WEIBULL

What it does

WEIBULL(x, alpha, beta, cumulative)

WEIBULL(x,alpha,beta,cumulative) evaluates a Weibull distribution at nonnegative x. Alpha is the positive shape; beta is the positive scale in the same units as x. TRUE returns a cumulative probability and FALSE a density. This is the compatibility spelling of WEIBULL.DIST.

Writing t=(x/beta)^alpha, the mathematical CDF is 1-exp(-t). In particular, at x=beta it is 1-exp(-1) for every positive shape. Shape one gives the exponential family with rate 1/beta.

Try it

=WEIBULL(4,2,2,TRUE) is about 0.9816843611: mathematically, (4/2)^2=4 and the CDF is 1-exp(-4). Changing TRUE to FALSE gives a density of about 0.03663127777746836.

The scale is not the exponential rate. In the shape-one case, a scale of two corresponds to rate one half, not rate two.

Quirks

Negative x or a nonpositive shape or scale gives #NUM!; nonnumeric numeric arguments give #VALUE!. Densities and cumulative probabilities are different quantities, even when they happen to have similar decimal values.

At zero, the mathematical CDF is zero. The density’s right-hand limit is zero for alpha>1, 1/beta for alpha=1, and unbounded for 0<alpha<1. The current helper returns zero at x=0 for both cumulative and density calls, for every admitted shape. For example, its WEIBULL(0,1,2,FALSE) result is zero rather than the mathematical one half. This is a code-level boundary behaviour, not a newly measured Excel answer; keep it separate from the positive-input reconstruction.

How to implement

For positive inputs, define a raw power helper P(a,b)=EXP(R(b*LN(a))), using the public stored LN and EXP leaves and the stored operation R. The current route uses this chain even for integer exponents and square roots; a host pow shortcut changes the candidate.

1. Validate finite inputs and the documented sign domain; handle the code’s zero-input branch separately.
2. Store r=R(x/beta), then obtain t=P(r,alpha).
3. For the CDF, compute -expm1_internal(-t). Here the Kahan correction uses ordinary binary64 arithmetic: with u=EXP(-t), return t if u==1; otherwise return -(u-1) for t>=1, or -((u-1)*(-t)/LN(u)) for t<1.
4. For density, obtain B=P(beta,alpha), X=P(x,alpha-1) and E=EXP(-t). Evaluate division first and multiply left to right: R(R(R(alpha/B)*X)*E).
5. The helper rejects a nonfinite result with #NUM!. Keep worksheet publication separate from the polynomial/power body.

For x=4,alpha=2,beta=2, the mathematical reduction has r=2, t=4, alpha/beta^alpha=0.5, and x^(alpha-1)=4; density is therefore 2*exp(-4). The raw power chain need not publish exact integer four at each intermediate stage, so the mathematical trace is not permission to remove those calls.

The cited source retains positive-input power-staging discriminants, including 113 controls separating the stored exponent product. Exact global branch boundaries and a whole-domain legacy-spelling result remain unestablished here. No Handbook suite is published for this function.

Rust code

62 weibull "WEIBULL" (4, 4) [V "x", V "alpha", V "beta", V "cumulative"] |s, a, sv| {
    let [x, alpha, beta] = nums::<3>(s, a)?;
    out(s, sd::weibull_kernel(x, alpha, beta, flag(s, a, 3)?))
}

pub fn weibull_kernel(
    x: f64,
    alpha: f64,
    beta: f64,
    cumulative: bool,
) -> Result<f64, WorksheetErrorCode> {
    weibull_dist_kernel(x, alpha, beta, cumulative)
}

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

Version differences

No same-input comparison between named Excel builds is recorded for this function here. The documented behaviour below is not a measurement of every Excel release.

Alternatives

WEIBULL.DIST is the modern spelling; EXPON.DIST is the shape-one special family; GAMMA.DIST supplies another positive-valued distribution.

Named variants

No named variants are recorded yet.