GD Function (Gnumeric)

• Introduced: Gnumeric 1.9.9 release notes explicitly record this function being added.
• Supported in: Gnumeric 1.12.62, with no retirement established.
• Scope: Desktop Gnumeric with the Mathematics plugin; English comma-separated formulas are shown.
• Checked: October 14, 2026. Reference and boundary tables plus the explicit advanced-check formula were recalculated through 1.12.57 ssconvert. Current 1.12.62 source was inspected; that later release was not executed.

GD evaluates the Gudermannian function, a relationship between circular and hyperbolic functions. The implementation uses 2×ATAN(TANH(x/2)). Its result is an angle in radians, not degrees and not a probability. Versioned function help.

=GD(x)

Exactly one numeric x is required. Both positive and negative real inputs are accepted. The function is odd, so GD(−x)=−GD(x), and GD(0)=0. For finite real x its exact mathematical value lies strictly between −PI()/2 and PI()/2. Function implementation.

Ordinary results are rounded to twelve decimal places; tiny nonzero values use scientific notation. The eight reference rows were independently checked.

At x=1, =GD(1) gives approximately 0.865769483240 radians. At −1 it gives the negative value. To compare a worksheet model with a degree-based report, convert the result deliberately using 180/PI() rather than changing the input arbitrarily.

Keep a column labeled “GD, radians” beside the original x values. If a later formula uses SIN or TAN, use the radian value directly; applying a degree conversion and then passing it into a radian function changes the calculation.

For real x, TAN(GD(x))=SINH(x) and SIN(GD(x))=TANH(x). The executed residual =TAN(GD(1))-SINH(1) returned 0. These identities connect the output to familiar hyperbolic functions and offer independent checks.

The inverse relation is x=ATANH(SIN(θ)) when θ lies strictly inside (−PI()/2,PI()/2). This domain condition matters: near either endpoint, a tiny angular change can correspond to a large change in x. It is not a safe numerical roundtrip for every extreme input.

The native values at ±1000 rounded to approximately ±1.570796326795, the floating-point representation of ±PI()/2. This is finite-precision saturation, even though the exact Gudermannian reaches those limits only as x tends to infinity.

Using such a rounded endpoint in an inverse formula can produce an infinite or invalid result. The source’s TANH-based formulation avoids explicitly computing EXP(1000), but it cannot eliminate rounding near the limiting angle. The empty-cell and Boolean examples below are coercion observations for this fixture, not an instruction to treat absent data as a meaningful zero.

F5 below is an actually empty cell. The listed errors are native error results, not mathematical zeroes.

• Worksheet entry
• Numeric or argument-handling source

The retained workbook results include a deliberately stale SUM cache and an uncached SUM formula, both recalculated correctly. Native evidence covers the named 1.12.57 build and listed cases; cell-reference layouts and other explanatory compositions are instructions or independent mathematical checks unless explicitly identified as executed. Other locales, direct GUI entry and later-release output were not tested.