1 // Copyright 2017 The Rust Project Developers. See the COPYRIGHT
2 // file at the top-level directory of this distribution and at
3 // http://rust-lang.org/COPYRIGHT.
5 // Licensed under the Apache License, Version 2.0 <LICENSE-APACHE or
6 // http://www.apache.org/licenses/LICENSE-2.0> or the MIT license
7 // <LICENSE-MIT or http://opensource.org/licenses/MIT>, at your
8 // option. This file may not be copied, modified, or distributed
9 // except according to those terms.
11 //! Port of LLVM's APFloat software floating-point implementation from the
12 //! following C++ sources (please update commit hash when backporting):
13 //! <https://github.com/llvm-mirror/llvm/tree/23efab2bbd424ed13495a420ad8641cb2c6c28f9>
15 //! * `include/llvm/ADT/APFloat.h` -> `Float` and `FloatConvert` traits
16 //! * `lib/Support/APFloat.cpp` -> `ieee` and `ppc` modules
17 //! * `unittests/ADT/APFloatTest.cpp` -> `tests` directory
19 //! The port contains no unsafe code, global state, or side-effects in general,
20 //! and the only allocations are in the conversion to/from decimal strings.
22 //! Most of the API and the testcases are intact in some form or another,
23 //! with some ergonomic changes, such as idiomatic short names, returning
24 //! new values instead of mutating the receiver, and having separate method
25 //! variants that take a non-default rounding mode (with the suffix `_r`).
26 //! Comments have been preserved where possible, only slightly adapted.
28 //! Instead of keeping a pointer to a configuration struct and inspecting it
29 //! dynamically on every operation, types (e.g., `ieee::Double`), traits
30 //! (e.g., `ieee::Semantics`) and associated constants are employed for
31 //! increased type safety and performance.
33 //! On-heap bigints are replaced everywhere (except in decimal conversion),
34 //! with short arrays of `type Limb = u128` elements (instead of `u64`),
35 //! This allows fitting the largest supported significands in one integer
36 //! (`ieee::Quad` and `ppc::Fallback` use slightly less than 128 bits).
37 //! All of the functions in the `ieee::sig` module operate on slices.
41 //! This API is completely unstable and subject to change.
43 #![doc(html_logo_url = "https://www.rust-lang.org/logos/rust-logo-128x128-blk-v2.png",
44 html_favicon_url = "https://doc.rust-lang.org/favicon.ico",
45 html_root_url = "https://doc.rust-lang.org/nightly/")]
46 #![forbid(unsafe_code)]
50 // See librustc_cratesio_shim/Cargo.toml for a comment explaining this.
51 #[allow(unused_extern_crates)]
52 extern crate rustc_cratesio_shim;
55 extern crate bitflags;
56 extern crate smallvec;
58 use std::cmp::Ordering;
60 use std::ops::{Neg, Add, Sub, Mul, Div, Rem};
61 use std::ops::{AddAssign, SubAssign, MulAssign, DivAssign, RemAssign};
62 use std::str::FromStr;
65 /// IEEE-754R 7: Default exception handling.
67 /// UNDERFLOW or OVERFLOW are always returned or-ed with INEXACT.
69 pub struct Status: u8 {
71 const INVALID_OP = 0x01;
72 const DIV_BY_ZERO = 0x02;
73 const OVERFLOW = 0x04;
74 const UNDERFLOW = 0x08;
80 #[derive(Copy, Clone, PartialEq, Eq, PartialOrd, Ord, Debug)]
81 pub struct StatusAnd<T> {
87 pub fn and<T>(self, value: T) -> StatusAnd<T> {
95 impl<T> StatusAnd<T> {
96 pub fn map<F: FnOnce(T) -> U, U>(self, f: F) -> StatusAnd<U> {
105 macro_rules! unpack {
106 ($status:ident|=, $e:expr) => {
108 $crate::StatusAnd { status, value } => {
114 ($status:ident=, $e:expr) => {
116 $crate::StatusAnd { status, value } => {
124 /// Category of internally-represented number.
125 #[derive(Copy, Clone, PartialEq, Eq, Debug)]
133 /// IEEE-754R 4.3: Rounding-direction attributes.
134 #[derive(Copy, Clone, PartialEq, Eq, Debug)]
145 fn neg(self) -> Round {
147 Round::TowardPositive => Round::TowardNegative,
148 Round::TowardNegative => Round::TowardPositive,
149 Round::NearestTiesToEven | Round::TowardZero | Round::NearestTiesToAway => self,
154 /// A signed type to represent a floating point number's unbiased exponent.
155 pub type ExpInt = i16;
157 // \c ilogb error results.
158 pub const IEK_INF: ExpInt = ExpInt::max_value();
159 pub const IEK_NAN: ExpInt = ExpInt::min_value();
160 pub const IEK_ZERO: ExpInt = ExpInt::min_value() + 1;
162 #[derive(Copy, Clone, PartialEq, Eq, Debug)]
163 pub struct ParseError(pub &'static str);
165 /// A self-contained host- and target-independent arbitrary-precision
166 /// floating-point software implementation.
168 /// `apfloat` uses significand bignum integer arithmetic as provided by functions
169 /// in the `ieee::sig`.
171 /// Written for clarity rather than speed, in particular with a view to use in
172 /// the front-end of a cross compiler so that target arithmetic can be correctly
173 /// performed on the host. Performance should nonetheless be reasonable,
174 /// particularly for its intended use. It may be useful as a base
175 /// implementation for a run-time library during development of a faster
176 /// target-specific one.
178 /// All 5 rounding modes in the IEEE-754R draft are handled correctly for all
179 /// implemented operations. Currently implemented operations are add, subtract,
180 /// multiply, divide, fused-multiply-add, conversion-to-float,
181 /// conversion-to-integer and conversion-from-integer. New rounding modes
182 /// (e.g., away from zero) can be added with three or four lines of code.
184 /// Four formats are built-in: IEEE single precision, double precision,
185 /// quadruple precision, and x87 80-bit extended double (when operating with
186 /// full extended precision). Adding a new format that obeys IEEE semantics
187 /// only requires adding two lines of code: a declaration and definition of the
190 /// All operations return the status of that operation as an exception bit-mask,
191 /// so multiple operations can be done consecutively with their results or-ed
192 /// together. The returned status can be useful for compiler diagnostics; e.g.,
193 /// inexact, underflow and overflow can be easily diagnosed on constant folding,
194 /// and compiler optimizers can determine what exceptions would be raised by
195 /// folding operations and optimize, or perhaps not optimize, accordingly.
197 /// At present, underflow tininess is detected after rounding; it should be
198 /// straight forward to add support for the before-rounding case too.
200 /// The library reads hexadecimal floating point numbers as per C99, and
201 /// correctly rounds if necessary according to the specified rounding mode.
202 /// Syntax is required to have been validated by the caller.
204 /// It also reads decimal floating point numbers and correctly rounds according
205 /// to the specified rounding mode.
207 /// Non-zero finite numbers are represented internally as a sign bit, a 16-bit
208 /// signed exponent, and the significand as an array of integer limbs. After
209 /// normalization of a number of precision P the exponent is within the range of
210 /// the format, and if the number is not denormal the P-th bit of the
211 /// significand is set as an explicit integer bit. For denormals the most
212 /// significant bit is shifted right so that the exponent is maintained at the
213 /// format's minimum, so that the smallest denormal has just the least
214 /// significant bit of the significand set. The sign of zeros and infinities
215 /// is significant; the exponent and significand of such numbers is not stored,
216 /// but has a known implicit (deterministic) value: 0 for the significands, 0
217 /// for zero exponent, all 1 bits for infinity exponent. For NaNs the sign and
218 /// significand are deterministic, although not really meaningful, and preserved
219 /// in non-conversion operations. The exponent is implicitly all 1 bits.
221 /// `apfloat` does not provide any exception handling beyond default exception
222 /// handling. We represent Signaling NaNs via IEEE-754R 2008 6.2.1 should clause
223 /// by encoding Signaling NaNs with the first bit of its trailing significand
229 /// Some features that may or may not be worth adding:
231 /// Optional ability to detect underflow tininess before rounding.
233 /// New formats: x87 in single and double precision mode (IEEE apart from
234 /// extended exponent range) (hard).
236 /// New operations: sqrt, nexttoward.
241 + FromStr<Err = ParseError>
250 + Add<Output = StatusAnd<Self>>
251 + Sub<Output = StatusAnd<Self>>
252 + Mul<Output = StatusAnd<Self>>
253 + Div<Output = StatusAnd<Self>>
254 + Rem<Output = StatusAnd<Self>> {
255 /// Total number of bits in the in-memory format.
258 /// Number of bits in the significand. This includes the integer bit.
259 const PRECISION: usize;
261 /// The largest E such that 2<sup>E</sup> is representable; this matches the
262 /// definition of IEEE 754.
263 const MAX_EXP: ExpInt;
265 /// The smallest E such that 2<sup>E</sup> is a normalized number; this
266 /// matches the definition of IEEE 754.
267 const MIN_EXP: ExpInt;
272 /// Positive Infinity.
273 const INFINITY: Self;
275 /// NaN (Not a Number).
276 // FIXME(eddyb) provide a default when qnan becomes const fn.
279 /// Factory for QNaN values.
280 // FIXME(eddyb) should be const fn.
281 fn qnan(payload: Option<u128>) -> Self;
283 /// Factory for SNaN values.
284 // FIXME(eddyb) should be const fn.
285 fn snan(payload: Option<u128>) -> Self;
287 /// Largest finite number.
288 // FIXME(eddyb) should be const (but FloatPair::largest is nontrivial).
289 fn largest() -> Self;
291 /// Smallest (by magnitude) finite number.
292 /// Might be denormalized, which implies a relative loss of precision.
293 const SMALLEST: Self;
295 /// Smallest (by magnitude) normalized finite number.
296 // FIXME(eddyb) should be const (but FloatPair::smallest_normalized is nontrivial).
297 fn smallest_normalized() -> Self;
301 fn add_r(self, rhs: Self, round: Round) -> StatusAnd<Self>;
302 fn sub_r(self, rhs: Self, round: Round) -> StatusAnd<Self> {
303 self.add_r(-rhs, round)
305 fn mul_r(self, rhs: Self, round: Round) -> StatusAnd<Self>;
306 fn mul_add_r(self, multiplicand: Self, addend: Self, round: Round) -> StatusAnd<Self>;
307 fn mul_add(self, multiplicand: Self, addend: Self) -> StatusAnd<Self> {
308 self.mul_add_r(multiplicand, addend, Round::NearestTiesToEven)
310 fn div_r(self, rhs: Self, round: Round) -> StatusAnd<Self>;
312 // This is not currently correct in all cases.
313 fn ieee_rem(self, rhs: Self) -> StatusAnd<Self> {
317 v = unpack!(status=, v / rhs);
318 if status == Status::DIV_BY_ZERO {
319 return status.and(self);
322 assert!(Self::PRECISION < 128);
325 let x = unpack!(status=, v.to_i128_r(128, Round::NearestTiesToEven, &mut false));
326 if status == Status::INVALID_OP {
327 return status.and(self);
331 let mut v = unpack!(status=, Self::from_i128(x));
332 assert_eq!(status, Status::OK); // should always work
335 v = unpack!(status=, v * rhs);
336 assert_eq!(status - Status::INEXACT, Status::OK); // should not overflow or underflow
339 v = unpack!(status=, self - v);
340 assert_eq!(status - Status::INEXACT, Status::OK); // likewise
343 status.and(v.copy_sign(self)) // IEEE754 requires this
348 /// C fmod, or llvm frem.
349 fn c_fmod(self, rhs: Self) -> StatusAnd<Self>;
350 fn round_to_integral(self, round: Round) -> StatusAnd<Self>;
352 /// IEEE-754R 2008 5.3.1: nextUp.
353 fn next_up(self) -> StatusAnd<Self>;
355 /// IEEE-754R 2008 5.3.1: nextDown.
357 /// *NOTE* since nextDown(x) = -nextUp(-x), we only implement nextUp with
358 /// appropriate sign switching before/after the computation.
359 fn next_down(self) -> StatusAnd<Self> {
360 (-self).next_up().map(|r| -r)
363 fn abs(self) -> Self {
364 if self.is_negative() { -self } else { self }
366 fn copy_sign(self, rhs: Self) -> Self {
367 if self.is_negative() != rhs.is_negative() {
375 fn from_bits(input: u128) -> Self;
376 fn from_i128_r(input: i128, round: Round) -> StatusAnd<Self> {
378 Self::from_u128_r(input.wrapping_neg() as u128, -round).map(|r| -r)
380 Self::from_u128_r(input as u128, round)
383 fn from_i128(input: i128) -> StatusAnd<Self> {
384 Self::from_i128_r(input, Round::NearestTiesToEven)
386 fn from_u128_r(input: u128, round: Round) -> StatusAnd<Self>;
387 fn from_u128(input: u128) -> StatusAnd<Self> {
388 Self::from_u128_r(input, Round::NearestTiesToEven)
390 fn from_str_r(s: &str, round: Round) -> Result<StatusAnd<Self>, ParseError>;
391 fn to_bits(self) -> u128;
393 /// Convert a floating point number to an integer according to the
394 /// rounding mode. In case of an invalid operation exception,
395 /// deterministic values are returned, namely zero for NaNs and the
396 /// minimal or maximal value respectively for underflow or overflow.
397 /// If the rounded value is in range but the floating point number is
398 /// not the exact integer, the C standard doesn't require an inexact
399 /// exception to be raised. IEEE-854 does require it so we do that.
401 /// Note that for conversions to integer type the C standard requires
402 /// round-to-zero to always be used.
404 /// The *is_exact output tells whether the result is exact, in the sense
405 /// that converting it back to the original floating point type produces
406 /// the original value. This is almost equivalent to result==Status::OK,
407 /// except for negative zeroes.
408 fn to_i128_r(self, width: usize, round: Round, is_exact: &mut bool) -> StatusAnd<i128> {
410 if self.is_negative() {
412 // Negative zero can't be represented as an int.
415 let r = unpack!(status=, (-self).to_u128_r(width, -round, is_exact));
417 // Check for values that don't fit in the signed integer.
418 if r > (1 << (width - 1)) {
419 // Return the most negative integer for the given width.
421 Status::INVALID_OP.and(-1 << (width - 1))
423 status.and(r.wrapping_neg() as i128)
426 // Positive case is simpler, can pretend it's a smaller unsigned
427 // integer, and `to_u128` will take care of all the edge cases.
428 self.to_u128_r(width - 1, round, is_exact).map(
433 fn to_i128(self, width: usize) -> StatusAnd<i128> {
434 self.to_i128_r(width, Round::TowardZero, &mut true)
436 fn to_u128_r(self, width: usize, round: Round, is_exact: &mut bool) -> StatusAnd<u128>;
437 fn to_u128(self, width: usize) -> StatusAnd<u128> {
438 self.to_u128_r(width, Round::TowardZero, &mut true)
441 fn cmp_abs_normal(self, rhs: Self) -> Ordering;
443 /// Bitwise comparison for equality (QNaNs compare equal, 0!=-0).
444 fn bitwise_eq(self, rhs: Self) -> bool;
446 // IEEE-754R 5.7.2 General operations.
448 /// Implements IEEE minNum semantics. Returns the smaller of the 2 arguments if
449 /// both are not NaN. If either argument is a NaN, returns the other argument.
450 fn min(self, other: Self) -> Self {
453 } else if other.is_nan() {
455 } else if other.partial_cmp(&self) == Some(Ordering::Less) {
462 /// Implements IEEE maxNum semantics. Returns the larger of the 2 arguments if
463 /// both are not NaN. If either argument is a NaN, returns the other argument.
464 fn max(self, other: Self) -> Self {
467 } else if other.is_nan() {
469 } else if self.partial_cmp(&other) == Some(Ordering::Less) {
476 /// IEEE-754R isSignMinus: Returns true if and only if the current value is
479 /// This applies to zeros and NaNs as well.
480 fn is_negative(self) -> bool;
482 /// IEEE-754R isNormal: Returns true if and only if the current value is normal.
484 /// This implies that the current value of the float is not zero, subnormal,
485 /// infinite, or NaN following the definition of normality from IEEE-754R.
486 fn is_normal(self) -> bool {
487 !self.is_denormal() && self.is_finite_non_zero()
490 /// Returns true if and only if the current value is zero, subnormal, or
493 /// This means that the value is not infinite or NaN.
494 fn is_finite(self) -> bool {
495 !self.is_nan() && !self.is_infinite()
498 /// Returns true if and only if the float is plus or minus zero.
499 fn is_zero(self) -> bool {
500 self.category() == Category::Zero
503 /// IEEE-754R isSubnormal(): Returns true if and only if the float is a
505 fn is_denormal(self) -> bool;
507 /// IEEE-754R isInfinite(): Returns true if and only if the float is infinity.
508 fn is_infinite(self) -> bool {
509 self.category() == Category::Infinity
512 /// Returns true if and only if the float is a quiet or signaling NaN.
513 fn is_nan(self) -> bool {
514 self.category() == Category::NaN
517 /// Returns true if and only if the float is a signaling NaN.
518 fn is_signaling(self) -> bool;
522 fn category(self) -> Category;
523 fn is_non_zero(self) -> bool {
526 fn is_finite_non_zero(self) -> bool {
527 self.is_finite() && !self.is_zero()
529 fn is_pos_zero(self) -> bool {
530 self.is_zero() && !self.is_negative()
532 fn is_neg_zero(self) -> bool {
533 self.is_zero() && self.is_negative()
536 /// Returns true if and only if the number has the smallest possible non-zero
537 /// magnitude in the current semantics.
538 fn is_smallest(self) -> bool {
539 Self::SMALLEST.copy_sign(self).bitwise_eq(self)
542 /// Returns true if and only if the number has the largest possible finite
543 /// magnitude in the current semantics.
544 fn is_largest(self) -> bool {
545 Self::largest().copy_sign(self).bitwise_eq(self)
548 /// Returns true if and only if the number is an exact integer.
549 fn is_integer(self) -> bool {
550 // This could be made more efficient; I'm going for obviously correct.
551 if !self.is_finite() {
554 self.round_to_integral(Round::TowardZero).value.bitwise_eq(
559 /// If this value has an exact multiplicative inverse, return it.
560 fn get_exact_inverse(self) -> Option<Self>;
562 /// Returns the exponent of the internal representation of the Float.
564 /// Because the radix of Float is 2, this is equivalent to floor(log2(x)).
565 /// For special Float values, this returns special error codes:
567 /// NaN -> \c IEK_NAN
569 /// Inf -> \c IEK_INF
571 fn ilogb(self) -> ExpInt;
573 /// Returns: self * 2<sup>exp</sup> for integral exponents.
574 fn scalbn_r(self, exp: ExpInt, round: Round) -> Self;
575 fn scalbn(self, exp: ExpInt) -> Self {
576 self.scalbn_r(exp, Round::NearestTiesToEven)
579 /// Equivalent of C standard library function.
581 /// While the C standard says exp is an unspecified value for infinity and nan,
582 /// this returns INT_MAX for infinities, and INT_MIN for NaNs (see `ilogb`).
583 fn frexp_r(self, exp: &mut ExpInt, round: Round) -> Self;
584 fn frexp(self, exp: &mut ExpInt) -> Self {
585 self.frexp_r(exp, Round::NearestTiesToEven)
589 pub trait FloatConvert<T: Float>: Float {
590 /// Convert a value of one floating point type to another.
591 /// The return value corresponds to the IEEE754 exceptions. *loses_info
592 /// records whether the transformation lost information, i.e., whether
593 /// converting the result back to the original type will produce the
594 /// original value (this is almost the same as return value==Status::OK,
595 /// but there are edge cases where this is not so).
596 fn convert_r(self, round: Round, loses_info: &mut bool) -> StatusAnd<T>;
597 fn convert(self, loses_info: &mut bool) -> StatusAnd<T> {
598 self.convert_r(Round::NearestTiesToEven, loses_info)
602 macro_rules! float_common_impls {
603 ($ty:ident<$t:tt>) => {
604 impl<$t> Default for $ty<$t> where Self: Float {
605 fn default() -> Self {
610 impl<$t> ::std::str::FromStr for $ty<$t> where Self: Float {
611 type Err = ParseError;
612 fn from_str(s: &str) -> Result<Self, ParseError> {
613 Self::from_str_r(s, Round::NearestTiesToEven).map(|x| x.value)
617 // Rounding ties to the nearest even, by default.
619 impl<$t> ::std::ops::Add for $ty<$t> where Self: Float {
620 type Output = StatusAnd<Self>;
621 fn add(self, rhs: Self) -> StatusAnd<Self> {
622 self.add_r(rhs, Round::NearestTiesToEven)
626 impl<$t> ::std::ops::Sub for $ty<$t> where Self: Float {
627 type Output = StatusAnd<Self>;
628 fn sub(self, rhs: Self) -> StatusAnd<Self> {
629 self.sub_r(rhs, Round::NearestTiesToEven)
633 impl<$t> ::std::ops::Mul for $ty<$t> where Self: Float {
634 type Output = StatusAnd<Self>;
635 fn mul(self, rhs: Self) -> StatusAnd<Self> {
636 self.mul_r(rhs, Round::NearestTiesToEven)
640 impl<$t> ::std::ops::Div for $ty<$t> where Self: Float {
641 type Output = StatusAnd<Self>;
642 fn div(self, rhs: Self) -> StatusAnd<Self> {
643 self.div_r(rhs, Round::NearestTiesToEven)
647 impl<$t> ::std::ops::Rem for $ty<$t> where Self: Float {
648 type Output = StatusAnd<Self>;
649 fn rem(self, rhs: Self) -> StatusAnd<Self> {
654 impl<$t> ::std::ops::AddAssign for $ty<$t> where Self: Float {
655 fn add_assign(&mut self, rhs: Self) {
656 *self = (*self + rhs).value;
660 impl<$t> ::std::ops::SubAssign for $ty<$t> where Self: Float {
661 fn sub_assign(&mut self, rhs: Self) {
662 *self = (*self - rhs).value;
666 impl<$t> ::std::ops::MulAssign for $ty<$t> where Self: Float {
667 fn mul_assign(&mut self, rhs: Self) {
668 *self = (*self * rhs).value;
672 impl<$t> ::std::ops::DivAssign for $ty<$t> where Self: Float {
673 fn div_assign(&mut self, rhs: Self) {
674 *self = (*self / rhs).value;
678 impl<$t> ::std::ops::RemAssign for $ty<$t> where Self: Float {
679 fn rem_assign(&mut self, rhs: Self) {
680 *self = (*self % rhs).value;