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+Instancing
+==========
+
+The attribute descriptor lets the attribute unit compute the address of an
+attribute given the vertex and instance ID. Unfortunately, the way this works is
+rather complicated when instancing is enabled.
+
+To explain this, first we need to explain how compute and vertex threads are
+dispatched.  When a quad is dispatched, it receives a single, linear index.
+However, we need to translate that index into a (vertex id, instance id) pair.
+One option would be to do:
+
+.. math::
+   \text{vertex id} = \text{linear id} \% \text{num vertices}
+
+   \text{instance id} = \text{linear id} / \text{num vertices}
+
+but this involves a costly division and modulus by an arbitrary number.
+Instead, we could pad num_vertices. We dispatch padded_num_vertices *
+num_instances threads instead of num_vertices * num_instances, which results
+in some "extra" threads with vertex_id >= num_vertices, which we have to
+discard.  The more we pad num_vertices, the more "wasted" threads we
+dispatch, but the division is potentially easier.
+
+One straightforward choice is to pad num_vertices to the next power of two,
+which means that the division and modulus are just simple bit shifts and
+masking. But the actual algorithm is a bit more complicated. The thread
+dispatcher has special support for dividing by 3, 5, 7, and 9, in addition
+to dividing by a power of two. As a result, padded_num_vertices can be
+1, 3, 5, 7, or 9 times a power of two. This results in less wasted threads,
+since we need less padding.
+
+padded_num_vertices is picked by the hardware. The driver just specifies the
+actual number of vertices. Note that padded_num_vertices is a multiple of four
+(presumably because threads are dispatched in groups of 4). Also,
+padded_num_vertices is always at least one more than num_vertices, which seems
+like a quirk of the hardware. For larger num_vertices, the hardware uses the
+following algorithm: using the binary representation of num_vertices, we look at
+the most significant set bit as well as the following 3 bits. Let n be the
+number of bits after those 4 bits. Then we set padded_num_vertices according to
+the following table:
+
+==========  =======================
+high bits   padded_num_vertices
+==========  =======================
+1000		   :math:`9 \cdot 2^n`
+1001		   :math:`5 \cdot 2^{n+1}`
+101x		   :math:`3 \cdot 2^{n+2}`
+110x		   :math:`7 \cdot 2^{n+1}`
+111x		   :math:`2^{n+4}`
+==========  =======================
+
+For example, if num_vertices = 70 is passed to glDraw(), its binary
+representation is 1000110, so n = 3 and the high bits are 1000, and
+therefore padded_num_vertices = :math:`9 \cdot 2^3` = 72.
+
+The attribute unit works in terms of the original linear_id. if
+num_instances = 1, then they are the same, and everything is simple.
+However, with instancing things get more complicated. There are four
+possible modes, two of them we can group together:
+
+1. Use the linear_id directly. Only used when there is no instancing.
+
+2. Use the linear_id modulo a constant. This is used for per-vertex
+attributes with instancing enabled by making the constant equal
+padded_num_vertices. Because the modulus is always padded_num_vertices, this
+mode only supports a modulus that is a power of 2 times 1, 3, 5, 7, or 9.
+The shift field specifies the power of two, while the extra_flags field
+specifies the odd number. If shift = n and extra_flags = m, then the modulus
+is :math:`(2m + 1) \cdot 2^n`. As an example, if num_vertices = 70, then as
+computed above, padded_num_vertices = :math:`9 \cdot 2^3`, so we should set
+extra_flags = 4 and shift = 3. Note that we must exactly follow the hardware
+algorithm used to get padded_num_vertices in order to correctly implement
+per-vertex attributes.
+
+3. Divide the linear_id by a constant. In order to correctly implement
+instance divisors, we have to divide linear_id by padded_num_vertices times
+to user-specified divisor. So first we compute padded_num_vertices, again
+following the exact same algorithm that the hardware uses, then multiply it
+by the GL-level divisor to get the hardware-level divisor. This case is
+further divided into two more cases. If the hardware-level divisor is a
+power of two, then we just need to shift. The shift amount is specified by
+the shift field, so that the hardware-level divisor is just
+:math:`2^\text{shift}`.
+
+If it isn't a power of two, then we have to divide by an arbitrary integer.
+For that, we use the well-known technique of multiplying by an approximation
+of the inverse. The driver must compute the magic multiplier and shift
+amount, and then the hardware does the multiplication and shift. The
+hardware and driver also use the "round-down" optimization as described in
+https://ridiculousfish.com/files/faster_unsigned_division_by_constants.pdf.
+The hardware further assumes the multiplier is between :math:`2^{31}` and
+:math:`2^{32}`, so the high bit is implicitly set to 1 even though it is set
+to 0 by the driver -- presumably this simplifies the hardware multiplier a
+little. The hardware first multiplies linear_id by the multiplier and
+takes the high 32 bits, then applies the round-down correction if
+extra_flags = 1, then finally shifts right by the shift field.
+
+There are some differences between ridiculousfish's algorithm and the Mali
+hardware algorithm, which means that the reference code from ridiculousfish
+doesn't always produce the right constants. Mali does not use the pre-shift
+optimization, since that would make a hardware implementation slower (it
+would have to always do the pre-shift, multiply, and post-shift operations).
+It also forces the multiplier to be at least :math:`2^{31}`, which means
+that the exponent is entirely fixed, so there is no trial-and-error.
+Altogether, given the divisor d, the algorithm the driver must follow is:
+
+1. Set shift = :math:`\lfloor \log_2(d) \rfloor`.
+2. Compute :math:`m = \lceil 2^{shift + 32} / d \rceil` and :math:`e = 2^{shift + 32} % d`.
+3. If :math:`e <= 2^{shift}`, then we need to use the round-down algorithm. Set
+   magic_divisor = m - 1 and extra_flags = 1.  4. Otherwise, set magic_divisor =
+   m and extra_flags = 0.