VictoriaMetrics/app/vmselect/promql/rollup.go

package promql

import (
	"flag"
	"fmt"
	"math"
	"strings"
	"sync"

	"github.com/VictoriaMetrics/VictoriaMetrics/lib/decimal"
	"github.com/VictoriaMetrics/VictoriaMetrics/lib/logger"
	"github.com/VictoriaMetrics/VictoriaMetrics/lib/storage"
	"github.com/VictoriaMetrics/metrics"
	"github.com/VictoriaMetrics/metricsql"
	"github.com/valyala/histogram"
)

var minStalenessInterval = flag.Duration("search.minStalenessInterval", 0, "The minimum interval for staleness calculations. "+
	"This flag could be useful for removing gaps on graphs generated from time series with irregular intervals between samples. "+
	"See also '-search.maxStalenessInterval'")

var rollupFuncs = map[string]newRollupFunc{
	// Standard rollup funcs from PromQL.
	// See funcs accepting range-vector on https://prometheus.io/docs/prometheus/latest/querying/functions/ .
	"changes":            newRollupFuncOneArg(rollupChanges),
	"delta":              newRollupFuncOneArg(rollupDelta),
	"deriv":              newRollupFuncOneArg(rollupDerivSlow),
	"deriv_fast":         newRollupFuncOneArg(rollupDerivFast),
	"holt_winters":       newRollupHoltWinters,
	"idelta":             newRollupFuncOneArg(rollupIdelta),
	"increase":           newRollupFuncOneArg(rollupDelta),  // + rollupFuncsRemoveCounterResets
	"irate":              newRollupFuncOneArg(rollupIderiv), // + rollupFuncsRemoveCounterResets
	"predict_linear":     newRollupPredictLinear,
	"rate":               newRollupFuncOneArg(rollupDerivFast), // + rollupFuncsRemoveCounterResets
	"resets":             newRollupFuncOneArg(rollupResets),
	"avg_over_time":      newRollupFuncOneArg(rollupAvg),
	"min_over_time":      newRollupFuncOneArg(rollupMin),
	"max_over_time":      newRollupFuncOneArg(rollupMax),
	"sum_over_time":      newRollupFuncOneArg(rollupSum),
	"count_over_time":    newRollupFuncOneArg(rollupCount),
	"quantile_over_time": newRollupQuantile,
	"stddev_over_time":   newRollupFuncOneArg(rollupStddev),
	"stdvar_over_time":   newRollupFuncOneArg(rollupStdvar),
	"absent_over_time":   newRollupFuncOneArg(rollupAbsent),
	"present_over_time":  newRollupFuncOneArg(rollupPresent),

	// Additional rollup funcs.
	"default_rollup":        newRollupFuncOneArg(rollupDefault), // default rollup func
	"range_over_time":       newRollupFuncOneArg(rollupRange),
	"sum2_over_time":        newRollupFuncOneArg(rollupSum2),
	"geomean_over_time":     newRollupFuncOneArg(rollupGeomean),
	"first_over_time":       newRollupFuncOneArg(rollupFirst),
	"last_over_time":        newRollupFuncOneArg(rollupLast),
	"distinct_over_time":    newRollupFuncOneArg(rollupDistinct),
	"increases_over_time":   newRollupFuncOneArg(rollupIncreases),
	"decreases_over_time":   newRollupFuncOneArg(rollupDecreases),
	"increase_pure":         newRollupFuncOneArg(rollupIncreasePure), // + rollupFuncsRemoveCounterResets
	"integrate":             newRollupFuncOneArg(rollupIntegrate),
	"ideriv":                newRollupFuncOneArg(rollupIderiv),
	"lifetime":              newRollupFuncOneArg(rollupLifetime),
	"lag":                   newRollupFuncOneArg(rollupLag),
	"scrape_interval":       newRollupFuncOneArg(rollupScrapeInterval),
	"tmin_over_time":        newRollupFuncOneArg(rollupTmin),
	"tmax_over_time":        newRollupFuncOneArg(rollupTmax),
	"tfirst_over_time":      newRollupFuncOneArg(rollupTfirst),
	"tlast_over_time":       newRollupFuncOneArg(rollupTlast),
	"share_le_over_time":    newRollupShareLE,
	"share_gt_over_time":    newRollupShareGT,
	"count_le_over_time":    newRollupCountLE,
	"count_gt_over_time":    newRollupCountGT,
	"count_eq_over_time":    newRollupCountEQ,
	"count_ne_over_time":    newRollupCountNE,
	"histogram_over_time":   newRollupFuncOneArg(rollupHistogram),
	"rollup":                newRollupFuncOneArg(rollupFake),
	"rollup_rate":           newRollupFuncOneArg(rollupFake), // + rollupFuncsRemoveCounterResets
	"rollup_deriv":          newRollupFuncOneArg(rollupFake),
	"rollup_delta":          newRollupFuncOneArg(rollupFake),
	"rollup_increase":       newRollupFuncOneArg(rollupFake), // + rollupFuncsRemoveCounterResets
	"rollup_candlestick":    newRollupFuncOneArg(rollupFake),
	"aggr_over_time":        newRollupFuncTwoArgs(rollupFake),
	"hoeffding_bound_upper": newRollupHoeffdingBoundUpper,
	"hoeffding_bound_lower": newRollupHoeffdingBoundLower,
	"ascent_over_time":      newRollupFuncOneArg(rollupAscentOverTime),
	"descent_over_time":     newRollupFuncOneArg(rollupDescentOverTime),
	"zscore_over_time":      newRollupFuncOneArg(rollupZScoreOverTime),

	// `timestamp` function must return timestamp for the last datapoint on the current window
	// in order to properly handle offset and timestamps unaligned to the current step.
	// See https://github.com/VictoriaMetrics/VictoriaMetrics/issues/415 for details.
	"timestamp": newRollupFuncOneArg(rollupTlast),

	// See https://en.wikipedia.org/wiki/Mode_(statistics)
	"mode_over_time": newRollupFuncOneArg(rollupModeOverTime),

	"rate_over_sum": newRollupFuncOneArg(rollupRateOverSum),
}

// rollupAggrFuncs are functions that can be passed to `aggr_over_time()`
var rollupAggrFuncs = map[string]rollupFunc{
	// Standard rollup funcs from PromQL.
	"changes":           rollupChanges,
	"delta":             rollupDelta,
	"deriv":             rollupDerivSlow,
	"deriv_fast":        rollupDerivFast,
	"idelta":            rollupIdelta,
	"increase":          rollupDelta,     // + rollupFuncsRemoveCounterResets
	"irate":             rollupIderiv,    // + rollupFuncsRemoveCounterResets
	"rate":              rollupDerivFast, // + rollupFuncsRemoveCounterResets
	"resets":            rollupResets,
	"avg_over_time":     rollupAvg,
	"min_over_time":     rollupMin,
	"max_over_time":     rollupMax,
	"sum_over_time":     rollupSum,
	"count_over_time":   rollupCount,
	"stddev_over_time":  rollupStddev,
	"stdvar_over_time":  rollupStdvar,
	"absent_over_time":  rollupAbsent,
	"present_over_time": rollupPresent,

	// Additional rollup funcs.
	"range_over_time":     rollupRange,
	"sum2_over_time":      rollupSum2,
	"geomean_over_time":   rollupGeomean,
	"first_over_time":     rollupFirst,
	"last_over_time":      rollupLast,
	"distinct_over_time":  rollupDistinct,
	"increases_over_time": rollupIncreases,
	"decreases_over_time": rollupDecreases,
	"increase_pure":       rollupIncreasePure,
	"integrate":           rollupIntegrate,
	"ideriv":              rollupIderiv,
	"lifetime":            rollupLifetime,
	"lag":                 rollupLag,
	"scrape_interval":     rollupScrapeInterval,
	"tmin_over_time":      rollupTmin,
	"tmax_over_time":      rollupTmax,
	"tfirst_over_time":    rollupTfirst,
	"tlast_over_time":     rollupTlast,
	"ascent_over_time":    rollupAscentOverTime,
	"descent_over_time":   rollupDescentOverTime,
	"zscore_over_time":    rollupZScoreOverTime,
	"timestamp":           rollupTlast,
	"mode_over_time":      rollupModeOverTime,
	"rate_over_sum":       rollupRateOverSum,
}

var rollupFuncsCannotAdjustWindow = map[string]bool{
	"changes":             true,
	"delta":               true,
	"holt_winters":        true,
	"idelta":              true,
	"increase":            true,
	"predict_linear":      true,
	"resets":              true,
	"avg_over_time":       true,
	"sum_over_time":       true,
	"count_over_time":     true,
	"quantile_over_time":  true,
	"stddev_over_time":    true,
	"stdvar_over_time":    true,
	"absent_over_time":    true,
	"present_over_time":   true,
	"sum2_over_time":      true,
	"geomean_over_time":   true,
	"distinct_over_time":  true,
	"increases_over_time": true,
	"decreases_over_time": true,
	"increase_pure":       true,
	"integrate":           true,
	"ascent_over_time":    true,
	"descent_over_time":   true,
	"zscore_over_time":    true,
}

var rollupFuncsRemoveCounterResets = map[string]bool{
	"increase":        true,
	"irate":           true,
	"rate":            true,
	"rollup_rate":     true,
	"rollup_increase": true,
	"increase_pure":   true,
}

var rollupFuncsKeepMetricGroup = map[string]bool{
	"holt_winters":          true,
	"predict_linear":        true,
	"default_rollup":        true,
	"avg_over_time":         true,
	"min_over_time":         true,
	"max_over_time":         true,
	"quantile_over_time":    true,
	"rollup":                true,
	"geomean_over_time":     true,
	"hoeffding_bound_lower": true,
	"hoeffding_bound_upper": true,
	"first_over_time":       true,
	"last_over_time":        true,
	"mode_over_time":        true,
}

func getRollupAggrFuncNames(expr metricsql.Expr) ([]string, error) {
	afe, ok := expr.(*metricsql.AggrFuncExpr)
	if ok {
		// This is for incremental aggregate function case:
		//
		//     sum(aggr_over_time(...))
		//
		// See aggr_incremental.go for details.
		expr = afe.Args[0]
	}
	fe, ok := expr.(*metricsql.FuncExpr)
	if !ok {
		logger.Panicf("BUG: unexpected expression; want metricsql.FuncExpr; got %T; value: %s", expr, expr.AppendString(nil))
	}
	if fe.Name != "aggr_over_time" {
		logger.Panicf("BUG: unexpected function name: %q; want `aggr_over_time`", fe.Name)
	}
	if len(fe.Args) != 2 {
		return nil, fmt.Errorf("unexpected number of args to aggr_over_time(); got %d; want %d", len(fe.Args), 2)
	}
	arg := fe.Args[0]
	var aggrFuncNames []string
	if se, ok := arg.(*metricsql.StringExpr); ok {
		aggrFuncNames = append(aggrFuncNames, se.S)
	} else {
		fe, ok := arg.(*metricsql.FuncExpr)
		if !ok || fe.Name != "" {
			return nil, fmt.Errorf("%s cannot be passed to aggr_over_time(); expecting quoted aggregate function name or a list of quoted aggregate function names",
				arg.AppendString(nil))
		}
		for _, e := range fe.Args {
			se, ok := e.(*metricsql.StringExpr)
			if !ok {
				return nil, fmt.Errorf("%s cannot be passed here; expecting quoted aggregate function name", e.AppendString(nil))
			}
			aggrFuncNames = append(aggrFuncNames, se.S)
		}
	}
	if len(aggrFuncNames) == 0 {
		return nil, fmt.Errorf("aggr_over_time() must contain at least a single aggregate function name")
	}
	for _, s := range aggrFuncNames {
		if rollupAggrFuncs[s] == nil {
			return nil, fmt.Errorf("%q cannot be used in `aggr_over_time` function; expecting quoted aggregate function name", s)
		}
	}
	return aggrFuncNames, nil
}

func getRollupArgIdx(funcName string) int {
	funcName = strings.ToLower(funcName)
	if rollupFuncs[funcName] == nil {
		logger.Panicf("BUG: getRollupArgIdx is called for non-rollup func %q", funcName)
	}
	switch funcName {
	case "quantile_over_time", "aggr_over_time",
		"hoeffding_bound_lower", "hoeffding_bound_upper":
		return 1
	default:
		return 0
	}
}

func getRollupConfigs(name string, rf rollupFunc, expr metricsql.Expr, start, end, step, window int64, lookbackDelta int64, sharedTimestamps []int64) (
	func(values []float64, timestamps []int64), []*rollupConfig, error) {
	preFunc := func(values []float64, timestamps []int64) {}
	if rollupFuncsRemoveCounterResets[name] {
		preFunc = func(values []float64, timestamps []int64) {
			removeCounterResets(values)
		}
	}
	newRollupConfig := func(rf rollupFunc, tagValue string) *rollupConfig {
		return &rollupConfig{
			TagValue:           tagValue,
			Func:               rf,
			Start:              start,
			End:                end,
			Step:               step,
			Window:             window,
			MayAdjustWindow:    !rollupFuncsCannotAdjustWindow[name],
			CanDropStalePoints: name == "default_rollup",
			LookbackDelta:      lookbackDelta,
			Timestamps:         sharedTimestamps,
		}
	}
	appendRollupConfigs := func(dst []*rollupConfig) []*rollupConfig {
		dst = append(dst, newRollupConfig(rollupMin, "min"))
		dst = append(dst, newRollupConfig(rollupMax, "max"))
		dst = append(dst, newRollupConfig(rollupAvg, "avg"))
		return dst
	}
	var rcs []*rollupConfig
	switch name {
	case "rollup":
		rcs = appendRollupConfigs(rcs)
	case "rollup_rate", "rollup_deriv":
		preFuncPrev := preFunc
		preFunc = func(values []float64, timestamps []int64) {
			preFuncPrev(values, timestamps)
			derivValues(values, timestamps)
		}
		rcs = appendRollupConfigs(rcs)
	case "rollup_increase", "rollup_delta":
		preFuncPrev := preFunc
		preFunc = func(values []float64, timestamps []int64) {
			preFuncPrev(values, timestamps)
			deltaValues(values)
		}
		rcs = appendRollupConfigs(rcs)
	case "rollup_candlestick":
		rcs = append(rcs, newRollupConfig(rollupOpen, "open"))
		rcs = append(rcs, newRollupConfig(rollupClose, "close"))
		rcs = append(rcs, newRollupConfig(rollupLow, "low"))
		rcs = append(rcs, newRollupConfig(rollupHigh, "high"))
	case "aggr_over_time":
		aggrFuncNames, err := getRollupAggrFuncNames(expr)
		if err != nil {
			return nil, nil, fmt.Errorf("invalid args to %s: %w", expr.AppendString(nil), err)
		}
		for _, aggrFuncName := range aggrFuncNames {
			if rollupFuncsRemoveCounterResets[aggrFuncName] {
				// There is no need to save the previous preFunc, since it is either empty or the same.
				preFunc = func(values []float64, timestamps []int64) {
					removeCounterResets(values)
				}
			}
			rf := rollupAggrFuncs[aggrFuncName]
			rcs = append(rcs, newRollupConfig(rf, aggrFuncName))
		}
	default:
		rcs = append(rcs, newRollupConfig(rf, ""))
	}
	return preFunc, rcs, nil
}

func getRollupFunc(funcName string) newRollupFunc {
	funcName = strings.ToLower(funcName)
	return rollupFuncs[funcName]
}

type rollupFuncArg struct {
	// The value preceeding values if it fits staleness interval.
	prevValue float64

	// The timestamp for prevValue.
	prevTimestamp int64

	// Values that fit window ending at currTimestamp.
	values []float64

	// Timestamps for values.
	timestamps []int64

	// Real value preceeding values without restrictions on staleness interval.
	realPrevValue float64

	// Real value which goes after values.
	realNextValue float64

	// Current timestamp for rollup evaluation.
	currTimestamp int64

	// Index for the currently evaluated point relative to time range for query evaluation.
	idx int

	// Time window for rollup calculations.
	window int64

	tsm *timeseriesMap
}

func (rfa *rollupFuncArg) reset() {
	rfa.prevValue = 0
	rfa.prevTimestamp = 0
	rfa.values = nil
	rfa.timestamps = nil
	rfa.currTimestamp = 0
	rfa.idx = 0
	rfa.window = 0
	rfa.tsm = nil
}

// rollupFunc must return rollup value for the given rfa.
//
// prevValue may be nan, values and timestamps may be empty.
type rollupFunc func(rfa *rollupFuncArg) float64

type rollupConfig struct {
	// This tag value must be added to "rollup" tag if non-empty.
	TagValue string

	Func   rollupFunc
	Start  int64
	End    int64
	Step   int64
	Window int64

	// Whether window may be adjusted to 2 x interval between data points.
	// This is needed for functions which have dt in the denominator
	// such as rate, deriv, etc.
	// Without the adjustment their value would jump in unexpected directions
	// when using window smaller than 2 x scrape_interval.
	MayAdjustWindow bool

	// Whether points after Prometheus stale marks can be dropped during rollup calculations.
	// Stale points can be dropped only if `default_rollup()` function is used.
	// See https://github.com/VictoriaMetrics/VictoriaMetrics/issues/1526 .
	CanDropStalePoints bool

	Timestamps []int64

	// LoookbackDelta is the analog to `-query.lookback-delta` from Prometheus world.
	LookbackDelta int64
}

var (
	nan = math.NaN()
	inf = math.Inf(1)
)

// The maximum interval without previous rows.
const maxSilenceInterval = 5 * 60 * 1000

type timeseriesMap struct {
	origin    *timeseries
	labelName string
	h         metrics.Histogram
	m         map[string]*timeseries
}

func newTimeseriesMap(funcName string, sharedTimestamps []int64, mnSrc *storage.MetricName) *timeseriesMap {
	if strings.ToLower(funcName) != "histogram_over_time" {
		return nil
	}

	values := make([]float64, len(sharedTimestamps))
	for i := range values {
		values[i] = nan
	}
	var origin timeseries
	origin.MetricName.CopyFrom(mnSrc)
	origin.MetricName.ResetMetricGroup()
	origin.Timestamps = sharedTimestamps
	origin.Values = values
	return &timeseriesMap{
		origin:    &origin,
		labelName: "vmrange",
		m:         make(map[string]*timeseries),
	}
}

func (tsm *timeseriesMap) AppendTimeseriesTo(dst []*timeseries) []*timeseries {
	for _, ts := range tsm.m {
		dst = append(dst, ts)
	}
	return dst
}

func (tsm *timeseriesMap) GetOrCreateTimeseries(labelValue string) *timeseries {
	ts := tsm.m[labelValue]
	if ts != nil {
		return ts
	}
	ts = &timeseries{}
	ts.CopyFromShallowTimestamps(tsm.origin)
	ts.MetricName.RemoveTag(tsm.labelName)
	ts.MetricName.AddTag(tsm.labelName, labelValue)
	tsm.m[labelValue] = ts
	return ts
}

// Do calculates rollups for the given timestamps and values, appends
// them to dstValues and returns results.
//
// rc.Timestamps are used as timestamps for dstValues.
//
// timestamps must cover time range [rc.Start - rc.Window - maxSilenceInterval ... rc.End].
//
// Do cannot be called from concurrent goroutines.
func (rc *rollupConfig) Do(dstValues []float64, values []float64, timestamps []int64) []float64 {
	return rc.doInternal(dstValues, nil, values, timestamps)
}

// DoTimeseriesMap calculates rollups for the given timestamps and values and puts them to tsm.
func (rc *rollupConfig) DoTimeseriesMap(tsm *timeseriesMap, values []float64, timestamps []int64) {
	ts := getTimeseries()
	ts.Values = rc.doInternal(ts.Values[:0], tsm, values, timestamps)
	putTimeseries(ts)
}

func (rc *rollupConfig) doInternal(dstValues []float64, tsm *timeseriesMap, values []float64, timestamps []int64) []float64 {
	// Sanity checks.
	if rc.Step <= 0 {
		logger.Panicf("BUG: Step must be bigger than 0; got %d", rc.Step)
	}
	if rc.Start > rc.End {
		logger.Panicf("BUG: Start cannot exceed End; got %d vs %d", rc.Start, rc.End)
	}
	if rc.Window < 0 {
		logger.Panicf("BUG: Window must be non-negative; got %d", rc.Window)
	}
	if err := ValidateMaxPointsPerTimeseries(rc.Start, rc.End, rc.Step); err != nil {
		logger.Panicf("BUG: %s; this must be validated before the call to rollupConfig.Do", err)
	}

	// Extend dstValues in order to remove mallocs below.
	dstValues = decimal.ExtendFloat64sCapacity(dstValues, len(rc.Timestamps))

	if !rc.CanDropStalePoints {
		// Remove Prometheus staleness marks from values, so rollup functions don't hit NaN values.
		values, timestamps = dropStaleNaNs(values, timestamps)
	}
	scrapeInterval := getScrapeInterval(timestamps)
	maxPrevInterval := getMaxPrevInterval(scrapeInterval)
	if rc.LookbackDelta > 0 && maxPrevInterval > rc.LookbackDelta {
		maxPrevInterval = rc.LookbackDelta
	}
	if *minStalenessInterval > 0 {
		if msi := minStalenessInterval.Milliseconds(); msi > 0 && maxPrevInterval < msi {
			maxPrevInterval = msi
		}
	}
	window := rc.Window
	if window <= 0 {
		window = rc.Step
		if rc.CanDropStalePoints && rc.LookbackDelta > 0 && window > rc.LookbackDelta {
			// Implicit window exceeds -search.maxStalenessInterval, so limit it to -search.maxStalenessInterval
			// according to https://github.com/VictoriaMetrics/VictoriaMetrics/issues/784
			window = rc.LookbackDelta
		}
	}
	if rc.MayAdjustWindow && window < maxPrevInterval {
		window = maxPrevInterval
	}
	rfa := getRollupFuncArg()
	rfa.idx = 0
	rfa.window = window
	rfa.tsm = tsm

	i := 0
	j := 0
	ni := 0
	nj := 0
	f := rc.Func
	for _, tEnd := range rc.Timestamps {
		tStart := tEnd - window
		ni = seekFirstTimestampIdxAfter(timestamps[i:], tStart, ni)
		i += ni
		if j < i {
			j = i
		}
		nj = seekFirstTimestampIdxAfter(timestamps[j:], tEnd, nj)
		j += nj

		rfa.prevValue = nan
		rfa.prevTimestamp = tStart - maxPrevInterval
		if i < len(timestamps) && i > 0 && timestamps[i-1] > rfa.prevTimestamp {
			rfa.prevValue = values[i-1]
			rfa.prevTimestamp = timestamps[i-1]
		}
		rfa.values = values[i:j]
		rfa.timestamps = timestamps[i:j]
		if i > 0 {
			rfa.realPrevValue = values[i-1]
		} else {
			rfa.realPrevValue = nan
		}
		if j < len(values) {
			rfa.realNextValue = values[j]
		} else {
			rfa.realNextValue = nan
		}
		rfa.currTimestamp = tEnd
		value := f(rfa)
		rfa.idx++
		dstValues = append(dstValues, value)
	}
	putRollupFuncArg(rfa)

	return dstValues
}

func dropStaleNaNs(values []float64, timestamps []int64) ([]float64, []int64) {
	hasStaleSamples := false
	for _, v := range values {
		if decimal.IsStaleNaN(v) {
			hasStaleSamples = true
			break
		}
	}
	if !hasStaleSamples {
		// Fast path: values have noe Prometheus staleness marks.
		return values, timestamps
	}
	// Slow path: drop Prometheus staleness marks from values.
	dstValues := make([]float64, 0, len(values))
	dstTimestamps := make([]int64, 0, len(timestamps))
	for i, v := range values {
		if decimal.IsStaleNaN(v) {
			continue
		}
		dstValues = append(dstValues, v)
		dstTimestamps = append(dstTimestamps, timestamps[i])
	}
	return dstValues, dstTimestamps
}

func seekFirstTimestampIdxAfter(timestamps []int64, seekTimestamp int64, nHint int) int {
	if len(timestamps) == 0 || timestamps[0] > seekTimestamp {
		return 0
	}
	startIdx := nHint - 2
	if startIdx < 0 {
		startIdx = 0
	}
	if startIdx >= len(timestamps) {
		startIdx = len(timestamps) - 1
	}
	endIdx := nHint + 2
	if endIdx > len(timestamps) {
		endIdx = len(timestamps)
	}
	if startIdx > 0 && timestamps[startIdx] <= seekTimestamp {
		timestamps = timestamps[startIdx:]
		endIdx -= startIdx
	} else {
		startIdx = 0
	}
	if endIdx < len(timestamps) && timestamps[endIdx] > seekTimestamp {
		timestamps = timestamps[:endIdx]
	}
	if len(timestamps) < 16 {
		// Fast path: the number of timestamps to search is small, so scan them all.
		for i, timestamp := range timestamps {
			if timestamp > seekTimestamp {
				return startIdx + i
			}
		}
		return startIdx + len(timestamps)
	}
	// Slow path: too big len(timestamps), so use binary search.
	i := binarySearchInt64(timestamps, seekTimestamp+1)
	return startIdx + int(i)
}

func binarySearchInt64(a []int64, v int64) uint {
	// Copy-pasted sort.Search from https://golang.org/src/sort/search.go?s=2246:2286#L49
	i, j := uint(0), uint(len(a))
	for i < j {
		h := (i + j) >> 1
		if h < uint(len(a)) && a[h] < v {
			i = h + 1
		} else {
			j = h
		}
	}
	return i
}

func getScrapeInterval(timestamps []int64) int64 {
	if len(timestamps) < 2 {
		return int64(maxSilenceInterval)
	}

	// Estimate scrape interval as 0.6 quantile for the first 20 intervals.
	h := histogram.GetFast()
	tsPrev := timestamps[0]
	timestamps = timestamps[1:]
	if len(timestamps) > 20 {
		timestamps = timestamps[:20]
	}
	for _, ts := range timestamps {
		h.Update(float64(ts - tsPrev))
		tsPrev = ts
	}
	scrapeInterval := int64(h.Quantile(0.6))
	histogram.PutFast(h)
	if scrapeInterval <= 0 {
		return int64(maxSilenceInterval)
	}
	return scrapeInterval
}

func getMaxPrevInterval(scrapeInterval int64) int64 {
	// Increase scrapeInterval more for smaller scrape intervals in order to hide possible gaps
	// when high jitter is present.
	// See https://github.com/VictoriaMetrics/VictoriaMetrics/issues/139 .
	if scrapeInterval <= 2*1000 {
		return scrapeInterval + 4*scrapeInterval
	}
	if scrapeInterval <= 4*1000 {
		return scrapeInterval + 2*scrapeInterval
	}
	if scrapeInterval <= 8*1000 {
		return scrapeInterval + scrapeInterval
	}
	if scrapeInterval <= 16*1000 {
		return scrapeInterval + scrapeInterval/2
	}
	if scrapeInterval <= 32*1000 {
		return scrapeInterval + scrapeInterval/4
	}
	return scrapeInterval + scrapeInterval/8
}

func removeCounterResets(values []float64) {
	if len(values) == 0 {
		return
	}
	var correction float64
	prevValue := values[0]
	for i, v := range values {
		d := v - prevValue
		if d < 0 {
			if (-d * 8) < prevValue {
				// This is likely jitter from `Prometheus HA pairs`.
				// Just substitute v with prevValue.
				v = prevValue
			} else {
				correction += prevValue
			}
		}
		prevValue = v
		if !decimal.IsStaleNaN(v) {
			values[i] = v + correction
		}
	}
}

func deltaValues(values []float64) {
	// There is no need in handling NaNs here, since they are impossible
	// on values from vmstorage.
	if len(values) == 0 {
		return
	}
	prevDelta := float64(0)
	prevValue := values[0]
	for i, v := range values[1:] {
		prevDelta = v - prevValue
		values[i] = prevDelta
		prevValue = v
	}
	values[len(values)-1] = prevDelta
}

func derivValues(values []float64, timestamps []int64) {
	// There is no need in handling NaNs here, since they are impossible
	// on values from vmstorage.
	if len(values) == 0 {
		return
	}
	prevDeriv := float64(0)
	prevValue := values[0]
	prevTs := timestamps[0]
	for i, v := range values[1:] {
		ts := timestamps[i+1]
		if ts == prevTs {
			// Use the previous value for duplicate timestamps.
			values[i] = prevDeriv
			continue
		}
		dt := float64(ts-prevTs) / 1e3
		prevDeriv = (v - prevValue) / dt
		values[i] = prevDeriv
		prevValue = v
		prevTs = ts
	}
	values[len(values)-1] = prevDeriv
}

type newRollupFunc func(args []interface{}) (rollupFunc, error)

func newRollupFuncOneArg(rf rollupFunc) newRollupFunc {
	return func(args []interface{}) (rollupFunc, error) {
		if err := expectRollupArgsNum(args, 1); err != nil {
			return nil, err
		}
		return rf, nil
	}
}

func newRollupFuncTwoArgs(rf rollupFunc) newRollupFunc {
	return func(args []interface{}) (rollupFunc, error) {
		if err := expectRollupArgsNum(args, 2); err != nil {
			return nil, err
		}
		return rf, nil
	}
}

func newRollupHoltWinters(args []interface{}) (rollupFunc, error) {
	if err := expectRollupArgsNum(args, 3); err != nil {
		return nil, err
	}
	sfs, err := getScalar(args[1], 1)
	if err != nil {
		return nil, err
	}
	tfs, err := getScalar(args[2], 2)
	if err != nil {
		return nil, err
	}
	rf := func(rfa *rollupFuncArg) float64 {
		// There is no need in handling NaNs here, since they must be cleaned up
		// before calling rollup funcs.
		values := rfa.values
		if len(values) == 0 {
			return rfa.prevValue
		}
		sf := sfs[rfa.idx]
		if sf <= 0 || sf >= 1 {
			return nan
		}
		tf := tfs[rfa.idx]
		if tf <= 0 || tf >= 1 {
			return nan
		}

		// See https://en.wikipedia.org/wiki/Exponential_smoothing#Double_exponential_smoothing .
		// TODO: determine whether this shit really works.
		s0 := rfa.prevValue
		if math.IsNaN(s0) {
			s0 = values[0]
			values = values[1:]
			if len(values) == 0 {
				return s0
			}
		}
		b0 := values[0] - s0
		for _, v := range values {
			s1 := sf*v + (1-sf)*(s0+b0)
			b1 := tf*(s1-s0) + (1-tf)*b0
			s0 = s1
			b0 = b1
		}
		return s0
	}
	return rf, nil
}

func newRollupPredictLinear(args []interface{}) (rollupFunc, error) {
	if err := expectRollupArgsNum(args, 2); err != nil {
		return nil, err
	}
	secs, err := getScalar(args[1], 1)
	if err != nil {
		return nil, err
	}
	rf := func(rfa *rollupFuncArg) float64 {
		v, k := linearRegression(rfa)
		if math.IsNaN(v) {
			return nan
		}
		sec := secs[rfa.idx]
		return v + k*sec
	}
	return rf, nil
}

func linearRegression(rfa *rollupFuncArg) (float64, float64) {
	// There is no need in handling NaNs here, since they must be cleaned up
	// before calling rollup funcs.
	values := rfa.values
	timestamps := rfa.timestamps
	if len(values) == 0 {
		return rfa.prevValue, 0
	}

	// See https://en.wikipedia.org/wiki/Simple_linear_regression#Numerical_example
	tFirst := rfa.prevTimestamp
	vSum := rfa.prevValue
	tSum := float64(0)
	tvSum := float64(0)
	ttSum := float64(0)
	n := 1.0
	if math.IsNaN(rfa.prevValue) {
		tFirst = timestamps[0]
		vSum = 0
		n = 0
	}
	for i, v := range values {
		dt := float64(timestamps[i]-tFirst) / 1e3
		vSum += v
		tSum += dt
		tvSum += dt * v
		ttSum += dt * dt
	}
	n += float64(len(values))
	if n == 1 {
		return vSum, 0
	}
	k := (n*tvSum - tSum*vSum) / (n*ttSum - tSum*tSum)
	v := (vSum - k*tSum) / n
	// Adjust v to the last timestamp on the given time range.
	v += k * (float64(timestamps[len(timestamps)-1]-tFirst) / 1e3)
	return v, k
}

func newRollupShareLE(args []interface{}) (rollupFunc, error) {
	return newRollupShareFilter(args, countFilterLE)
}

func countFilterLE(values []float64, le float64) int {
	n := 0
	for _, v := range values {
		if v <= le {
			n++
		}
	}
	return n
}

func newRollupShareGT(args []interface{}) (rollupFunc, error) {
	return newRollupShareFilter(args, countFilterGT)
}

func countFilterGT(values []float64, gt float64) int {
	n := 0
	for _, v := range values {
		if v > gt {
			n++
		}
	}
	return n
}

func countFilterEQ(values []float64, eq float64) int {
	n := 0
	for _, v := range values {
		if v == eq {
			n++
		}
	}
	return n
}

func countFilterNE(values []float64, ne float64) int {
	n := 0
	for _, v := range values {
		if v != ne {
			n++
		}
	}
	return n
}

func newRollupShareFilter(args []interface{}, countFilter func(values []float64, limit float64) int) (rollupFunc, error) {
	rf, err := newRollupCountFilter(args, countFilter)
	if err != nil {
		return nil, err
	}
	return func(rfa *rollupFuncArg) float64 {
		n := rf(rfa)
		return n / float64(len(rfa.values))
	}, nil
}

func newRollupCountLE(args []interface{}) (rollupFunc, error) {
	return newRollupCountFilter(args, countFilterLE)
}

func newRollupCountGT(args []interface{}) (rollupFunc, error) {
	return newRollupCountFilter(args, countFilterGT)
}

func newRollupCountEQ(args []interface{}) (rollupFunc, error) {
	return newRollupCountFilter(args, countFilterEQ)
}

func newRollupCountNE(args []interface{}) (rollupFunc, error) {
	return newRollupCountFilter(args, countFilterNE)
}

func newRollupCountFilter(args []interface{}, countFilter func(values []float64, limit float64) int) (rollupFunc, error) {
	if err := expectRollupArgsNum(args, 2); err != nil {
		return nil, err
	}
	limits, err := getScalar(args[1], 1)
	if err != nil {
		return nil, err
	}
	rf := func(rfa *rollupFuncArg) float64 {
		// There is no need in handling NaNs here, since they must be cleaned up
		// before calling rollup funcs.
		values := rfa.values
		if len(values) == 0 {
			return nan
		}
		limit := limits[rfa.idx]
		return float64(countFilter(values, limit))
	}
	return rf, nil
}

func newRollupHoeffdingBoundLower(args []interface{}) (rollupFunc, error) {
	if err := expectRollupArgsNum(args, 2); err != nil {
		return nil, err
	}
	phis, err := getScalar(args[0], 0)
	if err != nil {
		return nil, err
	}
	rf := func(rfa *rollupFuncArg) float64 {
		bound, avg := rollupHoeffdingBoundInternal(rfa, phis)
		return avg - bound
	}
	return rf, nil
}

func newRollupHoeffdingBoundUpper(args []interface{}) (rollupFunc, error) {
	if err := expectRollupArgsNum(args, 2); err != nil {
		return nil, err
	}
	phis, err := getScalar(args[0], 0)
	if err != nil {
		return nil, err
	}
	rf := func(rfa *rollupFuncArg) float64 {
		bound, avg := rollupHoeffdingBoundInternal(rfa, phis)
		return avg + bound
	}
	return rf, nil
}

func rollupHoeffdingBoundInternal(rfa *rollupFuncArg, phis []float64) (float64, float64) {
	// There is no need in handling NaNs here, since they must be cleaned up
	// before calling rollup funcs.
	values := rfa.values
	if len(values) == 0 {
		return nan, nan
	}
	if len(values) == 1 {
		return 0, values[0]
	}
	vMax := rollupMax(rfa)
	vMin := rollupMin(rfa)
	vAvg := rollupAvg(rfa)
	vRange := vMax - vMin
	if vRange <= 0 {
		return 0, vAvg
	}
	phi := phis[rfa.idx]
	if phi >= 1 {
		return inf, vAvg
	}
	if phi <= 0 {
		return 0, vAvg
	}
	// See https://en.wikipedia.org/wiki/Hoeffding%27s_inequality
	// and https://www.youtube.com/watch?v=6UwcqiNsZ8U&feature=youtu.be&t=1237
	bound := vRange * math.Sqrt(math.Log(1/(1-phi))/(2*float64(len(values))))
	return bound, vAvg
}

func newRollupQuantile(args []interface{}) (rollupFunc, error) {
	if err := expectRollupArgsNum(args, 2); err != nil {
		return nil, err
	}
	phis, err := getScalar(args[0], 0)
	if err != nil {
		return nil, err
	}
	rf := func(rfa *rollupFuncArg) float64 {
		// There is no need in handling NaNs here, since they must be cleaned up
		// before calling rollup funcs.
		values := rfa.values
		if len(values) == 0 {
			return rfa.prevValue
		}
		if len(values) == 1 {
			// Fast path - only a single value.
			return values[0]
		}
		hf := histogram.GetFast()
		for _, v := range values {
			hf.Update(v)
		}
		phi := phis[rfa.idx]
		qv := hf.Quantile(phi)
		histogram.PutFast(hf)
		return qv
	}
	return rf, nil
}

func rollupHistogram(rfa *rollupFuncArg) float64 {
	values := rfa.values
	tsm := rfa.tsm
	tsm.h.Reset()
	for _, v := range values {
		tsm.h.Update(v)
	}
	idx := rfa.idx
	tsm.h.VisitNonZeroBuckets(func(vmrange string, count uint64) {
		ts := tsm.GetOrCreateTimeseries(vmrange)
		ts.Values[idx] = float64(count)
	})
	return nan
}

func rollupAvg(rfa *rollupFuncArg) float64 {
	// Do not use `Rapid calculation methods` at https://en.wikipedia.org/wiki/Standard_deviation,
	// since it is slower and has no significant benefits in precision.

	// There is no need in handling NaNs here, since they must be cleaned up
	// before calling rollup funcs.
	values := rfa.values
	if len(values) == 0 {
		// Do not take into account rfa.prevValue, since it may lead
		// to inconsistent results comparing to Prometheus on broken time series
		// with irregular data points.
		return nan
	}
	var sum float64
	for _, v := range values {
		sum += v
	}
	return sum / float64(len(values))
}

func rollupMin(rfa *rollupFuncArg) float64 {
	// There is no need in handling NaNs here, since they must be cleaned up
	// before calling rollup funcs.
	values := rfa.values
	if len(values) == 0 {
		// Do not take into account rfa.prevValue, since it may lead
		// to inconsistent results comparing to Prometheus on broken time series
		// with irregular data points.
		return nan
	}
	minValue := values[0]
	for _, v := range values {
		if v < minValue {
			minValue = v
		}
	}
	return minValue
}

func rollupMax(rfa *rollupFuncArg) float64 {
	// There is no need in handling NaNs here, since they must be cleaned up
	// before calling rollup funcs.
	values := rfa.values
	if len(values) == 0 {
		// Do not take into account rfa.prevValue, since it may lead
		// to inconsistent results comparing to Prometheus on broken time series
		// with irregular data points.
		return nan
	}
	maxValue := values[0]
	for _, v := range values {
		if v > maxValue {
			maxValue = v
		}
	}
	return maxValue
}

func rollupTmin(rfa *rollupFuncArg) float64 {
	// There is no need in handling NaNs here, since they must be cleaned up
	// before calling rollup funcs.
	values := rfa.values
	timestamps := rfa.timestamps
	if len(values) == 0 {
		return nan
	}
	minValue := values[0]
	minTimestamp := timestamps[0]
	for i, v := range values {
		// Get the last timestamp for the minimum value as most users expect.
		if v <= minValue {
			minValue = v
			minTimestamp = timestamps[i]
		}
	}
	return float64(minTimestamp) / 1e3
}

func rollupTmax(rfa *rollupFuncArg) float64 {
	// There is no need in handling NaNs here, since they must be cleaned up
	// before calling rollup funcs.
	values := rfa.values
	timestamps := rfa.timestamps
	if len(values) == 0 {
		return nan
	}
	maxValue := values[0]
	maxTimestamp := timestamps[0]
	for i, v := range values {
		// Get the last timestamp for the maximum value as most users expect.
		if v >= maxValue {
			maxValue = v
			maxTimestamp = timestamps[i]
		}
	}
	return float64(maxTimestamp) / 1e3
}

func rollupTfirst(rfa *rollupFuncArg) float64 {
	// There is no need in handling NaNs here, since they must be cleaned up
	// before calling rollup funcs.
	timestamps := rfa.timestamps
	if len(timestamps) == 0 {
		// Do not take into account rfa.prevTimestamp, since it may lead
		// to inconsistent results comparing to Prometheus on broken time series
		// with irregular data points.
		return nan
	}
	return float64(timestamps[0]) / 1e3
}

func rollupTlast(rfa *rollupFuncArg) float64 {
	// There is no need in handling NaNs here, since they must be cleaned up
	// before calling rollup funcs.
	timestamps := rfa.timestamps
	if len(timestamps) == 0 {
		// Do not take into account rfa.prevTimestamp, since it may lead
		// to inconsistent results comparing to Prometheus on broken time series
		// with irregular data points.
		return nan
	}
	return float64(timestamps[len(timestamps)-1]) / 1e3
}

func rollupSum(rfa *rollupFuncArg) float64 {
	// There is no need in handling NaNs here, since they must be cleaned up
	// before calling rollup funcs.
	values := rfa.values
	if len(values) == 0 {
		// Do not take into account rfa.prevValue, since it may lead
		// to inconsistent results comparing to Prometheus on broken time series
		// with irregular data points.
		return nan
	}
	var sum float64
	for _, v := range values {
		sum += v
	}
	return sum
}

func rollupRateOverSum(rfa *rollupFuncArg) float64 {
	// There is no need in handling NaNs here, since they must be cleaned up
	// before calling rollup funcs.
	timestamps := rfa.timestamps
	if len(timestamps) == 0 {
		if math.IsNaN(rfa.prevValue) {
			return nan
		}
		// Assume that the value didn't change since rfa.prevValue.
		return 0
	}
	dt := rfa.window
	if !math.IsNaN(rfa.prevValue) {
		dt = timestamps[len(timestamps)-1] - rfa.prevTimestamp
	}
	sum := float64(0)
	for _, v := range rfa.values {
		sum += v
	}
	return sum / (float64(dt) / 1e3)
}

func rollupRange(rfa *rollupFuncArg) float64 {
	max := rollupMax(rfa)
	min := rollupMin(rfa)
	return max - min
}

func rollupSum2(rfa *rollupFuncArg) float64 {
	// There is no need in handling NaNs here, since they must be cleaned up
	// before calling rollup funcs.
	values := rfa.values
	if len(values) == 0 {
		return rfa.prevValue * rfa.prevValue
	}
	var sum2 float64
	for _, v := range values {
		sum2 += v * v
	}
	return sum2
}

func rollupGeomean(rfa *rollupFuncArg) float64 {
	// There is no need in handling NaNs here, since they must be cleaned up
	// before calling rollup funcs.
	values := rfa.values
	if len(values) == 0 {
		return rfa.prevValue
	}
	p := 1.0
	for _, v := range values {
		p *= v
	}
	return math.Pow(p, 1/float64(len(values)))
}

func rollupAbsent(rfa *rollupFuncArg) float64 {
	if len(rfa.values) == 0 {
		return 1
	}
	return nan
}

func rollupPresent(rfa *rollupFuncArg) float64 {
	// There is no need in handling NaNs here, since they must be cleaned up
	// before calling rollup funcs.
	if len(rfa.values) > 0 {
		return 1
	}
	return nan
}

func rollupCount(rfa *rollupFuncArg) float64 {
	// There is no need in handling NaNs here, since they must be cleaned up
	// before calling rollup funcs.
	values := rfa.values
	if len(values) == 0 {
		if math.IsNaN(rfa.prevValue) {
			return nan
		}
		return 0
	}
	return float64(len(values))
}

func rollupStddev(rfa *rollupFuncArg) float64 {
	stdvar := rollupStdvar(rfa)
	return math.Sqrt(stdvar)
}

func rollupStdvar(rfa *rollupFuncArg) float64 {
	// See `Rapid calculation methods` at https://en.wikipedia.org/wiki/Standard_deviation

	// There is no need in handling NaNs here, since they must be cleaned up
	// before calling rollup funcs.
	values := rfa.values
	if len(values) == 0 {
		if math.IsNaN(rfa.prevValue) {
			return nan
		}
		return 0
	}
	if len(values) == 1 {
		// Fast path.
		return values[0]
	}
	var avg float64
	var count float64
	var q float64
	for _, v := range values {
		count++
		avgNew := avg + (v-avg)/count
		q += (v - avg) * (v - avgNew)
		avg = avgNew
	}
	return q / count
}

func rollupIncreasePure(rfa *rollupFuncArg) float64 {
	// There is no need in handling NaNs here, since they must be cleaned up
	// before calling rollup funcs.
	values := rfa.values
	// restore to the real value because of potential staleness reset
	prevValue := rfa.realPrevValue
	if math.IsNaN(prevValue) {
		if len(values) == 0 {
			return nan
		}
		// Assume the counter starts from 0.
		prevValue = 0
	}
	if len(values) == 0 {
		// Assume the counter didsn't change since prevValue.
		return 0
	}
	return values[len(values)-1] - prevValue
}

func rollupDelta(rfa *rollupFuncArg) float64 {
	// There is no need in handling NaNs here, since they must be cleaned up
	// before calling rollup funcs.
	values := rfa.values
	prevValue := rfa.prevValue
	if math.IsNaN(prevValue) {
		if len(values) == 0 {
			return nan
		}
		if !math.IsNaN(rfa.realPrevValue) {
			// Assume that the value didn't change during the current gap.
			// This should fix high delta() and increase() values at the end of gaps.
			// See https://github.com/VictoriaMetrics/VictoriaMetrics/issues/894
			return values[len(values)-1] - rfa.realPrevValue
		}
		// Assume that the previous non-existing value was 0 only in the following cases:
		//
		// - If the delta with the next value equals to 0.
		//   This is the case for slow-changing counter - see https://github.com/VictoriaMetrics/VictoriaMetrics/issues/962
		// - If the first value doesn't exceed too much the delta with the next value.
		//
		// This should prevent from improper increase() results for os-level counters
		// such as cpu time or bytes sent over the network interface.
		// These counters may start long ago before the first value appears in the db.
		//
		// This also should prevent from improper increase() results when a part of label values are changed
		// without counter reset.
		var d float64
		if len(values) > 1 {
			d = values[1] - values[0]
		} else if !math.IsNaN(rfa.realNextValue) {
			d = rfa.realNextValue - values[0]
		}
		if d == 0 {
			d = 10
		}
		if math.Abs(values[0]) < 10*(math.Abs(d)+1) {
			prevValue = 0
		} else {
			prevValue = values[0]
			values = values[1:]
		}
	}
	if len(values) == 0 {
		// Assume that the value didn't change on the given interval.
		return 0
	}
	return values[len(values)-1] - prevValue
}

func rollupIdelta(rfa *rollupFuncArg) float64 {
	// There is no need in handling NaNs here, since they must be cleaned up
	// before calling rollup funcs.
	values := rfa.values
	if len(values) == 0 {
		if math.IsNaN(rfa.prevValue) {
			return nan
		}
		// Assume that the value didn't change on the given interval.
		return 0
	}
	lastValue := values[len(values)-1]
	values = values[:len(values)-1]
	if len(values) == 0 {
		prevValue := rfa.prevValue
		if math.IsNaN(prevValue) {
			// Assume that the previous non-existing value was 0.
			return lastValue
		}
		return lastValue - prevValue
	}
	return lastValue - values[len(values)-1]
}

func rollupDerivSlow(rfa *rollupFuncArg) float64 {
	// Use linear regression like Prometheus does.
	// See https://github.com/VictoriaMetrics/VictoriaMetrics/issues/73
	_, k := linearRegression(rfa)
	return k
}

func rollupDerivFast(rfa *rollupFuncArg) float64 {
	// There is no need in handling NaNs here, since they must be cleaned up
	// before calling rollup funcs.
	values := rfa.values
	timestamps := rfa.timestamps
	prevValue := rfa.prevValue
	prevTimestamp := rfa.prevTimestamp
	if math.IsNaN(prevValue) {
		if len(values) == 0 {
			return nan
		}
		if len(values) == 1 {
			// It is impossible to determine the duration during which the value changed
			// from 0 to the current value.
			// The following attempts didn't work well:
			// - using scrape interval as the duration. It fails on Prometheus restarts when it
			//   skips scraping for the counter. This results in too high rate() value for the first point
			//   after Prometheus restarts.
			// - using window or step as the duration. It results in too small rate() values for the first
			//   points of time series.
			//
			// So just return nan
			return nan
		}
		prevValue = values[0]
		prevTimestamp = timestamps[0]
	} else if len(values) == 0 {
		// Assume that the value didn't change on the given interval.
		return 0
	}
	vEnd := values[len(values)-1]
	tEnd := timestamps[len(timestamps)-1]
	dv := vEnd - prevValue
	dt := float64(tEnd-prevTimestamp) / 1e3
	return dv / dt
}

func rollupIderiv(rfa *rollupFuncArg) float64 {
	// There is no need in handling NaNs here, since they must be cleaned up
	// before calling rollup funcs.
	values := rfa.values
	timestamps := rfa.timestamps
	if len(values) < 2 {
		if len(values) == 0 {
			return nan
		}
		if math.IsNaN(rfa.prevValue) {
			// It is impossible to determine the duration during which the value changed
			// from 0 to the current value.
			// The following attempts didn't work well:
			// - using scrape interval as the duration. It fails on Prometheus restarts when it
			//   skips scraping for the counter. This results in too high rate() value for the first point
			//   after Prometheus restarts.
			// - using window or step as the duration. It results in too small rate() values for the first
			//   points of time series.
			//
			// So just return nan
			return nan
		}
		return (values[0] - rfa.prevValue) / (float64(timestamps[0]-rfa.prevTimestamp) / 1e3)
	}
	vEnd := values[len(values)-1]
	tEnd := timestamps[len(timestamps)-1]
	values = values[:len(values)-1]
	timestamps = timestamps[:len(timestamps)-1]
	// Skip data points with duplicate timestamps.
	for len(timestamps) > 0 && timestamps[len(timestamps)-1] >= tEnd {
		timestamps = timestamps[:len(timestamps)-1]
	}
	var tStart int64
	var vStart float64
	if len(timestamps) == 0 {
		if math.IsNaN(rfa.prevValue) {
			return 0
		}
		tStart = rfa.prevTimestamp
		vStart = rfa.prevValue
	} else {
		tStart = timestamps[len(timestamps)-1]
		vStart = values[len(timestamps)-1]
	}
	dv := vEnd - vStart
	dt := tEnd - tStart
	return dv / (float64(dt) / 1e3)
}

func rollupLifetime(rfa *rollupFuncArg) float64 {
	// Calculate the duration between the first and the last data points.
	timestamps := rfa.timestamps
	if math.IsNaN(rfa.prevValue) {
		if len(timestamps) < 2 {
			return nan
		}
		return float64(timestamps[len(timestamps)-1]-timestamps[0]) / 1e3
	}
	if len(timestamps) == 0 {
		return nan
	}
	return float64(timestamps[len(timestamps)-1]-rfa.prevTimestamp) / 1e3
}

func rollupLag(rfa *rollupFuncArg) float64 {
	// Calculate the duration between the current timestamp and the last data point.
	timestamps := rfa.timestamps
	if len(timestamps) == 0 {
		if math.IsNaN(rfa.prevValue) {
			return nan
		}
		return float64(rfa.currTimestamp-rfa.prevTimestamp) / 1e3
	}
	return float64(rfa.currTimestamp-timestamps[len(timestamps)-1]) / 1e3
}

func rollupScrapeInterval(rfa *rollupFuncArg) float64 {
	// Calculate the average interval between data points.
	timestamps := rfa.timestamps
	if math.IsNaN(rfa.prevValue) {
		if len(timestamps) < 2 {
			return nan
		}
		return (float64(timestamps[len(timestamps)-1]-timestamps[0]) / 1e3) / float64(len(timestamps)-1)
	}
	if len(timestamps) == 0 {
		return nan
	}
	return (float64(timestamps[len(timestamps)-1]-rfa.prevTimestamp) / 1e3) / float64(len(timestamps))
}

func rollupChanges(rfa *rollupFuncArg) float64 {
	// There is no need in handling NaNs here, since they must be cleaned up
	// before calling rollup funcs.
	values := rfa.values
	prevValue := rfa.prevValue
	n := 0
	if math.IsNaN(prevValue) {
		if len(values) == 0 {
			return nan
		}
		prevValue = values[0]
		values = values[1:]
		n++
	}
	for _, v := range values {
		if v != prevValue {
			n++
			prevValue = v
		}
	}
	return float64(n)
}

func rollupIncreases(rfa *rollupFuncArg) float64 {
	// There is no need in handling NaNs here, since they must be cleaned up
	// before calling rollup funcs.
	values := rfa.values
	if len(values) == 0 {
		if math.IsNaN(rfa.prevValue) {
			return nan
		}
		return 0
	}
	prevValue := rfa.prevValue
	if math.IsNaN(prevValue) {
		prevValue = values[0]
		values = values[1:]
	}
	if len(values) == 0 {
		return 0
	}
	n := 0
	for _, v := range values {
		if v > prevValue {
			n++
		}
		prevValue = v
	}
	return float64(n)
}

// `decreases_over_time` logic is the same as `resets` logic.
var rollupDecreases = rollupResets

func rollupResets(rfa *rollupFuncArg) float64 {
	// There is no need in handling NaNs here, since they must be cleaned up
	// before calling rollup funcs.
	values := rfa.values
	if len(values) == 0 {
		if math.IsNaN(rfa.prevValue) {
			return nan
		}
		return 0
	}
	prevValue := rfa.prevValue
	if math.IsNaN(prevValue) {
		prevValue = values[0]
		values = values[1:]
	}
	if len(values) == 0 {
		return 0
	}
	n := 0
	for _, v := range values {
		if v < prevValue {
			n++
		}
		prevValue = v
	}
	return float64(n)
}

// getCandlestickValues returns a subset of rfa.values suitable for rollup_candlestick
//
// See https://github.com/VictoriaMetrics/VictoriaMetrics/issues/309 for details.
func getCandlestickValues(rfa *rollupFuncArg) []float64 {
	currTimestamp := rfa.currTimestamp
	timestamps := rfa.timestamps
	for len(timestamps) > 0 && timestamps[len(timestamps)-1] >= currTimestamp {
		timestamps = timestamps[:len(timestamps)-1]
	}
	if len(timestamps) == 0 {
		return nil
	}
	return rfa.values[:len(timestamps)]
}

func getFirstValueForCandlestick(rfa *rollupFuncArg) float64 {
	if rfa.prevTimestamp+rfa.window >= rfa.currTimestamp {
		return rfa.prevValue
	}
	return nan
}

func rollupOpen(rfa *rollupFuncArg) float64 {
	v := getFirstValueForCandlestick(rfa)
	if !math.IsNaN(v) {
		return v
	}
	values := getCandlestickValues(rfa)
	if len(values) == 0 {
		return nan
	}
	return values[0]
}

func rollupClose(rfa *rollupFuncArg) float64 {
	values := getCandlestickValues(rfa)
	if len(values) == 0 {
		return getFirstValueForCandlestick(rfa)
	}
	return values[len(values)-1]
}

func rollupHigh(rfa *rollupFuncArg) float64 {
	values := getCandlestickValues(rfa)
	max := getFirstValueForCandlestick(rfa)
	if math.IsNaN(max) {
		if len(values) == 0 {
			return nan
		}
		max = values[0]
		values = values[1:]
	}
	for _, v := range values {
		if v > max {
			max = v
		}
	}
	return max
}

func rollupLow(rfa *rollupFuncArg) float64 {
	values := getCandlestickValues(rfa)
	min := getFirstValueForCandlestick(rfa)
	if math.IsNaN(min) {
		if len(values) == 0 {
			return nan
		}
		min = values[0]
		values = values[1:]
	}
	for _, v := range values {
		if v < min {
			min = v
		}
	}
	return min
}

func rollupModeOverTime(rfa *rollupFuncArg) float64 {
	// There is no need in handling NaNs here, since they must be cleaned up
	// before calling rollup funcs.

	// Copy rfa.values to a.A, since modeNoNaNs modifies a.A contents.
	a := float64sPool.Get().(*float64s)
	a.A = append(a.A[:0], rfa.values...)
	result := modeNoNaNs(rfa.prevValue, a.A)
	float64sPool.Put(a)
	return result
}

var float64sPool = &sync.Pool{
	New: func() interface{} {
		return &float64s{}
	},
}

type float64s struct {
	A []float64
}

func rollupAscentOverTime(rfa *rollupFuncArg) float64 {
	// There is no need in handling NaNs here, since they must be cleaned up
	// before calling rollup funcs.
	values := rfa.values
	prevValue := rfa.prevValue
	if math.IsNaN(prevValue) {
		if len(values) == 0 {
			return nan
		}
		prevValue = values[0]
		values = values[1:]
	}
	s := float64(0)
	for _, v := range values {
		d := v - prevValue
		if d > 0 {
			s += d
		}
		prevValue = v
	}
	return s
}

func rollupDescentOverTime(rfa *rollupFuncArg) float64 {
	// There is no need in handling NaNs here, since they must be cleaned up
	// before calling rollup funcs.
	values := rfa.values
	prevValue := rfa.prevValue
	if math.IsNaN(prevValue) {
		if len(values) == 0 {
			return nan
		}
		prevValue = values[0]
		values = values[1:]
	}
	s := float64(0)
	for _, v := range values {
		d := prevValue - v
		if d > 0 {
			s += d
		}
		prevValue = v
	}
	return s
}

func rollupZScoreOverTime(rfa *rollupFuncArg) float64 {
	// See https://about.gitlab.com/blog/2019/07/23/anomaly-detection-using-prometheus/#using-z-score-for-anomaly-detection
	scrapeInterval := rollupScrapeInterval(rfa)
	lag := rollupLag(rfa)
	if math.IsNaN(scrapeInterval) || math.IsNaN(lag) || lag > scrapeInterval {
		return nan
	}
	d := rollupLast(rfa) - rollupAvg(rfa)
	if d == 0 {
		return 0
	}
	return d / rollupStddev(rfa)
}

func rollupFirst(rfa *rollupFuncArg) float64 {
	// There is no need in handling NaNs here, since they must be cleaned up
	// before calling rollup funcs.
	values := rfa.values
	if len(values) == 0 {
		// Do not take into account rfa.prevValue, since it may lead
		// to inconsistent results comparing to Prometheus on broken time series
		// with irregular data points.
		return nan
	}
	return values[0]
}

func rollupDefault(rfa *rollupFuncArg) float64 {
	values := rfa.values
	if len(values) == 0 {
		// Do not take into account rfa.prevValue, since it may lead
		// to inconsistent results comparing to Prometheus on broken time series
		// with irregular data points.
		return nan
	}
	// Intentionally do not skip the possible last Prometheus staleness mark.
	// See https://github.com/VictoriaMetrics/VictoriaMetrics/issues/1526 .
	return values[len(values)-1]
}

func rollupLast(rfa *rollupFuncArg) float64 {
	values := rfa.values
	if len(values) == 0 {
		// Do not take into account rfa.prevValue, since it may lead
		// to inconsistent results comparing to Prometheus on broken time series
		// with irregular data points.
		return nan
	}
	return values[len(values)-1]
}

func rollupDistinct(rfa *rollupFuncArg) float64 {
	// There is no need in handling NaNs here, since they must be cleaned up
	// before calling rollup funcs.
	values := rfa.values
	if len(values) == 0 {
		if math.IsNaN(rfa.prevValue) {
			return nan
		}
		return 0
	}
	m := make(map[float64]struct{})
	for _, v := range values {
		m[v] = struct{}{}
	}
	return float64(len(m))
}

func rollupIntegrate(rfa *rollupFuncArg) float64 {
	// There is no need in handling NaNs here, since they must be cleaned up
	// before calling rollup funcs.
	values := rfa.values
	timestamps := rfa.timestamps
	prevValue := rfa.prevValue
	prevTimestamp := rfa.currTimestamp - rfa.window
	if math.IsNaN(prevValue) {
		if len(values) == 0 {
			return nan
		}
		prevValue = values[0]
		prevTimestamp = timestamps[0]
		values = values[1:]
		timestamps = timestamps[1:]
	}

	var sum float64
	for i, v := range values {
		timestamp := timestamps[i]
		dt := float64(timestamp-prevTimestamp) / 1e3
		sum += prevValue * dt
		prevTimestamp = timestamp
		prevValue = v
	}
	dt := float64(rfa.currTimestamp-prevTimestamp) / 1e3
	sum += prevValue * dt
	return sum
}

func rollupFake(rfa *rollupFuncArg) float64 {
	logger.Panicf("BUG: rollupFake shouldn't be called")
	return 0
}

func getScalar(arg interface{}, argNum int) ([]float64, error) {
	ts, ok := arg.([]*timeseries)
	if !ok {
		return nil, fmt.Errorf(`unexpected type for arg #%d; got %T; want %T`, argNum+1, arg, ts)
	}
	if len(ts) != 1 {
		return nil, fmt.Errorf(`arg #%d must contain a single timeseries; got %d timeseries`, argNum+1, len(ts))
	}
	return ts[0].Values, nil
}

func getString(tss []*timeseries, argNum int) (string, error) {
	if len(tss) != 1 {
		return "", fmt.Errorf(`arg #%d must contain a single timeseries; got %d timeseries`, argNum+1, len(tss))
	}
	ts := tss[0]
	for _, v := range ts.Values {
		if !math.IsNaN(v) {
			return "", fmt.Errorf(`arg #%d contains non-string timeseries`, argNum+1)
		}
	}
	return string(ts.MetricName.MetricGroup), nil
}

func expectRollupArgsNum(args []interface{}, expectedNum int) error {
	if len(args) == expectedNum {
		return nil
	}
	return fmt.Errorf(`unexpected number of args; got %d; want %d`, len(args), expectedNum)
}

func getRollupFuncArg() *rollupFuncArg {
	v := rfaPool.Get()
	if v == nil {
		return &rollupFuncArg{}
	}
	return v.(*rollupFuncArg)
}

func putRollupFuncArg(rfa *rollupFuncArg) {
	rfa.reset()
	rfaPool.Put(rfa)
}

var rfaPool sync.Pool