SECH

What it does

SECH(number)

SECH(number) returns the hyperbolic secant.

sech x = 1 / cosh x = 2 / (eˣ + e⁻ˣ)

Domain. All real x — there are no poles, because cosh x ≥ 1 everywhere. Range. (0, 1], attaining 1 only at x = 0 and approaching 0 as |x| → ∞. The function is even, smooth, strictly decreasing in |x|, and asymptotically

sech x ~ 2e^{−|x|}   as |x| → ∞

Its derivative is −sech x · tanh x, its integral is the Gudermannian function gd(x) = arctan(sinh x), and its Maclaurin series is again the Euler-number generating function, this time with alternating signs:

sech x = Σ_{n≥0} E_{2n} x^{2n} / (2n)!  =  1 − x²/2 + 5x⁴/24 − 61x⁶/720 + …

convergent for |x| < π/2 — the radius set by the nearest complex pole at iπ/2, which is the only trace the poles of SEC leave on this function. (A&S 4.5.65.)

sech² is also, up to normalisation, the density of the hyperbolic secant distribution and the derivative of tanh, which is why this function turns up in logistic modelling and in neural network derivatives far more often than its trigonometric namesake turns up anywhere.

The contrast with SEC is the point of the page. SEC is unbounded, has infinitely many poles, is periodic, and is refused above a large-argument threshold. SECH is bounded by 1, has no poles, no periodicity, and no threshold. They differ by an i.

Arguments

Microsoft's page describes it as "Angle in radians", which is the standard documentation phrasing inherited from the circular functions; the hyperbolic argument is not an angle and is not in radians — it is a dimensionless real number, or, if one insists on a geometric reading, twice the area of a hyperbolic sector. The wording is harmless and worth not taking literally.

Microsoft's VBA page states no remarks and no error conditions.

Try it

Quirks

Result and edge cases

Returns Number in (0, 1].

• SECH(0) is exactly 1, and it is the maximum.
• SECH(−x) = SECH(x) exactly, for every representable x: the function is even and any reasonable implementation is symmetric by construction.
• Arrays. Elementwise lift, element-local failures.

Errors

How to implement

Numerical notes

sech is a well-conditioned function, and that is unusual enough to say plainly. Its condition number is |x tanh x|, which grows only linearly in |x| — compare the unbounded amplification of SEC near its poles. There is no cancellation anywhere: the denominator eˣ + e⁻ˣ is a sum of positive terms, so no digits are lost forming it. Near zero the series is alternating but the terms decay fast and the value is near 1, so relative accuracy is easy. The only difficulty in the entire function is range, not accuracy.

The premature-underflow band, and the fix. For |x| beyond about 710 the correct formulation is not 1/cosh(x) but

sech x = 2·e^{−|x|} / (1 + e^{−2|x|})

in which every intermediate is bounded: e^{−|x|} underflows gracefully rather than overflowing, and the denominator tends to 1. This is the standard Cody & Waite treatment of the hyperbolic functions applied in the reciprocal direction, and it extends the usable domain by the full ~35 units and gives correctly rounded subnormals throughout. An even simpler variant for the far tail is exp(ln 2 − |x|), which is a single exponential and is accurate to within the error of exp until the result underflows to zero at the true boundary.

The remaining question — one an implementation must answer explicitly — is what to return once even that underflows: +0 is the correct rounding of a positive value smaller than the smallest subnormal, and it is also what the naive route returns 35 units too early. The two zeros are indistinguishable at the value level, which is why this defect is so easy to ship.

Small-argument accuracy. For |x| < 2⁻²⁶, sech x rounds to exactly 1, and any implementation that computes 1/cosh(x) gets that for free because cosh rounds to 1 first. No special case is needed, which is a pleasant contrast with almost every other function in this batch.

Evenness should be structural. Fold the argument to |x| at the top of the kernel. An implementation that does not, and that has any asymmetry in its exponential, will return different values for x and −x — a defect that no accuracy test looking only at magnitudes will find.

Rust code

17 sech "SECH" (1, 1) [V "number"] |s, a, sv| { unary(s, a, k::sech::sech_kernel) }

pub fn sech_kernel(n: f64) -> Result<f64, WorksheetErrorCode> {
    // Live Excel 16.0 b20326: SECH(x)=1/COSH(x) 40/40 on the COSH EXP-pair grid.
    let result = crate::ported::excel_numeric::excel_x87_recip(crate::ported::cosh::cosh_kernel(n));
    Ok(if result.abs() < f64::MIN_POSITIVE { 0.0 } else { result })
}

No implementation has been published with a Handbook test suite for this function.

Version differences

The recorded environments below distinguish a reading from a same-input comparison. Counts describe captured inputs in each environment; overlapping captures and comparisons are not totals of unique worksheet calls.

• XC-SECH-001: build 19929 versus build 20430; 2 same-input pairs, 0 differing answers. Compared axes: excel-build. Numeric answers were compared by their recorded binary64 bits; other answers by typed value.

Agreement on these compared inputs does not establish agreement on unmeasured inputs.

Recorded environments and comparison details.

Alternatives

Relationships

• COSH — the reciprocal. SECH(x) and 1/COSH(x) should agree; whether they do, and whether both are available where the other is not, is the natural consistency probe. COSH overflows to #NUM! under the no-infinities convention while SECH of the same argument is a perfectly ordinary small number — so SECH is strictly more capable than the expression it is shorthand for, in exactly the band where it matters.
• CSCH — the other reciprocal hyperbolic, 1/sinh. Unlike SECH it does have a pole, at x = 0, and it shares SECH's non_finite=allow declaration — which for CSCH is a substantive question rather than a benign one.
• TANH — related by sech²x + tanh²x = 1, the hyperbolic Pythagorean identity, which gives the alternative evaluation sech x = √(1 − tanh²x). That route is well conditioned for large |x| (where tanh → ±1 and the subtraction cancels catastrophically — so it is in fact the worse route there) and fine near zero. The identity is more useful in the other direction.
• SEC — the circular namesake, sech(x) = sec(ix). Beyond the name and the Euler numbers in their series they have almost nothing in common.
• There is no ASECH. The inverse must be written ACOSH(1/x), defined for 0 < x ≤ 1.
• 2/(EXP(x)+EXP(-x)) — the hand-written form, which overflows earlier than SECH and should not be used.

Named variants

No named variants are recorded yet.