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;
57 use std::cmp::Ordering;
59 use std::ops::{Neg, Add, Sub, Mul, Div, Rem};
60 use std::ops::{AddAssign, SubAssign, MulAssign, DivAssign, RemAssign};
61 use std::str::FromStr;
64 /// IEEE-754R 7: Default exception handling.
66 /// UNDERFLOW or OVERFLOW are always returned or-ed with INEXACT.
68 pub struct Status: u8 {
70 const INVALID_OP = 0x01;
71 const DIV_BY_ZERO = 0x02;
72 const OVERFLOW = 0x04;
73 const UNDERFLOW = 0x08;
79 #[derive(Copy, Clone, PartialEq, Eq, PartialOrd, Ord, Debug)]
80 pub struct StatusAnd<T> {
86 pub fn and<T>(self, value: T) -> StatusAnd<T> {
94 impl<T> StatusAnd<T> {
95 pub fn map<F: FnOnce(T) -> U, U>(self, f: F) -> StatusAnd<U> {
104 macro_rules! unpack {
105 ($status:ident|=, $e:expr) => {
107 $crate::StatusAnd { status, value } => {
113 ($status:ident=, $e:expr) => {
115 $crate::StatusAnd { status, value } => {
123 /// Category of internally-represented number.
124 #[derive(Copy, Clone, PartialEq, Eq, Debug)]
132 /// IEEE-754R 4.3: Rounding-direction attributes.
133 #[derive(Copy, Clone, PartialEq, Eq, Debug)]
144 fn neg(self) -> Round {
146 Round::TowardPositive => Round::TowardNegative,
147 Round::TowardNegative => Round::TowardPositive,
148 Round::NearestTiesToEven | Round::TowardZero | Round::NearestTiesToAway => self,
153 /// A signed type to represent a floating point number's unbiased exponent.
154 pub type ExpInt = i16;
156 // \c ilogb error results.
157 pub const IEK_INF: ExpInt = ExpInt::max_value();
158 pub const IEK_NAN: ExpInt = ExpInt::min_value();
159 pub const IEK_ZERO: ExpInt = ExpInt::min_value() + 1;
161 #[derive(Copy, Clone, PartialEq, Eq, Debug)]
162 pub struct ParseError(pub &'static str);
164 /// A self-contained host- and target-independent arbitrary-precision
165 /// floating-point software implementation.
167 /// `apfloat` uses significand bignum integer arithmetic as provided by functions
168 /// in the `ieee::sig`.
170 /// Written for clarity rather than speed, in particular with a view to use in
171 /// the front-end of a cross compiler so that target arithmetic can be correctly
172 /// performed on the host. Performance should nonetheless be reasonable,
173 /// particularly for its intended use. It may be useful as a base
174 /// implementation for a run-time library during development of a faster
175 /// target-specific one.
177 /// All 5 rounding modes in the IEEE-754R draft are handled correctly for all
178 /// implemented operations. Currently implemented operations are add, subtract,
179 /// multiply, divide, fused-multiply-add, conversion-to-float,
180 /// conversion-to-integer and conversion-from-integer. New rounding modes
181 /// (e.g. away from zero) can be added with three or four lines of code.
183 /// Four formats are built-in: IEEE single precision, double precision,
184 /// quadruple precision, and x87 80-bit extended double (when operating with
185 /// full extended precision). Adding a new format that obeys IEEE semantics
186 /// only requires adding two lines of code: a declaration and definition of the
189 /// All operations return the status of that operation as an exception bit-mask,
190 /// so multiple operations can be done consecutively with their results or-ed
191 /// together. The returned status can be useful for compiler diagnostics; e.g.,
192 /// inexact, underflow and overflow can be easily diagnosed on constant folding,
193 /// and compiler optimizers can determine what exceptions would be raised by
194 /// folding operations and optimize, or perhaps not optimize, accordingly.
196 /// At present, underflow tininess is detected after rounding; it should be
197 /// straight forward to add support for the before-rounding case too.
199 /// The library reads hexadecimal floating point numbers as per C99, and
200 /// correctly rounds if necessary according to the specified rounding mode.
201 /// Syntax is required to have been validated by the caller.
203 /// It also reads decimal floating point numbers and correctly rounds according
204 /// to the specified rounding mode.
206 /// Non-zero finite numbers are represented internally as a sign bit, a 16-bit
207 /// signed exponent, and the significand as an array of integer limbs. After
208 /// normalization of a number of precision P the exponent is within the range of
209 /// the format, and if the number is not denormal the P-th bit of the
210 /// significand is set as an explicit integer bit. For denormals the most
211 /// significant bit is shifted right so that the exponent is maintained at the
212 /// format's minimum, so that the smallest denormal has just the least
213 /// significant bit of the significand set. The sign of zeros and infinities
214 /// is significant; the exponent and significand of such numbers is not stored,
215 /// but has a known implicit (deterministic) value: 0 for the significands, 0
216 /// for zero exponent, all 1 bits for infinity exponent. For NaNs the sign and
217 /// significand are deterministic, although not really meaningful, and preserved
218 /// in non-conversion operations. The exponent is implicitly all 1 bits.
220 /// `apfloat` does not provide any exception handling beyond default exception
221 /// handling. We represent Signaling NaNs via IEEE-754R 2008 6.2.1 should clause
222 /// by encoding Signaling NaNs with the first bit of its trailing significand
228 /// Some features that may or may not be worth adding:
230 /// Optional ability to detect underflow tininess before rounding.
232 /// New formats: x87 in single and double precision mode (IEEE apart from
233 /// extended exponent range) (hard).
235 /// New operations: sqrt, nexttoward.
240 + FromStr<Err = ParseError>
249 + Add<Output = StatusAnd<Self>>
250 + Sub<Output = StatusAnd<Self>>
251 + Mul<Output = StatusAnd<Self>>
252 + Div<Output = StatusAnd<Self>>
253 + Rem<Output = StatusAnd<Self>> {
254 /// Total number of bits in the in-memory format.
257 /// Number of bits in the significand. This includes the integer bit.
258 const PRECISION: usize;
260 /// The largest E such that 2<sup>E</sup> is representable; this matches the
261 /// definition of IEEE 754.
262 const MAX_EXP: ExpInt;
264 /// The smallest E such that 2<sup>E</sup> is a normalized number; this
265 /// matches the definition of IEEE 754.
266 const MIN_EXP: ExpInt;
271 /// Positive Infinity.
272 const INFINITY: Self;
274 /// NaN (Not a Number).
275 // FIXME(eddyb) provide a default when qnan becomes const fn.
278 /// Factory for QNaN values.
279 // FIXME(eddyb) should be const fn.
280 fn qnan(payload: Option<u128>) -> Self;
282 /// Factory for SNaN values.
283 // FIXME(eddyb) should be const fn.
284 fn snan(payload: Option<u128>) -> Self;
286 /// Largest finite number.
287 // FIXME(eddyb) should be const (but FloatPair::largest is nontrivial).
288 fn largest() -> Self;
290 /// Smallest (by magnitude) finite number.
291 /// Might be denormalized, which implies a relative loss of precision.
292 const SMALLEST: Self;
294 /// Smallest (by magnitude) normalized finite number.
295 // FIXME(eddyb) should be const (but FloatPair::smallest_normalized is nontrivial).
296 fn smallest_normalized() -> Self;
300 fn add_r(self, rhs: Self, round: Round) -> StatusAnd<Self>;
301 fn sub_r(self, rhs: Self, round: Round) -> StatusAnd<Self> {
302 self.add_r(-rhs, round)
304 fn mul_r(self, rhs: Self, round: Round) -> StatusAnd<Self>;
305 fn mul_add_r(self, multiplicand: Self, addend: Self, round: Round) -> StatusAnd<Self>;
306 fn mul_add(self, multiplicand: Self, addend: Self) -> StatusAnd<Self> {
307 self.mul_add_r(multiplicand, addend, Round::NearestTiesToEven)
309 fn div_r(self, rhs: Self, round: Round) -> StatusAnd<Self>;
311 // This is not currently correct in all cases.
312 fn ieee_rem(self, rhs: Self) -> StatusAnd<Self> {
316 v = unpack!(status=, v / rhs);
317 if status == Status::DIV_BY_ZERO {
318 return status.and(self);
321 assert!(Self::PRECISION < 128);
324 let x = unpack!(status=, v.to_i128_r(128, Round::NearestTiesToEven, &mut false));
325 if status == Status::INVALID_OP {
326 return status.and(self);
330 let mut v = unpack!(status=, Self::from_i128(x));
331 assert_eq!(status, Status::OK); // should always work
334 v = unpack!(status=, v * rhs);
335 assert_eq!(status - Status::INEXACT, Status::OK); // should not overflow or underflow
338 v = unpack!(status=, self - v);
339 assert_eq!(status - Status::INEXACT, Status::OK); // likewise
342 status.and(v.copy_sign(self)) // IEEE754 requires this
347 /// C fmod, or llvm frem.
348 fn c_fmod(self, rhs: Self) -> StatusAnd<Self>;
349 fn round_to_integral(self, round: Round) -> StatusAnd<Self>;
351 /// IEEE-754R 2008 5.3.1: nextUp.
352 fn next_up(self) -> StatusAnd<Self>;
354 /// IEEE-754R 2008 5.3.1: nextDown.
356 /// *NOTE* since nextDown(x) = -nextUp(-x), we only implement nextUp with
357 /// appropriate sign switching before/after the computation.
358 fn next_down(self) -> StatusAnd<Self> {
359 (-self).next_up().map(|r| -r)
362 fn abs(self) -> Self {
363 if self.is_negative() { -self } else { self }
365 fn copy_sign(self, rhs: Self) -> Self {
366 if self.is_negative() != rhs.is_negative() {
374 fn from_bits(input: u128) -> Self;
375 fn from_i128_r(input: i128, round: Round) -> StatusAnd<Self> {
377 Self::from_u128_r(input.wrapping_neg() as u128, -round).map(|r| -r)
379 Self::from_u128_r(input as u128, round)
382 fn from_i128(input: i128) -> StatusAnd<Self> {
383 Self::from_i128_r(input, Round::NearestTiesToEven)
385 fn from_u128_r(input: u128, round: Round) -> StatusAnd<Self>;
386 fn from_u128(input: u128) -> StatusAnd<Self> {
387 Self::from_u128_r(input, Round::NearestTiesToEven)
389 fn from_str_r(s: &str, round: Round) -> Result<StatusAnd<Self>, ParseError>;
390 fn to_bits(self) -> u128;
392 /// Convert a floating point number to an integer according to the
393 /// rounding mode. In case of an invalid operation exception,
394 /// deterministic values are returned, namely zero for NaNs and the
395 /// minimal or maximal value respectively for underflow or overflow.
396 /// If the rounded value is in range but the floating point number is
397 /// not the exact integer, the C standard doesn't require an inexact
398 /// exception to be raised. IEEE-854 does require it so we do that.
400 /// Note that for conversions to integer type the C standard requires
401 /// round-to-zero to always be used.
403 /// The *is_exact output tells whether the result is exact, in the sense
404 /// that converting it back to the original floating point type produces
405 /// the original value. This is almost equivalent to result==Status::OK,
406 /// except for negative zeroes.
407 fn to_i128_r(self, width: usize, round: Round, is_exact: &mut bool) -> StatusAnd<i128> {
409 if self.is_negative() {
411 // Negative zero can't be represented as an int.
414 let r = unpack!(status=, (-self).to_u128_r(width, -round, is_exact));
416 // Check for values that don't fit in the signed integer.
417 if r > (1 << (width - 1)) {
418 // Return the most negative integer for the given width.
420 Status::INVALID_OP.and(-1 << (width - 1))
422 status.and(r.wrapping_neg() as i128)
425 // Positive case is simpler, can pretend it's a smaller unsigned
426 // integer, and `to_u128` will take care of all the edge cases.
427 self.to_u128_r(width - 1, round, is_exact).map(
432 fn to_i128(self, width: usize) -> StatusAnd<i128> {
433 self.to_i128_r(width, Round::TowardZero, &mut true)
435 fn to_u128_r(self, width: usize, round: Round, is_exact: &mut bool) -> StatusAnd<u128>;
436 fn to_u128(self, width: usize) -> StatusAnd<u128> {
437 self.to_u128_r(width, Round::TowardZero, &mut true)
440 fn cmp_abs_normal(self, rhs: Self) -> Ordering;
442 /// Bitwise comparison for equality (QNaNs compare equal, 0!=-0).
443 fn bitwise_eq(self, rhs: Self) -> bool;
445 // IEEE-754R 5.7.2 General operations.
447 /// Implements IEEE minNum semantics. Returns the smaller of the 2 arguments if
448 /// both are not NaN. If either argument is a NaN, returns the other argument.
449 fn min(self, other: Self) -> Self {
452 } else if other.is_nan() {
454 } else if other.partial_cmp(&self) == Some(Ordering::Less) {
461 /// Implements IEEE maxNum semantics. Returns the larger of the 2 arguments if
462 /// both are not NaN. If either argument is a NaN, returns the other argument.
463 fn max(self, other: Self) -> Self {
466 } else if other.is_nan() {
468 } else if self.partial_cmp(&other) == Some(Ordering::Less) {
475 /// IEEE-754R isSignMinus: Returns true if and only if the current value is
478 /// This applies to zeros and NaNs as well.
479 fn is_negative(self) -> bool;
481 /// IEEE-754R isNormal: Returns true if and only if the current value is normal.
483 /// This implies that the current value of the float is not zero, subnormal,
484 /// infinite, or NaN following the definition of normality from IEEE-754R.
485 fn is_normal(self) -> bool {
486 !self.is_denormal() && self.is_finite_non_zero()
489 /// Returns true if and only if the current value is zero, subnormal, or
492 /// This means that the value is not infinite or NaN.
493 fn is_finite(self) -> bool {
494 !self.is_nan() && !self.is_infinite()
497 /// Returns true if and only if the float is plus or minus zero.
498 fn is_zero(self) -> bool {
499 self.category() == Category::Zero
502 /// IEEE-754R isSubnormal(): Returns true if and only if the float is a
504 fn is_denormal(self) -> bool;
506 /// IEEE-754R isInfinite(): Returns true if and only if the float is infinity.
507 fn is_infinite(self) -> bool {
508 self.category() == Category::Infinity
511 /// Returns true if and only if the float is a quiet or signaling NaN.
512 fn is_nan(self) -> bool {
513 self.category() == Category::NaN
516 /// Returns true if and only if the float is a signaling NaN.
517 fn is_signaling(self) -> bool;
521 fn category(self) -> Category;
522 fn is_non_zero(self) -> bool {
525 fn is_finite_non_zero(self) -> bool {
526 self.is_finite() && !self.is_zero()
528 fn is_pos_zero(self) -> bool {
529 self.is_zero() && !self.is_negative()
531 fn is_neg_zero(self) -> bool {
532 self.is_zero() && self.is_negative()
535 /// Returns true if and only if the number has the smallest possible non-zero
536 /// magnitude in the current semantics.
537 fn is_smallest(self) -> bool {
538 Self::SMALLEST.copy_sign(self).bitwise_eq(self)
541 /// Returns true if and only if the number has the largest possible finite
542 /// magnitude in the current semantics.
543 fn is_largest(self) -> bool {
544 Self::largest().copy_sign(self).bitwise_eq(self)
547 /// Returns true if and only if the number is an exact integer.
548 fn is_integer(self) -> bool {
549 // This could be made more efficient; I'm going for obviously correct.
550 if !self.is_finite() {
553 self.round_to_integral(Round::TowardZero).value.bitwise_eq(
558 /// If this value has an exact multiplicative inverse, return it.
559 fn get_exact_inverse(self) -> Option<Self>;
561 /// Returns the exponent of the internal representation of the Float.
563 /// Because the radix of Float is 2, this is equivalent to floor(log2(x)).
564 /// For special Float values, this returns special error codes:
566 /// NaN -> \c IEK_NAN
568 /// Inf -> \c IEK_INF
570 fn ilogb(self) -> ExpInt;
572 /// Returns: self * 2<sup>exp</sup> for integral exponents.
573 fn scalbn_r(self, exp: ExpInt, round: Round) -> Self;
574 fn scalbn(self, exp: ExpInt) -> Self {
575 self.scalbn_r(exp, Round::NearestTiesToEven)
578 /// Equivalent of C standard library function.
580 /// While the C standard says exp is an unspecified value for infinity and nan,
581 /// this returns INT_MAX for infinities, and INT_MIN for NaNs (see `ilogb`).
582 fn frexp_r(self, exp: &mut ExpInt, round: Round) -> Self;
583 fn frexp(self, exp: &mut ExpInt) -> Self {
584 self.frexp_r(exp, Round::NearestTiesToEven)
588 pub trait FloatConvert<T: Float>: Float {
589 /// Convert a value of one floating point type to another.
590 /// The return value corresponds to the IEEE754 exceptions. *loses_info
591 /// records whether the transformation lost information, i.e. whether
592 /// converting the result back to the original type will produce the
593 /// original value (this is almost the same as return value==Status::OK,
594 /// but there are edge cases where this is not so).
595 fn convert_r(self, round: Round, loses_info: &mut bool) -> StatusAnd<T>;
596 fn convert(self, loses_info: &mut bool) -> StatusAnd<T> {
597 self.convert_r(Round::NearestTiesToEven, loses_info)
601 macro_rules! float_common_impls {
602 ($ty:ident<$t:tt>) => {
603 impl<$t> Default for $ty<$t> where Self: Float {
604 fn default() -> Self {
609 impl<$t> ::std::str::FromStr for $ty<$t> where Self: Float {
610 type Err = ParseError;
611 fn from_str(s: &str) -> Result<Self, ParseError> {
612 Self::from_str_r(s, Round::NearestTiesToEven).map(|x| x.value)
616 // Rounding ties to the nearest even, by default.
618 impl<$t> ::std::ops::Add for $ty<$t> where Self: Float {
619 type Output = StatusAnd<Self>;
620 fn add(self, rhs: Self) -> StatusAnd<Self> {
621 self.add_r(rhs, Round::NearestTiesToEven)
625 impl<$t> ::std::ops::Sub for $ty<$t> where Self: Float {
626 type Output = StatusAnd<Self>;
627 fn sub(self, rhs: Self) -> StatusAnd<Self> {
628 self.sub_r(rhs, Round::NearestTiesToEven)
632 impl<$t> ::std::ops::Mul for $ty<$t> where Self: Float {
633 type Output = StatusAnd<Self>;
634 fn mul(self, rhs: Self) -> StatusAnd<Self> {
635 self.mul_r(rhs, Round::NearestTiesToEven)
639 impl<$t> ::std::ops::Div for $ty<$t> where Self: Float {
640 type Output = StatusAnd<Self>;
641 fn div(self, rhs: Self) -> StatusAnd<Self> {
642 self.div_r(rhs, Round::NearestTiesToEven)
646 impl<$t> ::std::ops::Rem for $ty<$t> where Self: Float {
647 type Output = StatusAnd<Self>;
648 fn rem(self, rhs: Self) -> StatusAnd<Self> {
653 impl<$t> ::std::ops::AddAssign for $ty<$t> where Self: Float {
654 fn add_assign(&mut self, rhs: Self) {
655 *self = (*self + rhs).value;
659 impl<$t> ::std::ops::SubAssign for $ty<$t> where Self: Float {
660 fn sub_assign(&mut self, rhs: Self) {
661 *self = (*self - rhs).value;
665 impl<$t> ::std::ops::MulAssign for $ty<$t> where Self: Float {
666 fn mul_assign(&mut self, rhs: Self) {
667 *self = (*self * rhs).value;
671 impl<$t> ::std::ops::DivAssign for $ty<$t> where Self: Float {
672 fn div_assign(&mut self, rhs: Self) {
673 *self = (*self / rhs).value;
677 impl<$t> ::std::ops::RemAssign for $ty<$t> where Self: Float {
678 fn rem_assign(&mut self, rhs: Self) {
679 *self = (*self % rhs).value;