF reference.
• Editorial testing standards.

Expected outputs are independently calculated examples or documented rules, not captured native Excel results. Formulas use English function names and comma argument separators. Locale, rendering and boundary restrictions are stated where relevant.