freetype/truetype/hint.go

// Copyright 2012 The Freetype-Go Authors. All rights reserved.
// Use of this source code is governed by your choice of either the
// FreeType License or the GNU General Public License version 2 (or
// any later version), both of which can be found in the LICENSE file.

package truetype

// This file implements a Truetype bytecode interpreter.
// The opcodes are described at https://developer.apple.com/fonts/TTRefMan/RM05/Chap5.html

import (
	"errors"
	"math"
)

const (
	twilightZone = 0
	glyphZone    = 1
	numZone      = 2
)

type pointType uint32

const (
	current      pointType = 0
	unhinted     pointType = 1
	inFontUnits  pointType = 2
	numPointType           = 3
)

// callStackEntry is a bytecode call stack entry.
type callStackEntry struct {
	program   []byte
	pc        int
	loopCount int32
}

// hinter implements bytecode hinting. A hinter can be re-used to hint a series
// of glyphs from a Font.
type hinter struct {
	stack, store []int32

	// functions is a map from function number to bytecode.
	functions map[int32][]byte

	// font and scale are the font and scale last used for this hinter.
	// Changing the font will require running the new font's fpgm bytecode.
	// Changing either will require running the font's prep bytecode.
	font  *Font
	scale int32

	// gs and defaultGS are the current and default graphics state. The
	// default graphics state is the global default graphics state after
	// the font's fpgm and prep programs have been run.
	gs, defaultGS graphicsState

	// points and ends are the twilight zone's points, glyph's points
	// and glyph's contour boundaries.
	points [numZone][numPointType][]Point
	ends   []int

	// scaledCVT is the lazily initialized scaled Control Value Table.
	scaledCVTInitialized bool
	scaledCVT            []f26dot6
}

// graphicsState is described at https://developer.apple.com/fonts/TTRefMan/RM04/Chap4.html
type graphicsState struct {
	// Projection vector, freedom vector and dual projection vector.
	pv, fv, dv [2]f2dot14
	// Reference points and zone pointers.
	rp, zp [3]int32
	// Control Value / Single Width Cut-In.
	controlValueCutIn, singleWidthCutIn, singleWidth f26dot6
	// Delta base / shift.
	deltaBase, deltaShift int32
	// Minimum distance.
	minDist f26dot6
	// Loop count.
	loop int32
	// Rounding policy.
	roundPeriod, roundPhase, roundThreshold f26dot6
	roundSuper45                            bool
	// Auto-flip.
	autoFlip bool
}

var globalDefaultGS = graphicsState{
	pv:                [2]f2dot14{0x4000, 0}, // Unit vector along the X axis.
	fv:                [2]f2dot14{0x4000, 0},
	dv:                [2]f2dot14{0x4000, 0},
	zp:                [3]int32{1, 1, 1},
	controlValueCutIn: (17 << 6) / 16, // 17/16 as an f26dot6.
	deltaBase:         9,
	deltaShift:        3,
	minDist:           1 << 6, // 1 as an f26dot6.
	loop:              1,
	roundPeriod:       1 << 6, // 1 as an f26dot6.
	roundThreshold:    1 << 5, // 1/2 as an f26dot6.
	roundSuper45:      false,
	autoFlip:          true,
}

func resetTwilightPoints(f *Font, p []Point) []Point {
	if n := int(f.maxTwilightPoints) + 4; n <= cap(p) {
		p = p[:n]
		for i := range p {
			p[i] = Point{}
		}
	} else {
		p = make([]Point, n)
	}
	return p
}

func (h *hinter) init(f *Font, scale int32) error {
	h.points[twilightZone][0] = resetTwilightPoints(f, h.points[twilightZone][0])
	h.points[twilightZone][1] = resetTwilightPoints(f, h.points[twilightZone][1])
	h.points[twilightZone][2] = resetTwilightPoints(f, h.points[twilightZone][2])

	rescale := h.scale != scale
	if h.font != f {
		h.font, rescale = f, true
		if h.functions == nil {
			h.functions = make(map[int32][]byte)
		} else {
			for k := range h.functions {
				delete(h.functions, k)
			}
		}

		if x := int(f.maxStackElements); x > len(h.stack) {
			x += 255
			x &^= 255
			h.stack = make([]int32, x)
		}
		if x := int(f.maxStorage); x > len(h.store) {
			x += 15
			x &^= 15
			h.store = make([]int32, x)
		}
		if len(f.fpgm) != 0 {
			if err := h.run(f.fpgm, nil, nil, nil, nil); err != nil {
				return err
			}
		}
	}

	if rescale {
		h.scale = scale
		h.scaledCVTInitialized = false

		h.defaultGS = globalDefaultGS

		if len(f.prep) != 0 {
			if err := h.run(f.prep, nil, nil, nil, nil); err != nil {
				return err
			}
			h.defaultGS = h.gs
			// The MS rasterizer doesn't allow the following graphics state
			// variables to be modified by the CVT program.
			h.defaultGS.pv = globalDefaultGS.pv
			h.defaultGS.fv = globalDefaultGS.fv
			h.defaultGS.dv = globalDefaultGS.dv
			h.defaultGS.rp = globalDefaultGS.rp
			h.defaultGS.zp = globalDefaultGS.zp
			h.defaultGS.loop = globalDefaultGS.loop
		}
	}
	return nil
}

func (h *hinter) run(program []byte, pCurrent, pUnhinted, pInFontUnits []Point, ends []int) error {
	h.gs = h.defaultGS
	h.points[glyphZone][current] = pCurrent
	h.points[glyphZone][unhinted] = pUnhinted
	h.points[glyphZone][inFontUnits] = pInFontUnits
	h.ends = ends

	if len(program) > 50000 {
		return errors.New("truetype: hinting: too many instructions")
	}
	var (
		steps, pc, top int
		opcode         uint8

		callStack    [32]callStackEntry
		callStackTop int
	)

	for 0 <= pc && pc < len(program) {
		steps++
		if steps == 100000 {
			return errors.New("truetype: hinting: too many steps")
		}
		opcode = program[pc]
		if top < int(popCount[opcode]) {
			return errors.New("truetype: hinting: stack underflow")
		}
		switch opcode {

		case opSVTCA0:
			h.gs.pv = [2]f2dot14{0, 0x4000}
			h.gs.fv = [2]f2dot14{0, 0x4000}
			h.gs.dv = [2]f2dot14{0, 0x4000}

		case opSVTCA1:
			h.gs.pv = [2]f2dot14{0x4000, 0}
			h.gs.fv = [2]f2dot14{0x4000, 0}
			h.gs.dv = [2]f2dot14{0x4000, 0}

		case opSPVTCA0:
			h.gs.pv = [2]f2dot14{0, 0x4000}
			h.gs.dv = [2]f2dot14{0, 0x4000}

		case opSPVTCA1:
			h.gs.pv = [2]f2dot14{0x4000, 0}
			h.gs.dv = [2]f2dot14{0x4000, 0}

		case opSFVTCA0:
			h.gs.fv = [2]f2dot14{0, 0x4000}

		case opSFVTCA1:
			h.gs.fv = [2]f2dot14{0x4000, 0}

		case opSPVTL0, opSPVTL1, opSFVTL0, opSFVTL1:
			top -= 2
			p1 := h.point(0, current, h.stack[top+0])
			p2 := h.point(0, current, h.stack[top+1])
			if p1 == nil || p2 == nil {
				return errors.New("truetype: hinting: point out of range")
			}
			dx := f2dot14(p1.X - p2.X)
			dy := f2dot14(p1.Y - p2.Y)
			if dx == 0 && dy == 0 {
				dx = 0x4000
			} else if opcode&1 != 0 {
				// Counter-clockwise rotation.
				dx, dy = -dy, dx
			}
			v := normalize(dx, dy)
			if opcode < opSFVTL0 {
				h.gs.pv = v
				h.gs.dv = v
			} else {
				h.gs.fv = v
			}

		case opSPVFS:
			top -= 2
			h.gs.pv = normalize(f2dot14(h.stack[top]), f2dot14(h.stack[top+1]))
			h.gs.dv = h.gs.pv

		case opSFVFS:
			top -= 2
			h.gs.fv = normalize(f2dot14(h.stack[top]), f2dot14(h.stack[top+1]))

		case opGPV:
			if top+1 >= len(h.stack) {
				return errors.New("truetype: hinting: stack overflow")
			}
			h.stack[top+0] = int32(h.gs.pv[0])
			h.stack[top+1] = int32(h.gs.pv[1])
			top += 2

		case opGFV:
			if top+1 >= len(h.stack) {
				return errors.New("truetype: hinting: stack overflow")
			}
			h.stack[top+0] = int32(h.gs.fv[0])
			h.stack[top+1] = int32(h.gs.fv[1])
			top += 2

		case opSFVTPV:
			h.gs.fv = h.gs.pv

		case opISECT:
			top -= 5
			p := h.point(2, current, h.stack[top+0])
			a0 := h.point(1, current, h.stack[top+1])
			a1 := h.point(1, current, h.stack[top+2])
			b0 := h.point(0, current, h.stack[top+3])
			b1 := h.point(0, current, h.stack[top+4])
			if p == nil || a0 == nil || a1 == nil || b0 == nil || b1 == nil {
				return errors.New("truetype: hinting: point out of range")
			}

			dbx := b1.X - b0.X
			dby := b1.Y - b0.Y
			dax := a1.X - a0.X
			day := a1.Y - a0.Y
			dx := b0.X - a0.X
			dy := b0.Y - a0.Y
			discriminant := mulDiv(int64(dax), int64(-dby), 0x40) +
				mulDiv(int64(day), int64(dbx), 0x40)
			dotProduct := mulDiv(int64(dax), int64(dbx), 0x40) +
				mulDiv(int64(day), int64(dby), 0x40)
			// The discriminant above is actually a cross product of vectors
			// da and db. Together with the dot product, they can be used as
			// surrogates for sine and cosine of the angle between the vectors.
			// Indeed,
			//       dotproduct   = |da||db|cos(angle)
			//       discriminant = |da||db|sin(angle)
			// We use these equations to reject grazing intersections by
			// thresholding abs(tan(angle)) at 1/19, corresponding to 3 degrees.
			absDisc, absDotP := discriminant, dotProduct
			if absDisc < 0 {
				absDisc = -absDisc
			}
			if absDotP < 0 {
				absDotP = -absDotP
			}
			if 19*absDisc > absDotP {
				val := mulDiv(int64(dx), int64(-dby), 0x40) +
					mulDiv(int64(dy), int64(dbx), 0x40)
				rx := mulDiv(val, int64(dax), discriminant)
				ry := mulDiv(val, int64(day), discriminant)
				p.X = a0.X + int32(rx)
				p.Y = a0.Y + int32(ry)
			} else {
				p.X = (a0.X + a1.X + b0.X + b1.X) / 4
				p.Y = (a0.Y + a1.Y + b0.Y + b1.Y) / 4
			}
			p.Flags |= flagTouchedX | flagTouchedY

		case opSRP0, opSRP1, opSRP2:
			top--
			h.gs.rp[opcode-opSRP0] = h.stack[top]

		case opSZP0, opSZP1, opSZP2:
			top--
			h.gs.zp[opcode-opSZP0] = h.stack[top]

		case opSZPS:
			top--
			h.gs.zp[0] = h.stack[top]
			h.gs.zp[1] = h.stack[top]
			h.gs.zp[2] = h.stack[top]

		case opSLOOP:
			top--
			if h.stack[top] <= 0 {
				return errors.New("truetype: hinting: invalid data")
			}
			h.gs.loop = h.stack[top]

		case opRTG:
			h.gs.roundPeriod = 1 << 6
			h.gs.roundPhase = 0
			h.gs.roundThreshold = 1 << 5
			h.gs.roundSuper45 = false

		case opRTHG:
			h.gs.roundPeriod = 1 << 6
			h.gs.roundPhase = 1 << 5
			h.gs.roundThreshold = 1 << 5
			h.gs.roundSuper45 = false

		case opSMD:
			top--
			h.gs.minDist = f26dot6(h.stack[top])

		case opELSE:
			opcode = 1
			goto ifelse

		case opJMPR:
			top--
			pc += int(h.stack[top])
			continue

		case opSCVTCI:
			top--
			h.gs.controlValueCutIn = f26dot6(h.stack[top])

		case opSSWCI:
			top--
			h.gs.singleWidthCutIn = f26dot6(h.stack[top])

		case opSSW:
			top--
			h.gs.singleWidth = f26dot6(h.font.scale(h.scale * h.stack[top]))

		case opDUP:
			if top >= len(h.stack) {
				return errors.New("truetype: hinting: stack overflow")
			}
			h.stack[top] = h.stack[top-1]
			top++

		case opPOP:
			top--

		case opCLEAR:
			top = 0

		case opSWAP:
			h.stack[top-1], h.stack[top-2] = h.stack[top-2], h.stack[top-1]

		case opDEPTH:
			if top >= len(h.stack) {
				return errors.New("truetype: hinting: stack overflow")
			}
			h.stack[top] = int32(top)
			top++

		case opCINDEX, opMINDEX:
			x := int(h.stack[top-1])
			if x <= 0 || x >= top {
				return errors.New("truetype: hinting: invalid data")
			}
			h.stack[top-1] = h.stack[top-1-x]
			if opcode == opMINDEX {
				copy(h.stack[top-1-x:top-1], h.stack[top-x:top])
				top--
			}

		case opALIGNPTS:
			top -= 2
			p := h.point(1, current, h.stack[top])
			q := h.point(0, current, h.stack[top+1])
			if p == nil || q == nil {
				return errors.New("truetype: hinting: point out of range")
			}
			d := dotProduct(f26dot6(q.X-p.X), f26dot6(q.Y-p.Y), h.gs.pv) / 2
			h.move(p, +d, true)
			h.move(q, -d, true)

		case opUTP:
			top--
			p := h.point(0, current, h.stack[top])
			if p == nil {
				return errors.New("truetype: hinting: point out of range")
			}
			p.Flags &^= flagTouchedX | flagTouchedY

		case opLOOPCALL, opCALL:
			if callStackTop >= len(callStack) {
				return errors.New("truetype: hinting: call stack overflow")
			}
			top--
			f, ok := h.functions[h.stack[top]]
			if !ok {
				return errors.New("truetype: hinting: undefined function")
			}
			callStack[callStackTop] = callStackEntry{program, pc, 1}
			if opcode == opLOOPCALL {
				top--
				if h.stack[top] == 0 {
					break
				}
				callStack[callStackTop].loopCount = h.stack[top]
			}
			callStackTop++
			program, pc = f, 0
			continue

		case opFDEF:
			// Save all bytecode up until the next ENDF.
			startPC := pc + 1
		fdefloop:
			for {
				pc++
				if pc >= len(program) {
					return errors.New("truetype: hinting: unbalanced FDEF")
				}
				switch program[pc] {
				case opFDEF:
					return errors.New("truetype: hinting: nested FDEF")
				case opENDF:
					top--
					h.functions[h.stack[top]] = program[startPC : pc+1]
					break fdefloop
				default:
					var ok bool
					pc, ok = skipInstructionPayload(program, pc)
					if !ok {
						return errors.New("truetype: hinting: unbalanced FDEF")
					}
				}
			}

		case opENDF:
			if callStackTop == 0 {
				return errors.New("truetype: hinting: call stack underflow")
			}
			callStackTop--
			callStack[callStackTop].loopCount--
			if callStack[callStackTop].loopCount != 0 {
				callStackTop++
				pc = 0
				continue
			}
			program, pc = callStack[callStackTop].program, callStack[callStackTop].pc

		case opMDAP0, opMDAP1:
			top--
			i := h.stack[top]
			p := h.point(0, current, i)
			if p == nil {
				return errors.New("truetype: hinting: point out of range")
			}
			distance := f26dot6(0)
			if opcode == opMDAP1 {
				distance = dotProduct(f26dot6(p.X), f26dot6(p.Y), h.gs.pv)
				// TODO: metrics compensation.
				distance = h.round(distance) - distance
			}
			h.move(p, distance, true)
			h.gs.rp[0] = i
			h.gs.rp[1] = i

		case opIUP0, opIUP1:
			iupY, mask := opcode == opIUP0, uint32(flagTouchedX)
			if iupY {
				mask = flagTouchedY
			}
			prevEnd := 0
			for _, end := range h.ends {
				for i := prevEnd; i < end; i++ {
					for i < end && h.points[glyphZone][current][i].Flags&mask == 0 {
						i++
					}
					if i == end {
						break
					}
					firstTouched, curTouched := i, i
					i++
					for ; i < end; i++ {
						if h.points[glyphZone][current][i].Flags&mask != 0 {
							h.iupInterp(iupY, curTouched+1, i-1, curTouched, i)
							curTouched = i
						}
					}
					if curTouched == firstTouched {
						h.iupShift(iupY, prevEnd, end, curTouched)
					} else {
						h.iupInterp(iupY, curTouched+1, end-1, curTouched, firstTouched)
						if firstTouched > 0 {
							h.iupInterp(iupY, prevEnd, firstTouched-1, curTouched, firstTouched)
						}
					}
				}
				prevEnd = end
			}

		case opSHP0, opSHP1:
			if top < int(h.gs.loop) {
				return errors.New("truetype: hinting: stack underflow")
			}
			_, _, d, ok := h.displacement(opcode&1 == 0)
			if !ok {
				return errors.New("truetype: hinting: point out of range")
			}
			for ; h.gs.loop != 0; h.gs.loop-- {
				top--
				p := h.point(2, current, h.stack[top])
				if p == nil {
					return errors.New("truetype: hinting: point out of range")
				}
				h.move(p, d, true)
			}
			h.gs.loop = 1

		case opSHC0, opSHC1:
			top--
			zonePointer, i, d, ok := h.displacement(opcode&1 == 0)
			if !ok {
				return errors.New("truetype: hinting: point out of range")
			}
			if h.gs.zp[2] == 0 {
				// TODO: implement this when we have a glyph that does this.
				return errors.New("hinting: unimplemented SHC instruction")
			}
			contour := h.stack[top]
			if contour < 0 || len(ends) <= int(contour) {
				return errors.New("truetype: hinting: contour out of range")
			}
			j0, j1 := int32(0), int32(h.ends[contour])
			if contour > 0 {
				j0 = int32(h.ends[contour-1])
			}
			move := h.gs.zp[zonePointer] != h.gs.zp[2]
			for j := j0; j < j1; j++ {
				if move || j != i {
					h.move(h.point(2, current, j), d, true)
				}
			}

		case opSHZ0, opSHZ1:
			top--
			zonePointer, i, d, ok := h.displacement(opcode&1 == 0)
			if !ok {
				return errors.New("truetype: hinting: point out of range")
			}

			// As per C Freetype, SHZ doesn't move the phantom points, or mark
			// the points as touched.
			limit := int32(len(h.points[h.gs.zp[2]][current]))
			if h.gs.zp[2] == glyphZone {
				limit -= 4
			}
			for j := int32(0); j < limit; j++ {
				if i != j || h.gs.zp[zonePointer] != h.gs.zp[2] {
					h.move(h.point(2, current, j), d, false)
				}
			}

		case opSHPIX:
			top--
			d := f26dot6(h.stack[top])
			if top < int(h.gs.loop) {
				return errors.New("truetype: hinting: stack underflow")
			}
			for ; h.gs.loop != 0; h.gs.loop-- {
				top--
				p := h.point(2, current, h.stack[top])
				if p == nil {
					return errors.New("truetype: hinting: point out of range")
				}
				h.move(p, d, true)
			}
			h.gs.loop = 1

		case opIP:
			if top < int(h.gs.loop) {
				return errors.New("truetype: hinting: stack underflow")
			}
			pointType := inFontUnits
			twilight := h.gs.zp[0] == 0 || h.gs.zp[1] == 0 || h.gs.zp[2] == 0
			if twilight {
				pointType = unhinted
			}
			p := h.point(1, pointType, h.gs.rp[2])
			oldP := h.point(0, pointType, h.gs.rp[1])
			oldRange := dotProduct(f26dot6(p.X-oldP.X), f26dot6(p.Y-oldP.Y), h.gs.dv)

			p = h.point(1, current, h.gs.rp[2])
			curP := h.point(0, current, h.gs.rp[1])
			curRange := dotProduct(f26dot6(p.X-curP.X), f26dot6(p.Y-curP.Y), h.gs.pv)
			for ; h.gs.loop != 0; h.gs.loop-- {
				top--
				i := h.stack[top]
				p = h.point(2, pointType, i)
				oldDist := dotProduct(f26dot6(p.X-oldP.X), f26dot6(p.Y-oldP.Y), h.gs.dv)
				p = h.point(2, current, i)
				curDist := dotProduct(f26dot6(p.X-curP.X), f26dot6(p.Y-curP.Y), h.gs.pv)
				newDist := f26dot6(0)
				if oldDist != 0 {
					if oldRange != 0 {
						newDist = f26dot6(mulDiv(int64(oldDist), int64(curRange), int64(oldRange)))
					} else {
						newDist = -oldDist
					}
				}
				h.move(p, newDist-curDist, true)
			}
			h.gs.loop = 1

		case opMSIRP0, opMSIRP1:
			top -= 2
			i := h.stack[top]
			distance := f26dot6(h.stack[top+1])

			// TODO: special case h.gs.zp[1] == 0 in C Freetype.
			ref := h.point(0, current, h.gs.rp[0])
			p := h.point(1, current, i)
			if ref == nil || p == nil {
				return errors.New("truetype: hinting: point out of range")
			}
			curDist := dotProduct(f26dot6(p.X-ref.X), f26dot6(p.Y-ref.Y), h.gs.pv)

			// Set-RP0 bit.
			if opcode == opMSIRP1 {
				h.gs.rp[0] = i
			}
			h.gs.rp[1] = h.gs.rp[0]
			h.gs.rp[2] = i

			// Move the point.
			h.move(p, distance-curDist, true)

		case opALIGNRP:
			if top < int(h.gs.loop) {
				return errors.New("truetype: hinting: stack underflow")
			}
			ref := h.point(0, current, h.gs.rp[0])
			if ref == nil {
				return errors.New("truetype: hinting: point out of range")
			}
			for ; h.gs.loop != 0; h.gs.loop-- {
				top--
				p := h.point(1, current, h.stack[top])
				if p == nil {
					return errors.New("truetype: hinting: point out of range")
				}
				h.move(p, -dotProduct(f26dot6(p.X-ref.X), f26dot6(p.Y-ref.Y), h.gs.pv), true)
			}
			h.gs.loop = 1

		case opRTDG:
			h.gs.roundPeriod = 1 << 5
			h.gs.roundPhase = 0
			h.gs.roundThreshold = 1 << 4
			h.gs.roundSuper45 = false

		case opMIAP0, opMIAP1:
			top -= 2
			i := h.stack[top]
			distance := h.getScaledCVT(h.stack[top+1])
			if h.gs.zp[0] == 0 {
				p := h.point(0, unhinted, i)
				q := h.point(0, current, i)
				p.X = int32((int64(distance) * int64(h.gs.fv[0])) >> 14)
				p.Y = int32((int64(distance) * int64(h.gs.fv[1])) >> 14)
				*q = *p
			}
			p := h.point(0, current, i)
			oldDist := dotProduct(f26dot6(p.X), f26dot6(p.Y), h.gs.pv)
			if opcode == opMIAP1 {
				if (distance - oldDist).abs() > h.gs.controlValueCutIn {
					distance = oldDist
				}
				// TODO: metrics compensation.
				distance = h.round(distance)
			}
			h.move(p, distance-oldDist, true)
			h.gs.rp[0] = i
			h.gs.rp[1] = i

		case opNPUSHB:
			opcode = 0
			goto push

		case opNPUSHW:
			opcode = 0x80
			goto push

		case opWS:
			top -= 2
			i := int(h.stack[top])
			if i < 0 || len(h.store) <= i {
				return errors.New("truetype: hinting: invalid data")
			}
			h.store[i] = h.stack[top+1]

		case opRS:
			i := int(h.stack[top-1])
			if i < 0 || len(h.store) <= i {
				return errors.New("truetype: hinting: invalid data")
			}
			h.stack[top-1] = h.store[i]

		case opWCVTP:
			top -= 2
			h.setScaledCVT(h.stack[top], f26dot6(h.stack[top+1]))

		case opRCVT:
			h.stack[top-1] = int32(h.getScaledCVT(h.stack[top-1]))

		case opGC0, opGC1:
			i := h.stack[top-1]
			if opcode == opGC0 {
				p := h.point(2, current, i)
				h.stack[top-1] = int32(dotProduct(f26dot6(p.X), f26dot6(p.Y), h.gs.pv))
			} else {
				p := h.point(2, unhinted, i)
				// Using dv as per C Freetype.
				h.stack[top-1] = int32(dotProduct(f26dot6(p.X), f26dot6(p.Y), h.gs.dv))
			}

		case opSCFS:
			top -= 2
			i := h.stack[top]
			p := h.point(2, current, i)
			if p == nil {
				return errors.New("truetype: hinting: point out of range")
			}
			c := dotProduct(f26dot6(p.X), f26dot6(p.Y), h.gs.pv)
			h.move(p, f26dot6(h.stack[top+1])-c, true)
			if h.gs.zp[2] != 0 {
				break
			}
			q := h.point(2, unhinted, i)
			if q == nil {
				return errors.New("truetype: hinting: point out of range")
			}
			q.X = p.X
			q.Y = p.Y

		case opMD0, opMD1:
			top--
			pt, v, scale := pointType(0), [2]f2dot14{}, false
			if opcode == opMD0 {
				pt = current
				v = h.gs.pv
			} else if h.gs.zp[0] == 0 || h.gs.zp[1] == 0 {
				pt = unhinted
				v = h.gs.dv
			} else {
				pt = inFontUnits
				v = h.gs.dv
				scale = true
			}
			p := h.point(0, pt, h.stack[top-1])
			q := h.point(1, pt, h.stack[top])
			if p == nil || q == nil {
				return errors.New("truetype: hinting: point out of range")
			}
			d := int32(dotProduct(f26dot6(p.X-q.X), f26dot6(p.Y-q.Y), v))
			if scale {
				d = int32(int64(d*h.scale) / int64(h.font.fUnitsPerEm))
			}
			h.stack[top-1] = d

		case opMPPEM, opMPS:
			if top >= len(h.stack) {
				return errors.New("truetype: hinting: stack overflow")
			}
			// For MPS, point size should be irrelevant; we return the PPEM.
			h.stack[top] = h.scale >> 6
			top++

		case opFLIPON, opFLIPOFF:
			h.gs.autoFlip = opcode == opFLIPON

		case opDEBUG:
			// No-op.

		case opLT:
			top--
			h.stack[top-1] = bool2int32(h.stack[top-1] < h.stack[top])

		case opLTEQ:
			top--
			h.stack[top-1] = bool2int32(h.stack[top-1] <= h.stack[top])

		case opGT:
			top--
			h.stack[top-1] = bool2int32(h.stack[top-1] > h.stack[top])

		case opGTEQ:
			top--
			h.stack[top-1] = bool2int32(h.stack[top-1] >= h.stack[top])

		case opEQ:
			top--
			h.stack[top-1] = bool2int32(h.stack[top-1] == h.stack[top])

		case opNEQ:
			top--
			h.stack[top-1] = bool2int32(h.stack[top-1] != h.stack[top])

		case opODD, opEVEN:
			i := h.round(f26dot6(h.stack[top-1])) >> 6
			h.stack[top-1] = int32(i&1) ^ int32(opcode-opODD)

		case opIF:
			top--
			if h.stack[top] == 0 {
				opcode = 0
				goto ifelse
			}

		case opEIF:
			// No-op.

		case opAND:
			top--
			h.stack[top-1] = bool2int32(h.stack[top-1] != 0 && h.stack[top] != 0)

		case opOR:
			top--
			h.stack[top-1] = bool2int32(h.stack[top-1]|h.stack[top] != 0)

		case opNOT:
			h.stack[top-1] = bool2int32(h.stack[top-1] == 0)

		case opDELTAP1:
			goto delta

		case opSDB:
			top--
			h.gs.deltaBase = h.stack[top]

		case opSDS:
			top--
			h.gs.deltaShift = h.stack[top]

		case opADD:
			top--
			h.stack[top-1] += h.stack[top]

		case opSUB:
			top--
			h.stack[top-1] -= h.stack[top]

		case opDIV:
			top--
			if h.stack[top] == 0 {
				return errors.New("truetype: hinting: division by zero")
			}
			h.stack[top-1] = int32(f26dot6(h.stack[top-1]).div(f26dot6(h.stack[top])))

		case opMUL:
			top--
			h.stack[top-1] = int32(f26dot6(h.stack[top-1]).mul(f26dot6(h.stack[top])))

		case opABS:
			if h.stack[top-1] < 0 {
				h.stack[top-1] = -h.stack[top-1]
			}

		case opNEG:
			h.stack[top-1] = -h.stack[top-1]

		case opFLOOR:
			h.stack[top-1] &^= 63

		case opCEILING:
			h.stack[top-1] += 63
			h.stack[top-1] &^= 63

		case opROUND00, opROUND01, opROUND10, opROUND11:
			// The four flavors of opROUND are equivalent. See the comment below on
			// opNROUND for the rationale.
			h.stack[top-1] = int32(h.round(f26dot6(h.stack[top-1])))

		case opNROUND00, opNROUND01, opNROUND10, opNROUND11:
			// No-op. The spec says to add one of four "compensations for the engine
			// characteristics", to cater for things like "different dot-size printers".
			// https://developer.apple.com/fonts/TTRefMan/RM02/Chap2.html#engine_compensation
			// This code does not implement engine compensation, as we don't expect to
			// be used to output on dot-matrix printers.

		case opWCVTF:
			top -= 2
			h.setScaledCVT(h.stack[top], f26dot6(h.font.scale(h.scale*h.stack[top+1])))

		case opDELTAP2, opDELTAP3, opDELTAC1, opDELTAC2, opDELTAC3:
			goto delta

		case opSROUND, opS45ROUND:
			top--
			switch (h.stack[top] >> 6) & 0x03 {
			case 0:
				h.gs.roundPeriod = 1 << 5
			case 1, 3:
				h.gs.roundPeriod = 1 << 6
			case 2:
				h.gs.roundPeriod = 1 << 7
			}
			h.gs.roundSuper45 = opcode == opS45ROUND
			if h.gs.roundSuper45 {
				// The spec says to multiply by √2, but the C Freetype code says 1/√2.
				// We go with 1/√2.
				h.gs.roundPeriod *= 46341
				h.gs.roundPeriod /= 65536
			}
			h.gs.roundPhase = h.gs.roundPeriod * f26dot6((h.stack[top]>>4)&0x03) / 4
			if x := h.stack[top] & 0x0f; x != 0 {
				h.gs.roundThreshold = h.gs.roundPeriod * f26dot6(x-4) / 8
			} else {
				h.gs.roundThreshold = h.gs.roundPeriod - 1
			}

		case opJROT:
			top -= 2
			if h.stack[top+1] != 0 {
				pc += int(h.stack[top])
				continue
			}

		case opJROF:
			top -= 2
			if h.stack[top+1] == 0 {
				pc += int(h.stack[top])
				continue
			}

		case opROFF:
			h.gs.roundPeriod = 0
			h.gs.roundPhase = 0
			h.gs.roundThreshold = 0
			h.gs.roundSuper45 = false

		case opRUTG:
			h.gs.roundPeriod = 1 << 6
			h.gs.roundPhase = 0
			h.gs.roundThreshold = 1<<6 - 1
			h.gs.roundSuper45 = false

		case opRDTG:
			h.gs.roundPeriod = 1 << 6
			h.gs.roundPhase = 0
			h.gs.roundThreshold = 0
			h.gs.roundSuper45 = false

		case opSANGW, opAA:
			// These ops are "anachronistic" and no longer used.
			top--

		case opFLIPPT:
			if top < int(h.gs.loop) {
				return errors.New("truetype: hinting: stack underflow")
			}
			points := h.points[glyphZone][current]
			for ; h.gs.loop != 0; h.gs.loop-- {
				top--
				i := h.stack[top]
				if i < 0 || len(points) <= int(i) {
					return errors.New("truetype: hinting: point out of range")
				}
				points[i].Flags ^= flagOnCurve
			}
			h.gs.loop = 1

		case opFLIPRGON, opFLIPRGOFF:
			top -= 2
			i, j, points := h.stack[top], h.stack[top+1], h.points[glyphZone][current]
			if i < 0 || len(points) <= int(i) || j < 0 || len(points) <= int(j) {
				return errors.New("truetype: hinting: point out of range")
			}
			for ; i <= j; i++ {
				if opcode == opFLIPRGON {
					points[i].Flags |= flagOnCurve
				} else {
					points[i].Flags &^= flagOnCurve
				}
			}

		case opSCANCTRL:
			// We do not support dropout control, as we always rasterize grayscale glyphs.
			top--

		case opSDPVTL0, opSDPVTL1:
			top -= 2
			for i := 0; i < 2; i++ {
				pt := unhinted
				if i != 0 {
					pt = current
				}
				p := h.point(1, pt, h.stack[top])
				q := h.point(2, pt, h.stack[top+1])
				if p == nil || q == nil {
					return errors.New("truetype: hinting: point out of range")
				}
				dx := f2dot14(p.X - q.X)
				dy := f2dot14(p.Y - q.Y)
				if dx == 0 && dy == 0 {
					dx = 0x4000
				} else if opcode&1 != 0 {
					// Counter-clockwise rotation.
					dx, dy = -dy, dx
				}
				if i == 0 {
					h.gs.dv = normalize(dx, dy)
				} else {
					h.gs.pv = normalize(dx, dy)
				}
			}

		case opGETINFO:
			res := int32(0)
			if h.stack[top-1]&(1<<0) != 0 {
				// Set the engine version. We hard-code this to 35, the same as
				// the C freetype code, which says that "Version~35 corresponds
				// to MS rasterizer v.1.7 as used e.g. in Windows~98".
				res |= 35
			}
			if h.stack[top-1]&(1<<5) != 0 {
				// Set that we support grayscale.
				res |= 1 << 12
			}
			// We set no other bits, as we do not support rotated or stretched glyphs.
			h.stack[top-1] = res

		case opIDEF:
			// IDEF is for ancient versions of the bytecode interpreter, and is no longer used.
			return errors.New("truetype: hinting: unsupported IDEF instruction")

		case opROLL:
			h.stack[top-1], h.stack[top-3], h.stack[top-2] =
				h.stack[top-3], h.stack[top-2], h.stack[top-1]

		case opMAX:
			top--
			if h.stack[top-1] < h.stack[top] {
				h.stack[top-1] = h.stack[top]
			}

		case opMIN:
			top--
			if h.stack[top-1] > h.stack[top] {
				h.stack[top-1] = h.stack[top]
			}

		case opSCANTYPE:
			// We do not support dropout control, as we always rasterize grayscale glyphs.
			top--

		case opINSTCTRL:
			// TODO: support instruction execution control? It seems rare, and even when
			// nominally used (e.g. Source Sans Pro), it seems conditional on extreme or
			// unusual rasterization conditions. For example, the code snippet at
			// https://developer.apple.com/fonts/TTRefMan/RM05/Chap5.html#INSTCTRL
			// uses INSTCTRL when grid-fitting a rotated or stretched glyph, but
			// freetype-go does not support rotated or stretched glyphs.
			top -= 2

		default:
			if opcode < opPUSHB000 {
				return errors.New("truetype: hinting: unrecognized instruction")
			}

			if opcode < opMDRP00000 {
				// PUSHxxxx opcode.

				if opcode < opPUSHW000 {
					opcode -= opPUSHB000 - 1
				} else {
					opcode -= opPUSHW000 - 1 - 0x80
				}
				goto push
			}

			if opcode < opMIRP00000 {
				// MDRPxxxxx opcode.

				top--
				i := h.stack[top]
				ref := h.point(0, current, h.gs.rp[0])
				p := h.point(1, current, i)
				if ref == nil || p == nil {
					return errors.New("truetype: hinting: point out of range")
				}

				oldDist := f26dot6(0)
				if h.gs.zp[0] == 0 || h.gs.zp[1] == 0 {
					p0 := h.point(1, unhinted, i)
					p1 := h.point(0, unhinted, h.gs.rp[0])
					oldDist = dotProduct(f26dot6(p0.X-p1.X), f26dot6(p0.Y-p1.Y), h.gs.dv)
				} else {
					p0 := h.point(1, inFontUnits, i)
					p1 := h.point(0, inFontUnits, h.gs.rp[0])
					oldDist = dotProduct(f26dot6(p0.X-p1.X), f26dot6(p0.Y-p1.Y), h.gs.dv)
					oldDist = f26dot6(h.font.scale(h.scale * int32(oldDist)))
				}

				// Single-width cut-in test.
				if x := (oldDist - h.gs.singleWidth).abs(); x < h.gs.singleWidthCutIn {
					if oldDist >= 0 {
						oldDist = +h.gs.singleWidth
					} else {
						oldDist = -h.gs.singleWidth
					}
				}

				// Rounding bit.
				// TODO: metrics compensation.
				distance := oldDist
				if opcode&0x04 != 0 {
					distance = h.round(oldDist)
				}

				// Minimum distance bit.
				if opcode&0x08 != 0 {
					if oldDist >= 0 {
						if distance < h.gs.minDist {
							distance = h.gs.minDist
						}
					} else {
						if distance > -h.gs.minDist {
							distance = -h.gs.minDist
						}
					}
				}

				// Set-RP0 bit.
				h.gs.rp[1] = h.gs.rp[0]
				h.gs.rp[2] = i
				if opcode&0x10 != 0 {
					h.gs.rp[0] = i
				}

				// Move the point.
				oldDist = dotProduct(f26dot6(p.X-ref.X), f26dot6(p.Y-ref.Y), h.gs.pv)
				h.move(p, distance-oldDist, true)

			} else {
				// MIRPxxxxx opcode.

				top -= 2
				i := h.stack[top]
				cvtDist := h.getScaledCVT(h.stack[top+1])
				if (cvtDist - h.gs.singleWidth).abs() < h.gs.singleWidthCutIn {
					if cvtDist >= 0 {
						cvtDist = +h.gs.singleWidth
					} else {
						cvtDist = -h.gs.singleWidth
					}
				}

				if h.gs.zp[1] == 0 {
					// TODO: implement once we have a .ttf file that triggers
					// this, so that we can step through C's freetype.
					return errors.New("truetype: hinting: unimplemented twilight point adjustment")
				}

				ref := h.point(0, unhinted, h.gs.rp[0])
				p := h.point(1, unhinted, i)
				if ref == nil || p == nil {
					return errors.New("truetype: hinting: point out of range")
				}
				oldDist := dotProduct(f26dot6(p.X-ref.X), f26dot6(p.Y-ref.Y), h.gs.dv)

				ref = h.point(0, current, h.gs.rp[0])
				p = h.point(1, current, i)
				if ref == nil || p == nil {
					return errors.New("truetype: hinting: point out of range")
				}
				curDist := dotProduct(f26dot6(p.X-ref.X), f26dot6(p.Y-ref.Y), h.gs.pv)

				if h.gs.autoFlip && oldDist^cvtDist < 0 {
					cvtDist = -cvtDist
				}

				// Rounding bit.
				// TODO: metrics compensation.
				distance := cvtDist
				if opcode&0x04 != 0 {
					// The CVT value is only used if close enough to oldDist.
					if (h.gs.zp[0] == h.gs.zp[1]) &&
						((cvtDist - oldDist).abs() > h.gs.controlValueCutIn) {

						distance = oldDist
					}
					distance = h.round(distance)
				}

				// Minimum distance bit.
				if opcode&0x08 != 0 {
					if oldDist >= 0 {
						if distance < h.gs.minDist {
							distance = h.gs.minDist
						}
					} else {
						if distance > -h.gs.minDist {
							distance = -h.gs.minDist
						}
					}
				}

				// Set-RP0 bit.
				h.gs.rp[1] = h.gs.rp[0]
				h.gs.rp[2] = i
				if opcode&0x10 != 0 {
					h.gs.rp[0] = i
				}

				// Move the point.
				h.move(p, distance-curDist, true)
			}
		}
		pc++
		continue

	ifelse:
		// Skip past bytecode until the next ELSE (if opcode == 0) or the
		// next EIF (for all opcodes). Opcode == 0 means that we have come
		// from an IF. Opcode == 1 means that we have come from an ELSE.
		{
		ifelseloop:
			for depth := 0; ; {
				pc++
				if pc >= len(program) {
					return errors.New("truetype: hinting: unbalanced IF or ELSE")
				}
				switch program[pc] {
				case opIF:
					depth++
				case opELSE:
					if depth == 0 && opcode == 0 {
						break ifelseloop
					}
				case opEIF:
					depth--
					if depth < 0 {
						break ifelseloop
					}
				default:
					var ok bool
					pc, ok = skipInstructionPayload(program, pc)
					if !ok {
						return errors.New("truetype: hinting: unbalanced IF or ELSE")
					}
				}
			}
			pc++
			continue
		}

	push:
		// Push n elements from the program to the stack, where n is the low 7 bits of
		// opcode. If the low 7 bits are zero, then n is the next byte from the program.
		// The high bit being 0 means that the elements are zero-extended bytes.
		// The high bit being 1 means that the elements are sign-extended words.
		{
			width := 1
			if opcode&0x80 != 0 {
				opcode &^= 0x80
				width = 2
			}
			if opcode == 0 {
				pc++
				if pc >= len(program) {
					return errors.New("truetype: hinting: insufficient data")
				}
				opcode = program[pc]
			}
			pc++
			if top+int(opcode) > len(h.stack) {
				return errors.New("truetype: hinting: stack overflow")
			}
			if pc+width*int(opcode) > len(program) {
				return errors.New("truetype: hinting: insufficient data")
			}
			for ; opcode > 0; opcode-- {
				if width == 1 {
					h.stack[top] = int32(program[pc])
				} else {
					h.stack[top] = int32(int8(program[pc]))<<8 | int32(program[pc+1])
				}
				top++
				pc += width
			}
			continue
		}

	delta:
		{
			if opcode >= opDELTAC1 && !h.scaledCVTInitialized {
				h.initializeScaledCVT()
			}
			top--
			n := h.stack[top]
			if int32(top) < 2*n {
				return errors.New("truetype: hinting: stack underflow")
			}
			for ; n > 0; n-- {
				top -= 2
				b := h.stack[top]
				c := (b & 0xf0) >> 4
				switch opcode {
				case opDELTAP2, opDELTAC2:
					c += 16
				case opDELTAP3, opDELTAC3:
					c += 32
				}
				c += h.gs.deltaBase
				if ppem := (h.scale + 1<<5) >> 6; ppem != c {
					continue
				}
				b = (b & 0x0f) - 8
				if b >= 0 {
					b++
				}
				b = b * 64 / (1 << uint32(h.gs.deltaShift))
				if opcode >= opDELTAC1 {
					a := h.stack[top+1]
					if a < 0 || len(h.scaledCVT) <= int(a) {
						return errors.New("truetype: hinting: index out of range")
					}
					h.scaledCVT[a] += f26dot6(b)
				} else {
					p := h.point(0, current, h.stack[top+1])
					if p == nil {
						return errors.New("truetype: hinting: point out of range")
					}
					h.move(p, f26dot6(b), true)
				}
			}
			pc++
			continue
		}
	}
	return nil
}

func (h *hinter) initializeScaledCVT() {
	h.scaledCVTInitialized = true
	if n := len(h.font.cvt) / 2; n <= cap(h.scaledCVT) {
		h.scaledCVT = h.scaledCVT[:n]
	} else {
		if n < 32 {
			n = 32
		}
		h.scaledCVT = make([]f26dot6, len(h.font.cvt)/2, n)
	}
	for i := range h.scaledCVT {
		unscaled := uint16(h.font.cvt[2*i])<<8 | uint16(h.font.cvt[2*i+1])
		h.scaledCVT[i] = f26dot6(h.font.scale(h.scale * int32(int16(unscaled))))
	}
}

// getScaledCVT returns the scaled value from the font's Control Value Table.
func (h *hinter) getScaledCVT(i int32) f26dot6 {
	if !h.scaledCVTInitialized {
		h.initializeScaledCVT()
	}
	if i < 0 || len(h.scaledCVT) <= int(i) {
		return 0
	}
	return h.scaledCVT[i]
}

// setScaledCVT overrides the scaled value from the font's Control Value Table.
func (h *hinter) setScaledCVT(i int32, v f26dot6) {
	if !h.scaledCVTInitialized {
		h.initializeScaledCVT()
	}
	if i < 0 || len(h.scaledCVT) <= int(i) {
		return
	}
	h.scaledCVT[i] = v
}

func (h *hinter) point(zonePointer uint32, pt pointType, i int32) *Point {
	points := h.points[h.gs.zp[zonePointer]][pt]
	if i < 0 || len(points) <= int(i) {
		return nil
	}
	return &points[i]
}

func (h *hinter) move(p *Point, distance f26dot6, touch bool) {
	fvx := int64(h.gs.fv[0])
	pvx := int64(h.gs.pv[0])
	if fvx == 0x4000 && pvx == 0x4000 {
		p.X += int32(distance)
		if touch {
			p.Flags |= flagTouchedX
		}
		return
	}

	fvy := int64(h.gs.fv[1])
	pvy := int64(h.gs.pv[1])
	if fvy == 0x4000 && pvy == 0x4000 {
		p.Y += int32(distance)
		if touch {
			p.Flags |= flagTouchedY
		}
		return
	}

	fvDotPv := (fvx*pvx + fvy*pvy) >> 14

	if fvx != 0 {
		p.X += int32(mulDiv(fvx, int64(distance), fvDotPv))
		if touch {
			p.Flags |= flagTouchedX
		}
	}

	if fvy != 0 {
		p.Y += int32(mulDiv(fvy, int64(distance), fvDotPv))
		if touch {
			p.Flags |= flagTouchedY
		}
	}
}

func (h *hinter) iupInterp(interpY bool, p1, p2, ref1, ref2 int) {
	if p1 > p2 {
		return
	}
	if ref1 >= len(h.points[glyphZone][current]) ||
		ref2 >= len(h.points[glyphZone][current]) {
		return
	}

	var ifu1, ifu2 int32
	if interpY {
		ifu1 = h.points[glyphZone][inFontUnits][ref1].Y
		ifu2 = h.points[glyphZone][inFontUnits][ref2].Y
	} else {
		ifu1 = h.points[glyphZone][inFontUnits][ref1].X
		ifu2 = h.points[glyphZone][inFontUnits][ref2].X
	}
	if ifu1 > ifu2 {
		ifu1, ifu2 = ifu2, ifu1
		ref1, ref2 = ref2, ref1
	}

	var unh1, unh2, delta1, delta2 int32
	if interpY {
		unh1 = h.points[glyphZone][unhinted][ref1].Y
		unh2 = h.points[glyphZone][unhinted][ref2].Y
		delta1 = h.points[glyphZone][current][ref1].Y - unh1
		delta2 = h.points[glyphZone][current][ref2].Y - unh2
	} else {
		unh1 = h.points[glyphZone][unhinted][ref1].X
		unh2 = h.points[glyphZone][unhinted][ref2].X
		delta1 = h.points[glyphZone][current][ref1].X - unh1
		delta2 = h.points[glyphZone][current][ref2].X - unh2
	}

	var xy, ifuXY int32
	if ifu1 == ifu2 {
		for i := p1; i <= p2; i++ {
			if interpY {
				xy = h.points[glyphZone][unhinted][i].Y
			} else {
				xy = h.points[glyphZone][unhinted][i].X
			}

			if xy <= unh1 {
				xy += delta1
			} else {
				xy += delta2
			}

			if interpY {
				h.points[glyphZone][current][i].Y = xy
			} else {
				h.points[glyphZone][current][i].X = xy
			}
		}
		return
	}

	scale, scaleOK := int64(0), false
	for i := p1; i <= p2; i++ {
		if interpY {
			xy = h.points[glyphZone][unhinted][i].Y
			ifuXY = h.points[glyphZone][inFontUnits][i].Y
		} else {
			xy = h.points[glyphZone][unhinted][i].X
			ifuXY = h.points[glyphZone][inFontUnits][i].X
		}

		if xy <= unh1 {
			xy += delta1
		} else if xy >= unh2 {
			xy += delta2
		} else {
			if !scaleOK {
				scaleOK = true
				scale = mulDiv(int64(unh2+delta2-unh1-delta1), 0x10000, int64(ifu2-ifu1))
			}
			numer := int64(ifuXY-ifu1) * scale
			if numer >= 0 {
				numer += 0x8000
			} else {
				numer -= 0x8000
			}
			xy = unh1 + delta1 + int32(numer/0x10000)
		}

		if interpY {
			h.points[glyphZone][current][i].Y = xy
		} else {
			h.points[glyphZone][current][i].X = xy
		}
	}
}

func (h *hinter) iupShift(interpY bool, p1, p2, p int) {
	var delta int32
	if interpY {
		delta = h.points[glyphZone][current][p].Y - h.points[glyphZone][unhinted][p].Y
	} else {
		delta = h.points[glyphZone][current][p].X - h.points[glyphZone][unhinted][p].X
	}
	if delta == 0 {
		return
	}
	for i := p1; i < p2; i++ {
		if i == p {
			continue
		}
		if interpY {
			h.points[glyphZone][current][i].Y += delta
		} else {
			h.points[glyphZone][current][i].X += delta
		}
	}
}

func (h *hinter) displacement(useZP1 bool) (zonePointer uint32, i int32, d f26dot6, ok bool) {
	zonePointer, i = uint32(0), h.gs.rp[1]
	if useZP1 {
		zonePointer, i = 1, h.gs.rp[2]
	}
	p := h.point(zonePointer, current, i)
	q := h.point(zonePointer, unhinted, i)
	if p == nil || q == nil {
		return 0, 0, 0, false
	}
	d = dotProduct(f26dot6(p.X-q.X), f26dot6(p.Y-q.Y), h.gs.pv)
	return zonePointer, i, d, true
}

// skipInstructionPayload increments pc by the extra data that follows a
// variable length PUSHB or PUSHW instruction.
func skipInstructionPayload(program []byte, pc int) (newPC int, ok bool) {
	switch program[pc] {
	case opNPUSHB:
		pc++
		if pc >= len(program) {
			return 0, false
		}
		pc += int(program[pc])
	case opNPUSHW:
		pc++
		if pc >= len(program) {
			return 0, false
		}
		pc += 2 * int(program[pc])
	case opPUSHB000, opPUSHB001, opPUSHB010, opPUSHB011,
		opPUSHB100, opPUSHB101, opPUSHB110, opPUSHB111:
		pc += int(program[pc] - (opPUSHB000 - 1))
	case opPUSHW000, opPUSHW001, opPUSHW010, opPUSHW011,
		opPUSHW100, opPUSHW101, opPUSHW110, opPUSHW111:
		pc += 2 * int(program[pc]-(opPUSHW000-1))
	}
	return pc, true
}

// f2dot14 is a 2.14 fixed point number.
type f2dot14 int16

func normalize(x, y f2dot14) [2]f2dot14 {
	fx, fy := float64(x), float64(y)
	l := 0x4000 / math.Hypot(fx, fy)
	fx *= l
	if fx >= 0 {
		fx += 0.5
	} else {
		fx -= 0.5
	}
	fy *= l
	if fy >= 0 {
		fy += 0.5
	} else {
		fy -= 0.5
	}
	return [2]f2dot14{f2dot14(fx), f2dot14(fy)}
}

// f26dot6 is a 26.6 fixed point number.
type f26dot6 int32

// abs returns abs(x) in 26.6 fixed point arithmetic.
func (x f26dot6) abs() f26dot6 {
	if x < 0 {
		return -x
	}
	return x
}

// div returns x/y in 26.6 fixed point arithmetic.
func (x f26dot6) div(y f26dot6) f26dot6 {
	return f26dot6((int64(x) << 6) / int64(y))
}

// mul returns x*y in 26.6 fixed point arithmetic.
func (x f26dot6) mul(y f26dot6) f26dot6 {
	return f26dot6((int64(x)*int64(y) + 1<<5) >> 6)
}

// dotProduct returns the dot product of [x, y] and q. It is almost the same as
//	px := int64(x)
//	py := int64(y)
//	qx := int64(q[0])
//	qy := int64(q[1])
//	return f26dot6((px*qx + py*qy + 1<<13) >> 14)
// except that the computation is done with 32-bit integers to produce exactly
// the same rounding behavior as C Freetype.
func dotProduct(x, y f26dot6, q [2]f2dot14) f26dot6 {
	// Compute x*q[0] as 64-bit value.
	l := uint32((int32(x) & 0xFFFF) * int32(q[0]))
	m := (int32(x) >> 16) * int32(q[0])

	lo1 := l + (uint32(m) << 16)
	hi1 := (m >> 16) + (int32(l) >> 31) + bool2int32(lo1 < l)

	// Compute y*q[1] as 64-bit value.
	l = uint32((int32(y) & 0xFFFF) * int32(q[1]))
	m = (int32(y) >> 16) * int32(q[1])

	lo2 := l + (uint32(m) << 16)
	hi2 := (m >> 16) + (int32(l) >> 31) + bool2int32(lo2 < l)

	// Add them.
	lo := lo1 + lo2
	hi := hi1 + hi2 + bool2int32(lo < lo1)

	// Divide the result by 2^14 with rounding.
	s := hi >> 31
	l = lo + uint32(s)
	hi += s + bool2int32(l < lo)
	lo = l

	l = lo + 0x2000
	hi += bool2int32(l < lo)

	return f26dot6((uint32(hi) << 18) | (l >> 14))
}

// mulDiv returns x*y/z, rounded to the nearest integer.
func mulDiv(x, y, z int64) int64 {
	xy := x * y
	if z < 0 {
		xy, z = -xy, -z
	}
	if xy >= 0 {
		xy += z / 2
	} else {
		xy -= z / 2
	}
	return xy / z
}

// round rounds the given number. The rounding algorithm is described at
// https://developer.apple.com/fonts/TTRefMan/RM02/Chap2.html#rounding
func (h *hinter) round(x f26dot6) f26dot6 {
	if h.gs.roundPeriod == 0 {
		// Rounding is off.
		return x
	}
	if x >= 0 {
		ret := x - h.gs.roundPhase + h.gs.roundThreshold
		if h.gs.roundSuper45 {
			ret /= h.gs.roundPeriod
			ret *= h.gs.roundPeriod
		} else {
			ret &= -h.gs.roundPeriod
		}
		if x != 0 && ret < 0 {
			ret = 0
		}
		return ret + h.gs.roundPhase
	}
	ret := -x - h.gs.roundPhase + h.gs.roundThreshold
	if h.gs.roundSuper45 {
		ret /= h.gs.roundPeriod
		ret *= h.gs.roundPeriod
	} else {
		ret &= -h.gs.roundPeriod
	}
	if ret < 0 {
		ret = 0
	}
	return -ret - h.gs.roundPhase
}

func bool2int32(b bool) int32 {
	if b {
		return 1
	}
	return 0
}