probl_engine/

report.rs

//! Collecting `report` values across worlds, and printing them
//! (docs/semantics.md, sections 9, 10 and 14).

use crate::value::{Value, fmt_float};
use crate::weight::Weight;
use probl_sema::ir::{ReportKind, ReportSite};
use rustc_hash::FxHashMap;
use std::collections::BTreeMap;
use std::fmt::Write;

mod results;

pub use results::{Fields, GroupResult, Numeric, Quantity, Reach, ReportResult, Status, Support, Uncertainty, results};

/// Everything one report site saw for one key.
#[derive(Clone, Debug)]
pub struct Acc {
    /// Weight of the worlds that reached the report.
    pub total: Weight,
    /// Of the reported facts, the weight where they're true.
    pub yes: Weight,
    /// Weight of the reported facts.
    pub facts: Weight,
    /// Weight of every other reported value.
    pub values: FxHashMap<Value, Weight>,
    /// Weight of reported distributions' missing mass.
    pub missing: Weight,
    /// When sampling: what each run of the current batch reported.
    runs: FxHashMap<u32, RunStat>,
    /// When sampling: sums over the finished runs, for standard errors.
    moments: Option<Moments>,
    continuous: bool,
    nonnumeric: bool,
    /// The values are records or lists that hold outcomes of draws, all
    /// with the same fields: the report summarizes each field on its own.
    structured: bool,
}

impl Default for Acc {
    fn default() -> Acc {
        Acc {
            total: Weight::ZERO,
            yes: Weight::ZERO,
            facts: Weight::ZERO,
            values: FxHashMap::default(),
            missing: Weight::ZERO,
            runs: FxHashMap::default(),
            moments: None,
            continuous: false,
            nonnumeric: false,
            structured: false,
        }
    }
}

/// Whether a value is a record, list or map, whose fields a structured
/// report summarizes.
fn is_aggregate(v: &Value) -> bool {
    matches!(v, Value::Record(_) | Value::List(_) | Value::Map(_))
}

/// The fields of a reported record, list or map, by path (`.name`, `[0]`,
/// `["key"]`), each with its value; anything else is one field, `""`.
pub(crate) fn fields(v: &Value) -> Vec<(String, Value)> {
    fn walk(v: &Value, path: &mut String, out: &mut Vec<(String, Value)>) {
        let mut child = |path: &mut String, name: String, x: &Value| {
            let len = path.len();
            path.push_str(&name);
            walk(x, path, out);
            path.truncate(len);
        };
        match v {
            Value::Record(r) => {
                for (k, x) in &r.fields {
                    child(path, format!(".{k}"), x);
                }
            }
            Value::List(xs) => {
                for (i, x) in xs.iter().enumerate() {
                    child(path, format!("[{i}]"), x);
                }
            }
            Value::Map(m) => {
                for (k, x) in m.iter() {
                    child(path, format!("[{}]", display(k)), x);
                }
            }
            leaf => out.push((path.clone(), leaf.clone())),
        }
    }
    let mut out = Vec::new();
    walk(v, &mut String::new(), &mut out);
    out
}

fn shape(v: &Value) -> Vec<String> {
    fields(v).into_iter().map(|(path, _)| path).collect()
}

/// Sums over sampled runs, for the standard error of an estimate
/// Σ wᵢaᵢ / Σ wᵢbᵢ by the delta method (docs/semantics.md, section 14):
/// Σ wᵢ²aᵢ² and Σ wᵢ²aᵢbᵢ, the second split into its positive and negative
/// terms (weights can't be negative).
#[derive(Clone, Copy, Debug, Default)]
struct Sums {
    aa: Weight,
    ab: Weight,
    ab_negative: Weight,
}

/// Σ wᵢbᵢ and Σ wᵢ²bᵢ², shared by the estimates with the same bᵢ.
#[derive(Clone, Copy, Debug, Default)]
struct Base {
    b: Weight,
    bb: Weight,
}

impl Base {
    fn add(&mut self, w: Weight, b: f64) {
        self.b += w.scale(b);
        self.bb += (w * w).scale(b * b);
    }

    fn absorb(&mut self, other: Base) {
        self.b += other.b;
        self.bb += other.bb;
    }
}

impl Sums {
    fn absorb(&mut self, other: Sums) {
        self.aa += other.aa;
        self.ab += other.ab;
        self.ab_negative += other.ab_negative;
    }

    fn add(&mut self, w: Weight, a: f64, b: f64) {
        let w2 = w * w;
        self.aa += w2.scale(a * a);
        if a * b >= 0.0 {
            self.ab += w2.scale(a * b);
        } else {
            self.ab_negative += w2.scale(-a * b);
        }
    }

    /// √(Σ wᵢ²(aᵢ − p bᵢ)²) / Σ wᵢbᵢ, expanded into the sums.
    fn standard_error(&self, base: &Base, p: f64) -> f64 {
        if base.b.is_zero() {
            return 0.0;
        }
        let d = base.b * base.b;
        let (aa, bb) = (self.aa.ratio(d), p * p * base.bb.ratio(d));
        let ab = self.ab.ratio(d) - self.ab_negative.ratio(d);
        let v = aa - 2.0 * p * ab + bb;
        // Below the rounding error of its terms, the variance is zero: every
        // run reported the same thing.
        if v <= 1e-12 * (aa + bb) {
            return 0.0;
        }
        v.sqrt()
    }
}

/// What the finished runs reported for one key, summed.
#[derive(Clone, Debug, Default)]
struct Moments {
    contributing_runs: u64,
    /// Wilson intervals require one ordinary Bernoulli observation per run
    /// and equal weights, rather than integrated or weighted contributions.
    not_bernoulli: bool,
    bernoulli_weight: Option<Weight>,
    successes: u64,
    integrated: bool,
    /// Facts: a is P(true), b the probability of a fact.
    facts: Base,
    yes: Sums,
    /// Numbers: a is the value (times its probability), b the probability of
    /// a number; Σ wᵢaᵢ in positive and negative parts, for the mean.
    numbers: Base,
    sum: Sums,
    sum_positive: Weight,
    sum_negative: Weight,
    /// Each value's probability: a is its share of a run's visits, b the
    /// number of visits.
    visits: Base,
    values: Vec<(Value, Sums)>,
}

impl Moments {
    fn absorb(&mut self, other: Moments) {
        self.not_bernoulli |= other.not_bernoulli
            || matches!((self.bernoulli_weight, other.bernoulli_weight), (Some(a), Some(b)) if a != b);
        self.bernoulli_weight = self.bernoulli_weight.or(other.bernoulli_weight);
        self.contributing_runs += other.contributing_runs;
        self.successes += other.successes;
        self.integrated |= other.integrated;
        self.facts.absorb(other.facts);
        self.yes.absorb(other.yes);
        self.numbers.absorb(other.numbers);
        self.sum.absorb(other.sum);
        self.sum_positive += other.sum_positive;
        self.sum_negative += other.sum_negative;
        self.visits.absorb(other.visits);
        for (v, sums) in other.values {
            match self.values.iter_mut().find(|(x, _)| *x == v) {
                Some((_, mine)) => mine.absorb(sums),
                None => self.values.push((v, sums)),
            }
        }
    }
}

/// What one sampled run reported for one key, added up over its visits: the
/// aᵢ and bᵢ of docs/semantics.md, section 14.
#[derive(Clone, Debug)]
pub struct RunStat {
    /// The run's weight (its final weight: no observation follows a report).
    pub weight: Weight,
    pub visits: u32,
    /// At least one visit supplied a distribution whose outcomes were
    /// integrated, instead of an ordinary sampled value.
    integrated: bool,
    /// The probability that the reported fact was true.
    pub yes: f64,
    /// The probability that the reported value was a fact.
    pub facts: f64,
    /// The reported numbers, times their probabilities.
    pub sum: f64,
    /// The probability that the reported value was a number.
    pub numbers: f64,
    /// The probability of each other value (dates excluded).
    pub others: Vec<(Value, f64)>,
}

impl RunStat {
    fn record(&mut self, value: &Value, share: f64) {
        match value {
            Value::Bool(b) => {
                self.facts += share;
                if *b {
                    self.yes += share;
                }
            }
            Value::Dist(d) => {
                for (x, q) in &d.outcomes {
                    self.record(x, share * q);
                }
            }
            Value::Int(_) | Value::Float(_) | Value::Prob(_) if value.as_f64().is_some() => {
                self.numbers += share;
                self.sum += share * value.as_f64().unwrap();
            }
            Value::Date(_) => {}
            other => match self.others.iter_mut().find(|(v, _)| v == other) {
                Some((_, s)) => *s += share,
                None => self.others.push((other.clone(), share)),
            },
        }
    }
}

impl Acc {
    fn add(&mut self, value: &Value, weight: Weight) {
        // A discrete outcome, like `floor(x)`, is each of its values.
        if let Value::Analytic(a) = value {
            if a.is_discrete() {
                for (v, _, share) in a.atoms() {
                    self.add(&v, weight.scale(share));
                }
                return;
            }
        }
        self.continuous |= matches!(value, Value::Analytic(_) | Value::Continuous(_));
        self.nonnumeric |= !matches!(
            value,
            Value::Int(_)
                | Value::Float(_)
                | Value::Prob(_)
                | Value::Analytic(_)
                | Value::Continuous(_)
                | Value::Dist(_)
                | Value::Joint(_)
        );
        match value {
            Value::Analytic(a) => {
                // Reports retain marginals only; merging identical laws here
                // avoids keeping one copy per independent draw/branch.
                let mut marginal = (**a).clone();
                marginal.id = 0;
                marginal.form = None;
                *self
                    .values
                    .entry(Value::Analytic(std::sync::Arc::new(marginal)))
                    .or_insert(Weight::ZERO) += weight;
            }
            Value::Event(e) => {
                self.facts += weight;
                self.yes += weight.scale(e.probability());
            }
            Value::Bool(b) => {
                self.facts += weight;
                if *b {
                    self.yes += weight;
                }
            }
            Value::Dist(d) | Value::Joint(d) => {
                self.missing += weight.scale(d.missing);
                for (x, q) in &d.outcomes {
                    self.add(x, weight.scale(*q));
                }
            }
            other if is_aggregate(other) && crate::analytic::contains(other) => {
                self.structured = true;
                let slot = self
                    .values
                    .entry(crate::analytic::marginal(other))
                    .or_insert(Weight::ZERO);
                *slot += weight;
            }
            other => {
                let slot = self.values.entry(other.clone()).or_insert(Weight::ZERO);
                *slot += weight;
            }
        }
    }

    /// Whether every reported value was a fact (or a distribution of facts).
    pub fn is_event(&self) -> bool {
        self.values.is_empty() && !self.facts.is_zero()
    }

    /// The probability that the reported fact is true, among the resolved
    /// worlds that reached the report.
    pub fn chance(&self) -> f64 {
        self.yes.ratio(self.facts)
    }

    /// Bounds on that probability, given `unresolved` weight elsewhere
    /// (docs/semantics.md, section 10).
    pub fn chance_bounds(&self, unresolved: Weight) -> (f64, f64) {
        let u = unresolved + self.missing;
        let denom = self.facts + u;
        (self.yes.ratio(denom), (self.yes + u).ratio(denom).min(1.0))
    }

    /// The reported values as a sorted distribution, conditional on being
    /// resolved (facts count as `true`/`false` when mixed with other values).
    pub fn distribution(&self) -> Vec<(Value, f64)> {
        let mut pairs: Vec<(Value, Weight)> = self.values.iter().map(|(v, w)| (v.clone(), *w)).collect();
        if !self.facts.is_zero() {
            pairs.push((Value::Bool(true), self.yes));
            pairs.push((Value::Bool(false), self.facts.saturating_sub(self.yes)));
        }
        let total = Weight::sum(pairs.iter().map(|(_, w)| *w));
        let mut out: Vec<(Value, f64)> = pairs.into_iter().map(|(v, w)| (v, w.ratio(total))).collect();
        out.retain(|(_, p)| *p > 0.0);
        out.sort_by(|a, b| a.0.cmp(&b.0));
        out
    }

    /// The share of this report's weight that isn't resolved.
    fn unresolved_share(&self, unresolved: Weight) -> f64 {
        let u = unresolved + self.missing;
        u.ratio(self.total + unresolved)
    }

    /// Add what another batch of runs reported for this key.
    fn absorb(&mut self, other: Acc) {
        debug_assert!(
            self.runs.is_empty() && other.runs.is_empty(),
            "batches end before they're combined"
        );
        self.total += other.total;
        self.yes += other.yes;
        self.facts += other.facts;
        self.missing += other.missing;
        self.continuous |= other.continuous;
        self.nonnumeric |= other.nonnumeric;
        self.structured |= other.structured;
        for (v, w) in other.values {
            *self.values.entry(v).or_insert(Weight::ZERO) += w;
        }
        match (&mut self.moments, other.moments) {
            (Some(mine), Some(theirs)) => mine.absorb(theirs),
            (mine @ None, theirs) => *mine = theirs,
            (Some(_), None) => {}
        }
    }

    /// Whether the values come from sampled runs.
    pub fn sampled(&self) -> bool {
        self.moments.is_some() || !self.runs.is_empty()
    }

    /// Fold the current batch's runs into the sums: they're finished.
    fn end_batch(&mut self) {
        if self.runs.is_empty() {
            return;
        }
        let m = self.moments.get_or_insert_with(Moments::default);
        for (_, run) in self.runs.drain() {
            let w = run.weight;
            m.contributing_runs += 1;
            m.not_bernoulli |= run.visits != 1
                || run.facts != 1.0
                || run.integrated
                || (run.yes != 0.0 && run.yes != 1.0)
                || m.bernoulli_weight.is_some_and(|previous| previous != w);
            m.bernoulli_weight.get_or_insert(w);
            m.successes += (run.yes == 1.0) as u64;
            m.integrated |= run.integrated;
            m.facts.add(w, run.facts);
            m.yes.add(w, run.yes, run.facts);
            m.numbers.add(w, run.numbers);
            m.sum.add(w, run.sum, run.numbers);
            if run.sum >= 0.0 {
                m.sum_positive += w.scale(run.sum);
            } else {
                m.sum_negative += w.scale(-run.sum);
            }
            let visits = run.visits as f64;
            m.visits.add(w, visits);
            let mut shares = run.others;
            if run.facts > 0.0 {
                shares.push((Value::Bool(true), run.yes));
                shares.push((Value::Bool(false), run.facts - run.yes));
            }
            for (v, share) in shares {
                match m.values.iter_mut().find(|(x, _)| *x == v) {
                    Some((_, sums)) => sums.add(w, share, visits),
                    None => {
                        let mut sums = Sums::default();
                        sums.add(w, share, visits);
                        m.values.push((v, sums));
                    }
                }
            }
        }
    }

    fn moments(&self) -> Moments {
        self.moments.clone().unwrap_or_default()
    }

    /// Number of independent runs contributing to this report/key. Multiple
    /// visits in one run count once.
    pub fn contributing_runs(&self) -> u64 {
        self.moments.as_ref().map_or(0, |m| m.contributing_runs)
    }

    /// When sampling: effective sample size using the report denominator's
    /// per-run contributions, including visit multiplicity.
    pub fn effective(&self) -> f64 {
        let m = self.moments();
        let base = if self.is_event() {
            m.facts
        } else if self.facts.is_zero()
            && !m.numbers.b.is_zero()
            && self
                .values
                .keys()
                .all(|v| matches!(v, Value::Int(_) | Value::Float(_) | Value::Prob(_)))
        {
            m.numbers
        } else {
            m.visits
        };
        if base.bb.is_zero() {
            return 0.0;
        }
        (base.b * base.b).ratio(base.bb).min(m.contributing_runs as f64)
    }

    fn chance_interval95(&self, kind: ReportKind) -> Option<(f64, f64)> {
        let m = self.moments.as_ref()?;
        if kind == ReportKind::PerVisit || m.not_bernoulli || m.contributing_runs == 0 {
            return None;
        }
        Some(wilson95(m.successes, m.contributing_runs))
    }

    /// When sampling: the standard error of `chance`.
    pub fn chance_se(&self) -> f64 {
        let m = self.moments();
        m.yes.standard_error(&m.facts, self.chance())
    }

    /// When sampling: the mean of the reported numbers and its standard error.
    pub fn mean_se(&self) -> (f64, f64) {
        let m = self.moments();
        let mean = m.sum_positive.ratio(m.numbers.b) - m.sum_negative.ratio(m.numbers.b);
        (mean, m.sum.standard_error(&m.numbers, mean))
    }

    /// When sampling: the standard error of a value's probability `p`.
    pub fn value_se(&self, value: &Value, p: f64) -> f64 {
        let m = self.moments();
        let sums = m
            .values
            .iter()
            .find(|(x, _)| x == value)
            .map_or(Sums::default(), |(_, s)| *s);
        sums.standard_error(&m.visits, p)
    }
}

/// All values reported at one site, per `by` key (`()` without `by`).
#[derive(Clone, Debug, Default)]
pub struct Sink {
    pub groups: BTreeMap<Value, Acc>,
    /// Whether a group's values are records or lists of outcomes of draws.
    structured: bool,
    /// When sampling: Σ w and Σ w² over the runs that reached the report,
    /// each counted once.
    pub reached: Weight,
    pub reached_squares: Weight,
}

impl Sink {
    pub(crate) fn validate_analytic(&self, key: &Value, value: &Value) -> crate::error::OpResult<()> {
        fn kinds(v: &Value) -> (bool, bool) {
            match v {
                Value::Analytic(a) if a.is_discrete() => (false, true),
                Value::Analytic(_) | Value::Continuous(_) => (true, true),
                Value::Int(_) | Value::Float(_) | Value::Prob(_) => (false, true),
                Value::Dist(d) | Value::Joint(d) => d
                    .outcomes
                    .iter()
                    .map(|(v, _)| kinds(v))
                    .fold((false, true), |(a, b), (c, d)| (a || c, b && d)),
                _ => (false, false),
            }
        }
        let (mut analytic, mut numeric) = kinds(value);
        if let Some(acc) = self.groups.get(key) {
            analytic |= acc.continuous;
            numeric &= !acc.nonnumeric;
        }
        if analytic && !numeric {
            return Err(crate::analytic::unsupported(
                "mixing a continuous report with nonnumeric outcomes",
            ));
        }
        Ok(())
    }

    /// When enumerating: a record or list of outcomes of draws can only be
    /// reported with others of its shape, whose fields are summarized each
    /// on its own.
    #[inline(never)]
    pub(crate) fn validate_structure(&self, key: &Value, value: &Value) -> crate::error::OpResult<()> {
        let holds = |v: &Value| is_aggregate(v) && crate::analytic::contains(v);
        let structured = match value {
            Value::Dist(d) | Value::Joint(d) => d.outcomes.iter().any(|(v, _)| holds(v)),
            v => holds(v),
        };
        if !self.structured && !structured {
            return Ok(());
        }
        self.check_structure(key, value)
    }

    #[cold]
    #[inline(never)]
    fn check_structure(&self, key: &Value, value: &Value) -> crate::error::OpResult<()> {
        let items: Vec<&Value> = match value {
            Value::Dist(d) | Value::Joint(d) => d.outcomes.iter().map(|(v, _)| v).collect(),
            v => vec![v],
        };
        let acc = self.groups.get(key);
        // Other keys' values may be structured, and these not.
        let holds = |v: &&Value| is_aggregate(v) && crate::analytic::contains(v);
        if !acc.is_some_and(|a| a.structured) && !items.iter().any(holds) {
            return Ok(());
        }
        let mixed = || {
            crate::analytic::unsupported("reporting a record or list of continuous outcomes together with other values")
        };
        if acc.is_some_and(|a| !a.facts.is_zero()) {
            return Err(mixed());
        }
        // What was reported before, all of one shape once structured, must
        // fit too.
        let earlier: Vec<&Value> = match acc {
            Some(a) if !a.structured => a.values.keys().collect(),
            Some(a) => a.values.keys().take(1).collect(),
            None => Vec::new(),
        };
        let mut reference = None;
        for v in earlier.into_iter().chain(items) {
            if !is_aggregate(v) {
                return Err(mixed());
            }
            let s = shape(v);
            match &reference {
                Some(r) if *r != s => {
                    return Err(crate::analytic::unsupported(
                        "reporting records or lists of continuous outcomes with different fields",
                    ));
                }
                Some(_) => {}
                None => reference = Some(s),
            }
        }
        Ok(())
    }

    /// Record a value reported in a world, and when sampling, by which run.
    pub fn add(&mut self, key: Value, value: &Value, weight: Weight, run: Option<u32>) {
        let acc = self.groups.entry(key).or_default();
        acc.total += weight;
        acc.add(value, weight);
        self.structured |= acc.structured;
        if let Some(run) = run {
            let stat = acc.runs.entry(run).or_insert_with(|| RunStat {
                weight,
                visits: 0,
                integrated: false,
                yes: 0.0,
                facts: 0.0,
                sum: 0.0,
                numbers: 0.0,
                others: Vec::new(),
            });
            stat.visits += 1;
            stat.integrated |= matches!(value, Value::Dist(_));
            stat.record(value, 1.0);
        }
    }

    /// When sampling: fold the batch's runs into the sums. They're finished,
    /// so what's kept doesn't grow with the number of runs.
    pub fn end_batch(&mut self) {
        let mut runs: FxHashMap<u32, Weight> = FxHashMap::default();
        for acc in self.groups.values() {
            runs.extend(acc.runs.iter().map(|(id, r)| (*id, r.weight)));
        }
        for w in runs.into_values() {
            self.reached += w;
            self.reached_squares += w * w;
        }
        for acc in self.groups.values_mut() {
            acc.end_batch();
        }
    }

    /// Add what another batch of runs reported. Batches are combined in
    /// order, so the sums don't depend on which thread ran which batch.
    pub fn absorb(&mut self, other: Sink) {
        self.structured |= other.structured;
        self.reached += other.reached;
        self.reached_squares += other.reached_squares;
        for (key, acc) in other.groups {
            match self.groups.entry(key) {
                std::collections::btree_map::Entry::Occupied(mut mine) => mine.get_mut().absorb(acc),
                std::collections::btree_map::Entry::Vacant(slot) => {
                    slot.insert(acc);
                }
            }
        }
    }

    /// For a report without `by` of facts: the probability they're true.
    pub fn chance(&self) -> Option<f64> {
        let acc = self.groups.get(&Value::Unit)?;
        acc.is_event().then(|| acc.chance())
    }

    /// For a report without `by`: the reported values and their probabilities.
    pub fn distribution(&self) -> Vec<(Value, f64)> {
        self.groups.get(&Value::Unit).map(Acc::distribution).unwrap_or_default()
    }

    /// Total weight that reached the report.
    pub fn reach(&self) -> Weight {
        Weight::sum(self.groups.values().map(|a| a.total))
    }
}

#[derive(Clone, Copy, Debug)]
pub struct Format {
    /// Also print the simplest nearby fraction of each probability.
    pub fractions: bool,
    /// Weight left unresolved by the whole run.
    pub unresolved: Weight,
    /// The total weight of the worlds that finished the program.
    pub program_total: Weight,
    /// When sampling: the sum of the runs' squared weights.
    pub run_squares: Option<Weight>,
    /// The model may condition outer worlds. Equal weights observed so far
    /// alone do not establish an ordinary binomial sampling scheme.
    pub weighted: bool,
    /// Whether `program_total` is a denominator for each report's reach.
    /// It isn't when worlds failed before evidence they'd have met
    /// (docs/semantics.md, section 11): their weight doesn't count it.
    pub reach_known: bool,
    /// With `@integrate`: the tolerance to compute means and spreads that
    /// have no formula to, by numerical integration.
    pub integrate: Option<f64>,
}

/// Render reports, each with what its sink collected, in source order.
pub fn render(sites: &[ReportSite], sinks: &[Sink], format: Format) -> String {
    render_results(sites, &results(sites, sinks, format, format.unresolved), format)
}

/// Render reports from their [`results`], in source order: the renderer
/// only formats the numbers they hold.
pub fn render_results(sites: &[ReportSite], results: &[ReportResult], format: Format) -> String {
    let mut out = String::new();
    let mut simple: Vec<(String, String)> = Vec::new();
    let flush = |simple: &mut Vec<(String, String)>, out: &mut String| {
        let width = simple.iter().map(|(l, _)| l.chars().count()).max().unwrap_or(0);
        for (label, text) in simple.drain(..) {
            let pad = width - label.chars().count();
            writeln!(out, "{label}{}    {text}", " ".repeat(pad)).unwrap();
        }
    };
    for (site, result) in sites.iter().zip(results) {
        let mut label = site.label.clone();
        if site.kind == ReportKind::PerVisit {
            label.push_str(" (per visit)");
        }
        let reach = reach_note(result.reach, format);
        let structured = result.groups.iter().any(|g| g.fields.is_some());
        if site.key_label.is_none() && !structured {
            let text = match result.groups.first() {
                None => "(never reached)".to_string(),
                Some(group) => format!("{}{reach}", value_text(group, format)),
            };
            simple.push((label, text));
            continue;
        }
        if !simple.is_empty() {
            flush(&mut simple, &mut out);
        }
        // A blank line before the table, unless one ends the table before.
        if !out.is_empty() && !out.ends_with("\n\n") {
            out.push('\n');
        }
        if structured {
            out.push_str(&fields_text(&label, &reach, site.key_label.as_deref(), result, format));
            out.push('\n');
            continue;
        }
        writeln!(out, "{label}{reach}").unwrap();
        if result.groups.is_empty() {
            writeln!(out, "  (never reached)").unwrap();
        } else {
            out.push_str(&table(site.key_label.as_deref().unwrap_or(""), result, format));
        }
        out.push('\n');
    }
    flush(&mut simple, &mut out);
    while out.ends_with("\n\n") {
        out.pop();
    }
    out
}

/// A report of records or lists of outcomes of draws: under its label,
/// each field's own summary, for each `by` key.
fn fields_text(label: &str, reach: &str, key_label: Option<&str>, result: &ReportResult, format: Format) -> String {
    let mut out = String::new();
    writeln!(out, "{label} (marginal of each field){reach}").unwrap();
    for group in &result.groups {
        let indent = match key_label {
            Some(k) => {
                writeln!(out, "  {k} = {}", display(&group.key)).unwrap();
                "    "
            }
            None => "  ",
        };
        let Some(fields) = &group.fields else {
            writeln!(out, "{indent}{}", value_text(group, format)).unwrap();
            continue;
        };
        let width = fields.0.iter().map(|(path, _)| path.chars().count()).max().unwrap_or(0);
        for (path, field) in &fields.0 {
            let pad = " ".repeat(width - path.chars().count());
            writeln!(out, "{indent}{path}{pad}    {}", value_text(field, format)).unwrap();
        }
    }
    out
}

/// " (reached in 1.00% of worlds)" when a report sees part of the weight.
fn reach_note(reach: Option<Reach>, format: Format) -> String {
    let Some(reach) = reach else {
        return String::new();
    };
    if reach.share >= 0.99995 {
        return String::new();
    }
    if let Some(se) = reach.se {
        return format!(" (reached in {} of runs)", estimate(reach.share, se));
    }
    format!(
        " (reached in {} of worlds)",
        pct(
            reach.share,
            Format {
                fractions: false,
                ..format
            }
        )
    )
}

fn value_text(group: &GroupResult, format: Format) -> String {
    if let Some(fact) = &group.fact {
        return chance_text(fact, format);
    }
    let dist = &group.distribution;
    let mut text = if let Some(numeric) = group.numeric.as_ref().filter(|n| n.mixture().is_some()) {
        analytic_stats(numeric)
    } else if dist.len() == 1 {
        display(&dist[0].0)
    } else if let Some(numeric) = &group.numeric {
        numeric_stats(numeric, dist)
    } else if dist.iter().all(|(v, _)| matches!(v, Value::Date(_))) {
        let [a, b, c] =
            [0.05, 0.5, 0.95].map(|q| summary_quantile(dist, q).map_or_else(|| "out of range".into(), |v| display(&v)));
        format!("5% {a} · median {b} · 95% {c}")
    } else if group.support.is_some() {
        categorical_sampled(group)
    } else {
        categorical(dist, format)
    };
    let share = group.unresolved_share;
    if share >= 0.00005 {
        write!(
            text,
            " · {} unresolved",
            pct(
                share,
                Format {
                    fractions: false,
                    ..format
                }
            )
        )
        .unwrap();
    }
    text.push_str(&reliability_note(group.support));
    text
}

/// A probability, or the range it lies in when unresolved weight is visible;
/// when sampling, an estimate and its standard error.
fn chance_text(fact: &Quantity, format: Format) -> String {
    let p = fact.point.unwrap_or(f64::NAN);
    if let Some(sampling) = fact.sampling {
        let runs = thousands(sampling.support.contributing_runs as i64);
        let mut text = if let Some((lo, hi)) = sampling.wilson {
            format!(
                "{} (95% Wilson interval {}–{}; {runs} contributing runs)",
                pct(p, format),
                pct(lo, format),
                pct(hi, format),
            )
        } else if let (Status::Estimated, Some(se)) = (sampling.status, sampling.se) {
            estimate(p, se)
        } else {
            let note = if sampling.status == Status::IntegratedZero {
                "zero empirical MC error; integrated outcomes"
            } else {
                "MC error not estimable"
            };
            format!("{} ({note}; {runs} contributing runs)", pct(p, format))
        };
        text.push_str(&reliability_note(Some(sampling.support)));
        return text;
    }
    if let Some((lo, hi)) = fact.bounds.filter(|(lo, hi)| hi - lo >= 0.00005) {
        let plain = Format {
            fractions: false,
            ..format
        };
        return format!("{}–{}", pct(lo, plain), pct(hi, plain));
    }
    pct(p, format)
}

/// Two-sided 95% Wilson score interval for ordinary independent Bernoulli
/// observations. Not an interval for importance-weighted or integrated data.
fn wilson95(successes: u64, trials: u64) -> (f64, f64) {
    let n = trials as f64;
    let p = successes as f64 / n;
    let z2 = 1.959963984540054_f64.powi(2);
    let denominator = 1.0 + z2 / n;
    let center = (p + z2 / (2.0 * n)) / denominator;
    let half = (z2 * (p * (1.0 - p) / n + z2 / (4.0 * n * n))).sqrt() / denominator;
    (
        if successes == 0 { 0.0 } else { (center - half).max(0.0) },
        if successes == trials {
            1.0
        } else {
            (center + half).min(1.0)
        },
    )
}

fn reliability_note(support: Option<Support>) -> String {
    let Some(support) = support.filter(|s| s.effective < 30.0) else {
        return String::new();
    };
    format!(
        " (low sample support: {} contributing runs; effective sample size {})",
        thousands(support.contributing_runs as i64),
        fixed(support.effective, 1)
    )
}

fn categorical(dist: &[(Value, f64)], format: Format) -> String {
    let mut by_chance: Vec<&(Value, f64)> = dist.iter().collect();
    by_chance.sort_by(|a, b| b.1.total_cmp(&a.1).then_with(|| a.0.cmp(&b.0)));
    let shown = by_chance.len().min(12);
    let mut parts: Vec<String> = by_chance[..shown]
        .iter()
        .map(|(v, p)| format!("{} {}", display(v), pct(*p, format)))
        .collect();
    if by_chance.len() > shown {
        parts.push(format!("… {} more", by_chance.len() - shown));
    }
    parts.join(" · ")
}

/// Sampled values with their probabilities and standard errors.
fn categorical_sampled(group: &GroupResult) -> String {
    let mut by_chance: Vec<&(Value, f64)> = group.distribution.iter().collect();
    by_chance.sort_by(|a, b| b.1.total_cmp(&a.1).then_with(|| a.0.cmp(&b.0)));
    let shown = by_chance.len().min(12);
    let mut parts: Vec<String> = by_chance[..shown]
        .iter()
        .map(|(v, p)| format!("{} {}", display(v), sampled_value_text(group, v, *p)))
        .collect();
    if by_chance.len() > shown {
        parts.push(format!("… {} more", by_chance.len() - shown));
    }
    parts.join(" · ")
}

/// A sampled value's probability `p` (zero for a value the group never
/// reported), with its standard error.
fn sampled_value_text(group: &GroupResult, value: &Value, p: f64) -> String {
    let quantity = group.values.iter().flatten().find(|(v, _)| v == value).map(|(_, q)| q);
    match quantity
        .and_then(|q| q.sampling)
        .filter(|s| s.status == Status::Estimated)
    {
        Some(Uncertainty { se: Some(se), .. }) => estimate(p, se),
        _ => format!("{}% (MC error not estimable)", fixed(p * 100.0, 2)),
    }
}

/// A sampled probability and its standard error, rounded to the error's
/// precision: `46.1% ± 0.3%` (docs/semantics.md, section 14).
pub fn estimate(p: f64, se: f64) -> String {
    let se = se * 100.0;
    let decimals = if se < 0.005 {
        2
    } else {
        (-libm::log10(se).floor()).clamp(0.0, 2.0) as usize
    };
    format!("{} ± {}%", fmt_percent(p, decimals), fixed(se, decimals))
}

/// Numeric components of a report containing a continuous marginal.
/// These are report summaries, not conversions of scalar outcomes in programs.
pub fn analytic_mixture(dist: &[(Value, f64)]) -> Option<crate::continuous::Mixture> {
    use crate::continuous::{Mixture, Part};
    if !dist
        .iter()
        .any(|(v, _)| matches!(v, Value::Analytic(_) | Value::Continuous(_)))
    {
        return None;
    }
    let parts = dist
        .iter()
        .map(|(v, p)| {
            let part = match v {
                Value::Analytic(a) => Part::Analytic((**a).clone()),
                Value::Continuous(f) => Part::Continuous(**f),
                v => Part::Point(v.as_f64()?),
            };
            Some((part, *p))
        })
        .collect::<Option<_>>()?;
    Some(Mixture { parts })
}

/// A continuous marginal's quantile, by the report convention.
fn mixture_quantile(numeric: &Numeric, q: f64, decimals: usize) -> String {
    fixed(numeric.quantile(q).and_then(|x| x.point).unwrap_or(f64::NAN), decimals)
}

fn analytic_stats(numeric: &Numeric) -> String {
    let (mean, sd) = (numeric.mean.point, numeric.sd.point);
    let scale = match (mean, sd) {
        (Some(m), Some(s)) => m.abs().max(s),
        _ => numeric.quantile(0.95).and_then(|x| x.point).unwrap_or(0.0).abs(),
    };
    let decimals = if scale < 100.0 { 2 } else { 0 };
    let [a, b, c] = [0.05, 0.5, 0.95].map(|q| mixture_quantile(numeric, q, decimals));
    // Some outcomes' moments have no formula yet: their quantiles do.
    let shown = |x: Option<f64>| x.map_or("unavailable".to_string(), |x| fixed(x, decimals));
    format!(
        "mean {} · sd {} · 5% {a} · median {b} · 95% {c}",
        shown(mean),
        shown(sd)
    )
}

/// `mean · sd · 5% · median · 95%`, and a sparkline for small integer ranges.
/// Probabilities (as values, not facts) are shown as percentages. A sampled
/// mean shows its standard error when it's visible at the printed precision.
fn numeric_stats(numeric: &Numeric, dist: &[(Value, f64)]) -> String {
    let nums = numeric.points().unwrap_or_default();
    let percent = numeric.percent;
    let total: f64 = nums.iter().map(|(_, p)| p).sum();
    let mean = numeric.mean.point.unwrap_or(f64::NAN);
    let sd = numeric.sd.point.unwrap_or(f64::NAN);
    let mean_se = numeric.mean.sampling.and_then(|s| s.se);
    let show = |x: f64, decimals: usize| {
        if percent { fmt_percent(x, 2) } else { fixed(x, decimals) }
    };
    let decimals = if mean.abs().max(sd) < 100.0 { 2 } else { 0 };
    // Mixed-sign sums can leave a platform-dependent residue below their
    // floating-point summation resolution. Mark that summary approximately
    // zero, without changing the value or hiding genuinely small-scale data.
    let roundoff_factor = (nums.len() as f64 + 2.0) * f64::EPSILON;
    let absolute_mean = nums.iter().map(|(x, p)| x.abs() * p).sum::<f64>() / total;
    let cancellation = nums.iter().any(|(x, _)| *x < 0.0)
        && nums.iter().any(|(x, _)| *x > 0.0)
        && mean.abs() <= absolute_mean * roundoff_factor / (1.0 - roundoff_factor);
    let shown_mean = if cancellation {
        if percent { "≈0%" } else { "≈0" }.to_string()
    } else {
        show(mean, decimals)
    };
    let [a, b, c] = [0.05, 0.5, 0.95].map(|q| {
        let Some(v) = numeric.quantile_value(q) else {
            return "out of range".to_string();
        };
        if percent {
            show(v.as_f64().unwrap(), 2)
        } else {
            number(&v, decimals)
        }
    });
    let mean_text = match mean_se {
        Some(se)
            if se > 0.0
                && (se * if percent { 100.0 } else { 1.0 } >= 0.5 * libm::pow(10.0, -(decimals as f64))
                    || (mean != 0.0 && mean.abs() * if percent { 100.0 } else { 1.0 } < 0.005)) =>
        {
            format!("{} ± {}", shown_mean, show(se, decimals))
        }
        _ => shown_mean,
    };
    let mut text = format!(
        "mean {mean_text} · sd {} · 5% {a} · median {b} · 95% {c}",
        show(sd, decimals)
    );
    if let Some(spark) = sparkline(dist) {
        write!(text, " · {spark}").unwrap();
    }
    text
}

const BLOCKS: [char; 8] = ['▁', '▂', '▃', '▄', '▅', '▆', '▇', '█'];

/// `2 ▂▃▄▆▇█▇▆▄▃▂ 12` for integers spanning at most 25 values.
fn sparkline(dist: &[(Value, f64)]) -> Option<String> {
    let ints: Vec<(i64, f64)> = dist
        .iter()
        .map(|(v, p)| match v {
            Value::Int(i) => i.to_i64().map(|n| (n, *p)),
            _ => None,
        })
        .collect::<Option<_>>()?;
    let (lo, hi) = (ints.first()?.0, ints.last()?.0);
    if hi as i128 - lo as i128 + 1 > 25 || hi == lo {
        return None;
    }
    let max = ints.iter().map(|(_, p)| *p).fold(0.0, f64::max);
    let by_value: BTreeMap<i64, f64> = ints.into_iter().collect();
    let bars: String = (lo..=hi)
        .map(|i| match by_value.get(&i) {
            Some(&p) if p > 0.0 => BLOCKS[((p / max * 8.0).ceil() as usize).clamp(1, 8) - 1],
            _ => ' ',
        })
        .collect();
    Some(format!("{lo} {bars} {hi}"))
}

/// The `q` quantile of a report's values, for its summary. The median uses
/// the same midpoint rule as `median()`; the other columns select outcomes.
/// An unrepresentable fractional bigint midpoint is labelled rather than
/// silently replaced with the lower bound.
fn summary_quantile(dist: &[(Value, f64)], q: f64) -> Option<Value> {
    if q != 0.5 {
        return Some(quantile(dist, q));
    }
    let (lo, hi) = crate::stats::median_bounds(dist)?;
    if lo == hi {
        return Some(lo.clone());
    }
    crate::builtins::midpoint(lo, hi, &mut crate::dist::Budget::unlimited()).ok()
}

fn quantile(dist: &[(Value, f64)], q: f64) -> Value {
    crate::stats::quantile(dist, q)
        .expect("nonempty report population")
        .clone()
}

/// A report table: one row per `by` key.
fn table(key_label: &str, result: &ReportResult, format: Format) -> String {
    let groups = &result.groups;
    let keys: Vec<String> = groups.iter().map(|g| display(&g.key)).collect();
    let mut rows: Vec<Vec<String>> = Vec::new();
    let header: Vec<String>;
    let all_facts = groups.iter().all(|g| g.fact.is_some());

    if all_facts {
        header = Vec::new();
        for (key, group) in keys.iter().zip(groups) {
            rows.push(vec![key.clone(), value_text(group, format)]);
        }
    } else {
        let numeric = groups.iter().all(|g| {
            g.distribution.iter().all(|(v, _)| {
                matches!(
                    v,
                    Value::Int(_) | Value::Float(_) | Value::Analytic(_) | Value::Continuous(_)
                )
            })
        });
        if numeric {
            header = ["5%", "25%", "median", "75%", "95%"].map(String::from).to_vec();
            for (key, group) in keys.iter().zip(groups) {
                if let Some(numeric) = group.numeric.as_ref().filter(|n| n.mixture().is_some()) {
                    let (mean, sd) = (
                        numeric.mean.point.unwrap_or(f64::NAN),
                        numeric.sd.point.unwrap_or(f64::NAN),
                    );
                    let decimals = if mean.abs().max(sd) < 100.0 { 2 } else { 0 };
                    let mut row = vec![key.clone()];
                    row.extend([0.05, 0.25, 0.5, 0.75, 0.95].map(|q| mixture_quantile(numeric, q, decimals)));
                    rows.push(row);
                    continue;
                }
                let d = &group.distribution;
                let scale = d
                    .iter()
                    .filter_map(|(v, _)| v.as_f64())
                    .fold(0.0f64, |m, x| m.max(x.abs()));
                let decimals = if scale < 100.0 { 2 } else { 0 };
                let mut row = vec![key.clone()];
                row.extend(
                    [0.05, 0.25, 0.5, 0.75, 0.95].map(|q| {
                        summary_quantile(d, q).map_or_else(|| "out of range".into(), |v| number(&v, decimals))
                    }),
                );
                rows.push(row);
            }
        } else {
            let mut columns: Vec<Value> = groups
                .iter()
                .flat_map(|g| g.distribution.iter().map(|(v, _)| v.clone()))
                .collect();
            columns.sort();
            columns.dedup();
            header = columns.iter().map(display).collect();
            for (key, group) in keys.iter().zip(groups) {
                let mut row = vec![key.clone()];
                for c in &columns {
                    let p = group.distribution.iter().find(|(v, _)| v == c).map_or(0.0, |(_, p)| *p);
                    row.push(if group.support.is_some() {
                        sampled_value_text(group, c, p)
                    } else {
                        pct(p, format)
                    });
                }
                rows.push(row);
            }
        }
    }

    // Reliability belongs to a key, not to the whole table. Include it for
    // numeric/date/categorical rows too, where no probability cell carries it.
    let mut header = header;
    if !all_facts && groups.iter().any(|g| !reliability_note(g.support).is_empty()) {
        if !header.is_empty() {
            header.push("reliability".to_string());
        }
        for (row, group) in rows.iter_mut().zip(groups) {
            row.push(reliability_note(group.support).trim().to_string());
        }
    }
    let mut all = Vec::new();
    if !header.is_empty() {
        let mut h = vec![key_label.to_string()];
        h.extend(header);
        all.push(h);
    }
    all.extend(rows);
    let ncols = all.iter().map(Vec::len).max().unwrap_or(0);
    let widths: Vec<usize> = (0..ncols)
        .map(|c| {
            all.iter()
                .filter_map(|r| r.get(c))
                .map(|s| s.chars().count())
                .max()
                .unwrap_or(0)
        })
        .collect();
    let mut out = String::new();
    for row in all {
        out.push_str("  ");
        let cells: Vec<String> = row
            .iter()
            .enumerate()
            .map(|(c, cell)| format!("{}{cell}", " ".repeat(widths[c] - cell.chars().count())))
            .collect();
        out.push_str(&cells.join("   "));
        out.push('\n');
    }
    out
}

// ── Number formatting ────────────────────────────────────────────────────

/// A probability as a percentage with two decimals.
pub fn pct(p: f64, format: Format) -> String {
    let text = fmt_percent(p, 2);
    let text = if text == "-0.00%" { "0.00%".to_string() } else { text };
    if format.fractions {
        if let Some((n, d)) = fraction(p) {
            if d > 1 {
                return format!("{text} (≈ {n}/{d})");
            }
        }
    }
    text
}

fn fmt_percent(p: f64, decimals: usize) -> String {
    // Keep a nonzero tail visible rather than rounding an interval endpoint
    // or a nearly certain event to exactly 100%.
    let decimals = if p > 0.0 && p < 1.0 && (1.0 - p) * 100.0 < 0.5 * libm::pow(10.0, -(decimals as f64)) {
        (-libm::log10((1.0 - p) * 100.0).floor() + 2.0).clamp(decimals as f64, 14.0) as usize
    } else {
        decimals
    };
    format!("{}%", fixed(p * 100.0, decimals))
}

/// The simplest fraction within 1e-13 of `x` with a denominator of at most a
/// million. It helps recognize an answer; it doesn't prove the answer is
/// exactly that fraction (docs/semantics.md, section 10).
pub fn fraction(x: f64) -> Option<(u64, u64)> {
    if !(0.0..=1.0).contains(&x) {
        return None;
    }
    let (mut h0, mut h1, mut k0, mut k1) = (0f64, 1f64, 1f64, 0f64);
    let mut v = x;
    for _ in 0..64 {
        let a = v.floor();
        let (h2, k2) = (a * h1 + h0, a * k1 + k0);
        if k2 > 1e6 {
            break;
        }
        (h0, h1, k0, k1) = (h1, h2, k1, k2);
        if (h1 / k1 - x).abs() < 1e-13 {
            return Some((h1 as u64, k1 as u64));
        }
        let frac = v - a;
        if frac < 1e-15 {
            break;
        }
        v = 1.0 / frac;
    }
    None
}

/// A value as reports print it.
pub fn display(v: &Value) -> String {
    match v {
        Value::Int(i) => integer_text(i),
        // Twelve significant digits hide rounding like 0.30000000000000004.
        Value::Float(f) if f.is_finite() => fmt_float(format!("{f:.11e}").parse().unwrap_or(*f)),
        Value::Float(f) => fmt_float(*f),
        other => other.to_string(),
    }
}

/// Integers with separators; floats with `decimals` decimals.
fn number(v: &Value, decimals: usize) -> String {
    match v {
        Value::Int(i) => integer_text(i),
        Value::Float(f) => fixed(*f, decimals),
        other => other.to_string(),
    }
}

fn fixed(x: f64, decimals: usize) -> String {
    if !x.is_finite() {
        return "out of range".into();
    }
    if x != 0.0 && x.abs() < 0.5 * libm::pow(10.0, -(decimals as f64)) {
        return format!("{x:.2e}");
    }
    let text = format!("{:.*}", decimals, x);
    let text = if text.starts_with('-') && text.trim_start_matches(['-', '0', '.']).is_empty() {
        text[1..].to_string()
    } else {
        text
    };
    let (int, frac) = match text.find('.') {
        Some(i) => (&text[..i], &text[i..]),
        None => (text.as_str(), ""),
    };
    let (sign, digits) = int.strip_prefix('-').map_or(("", int), |d| ("-", d));
    format!("{sign}{}{frac}", group(digits))
}

/// An integer with thousands separators: `50,000`.
pub fn thousands(i: i64) -> String {
    let digits = i.unsigned_abs().to_string();
    format!("{}{}", if i < 0 { "-" } else { "" }, group(&digits))
}

fn group(digits: &str) -> String {
    if digits.len() <= 3 {
        return digits.to_string();
    }
    let mut out = String::new();
    for (i, c) in digits.chars().enumerate() {
        if i > 0 && (digits.len() - i) % 3 == 0 {
            out.push(',');
        }
        out.push(c);
    }
    out
}

fn integer_text(i: &probl_number::Integer) -> String {
    let text = i.to_string();
    let (sign, digits) = text.strip_prefix('-').map_or(("", text.as_str()), |d| ("-", d));
    format!("{sign}{}", group(digits))
}

#[cfg(test)]
mod tests {
    use super::*;

    fn plain() -> Format {
        Format {
            fractions: false,
            unresolved: Weight::ZERO,
            program_total: Weight::ONE,
            run_squares: None,
            weighted: false,
            reach_known: true,
            integrate: None,
        }
    }

    #[test]
    fn fractions() {
        assert_eq!(fraction(244.0 / 495.0), Some((244, 495)));
        assert_eq!(fraction(1.0 / 6.0), Some((1, 6)));
        assert_eq!(fraction(0.5), Some((1, 2)));
        assert_eq!(fraction(std::f64::consts::FRAC_1_SQRT_2), None);
        let with = Format {
            fractions: true,
            ..plain()
        };
        assert_eq!(pct(244.0 / 495.0, with), "49.29% (≈ 244/495)");
    }

    #[test]
    fn numbers() {
        assert_eq!(thousands(1234567), "1,234,567");
        assert_eq!(thousands(-1000), "-1,000");
        assert_eq!(thousands(i64::MIN), "-9,223,372,036,854,775,808");
        assert_eq!(fixed(3.375, 2), "3.38");
        assert_eq!(fixed(-0.001, 2), "-1.00e-3");
        assert_eq!(fixed(-0.0, 2), "0.00");
        assert_eq!(fixed(92282.9, 0), "92,283");
        assert_eq!(pct(0.4929292929, plain()), "49.29%");
        assert_eq!(pct(1e-10, plain()), "1.00e-8%");
        assert_eq!(estimate(1e-10, 1e-12), "1.00e-8% ± 1.00e-10%");
        assert_eq!(number(&Value::Float(-1e-10), 2), "-1.00e-10");
        assert_ne!(pct(1.0 - 1e-10, plain()), "100.00%");
        assert_eq!(pct(1.0, plain()), "100.00%");
    }

    #[test]
    fn wilson_intervals_cover_boundaries_and_an_interior_reference() {
        let close = |a: f64, b: f64| assert!((a - b).abs() < 1e-12, "{a} != {b}");
        let (lo, hi) = wilson95(2, 2);
        close(lo, 0.342380227506653);
        assert_eq!(hi, 1.0);
        let (lo, hi) = wilson95(0, 1000);
        assert_eq!(lo, 0.0);
        close(hi, 0.0038267584855551234);
        let (lo, hi) = wilson95(50, 100);
        close(lo, 0.4038315303659956);
        close(hi, 0.5961684696340044);
        assert!(wilson95(0, 1).1 > 0.79);
    }

    #[test]
    fn report_counts_and_intervals_merge_across_batches() {
        let mut combined = Sink::default();
        for _ in 0..3 {
            let mut batch = Sink::default();
            for run in 0..1000 {
                batch.add(Value::Unit, &Value::Bool(true), Weight::ONE, Some(run));
            }
            batch.end_batch();
            combined.absorb(batch);
        }
        let acc = &combined.groups[&Value::Unit];
        assert_eq!(acc.contributing_runs(), 3000);
        assert!((acc.effective() - 3000.0).abs() < 1e-8);
        assert_eq!(acc.chance_interval95(ReportKind::Once), Some(wilson95(3000, 3000)));
        assert_eq!(acc.chance_interval95(ReportKind::PerVisit), None);

        // Each batch has equal weights, but their union does not.
        let mut weighted = Sink::default();
        weighted.add(Value::Unit, &Value::Bool(true), Weight::new(0.1), Some(0));
        weighted.end_batch();
        combined.absorb(weighted);
        let acc = &combined.groups[&Value::Unit];
        assert_eq!(acc.contributing_runs(), 3001);
        assert_eq!(acc.chance_interval95(ReportKind::Once), None);
    }

    #[test]
    fn repeated_visits_do_not_inflate_independent_sample_counts() {
        let mut sink = Sink::default();
        sink.add(Value::Unit, &Value::Bool(false), Weight::ONE, Some(0));
        for _ in 0..9 {
            sink.add(Value::Unit, &Value::Bool(true), Weight::ONE, Some(1));
        }
        sink.end_batch();
        let acc = &sink.groups[&Value::Unit];
        assert_eq!(acc.contributing_runs(), 2);
        assert!((acc.effective() - 100.0 / 82.0).abs() < 1e-12);
        assert_eq!(acc.chance_interval95(ReportKind::Once), None);
        let support = Support {
            contributing_runs: acc.contributing_runs(),
            effective: acc.effective(),
        };
        assert!(reliability_note(Some(support)).contains("2 contributing runs"));

        // A partially resolved numeric distribution contributes its resolved
        // mass to the mean's denominator, rather than one whole visit.
        let partial = crate::dist::Dist::from_pairs(vec![(Value::Float(1.0), 0.1)], 0.9).into_value();
        let mut numeric = Sink::default();
        numeric.add(Value::Unit, &partial, Weight::ONE, Some(0));
        numeric.add(Value::Unit, &Value::Float(1.0), Weight::ONE, Some(1));
        numeric.end_batch();
        assert!((numeric.groups[&Value::Unit].effective() - 1.21 / 1.01).abs() < 1e-12);
    }

    #[test]
    fn bounds_widen_with_unresolved_weight() {
        let mut sink = Sink::default();
        sink.add(Value::Unit, &Value::Bool(false), Weight::new(1e-5), None);
        let acc = &sink.groups[&Value::Unit];
        // Half a percent of the weight was cut before it could reach the report.
        let (lo, hi) = acc.chance_bounds(Weight::new(0.005));
        assert!(lo == 0.0 && (hi - 0.005 / (1e-5 + 0.005)).abs() < 1e-12);
        let format = Format {
            unresolved: Weight::new(0.005),
            ..plain()
        };
        let group = results::group(&Value::Unit, acc, format, ReportKind::Once, format.unresolved);
        let fact = group.fact.unwrap();
        assert!(!fact.complete && fact.bounds == Some((lo, hi)));
        assert_eq!(chance_text(&fact, format), "0.00%–99.80%");
    }
}