File size: 32,419 Bytes

9b141ed

from fontTools.misc.roundTools import noRound, otRound
from fontTools.misc.intTools import bit_count
from fontTools.ttLib.tables import otTables as ot
from fontTools.varLib.models import supportScalar
from fontTools.varLib.builder import (
    buildVarRegionList,
    buildVarStore,
    buildVarRegion,
    buildVarData,
)
from functools import partial
from collections import defaultdict
from heapq import heappush, heappop
import itertools

NO_VARIATION_INDEX = ot.NO_VARIATION_INDEX
ot.VarStore.NO_VARIATION_INDEX = NO_VARIATION_INDEX


def _getLocationKey(loc):
    return tuple(sorted(loc.items(), key=lambda kv: kv[0]))


class OnlineVarStoreBuilder(object):
    def __init__(self, axisTags):
        self._axisTags = axisTags
        self._regionMap = {}
        self._regionList = buildVarRegionList([], axisTags)
        self._store = buildVarStore(self._regionList, [])
        self._data = None
        self._model = None
        self._supports = None
        self._varDataIndices = {}
        self._varDataCaches = {}
        self._cache = None

    def setModel(self, model):
        self.setSupports(model.supports)
        self._model = model

    def setSupports(self, supports):
        self._model = None
        self._supports = list(supports)
        if self._supports and not self._supports[0]:
            del self._supports[0]  # Drop base master support
        self._cache = None
        self._data = None

    def finish(self, optimize=True):
        self._regionList.RegionCount = len(self._regionList.Region)
        self._store.VarDataCount = len(self._store.VarData)
        for data in self._store.VarData:
            data.ItemCount = len(data.Item)
            data.calculateNumShorts(optimize=optimize)
        return self._store

    def _add_VarData(self, num_items=1):
        regionMap = self._regionMap
        regionList = self._regionList

        regions = self._supports
        regionIndices = []
        for region in regions:
            key = _getLocationKey(region)
            idx = regionMap.get(key)
            if idx is None:
                varRegion = buildVarRegion(region, self._axisTags)
                idx = regionMap[key] = len(regionList.Region)
                regionList.Region.append(varRegion)
            regionIndices.append(idx)

        # Check if we have one already...
        key = tuple(regionIndices)
        varDataIdx = self._varDataIndices.get(key)
        if varDataIdx is not None:
            self._outer = varDataIdx
            self._data = self._store.VarData[varDataIdx]
            self._cache = self._varDataCaches[key]
            if len(self._data.Item) + num_items > 0xFFFF:
                # This is full.  Need new one.
                varDataIdx = None

        if varDataIdx is None:
            self._data = buildVarData(regionIndices, [], optimize=False)
            self._outer = len(self._store.VarData)
            self._store.VarData.append(self._data)
            self._varDataIndices[key] = self._outer
            if key not in self._varDataCaches:
                self._varDataCaches[key] = {}
            self._cache = self._varDataCaches[key]

    def storeMasters(self, master_values, *, round=round):
        deltas = self._model.getDeltas(master_values, round=round)
        base = deltas.pop(0)
        return base, self.storeDeltas(deltas, round=noRound)

    def storeMastersMany(self, master_values_list, *, round=round):
        deltas_list = [
            self._model.getDeltas(master_values, round=round)
            for master_values in master_values_list
        ]
        base_list = [deltas.pop(0) for deltas in deltas_list]
        return base_list, self.storeDeltasMany(deltas_list, round=noRound)

    def storeDeltas(self, deltas, *, round=round):
        deltas = [round(d) for d in deltas]
        if len(deltas) == len(self._supports) + 1:
            deltas = tuple(deltas[1:])
        else:
            assert len(deltas) == len(self._supports)
            deltas = tuple(deltas)

        if not self._data:
            self._add_VarData()

        varIdx = self._cache.get(deltas)
        if varIdx is not None:
            return varIdx

        inner = len(self._data.Item)
        if inner == 0xFFFF:
            # Full array. Start new one.
            self._add_VarData()
            return self.storeDeltas(deltas, round=noRound)
        self._data.addItem(deltas, round=noRound)

        varIdx = (self._outer << 16) + inner
        self._cache[deltas] = varIdx
        return varIdx

    def storeDeltasMany(self, deltas_list, *, round=round):
        deltas_list = [[round(d) for d in deltas] for deltas in deltas_list]
        deltas_list = tuple(tuple(deltas) for deltas in deltas_list)

        if not self._data:
            self._add_VarData(len(deltas_list))

        varIdx = self._cache.get(deltas_list)
        if varIdx is not None:
            return varIdx

        inner = len(self._data.Item)
        if inner + len(deltas_list) > 0xFFFF:
            # Full array. Start new one.
            self._add_VarData(len(deltas_list))
            return self.storeDeltasMany(deltas_list, round=noRound)
        for i, deltas in enumerate(deltas_list):
            self._data.addItem(deltas, round=noRound)

            varIdx = (self._outer << 16) + inner + i
            self._cache[deltas] = varIdx

        varIdx = (self._outer << 16) + inner
        self._cache[deltas_list] = varIdx

        return varIdx


def VarData_addItem(self, deltas, *, round=round):
    deltas = [round(d) for d in deltas]

    countUs = self.VarRegionCount
    countThem = len(deltas)
    if countUs + 1 == countThem:
        deltas = list(deltas[1:])
    else:
        assert countUs == countThem, (countUs, countThem)
        deltas = list(deltas)
    self.Item.append(deltas)
    self.ItemCount = len(self.Item)


ot.VarData.addItem = VarData_addItem


def VarRegion_get_support(self, fvar_axes):
    return {
        fvar_axes[i].axisTag: (reg.StartCoord, reg.PeakCoord, reg.EndCoord)
        for i, reg in enumerate(self.VarRegionAxis)
        if reg.PeakCoord != 0
    }


ot.VarRegion.get_support = VarRegion_get_support


def VarStore___bool__(self):
    return bool(self.VarData)


ot.VarStore.__bool__ = VarStore___bool__


class VarStoreInstancer(object):
    """Evaluate a VarStore's deltas at a (mutable via setLocation) location.

    Region supports are cached for the lifetime of the instancer: after
    mutating the store's regions (or the fvar axes), construct a new
    instancer rather than reusing this one via setLocation.
    """

    def __init__(self, varstore, fvar_axes, location={}):
        self.fvar_axes = fvar_axes
        assert varstore is None or varstore.Format == 1
        self._varData = varstore.VarData if varstore else []
        self._regions = varstore.VarRegionList.Region if varstore else []
        # Region supports are location-independent, so cache them for the
        # lifetime of the instancer (they survive setLocation); only the
        # per-region scalars, which depend on the location, are recomputed.
        # This makes reusing one instancer across many locations cheap.
        self._supports = {}
        self.setLocation(location)

    def setLocation(self, location):
        self.location = dict(location)
        self._clearCaches()

    def _clearCaches(self):
        self._scalars = {}

    def _getSupport(self, regionIdx):
        support = self._supports.get(regionIdx)
        if support is None:
            support = self._regions[regionIdx].get_support(self.fvar_axes)
            self._supports[regionIdx] = support
        return support

    def _getScalar(self, regionIdx):
        scalar = self._scalars.get(regionIdx)
        if scalar is None:
            scalar = supportScalar(self.location, self._getSupport(regionIdx))
            self._scalars[regionIdx] = scalar
        return scalar

    @staticmethod
    def interpolateFromDeltasAndScalars(deltas, scalars):
        delta = 0.0
        for d, s in zip(deltas, scalars):
            if not s:
                continue
            delta += d * s
        return delta

    def __getitem__(self, varidx):
        major, minor = varidx >> 16, varidx & 0xFFFF
        if varidx == NO_VARIATION_INDEX:
            return 0.0
        varData = self._varData
        # Keep full instancing consistent with _effectiveAvar2VarIdx: out-of-range
        # delta-set indices contribute zero. HarfBuzz uses the same policy:
        # https://github.com/harfbuzz/harfbuzz/blob/d2df3cdcc0836299a163dc0399a6b047d19ac56c/src/hb-ot-layout-common.hh#L2912-L2918
        # https://github.com/harfbuzz/harfbuzz/blob/d2df3cdcc0836299a163dc0399a6b047d19ac56c/src/hb-ot-layout-common.hh#L3354-L3363
        if major >= len(varData) or minor >= len(varData[major].Item):
            return 0.0
        scalars = [self._getScalar(ri) for ri in varData[major].VarRegionIndex]
        deltas = varData[major].Item[minor]
        return self.interpolateFromDeltasAndScalars(deltas, scalars)

    def interpolateFromDeltas(self, varDataIndex, deltas):
        varData = self._varData
        scalars = [self._getScalar(ri) for ri in varData[varDataIndex].VarRegionIndex]
        return self.interpolateFromDeltasAndScalars(deltas, scalars)


#
# Optimizations
#
# retainFirstMap - If true, major 0 mappings are retained. Deltas for unused indices are zeroed
# advIdxes - Set of major 0 indices for advance deltas to be listed first. Other major 0 indices follow.


def VarStore_subset_varidxes(
    self,
    varIdxes,
    optimize=True,
    retainFirstMap=False,
    advIdxes=set(),
    *,
    VarData="VarData",
):
    # Sort out used varIdxes by major/minor.
    used = defaultdict(set)
    for varIdx in varIdxes:
        if varIdx == NO_VARIATION_INDEX:
            continue
        major = varIdx >> 16
        minor = varIdx & 0xFFFF
        used[major].add(minor)
    del varIdxes

    #
    # Subset VarData
    #

    varData = getattr(self, VarData)
    newVarData = []
    varDataMap = {NO_VARIATION_INDEX: NO_VARIATION_INDEX}
    for major, data in enumerate(varData):
        usedMinors = used.get(major)
        if usedMinors is None:
            continue
        newMajor = len(newVarData)
        newVarData.append(data)

        items = data.Item
        newItems = []
        if major == 0 and retainFirstMap:
            for minor in range(len(items)):
                newItems.append(
                    items[minor] if minor in usedMinors else [0] * len(items[minor])
                )
                varDataMap[minor] = minor
        else:
            if major == 0:
                minors = sorted(advIdxes) + sorted(usedMinors - advIdxes)
            else:
                minors = sorted(usedMinors)
            for minor in minors:
                newMinor = len(newItems)
                newItems.append(items[minor])
                varDataMap[(major << 16) + minor] = (newMajor << 16) + newMinor

        data.Item = newItems
        data.ItemCount = len(data.Item)

        if VarData == "VarData":
            data.calculateNumShorts(optimize=optimize)

    setattr(self, VarData, newVarData)
    setattr(self, VarData + "Count", len(newVarData))

    self.prune_regions()

    return varDataMap


ot.VarStore.subset_varidxes = VarStore_subset_varidxes


def VarStore_prune_regions(self, *, VarData="VarData", VarRegionList="VarRegionList"):
    """Remove unused VarRegions."""
    #
    # Subset VarRegionList
    #

    # Collect.
    usedRegions = set()
    for data in getattr(self, VarData):
        usedRegions.update(data.VarRegionIndex)
    # Subset.
    regionList = getattr(self, VarRegionList)
    regions = regionList.Region
    newRegions = []
    regionMap = {}
    for i in sorted(usedRegions):
        regionMap[i] = len(newRegions)
        newRegions.append(regions[i])
    regionList.Region = newRegions
    regionList.RegionCount = len(regionList.Region)
    # Map.
    for data in getattr(self, VarData):
        data.VarRegionIndex = [regionMap[i] for i in data.VarRegionIndex]


ot.VarStore.prune_regions = VarStore_prune_regions


def _visit(self, func):
    """Recurse down from self, if type of an object is ot.Device,
    call func() on it.  Works on otData-style classes."""

    if type(self) == ot.Device:
        func(self)

    elif isinstance(self, list):
        for that in self:
            _visit(that, func)

    elif hasattr(self, "getConverters") and not hasattr(self, "postRead"):
        for conv in self.getConverters():
            that = getattr(self, conv.name, None)
            if that is not None:
                _visit(that, func)

    elif isinstance(self, ot.ValueRecord):
        for that in self.__dict__.values():
            _visit(that, func)


def _Device_recordVarIdx(self, s):
    """Add VarIdx in this Device table (if any) to the set s."""
    if self.DeltaFormat == 0x8000:
        s.add((self.StartSize << 16) + self.EndSize)


def Object_collect_device_varidxes(self, varidxes):
    adder = partial(_Device_recordVarIdx, s=varidxes)
    _visit(self, adder)


ot.GDEF.collect_device_varidxes = Object_collect_device_varidxes
ot.GPOS.collect_device_varidxes = Object_collect_device_varidxes


def _Device_mapVarIdx(self, mapping, done):
    """Map VarIdx in this Device table (if any) through mapping."""
    if id(self) in done:
        return
    done.add(id(self))
    if self.DeltaFormat == 0x8000:
        varIdx = mapping[(self.StartSize << 16) + self.EndSize]
        self.StartSize = varIdx >> 16
        self.EndSize = varIdx & 0xFFFF


def Object_remap_device_varidxes(self, varidxes_map):
    mapper = partial(_Device_mapVarIdx, mapping=varidxes_map, done=set())
    _visit(self, mapper)


ot.GDEF.remap_device_varidxes = Object_remap_device_varidxes
ot.GPOS.remap_device_varidxes = Object_remap_device_varidxes
ot.BASE.remap_device_varidxes = Object_remap_device_varidxes


class _Encoding(object):
    def __init__(self, chars):
        self.chars = chars
        self.width = bit_count(chars)
        self.columns = self._columns(chars)
        self.overhead = self._characteristic_overhead(self.columns)
        self.items = set()

    def append(self, row):
        self.items.add(row)

    def extend(self, lst):
        self.items.update(lst)

    def width_sort_key(self):
        return self.width, self.chars

    @staticmethod
    def _characteristic_overhead(columns):
        """Returns overhead in bytes of encoding this characteristic
        as a VarData."""
        c = 4 + 6  # 4 bytes for LOffset, 6 bytes for VarData header
        c += bit_count(columns) * 2
        return c

    @staticmethod
    def _columns(chars):
        cols = 0
        i = 1
        while chars:
            if chars & 0b1111:
                cols |= i
            chars >>= 4
            i <<= 1
        return cols

    def gain_from_merging(self, other_encoding):
        combined_chars = other_encoding.chars | self.chars
        combined_width = bit_count(combined_chars)
        combined_columns = self.columns | other_encoding.columns
        combined_overhead = _Encoding._characteristic_overhead(combined_columns)
        combined_gain = (
            +self.overhead
            + other_encoding.overhead
            - combined_overhead
            - (combined_width - self.width) * len(self.items)
            - (combined_width - other_encoding.width) * len(other_encoding.items)
        )
        return combined_gain


class _EncodingDict(dict):
    def __missing__(self, chars):
        r = self[chars] = _Encoding(chars)
        return r

    def add_row(self, row):
        chars = self._row_characteristics(row)
        self[chars].append(row)

    @staticmethod
    def _row_characteristics(row):
        """Returns encoding characteristics for a row."""
        longWords = False

        chars = 0
        i = 1
        for v in row:
            if v:
                chars += i
            if not (-128 <= v <= 127):
                chars += i * 0b0010
            if not (-32768 <= v <= 32767):
                longWords = True
                break
            i <<= 4

        if longWords:
            # Redo; only allow 2byte/4byte encoding
            chars = 0
            i = 1
            for v in row:
                if v:
                    chars += i * 0b0011
                if not (-32768 <= v <= 32767):
                    chars += i * 0b1100
                i <<= 4

        return chars


def VarStore_optimize(self, use_NO_VARIATION_INDEX=True, quantization=1):
    """Optimize storage. Returns mapping from old VarIdxes to new ones."""

    # Overview:
    #
    # For each VarData row, we first extend it with zeroes to have
    # one column per region in VarRegionList. We then group the
    # rows into _Encoding objects, by their "characteristic" bitmap.
    # The characteristic bitmap is a binary number representing how
    # many bytes each column of the data takes up to encode. Each
    # column is encoded in four bits. For example, if a column has
    # only values in the range -128..127, it would only have a single
    # bit set in the characteristic bitmap for that column. If it has
    # values in the range -32768..32767, it would have two bits set.
    # The number of ones in the characteristic bitmap is the "width"
    # of the encoding.
    #
    # Each encoding as such has a number of "active" (ie. non-zero)
    # columns. The overhead of encoding the characteristic bitmap
    # is 10 bytes, plus 2 bytes per active column.
    #
    # When an encoding is merged into another one, if the characteristic
    # of the old encoding is a subset of the new one, then the overhead
    # of the old encoding is completely eliminated. However, each row
    # now would require more bytes to encode, to the tune of one byte
    # per characteristic bit that is active in the new encoding but not
    # in the old one.
    #
    # The "gain" of merging two encodings is how many bytes we save by doing so.
    #
    # High-level algorithm:
    #
    # - Each encoding has a minimal way to encode it. However, because
    #   of the overhead of encoding the characteristic bitmap, it may
    #   be beneficial to merge two encodings together, if there is
    #   gain in doing so. As such, we need to search for the best
    #   such successive merges.
    #
    # Algorithm:
    #
    # - Put all encodings into a "todo" list.
    #
    # - Sort todo list (for stability) by width_sort_key(), which is a tuple
    #   of the following items:
    #   * The "width" of the encoding.
    #   * The characteristic bitmap of the encoding, with higher-numbered
    #     columns compared first.
    #
    # - Make a priority-queue of the gain from combining each two
    #   encodings in the todo list. The priority queue is sorted by
    #   decreasing gain. Only positive gains are included.
    #
    # - While priority queue is not empty:
    #   - Pop the first item from the priority queue,
    #   - Merge the two encodings it represents,
    #   - Remove the two encodings from the todo list,
    #   - Insert positive gains from combining the new encoding with
    #     all existing todo list items into the priority queue,
    #   - If a todo list item with the same characteristic bitmap as
    #     the new encoding exists, remove it from the todo list and
    #     merge it into the new encoding.
    #   - Insert the new encoding into the todo list,
    #
    # - Encode all remaining items in the todo list.
    #
    # The output is then sorted for stability, in the following way:
    # - The VarRegionList of the input is kept intact.
    # - The VarData is sorted by the same width_sort_key() used at the beginning.
    # - Within each VarData, the items are sorted as vectors of numbers.
    #
    # Finally, each VarData is optimized to remove the empty columns and
    # reorder columns as needed.

    # TODO
    # Check that no two VarRegions are the same; if they are, fold them.

    n = len(self.VarRegionList.Region)  # Number of columns
    zeroes = [0] * n

    front_mapping = {}  # Map from old VarIdxes to full row tuples

    encodings = _EncodingDict()

    # Collect all items into a set of full rows (with lots of zeroes.)
    for major, data in enumerate(self.VarData):
        regionIndices = data.VarRegionIndex

        for minor, item in enumerate(data.Item):
            row = list(zeroes)

            if quantization == 1:
                for regionIdx, v in zip(regionIndices, item):
                    row[regionIdx] += v
            else:
                for regionIdx, v in zip(regionIndices, item):
                    row[regionIdx] += (
                        round(v / quantization) * quantization
                    )  # TODO https://github.com/fonttools/fonttools/pull/3126#discussion_r1205439785

            row = tuple(row)

            if use_NO_VARIATION_INDEX and not any(row):
                front_mapping[(major << 16) + minor] = None
                continue

            encodings.add_row(row)
            front_mapping[(major << 16) + minor] = row

    # Prepare for the main algorithm.
    todo = sorted(encodings.values(), key=_Encoding.width_sort_key)
    del encodings

    # Repeatedly pick two best encodings to combine, and combine them.

    heap = []
    for i, encoding in enumerate(todo):
        for j in range(i + 1, len(todo)):
            other_encoding = todo[j]
            combining_gain = encoding.gain_from_merging(other_encoding)
            if combining_gain > 0:
                heappush(heap, (-combining_gain, i, j))

    while heap:
        _, i, j = heappop(heap)
        if todo[i] is None or todo[j] is None:
            continue

        encoding, other_encoding = todo[i], todo[j]
        todo[i], todo[j] = None, None

        # Combine the two encodings
        combined_chars = other_encoding.chars | encoding.chars
        combined_encoding = _Encoding(combined_chars)
        combined_encoding.extend(encoding.items)
        combined_encoding.extend(other_encoding.items)

        for k, enc in enumerate(todo):
            if enc is None:
                continue

            # In the unlikely event that the same encoding exists already,
            # combine it.
            if enc.chars == combined_chars:
                combined_encoding.extend(enc.items)
                todo[k] = None
                continue

            combining_gain = combined_encoding.gain_from_merging(enc)
            if combining_gain > 0:
                heappush(heap, (-combining_gain, k, len(todo)))

        todo.append(combined_encoding)

    encodings = [encoding for encoding in todo if encoding is not None]

    # Assemble final store.
    back_mapping = {}  # Mapping from full rows to new VarIdxes
    encodings.sort(key=_Encoding.width_sort_key)
    self.VarData = []
    for encoding in encodings:
        items = sorted(encoding.items)

        while items:
            major = len(self.VarData)
            data = ot.VarData()
            self.VarData.append(data)
            data.VarRegionIndex = range(n)
            data.VarRegionCount = len(data.VarRegionIndex)

            # Each major can only encode up to 0xFFFF entries.
            data.Item, items = items[:0xFFFF], items[0xFFFF:]

            for minor, item in enumerate(data.Item):
                back_mapping[item] = (major << 16) + minor

    # Compile final mapping.
    varidx_map = {NO_VARIATION_INDEX: NO_VARIATION_INDEX}
    for k, v in front_mapping.items():
        varidx_map[k] = back_mapping[v] if v is not None else NO_VARIATION_INDEX

    # Recalculate things and go home.
    self.VarRegionList.RegionCount = len(self.VarRegionList.Region)
    self.VarDataCount = len(self.VarData)
    for data in self.VarData:
        data.ItemCount = len(data.Item)
        data.optimize()

    # Remove unused regions.
    self.prune_regions()

    return varidx_map


ot.VarStore.optimize = VarStore_optimize


def _tentValue(v, start, peak, end):
    """Per-axis support-scalar factor at coordinate v (supportScalar semantics)."""
    if v == peak:
        return 1.0
    if v <= start or end <= v:
        return 0.0
    if v < peak:
        return (v - start) / (peak - start)
    return (end - v) / (end - peak)


# Maximum number of grid vertices VarStore_getExtremes enumerates for the
# exact answer before falling back to the (looser but conservative and cheap)
# interval bound. The grid is authoring-controlled, so a cap is needed to keep
# crafted fonts from hanging the instancer.
_GET_EXTREMES_MAX_GRID = 1 << 14
_F2DOT14_UNIT = 1.0 / (1 << 14)


def VarStore_getExtremes(self, varIdx, fvarAxes, axisLimits, identityAxisIndex=None):
    """Conservatively bound the value this VarStore produces for varIdx.

    Returns (minV, maxV) in F2Dot14 delta units, guaranteed to cover the
    truly reachable range as the input axes vary over their extents:
    [-1, +1] per axis, or the [min, max] interval from axisLimits (a dict of
    axis tag to a (min, default, max, ...) sequence) for axes listed there.
    If identityAxisIndex is not None, that axis's own identity contribution
    (its coordinate, in F2Dot14 units) is added to the bound.

    Before the final OpenType rounding, the produced value is piecewise
    multilinear in the axis coordinates (each region's scalar is a product
    of independent per-axis tents), so over the box its exact extremes are
    attained at vertices of the grid formed by each axis's tent
    breakpoints. One-sided tents (start == peak or peak == end) are
    discontinuous at their peak, so the grid also includes the adjacent
    F2Dot14 coordinate on the open side. When that grid is small enough
    (real fonts) it is enumerated and the result is exact; otherwise the
    result falls back to a per-region interval bound, which ignores
    correlations between regions and may be wider than the true range —
    never narrower. Callers use this to cull provably-unreachable
    variation data, so conservativeness is a soundness requirement, not a
    nicety. The fallback costs O(regions × axes); the cap on the exact
    path keeps crafted fonts from hanging the instancer.

    Raw extrema are rounded only after summing the identity and delta terms,
    matching avar2 processing.
    """

    def axisInterval(axisIdx):
        tag = fvarAxes[axisIdx].axisTag
        if tag in axisLimits:
            limits = axisLimits[tag]
            return limits[0], limits[2]
        return -1.0, +1.0

    if varIdx == NO_VARIATION_INDEX:
        activeRegions = []
    else:
        major = varIdx >> 16
        minor = varIdx & 0xFFFF
        varData = self.VarData[major]
        regions = self.VarRegionList.Region
        deltas = varData.Item[minor]
        # Per region: (delta, [(axisIndex, start, peak, end), ...]) with the
        # axes supportScalar would ignore already dropped.
        activeRegions = []
        for regionIndex, delta in zip(varData.VarRegionIndex, deltas):
            if not delta:
                continue
            region = regions[regionIndex]
            tents = []
            for i, regionAxis in enumerate(region.VarRegionAxis):
                start = regionAxis.StartCoord
                peak = regionAxis.PeakCoord
                end = regionAxis.EndCoord
                # Mirror supportScalar's OT rules for axes it ignores.
                if peak == 0 or start > peak or peak > end:
                    continue
                if start < 0 and end > 0:
                    continue
                tents.append((i, start, peak, end))
            activeRegions.append((delta, tents))

    # Try the exact path: enumerate the breakpoint grid.
    breakpoints = {}  # axisIndex -> set of coordinates
    if identityAxisIndex is not None:
        lo, hi = axisInterval(identityAxisIndex)
        breakpoints[identityAxisIndex] = {lo, hi}
    for delta, tents in activeRegions:
        for i, start, peak, end in tents:
            if i not in breakpoints:
                lo, hi = axisInterval(i)
                breakpoints[i] = {lo, hi}
            lo, hi = axisInterval(i)
            breakpoints[i].update(min(max(v, lo), hi) for v in (start, peak, end))
            if start == peak:
                breakpoints[i].add(min(max(peak - _F2DOT14_UNIT, lo), hi))
            if peak == end:
                breakpoints[i].add(min(max(peak + _F2DOT14_UNIT, lo), hi))
    gridSize = 1
    for points in breakpoints.values():
        gridSize *= len(points)
        if gridSize > _GET_EXTREMES_MAX_GRID:
            break

    if gridSize <= _GET_EXTREMES_MAX_GRID:
        # Exact: evaluate at every grid vertex. Within each grid cell every
        # region scalar is multilinear, so the extremes lie on vertices.
        axisIndices = sorted(breakpoints)
        axisPos = {i: n for n, i in enumerate(axisIndices)}
        minV = maxV = None
        for vertex in itertools.product(*(sorted(breakpoints[i]) for i in axisIndices)):
            v = 0.0
            for delta, tents in activeRegions:
                scalar = 1.0
                for i, start, peak, end in tents:
                    scalar *= _tentValue(vertex[axisPos[i]], start, peak, end)
                    if scalar == 0:
                        break
                if scalar:
                    v += delta * scalar
            if identityAxisIndex is not None:
                v += otRound(vertex[axisPos[identityAxisIndex]] * 16384)
            if minV is None:
                minV = maxV = v
            else:
                minV = min(minV, v)
                maxV = max(maxV, v)
        return otRound(minV), otRound(maxV)

    # Fallback: per-region interval arithmetic. Each region's scalar range
    # over the box is exact (per-axis tent factors are independent), but
    # correlations between regions (and with the identity term) are ignored.
    minV = maxV = 0.0
    for delta, tents in activeRegions:
        scalarMin = scalarMax = 1.0
        for i, start, peak, end in tents:
            lo, hi = axisInterval(i)
            # The tent rises to the peak then falls, so over [lo, hi] the
            # maximum is at the clamped peak, the minimum at an endpoint.
            scalarMax *= _tentValue(min(max(peak, lo), hi), start, peak, end)
            scalarMin *= min(
                _tentValue(lo, start, peak, end),
                _tentValue(hi, start, peak, end),
            )
            if scalarMax == 0:
                break
        if scalarMax == 0:
            continue
        minV += min(delta * scalarMin, delta * scalarMax)
        maxV += max(delta * scalarMin, delta * scalarMax)

    if identityAxisIndex is not None:
        lo, hi = axisInterval(identityAxisIndex)
        minV += otRound(lo * 16384)
        maxV += otRound(hi * 16384)

    return otRound(minV), otRound(maxV)


ot.VarStore.getExtremes = VarStore_getExtremes


def main(args=None):
    """Optimize a font's GDEF variation store"""
    from argparse import ArgumentParser
    from fontTools import configLogger
    from fontTools.ttLib import TTFont
    from fontTools.ttLib.tables.otBase import OTTableWriter

    parser = ArgumentParser(prog="varLib.varStore", description=main.__doc__)
    parser.add_argument("--quantization", type=int, default=1)
    parser.add_argument("fontfile")
    parser.add_argument("outfile", nargs="?")
    options = parser.parse_args(args)

    # TODO: allow user to configure logging via command-line options
    configLogger(level="INFO")

    quantization = options.quantization
    fontfile = options.fontfile
    outfile = options.outfile

    font = TTFont(fontfile)
    gdef = font["GDEF"]
    store = gdef.table.VarStore

    writer = OTTableWriter()
    store.compile(writer, font)
    size = len(writer.getAllData())
    print("Before: %7d bytes" % size)

    varidx_map = store.optimize(quantization=quantization)

    writer = OTTableWriter()
    store.compile(writer, font)
    size = len(writer.getAllData())
    print("After:  %7d bytes" % size)

    if outfile is not None:
        gdef.table.remap_device_varidxes(varidx_map)
        if "GPOS" in font:
            font["GPOS"].table.remap_device_varidxes(varidx_map)

        font.save(outfile)


if __name__ == "__main__":
    import sys

    if len(sys.argv) > 1:
        sys.exit(main())
    import doctest

    sys.exit(doctest.testmod().failed)