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
14 //! * `include/llvm/ADT/APFloat.h` -> `Float` and `FloatConvert` traits
15 //! * `lib/Support/APFloat.cpp` -> `ieee` and `ppc` modules
16 //! * `unittests/ADT/APFloatTest.cpp` -> `tests` directory
18 //! The port contains no unsafe code, global state, or side-effects in general,
19 //! and the only allocations are in the conversion to/from decimal strings.
21 //! Most of the API and the testcases are intact in some form or another,
22 //! with some ergonomic changes, such as idiomatic short names, returning
23 //! new values instead of mutating the receiver, and having separate method
24 //! variants that take a non-default rounding mode (with the suffix `_r`).
25 //! Comments have been preserved where possible, only slightly adapted.
27 //! Instead of keeping a pointer to a configuration struct and inspecting it
28 //! dynamically on every operation, types (e.g. `ieee::Double`), traits
29 //! (e.g. `ieee::Semantics`) and associated constants are employed for
30 //! increased type safety and performance.
32 //! On-heap bigints are replaced everywhere (except in decimal conversion),
33 //! with short arrays of `type Limb = u128` elements (instead of `u64`),
34 //! This allows fitting the largest supported significands in one integer
35 //! (`ieee::Quad` and `ppc::Fallback` use slightly less than 128 bits).
36 //! All of the functions in the `ieee::sig` module operate on slices.
40 //! This API is completely unstable and subject to change.
42 #![doc(html_logo_url = "https://www.rust-lang.org/logos/rust-logo-128x128-blk-v2.png",
43 html_favicon_url = "https://doc.rust-lang.org/favicon.ico",
44 html_root_url = "https://doc.rust-lang.org/nightly/")]
46 #![forbid(unsafe_code)]
49 #![feature(i128_type)]
50 #![feature(slice_patterns)]
54 extern crate rustc_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, BitOrAssign};
60 use std::str::FromStr;
63 /// IEEE-754R 7: Default exception handling.
65 /// UNDERFLOW or OVERFLOW are always returned or-ed with INEXACT.
70 const INVALID_OP = 0x01,
71 const DIV_BY_ZERO = 0x02,
72 const OVERFLOW = 0x04,
73 const UNDERFLOW = 0x08,
78 impl BitOrAssign for Status {
79 fn bitor_assign(&mut self, rhs: Self) {
85 #[derive(Copy, Clone, PartialEq, Eq, PartialOrd, Ord, Debug)]
86 pub struct StatusAnd<T> {
92 pub fn and<T>(self, value: T) -> StatusAnd<T> {
100 impl<T> StatusAnd<T> {
101 fn map<F: FnOnce(T) -> U, U>(self, f: F) -> StatusAnd<U> {
104 value: f(self.value),
110 macro_rules! unpack {
111 ($status:ident|=, $e:expr) => {
113 $crate::StatusAnd { status, value } => {
119 ($status:ident=, $e:expr) => {
121 $crate::StatusAnd { status, value } => {
129 /// Category of internally-represented number.
130 #[derive(Copy, Clone, PartialEq, Eq, Debug)]
138 /// IEEE-754R 4.3: Rounding-direction attributes.
139 #[derive(Copy, Clone, PartialEq, Eq, Debug)]
150 fn neg(self) -> Round {
152 Round::TowardPositive => Round::TowardNegative,
153 Round::TowardNegative => Round::TowardPositive,
154 Round::NearestTiesToEven | Round::TowardZero | Round::NearestTiesToAway => self,
159 /// A signed type to represent a floating point number's unbiased exponent.
160 pub type ExpInt = i16;
162 // \c ilogb error results.
163 pub const IEK_INF: ExpInt = ExpInt::max_value();
164 pub const IEK_NAN: ExpInt = ExpInt::min_value();
165 pub const IEK_ZERO: ExpInt = ExpInt::min_value() + 1;
167 #[derive(Copy, Clone, PartialEq, Eq, Debug)]
168 pub struct ParseError(pub &'static str);
170 /// A self-contained host- and target-independent arbitrary-precision
171 /// floating-point software implementation.
173 /// `apfloat` uses significand bignum integer arithmetic as provided by functions
174 /// in the `ieee::sig`.
176 /// Written for clarity rather than speed, in particular with a view to use in
177 /// the front-end of a cross compiler so that target arithmetic can be correctly
178 /// performed on the host. Performance should nonetheless be reasonable,
179 /// particularly for its intended use. It may be useful as a base
180 /// implementation for a run-time library during development of a faster
181 /// target-specific one.
183 /// All 5 rounding modes in the IEEE-754R draft are handled correctly for all
184 /// implemented operations. Currently implemented operations are add, subtract,
185 /// multiply, divide, fused-multiply-add, conversion-to-float,
186 /// conversion-to-integer and conversion-from-integer. New rounding modes
187 /// (e.g. away from zero) can be added with three or four lines of code.
189 /// Four formats are built-in: IEEE single precision, double precision,
190 /// quadruple precision, and x87 80-bit extended double (when operating with
191 /// full extended precision). Adding a new format that obeys IEEE semantics
192 /// only requires adding two lines of code: a declaration and definition of the
195 /// All operations return the status of that operation as an exception bit-mask,
196 /// so multiple operations can be done consecutively with their results or-ed
197 /// together. The returned status can be useful for compiler diagnostics; e.g.,
198 /// inexact, underflow and overflow can be easily diagnosed on constant folding,
199 /// and compiler optimizers can determine what exceptions would be raised by
200 /// folding operations and optimize, or perhaps not optimize, accordingly.
202 /// At present, underflow tininess is detected after rounding; it should be
203 /// straight forward to add support for the before-rounding case too.
205 /// The library reads hexadecimal floating point numbers as per C99, and
206 /// correctly rounds if necessary according to the specified rounding mode.
207 /// Syntax is required to have been validated by the caller.
209 /// It also reads decimal floating point numbers and correctly rounds according
210 /// to the specified rounding mode.
212 /// Non-zero finite numbers are represented internally as a sign bit, a 16-bit
213 /// signed exponent, and the significand as an array of integer limbs. After
214 /// normalization of a number of precision P the exponent is within the range of
215 /// the format, and if the number is not denormal the P-th bit of the
216 /// significand is set as an explicit integer bit. For denormals the most
217 /// significant bit is shifted right so that the exponent is maintained at the
218 /// format's minimum, so that the smallest denormal has just the least
219 /// significant bit of the significand set. The sign of zeros and infinities
220 /// is significant; the exponent and significand of such numbers is not stored,
221 /// but has a known implicit (deterministic) value: 0 for the significands, 0
222 /// for zero exponent, all 1 bits for infinity exponent. For NaNs the sign and
223 /// significand are deterministic, although not really meaningful, and preserved
224 /// in non-conversion operations. The exponent is implicitly all 1 bits.
226 /// `apfloat` does not provide any exception handling beyond default exception
227 /// handling. We represent Signaling NaNs via IEEE-754R 2008 6.2.1 should clause
228 /// by encoding Signaling NaNs with the first bit of its trailing significand as
234 /// Some features that may or may not be worth adding:
236 /// Optional ability to detect underflow tininess before rounding.
238 /// New formats: x87 in single and double precision mode (IEEE apart from
239 /// extended exponent range) (hard).
241 /// New operations: sqrt, nexttoward.
246 + FromStr<Err = ParseError>
255 + Add<Output = StatusAnd<Self>>
256 + Sub<Output = StatusAnd<Self>>
257 + Mul<Output = StatusAnd<Self>>
258 + Div<Output = StatusAnd<Self>>
259 + Rem<Output = StatusAnd<Self>> {
260 /// Total number of bits in the in-memory format.
263 /// Number of bits in the significand. This includes the integer bit.
264 const PRECISION: usize;
266 /// The largest E such that 2^E is representable; this matches the
267 /// definition of IEEE 754.
268 const MAX_EXP: ExpInt;
270 /// The smallest E such that 2^E is a normalized number; this
271 /// matches the definition of IEEE 754.
272 const MIN_EXP: ExpInt;
277 /// Positive Infinity.
278 const INFINITY: Self;
280 /// NaN (Not a Number).
281 // FIXME(eddyb) provide a default when qnan becomes const fn.
284 /// Factory for QNaN values.
285 // FIXME(eddyb) should be const fn.
286 fn qnan(payload: Option<u128>) -> Self;
288 /// Factory for SNaN values.
289 // FIXME(eddyb) should be const fn.
290 fn snan(payload: Option<u128>) -> Self;
292 /// Largest finite number.
293 // FIXME(eddyb) should be const (but FloatPair::largest is nontrivial).
294 fn largest() -> Self;
296 /// Smallest (by magnitude) finite number.
297 /// Might be denormalized, which implies a relative loss of precision.
298 const SMALLEST: Self;
300 /// Smallest (by magnitude) normalized finite number.
301 // FIXME(eddyb) should be const (but FloatPair::smallest_normalized is nontrivial).
302 fn smallest_normalized() -> Self;
306 fn add_r(self, rhs: Self, round: Round) -> StatusAnd<Self>;
307 fn sub_r(self, rhs: Self, round: Round) -> StatusAnd<Self> {
308 self.add_r(-rhs, round)
310 fn mul_r(self, rhs: Self, round: Round) -> StatusAnd<Self>;
311 fn mul_add_r(self, multiplicand: Self, addend: Self, round: Round) -> StatusAnd<Self>;
312 fn mul_add(self, multiplicand: Self, addend: Self) -> StatusAnd<Self> {
313 self.mul_add_r(multiplicand, addend, Round::NearestTiesToEven)
315 fn div_r(self, rhs: Self, round: Round) -> StatusAnd<Self>;
317 // This is not currently correct in all cases.
318 fn ieee_rem(self, rhs: Self) -> StatusAnd<Self> {
322 v = unpack!(status=, v / rhs);
323 if status == Status::DIV_BY_ZERO {
324 return status.and(self);
327 assert!(Self::PRECISION < 128);
330 let x = unpack!(status=, v.to_i128_r(128, Round::NearestTiesToEven, &mut false));
331 if status == Status::INVALID_OP {
332 return status.and(self);
336 let mut v = unpack!(status=, Self::from_i128(x));
337 assert_eq!(status, Status::OK); // should always work
340 v = unpack!(status=, v * rhs);
341 assert_eq!(status - Status::INEXACT, Status::OK); // should not overflow or underflow
344 v = unpack!(status=, self - v);
345 assert_eq!(status - Status::INEXACT, Status::OK); // likewise
348 status.and(v.copy_sign(self)) // IEEE754 requires this
353 /// C fmod, or llvm frem.
354 fn c_fmod(self, rhs: Self) -> StatusAnd<Self>;
355 fn round_to_integral(self, round: Round) -> StatusAnd<Self>;
357 /// IEEE-754R 2008 5.3.1: nextUp.
358 fn next_up(self) -> StatusAnd<Self>;
360 /// IEEE-754R 2008 5.3.1: nextDown.
362 /// *NOTE* since nextDown(x) = -nextUp(-x), we only implement nextUp with
363 /// appropriate sign switching before/after the computation.
364 fn next_down(self) -> StatusAnd<Self> {
365 (-self).next_up().map(|r| -r)
368 fn abs(self) -> Self {
369 if self.is_negative() { -self } else { self }
371 fn copy_sign(self, rhs: Self) -> Self {
372 if self.is_negative() != rhs.is_negative() {
380 fn from_bits(input: u128) -> Self;
381 fn from_i128_r(input: i128, round: Round) -> StatusAnd<Self> {
383 Self::from_u128_r(-input as u128, -round).map(|r| -r)
385 Self::from_u128_r(input as u128, round)
388 fn from_i128(input: i128) -> StatusAnd<Self> {
389 Self::from_i128_r(input, Round::NearestTiesToEven)
391 fn from_u128_r(input: u128, round: Round) -> StatusAnd<Self>;
392 fn from_u128(input: u128) -> StatusAnd<Self> {
393 Self::from_u128_r(input, Round::NearestTiesToEven)
395 fn from_str_r(s: &str, round: Round) -> Result<StatusAnd<Self>, ParseError>;
396 fn to_bits(self) -> u128;
398 /// Convert a floating point number to an integer according to the
399 /// rounding mode. In case of an invalid operation exception,
400 /// deterministic values are returned, namely zero for NaNs and the
401 /// minimal or maximal value respectively for underflow or overflow.
402 /// If the rounded value is in range but the floating point number is
403 /// not the exact integer, the C standard doesn't require an inexact
404 /// exception to be raised. IEEE-854 does require it so we do that.
406 /// Note that for conversions to integer type the C standard requires
407 /// round-to-zero to always be used.
409 /// The *is_exact output tells whether the result is exact, in the sense
410 /// that converting it back to the original floating point type produces
411 /// the original value. This is almost equivalent to result==Status::OK,
412 /// except for negative zeroes.
413 fn to_i128_r(self, width: usize, round: Round, is_exact: &mut bool) -> StatusAnd<i128> {
415 if self.is_negative() {
417 // Negative zero can't be represented as an int.
420 let r = unpack!(status=, (-self).to_u128_r(width, -round, is_exact));
422 // Check for values that don't fit in the signed integer.
423 if r > (1 << (width - 1)) {
424 // Return the most negative integer for the given width.
426 Status::INVALID_OP.and(-1 << (width - 1))
428 status.and(r.wrapping_neg() as i128)
431 // Positive case is simpler, can pretend it's a smaller unsigned
432 // integer, and `to_u128` will take care of all the edge cases.
433 self.to_u128_r(width - 1, round, is_exact).map(
438 fn to_i128(self, width: usize) -> StatusAnd<i128> {
439 self.to_i128_r(width, Round::TowardZero, &mut true)
441 fn to_u128_r(self, width: usize, round: Round, is_exact: &mut bool) -> StatusAnd<u128>;
442 fn to_u128(self, width: usize) -> StatusAnd<u128> {
443 self.to_u128_r(width, Round::TowardZero, &mut true)
446 fn cmp_abs_normal(self, rhs: Self) -> Ordering;
448 /// Bitwise comparison for equality (QNaNs compare equal, 0!=-0).
449 fn bitwise_eq(self, rhs: Self) -> bool;
451 // IEEE-754R 5.7.2 General operations.
453 /// Implements IEEE minNum semantics. Returns the smaller of the 2 arguments if
454 /// both are not NaN. If either argument is a NaN, returns the other argument.
455 fn min(self, other: Self) -> Self {
458 } else if other.is_nan() {
460 } else if other.partial_cmp(&self) == Some(Ordering::Less) {
467 /// Implements IEEE maxNum semantics. Returns the larger of the 2 arguments if
468 /// both are not NaN. If either argument is a NaN, returns the other argument.
469 fn max(self, other: Self) -> Self {
472 } else if other.is_nan() {
474 } else if self.partial_cmp(&other) == Some(Ordering::Less) {
481 /// IEEE-754R isSignMinus: Returns true if and only if the current value is
484 /// This applies to zeros and NaNs as well.
485 fn is_negative(self) -> bool;
487 /// IEEE-754R isNormal: Returns true if and only if the current value is normal.
489 /// This implies that the current value of the float is not zero, subnormal,
490 /// infinite, or NaN following the definition of normality from IEEE-754R.
491 fn is_normal(self) -> bool {
492 !self.is_denormal() && self.is_finite_non_zero()
495 /// Returns true if and only if the current value is zero, subnormal, or
498 /// This means that the value is not infinite or NaN.
499 fn is_finite(self) -> bool {
500 !self.is_nan() && !self.is_infinite()
503 /// Returns true if and only if the float is plus or minus zero.
504 fn is_zero(self) -> bool {
505 self.category() == Category::Zero
508 /// IEEE-754R isSubnormal(): Returns true if and only if the float is a
510 fn is_denormal(self) -> bool;
512 /// IEEE-754R isInfinite(): Returns true if and only if the float is infinity.
513 fn is_infinite(self) -> bool {
514 self.category() == Category::Infinity
517 /// Returns true if and only if the float is a quiet or signaling NaN.
518 fn is_nan(self) -> bool {
519 self.category() == Category::NaN
522 /// Returns true if and only if the float is a signaling NaN.
523 fn is_signaling(self) -> bool;
527 fn category(self) -> Category;
528 fn is_non_zero(self) -> bool {
531 fn is_finite_non_zero(self) -> bool {
532 self.is_finite() && !self.is_zero()
534 fn is_pos_zero(self) -> bool {
535 self.is_zero() && !self.is_negative()
537 fn is_neg_zero(self) -> bool {
538 self.is_zero() && self.is_negative()
541 /// Returns true if and only if the number has the smallest possible non-zero
542 /// magnitude in the current semantics.
543 fn is_smallest(self) -> bool {
544 Self::SMALLEST.copy_sign(self).bitwise_eq(self)
547 /// Returns true if and only if the number has the largest possible finite
548 /// magnitude in the current semantics.
549 fn is_largest(self) -> bool {
550 Self::largest().copy_sign(self).bitwise_eq(self)
553 /// Returns true if and only if the number is an exact integer.
554 fn is_integer(self) -> bool {
555 // This could be made more efficient; I'm going for obviously correct.
556 if !self.is_finite() {
559 self.round_to_integral(Round::TowardZero).value.bitwise_eq(
564 /// If this value has an exact multiplicative inverse, return it.
565 fn get_exact_inverse(self) -> Option<Self>;
567 /// Returns the exponent of the internal representation of the Float.
569 /// Because the radix of Float is 2, this is equivalent to floor(log2(x)).
570 /// For special Float values, this returns special error codes:
572 /// NaN -> \c IEK_NAN
574 /// Inf -> \c IEK_INF
576 fn ilogb(self) -> ExpInt;
578 /// Returns: self * 2^exp for integral exponents.
579 fn scalbn_r(self, exp: ExpInt, round: Round) -> Self;
580 fn scalbn(self, exp: ExpInt) -> Self {
581 self.scalbn_r(exp, Round::NearestTiesToEven)
584 /// Equivalent of C standard library function.
586 /// While the C standard says exp is an unspecified value for infinity and nan,
587 /// this returns INT_MAX for infinities, and INT_MIN for NaNs (see `ilogb`).
588 fn frexp_r(self, exp: &mut ExpInt, round: Round) -> Self;
589 fn frexp(self, exp: &mut ExpInt) -> Self {
590 self.frexp_r(exp, Round::NearestTiesToEven)
594 pub trait FloatConvert<T: Float>: Float {
595 /// Convert a value of one floating point type to another.
596 /// The return value corresponds to the IEEE754 exceptions. *loses_info
597 /// records whether the transformation lost information, i.e. whether
598 /// converting the result back to the original type will produce the
599 /// original value (this is almost the same as return value==Status::OK,
600 /// but there are edge cases where this is not so).
601 fn convert_r(self, round: Round, loses_info: &mut bool) -> StatusAnd<T>;
602 fn convert(self, loses_info: &mut bool) -> StatusAnd<T> {
603 self.convert_r(Round::NearestTiesToEven, loses_info)
607 macro_rules! float_common_impls {
608 ($ty:ident<$t:tt>) => {
609 impl<$t> Default for $ty<$t> where Self: Float {
610 fn default() -> Self {
615 impl<$t> ::std::str::FromStr for $ty<$t> where Self: Float {
616 type Err = ParseError;
617 fn from_str(s: &str) -> Result<Self, ParseError> {
618 Self::from_str_r(s, Round::NearestTiesToEven).map(|x| x.value)
622 // Rounding ties to the nearest even, by default.
624 impl<$t> ::std::ops::Add for $ty<$t> where Self: Float {
625 type Output = StatusAnd<Self>;
626 fn add(self, rhs: Self) -> StatusAnd<Self> {
627 self.add_r(rhs, Round::NearestTiesToEven)
631 impl<$t> ::std::ops::Sub for $ty<$t> where Self: Float {
632 type Output = StatusAnd<Self>;
633 fn sub(self, rhs: Self) -> StatusAnd<Self> {
634 self.sub_r(rhs, Round::NearestTiesToEven)
638 impl<$t> ::std::ops::Mul for $ty<$t> where Self: Float {
639 type Output = StatusAnd<Self>;
640 fn mul(self, rhs: Self) -> StatusAnd<Self> {
641 self.mul_r(rhs, Round::NearestTiesToEven)
645 impl<$t> ::std::ops::Div for $ty<$t> where Self: Float {
646 type Output = StatusAnd<Self>;
647 fn div(self, rhs: Self) -> StatusAnd<Self> {
648 self.div_r(rhs, Round::NearestTiesToEven)
652 impl<$t> ::std::ops::Rem for $ty<$t> where Self: Float {
653 type Output = StatusAnd<Self>;
654 fn rem(self, rhs: Self) -> StatusAnd<Self> {
659 impl<$t> ::std::ops::AddAssign for $ty<$t> where Self: Float {
660 fn add_assign(&mut self, rhs: Self) {
661 *self = (*self + rhs).value;
665 impl<$t> ::std::ops::SubAssign for $ty<$t> where Self: Float {
666 fn sub_assign(&mut self, rhs: Self) {
667 *self = (*self - rhs).value;
671 impl<$t> ::std::ops::MulAssign for $ty<$t> where Self: Float {
672 fn mul_assign(&mut self, rhs: Self) {
673 *self = (*self * rhs).value;
677 impl<$t> ::std::ops::DivAssign for $ty<$t> where Self: Float {
678 fn div_assign(&mut self, rhs: Self) {
679 *self = (*self / rhs).value;
683 impl<$t> ::std::ops::RemAssign for $ty<$t> where Self: Float {
684 fn rem_assign(&mut self, rhs: Self) {
685 *self = (*self % rhs).value;