Save an instruction in EntityHasher (#10648)

# Objective

Keep essentially the same structure of `EntityHasher` from #9903, but
rephrase the multiplication slightly to save an instruction.

cc @superdump 
Discord thread:
https://discord.com/channels/691052431525675048/1172033156845674507/1174969772522356756

## Solution

Today, the hash is
```rust
        self.hash = i | (i.wrapping_mul(FRAC_U64MAX_PI) << 32);
```
with `i` being `(generation << 32) | index`.

Expanding things out, we get
```rust
i | ( (i * CONST) << 32 )
= (generation << 32) | index | ((((generation << 32) | index) * CONST) << 32)
= (generation << 32) | index | ((index * CONST) << 32)  // because the generation overflowed
= (index * CONST | generation) << 32 | index
```

What if we do the same thing, but with `+` instead of `|`? That's almost
the same thing, except that it has carries, which are actually often
better in a hash function anyway, since it doesn't saturate. (`|` can be
dangerous, since once something becomes `-1` it'll stay that, and
there's no mixing available.)

```rust
(index * CONST + generation) << 32 + index
= (CONST << 32 + 1) * index + generation << 32
= (CONST << 32 + 1) * index + (WHATEVER << 32 + generation) << 32 // because the extra overflows and thus can be anything
= (CONST << 32 + 1) * index + ((CONST * generation) << 32 + generation) << 32 // pick "whatever" to be something convenient
= (CONST << 32 + 1) * index + ((CONST << 32 + 1) * generation) << 32
= (CONST << 32 + 1) * index +((CONST << 32 + 1) * (generation << 32)
= (CONST << 32 + 1) * (index + generation << 32)
= (CONST << 32 + 1) * (generation << 32 | index)
= (CONST << 32 + 1) * i
```

So we can do essentially the same thing using a single multiplication
instead of doing multiply-shift-or.

LLVM was already smart enough to merge the shifting into a
multiplication, but this saves the extra `or`:

![image](https://github.com/bevyengine/bevy/assets/18526288/d9396614-2326-4730-abbe-4908c01b5ace)
<https://rust.godbolt.org/z/MEvbz4eo4>

It's a very small change, and often will disappear in load latency
anyway, but it's a couple percent faster in lookups:

![image](https://github.com/bevyengine/bevy/assets/18526288/c365ec85-6adc-4f6d-8fa6-a65146f55a75)

(There was more of an improvement here before #10558, but with `to_bits`
being a single `qword` load now, keeping things mostly as it is turned
out to be better than the bigger changes I'd tried in #10605.)

---

## Changelog

(Probably skip it)

## Migration Guide

(none needed)

crates/bevy_ecs/src/entity/mod.rs

@@ -998,4 +998,36 @@ mod tests {
         assert!(Entity::new(2, 2) > Entity::new(1, 2));
         assert!(Entity::new(2, 2) >= Entity::new(1, 2));
     }
+
+    // Feel free to change this test if needed, but it seemed like an important
+    // part of the best-case performance changes in PR#9903.
+    #[test]
+    fn entity_hash_keeps_similar_ids_together() {
+        use std::hash::BuildHasher;
+        let hash = bevy_utils::EntityHash;
+
+        let first_id = 0xC0FFEE << 8;
+        let first_hash = hash.hash_one(Entity::from_raw(first_id));
+
+        for i in 1..=255 {
+            let id = first_id + i;
+            let hash = hash.hash_one(Entity::from_raw(id));
+            assert_eq!(hash.wrapping_sub(first_hash) as u32, i);
+        }
+    }
+
+    #[test]
+    fn entity_hash_id_bitflip_affects_high_7_bits() {
+        use std::hash::BuildHasher;
+        let hash = bevy_utils::EntityHash;
+
+        let first_id = 0xC0FFEE;
+        let first_hash = hash.hash_one(Entity::from_raw(first_id)) >> 57;
+
+        for bit in 0..u32::BITS {
+            let id = first_id ^ (1 << bit);
+            let hash = hash.hash_one(Entity::from_raw(id)) >> 57;
+            assert_ne!(hash, first_hash);
+        }
+    }
 }

crates/bevy_utils/src/lib.rs

@@ -267,29 +267,54 @@ impl BuildHasher for EntityHash {
 /// A very fast hash that is only designed to work on generational indices
 /// like `Entity`. It will panic if attempting to hash a type containing
 /// non-u64 fields.
+///
+/// This is heavily optimized for typical cases, where you have mostly live
+/// entities, and works particularly well for contiguous indices.
+///
+/// If you have an unusual case -- say all your indices are multiples of 256
+/// or most of the entities are dead generations -- then you might want also to
+/// try [`AHasher`] for a slower hash computation but fewer lookup conflicts.
 #[derive(Debug, Default)]
 pub struct EntityHasher {
     hash: u64,
 }
 
-// This value comes from rustc-hash (also known as FxHasher) which in turn got
-// it from Firefox. It is something like `u64::MAX / N` for an N that gives a
-// value close to π and works well for distributing bits for hashing when using
-// with a wrapping multiplication.
-const FRAC_U64MAX_PI: u64 = 0x517cc1b727220a95;
-
 impl Hasher for EntityHasher {
     fn write(&mut self, _bytes: &[u8]) {
         panic!("can only hash u64 using EntityHasher");
     }
 
     #[inline]
-    fn write_u64(&mut self, i: u64) {
-        // Apparently hashbrown's hashmap uses the upper 7 bits for some SIMD
-        // optimisation that uses those bits for binning. This hash function
-        // was faster than i | (i << (64 - 7)) in the worst cases, and was
-        // faster than PassHasher for all cases tested.
-        self.hash = i | (i.wrapping_mul(FRAC_U64MAX_PI) << 32);
+    fn write_u64(&mut self, bits: u64) {
+        // SwissTable (and thus `hashbrown`) cares about two things from the hash:
+        // - H1: low bits (masked by `2ⁿ-1`) to pick the slot in which to store the item
+        // - H2: high 7 bits are used to SIMD optimize hash collision probing
+        // For more see <https://abseil.io/about/design/swisstables#metadata-layout>
+
+        // This hash function assumes that the entity ids are still well-distributed,
+        // so for H1 leaves the entity id alone in the low bits so that id locality
+        // will also give memory locality for things spawned together.
+        // For H2, take advantage of the fact that while multiplication doesn't
+        // spread entropy to the low bits, it's incredibly good at spreading it
+        // upward, which is exactly where we need it the most.
+
+        // While this does include the generation in the output, it doesn't do so
+        // *usefully*.  H1 won't care until you have over 3 billion entities in
+        // the table, and H2 won't care until something hits generation 33 million.
+        // Thus the comment suggesting that this is best for live entities,
+        // where there won't be generation conflicts where it would matter.
+
+        // The high 32 bits of this are ⅟φ for Fibonacci hashing.  That works
+        // particularly well for hashing for the same reason as described in
+        // <https://extremelearning.com.au/unreasonable-effectiveness-of-quasirandom-sequences/>
+        // It loses no information because it has a modular inverse.
+        // (Specifically, `0x144c_bc89_u32 * 0x9e37_79b9_u32 == 1`.)
+        //
+        // The low 32 bits make that part of the just product a pass-through.
+        const UPPER_PHI: u64 = 0x9e37_79b9_0000_0001;
+
+        // This is `(MAGIC * index + generation) << 32 + index`, in a single instruction.
+        self.hash = bits.wrapping_mul(UPPER_PHI);
     }
 
     #[inline]