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//! Port of LLVM's APFloat software floating-point implementation from the
//! following C++ sources (please update commit hash when backporting):
//! <https://github.com/llvm-mirror/llvm/tree/23efab2bbd424ed13495a420ad8641cb2c6c28f9>
//!
//! * `include/llvm/ADT/APFloat.h` -> `Float` and `FloatConvert` traits
//! * `lib/Support/APFloat.cpp` -> `ieee` and `ppc` modules
//! * `unittests/ADT/APFloatTest.cpp` -> `tests` directory
//!
//! The port contains no unsafe code, global state, or side-effects in general,
//! and the only allocations are in the conversion to/from decimal strings.
//!
//! Most of the API and the testcases are intact in some form or another,
//! with some ergonomic changes, such as idiomatic short names, returning
//! new values instead of mutating the receiver, and having separate method
//! variants that take a non-default rounding mode (with the suffix `_r`).
//! Comments have been preserved where possible, only slightly adapted.
//!
//! Instead of keeping a pointer to a configuration struct and inspecting it
//! dynamically on every operation, types (e.g., `ieee::Double`), traits
//! (e.g., `ieee::Semantics`) and associated constants are employed for
//! increased type safety and performance.
//!
//! On-heap bigints are replaced everywhere (except in decimal conversion),
//! with short arrays of `type Limb = u128` elements (instead of `u64`),
//! This allows fitting the largest supported significands in one integer
//! (`ieee::Quad` and `ppc::Fallback` use slightly less than 128 bits).
//! All of the functions in the `ieee::sig` module operate on slices.
//!
//! # Note
//!
//! This API is completely unstable and subject to change.
#![doc(html_root_url = "https://doc.rust-lang.org/nightly/")]
#![forbid(unsafe_code)]
#![deny(rust_2018_idioms)]
#![feature(nll)]
use std::cmp::Ordering;
use std::fmt;
use std::ops::{Neg, Add, Sub, Mul, Div, Rem};
use std::ops::{AddAssign, SubAssign, MulAssign, DivAssign, RemAssign};
use std::str::FromStr;
bitflags::bitflags! {
/// IEEE-754R 7: Default exception handling.
///
/// UNDERFLOW or OVERFLOW are always returned or-ed with INEXACT.
#[must_use]
pub struct Status: u8 {
const OK = 0x00;
const INVALID_OP = 0x01;
const DIV_BY_ZERO = 0x02;
const OVERFLOW = 0x04;
const UNDERFLOW = 0x08;
const INEXACT = 0x10;
}
}
#[must_use]
#[derive(Copy, Clone, PartialEq, Eq, PartialOrd, Ord, Debug)]
pub struct StatusAnd<T> {
pub status: Status,
pub value: T,
}
impl Status {
pub fn and<T>(self, value: T) -> StatusAnd<T> {
StatusAnd {
status: self,
value,
}
}
}
impl<T> StatusAnd<T> {
pub fn map<F: FnOnce(T) -> U, U>(self, f: F) -> StatusAnd<U> {
StatusAnd {
status: self.status,
value: f(self.value),
}
}
}
#[macro_export]
macro_rules! unpack {
($status:ident|=, $e:expr) => {
match $e {
$crate::StatusAnd { status, value } => {
$status |= status;
value
}
}
};
($status:ident=, $e:expr) => {
match $e {
$crate::StatusAnd { status, value } => {
$status = status;
value
}
}
}
}
/// Category of internally-represented number.
#[derive(Copy, Clone, PartialEq, Eq, Debug)]
pub enum Category {
Infinity,
NaN,
Normal,
Zero,
}
/// IEEE-754R 4.3: Rounding-direction attributes.
#[derive(Copy, Clone, PartialEq, Eq, Debug)]
pub enum Round {
NearestTiesToEven,
TowardPositive,
TowardNegative,
TowardZero,
NearestTiesToAway,
}
impl Neg for Round {
type Output = Round;
fn neg(self) -> Round {
match self {
Round::TowardPositive => Round::TowardNegative,
Round::TowardNegative => Round::TowardPositive,
Round::NearestTiesToEven | Round::TowardZero | Round::NearestTiesToAway => self,
}
}
}
/// A signed type to represent a floating point number's unbiased exponent.
pub type ExpInt = i16;
// \c ilogb error results.
pub const IEK_INF: ExpInt = ExpInt::max_value();
pub const IEK_NAN: ExpInt = ExpInt::min_value();
pub const IEK_ZERO: ExpInt = ExpInt::min_value() + 1;
#[derive(Copy, Clone, PartialEq, Eq, Debug)]
pub struct ParseError(pub &'static str);
/// A self-contained host- and target-independent arbitrary-precision
/// floating-point software implementation.
///
/// `apfloat` uses significand bignum integer arithmetic as provided by functions
/// in the `ieee::sig`.
///
/// Written for clarity rather than speed, in particular with a view to use in
/// the front-end of a cross compiler so that target arithmetic can be correctly
/// performed on the host. Performance should nonetheless be reasonable,
/// particularly for its intended use. It may be useful as a base
/// implementation for a run-time library during development of a faster
/// target-specific one.
///
/// All 5 rounding modes in the IEEE-754R draft are handled correctly for all
/// implemented operations. Currently implemented operations are add, subtract,
/// multiply, divide, fused-multiply-add, conversion-to-float,
/// conversion-to-integer and conversion-from-integer. New rounding modes
/// (e.g., away from zero) can be added with three or four lines of code.
///
/// Four formats are built-in: IEEE single precision, double precision,
/// quadruple precision, and x87 80-bit extended double (when operating with
/// full extended precision). Adding a new format that obeys IEEE semantics
/// only requires adding two lines of code: a declaration and definition of the
/// format.
///
/// All operations return the status of that operation as an exception bit-mask,
/// so multiple operations can be done consecutively with their results or-ed
/// together. The returned status can be useful for compiler diagnostics; e.g.,
/// inexact, underflow and overflow can be easily diagnosed on constant folding,
/// and compiler optimizers can determine what exceptions would be raised by
/// folding operations and optimize, or perhaps not optimize, accordingly.
///
/// At present, underflow tininess is detected after rounding; it should be
/// straight forward to add support for the before-rounding case too.
///
/// The library reads hexadecimal floating point numbers as per C99, and
/// correctly rounds if necessary according to the specified rounding mode.
/// Syntax is required to have been validated by the caller.
///
/// It also reads decimal floating point numbers and correctly rounds according
/// to the specified rounding mode.
///
/// Non-zero finite numbers are represented internally as a sign bit, a 16-bit
/// signed exponent, and the significand as an array of integer limbs. After
/// normalization of a number of precision P the exponent is within the range of
/// the format, and if the number is not denormal the P-th bit of the
/// significand is set as an explicit integer bit. For denormals the most
/// significant bit is shifted right so that the exponent is maintained at the
/// format's minimum, so that the smallest denormal has just the least
/// significant bit of the significand set. The sign of zeros and infinities
/// is significant; the exponent and significand of such numbers is not stored,
/// but has a known implicit (deterministic) value: 0 for the significands, 0
/// for zero exponent, all 1 bits for infinity exponent. For NaNs the sign and
/// significand are deterministic, although not really meaningful, and preserved
/// in non-conversion operations. The exponent is implicitly all 1 bits.
///
/// `apfloat` does not provide any exception handling beyond default exception
/// handling. We represent Signaling NaNs via IEEE-754R 2008 6.2.1 should clause
/// by encoding Signaling NaNs with the first bit of its trailing significand
/// as 0.
///
/// Future work
/// ===========
///
/// Some features that may or may not be worth adding:
///
/// Optional ability to detect underflow tininess before rounding.
///
/// New formats: x87 in single and double precision mode (IEEE apart from
/// extended exponent range) (hard).
///
/// New operations: sqrt, nexttoward.
///
pub trait Float
: Copy
+ Default
+ FromStr<Err = ParseError>
+ PartialOrd
+ fmt::Display
+ Neg<Output = Self>
+ AddAssign
+ SubAssign
+ MulAssign
+ DivAssign
+ RemAssign
+ Add<Output = StatusAnd<Self>>
+ Sub<Output = StatusAnd<Self>>
+ Mul<Output = StatusAnd<Self>>
+ Div<Output = StatusAnd<Self>>
+ Rem<Output = StatusAnd<Self>> {
/// Total number of bits in the in-memory format.
const BITS: usize;
/// Number of bits in the significand. This includes the integer bit.
const PRECISION: usize;
/// The largest E such that 2<sup>E</sup> is representable; this matches the
/// definition of IEEE 754.
const MAX_EXP: ExpInt;
/// The smallest E such that 2<sup>E</sup> is a normalized number; this
/// matches the definition of IEEE 754.
const MIN_EXP: ExpInt;
/// Positive Zero.
const ZERO: Self;
/// Positive Infinity.
const INFINITY: Self;
/// NaN (Not a Number).
// FIXME(eddyb) provide a default when qnan becomes const fn.
const NAN: Self;
/// Factory for QNaN values.
// FIXME(eddyb) should be const fn.
fn qnan(payload: Option<u128>) -> Self;
/// Factory for SNaN values.
// FIXME(eddyb) should be const fn.
fn snan(payload: Option<u128>) -> Self;
/// Largest finite number.
// FIXME(eddyb) should be const (but FloatPair::largest is nontrivial).
fn largest() -> Self;
/// Smallest (by magnitude) finite number.
/// Might be denormalized, which implies a relative loss of precision.
const SMALLEST: Self;
/// Smallest (by magnitude) normalized finite number.
// FIXME(eddyb) should be const (but FloatPair::smallest_normalized is nontrivial).
fn smallest_normalized() -> Self;
// Arithmetic
fn add_r(self, rhs: Self, round: Round) -> StatusAnd<Self>;
fn sub_r(self, rhs: Self, round: Round) -> StatusAnd<Self> {
self.add_r(-rhs, round)
}
fn mul_r(self, rhs: Self, round: Round) -> StatusAnd<Self>;
fn mul_add_r(self, multiplicand: Self, addend: Self, round: Round) -> StatusAnd<Self>;
fn mul_add(self, multiplicand: Self, addend: Self) -> StatusAnd<Self> {
self.mul_add_r(multiplicand, addend, Round::NearestTiesToEven)
}
fn div_r(self, rhs: Self, round: Round) -> StatusAnd<Self>;
/// IEEE remainder.
// This is not currently correct in all cases.
fn ieee_rem(self, rhs: Self) -> StatusAnd<Self> {
let mut v = self;
let status;
v = unpack!(status=, v / rhs);
if status == Status::DIV_BY_ZERO {
return status.and(self);
}
assert!(Self::PRECISION < 128);
let status;
let x = unpack!(status=, v.to_i128_r(128, Round::NearestTiesToEven, &mut false));
if status == Status::INVALID_OP {
return status.and(self);
}
let status;
let mut v = unpack!(status=, Self::from_i128(x));
assert_eq!(status, Status::OK); // should always work
let status;
v = unpack!(status=, v * rhs);
assert_eq!(status - Status::INEXACT, Status::OK); // should not overflow or underflow
let status;
v = unpack!(status=, self - v);
assert_eq!(status - Status::INEXACT, Status::OK); // likewise
if v.is_zero() {
status.and(v.copy_sign(self)) // IEEE754 requires this
} else {
status.and(v)
}
}
/// C fmod, or llvm frem.
fn c_fmod(self, rhs: Self) -> StatusAnd<Self>;
fn round_to_integral(self, round: Round) -> StatusAnd<Self>;
/// IEEE-754R 2008 5.3.1: nextUp.
fn next_up(self) -> StatusAnd<Self>;
/// IEEE-754R 2008 5.3.1: nextDown.
///
/// *NOTE* since nextDown(x) = -nextUp(-x), we only implement nextUp with
/// appropriate sign switching before/after the computation.
fn next_down(self) -> StatusAnd<Self> {
(-self).next_up().map(|r| -r)
}
fn abs(self) -> Self {
if self.is_negative() { -self } else { self }
}
fn copy_sign(self, rhs: Self) -> Self {
if self.is_negative() != rhs.is_negative() {
-self
} else {
self
}
}
// Conversions
fn from_bits(input: u128) -> Self;
fn from_i128_r(input: i128, round: Round) -> StatusAnd<Self> {
if input < 0 {
Self::from_u128_r(input.wrapping_neg() as u128, -round).map(|r| -r)
} else {
Self::from_u128_r(input as u128, round)
}
}
fn from_i128(input: i128) -> StatusAnd<Self> {
Self::from_i128_r(input, Round::NearestTiesToEven)
}
fn from_u128_r(input: u128, round: Round) -> StatusAnd<Self>;
fn from_u128(input: u128) -> StatusAnd<Self> {
Self::from_u128_r(input, Round::NearestTiesToEven)
}
fn from_str_r(s: &str, round: Round) -> Result<StatusAnd<Self>, ParseError>;
fn to_bits(self) -> u128;
/// Converts a floating point number to an integer according to the
/// rounding mode. In case of an invalid operation exception,
/// deterministic values are returned, namely zero for NaNs and the
/// minimal or maximal value respectively for underflow or overflow.
/// If the rounded value is in range but the floating point number is
/// not the exact integer, the C standard doesn't require an inexact
/// exception to be raised. IEEE-854 does require it so we do that.
///
/// Note that for conversions to integer type the C standard requires
/// round-to-zero to always be used.
///
/// The *is_exact output tells whether the result is exact, in the sense
/// that converting it back to the original floating point type produces
/// the original value. This is almost equivalent to `result == Status::OK`,
/// except for negative zeroes.
fn to_i128_r(self, width: usize, round: Round, is_exact: &mut bool) -> StatusAnd<i128> {
let status;
if self.is_negative() {
if self.is_zero() {
// Negative zero can't be represented as an int.
*is_exact = false;
}
let r = unpack!(status=, (-self).to_u128_r(width, -round, is_exact));
// Check for values that don't fit in the signed integer.
if r > (1 << (width - 1)) {
// Return the most negative integer for the given width.
*is_exact = false;
Status::INVALID_OP.and(-1 << (width - 1))
} else {
status.and(r.wrapping_neg() as i128)
}
} else {
// Positive case is simpler, can pretend it's a smaller unsigned
// integer, and `to_u128` will take care of all the edge cases.
self.to_u128_r(width - 1, round, is_exact).map(
|r| r as i128,
)
}
}
fn to_i128(self, width: usize) -> StatusAnd<i128> {
self.to_i128_r(width, Round::TowardZero, &mut true)
}
fn to_u128_r(self, width: usize, round: Round, is_exact: &mut bool) -> StatusAnd<u128>;
fn to_u128(self, width: usize) -> StatusAnd<u128> {
self.to_u128_r(width, Round::TowardZero, &mut true)
}
fn cmp_abs_normal(self, rhs: Self) -> Ordering;
/// Bitwise comparison for equality (QNaNs compare equal, 0!=-0).
fn bitwise_eq(self, rhs: Self) -> bool;
// IEEE-754R 5.7.2 General operations.
/// Implements IEEE minNum semantics. Returns the smaller of the 2 arguments if
/// both are not NaN. If either argument is a NaN, returns the other argument.
fn min(self, other: Self) -> Self {
if self.is_nan() {
other
} else if other.is_nan() {
self
} else if other.partial_cmp(&self) == Some(Ordering::Less) {
other
} else {
self
}
}
/// Implements IEEE maxNum semantics. Returns the larger of the 2 arguments if
/// both are not NaN. If either argument is a NaN, returns the other argument.
fn max(self, other: Self) -> Self {
if self.is_nan() {
other
} else if other.is_nan() {
self
} else if self.partial_cmp(&other) == Some(Ordering::Less) {
other
} else {
self
}
}
/// IEEE-754R isSignMinus: Returns whether the current value is
/// negative.
///
/// This applies to zeros and NaNs as well.
fn is_negative(self) -> bool;
/// IEEE-754R isNormal: Returns whether the current value is normal.
///
/// This implies that the current value of the float is not zero, subnormal,
/// infinite, or NaN following the definition of normality from IEEE-754R.
fn is_normal(self) -> bool {
!self.is_denormal() && self.is_finite_non_zero()
}
/// Returns `true` if the current value is zero, subnormal, or
/// normal.
///
/// This means that the value is not infinite or NaN.
fn is_finite(self) -> bool {
!self.is_nan() && !self.is_infinite()
}
/// Returns `true` if the float is plus or minus zero.
fn is_zero(self) -> bool {
self.category() == Category::Zero
}
/// IEEE-754R isSubnormal(): Returns whether the float is a
/// denormal.
fn is_denormal(self) -> bool;
/// IEEE-754R isInfinite(): Returns whether the float is infinity.
fn is_infinite(self) -> bool {
self.category() == Category::Infinity
}
/// Returns `true` if the float is a quiet or signaling NaN.
fn is_nan(self) -> bool {
self.category() == Category::NaN
}
/// Returns `true` if the float is a signaling NaN.
fn is_signaling(self) -> bool;
// Simple Queries
fn category(self) -> Category;
fn is_non_zero(self) -> bool {
!self.is_zero()
}
fn is_finite_non_zero(self) -> bool {
self.is_finite() && !self.is_zero()
}
fn is_pos_zero(self) -> bool {
self.is_zero() && !self.is_negative()
}
fn is_neg_zero(self) -> bool {
self.is_zero() && self.is_negative()
}
/// Returns `true` if the number has the smallest possible non-zero
/// magnitude in the current semantics.
fn is_smallest(self) -> bool {
Self::SMALLEST.copy_sign(self).bitwise_eq(self)
}
/// Returns `true` if the number has the largest possible finite
/// magnitude in the current semantics.
fn is_largest(self) -> bool {
Self::largest().copy_sign(self).bitwise_eq(self)
}
/// Returns `true` if the number is an exact integer.
fn is_integer(self) -> bool {
// This could be made more efficient; I'm going for obviously correct.
if !self.is_finite() {
return false;
}
self.round_to_integral(Round::TowardZero).value.bitwise_eq(
self,
)
}
/// If this value has an exact multiplicative inverse, return it.
fn get_exact_inverse(self) -> Option<Self>;
/// Returns the exponent of the internal representation of the Float.
///
/// Because the radix of Float is 2, this is equivalent to floor(log2(x)).
/// For special Float values, this returns special error codes:
///
/// NaN -> \c IEK_NAN
/// 0 -> \c IEK_ZERO
/// Inf -> \c IEK_INF
///
fn ilogb(self) -> ExpInt;
/// Returns: self * 2<sup>exp</sup> for integral exponents.
fn scalbn_r(self, exp: ExpInt, round: Round) -> Self;
fn scalbn(self, exp: ExpInt) -> Self {
self.scalbn_r(exp, Round::NearestTiesToEven)
}
/// Equivalent of C standard library function.
///
/// While the C standard says exp is an unspecified value for infinity and nan,
/// this returns INT_MAX for infinities, and INT_MIN for NaNs (see `ilogb`).
fn frexp_r(self, exp: &mut ExpInt, round: Round) -> Self;
fn frexp(self, exp: &mut ExpInt) -> Self {
self.frexp_r(exp, Round::NearestTiesToEven)
}
}
pub trait FloatConvert<T: Float>: Float {
/// Converts a value of one floating point type to another.
/// The return value corresponds to the IEEE754 exceptions. *loses_info
/// records whether the transformation lost information, i.e., whether
/// converting the result back to the original type will produce the
/// original value (this is almost the same as return `value == Status::OK`,
/// but there are edge cases where this is not so).
fn convert_r(self, round: Round, loses_info: &mut bool) -> StatusAnd<T>;
fn convert(self, loses_info: &mut bool) -> StatusAnd<T> {
self.convert_r(Round::NearestTiesToEven, loses_info)
}
}
macro_rules! float_common_impls {
($ty:ident<$t:tt>) => {
impl<$t> Default for $ty<$t> where Self: Float {
fn default() -> Self {
Self::ZERO
}
}
impl<$t> ::std::str::FromStr for $ty<$t> where Self: Float {
type Err = ParseError;
fn from_str(s: &str) -> Result<Self, ParseError> {
Self::from_str_r(s, Round::NearestTiesToEven).map(|x| x.value)
}
}
// Rounding ties to the nearest even, by default.
impl<$t> ::std::ops::Add for $ty<$t> where Self: Float {
type Output = StatusAnd<Self>;
fn add(self, rhs: Self) -> StatusAnd<Self> {
self.add_r(rhs, Round::NearestTiesToEven)
}
}
impl<$t> ::std::ops::Sub for $ty<$t> where Self: Float {
type Output = StatusAnd<Self>;
fn sub(self, rhs: Self) -> StatusAnd<Self> {
self.sub_r(rhs, Round::NearestTiesToEven)
}
}
impl<$t> ::std::ops::Mul for $ty<$t> where Self: Float {
type Output = StatusAnd<Self>;
fn mul(self, rhs: Self) -> StatusAnd<Self> {
self.mul_r(rhs, Round::NearestTiesToEven)
}
}
impl<$t> ::std::ops::Div for $ty<$t> where Self: Float {
type Output = StatusAnd<Self>;
fn div(self, rhs: Self) -> StatusAnd<Self> {
self.div_r(rhs, Round::NearestTiesToEven)
}
}
impl<$t> ::std::ops::Rem for $ty<$t> where Self: Float {
type Output = StatusAnd<Self>;
fn rem(self, rhs: Self) -> StatusAnd<Self> {
self.c_fmod(rhs)
}
}
impl<$t> ::std::ops::AddAssign for $ty<$t> where Self: Float {
fn add_assign(&mut self, rhs: Self) {
*self = (*self + rhs).value;
}
}
impl<$t> ::std::ops::SubAssign for $ty<$t> where Self: Float {
fn sub_assign(&mut self, rhs: Self) {
*self = (*self - rhs).value;
}
}
impl<$t> ::std::ops::MulAssign for $ty<$t> where Self: Float {
fn mul_assign(&mut self, rhs: Self) {
*self = (*self * rhs).value;
}
}
impl<$t> ::std::ops::DivAssign for $ty<$t> where Self: Float {
fn div_assign(&mut self, rhs: Self) {
*self = (*self / rhs).value;
}
}
impl<$t> ::std::ops::RemAssign for $ty<$t> where Self: Float {
fn rem_assign(&mut self, rhs: Self) {
*self = (*self % rhs).value;
}
}
}
}
pub mod ieee;
pub mod ppc;
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