TANH

What it does

TANH(number)

TANH(number) is the hyperbolic tangent:

tanh x  =  sinh x / cosh x  =  (e^x - e^(-x)) / (e^x + e^(-x))  =  (e^(2x) - 1) / (e^(2x) + 1)

• Domain: all real numbers. No poles, no branch cuts, no excluded points. tanh is meromorphic with poles only at odd multiples of i*pi/2 — nowhere on the real line. This is the sharpest contrast with TAN, which has a pole every pi.
• Range: the open interval (-1, 1). Strictly increasing, hence a bijection from R onto (-1, 1), which is why ATANH has domain (-1, 1) and blows up at both ends.
• Parity: odd.
• Asymptotes: tanh x -> +1 as x -> +infinity and -> -1 as x -> -infinity, and the approach is exponential: 1 - tanh x ~ 2 e^(-2x).
• Series about zero: tanh x = x - x^3/3 + 2 x^5/15 - 17 x^7/315 + ..., radius of convergence pi/2 (the distance to the nearest complex pole). The coefficients are those of tan with alternating signs, which is the series form of tanh x = -i tan(ix).
• Identity: 1 - tanh^2 x = sech^2 x, the hyperbolic counterpart of the derivative relation for tan.

Microsoft's article gives the signature and states the defining formula. Retrieval of that page for this entry was blocked by the upstream host (HTTP 403), so the documented content is stated here as documented behaviour and should be re-checked against the live article. The mathematics above is from the standard literature, not from the article.

Arguments

No units. TANH is not an angle function and nothing in the RADIANS/DEGREES apparatus applies to it.

Try it

Quirks

Result and edge cases

Returns Number, always in (-1, 1) mathematically — and in floating point, exactly 1 or -1 for arguments beyond the saturation point, because 1 - 2e^(-2x) rounds to 1 once the correction falls below half an ULP of 1.

Small arguments round-trip: for |x| small enough that x^3/3 is negligible, the correctly rounded tanh x is x, so subnormals pass through.

Errors

How to implement

Numerical notes

TANH is the well-conditioned member of its family, but there is exactly one way to get it badly wrong and it is the obvious way.

Do not compute TANH as SINH(x)/COSH(x). Both intermediates overflow at a moderate argument while their ratio remains a perfectly ordinary number near 1. A naive quotient produces #NUM!, an infinity, or a NaN — depending on the host's overflow policy — for arguments where the answer is simply 1. The same objection applies to (exp(x) - exp(-x))/(exp(x) + exp(-x)).

The standard three-band scheme (Cody & Waite; fdlibm's tanh):

1. Tiny |x|: return x (with the inexact flag raised where the platform cares), because the correction is below rounding.
2. Small to moderate |x|: use expm1. With t = expm1(2|x|), tanh|x| = t/(t + 2); or with t = expm1(-2|x|), tanh|x| = -t/(t + 2). The expm1 form is what keeps relative accuracy near zero, where the direct exponential quotient would cancel 1 against 1.
3. Large |x|: return 1 (with the sign applied). There is a well-defined threshold above which the correctly rounded result is exactly 1, and computing anything at all beyond it is wasted work — and, if done naively, dangerous.

Near-zero cancellation, stated precisely. In (e^x - e^(-x))/(e^x + e^(-x)) the numerator suffers the same total cancellation as in SINH: both exponentials tend to 1 and their difference is 2x. The denominator is fine. So the relative error of the naive form grows without bound as x -> 0, exactly where the function is most linear and a user is most likely to trust it. The expm1 reformulation is not an optimization; it is the difference between a correct and an incorrect function on a large part of its useful domain.

Symmetry. Odd function: compute on |x|, apply the sign. An implementation that does so gets exact antisymmetry for free.

Why TANH is easy where TAN is hard. There is no argument reduction. There is no pi. There is no cancellation in the reduced argument because there is no reduced argument. Every difficulty TAN has comes from mapping the input into a fundamental domain, and TANH has no fundamental domain to map into. This is worth stating because the two functions are usually taught as siblings, and numerically they have almost nothing in common.

Rust code

21 tanh "TANH" (1, 1) [V "number"] |s, a, sv| {
    raw(s, a, ExcelRealPolicy::SATURATE_SIGN, k::tanh::tanh_kernel)
}

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-TANH-001: build 19929 versus build 20430; 1 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

• SINH and COSH — the numerator and denominator of the defining quotient. Note that TANH is well behaved precisely where SINH overflows: the two exponentials that overflow individually cancel in the ratio.
• ATANH — the inverse, ATANH(y) = ln((1+y)/(1-y))/2, with domain (-1, 1) and poles at both endpoints. The saturation of TANH and the singularity of ATANH are the same fact seen from two sides: once TANH has rounded to 1, the round trip through ATANH is lost.
• TAN — the circular sibling, tanh x = -i tan(ix). Bounded and pole-free versus unbounded and pole-ridden.
• SECH — 1 - TANH^2, and the reciprocal of COSH.
• Confused with: TAN, by name; and SIGN, by shape, since TANH of a large argument is a smooth saturating stand-in for the sign function. That resemblance is the reason TANH is the classical neural-network activation.

Named variants

No named variants are recorded yet.