RAYLEIGHTAIL Function (Gnumeric)

• Introduced: Introduced in Gnumeric 1.1.7 release notes.
• Supported in: Gnumeric 1.12.62; no retirement is established.
• Scope: Desktop Gnumeric with its Statistics plugin. English comma-separated formulas are shown.
• Checked: October 14, 2026. Tables and the named advanced check use 1.12.57 ssconvert. Current 1.12.62 source was inspected; that release was not executed.

RAYLEIGHTAIL(x,threshold,scale) gives the Rayleigh density conditional on the value being at least the nonnegative threshold. Below the threshold it is zero; at or above it, the density is x/scale^2*EXP((threshold^2-x^2)/(2*scale^2)). It renormalizes the retained tail to total probability one. Versioned function help.

=RAYLEIGHTAIL(x,threshold,scale)

All three arguments are required. The threshold and scale are different parameters.

Current worksheet implementation.

The native column comes from saved recalculated formulas. The separate reference column evaluates the stated mathematical model, preserving approximation differences. Results are rounded to twelve decimal places, or thirteen significant digits for small nonzero values; printed precision is not a universal accuracy guarantee.

Suppose a magnitude model has scale 1 and only values at least 1 are retained. =RAYLEIGHTAIL(2,1,1) gives about 0.446260320297. The ordinary Rayleigh density at 2 is about 0.270670566473; conditioning increases it because the excluded portion has been removed and the remainder renormalized.

With magnitude A2, threshold B2 and scale C2, enter =RAYLEIGHTAIL(A2,B2,C2). Validate B2>=0 and C2>0 before using the result. A value below the threshold has zero conditional density, as the x = 0.5 example shows.

For x ≥ threshold, the conditional survival probability is EXP((threshold^2-x^2)/(2*scale^2)). At x = 2, threshold 1 and scale 1 this is about 0.223130160148, and the conditional CDF is about 0.776869839852. The density at that point is twice the survival probability, because x/scale^2 = 2.

Multiplying the conditional density by the original probability of passing the threshold recovers the ordinary density. The native check =RAYLEIGHTAIL(2,1,1)*EXP(-1/2)-RAYLEIGH(2,1) gave a tiny binary-rounding residual, about −1.110223024625E-16. The mathematical difference is zero. A threshold of zero reduces the distribution to ordinary Rayleigh.

The saved advanced-check result for =RAYLEIGHTAIL(2,1,1)*EXP(-1/2)-RAYLEIGH(2,1) was -1.110223024625E-16.

The negative-threshold rows are diagnostic observations outside the intended model. =RAYLEIGHTAIL(-1,-2,1) returned about −4.481689070338, which cannot be a valid density. Other invalid negative-threshold cases happened to return zero or one; apparently reasonable output does not repair the invalid assumptions.

Nonpositive scale produced #NUM!. A large positive x underflowed to zero. Numeric text was converted, nonnumeric text produced #VALUE!, and the empty x reference acted as zero. Missing or additional arguments produced #N/A. The current source retains the unchecked threshold path; newer native execution was not performed.

F5 is genuinely empty, rather than a cell containing an empty string.

• Current worksheet callback
• Tested-version worksheet callback
• Current random_rayleigh_tail_pdf calculation
• Tested-version random_rayleigh_tail_pdf calculation
• Nonfinite worksheet-value handling
• Statistics plugin declaration

The complete callback and inspected calculation helpers are unchanged between these source versions. Both stale and uncached SUM controls recalculated correctly. Native XML retains the function identities; secondary ODF export does not establish execution in another product. GUI entry, other locales and newer-release numerical behavior were not tested.

R.DRAYLEIGH, R.PRAYLEIGH, R.QRAYLEIGH.