diff --git a/CHANGELOG.md b/CHANGELOG.md
index 2e55a82..bbbdada 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -4,9 +4,16 @@ All notable changes to this project will be documented in this file.
 The format is based on [Keep a Changelog](https://keepachangelog.com/en/1.0.0/),
 and this project adheres to [Semantic Versioning](https://semver.org/spec/v2.0.0.html).
 
-## [Unreleased]
+## [2.0.0] - 2021-02-13
 ### Added
 - Added `ErrorDiffusionStrength` to set the strength of error diffusion dithering (#4)
+- `RoundClamp` function for making your own `PixelMappers` that round correctly
+
+### Changed
+- All linear RGB values are represented using `uint16` instead of `uint8` now, because 8-bits is not enough to accurately hold a linearized value. This is a breaking change, hence the new major version.
+
+### Fixed
+- Rounding is no longer biased, because ties are rounded to the nearest even number
 
 
 ## [1.0.0] - 2021-02-11
diff --git a/README.md b/README.md
index 73f7843..13e78fc 100644
--- a/README.md
+++ b/README.md
@@ -41,10 +41,12 @@ More methods of dithering are being worked on, such as Riemersma, Yuliluoma, and
 In your project, run
 
 ```
-go get github.com/makeworld-the-better-one/dither@latest
+go get github.com/makeworld-the-better-one/dither/v2@latest
 go mod tidy
 ```
 
+You can import it as `"github.com/makeworld-the-better-one/dither/v2"` and use it as `dither`.
+
 ## Usage
 
 Here's a simple example using Floyd-Steinberg dithering.
@@ -153,8 +155,6 @@ you should definitely be using those functions over the color ones.
 
 All the `[][]uint` matrices are supposed to be applied with `PixelMapperFromMatrix`.
 
-If you ever find yourself multiplying a number by `255.0` (for example to scale a float in the range [0, 1]), make sure to add 0.5 before returning. Only by doing this can you get a correct answer in all cases. For example, `255.0 * n` should become `255.0 * n + 0.5`.
-
 
 ## Projects using `dither`
 
diff --git a/color_spaces.go b/color_spaces.go
index 594cbb6..3594590 100644
--- a/color_spaces.go
+++ b/color_spaces.go
@@ -7,43 +7,43 @@ import (
 
 // linearize1 linearizes an R, G, or B channel value from an sRGB color.
 // Must be in the range [0, 1].
-func linearize1(v float32) float32 {
+func linearize1(v float64) float64 {
 	if v <= 0.04045 {
 		return v / 12.92
 	}
-	return float32(math.Pow((float64(v)+0.055)/1.055, 2.4))
+	return math.Pow((v+0.055)/1.055, 2.4)
 }
 
-func linearize65535to255(i uint16) uint8 {
-	v := float32(i) / 65535.0
-	return uint8(linearize1(v)*255.0 + 0.5)
+func linearize65535(i uint16) uint16 {
+	v := float64(i) / 65535.0
+	return uint16(math.RoundToEven(linearize1(v) * 65535.0))
 }
 
-func linearize255(i uint8) uint8 {
-	v := float32(i) / 255.0
-	return uint8(linearize1(v)*255.0 + 0.5)
+func linearize255to65535(i uint8) uint16 {
+	v := float64(i) / 255.0
+	return uint16(math.RoundToEven(linearize1(v) * 65535.0))
 }
 
 // toLinearRGB converts a non-linear sRGB color to a linear RGB color space.
-func toLinearRGB(c color.Color) (uint8, uint8, uint8) {
+func toLinearRGB(c color.Color) (uint16, uint16, uint16) {
 	// Optimize for different color types
 	switch v := c.(type) {
 	case color.Gray:
-		g := linearize255(v.Y)
+		g := linearize255to65535(v.Y)
 		return g, g, g
 	case color.Gray16:
-		g := linearize65535to255(v.Y)
+		g := linearize65535(v.Y)
 		return g, g, g
 	case color.NRGBA:
-		return linearize255(v.R), linearize255(v.G), linearize255(v.B)
+		return linearize255to65535(v.R), linearize255to65535(v.G), linearize255to65535(v.B)
 	case color.NRGBA64:
-		return linearize65535to255(v.R), linearize65535to255(v.G), linearize65535to255(v.B)
+		return linearize65535(v.R), linearize65535(v.G), linearize65535(v.B)
 	case color.RGBA:
-		return linearize255(v.R), linearize255(v.G), linearize255(v.B)
+		return linearize255to65535(v.R), linearize255to65535(v.G), linearize255to65535(v.B)
 	case color.RGBA64:
-		return linearize65535to255(v.R), linearize65535to255(v.G), linearize65535to255(v.B)
+		return linearize65535(v.R), linearize65535(v.G), linearize65535(v.B)
 	}
 
 	r, g, b, _ := c.RGBA()
-	return linearize65535to255(uint16(r)), linearize65535to255(uint16(g)), linearize65535to255(uint16(b))
+	return linearize65535(uint16(r)), linearize65535(uint16(g)), linearize65535(uint16(b))
 }
diff --git a/dither.go b/dither.go
index e7760e5..bb7f53e 100644
--- a/dither.go
+++ b/dither.go
@@ -53,7 +53,7 @@ type Ditherer struct {
 	palette []color.Color
 
 	// linearPalette holds all the palette colors, but in linear RGB space.
-	linearPalette [][3]uint8
+	linearPalette [][3]uint16
 }
 
 // NewDitherer creates a new Ditherer that uses a copy of the provided palette.
@@ -70,10 +70,10 @@ func NewDitherer(palette []color.Color) *Ditherer {
 	copy(d.palette, palette)
 
 	// Create linear RGB version of the palette
-	d.linearPalette = make([][3]uint8, len(d.palette))
+	d.linearPalette = make([][3]uint16, len(d.palette))
 	for i := range d.linearPalette {
 		r, g, b := toLinearRGB(d.palette[i])
-		d.linearPalette[i] = [3]uint8{r, g, b}
+		d.linearPalette[i] = [3]uint16{r, g, b}
 	}
 
 	return d
@@ -104,20 +104,20 @@ func (d *Ditherer) GetPalette() []color.Color {
 	return p
 }
 
-func sqDiff(v1 uint8, v2 uint8) uint16 {
+func sqDiff(v1 uint16, v2 uint16) uint32 {
 	// This optimization is copied from Go stdlib, see
 	// https://github.com/golang/go/blob/go1.15.7/src/image/color/color.go#L314
 
-	d := uint16(v1) - uint16(v2)
+	d := uint32(v1) - uint32(v2)
 	return (d * d) >> 2
 }
 
 // closestColor returns the index of the color in the palette that's closest to
 // the provided one, using Euclidean distance in linear RGB space. The provided
 // RGB values must be linear RGB.
-func (d *Ditherer) closestColor(r, g, b uint8) int {
+func (d *Ditherer) closestColor(r, g, b uint16) int {
 	// Go through each color and find the closest one
-	color, best := 0, uint16(math.MaxUint16)
+	color, best := 0, uint32(math.MaxUint32)
 	for i, c := range d.linearPalette {
 		// Euclidean distance, but the square root part is removed
 		dist := sqDiff(r, c[0]) + sqDiff(g, c[1]) + sqDiff(b, c[2])
@@ -189,16 +189,16 @@ func (d *Ditherer) Dither(src image.Image) image.Image {
 	// Store linear values here instead of converting back and forth and storing
 	// sRGB values inside the image.
 	// Pointers are used to differentiate between a zero value and an unset value
-	lins := make([][][3]*uint8, b.Dy())
+	lins := make([][][3]*uint16, b.Dy())
 	for i := 0; i < len(lins); i++ {
-		lins[i] = make([][3]*uint8, b.Dx())
+		lins[i] = make([][3]*uint16, b.Dx())
 	}
 
 	// Setters and getters for that linear storage
-	linearSet := func(x, y int, r, g, b uint8) {
-		lins[y][x] = [3]*uint8{&r, &g, &b}
+	linearSet := func(x, y int, r, g, b uint16) {
+		lins[y][x] = [3]*uint16{&r, &g, &b}
 	}
-	linearAt := func(x, y int) (uint8, uint8, uint8) {
+	linearAt := func(x, y int) (uint16, uint16, uint16) {
 		c := lins[y][x]
 		if c[0] == nil {
 			// This pixel hasn't been linearized yet
@@ -225,11 +225,16 @@ func (d *Ditherer) Dither(src image.Image) image.Image {
 
 			new := d.linearPalette[newColorIdx]
 			// Quant errors in each channel
-			er, eg, eb := int16(oldR)-int16(new[0]), int16(oldG)-int16(new[1]), int16(oldB)-int16(new[2])
+			er, eg, eb := int32(oldR)-int32(new[0]), int32(oldG)-int32(new[1]), int32(oldB)-int32(new[2])
 
 			// Diffuse error in two dimensions
 			for yy := range d.Matrix {
 				for xx := range d.Matrix[yy] {
+					if d.Matrix[yy][xx] == 0 {
+						// Skip, because it won't affect anything
+						continue
+					}
+
 					// Get the coords of the pixel the error is being applied to
 					deltaX, deltaY := d.Matrix.Offset(xx, yy, curPx)
 					if d.Serpentine && y%2 == 0 {
@@ -247,12 +252,9 @@ func (d *Ditherer) Dither(src image.Image) image.Image {
 
 					r, g, b := linearAt(pxX, pxY)
 					linearSet(pxX, pxY,
-						// +0.5 is important to prevent pure white from being quantized to black due
-						// to the addition of error being rounded. This can be seen if you remove
-						// the +0.5 and test Floyd-Steinberg dithering on the gradient.png image.
-						clamp(float32(r)+float32(er)*d.Matrix[yy][xx]+0.5),
-						clamp(float32(g)+float32(eg)*d.Matrix[yy][xx]+0.5),
-						clamp(float32(b)+float32(eb)*d.Matrix[yy][xx]+0.5),
+						RoundClamp(float32(r)+float32(er)*d.Matrix[yy][xx]),
+						RoundClamp(float32(g)+float32(eg)*d.Matrix[yy][xx]),
+						RoundClamp(float32(b)+float32(eb)*d.Matrix[yy][xx]),
 					)
 				}
 			}
@@ -328,15 +330,16 @@ func (d *Ditherer) DitherPalettedConfig(src image.Image) (*image.Paletted, image
 	}
 }
 
-// clamp clamps i to the interval [0, 255].
-func clamp(i float32) uint8 {
+// RoundClamp clamps the number and rounds it, rounding ties to the nearest even number.
+// This should be used if you're writing your own PixelMapper.
+func RoundClamp(i float32) uint16 {
 	if i < 0 {
 		return 0
 	}
-	if i > 255 {
-		return 255
+	if i > 65535 {
+		return 65535
 	}
-	return uint8(i)
+	return uint16(math.RoundToEven(float64(i)))
 }
 
 // copyImage copies src's pixels into dst.
diff --git a/examples/gif_animation.go b/examples/gif_animation.go
index b4da935..41558aa 100644
--- a/examples/gif_animation.go
+++ b/examples/gif_animation.go
@@ -10,7 +10,7 @@ import (
 	_ "image/png" // For frame decoding
 	"os"
 
-	"github.com/makeworld-the-better-one/dither"
+	"github.com/makeworld-the-better-one/dither/v2"
 )
 
 const numFrames = 20
diff --git a/examples/gif_image.go b/examples/gif_image.go
index c9af1d7..10bba12 100644
--- a/examples/gif_image.go
+++ b/examples/gif_image.go
@@ -10,7 +10,7 @@ import (
 	_ "image/png" // Imported for decoding of the input image
 	"os"
 
-	"github.com/makeworld-the-better-one/dither"
+	"github.com/makeworld-the-better-one/dither/v2"
 )
 
 func main() {
diff --git a/examples/output/gif_animation.gif b/examples/output/gif_animation.gif
index c8a664b..57a032c 100644
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diff --git a/examples/output/gif_image.gif b/examples/output/gif_image.gif
index 95a7d55..d72ace5 100644
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diff --git a/go.mod b/go.mod
index c3e6480..ba11e35 100644
--- a/go.mod
+++ b/go.mod
@@ -1,4 +1,4 @@
-module github.com/makeworld-the-better-one/dither
+module github.com/makeworld-the-better-one/dither/v2
 
 go 1.15
 
diff --git a/images/output/ClusteredDot4x4.png b/images/output/ClusteredDot4x4.png
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diff --git a/images/output/random_noise_rgb_red-green-black.png b/images/output/random_noise_rgb_red-green-black.png
index d12bfad..2bfd7eb 100644
Binary files a/images/output/random_noise_rgb_red-green-black.png and b/images/output/random_noise_rgb_red-green-black.png differ
diff --git a/images/output/random_noise_rgb_red-green-yellow-black.png b/images/output/random_noise_rgb_red-green-yellow-black.png
index 80c8639..94e835c 100644
Binary files a/images/output/random_noise_rgb_red-green-yellow-black.png and b/images/output/random_noise_rgb_red-green-yellow-black.png differ
diff --git a/ordered_ditherers.go b/ordered_ditherers.go
index cae73cc..4af07b3 100644
--- a/ordered_ditherers.go
+++ b/ordered_ditherers.go
@@ -219,11 +219,12 @@ var ClusteredDotVerticalLine = OrderedDitherMatrix{
 var ClusteredDot8x8 = OrderedDitherMatrix{
 	Matrix: [][]uint{
 		// For some reason the values in the book were in the range of 0-64
-		// instead of 0-63.
+		// instead of 0-63. I changed the 64 value to 63, so that pure black
+		// didn't end up with occasional white dots.
 		{3, 9, 17, 27, 25, 15, 7, 1},
 		{11, 29, 38, 46, 44, 36, 23, 5},
 		{19, 40, 52, 58, 56, 50, 34, 13},
-		{31, 48, 60, 64, 62, 54, 42, 21},
+		{31, 48, 60, 63, 62, 54, 42, 21},
 		{30, 47, 59, 63, 61, 53, 41, 20},
 		{18, 39, 51, 57, 55, 49, 33, 12},
 		{10, 28, 37, 45, 43, 35, 22, 4},
diff --git a/pixelmappers.go b/pixelmappers.go
index b84f64b..2de6d53 100644
--- a/pixelmappers.go
+++ b/pixelmappers.go
@@ -14,10 +14,11 @@ import (
 //
 // The provided RGB values are in the linear RGB space, and the returned values
 // must be as well. All dithering operations should be happening in this space
-// anyway, so this is done as a convenience.
+// anyway, so this is done as a convenience. The RGB values are in the range
+// [0, 65535], and must be returned in the same range.
 //
 // It must be thread-safe, as it will be called concurrently.
-type PixelMapper func(x, y int, r, g, b uint8) (uint8, uint8, uint8)
+type PixelMapper func(x, y int, r, g, b uint16) (uint16, uint16, uint16)
 
 // RandomNoiseGrayscale returns a PixelMapper that adds random noise to the
 // color before returning. This is the simplest form of dithering.
@@ -49,17 +50,17 @@ type PixelMapper func(x, y int, r, g, b uint8) (uint8, uint8, uint8)
 // not wrapped. Basically, don't worry about the values of your min and max
 // distorting the image in an unexpected way.
 func RandomNoiseGrayscale(min, max float32) PixelMapper {
-	return PixelMapper(func(x, y int, r, g, b uint8) (uint8, uint8, uint8) {
+	return PixelMapper(func(x, y int, r, g, b uint16) (uint16, uint16, uint16) {
 		// These values were taken from Wikipedia:
 		// https://en.wikipedia.org/wiki/Grayscale#Colorimetric_(perceptual_luminance-preserving)_conversion_to_grayscale
 		// 0.2126, 0.7152, 0.0722
-		// Then multiplied by 255, to scale them for uint8 color.
-		// Note that 54 + 183 + 19 = 256.
+		// Then multiplied by 65535, to scale them for 16-bit color.
+		// Note that 13933 + 46871 + 4732 = 65536
 		//
 		// Basically, this takes linear RGB and gives a linear gray.
-		gray := (54*uint16(r) + 183*uint16(g) + 19*uint16(b) + 1<<7) >> 8
+		gray := (13933*uint32(r) + 46871*uint32(g) + 4732*uint32(b) + 1<<15) >> 16
 
-		new := clamp(float32(gray) + 255.0*(rand.Float32()*(max-min)+min) + 0.5)
+		new := RoundClamp(float32(gray) + 65535.0*(rand.Float32()*(max-min)+min))
 		return new, new, new
 	})
 }
@@ -74,10 +75,10 @@ func RandomNoiseGrayscale(min, max float32) PixelMapper {
 // See RandomNoiseGrayscale for more details about values and how this function
 // works.
 func RandomNoiseRGB(minR, maxR, minG, maxG, minB, maxB float32) PixelMapper {
-	return PixelMapper(func(x, y int, r, g, b uint8) (uint8, uint8, uint8) {
-		return clamp(float32(r) + 255.0*(rand.Float32()*(maxR-minR)+minR) + 0.5),
-			clamp(float32(g) + 255.0*(rand.Float32()*(maxG-minG)+minG) + 0.5),
-			clamp(float32(b) + 255.0*(rand.Float32()*(maxB-minB)+minB) + 0.5)
+	return PixelMapper(func(x, y int, r, g, b uint16) (uint16, uint16, uint16) {
+		return RoundClamp(float32(r) + 65535.0*(rand.Float32()*(maxR-minR)+minR)),
+			RoundClamp(float32(g) + 65535.0*(rand.Float32()*(maxG-minG)+minG)),
+			RoundClamp(float32(b) + 65535.0*(rand.Float32()*(maxB-minB)+minB))
 	})
 }
 
@@ -165,14 +166,18 @@ func bayerMatrix(xdim, ydim uint) [][]uint {
 // and returns a value that can be added to a color instead of thresholded.
 //
 // scale is the number that's multiplied at the end, usually you want this to be
-// 255 to scale to match the color value range. value is the cell of the matrix.
+// 65535 to scale to match the color value range. value is the cell of the matrix.
 // max is the divisor of the cell value, usually this is the product of the matrix
 // dimensions.
 func convThresholdToAddition(scale float32, value uint, max uint) float32 {
 	// See:
 	// https://en.wikipedia.org/wiki/Ordered_dithering
 	// https://en.wikipedia.org/wiki/Talk:Ordered_dithering#Sources
-	return scale * (float32(value+1.0)/float32(max) - 0.5)
+
+	// 0.50000006 is next possible float32 value after 0.5. This is to correct
+	// a rounding error that occurs when the number is exactly 0.5, which results
+	// in pure black being dithered when it should be left alone.
+	return scale * (float32(value+1.0)/float32(max) - 0.50000006)
 }
 
 // Bayer returns a PixelMapper that applies a Bayer matrix with the specified size.
@@ -191,8 +196,8 @@ func convThresholdToAddition(scale float32, value uint, max uint) float32 {
 // Source:
 //     https://bisqwit.iki.fi/story/howto/dither/jy/#Appendix%202ThresholdMatrix
 //
-// strength should be in the range [-1, 1]. It is multiplied with 255, which is
-// then multiplied with the matrix.
+// strength should be in the range [-1, 1]. It is multiplied with 65535
+// (the max color value), which is then multiplied with the matrix.
 //
 // You can use this to change the amount the matrix is applied to the image, the
 // "strength" of the dithering matrix. Usually just keeping it at 1.0 is fine.
@@ -281,7 +286,7 @@ func Bayer(x, y uint, strength float32) PixelMapper {
 	}
 
 	// Create precalculated matrix
-	scale := 255.0 * strength
+	scale := 65535.0 * strength
 	max := x * y
 
 	precalc := make([][]float32, y)
@@ -292,10 +297,10 @@ func Bayer(x, y uint, strength float32) PixelMapper {
 		}
 	}
 
-	return PixelMapper(func(xx, yy int, r, g, b uint8) (uint8, uint8, uint8) {
-		return clamp(float32(r) + precalc[yy%int(y)][xx%int(x)]),
-			clamp(float32(g) + precalc[yy%int(y)][xx%int(x)]),
-			clamp(float32(b) + precalc[yy%int(y)][xx%int(x)])
+	return PixelMapper(func(xx, yy int, r, g, b uint16) (uint16, uint16, uint16) {
+		return RoundClamp(float32(r) + precalc[yy%int(y)][xx%int(x)]),
+			RoundClamp(float32(g) + precalc[yy%int(y)][xx%int(x)]),
+			RoundClamp(float32(b) + precalc[yy%int(y)][xx%int(x)])
 	})
 }
 
@@ -313,7 +318,7 @@ func Bayer(x, y uint, strength float32) PixelMapper {
 func PixelMapperFromMatrix(odm OrderedDitherMatrix, strength float32) PixelMapper {
 	ydim := len(odm.Matrix)
 	xdim := len(odm.Matrix[0])
-	scale := 255.0 * strength
+	scale := 65535.0 * strength
 
 	// Create precalculated matrix
 	precalc := make([][]float32, ydim)
@@ -324,9 +329,9 @@ func PixelMapperFromMatrix(odm OrderedDitherMatrix, strength float32) PixelMappe
 		}
 	}
 
-	return PixelMapper(func(xx, yy int, r, g, b uint8) (uint8, uint8, uint8) {
-		return clamp(float32(r) + precalc[yy%ydim][xx%xdim]),
-			clamp(float32(g) + precalc[yy%ydim][xx%xdim]),
-			clamp(float32(b) + precalc[yy%ydim][xx%xdim])
+	return PixelMapper(func(xx, yy int, r, g, b uint16) (uint16, uint16, uint16) {
+		return RoundClamp(float32(r) + precalc[yy%ydim][xx%xdim]),
+			RoundClamp(float32(g) + precalc[yy%ydim][xx%xdim]),
+			RoundClamp(float32(b) + precalc[yy%ydim][xx%xdim])
 	})
 }