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)]
49 // See librustc_cratesio_shim/Cargo.toml for a comment explaining this.
50 #[allow(unused_extern_crates)]
51 extern crate rustc_cratesio_shim;
54 extern crate bitflags;
56 use std::cmp::Ordering;
58 use std::ops::{Neg, Add, Sub, Mul, Div, Rem};
59 use std::ops::{AddAssign, SubAssign, MulAssign, DivAssign, RemAssign};
60 use std::str::FromStr;
63 /// IEEE-754R 7: Default exception handling.
65 /// UNDERFLOW or OVERFLOW are always returned or-ed with INEXACT.
67 pub struct Status: u8 {
69 const INVALID_OP = 0x01;
70 const DIV_BY_ZERO = 0x02;
71 const OVERFLOW = 0x04;
72 const UNDERFLOW = 0x08;
78 #[derive(Copy, Clone, PartialEq, Eq, PartialOrd, Ord, Debug)]
79 pub struct StatusAnd<T> {
85 pub fn and<T>(self, value: T) -> StatusAnd<T> {
93 impl<T> StatusAnd<T> {
94 pub fn map<F: FnOnce(T) -> U, U>(self, f: F) -> StatusAnd<U> {
103 macro_rules! unpack {
104 ($status:ident|=, $e:expr) => {
106 $crate::StatusAnd { status, value } => {
112 ($status:ident=, $e:expr) => {
114 $crate::StatusAnd { status, value } => {
122 /// Category of internally-represented number.
123 #[derive(Copy, Clone, PartialEq, Eq, Debug)]
131 /// IEEE-754R 4.3: Rounding-direction attributes.
132 #[derive(Copy, Clone, PartialEq, Eq, Debug)]
143 fn neg(self) -> Round {
145 Round::TowardPositive => Round::TowardNegative,
146 Round::TowardNegative => Round::TowardPositive,
147 Round::NearestTiesToEven | Round::TowardZero | Round::NearestTiesToAway => self,
152 /// A signed type to represent a floating point number's unbiased exponent.
153 pub type ExpInt = i16;
155 // \c ilogb error results.
156 pub const IEK_INF: ExpInt = ExpInt::max_value();
157 pub const IEK_NAN: ExpInt = ExpInt::min_value();
158 pub const IEK_ZERO: ExpInt = ExpInt::min_value() + 1;
160 #[derive(Copy, Clone, PartialEq, Eq, Debug)]
161 pub struct ParseError(pub &'static str);
163 /// A self-contained host- and target-independent arbitrary-precision
164 /// floating-point software implementation.
166 /// `apfloat` uses significand bignum integer arithmetic as provided by functions
167 /// in the `ieee::sig`.
169 /// Written for clarity rather than speed, in particular with a view to use in
170 /// the front-end of a cross compiler so that target arithmetic can be correctly
171 /// performed on the host. Performance should nonetheless be reasonable,
172 /// particularly for its intended use. It may be useful as a base
173 /// implementation for a run-time library during development of a faster
174 /// target-specific one.
176 /// All 5 rounding modes in the IEEE-754R draft are handled correctly for all
177 /// implemented operations. Currently implemented operations are add, subtract,
178 /// multiply, divide, fused-multiply-add, conversion-to-float,
179 /// conversion-to-integer and conversion-from-integer. New rounding modes
180 /// (e.g. away from zero) can be added with three or four lines of code.
182 /// Four formats are built-in: IEEE single precision, double precision,
183 /// quadruple precision, and x87 80-bit extended double (when operating with
184 /// full extended precision). Adding a new format that obeys IEEE semantics
185 /// only requires adding two lines of code: a declaration and definition of the
188 /// All operations return the status of that operation as an exception bit-mask,
189 /// so multiple operations can be done consecutively with their results or-ed
190 /// together. The returned status can be useful for compiler diagnostics; e.g.,
191 /// inexact, underflow and overflow can be easily diagnosed on constant folding,
192 /// and compiler optimizers can determine what exceptions would be raised by
193 /// folding operations and optimize, or perhaps not optimize, accordingly.
195 /// At present, underflow tininess is detected after rounding; it should be
196 /// straight forward to add support for the before-rounding case too.
198 /// The library reads hexadecimal floating point numbers as per C99, and
199 /// correctly rounds if necessary according to the specified rounding mode.
200 /// Syntax is required to have been validated by the caller.
202 /// It also reads decimal floating point numbers and correctly rounds according
203 /// to the specified rounding mode.
205 /// Non-zero finite numbers are represented internally as a sign bit, a 16-bit
206 /// signed exponent, and the significand as an array of integer limbs. After
207 /// normalization of a number of precision P the exponent is within the range of
208 /// the format, and if the number is not denormal the P-th bit of the
209 /// significand is set as an explicit integer bit. For denormals the most
210 /// significant bit is shifted right so that the exponent is maintained at the
211 /// format's minimum, so that the smallest denormal has just the least
212 /// significant bit of the significand set. The sign of zeros and infinities
213 /// is significant; the exponent and significand of such numbers is not stored,
214 /// but has a known implicit (deterministic) value: 0 for the significands, 0
215 /// for zero exponent, all 1 bits for infinity exponent. For NaNs the sign and
216 /// significand are deterministic, although not really meaningful, and preserved
217 /// in non-conversion operations. The exponent is implicitly all 1 bits.
219 /// `apfloat` does not provide any exception handling beyond default exception
220 /// handling. We represent Signaling NaNs via IEEE-754R 2008 6.2.1 should clause
221 /// by encoding Signaling NaNs with the first bit of its trailing significand
227 /// Some features that may or may not be worth adding:
229 /// Optional ability to detect underflow tininess before rounding.
231 /// New formats: x87 in single and double precision mode (IEEE apart from
232 /// extended exponent range) (hard).
234 /// New operations: sqrt, nexttoward.
239 + FromStr<Err = ParseError>
248 + Add<Output = StatusAnd<Self>>
249 + Sub<Output = StatusAnd<Self>>
250 + Mul<Output = StatusAnd<Self>>
251 + Div<Output = StatusAnd<Self>>
252 + Rem<Output = StatusAnd<Self>> {
253 /// Total number of bits in the in-memory format.
256 /// Number of bits in the significand. This includes the integer bit.
257 const PRECISION: usize;
259 /// The largest E such that 2<sup>E</sup> is representable; this matches the
260 /// definition of IEEE 754.
261 const MAX_EXP: ExpInt;
263 /// The smallest E such that 2<sup>E</sup> is a normalized number; this
264 /// matches the definition of IEEE 754.
265 const MIN_EXP: ExpInt;
270 /// Positive Infinity.
271 const INFINITY: Self;
273 /// NaN (Not a Number).
274 // FIXME(eddyb) provide a default when qnan becomes const fn.
277 /// Factory for QNaN values.
278 // FIXME(eddyb) should be const fn.
279 fn qnan(payload: Option<u128>) -> Self;
281 /// Factory for SNaN values.
282 // FIXME(eddyb) should be const fn.
283 fn snan(payload: Option<u128>) -> Self;
285 /// Largest finite number.
286 // FIXME(eddyb) should be const (but FloatPair::largest is nontrivial).
287 fn largest() -> Self;
289 /// Smallest (by magnitude) finite number.
290 /// Might be denormalized, which implies a relative loss of precision.
291 const SMALLEST: Self;
293 /// Smallest (by magnitude) normalized finite number.
294 // FIXME(eddyb) should be const (but FloatPair::smallest_normalized is nontrivial).
295 fn smallest_normalized() -> Self;
299 fn add_r(self, rhs: Self, round: Round) -> StatusAnd<Self>;
300 fn sub_r(self, rhs: Self, round: Round) -> StatusAnd<Self> {
301 self.add_r(-rhs, round)
303 fn mul_r(self, rhs: Self, round: Round) -> StatusAnd<Self>;
304 fn mul_add_r(self, multiplicand: Self, addend: Self, round: Round) -> StatusAnd<Self>;
305 fn mul_add(self, multiplicand: Self, addend: Self) -> StatusAnd<Self> {
306 self.mul_add_r(multiplicand, addend, Round::NearestTiesToEven)
308 fn div_r(self, rhs: Self, round: Round) -> StatusAnd<Self>;
310 // This is not currently correct in all cases.
311 fn ieee_rem(self, rhs: Self) -> StatusAnd<Self> {
315 v = unpack!(status=, v / rhs);
316 if status == Status::DIV_BY_ZERO {
317 return status.and(self);
320 assert!(Self::PRECISION < 128);
323 let x = unpack!(status=, v.to_i128_r(128, Round::NearestTiesToEven, &mut false));
324 if status == Status::INVALID_OP {
325 return status.and(self);
329 let mut v = unpack!(status=, Self::from_i128(x));
330 assert_eq!(status, Status::OK); // should always work
333 v = unpack!(status=, v * rhs);
334 assert_eq!(status - Status::INEXACT, Status::OK); // should not overflow or underflow
337 v = unpack!(status=, self - v);
338 assert_eq!(status - Status::INEXACT, Status::OK); // likewise
341 status.and(v.copy_sign(self)) // IEEE754 requires this
346 /// C fmod, or llvm frem.
347 fn c_fmod(self, rhs: Self) -> StatusAnd<Self>;
348 fn round_to_integral(self, round: Round) -> StatusAnd<Self>;
350 /// IEEE-754R 2008 5.3.1: nextUp.
351 fn next_up(self) -> StatusAnd<Self>;
353 /// IEEE-754R 2008 5.3.1: nextDown.
355 /// *NOTE* since nextDown(x) = -nextUp(-x), we only implement nextUp with
356 /// appropriate sign switching before/after the computation.
357 fn next_down(self) -> StatusAnd<Self> {
358 (-self).next_up().map(|r| -r)
361 fn abs(self) -> Self {
362 if self.is_negative() { -self } else { self }
364 fn copy_sign(self, rhs: Self) -> Self {
365 if self.is_negative() != rhs.is_negative() {
373 fn from_bits(input: u128) -> Self;
374 fn from_i128_r(input: i128, round: Round) -> StatusAnd<Self> {
376 Self::from_u128_r(input.wrapping_neg() as u128, -round).map(|r| -r)
378 Self::from_u128_r(input as u128, round)
381 fn from_i128(input: i128) -> StatusAnd<Self> {
382 Self::from_i128_r(input, Round::NearestTiesToEven)
384 fn from_u128_r(input: u128, round: Round) -> StatusAnd<Self>;
385 fn from_u128(input: u128) -> StatusAnd<Self> {
386 Self::from_u128_r(input, Round::NearestTiesToEven)
388 fn from_str_r(s: &str, round: Round) -> Result<StatusAnd<Self>, ParseError>;
389 fn to_bits(self) -> u128;
391 /// Convert a floating point number to an integer according to the
392 /// rounding mode. In case of an invalid operation exception,
393 /// deterministic values are returned, namely zero for NaNs and the
394 /// minimal or maximal value respectively for underflow or overflow.
395 /// If the rounded value is in range but the floating point number is
396 /// not the exact integer, the C standard doesn't require an inexact
397 /// exception to be raised. IEEE-854 does require it so we do that.
399 /// Note that for conversions to integer type the C standard requires
400 /// round-to-zero to always be used.
402 /// The *is_exact output tells whether the result is exact, in the sense
403 /// that converting it back to the original floating point type produces
404 /// the original value. This is almost equivalent to result==Status::OK,
405 /// except for negative zeroes.
406 fn to_i128_r(self, width: usize, round: Round, is_exact: &mut bool) -> StatusAnd<i128> {
408 if self.is_negative() {
410 // Negative zero can't be represented as an int.
413 let r = unpack!(status=, (-self).to_u128_r(width, -round, is_exact));
415 // Check for values that don't fit in the signed integer.
416 if r > (1 << (width - 1)) {
417 // Return the most negative integer for the given width.
419 Status::INVALID_OP.and(-1 << (width - 1))
421 status.and(r.wrapping_neg() as i128)
424 // Positive case is simpler, can pretend it's a smaller unsigned
425 // integer, and `to_u128` will take care of all the edge cases.
426 self.to_u128_r(width - 1, round, is_exact).map(
431 fn to_i128(self, width: usize) -> StatusAnd<i128> {
432 self.to_i128_r(width, Round::TowardZero, &mut true)
434 fn to_u128_r(self, width: usize, round: Round, is_exact: &mut bool) -> StatusAnd<u128>;
435 fn to_u128(self, width: usize) -> StatusAnd<u128> {
436 self.to_u128_r(width, Round::TowardZero, &mut true)
439 fn cmp_abs_normal(self, rhs: Self) -> Ordering;
441 /// Bitwise comparison for equality (QNaNs compare equal, 0!=-0).
442 fn bitwise_eq(self, rhs: Self) -> bool;
444 // IEEE-754R 5.7.2 General operations.
446 /// Implements IEEE minNum semantics. Returns the smaller of the 2 arguments if
447 /// both are not NaN. If either argument is a NaN, returns the other argument.
448 fn min(self, other: Self) -> Self {
451 } else if other.is_nan() {
453 } else if other.partial_cmp(&self) == Some(Ordering::Less) {
460 /// Implements IEEE maxNum semantics. Returns the larger of the 2 arguments if
461 /// both are not NaN. If either argument is a NaN, returns the other argument.
462 fn max(self, other: Self) -> Self {
465 } else if other.is_nan() {
467 } else if self.partial_cmp(&other) == Some(Ordering::Less) {
474 /// IEEE-754R isSignMinus: Returns true if and only if the current value is
477 /// This applies to zeros and NaNs as well.
478 fn is_negative(self) -> bool;
480 /// IEEE-754R isNormal: Returns true if and only if the current value is normal.
482 /// This implies that the current value of the float is not zero, subnormal,
483 /// infinite, or NaN following the definition of normality from IEEE-754R.
484 fn is_normal(self) -> bool {
485 !self.is_denormal() && self.is_finite_non_zero()
488 /// Returns true if and only if the current value is zero, subnormal, or
491 /// This means that the value is not infinite or NaN.
492 fn is_finite(self) -> bool {
493 !self.is_nan() && !self.is_infinite()
496 /// Returns true if and only if the float is plus or minus zero.
497 fn is_zero(self) -> bool {
498 self.category() == Category::Zero
501 /// IEEE-754R isSubnormal(): Returns true if and only if the float is a
503 fn is_denormal(self) -> bool;
505 /// IEEE-754R isInfinite(): Returns true if and only if the float is infinity.
506 fn is_infinite(self) -> bool {
507 self.category() == Category::Infinity
510 /// Returns true if and only if the float is a quiet or signaling NaN.
511 fn is_nan(self) -> bool {
512 self.category() == Category::NaN
515 /// Returns true if and only if the float is a signaling NaN.
516 fn is_signaling(self) -> bool;
520 fn category(self) -> Category;
521 fn is_non_zero(self) -> bool {
524 fn is_finite_non_zero(self) -> bool {
525 self.is_finite() && !self.is_zero()
527 fn is_pos_zero(self) -> bool {
528 self.is_zero() && !self.is_negative()
530 fn is_neg_zero(self) -> bool {
531 self.is_zero() && self.is_negative()
534 /// Returns true if and only if the number has the smallest possible non-zero
535 /// magnitude in the current semantics.
536 fn is_smallest(self) -> bool {
537 Self::SMALLEST.copy_sign(self).bitwise_eq(self)
540 /// Returns true if and only if the number has the largest possible finite
541 /// magnitude in the current semantics.
542 fn is_largest(self) -> bool {
543 Self::largest().copy_sign(self).bitwise_eq(self)
546 /// Returns true if and only if the number is an exact integer.
547 fn is_integer(self) -> bool {
548 // This could be made more efficient; I'm going for obviously correct.
549 if !self.is_finite() {
552 self.round_to_integral(Round::TowardZero).value.bitwise_eq(
557 /// If this value has an exact multiplicative inverse, return it.
558 fn get_exact_inverse(self) -> Option<Self>;
560 /// Returns the exponent of the internal representation of the Float.
562 /// Because the radix of Float is 2, this is equivalent to floor(log2(x)).
563 /// For special Float values, this returns special error codes:
565 /// NaN -> \c IEK_NAN
567 /// Inf -> \c IEK_INF
569 fn ilogb(self) -> ExpInt;
571 /// Returns: self * 2<sup>exp</sup> for integral exponents.
572 fn scalbn_r(self, exp: ExpInt, round: Round) -> Self;
573 fn scalbn(self, exp: ExpInt) -> Self {
574 self.scalbn_r(exp, Round::NearestTiesToEven)
577 /// Equivalent of C standard library function.
579 /// While the C standard says exp is an unspecified value for infinity and nan,
580 /// this returns INT_MAX for infinities, and INT_MIN for NaNs (see `ilogb`).
581 fn frexp_r(self, exp: &mut ExpInt, round: Round) -> Self;
582 fn frexp(self, exp: &mut ExpInt) -> Self {
583 self.frexp_r(exp, Round::NearestTiesToEven)
587 pub trait FloatConvert<T: Float>: Float {
588 /// Convert a value of one floating point type to another.
589 /// The return value corresponds to the IEEE754 exceptions. *loses_info
590 /// records whether the transformation lost information, i.e. whether
591 /// converting the result back to the original type will produce the
592 /// original value (this is almost the same as return value==Status::OK,
593 /// but there are edge cases where this is not so).
594 fn convert_r(self, round: Round, loses_info: &mut bool) -> StatusAnd<T>;
595 fn convert(self, loses_info: &mut bool) -> StatusAnd<T> {
596 self.convert_r(Round::NearestTiesToEven, loses_info)
600 macro_rules! float_common_impls {
601 ($ty:ident<$t:tt>) => {
602 impl<$t> Default for $ty<$t> where Self: Float {
603 fn default() -> Self {
608 impl<$t> ::std::str::FromStr for $ty<$t> where Self: Float {
609 type Err = ParseError;
610 fn from_str(s: &str) -> Result<Self, ParseError> {
611 Self::from_str_r(s, Round::NearestTiesToEven).map(|x| x.value)
615 // Rounding ties to the nearest even, by default.
617 impl<$t> ::std::ops::Add for $ty<$t> where Self: Float {
618 type Output = StatusAnd<Self>;
619 fn add(self, rhs: Self) -> StatusAnd<Self> {
620 self.add_r(rhs, Round::NearestTiesToEven)
624 impl<$t> ::std::ops::Sub for $ty<$t> where Self: Float {
625 type Output = StatusAnd<Self>;
626 fn sub(self, rhs: Self) -> StatusAnd<Self> {
627 self.sub_r(rhs, Round::NearestTiesToEven)
631 impl<$t> ::std::ops::Mul for $ty<$t> where Self: Float {
632 type Output = StatusAnd<Self>;
633 fn mul(self, rhs: Self) -> StatusAnd<Self> {
634 self.mul_r(rhs, Round::NearestTiesToEven)
638 impl<$t> ::std::ops::Div for $ty<$t> where Self: Float {
639 type Output = StatusAnd<Self>;
640 fn div(self, rhs: Self) -> StatusAnd<Self> {
641 self.div_r(rhs, Round::NearestTiesToEven)
645 impl<$t> ::std::ops::Rem for $ty<$t> where Self: Float {
646 type Output = StatusAnd<Self>;
647 fn rem(self, rhs: Self) -> StatusAnd<Self> {
652 impl<$t> ::std::ops::AddAssign for $ty<$t> where Self: Float {
653 fn add_assign(&mut self, rhs: Self) {
654 *self = (*self + rhs).value;
658 impl<$t> ::std::ops::SubAssign for $ty<$t> where Self: Float {
659 fn sub_assign(&mut self, rhs: Self) {
660 *self = (*self - rhs).value;
664 impl<$t> ::std::ops::MulAssign for $ty<$t> where Self: Float {
665 fn mul_assign(&mut self, rhs: Self) {
666 *self = (*self * rhs).value;
670 impl<$t> ::std::ops::DivAssign for $ty<$t> where Self: Float {
671 fn div_assign(&mut self, rhs: Self) {
672 *self = (*self / rhs).value;
676 impl<$t> ::std::ops::RemAssign for $ty<$t> where Self: Float {
677 fn rem_assign(&mut self, rhs: Self) {
678 *self = (*self % rhs).value;