Conversions: FromLossy and TryFromLossy traits

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+- Feature Name: from-lossy
+- Start Date: 2018-06-22
+- RFC PR: (leave this empty)
+- Rust Issue: (leave this empty)
+
+# Summary
+[summary]: #summary
+
+Add `FromLossy`, `TryFromLossy` traits.
+
+Discuss the bigger picture of conversions and the `as` keyword.
+
+Specify that `From` implementations must not only be *safe*, but also *exact*
+(homomorphisms).
+
+# Motivation
+[motivation]: #motivation
+
+Currently many numeric conversions can only be done via the `as` keyword or third-party libraries. To [quote @scottmcm](https://internals.rust-lang.org/t/lossy-conversion-trait-as/7672/4?u=dhardy):
+
+> I’m strongly against [a trait] that’s just “whatever as does”, since I find it an unfortunate mess of conversions right now – some coercions, some extensions, some truncations, …
+
+This has several problems:
+
+- `as` can perform several different types of conversion (as noted) and is therefore error-prone
+- `as` conversions must be infallible, which allows no correct result for some
+  operations, for example when converting very large floats to integers (see
+  [this issue](https://github.com/rust-lang/rust/issues/10184))
+- since `as` is not a trait, it cannot be used as a bound in generic code (excepting via `num_traits::cast::AsPrimitive`)
+- these conversions are mostly primitive instructions and are very widely used, so requiring users to use another crate like `num` is unexpected
+
+Several types of conversion are possible:
+
+- safe, exact conversions (e.g. `u32` → `u64`) are handled by the `From` trait ([RFC 529](https://github.com/rust-lang/rfcs/pull/529))
+- fallible conversions (e.g. `u32` → `i8`) are handled by the `TryFrom` trait ([RFC 1542](https://github.com/rust-lang/rfcs/pull/1542))
+- lossy conversions (e.g. `i64` → `f32`)
+- lossy fallible conversions (e.g. `f64` → `u64`)
+- truncations on unsigned integers (e.g. `u64` → `u32`, dropping unused high bits)
+- sign-ignoring coercions/transmutations (e.g. `i8` → `u8`, `i64` → `i32`);
+  these can yield totally different values due to interpretation of the sign
+  bit (e.g. `3i32 << 14` is 49152=2^14+2^15; converting to `i16` yields -16384=2^14-2^15)
+- conversions between types with platform-dependent size (i.e. `usize` and `isize`)
+
+This RFC is principally concerned with lossy conversions, but considers other
+conversions for a broader picture.
+
+It is *my opinion* that we should cover all *common* conversions performed by
+the `as` keyword redundant with `std` library functionality, and *consider*
+deprecating `as` eventually. Whether this entails traits covering all the above
+conversions or methods on primitive types for some conversions, or even leaves
+no alternative to `transmute` in some cases is beyond the scope of this RFC.
+
+# Guide-level explanation
+[guide-level-explanation]: #guide-level-explanation
+
+Type conversions can be handled by the following traits:
+
+- `From` for infallible, exact conversions (e.g. widening, `u8` → `u16`)
+- `TryFrom` for fallible, exact conversions (e.g. narrowing, `u16` → `u8`, and signed, `i16` → `u16`)
+- `FromLossy` for infallible, lossy conversions (mostly concerning floating-point types, e.g. `u32` → `f32`)
+- `TryFromLossy` for fallible, inexact conversions (e.g. `f32` → `u32`)
+- `TruncateFrom` for truncations (e.g. `u16` → `u8` which drops high bits)
+
+# Reference-level explanation
+[reference-level-explanation]: #reference-level-explanation
+
+### `From` trait
+
+Tweak the documentation to clarify that `From` implementations should be
+monomorphisms with regards to the `==` operation:
+
+```rust
+/// Simple and safe type conversions in to Self. It is the reciprocal of Into.
+/// 
+/// Implementations are expected to be injective homomorphisms with respect to
+/// the `PartialEq` implementations of source and target types, as well as the
+/// inverse of `PartialEq`: that is, with target type `T` and any `x, y` of
+/// source type `S`, then `x == y` if and only if `T::from(x) == T::from(y)`.
+..
+pub trait From {
+```
+
+Note that monomorphisms are not isomorphisms since it is not required that the
+functions have injective inverse (i.e. we may define `From<u8> for u16` and an
+inverse `g` such that `g(u16::from(x)) == x` for `x: u8`, but it does not hold
+that `u16::from(g(y)) == y` for `y: u16`).
+
+Note also that for the purposes of this requirement we consider the `PartialEq`
+implementation (if available) and not binary value; for example we do not
+require that `0f32` and `-0f32` map to different binary values.
+
+Nightly rust currently has several implementations of `From` on SIMD types
+which should be removed (e.g. `f32x4` → `i8x4` (fallible) and `u64x4` → `f32x4`
+(lossy i.e. not injective)).
+
+### `FromLossy` trait
+
+Add `std::convert::FromLossy`:
+
+```rust
+/// A trait for conversions which are potentially inexact.
+/// 
+/// It is required that conversions do not fail, and required that results are
+/// either approximately equivalent or are an appropriate special value.
+/// For example, it is reasonable for an implementation converting `f64` to
+/// `f32` to lose precision and to return positive or negative infinity for
+/// out-of-range results, as well as to drop payloads from NaN values.
+pub trait FromLossy {
+    fn from_lossy(x: T) -> Self;
+}
+```
+
+Add implementations to `f64` from all of:
+
+- `u64, i64, u128, i128`
+
+and to `f32` from all of:
+
+- `f64, u32, i32, u64, i64, u128, i128`
+
+(Note: other integer → float conversions are already handled by `From` since
+they are loss-less. There is a question below about trait overlap.)
+
+These conversions should round to the nearest representable value, with ties to
+even (as is commonly used for floating-point operations).
+
+### `TryFromLossy` trait
+
+Add `std::convert::TryFromLossy`:
+
+```rust
+/// A trait for conversions which may fail and may be inexact.
+/// 
+/// Implementations should fail when the output type has no suitable
+/// corresponding or general-purpose value, and return an approximately
+/// equivalent value when possible.
+pub trait TryFromLossy {
+    type Error;
+    fn try_from_lossy(x: T) -> Result<Self, Self::Error>;
+}
+```
+
+Add implementations from all of:
+
+- `f32, f64`
+
+to all of:
+
+- `u8, u16, u32, u64, u128`
+- `i8, i16, i32, i64, i128`
+
+(Note: `f32` → `u128` is infallible (but still lossy) *if* the input is not
+negative, infinite or an NaN. So even though the output type has large enough
+range, this conversion trait is still applicable.)
+
+The implementations should fail on NaN, Inf, and values whose integer
+approximation is not representable. The integer approximation should be the
+value rounded towards zero. E.g.:
+
+- 1.6f32 → u32: 1
+- -1f32 → u32: error
+- -0.2f32 → u32: 0
+- 100_000f32 → u16: error
+
+# Related problems
+
+These problems are discussed in the search for a complete solution; however it
+is not currently proposed to solve them within this RFC.
+
+## Integer transmutations
+
+There are several types of transmutation, discussed here separately, although
+they are all transmutations.
+
+Note that there is less insentive against usage of `as` in these cases since the
+conversions do not preserve "value", although alternatives still have some use
+(e.g. to clarify that a conversion is a truncation).
+
+### Sign transmutations
+
+The conversions done by `as` between signed and unsigned types of the same size
+are simply transmutations (reinterpretations of the underlying bits), e.g.
+`0x80u8 as i8 == -128`.
+
+As these are not simple mathematical operations we could simply not provide any
+alternative to `as`, and suggest usage of `mem::transmute` if necessary.
+
+Alternatively, we could add `transmute_sign` methods to all primitive integer
+types, e.g.:
+```rust
+impl i32 {
+    ...
+    fn transmute_sign(self) -> u32 { ... }
+}
+```
+
+### Unsigned truncation
+
+We could add `std::convert::TruncateFrom`:
+
+```rust
+/// A trait for conversions which are truncate values by dropping unused high
+/// bits.
+/// 
+/// Note that this is distinct from other types of conversion since high bits
+/// are explicitly ignored and results are thus not numerically equivalent to
+/// input values.
+pub trait TruncateFrom {
+    fn truncate_from(x: T) -> Self;
+}
+```
+
+Add implementations for each unsigned integer type to each smaller unsigned
+integer type. (See below regarding signed types.)
+
+Note that we *could* suggest users drop unwanted high bits (via masks or
+bit-shifting) *then* use `TryFrom`, but this is a very unergonomic approach to
+what is a simple and commonly used operation.
+
+### Signed truncation
+
+Bitwise operations on signed integers can have "unintuitive" results. For example,
+```rust
+fn main() {
+    let x = 3i32 << 14;
+    println!("{}", x);
+    println!("{}", x as i16);
+}
+```
+prints:
+```
+49152
+-16384
+```
+since the 16th bit is later interpreted as a -2<sup>15</sup> in the Two's
+Complement representation, the numeric value on conversion to `i16` is quite
+different despite all the dropped bits being 0.
+
+Essentially, operations like `i32` → `i16` are *shorten-and-transmute*.
+
+Since these operations are not intuitive and not so widely useful, it may not
+be necessary to implement traits over them.
+
+Instead, we could suggest users implement signed truncations like this:
+`x.transmute_sign().truncate_into::<u16>().transmute_sign()`.
+
+## Platform-dependent types
+
+### isize / usize
+
+The `isize` and `usize` types have undefined size, though there appears to be
+an assumption that they are at least 16 bits (existing `From` implementations)
+and that they could be larger than 64 bits.
+[Discussion thread on internals](https://internals.rust-lang.org/t/numeric-into-should-not-require-everyone-to-support-16-bit-and-128-bit-usize/3609/7).
+[Discussion about `usize` → `u64` on users forum](https://users.rust-lang.org/t/cant-convert-usize-to-u64/6243).
+
+Checked integer conversions using `TryFrom` [are being reintroduced](https://github.com/rust-lang/rust/issues/49415).
+
+It is possible that unchecked conversions could be added, perhaps using
+`TruncateFrom` (or some other trait allowing both truncation and
+zero-extension).
+
+# Drawbacks
+[drawbacks]: #drawbacks
+
+This RFC proposes at least two new conversion traits and *still* doesn't solve
+all conversion problems (notably, platform-depedent types).
+
+It also makes numeric conversions complex, and *deliberately does so*. This does
+of course make the language more difficult to learn. Is there a simpler option?
+The current undefined behaviour when converting from floating-point types
+highlights the problems with a very generic solution.
+
+# Rationale and alternatives
+[alternatives]: #alternatives
+
+As mentioned, the `as` keyword has multiple problems, is not Rustic, and we
+should aim to provide users with *safer* alternatives.
+
+This RFC mostly approaches the problem via conversion traits, though as
+highlighted with the suggested `transmute_sign` method, traits are not the only
+approach. Since multiple types of conversion have multiple possible target types
+for each initial type, traits are however appropriate.
+
+There is also the possibility to leave this type of conversion trait to external
+libraries. Given how widely useful some of these conversions are (especially
+those to/from floating-point types), it would be desireable to have a `std`-lib
+solution.
+
+As a simplification, all integer transmutation conversions discussed above could
+be handled by a single trait, as a "safe" alternative to `mem::transmute`.
+
+# Prior art
+[prior-art]: #prior-art
+
+Rust-specific:
+
+- Conversion traits: [RFC 529](https://github.com/rust-lang/rfcs/pull/529)
+- `TryFrom` trait: [RFC 1542](https://github.com/rust-lang/rfcs/pull/1542)
+- internals thread: https://internals.rust-lang.org/t/lossy-conversion-trait-as/7672/4
+- internals thread: https://internals.rust-lang.org/t/numeric-into-should-not-require-everyone-to-support-16-bit-and-128-bit-usize/3609/7
+- users thread: https://users.rust-lang.org/t/cant-convert-usize-to-u64/6243
+- [`num_traits::cast::cast`](https://docs.rs/num-traits/0.2.4/num_traits/cast/fn.cast.html)
+
+C++ tries to add some degree of explicitness with [`static_cast`](https://en.cppreference.com/w/cpp/language/static_cast) etc., but with much less granularity than discussed here.
+
+# Unresolved questions
+[unresolved]: #unresolved-questions
+
+**Should we add `FromLossy` implementations from `usize`/`isize` types?**
+
+I suspect it is better to leave this decision until another RFC solves
+conversions on those types.
+
+**Should the infallible traits be special cases of the fallible traits?**
+
+@newpavlov points out that `From<T>` is equivalent to `TryFrom<T, Error=!>`
+and similarly for `FromLossy` and `TryFromLossy`. Should the former traits
+therefore just be aliases of the latter?
+
+Since `From` is already stable we cannot tweak it in backward incompatible ways,
+so the first question should be whether this change can be made without
+breakage.
+
+Probably it will be better to revisit this later (but there are implications on
+the next question).
+
+**Should we allow overlapping conversion implementations?**
+
+E.g. `FromLossy` could have a blanket implementation for all `From`
+implementations, and similarly for `TryFromLossy` and `TryFrom`.
+Similarly, `TryFrom` could be implemented for all `From` implementations,
+and the same for `TryFromLossy` for `FromLossy` implementations — except this
+would have rules to implement `TryFromLossy` for `From` implementations through
+both `FromLossy` and `TryFrom`, which is a type error (possibly solvable with
+specialization).
+
+This would be useful in generic code, but isn't strictly necessary.
+Unfortunately without these overlapping implementations generic code wanting to
+support types over multiple conversion traits requires a local re-implementation;
+worse, this is a technique used today, thus adding overlap today will cause
+breakage (although again specialization may solve this).
+
+**Should the new traits be added to the prelude?**
+
+If the ultimate aim is to make the `as` keyword redundant, then probably the answer is yes, but I will refer you to [this post by @alexcrichton](https://github.com/rust-lang/rfcs/pull/1542#issuecomment-211592332):
+
+> On question I'd have is whether we'd want to add these traits to the prelude? One of the reasons Into and From are so easy to use is because of their prominent placement and lack of a need to import them. I'd be of the opinion that these likely want to go into the prelude at some point, but we may not have all of the resolve changes in place to do that without unduly breaking crates. To me though one of the major motivations for this RFC is integer conversions, which I believe aren't gonna work out unless they're in the prelude, so I'd be fine adding it sooner rather than later if we can.
+
+**Should we eventually remove the `as` keyword?**
+
+I suspect it is too late to make such a change for Rust 2018, and do not see a
+need to rush this (at the very least, conversions involving `usize` and `isize`
+also need solving before removal of `as`).
+
+Removing this keyword would no doubt be contraversial and would further distance
+Rust from *convenient* languages like Python, so this is not a decision to make
+lightly.
+
+Alternatively we could simply rely on Clippy to lint against "unnecessary" uses of `as`.