Adding upstream version 110.0.1.upstream/110.0.1upstream

Signed-off-by: Daniel Baumann <daniel.baumann@progress-linux.org>

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+# Text Rendering
+
+This document describes the details of how WebRender renders text, particularly the blending stage of text rendering.
+We will go into grayscale text blending, subpixel text blending, and "subpixel text with background color" blending.
+
+### Prerequisites
+
+The description below assumes you're familiar with regular rgba compositing, operator over,
+and the concept of premultiplied alpha.
+
+### Not covered in this document
+
+We are going to treat the origin of the text mask as a black box.
+We're also going to assume we can blend text in the device color space and will not go into the gamma correction and linear pre-blending that happens in some of the backends that produce the text masks.
+
+## Grayscale Text Blending
+
+Grayscale text blending is the simplest form of text blending. Our blending function has three inputs:
+
+ - The text color, as a premultiplied rgba color.
+ - The text mask, as a single-channel alpha texture.
+ - The existing contents of the framebuffer that we're rendering to, the "destination". This is also a premultiplied rgba buffer.
+
+Note: The word "grayscale" here does *not* mean that we can only draw gray text.
+It means that the mask only has a single alpha value per pixel, so we can visualize
+the mask in our minds as a grayscale image.
+
+### Deriving the math
+
+We want to mask our text color using the single-channel mask, and composite that to the destination.
+This compositing step uses operator "over", just like regular compositing of rgba images.
+
+I'll be using GLSL syntax to describe the blend equations, but please consider most of the code below pseudocode.
+
+We can express the blending described above as the following blend equation:
+
+```glsl
+vec4 textblend(vec4 text_color, vec4 mask, vec4 dest) {
+  return over(in(text_color, mask), dest);
+}
+```
+
+with `over` being the blend function for (premultiplied) operator "over":
+
+```glsl
+vec4 over(vec4 src, vec4 dest) {
+  return src + (1.0 - src.a) * dest;
+}
+```
+
+and `in` being the blend function for (premultiplied) operator "in", i.e. the masking operator:
+
+```glsl
+vec4 in(vec4 src, vec4 mask) {
+  return src * mask.a;
+}
+```
+
+So the complete blending function is:
+
+```glsl
+result.r = text_color.r * mask.a + (1.0 - text_color.a * mask.a) * dest.r;
+result.g = text_color.g * mask.a + (1.0 - text_color.a * mask.a) * dest.g;
+result.b = text_color.b * mask.a + (1.0 - text_color.a * mask.a) * dest.b;
+result.a = text_color.a * mask.a + (1.0 - text_color.a * mask.a) * dest.a;
+```
+
+### Rendering this with OpenGL
+
+In general, a fragment shader does not have access to the destination.
+So the full blend equation needs to be expressed in a way that the shader only computes values that are independent of the destination,
+and the parts of the equation that use the destination values need to be applied by the OpenGL blend pipeline itself.
+The OpenGL blend pipeline can be tweaked using the functions `glBlendEquation` and `glBlendFunc`.
+
+In our example, the fragment shader can output just `text_color * mask.a`:
+
+```glsl
+  oFragColor = text_color * mask.a;
+```
+
+and the OpenGL blend pipeline can be configured like so:
+
+```rust
+    pub fn set_blend_mode_premultiplied_alpha(&self) {
+        self.gl.blend_func(gl::ONE, gl::ONE_MINUS_SRC_ALPHA);
+        self.gl.blend_equation(gl::FUNC_ADD);
+    }
+```
+
+This results in an overall blend equation of
+
+```
+result.r = 1 * oFragColor.r + (1 - oFragColor.a) * dest.r;
+           ^                ^  ^^^^^^^^^^^^^^^^^
+           |                |         |
+           +--gl::ONE       |         +-- gl::ONE_MINUS_SRC_ALPHA
+                            |
+                            +-- gl::FUNC_ADD
+
+         = 1 * (text_color.r * mask.a) + (1 - (text_color.a * mask.a)) * dest.r
+         = text_color.r * mask.a + (1 - text_color.a * mask.a) * dest.r
+```
+
+which is exactly what we wanted.
+
+### Differences to the actual WebRender code
+
+There are two minor differences between the shader code above and the actual code in the text run shader in WebRender:
+
+```glsl
+oFragColor = text_color * mask.a;    // (shown above)
+// vs.
+oFragColor = vColor * mask * alpha;  // (actual webrender code)
+```
+
+`vColor` is set to the text color. The differences are:
+
+ - WebRender multiplies with all components of `mask` instead of just with `mask.a`.
+   However, our font rasterization code fills the rgb values of `mask` with the value of `mask.a`,
+   so this is completely equivalent.
+ - WebRender applies another alpha to the text. This is coming from the clip.
+   You can think of this alpha to be a pre-adjustment of the text color for that pixel, or as an
+   additional mask that gets applied to the mask.
+
+## Subpixel Text Blending
+
+Now that we have the blend equation for single-channel text blending, we can look at subpixel text blending.
+
+The main difference between subpixel text blending and grayscale text blending is the fact that,
+for subpixel text, the text mask contains a separate alpha value for each color component.
+
+### Component alpha
+
+Regular painting uses four values per pixel: three color values, and one alpha value. The alpha value applies to all components of the pixel equally.
+
+Imagine for a second a world in which you have *three alpha values per pixel*, one for each color component.
+
+ - Old world: Each pixel has four values: `color.r`, `color.g`, `color.b`, and `color.a`.
+ - New world: Each pixel has *six* values: `color.r`, `color.a_r`, `color.g`, `color.a_g`, `color.b`, and `color.a_b`.
+
+In such a world we can define a component-alpha-aware operator "over":
+
+```glsl
+vec6 over_comp(vec6 src, vec6 dest) {
+  vec6 result;
+  result.r = src.r + (1.0 - src.a_r) * dest.r;
+  result.g = src.g + (1.0 - src.a_g) * dest.g;
+  result.b = src.b + (1.0 - src.a_b) * dest.b;
+  result.a_r = src.a_r + (1.0 - src.a_r) * dest.a_r;
+  result.a_g = src.a_g + (1.0 - src.a_g) * dest.a_g;
+  result.a_b = src.a_b + (1.0 - src.a_b) * dest.a_b;
+  return result;
+}
+```
+
+and a component-alpha-aware operator "in":
+
+```glsl
+vec6 in_comp(vec6 src, vec6 mask) {
+  vec6 result;
+  result.r = src.r * mask.a_r;
+  result.g = src.g * mask.a_g;
+  result.b = src.b * mask.a_b;
+  result.a_r = src.a_r * mask.a_r;
+  result.a_g = src.a_g * mask.a_g;
+  result.a_b = src.a_b * mask.a_b;
+  return result;
+}
+```
+
+and even a component-alpha-aware version of `textblend`:
+
+```glsl
+vec6 textblend_comp(vec6 text_color, vec6 mask, vec6 dest) {
+  return over_comp(in_comp(text_color, mask), dest);
+}
+```
+
+This results in the following set of equations:
+
+```glsl
+result.r = text_color.r * mask.a_r + (1.0 - text_color.a_r * mask.a_r) * dest.r;
+result.g = text_color.g * mask.a_g + (1.0 - text_color.a_g * mask.a_g) * dest.g;
+result.b = text_color.b * mask.a_b + (1.0 - text_color.a_b * mask.a_b) * dest.b;
+result.a_r = text_color.a_r * mask.a_r + (1.0 - text_color.a_r * mask.a_r) * dest.a_r;
+result.a_g = text_color.a_g * mask.a_g + (1.0 - text_color.a_g * mask.a_g) * dest.a_g;
+result.a_b = text_color.a_b * mask.a_b + (1.0 - text_color.a_b * mask.a_b) * dest.a_b;
+```
+
+### Back to the real world
+
+If we want to transfer the component alpha blend equation into the real world, we need to make a few small changes:
+
+ - Our text color only needs one alpha value.
+   So we'll replace all instances of `text_color.a_r/g/b` with `text_color.a`.
+ - We're currently not making use of the mask's `r`, `g` and `b` values, only of the `a_r`, `a_g` and `a_b` values.
+   So in the real world, we can use the rgb channels of `mask` to store those component alphas and
+   replace `mask.a_r/g/b` with `mask.r/g/b`.
+
+These two changes give us:
+
+```glsl
+result.r = text_color.r * mask.r + (1.0 - text_color.a * mask.r) * dest.r;
+result.g = text_color.g * mask.g + (1.0 - text_color.a * mask.g) * dest.g;
+result.b = text_color.b * mask.b + (1.0 - text_color.a * mask.b) * dest.b;
+result.a_r = text_color.a * mask.r + (1.0 - text_color.a * mask.r) * dest.a_r;
+result.a_g = text_color.a * mask.g + (1.0 - text_color.a * mask.g) * dest.a_g;
+result.a_b = text_color.a * mask.b + (1.0 - text_color.a * mask.b) * dest.a_b;
+```
+
+There's a third change we need to make:
+
+ - We're rendering to a destination surface that only has one alpha channel instead of three.
+   So `dest.a_r/g/b` and `result.a_r/g/b` will need to become `dest.a` and `result.a`.
+
+This creates a problem: We're currently assigning different values to `result.a_r`, `result.a_g` and `result.a_b`.
+Which of them should we use to compute `result.a`?
+
+This question does not have an answer. One alpha value per pixel is simply not sufficient
+to express the same information as three alpha values.
+
+However, see what happens if the destination is already opaque:
+
+We have `dest.a_r == 1`, `dest.a_g == 1`, and `dest.a_b == 1`.
+
+```
+result.a_r = text_color.a * mask.r + (1 - text_color.a * mask.r) * dest.a_r
+           = text_color.a * mask.r + (1 - text_color.a * mask.r) * 1
+           = text_color.a * mask.r + 1 - text_color.a * mask.r
+           = 1
+same for result.a_g and result.a_b
+```
+
+In other words, for opaque destinations, it doesn't matter what which channel of the mask we use when computing `result.a`, the result will always be completely opaque anyways. In WebRender we just pick `mask.g` (or rather,
+have font rasterization set `mask.a` to the value of `mask.g`) because it's as good as any.
+
+The takeaway here is: **Subpixel text blending is only supported for opaque destinations.** Attempting to render subpixel
+text into partially transparent destinations will result in bad alpha values. Or rather, it will result in alpha values which
+are not anticipated by the r, g, and b values in the same pixel, so that subsequent blend operations, which will mix r and a values
+from the same pixel, will produce incorrect colors.
+
+Here's the final subpixel blend function:
+
+```glsl
+vec4 subpixeltextblend(vec4 text_color, vec4 mask, vec4 dest) {
+  vec4 result;
+  result.r = text_color.r * mask.r + (1.0 - text_color.a * mask.r) * dest.r;
+  result.g = text_color.g * mask.g + (1.0 - text_color.a * mask.g) * dest.g;
+  result.b = text_color.b * mask.b + (1.0 - text_color.a * mask.b) * dest.b;
+  result.a = text_color.a * mask.a + (1.0 - text_color.a * mask.a) * dest.a;
+  return result;
+}
+```
+
+or for short:
+
+```glsl
+vec4 subpixeltextblend(vec4 text_color, vec4 mask, vec4 dest) {
+  return text_color * mask + (1.0 - text_color.a * mask) * dest;
+}
+```
+
+To recap, here's what we gained and lost by making the transition from the full-component-alpha world to the
+regular rgba world: All colors and textures now only need four values to be represented, we still use a
+component alpha mask, and the results are equivalent to the full-component-alpha result assuming that the
+destination is opaque. We lost the ability to draw to partially transparent destinations.
+
+### Making this work in OpenGL
+
+We have the complete subpixel blend function.
+Now we need to cut it into pieces and mix it with the OpenGL blend pipeline in such a way that
+the fragment shader does not need to know about the destination.
+
+Compare the equation for the red channel and the alpha channel between the two ways of text blending:
+
+```
+  single-channel alpha:
+    result.r = text_color.r * mask.a + (1.0 - text_color.a * mask.a) * dest.r
+    result.a = text_color.a * mask.a + (1.0 - text_color.a * mask.a) * dest.r
+
+  component alpha:
+    result.r = text_color.r * mask.r + (1.0 - text_color.a * mask.r) * dest.r
+    result.a = text_color.a * mask.a + (1.0 - text_color.a * mask.a) * dest.r
+```
+
+Notably, in the single-channel alpha case, all three destination color channels are multiplied with the same thing:
+`(1.0 - text_color.a * mask.a)`. This factor also happens to be "one minus `oFragColor.a`".
+So we were able to take advantage of OpenGL's `ONE_MINUS_SRC_ALPHA` blend func.
+
+In the component alpha case, we're not so lucky: Each destination color channel
+is multiplied with a different factor. We can use `ONE_MINUS_SRC_COLOR` instead,
+and output `text_color.a * mask` from our fragment shader.
+But then there's still the problem that the first summand of the computation for `result.r` uses
+`text_color.r * mask.r` and the second summand uses `text_color.a * mask.r`.
+
+There are multiple ways to deal with this. They are:
+
+ 1. Making use of `glBlendColor` and the `GL_CONSTANT_COLOR` blend func.
+ 2. Using a two-pass method.
+ 3. Using "dual source blending".
+
+Let's look at them in order.
+
+#### 1. Subpixel text blending in OpenGL using `glBlendColor`
+
+In this approach we return `text_color.a * mask` from the shader.
+Then we set the blend color to `text_color / text_color.a` and use `GL_CONSTANT_COLOR` as the source blendfunc.
+This results in the following blend equation:
+
+```
+result.r = (text_color.r / text_color.a) * oFragColor.r + (1 - oFragColor.r) * dest.r;
+           ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^                ^  ^^^^^^^^^^^^^^^^^
+                         |                              |      |
+                         +--gl::CONSTANT_COLOR          |      +-- gl::ONE_MINUS_SRC_COLOR
+                                                        |
+                                                        +-- gl::FUNC_ADD
+
+         = (text_color.r / text_color.a) * (text_color.a * mask.r) + (1 - (text_color.a * mask.r)) * dest.r
+         = text_color.r * mask.r + (1 - text_color.a * mask.r) * dest.r
+```
+
+At the very beginning of this document, we defined `text_color` as the *premultiplied* text color.
+So instead of actually doing the calculation `text_color.r / text_color.a` when specifying the blend color,
+we really just want to use the *unpremultiplied* text color in that place.
+That's usually the representation we start with anyway.
+
+#### 2. Two-pass subpixel blending in OpenGL
+
+The `glBlendColor` method has the disadvantage that the text color is part of the OpenGL state.
+So if we want to draw text with different colors, we have two use separate batches / draw calls
+to draw the differently-colored parts of text.
+
+Alternatively, we can use a two-pass method which avoids the need to use the `GL_CONSTANT_COLOR` blend func:
+
+ - The first pass outputs `text_color.a * mask` from the fragment shader and
+   uses `gl::ZERO, gl::ONE_MINUS_SRC_COLOR` as the glBlendFuncs. This achieves:
+
+```
+oFragColor = text_color.a * mask;
+
+result_after_pass0.r = 0 * oFragColor.r + (1 - oFragColor.r) * dest.r
+                     = (1 - text_color.a * mask.r) * dest.r
+
+result_after_pass0.g = 0 * oFragColor.g + (1 - oFragColor.g) * dest.r
+                     = (1 - text_color.a * mask.r) * dest.r
+
+...
+```
+
+ - The second pass outputs `text_color * mask` from the fragment shader and uses
+   `gl::ONE, gl::ONE` as the glBlendFuncs. This results in the correct overall blend equation.
+
+```
+oFragColor = text_color * mask;
+
+result_after_pass1.r
+ = 1 * oFragColor.r + 1 * result_after_pass0.r
+ = text_color.r * mask.r + result_after_pass0.r
+ = text_color.r * mask.r + (1 - text_color.a * mask.r) * dest.r
+```
+
+#### 3. Dual source subpixel blending in OpenGL
+
+The third approach is similar to the second approach, but makes use of the [`ARB_blend_func_extended`](https://www.khronos.org/registry/OpenGL/extensions/ARB/ARB_blend_func_extended.txt) extension
+in order to fold the two passes into one:
+Instead of outputting the two different colors in two separate passes, we output them from the same pass,
+as two separate fragment shader outputs.
+Those outputs can then be treated as two different sources in the blend equation.
+
+## Subpixel Text Rendering to Transparent Destinations with a Background Color Hint
+
+### Motivation
+
+As we've seen in the previous section, subpixel text drawing has the limitation that it only works on opaque destinations.
+
+In other words, if you use the `subpixeltextblend` function to draw something to a transparent surface,
+and then composite that surface onto on opaque background,
+the result will generally be different from drawing the text directly onto the opaque background.
+
+Let's express that inequality in code.
+
+```
+ - vec4 text_color
+ - vec4 mask
+ - vec4 transparency = vec4(0.0, 0.0, 0.0, 0.0)
+ - vec4 background with background.a == 1.0
+
+over(subpixeltextblend(text_color, mask, transparency), background).rgb
+ is, in general, not equal to
+subpixeltextblend(text_color, mask, background).rgb
+```
+
+However, one interesting observation is that if the background is black, the two *are* equal:
+
+```
+vec4 black = vec4(0.0, 0.0, 0.0, 1.0);
+
+over(subpixeltextblend(text_color, mask, transparency), black).r
+ = subpixeltextblend(text_color, mask, transparency).r +
+     (1 - subpixeltextblend(text_color, mask, transparency).a) * black.r
+ = subpixeltextblend(text_color, mask, transparency).r +
+     (1 - subpixeltextblend(text_color, mask, transparency).a) * 0
+ = subpixeltextblend(text_color, mask, transparency).r
+ = text_color.r * mask.r + (1 - text_color.a * mask.r) * transparency.r
+ = text_color.r * mask.r + (1 - text_color.a * mask.r) * 0
+ = text_color.r * mask.r + (1 - text_color.a * mask.r) * black.r
+ = subpixeltextblend(text_color, mask, black).r
+```
+
+So it works out for black backgrounds. The further your *actual* background color gets away from black,
+the more incorrect your result will be.
+
+If it works for black, is there a way to make it work for other colors?
+This is the motivating question for this third way of text blending:
+
+We want to be able to specify an *estimated background color*, and have a blending function
+`vec4 subpixeltextblend_withbgcolor(vec4 text_color, vec4 mask, vec4 bg_color, vec4 dest)`,
+in such a way that the error we get by using an intermediate surface is somehow in relation
+to the error we made when estimating the background color. In particular, if we estimated
+the background color perfectly, we want the intermediate surface to go unnoticed.
+
+Expressed as code:
+
+```
+over(subpixeltextblend_withbgcolor(text_color, mask, bg_color, transparency), bg_color)
+ should always be equal to
+subpixeltextblend(text_color, mask, bg_color)
+```
+
+This is one of three constraints we'd like `subpixeltextblend_withbgcolor` to satisfy.
+
+The next constraint is the following: If `dest` is already opaque, `subpixeltextblend_withbgcolor`
+should have the same results as `subpixeltextblend`, and the background color hint should be ignored.
+
+```
+ If dest.a == 1.0,
+subpixeltextblend_withbgcolor(text_color, mask, bg_color, dest)
+ should always be equal to
+subpixeltextblend(text_color, mask, dest)
+```
+
+And there's a third condition we'd like it to fulfill:
+In places where the mask is zero, the destination should be unaffected.
+
+```
+subpixeltextblend_withbgcolor(text_color, transparency, bg_color, dest)
+ should always be equal to
+dest
+```
+
+### Use cases
+
+The primary use case for such a blend method is text on top of vibrant areas of a window on macOS.
+
+Vibrant backgrounds with behind-window blending are computed by the window server, and they are tinted
+in a color that's based on the chosen vibrancy type.
+
+The window's rgba buffer is transparent in the vibrant areas. Window contents, even text, are drawn onto
+that transparent rgba buffer. Then the window server composites the window onto an opaque backdrop.
+So the results on the screen are computed as follows:
+
+```glsl
+window_buffer_pixel = subpixeltextblend_withbgcolor(text_color, mask, bg_color, transparency);
+screen_pixel = over(window_buffer_pixel, window_backdrop);
+```
+
+### Prior art
+
+Apple has implemented such a method of text blending in CoreGraphics, specifically for rendering text onto vibrant backgrounds.
+It's hidden behind the private API `CGContextSetFontSmoothingBackgroundColor` and is called by AppKit internally before
+calling the `-[NSView drawRect:]` method of your `NSVisualEffectView`, with the appropriate font smoothing background color
+for the vibrancy type of that view.
+
+I'm not aware of any public documentation of this way of text blending.
+It seems to be considered an implementation detail by Apple, and is probably hidden by default because it can be a footgun:
+If the font smoothing background color you specify is very different from the actual background that our surface is placed
+on top of, the text will look glitchy.
+
+### Deriving the blending function from first principles
+
+Before we dive into the math, let's repeat our goal once more.
+
+We want to create a blending function of the form
+`vec4 subpixeltextblend_withbgcolor(vec4 text_color, vec4 mask, vec4 bg_color, vec4 dest)`
+(with `bg_color` being an opaque color)
+which satisfies the following three constraints:
+
+```
+Constraint I:
+  over(subpixeltextblend_withbgcolor(text_color, mask, bg_color, transparency), bg_color)
+   should always be equal to
+  subpixeltextblend(text_color, mask, bg_color)
+
+Constraint II:
+   If dest.a == 1.0,
+  subpixeltextblend_withbgcolor(text_color, mask, bg_color, dest)
+   should always be equal to
+  subpixeltextblend(text_color, mask, dest)
+
+Constraint II:
+  subpixeltextblend_withbgcolor(text_color, transparency, bg_color, dest)
+   should always be equal to
+  dest
+```
+
+Constraint I and constraint II are about what happens depending on the destination's alpha.
+In particular: If the destination is completely transparent, we should blend into the
+estimated background color, and if it's completely opaque, we should blend into the destination color.
+In fact, we really want to blend into `over(dest, bg_color)`: we want `bg_color` to be used
+as a backdrop *behind* the current destination. So let's combine constraints I and II into a new
+constraint IV:
+
+```
+Constraint IV:
+  over(subpixeltextblend_withbgcolor(text_color, mask, bg_color, dest), bg_color)
+   should always be equal to
+  subpixeltextblend(text_color, mask, over(dest, bg_color))
+```
+
+Let's look at just the left side of that equation and rejiggle it a bit:
+
+```
+over(subpixeltextblend_withbgcolor(text_color, mask, bg_color, dest), bg_color).r
+ = subpixeltextblend_withbgcolor(text_color, mask, bg_color, dest).r +
+   (1 - subpixeltextblend_withbgcolor(text_color, mask, bg_color, dest).a) * bg_color.r
+
+<=>
+
+over(subpixeltextblend_withbgcolor(text_color, mask, bg_color, dest), bg_color).r -
+(1 - subpixeltextblend_withbgcolor(text_color, mask, bg_color, dest).a) * bg_color.r
+ = subpixeltextblend_withbgcolor(text_color, mask, bg_color, dest).r
+```
+
+Now insert the right side of constraint IV:
+
+```
+subpixeltextblend_withbgcolor(text_color, mask, bg_color, dest).r
+ = over(subpixeltextblend_withbgcolor(text_color, mask, bg_color, dest), bg_color).r -
+   (1 - subpixeltextblend_withbgcolor(text_color, mask, bg_color, dest).a) * bg_color.r
+ = subpixeltextblend(text_color, mask, over(dest, bg_color)).r -
+   (1 - subpixeltextblend_withbgcolor(text_color, mask, bg_color, dest).a) * bg_color.r
+```
+
+Our blend function is almost finished. We just need select an alpha for our result.
+Constraints I, II and IV don't really care about the alpha value. But constraint III requires that:
+
+```
+  subpixeltextblend_withbgcolor(text_color, transparency, bg_color, dest).a
+   should always be equal to
+  dest.a
+```
+
+so the computation of the alpha value somehow needs to take into account the mask.
+
+Let's say we have an unknown function `make_alpha(text_color.a, mask)` which returns
+a number between 0 and 1 and which is 0 if the mask is entirely zero, and let's defer
+the actual implementation of that function until later.
+
+Now we can define the alpha of our overall function using the `over` function:
+
+```
+subpixeltextblend_withbgcolor(text_color, mask, bg_color, dest).a
+ := make_alpha(text_color.a, mask) + (1 - make_alpha(text_color.a, mask)) * dest.a
+```
+
+We can plug this in to our previous result:
+
+```
+subpixeltextblend_withbgcolor(text_color, mask, bg_color, dest).r
+ = subpixeltextblend(text_color, mask, over(dest, bg_color)).r
+   - (1 - subpixeltextblend_withbgcolor(text_color, mask, bg_color, dest).a) * bg_color.r
+ = subpixeltextblend(text_color, mask, over(dest, bg_color)).r
+   - (1 - (make_alpha(text_color.a, mask) +
+           (1 - make_alpha(text_color.a, mask)) * dest.a)) * bg_color.r
+ = text_color.r * mask.r + (1 - text_color.a * mask.r) * over(dest, bg_color).r
+   - (1 - (make_alpha(text_color.a, mask)
+           + (1 - make_alpha(text_color.a, mask)) * dest.a)) * bg_color.r
+ = text_color.r * mask.r
+   + (1 - text_color.a * mask.r) * (dest.r + (1 - dest.a) * bg_color.r)
+   - (1 - (make_alpha(text_color.a, mask)
+           + (1 - make_alpha(text_color.a, mask)) * dest.a)) * bg_color.r
+ = text_color.r * mask.r
+   + (1 - text_color.a * mask.r) * (dest.r + (1 - dest.a) * bg_color.r)
+   - (1 - (make_alpha(text_color.a, mask)
+           + (1 - make_alpha(text_color.a, mask)) * dest.a)) * bg_color.r
+ = text_color.r * mask.r
+   + (dest.r + (1 - dest.a) * bg_color.r)
+   - (text_color.a * mask.r) * (dest.r + (1 - dest.a) * bg_color.r)
+   - (1 - make_alpha(text_color.a, mask)
+      - (1 - make_alpha(text_color.a, mask)) * dest.a) * bg_color.r
+ = text_color.r * mask.r
+   + dest.r + (1 - dest.a) * bg_color.r
+   - text_color.a * mask.r * dest.r
+   - text_color.a * mask.r * (1 - dest.a) * bg_color.r
+   - (1 - make_alpha(text_color.a, mask)
+      - (1 - make_alpha(text_color.a, mask)) * dest.a) * bg_color.r
+ = text_color.r * mask.r
+   + dest.r + (1 - dest.a) * bg_color.r
+   - text_color.a * mask.r * dest.r
+   - text_color.a * mask.r * (1 - dest.a) * bg_color.r
+   - ((1 - make_alpha(text_color.a, mask)) * 1
+      - (1 - make_alpha(text_color.a, mask)) * dest.a) * bg_color.r
+ = text_color.r * mask.r
+   + dest.r + (1 - dest.a) * bg_color.r
+   - text_color.a * mask.r * dest.r
+   - text_color.a * mask.r * (1 - dest.a) * bg_color.r
+   - ((1 - make_alpha(text_color.a, mask)) * (1 - dest.a)) * bg_color.r
+ = text_color.r * mask.r
+   + dest.r - text_color.a * mask.r * dest.r
+   + (1 - dest.a) * bg_color.r
+   - text_color.a * mask.r * (1 - dest.a) * bg_color.r
+   - (1 - make_alpha(text_color.a, mask)) * (1 - dest.a) * bg_color.r
+ = text_color.r * mask.r
+   + (1 - text_color.a * mask.r) * dest.r
+   + (1 - dest.a) * bg_color.r
+   - text_color.a * mask.r * (1 - dest.a) * bg_color.r
+   - (1 - make_alpha(text_color.a, mask)) * (1 - dest.a) * bg_color.r
+ = text_color.r * mask.r
+   + (1 - text_color.a * mask.r) * dest.r
+   + (1 - text_color.a * mask.r) * (1 - dest.a) * bg_color.r
+   - (1 - make_alpha(text_color.a, mask)) * (1 - dest.a) * bg_color.r
+ = text_color.r * mask.r
+   + (1 - text_color.a * mask.r) * dest.r
+   + ((1 - text_color.a * mask.r)
+      - (1 - make_alpha(text_color.a, mask))) * (1 - dest.a) * bg_color.r
+ = text_color.r * mask.r
+   + (1 - text_color.a * mask.r) * dest.r
+   + (1 - text_color.a * mask.r
+      - 1 + make_alpha(text_color.a, mask)) * (1 - dest.a) * bg_color.r
+ = text_color.r * mask.r
+   + (1 - text_color.a * mask.r) * dest.r
+   + (make_alpha(text_color.a, mask) - text_color.a * mask.r) * (1 - dest.a) * bg_color.r
+```
+
+We now have a term of the form `A + B + C`, with `A` and `B` being guaranteed to
+be between zero and one.
+
+We also want `C` to be between zero and one.
+We can use this restriction to help us decide on an implementation of `make_alpha`.
+
+If we define `make_alpha` as
+
+```glsl
+float make_alpha(text_color_a, mask) {
+  float max_rgb = max(max(mask.r, mask.g), mask.b);
+  return text_color_a * max_rgb;
+}
+```
+
+, then `(make_alpha(text_color.a, mask) - text_color.a * mask.r)` becomes
+`(text_color.a * max(max(mask.r, mask.g), mask.b) - text_color.a * mask.r)`, which is
+`text_color.a * (max(max(mask.r, mask.g), mask.b) - mask.r)`, and the subtraction will
+always yield something that's greater or equal to zero for r, g, and b,
+because we will subtract each channel from the maximum of the channels.
+
+Putting this all together, we have:
+
+```glsl
+vec4 subpixeltextblend_withbgcolor(vec4 text_color, vec4 mask, vec4 bg_color, vec4 dest) {
+  float max_rgb = max(max(mask.r, mask.g), mask.b);
+  vec4 result;
+  result.r = text_color.r * mask.r + (1 - text_color.a * mask.r) * dest.r +
+             text_color.a * bg_color.r * (max_rgb - mask.r) * (1 - dest.a);
+  result.g = text_color.g * mask.g + (1 - text_color.a * mask.g) * dest.g +
+             text_color.a * bg_color.g * (max_rgb - mask.g) * (1 - dest.a);
+  result.b = text_color.b * mask.b + (1 - text_color.a * mask.b) * dest.b +
+             text_color.a * bg_color.b * (max_rgb - mask.b) * (1 - dest.a);
+  result.a = text_color.a * max_rgb + (1 - text_color.a * max_rgb) * dest.a;
+  return result;
+}
+```
+
+This is the final form of this blend function. It satisfies all of the four constraints.
+
+### Implementing it with OpenGL
+
+Our color channel equations consist of three pieces:
+
+ - `text_color.r * mask.r`, which simply gets added to the rest.
+ - `(1 - text_color.a * mask.r) * dest.r`, a factor which gets multiplied with the destination color.
+ - `text_color.a * bg_color.r * (max_rgb - mask.r) * (1 - dest.a)`, a factor which gets multiplied
+   with "one minus destination alpha".
+
+We will need three passes. Each pass modifies the color channels in the destination.
+This means that the part that uses `dest.r` needs to be applied first.
+Then we can apply the part that uses `1 - dest.a`.
+(This means that the first pass needs to leave `dest.a` untouched.)
+And the final pass can apply the `result.a` equation and modify `dest.a`.
+
+```
+pub fn set_blend_mode_subpixel_with_bg_color_pass0(&self) {
+    self.gl.blend_func_separate(gl::ZERO, gl::ONE_MINUS_SRC_COLOR, gl::ZERO, gl::ONE);
+}
+pub fn set_blend_mode_subpixel_with_bg_color_pass1(&self) {
+    self.gl.blend_func_separate(gl::ONE_MINUS_DST_ALPHA, gl::ONE, gl::ZERO, gl::ONE);
+}
+pub fn set_blend_mode_subpixel_with_bg_color_pass2(&self) {
+    self.gl.blend_func_separate(gl::ONE, gl::ONE, gl::ONE, gl::ONE_MINUS_SRC_ALPHA);
+}
+
+Pass0:
+    oFragColor = vec4(text.color.a) * mask;
+Pass1:
+    oFragColor = vec4(text.color.a) * text.bg_color * (vec4(mask.a) - mask);
+Pass2:
+    oFragColor = text.color * mask;
+
+result_after_pass0.r = 0 * (text_color.a * mask.r) + (1 - text_color.a * mask.r) * dest.r
+result_after_pass0.a = 0 * (text_color.a * mask.a) + 1 * dest.a
+
+result_after_pass1.r = (1 - result_after_pass0.a) * (text_color.a * (mask.max_rgb - mask.r) * bg_color.r) + 1 * result_after_pass0.r
+result_after_pass1.a = 0 * (text_color.a * (mask.max_rgb - mask.a) * bg_color.a) + 1 * result_after_pass0.a
+
+result_after_pass2.r = 1 * (text_color.r * mask.r) + 1 * result_after_pass1.r
+result_after_pass2.a = 1 * (text_color.a * mask.max_rgb) + (1 - text_color.a * mask.max_rgb) * result_after_pass1.a
+```
+
+Instead of computing `max_rgb` in the shader, we can just require the font rasterization code to fill
+`mask.a` with the `max_rgb` value.
+