This package is initialized using the reviewed mapping functions

contained in these OpenTelemetry-Go PRs:

open-telemetry/opentelemetry-go#2982
open-telemetry/opentelemetry-go#2502

The data structure was reviewed by Lightstep engineers for inclusion
in otel-launcher-go:

lightstep/otel-launcher-go#174
lightstep/otel-launcher-go#215
lightstep/otel-launcher-go#222

Author: Joshua MacDonald

README.md

@@ -0,0 +1,215 @@
+# Base-2 Exponential Histogram
+
+## Design
+
+This is a fixed-size data structure for aggregating the OpenTelemetry
+base-2 exponential histogram introduced in [OTEP
+149](https://github.com/open-telemetry/oteps/blob/main/text/0149-exponential-histogram.md)
+and [described in the metrics data
+model](https://github.com/open-telemetry/opentelemetry-specification/blob/main/specification/metrics/datamodel.md#exponentialhistogram).
+The exponential histogram data point is characterized by a `scale`
+factor that determines resolution.  Positive scales correspond with
+more resolution, and negatives scales correspond with less resolution.
+
+Given a maximum size, in terms of the number of buckets, the
+implementation determines the best scale possible given the set of
+measurements received.  The size of the histogram is configured using
+the `WithMaxSize()` option, which defaults to 160.
+
+The implementation here maintains the best resolution possible.  Since
+the scale parameter is shared by the positive and negative ranges, the
+best value of the scale parameter is determined by the range with the
+greater difference between minimum and maximum bucket index:
+
+```golang
+func bucketsNeeded(minValue, maxValue float64, scale int32) int32 {
+	return bucketIndex(maxValue, scale) - bucketIndex(minValue, scale) + 1
+}
+
+func bucketIndex(value float64, scale int32) int32 {
+	return math.Log(value) * math.Ldexp(math.Log2E, scale)
+}
+```
+
+The best scale is uniquely determined when `maxSize/2 <
+bucketsNeeded(minValue, maxValue, scale) <= maxSize`.  This
+implementation maintains the best scale by rescaling as needed to stay
+within the maximum size.
+
+## Layout
+
+### Mapping function
+
+The `mapping` sub-package contains the equations specified in the [data
+model for Exponential Histogram data
+points](https://github.com/open-telemetry/opentelemetry-specification/blob/main/specification/metrics/data-model.md#exponentialhistogram).
+
+There are two mapping functions used, depending on the sign of the
+scale.  Negative and zero scales use the `mapping/exponent` mapping
+function, which computes the bucket index directly from the bits of
+the `float64` exponent.  This mapping function is used with scale `-10
+<= scale <= 0`.  Scales smaller than -10 map the entire normal
+`float64` number range into a single bucket, thus are not considered
+useful.
+
+The `mapping/logarithm` mapping function uses `math.Log(value)` times
+the scaling factor `math.Ldexp(math.Log2E, scale)`.  This mapping
+function is used with `0 < scale <= 20`.  The maximum scale is
+selected because at scale 21, simply, it becomes difficult to test
+correctness--at this point `math.MaxFloat64` maps to index
+`math.MaxInt32` and the `math/big` logic used in testing breaks down.
+
+### Data structure
+
+The `structure` sub-package contains a Histogram aggregator for use by
+the OpenTelemetry-Go Metrics SDK as well as OpenTelemetry Collector
+receivers, processors, and exporters.
+
+## Implementation
+
+The implementation maintains a slice of buckets and grows the array in
+size only as necessary given the actual range of values, up to the
+maximum size.  The structure of a single range of buckets is:
+
+```golang
+type buckets struct {
+	backing    bucketsVarwidth[T]  // for T = uint8 | uint16 | uint32 | uint64
+	indexBase  int32
+	indexStart int32
+	indexEnd   int32
+}
+```
+
+The `backing` field is a generic slice of `[]uint8`, `[]uint16`,
+`[]uint32`, or `[]uint64`.
+
+The positive and negative backing arrays are independent, so the
+maximum space used for `buckets` by one `Aggregator` is twice the
+configured maximum size.
+
+### Backing array
+
+The backing array is circular.  The first observation is counted in
+the 0th index of the backing array and the initial bucket number is
+stored in `indexBase`.  After the initial observation, the backing
+array grows in either direction (i.e., larger or smaller bucket
+numbers), until rescaling is necessary.  This mechanism allows the
+histogram to maintain the ideal scale without shifting values inside
+the array.
+
+The `indexStart` and `indexEnd` fields store the current minimum and
+maximum bucket number.  The initial condition is `indexBase ==
+indexStart == indexEnd`, representing a single bucket.
+
+Following the first observation, new observations may fall into a
+bucket up to `size-1` in either direction.  Growth is possible by
+adjusting either `indexEnd` or `indexStart` as long as the constraint
+`indexEnd-indexStart < size` remains true.
+
+Bucket numbers in the range `[indexBase, indexEnd]` are stored in the
+interval `[0, indexEnd-indexBase]` of the backing array.  Buckets in
+the range `[indexStart, indexBase-1]` are stored in the interval
+`[size+indexStart-indexBase, size-1]` of the backing array.
+
+Considering the `aggregation.Buckets` interface, `Offset()` returns
+`indexStart`, `Len()` returns `indexEnd-indexStart+1`, and `At()`
+locates the correct bucket in the circular array.
+
+### Determining change of scale
+
+The algorithm used to determine the (best) change of scale when a new
+value arrives is:
+
+```golang
+func newScale(minIndex, maxIndex, scale, maxSize int32) int32 {
+    return scale - changeScale(minIndex, maxIndex, scale, maxSize)
+}
+
+func changeScale(minIndex, maxIndex, scale, maxSize int32) int32 {
+    var change int32
+    for maxIndex - minIndex >= maxSize {
+	   maxIndex >>= 1
+	   minIndex >>= 1
+	   change++
+    }
+	return change
+}
+```
+
+The `changeScale` function is also used to determine how many bits to
+shift during `Merge`.
+
+### Downscale function
+
+The downscale function rotates the circular backing array so that
+`indexStart == indexBase`, using the "3 reversals" method, before
+combining the buckets in place.
+
+### Merge function
+
+`Merge` first calculates the correct final scale by comparing the
+combined positive and negative ranges.  The destination aggregator is
+then downscaled, if necessary, and the `UpdateByIncr` code path to add
+the source buckets to the destination buckets.
+
+### Scale function
+
+The `Scale` function returns the current scale of the histogram.
+
+If the scale is variable and there are no non-zero values in the
+histogram, the scale is zero by definition; when there is only a
+single value in this case, its scale is MinScale (20) by definition.
+
+If the scale is fixed because of range limits, the fixed scale will be
+returned even for any size histogram.
+
+### Handling subnormal values
+
+Subnormal values are those in the range [0x1p-1074, 0x1p-1022), these
+being numbers that "gradually underflow" and use less than 52 bits of
+precision in the significand at the smallest representable exponent
+(i.e., -1022).  Subnormal numbers present special challenges for both
+the exponent- and logarithm-based mapping function, and to avoid
+additional complexity induced by corner cases, subnormal numbers are
+rounded up to 0x1p-1022 in this implementation.
+
+Handling subnormal numbers is difficult for the logarithm mapping
+function because Golang's `math.Log()` function rounds subnormal
+numbers up to 0x1p-1022.  Handling subnormal numbers is difficult for
+the exponent mapping function because Golang's `math.Frexp()`, the
+natural API for extracting a value's base-2 exponent, also rounds
+subnormal numbers up to 0x1p-1022.
+
+While the additional complexity needed to correctly map subnormal
+numbers is small in both cases, there are few real benefits in doing
+so because of the inherent loss of precision.  As secondary
+motivation, clamping values to the range [0x1p-1022, math.MaxFloat64]
+increases symmetry. This limit means that minimum bucket index and the
+maximum bucket index have similar magnitude, which helps support
+greater maximum scale.  Supporting numbers smaller than 0x1p-1022
+would mean changing the valid scale interval to [-11,19] compared with
+[-10,20].
+
+### UpdateByIncr interface
+
+The OpenTelemetry metrics SDK `Aggregator` type supports an `Update()`
+interface which implies updating the histogram by a count of 1.  This
+implementation also supports `UpdateByIncr()`, which makes it possible
+to support counting multiple observations in a single API call.  This
+extension is useful in applying `Histogram` aggregation to _sampled_
+metric events (e.g. in the [OpenTelemetry statsd
+receiver](https://github.com/open-telemetry/opentelemetry-collector-contrib/tree/main/receiver/statsdreceiver)).
+
+Another use for `UpdateByIncr` is in a Span-to-metrics pipeline
+following [probability sampling in OpenTelemetry tracing
+(WIP)](https://github.com/open-telemetry/opentelemetry-specification/pull/2047).
+
+## Acknowledgements
+
+This implementation is based on work by [Yuke
+Zhuge](https://github.com/yzhuge) and [Otmar
+Ertl](https://github.com/oertl).  See
+[NrSketch](https://github.com/newrelic-experimental/newrelic-sketch-java/blob/1ce245713603d61ba3a4510f6df930a5479cd3f6/src/main/java/com/newrelic/nrsketch/indexer/LogIndexer.java)
+and
+[DynaHist](https://github.com/dynatrace-oss/dynahist/blob/9a6003fd0f661a9ef9dfcced0b428a01e303805e/src/main/java/com/dynatrace/dynahist/layout/OpenTelemetryExponentialBucketsLayout.java)
+repositories for more detail.

doc.go

@@ -0,0 +1,19 @@
+// Copyright The OpenTelemetry Authors
+//
+// Licensed under the Apache License, Version 2.0 (the "License");
+// you may not use this file except in compliance with the License.
+// You may obtain a copy of the License at
+//
+//       http://www.apache.org/licenses/LICENSE-2.0
+//
+// Unless required by applicable law or agreed to in writing, software
+// distributed under the License is distributed on an "AS IS" BASIS,
+// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+// See the License for the specific language governing permissions and
+// limitations under the License.
+
+// expohisto contains two sub-packages: (1) the `mapping` package
+// includes ways to convert between values and bucket index numbers as
+// a function of scale, (2) the `structure` package contains a generic
+// data structure.
+package expohisto // import "github.com/lightstep/go-expohisto"

go.mod

@@ -0,0 +1,11 @@
+module github.com/lightstep/go-expohisto
+
+go 1.19
+
+require github.com/stretchr/testify v1.8.0
+
+require (
+	github.com/davecgh/go-spew v1.1.1 // indirect
+	github.com/pmezard/go-difflib v1.0.0 // indirect
+	gopkg.in/yaml.v3 v3.0.1 // indirect
+)

go.sum

@@ -0,0 +1,15 @@
+github.com/davecgh/go-spew v1.1.0/go.mod h1:J7Y8YcW2NihsgmVo/mv3lAwl/skON4iLHjSsI+c5H38=
+github.com/davecgh/go-spew v1.1.1 h1:vj9j/u1bqnvCEfJOwUhtlOARqs3+rkHYY13jYWTU97c=
+github.com/davecgh/go-spew v1.1.1/go.mod h1:J7Y8YcW2NihsgmVo/mv3lAwl/skON4iLHjSsI+c5H38=
+github.com/pmezard/go-difflib v1.0.0 h1:4DBwDE0NGyQoBHbLQYPwSUPoCMWR5BEzIk/f1lZbAQM=
+github.com/pmezard/go-difflib v1.0.0/go.mod h1:iKH77koFhYxTK1pcRnkKkqfTogsbg7gZNVY4sRDYZ/4=
+github.com/stretchr/objx v0.1.0/go.mod h1:HFkY916IF+rwdDfMAkV7OtwuqBVzrE8GR6GFx+wExME=
+github.com/stretchr/objx v0.4.0/go.mod h1:YvHI0jy2hoMjB+UWwv71VJQ9isScKT/TqJzVSSt89Yw=
+github.com/stretchr/testify v1.7.1/go.mod h1:6Fq8oRcR53rry900zMqJjRRixrwX3KX962/h/Wwjteg=
+github.com/stretchr/testify v1.8.0 h1:pSgiaMZlXftHpm5L7V1+rVB+AZJydKsMxsQBIJw4PKk=
+github.com/stretchr/testify v1.8.0/go.mod h1:yNjHg4UonilssWZ8iaSj1OCr/vHnekPRkoO+kdMU+MU=
+gopkg.in/check.v1 v0.0.0-20161208181325-20d25e280405 h1:yhCVgyC4o1eVCa2tZl7eS0r+SDo693bJlVdllGtEeKM=
+gopkg.in/check.v1 v0.0.0-20161208181325-20d25e280405/go.mod h1:Co6ibVJAznAaIkqp8huTwlJQCZ016jof/cbN4VW5Yz0=
+gopkg.in/yaml.v3 v3.0.0-20200313102051-9f266ea9e77c/go.mod h1:K4uyk7z7BCEPqu6E+C64Yfv1cQ7kz7rIZviUmN+EgEM=
+gopkg.in/yaml.v3 v3.0.1 h1:fxVm/GzAzEWqLHuvctI91KS9hhNmmWOoWu0XTYJS7CA=
+gopkg.in/yaml.v3 v3.0.1/go.mod h1:K4uyk7z7BCEPqu6E+C64Yfv1cQ7kz7rIZviUmN+EgEM=

mapping/exponent/exponent.go

@@ -0,0 +1,127 @@
+// Copyright The OpenTelemetry Authors
+//
+// Licensed under the Apache License, Version 2.0 (the "License");
+// you may not use this file except in compliance with the License.
+// You may obtain a copy of the License at
+//
+//       http://www.apache.org/licenses/LICENSE-2.0
+//
+// Unless required by applicable law or agreed to in writing, software
+// distributed under the License is distributed on an "AS IS" BASIS,
+// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+// See the License for the specific language governing permissions and
+// limitations under the License.
+
+package exponent // import "github.com/lightstep/go-expohisto/mapping/exponent"
+
+import (
+	"fmt"
+	"math"
+
+	"github.com/lightstep/go-expohisto/mapping"
+	"github.com/lightstep/go-expohisto/mapping/internal"
+)
+
+const (
+	// MinScale defines the point at which the exponential mapping
+	// function becomes useless for float64.  With scale -10, ignoring
+	// subnormal values, bucket indices range from -1 to 1.
+	MinScale int32 = -10
+
+	// MaxScale is the largest scale supported in this code.  Use
+	// ../logarithm for larger scales.
+	MaxScale int32 = 0
+)
+
+type exponentMapping struct {
+	shift uint8 // equals negative scale
+}
+
+// exponentMapping is used for negative scales, effectively a
+// mapping of the base-2 logarithm of the exponent.
+var prebuiltMappings = [-MinScale + 1]exponentMapping{
+	{10},
+	{9},
+	{8},
+	{7},
+	{6},
+	{5},
+	{4},
+	{3},
+	{2},
+	{1},
+	{0},
+}
+
+// NewMapping constructs an exponential mapping function, used for scales <= 0.
+func NewMapping(scale int32) (mapping.Mapping, error) {
+	if scale > MaxScale {
+		return nil, fmt.Errorf("exponent mapping requires scale <= 0")
+	}
+	if scale < MinScale {
+		return nil, fmt.Errorf("scale too low")
+	}
+	return &prebuiltMappings[scale-MinScale], nil
+}
+
+// minNormalLowerBoundaryIndex is the largest index such that
+// base**index is <= MinValue.  A histogram bucket with this index
+// covers the range (base**index, base**(index+1)], including
+// MinValue.
+func (e *exponentMapping) minNormalLowerBoundaryIndex() int32 {
+	idx := int32(internal.MinNormalExponent) >> e.shift
+	if e.shift < 2 {
+		// For scales -1 and 0 the minimum value 2**-1022
+		// is a power-of-two multiple, meaning it belongs
+		// to the index one less.
+		idx--
+	}
+	return idx
+}
+
+// maxNormalLowerBoundaryIndex is the index such that base**index
+// equals the largest representable boundary.  A histogram bucket with this
+// index covers the range (0x1p+1024/base, 0x1p+1024], which includes
+// MaxValue; note that this bucket is incomplete, since the upper
+// boundary cannot be represented.  One greater than this index
+// corresponds with the bucket containing values > 0x1p1024.
+func (e *exponentMapping) maxNormalLowerBoundaryIndex() int32 {
+	return int32(internal.MaxNormalExponent) >> e.shift
+}
+
+// MapToIndex implements mapping.Mapping.
+func (e *exponentMapping) MapToIndex(value float64) int32 {
+	// Note: we can assume not a 0, Inf, or NaN; positive sign bit.
+	if value < internal.MinValue {
+		return e.minNormalLowerBoundaryIndex()
+	}
+
+	// Extract the raw exponent.
+	rawExp := internal.GetNormalBase2(value)
+
+	// In case the value is an exact power of two, compute a
+	// correction of -1:
+	correction := int32((internal.GetSignificand(value) - 1) >> internal.SignificandWidth)
+
+	// Note: bit-shifting does the right thing for negative
+	// exponents, e.g., -1 >> 1 == -1.
+	return (rawExp + correction) >> e.shift
+}
+
+// LowerBoundary implements mapping.Mapping.
+func (e *exponentMapping) LowerBoundary(index int32) (float64, error) {
+	if min := e.minNormalLowerBoundaryIndex(); index < min {
+		return 0, mapping.ErrUnderflow
+	}
+
+	if max := e.maxNormalLowerBoundaryIndex(); index > max {
+		return 0, mapping.ErrOverflow
+	}
+
+	return math.Ldexp(1, int(index<<e.shift)), nil
+}
+
+// Scale implements mapping.Mapping.
+func (e *exponentMapping) Scale() int32 {
+	return -int32(e.shift)
+}