1 //! Port of LLVM's APFloat software floating-point implementation from the
2 //! following C++ sources (please update commit hash when backporting):
3 //! <https://github.com/llvm-mirror/llvm/tree/23efab2bbd424ed13495a420ad8641cb2c6c28f9>
5 //! * `include/llvm/ADT/APFloat.h` -> `Float` and `FloatConvert` traits
6 //! * `lib/Support/APFloat.cpp` -> `ieee` and `ppc` modules
7 //! * `unittests/ADT/APFloatTest.cpp` -> `tests` directory
9 //! The port contains no unsafe code, global state, or side-effects in general,
10 //! and the only allocations are in the conversion to/from decimal strings.
12 //! Most of the API and the testcases are intact in some form or another,
13 //! with some ergonomic changes, such as idiomatic short names, returning
14 //! new values instead of mutating the receiver, and having separate method
15 //! variants that take a non-default rounding mode (with the suffix `_r`).
16 //! Comments have been preserved where possible, only slightly adapted.
18 //! Instead of keeping a pointer to a configuration struct and inspecting it
19 //! dynamically on every operation, types (e.g., `ieee::Double`), traits
20 //! (e.g., `ieee::Semantics`) and associated constants are employed for
21 //! increased type safety and performance.
23 //! On-heap bigints are replaced everywhere (except in decimal conversion),
24 //! with short arrays of `type Limb = u128` elements (instead of `u64`),
25 //! This allows fitting the largest supported significands in one integer
26 //! (`ieee::Quad` and `ppc::Fallback` use slightly less than 128 bits).
27 //! All of the functions in the `ieee::sig` module operate on slices.
31 //! This API is completely unstable and subject to change.
33 #![doc(html_root_url = "https://doc.rust-lang.org/nightly/")]
34 #![forbid(unsafe_code)]
38 use std::cmp::Ordering;
40 use std::ops::{Neg, Add, Sub, Mul, Div, Rem};
41 use std::ops::{AddAssign, SubAssign, MulAssign, DivAssign, RemAssign};
42 use std::str::FromStr;
45 /// IEEE-754R 7: Default exception handling.
47 /// UNDERFLOW or OVERFLOW are always returned or-ed with INEXACT.
49 pub struct Status: u8 {
51 const INVALID_OP = 0x01;
52 const DIV_BY_ZERO = 0x02;
53 const OVERFLOW = 0x04;
54 const UNDERFLOW = 0x08;
60 #[derive(Copy, Clone, PartialEq, Eq, PartialOrd, Ord, Debug)]
61 pub struct StatusAnd<T> {
67 pub fn and<T>(self, value: T) -> StatusAnd<T> {
75 impl<T> StatusAnd<T> {
76 pub fn map<F: FnOnce(T) -> U, U>(self, f: F) -> StatusAnd<U> {
86 ($status:ident|=, $e:expr) => {
88 $crate::StatusAnd { status, value } => {
94 ($status:ident=, $e:expr) => {
96 $crate::StatusAnd { status, value } => {
104 /// Category of internally-represented number.
105 #[derive(Copy, Clone, PartialEq, Eq, Debug)]
113 /// IEEE-754R 4.3: Rounding-direction attributes.
114 #[derive(Copy, Clone, PartialEq, Eq, Debug)]
125 fn neg(self) -> Round {
127 Round::TowardPositive => Round::TowardNegative,
128 Round::TowardNegative => Round::TowardPositive,
129 Round::NearestTiesToEven | Round::TowardZero | Round::NearestTiesToAway => self,
134 /// A signed type to represent a floating point number's unbiased exponent.
135 pub type ExpInt = i16;
137 // \c ilogb error results.
138 pub const IEK_INF: ExpInt = ExpInt::max_value();
139 pub const IEK_NAN: ExpInt = ExpInt::min_value();
140 pub const IEK_ZERO: ExpInt = ExpInt::min_value() + 1;
142 #[derive(Copy, Clone, PartialEq, Eq, Debug)]
143 pub struct ParseError(pub &'static str);
145 /// A self-contained host- and target-independent arbitrary-precision
146 /// floating-point software implementation.
148 /// `apfloat` uses significand bignum integer arithmetic as provided by functions
149 /// in the `ieee::sig`.
151 /// Written for clarity rather than speed, in particular with a view to use in
152 /// the front-end of a cross compiler so that target arithmetic can be correctly
153 /// performed on the host. Performance should nonetheless be reasonable,
154 /// particularly for its intended use. It may be useful as a base
155 /// implementation for a run-time library during development of a faster
156 /// target-specific one.
158 /// All 5 rounding modes in the IEEE-754R draft are handled correctly for all
159 /// implemented operations. Currently implemented operations are add, subtract,
160 /// multiply, divide, fused-multiply-add, conversion-to-float,
161 /// conversion-to-integer and conversion-from-integer. New rounding modes
162 /// (e.g., away from zero) can be added with three or four lines of code.
164 /// Four formats are built-in: IEEE single precision, double precision,
165 /// quadruple precision, and x87 80-bit extended double (when operating with
166 /// full extended precision). Adding a new format that obeys IEEE semantics
167 /// only requires adding two lines of code: a declaration and definition of the
170 /// All operations return the status of that operation as an exception bit-mask,
171 /// so multiple operations can be done consecutively with their results or-ed
172 /// together. The returned status can be useful for compiler diagnostics; e.g.,
173 /// inexact, underflow and overflow can be easily diagnosed on constant folding,
174 /// and compiler optimizers can determine what exceptions would be raised by
175 /// folding operations and optimize, or perhaps not optimize, accordingly.
177 /// At present, underflow tininess is detected after rounding; it should be
178 /// straight forward to add support for the before-rounding case too.
180 /// The library reads hexadecimal floating point numbers as per C99, and
181 /// correctly rounds if necessary according to the specified rounding mode.
182 /// Syntax is required to have been validated by the caller.
184 /// It also reads decimal floating point numbers and correctly rounds according
185 /// to the specified rounding mode.
187 /// Non-zero finite numbers are represented internally as a sign bit, a 16-bit
188 /// signed exponent, and the significand as an array of integer limbs. After
189 /// normalization of a number of precision P the exponent is within the range of
190 /// the format, and if the number is not denormal the P-th bit of the
191 /// significand is set as an explicit integer bit. For denormals the most
192 /// significant bit is shifted right so that the exponent is maintained at the
193 /// format's minimum, so that the smallest denormal has just the least
194 /// significant bit of the significand set. The sign of zeros and infinities
195 /// is significant; the exponent and significand of such numbers is not stored,
196 /// but has a known implicit (deterministic) value: 0 for the significands, 0
197 /// for zero exponent, all 1 bits for infinity exponent. For NaNs the sign and
198 /// significand are deterministic, although not really meaningful, and preserved
199 /// in non-conversion operations. The exponent is implicitly all 1 bits.
201 /// `apfloat` does not provide any exception handling beyond default exception
202 /// handling. We represent Signaling NaNs via IEEE-754R 2008 6.2.1 should clause
203 /// by encoding Signaling NaNs with the first bit of its trailing significand
209 /// Some features that may or may not be worth adding:
211 /// Optional ability to detect underflow tininess before rounding.
213 /// New formats: x87 in single and double precision mode (IEEE apart from
214 /// extended exponent range) (hard).
216 /// New operations: sqrt, nexttoward.
221 + FromStr<Err = ParseError>
230 + Add<Output = StatusAnd<Self>>
231 + Sub<Output = StatusAnd<Self>>
232 + Mul<Output = StatusAnd<Self>>
233 + Div<Output = StatusAnd<Self>>
234 + Rem<Output = StatusAnd<Self>> {
235 /// Total number of bits in the in-memory format.
238 /// Number of bits in the significand. This includes the integer bit.
239 const PRECISION: usize;
241 /// The largest E such that 2<sup>E</sup> is representable; this matches the
242 /// definition of IEEE 754.
243 const MAX_EXP: ExpInt;
245 /// The smallest E such that 2<sup>E</sup> is a normalized number; this
246 /// matches the definition of IEEE 754.
247 const MIN_EXP: ExpInt;
252 /// Positive Infinity.
253 const INFINITY: Self;
255 /// NaN (Not a Number).
256 // FIXME(eddyb) provide a default when qnan becomes const fn.
259 /// Factory for QNaN values.
260 // FIXME(eddyb) should be const fn.
261 fn qnan(payload: Option<u128>) -> Self;
263 /// Factory for SNaN values.
264 // FIXME(eddyb) should be const fn.
265 fn snan(payload: Option<u128>) -> Self;
267 /// Largest finite number.
268 // FIXME(eddyb) should be const (but FloatPair::largest is nontrivial).
269 fn largest() -> Self;
271 /// Smallest (by magnitude) finite number.
272 /// Might be denormalized, which implies a relative loss of precision.
273 const SMALLEST: Self;
275 /// Smallest (by magnitude) normalized finite number.
276 // FIXME(eddyb) should be const (but FloatPair::smallest_normalized is nontrivial).
277 fn smallest_normalized() -> Self;
281 fn add_r(self, rhs: Self, round: Round) -> StatusAnd<Self>;
282 fn sub_r(self, rhs: Self, round: Round) -> StatusAnd<Self> {
283 self.add_r(-rhs, round)
285 fn mul_r(self, rhs: Self, round: Round) -> StatusAnd<Self>;
286 fn mul_add_r(self, multiplicand: Self, addend: Self, round: Round) -> StatusAnd<Self>;
287 fn mul_add(self, multiplicand: Self, addend: Self) -> StatusAnd<Self> {
288 self.mul_add_r(multiplicand, addend, Round::NearestTiesToEven)
290 fn div_r(self, rhs: Self, round: Round) -> StatusAnd<Self>;
292 // This is not currently correct in all cases.
293 fn ieee_rem(self, rhs: Self) -> StatusAnd<Self> {
297 v = unpack!(status=, v / rhs);
298 if status == Status::DIV_BY_ZERO {
299 return status.and(self);
302 assert!(Self::PRECISION < 128);
305 let x = unpack!(status=, v.to_i128_r(128, Round::NearestTiesToEven, &mut false));
306 if status == Status::INVALID_OP {
307 return status.and(self);
311 let mut v = unpack!(status=, Self::from_i128(x));
312 assert_eq!(status, Status::OK); // should always work
315 v = unpack!(status=, v * rhs);
316 assert_eq!(status - Status::INEXACT, Status::OK); // should not overflow or underflow
319 v = unpack!(status=, self - v);
320 assert_eq!(status - Status::INEXACT, Status::OK); // likewise
323 status.and(v.copy_sign(self)) // IEEE754 requires this
328 /// C fmod, or llvm frem.
329 fn c_fmod(self, rhs: Self) -> StatusAnd<Self>;
330 fn round_to_integral(self, round: Round) -> StatusAnd<Self>;
332 /// IEEE-754R 2008 5.3.1: nextUp.
333 fn next_up(self) -> StatusAnd<Self>;
335 /// IEEE-754R 2008 5.3.1: nextDown.
337 /// *NOTE* since nextDown(x) = -nextUp(-x), we only implement nextUp with
338 /// appropriate sign switching before/after the computation.
339 fn next_down(self) -> StatusAnd<Self> {
340 (-self).next_up().map(|r| -r)
343 fn abs(self) -> Self {
344 if self.is_negative() { -self } else { self }
346 fn copy_sign(self, rhs: Self) -> Self {
347 if self.is_negative() != rhs.is_negative() {
355 fn from_bits(input: u128) -> Self;
356 fn from_i128_r(input: i128, round: Round) -> StatusAnd<Self> {
358 Self::from_u128_r(input.wrapping_neg() as u128, -round).map(|r| -r)
360 Self::from_u128_r(input as u128, round)
363 fn from_i128(input: i128) -> StatusAnd<Self> {
364 Self::from_i128_r(input, Round::NearestTiesToEven)
366 fn from_u128_r(input: u128, round: Round) -> StatusAnd<Self>;
367 fn from_u128(input: u128) -> StatusAnd<Self> {
368 Self::from_u128_r(input, Round::NearestTiesToEven)
370 fn from_str_r(s: &str, round: Round) -> Result<StatusAnd<Self>, ParseError>;
371 fn to_bits(self) -> u128;
373 /// Converts a floating point number to an integer according to the
374 /// rounding mode. In case of an invalid operation exception,
375 /// deterministic values are returned, namely zero for NaNs and the
376 /// minimal or maximal value respectively for underflow or overflow.
377 /// If the rounded value is in range but the floating point number is
378 /// not the exact integer, the C standard doesn't require an inexact
379 /// exception to be raised. IEEE-854 does require it so we do that.
381 /// Note that for conversions to integer type the C standard requires
382 /// round-to-zero to always be used.
384 /// The *is_exact output tells whether the result is exact, in the sense
385 /// that converting it back to the original floating point type produces
386 /// the original value. This is almost equivalent to `result == Status::OK`,
387 /// except for negative zeroes.
388 fn to_i128_r(self, width: usize, round: Round, is_exact: &mut bool) -> StatusAnd<i128> {
390 if self.is_negative() {
392 // Negative zero can't be represented as an int.
395 let r = unpack!(status=, (-self).to_u128_r(width, -round, is_exact));
397 // Check for values that don't fit in the signed integer.
398 if r > (1 << (width - 1)) {
399 // Return the most negative integer for the given width.
401 Status::INVALID_OP.and(-1 << (width - 1))
403 status.and(r.wrapping_neg() as i128)
406 // Positive case is simpler, can pretend it's a smaller unsigned
407 // integer, and `to_u128` will take care of all the edge cases.
408 self.to_u128_r(width - 1, round, is_exact).map(
413 fn to_i128(self, width: usize) -> StatusAnd<i128> {
414 self.to_i128_r(width, Round::TowardZero, &mut true)
416 fn to_u128_r(self, width: usize, round: Round, is_exact: &mut bool) -> StatusAnd<u128>;
417 fn to_u128(self, width: usize) -> StatusAnd<u128> {
418 self.to_u128_r(width, Round::TowardZero, &mut true)
421 fn cmp_abs_normal(self, rhs: Self) -> Ordering;
423 /// Bitwise comparison for equality (QNaNs compare equal, 0!=-0).
424 fn bitwise_eq(self, rhs: Self) -> bool;
426 // IEEE-754R 5.7.2 General operations.
428 /// Implements IEEE minNum semantics. Returns the smaller of the 2 arguments if
429 /// both are not NaN. If either argument is a NaN, returns the other argument.
430 fn min(self, other: Self) -> Self {
433 } else if other.is_nan() {
435 } else if other.partial_cmp(&self) == Some(Ordering::Less) {
442 /// Implements IEEE maxNum semantics. Returns the larger of the 2 arguments if
443 /// both are not NaN. If either argument is a NaN, returns the other argument.
444 fn max(self, other: Self) -> Self {
447 } else if other.is_nan() {
449 } else if self.partial_cmp(&other) == Some(Ordering::Less) {
456 /// IEEE-754R isSignMinus: Returns whether the current value is
459 /// This applies to zeros and NaNs as well.
460 fn is_negative(self) -> bool;
462 /// IEEE-754R isNormal: Returns whether the current value is normal.
464 /// This implies that the current value of the float is not zero, subnormal,
465 /// infinite, or NaN following the definition of normality from IEEE-754R.
466 fn is_normal(self) -> bool {
467 !self.is_denormal() && self.is_finite_non_zero()
470 /// Returns `true` if the current value is zero, subnormal, or
473 /// This means that the value is not infinite or NaN.
474 fn is_finite(self) -> bool {
475 !self.is_nan() && !self.is_infinite()
478 /// Returns `true` if the float is plus or minus zero.
479 fn is_zero(self) -> bool {
480 self.category() == Category::Zero
483 /// IEEE-754R isSubnormal(): Returns whether the float is a
485 fn is_denormal(self) -> bool;
487 /// IEEE-754R isInfinite(): Returns whether the float is infinity.
488 fn is_infinite(self) -> bool {
489 self.category() == Category::Infinity
492 /// Returns `true` if the float is a quiet or signaling NaN.
493 fn is_nan(self) -> bool {
494 self.category() == Category::NaN
497 /// Returns `true` if the float is a signaling NaN.
498 fn is_signaling(self) -> bool;
502 fn category(self) -> Category;
503 fn is_non_zero(self) -> bool {
506 fn is_finite_non_zero(self) -> bool {
507 self.is_finite() && !self.is_zero()
509 fn is_pos_zero(self) -> bool {
510 self.is_zero() && !self.is_negative()
512 fn is_neg_zero(self) -> bool {
513 self.is_zero() && self.is_negative()
516 /// Returns `true` if the number has the smallest possible non-zero
517 /// magnitude in the current semantics.
518 fn is_smallest(self) -> bool {
519 Self::SMALLEST.copy_sign(self).bitwise_eq(self)
522 /// Returns `true` if the number has the largest possible finite
523 /// magnitude in the current semantics.
524 fn is_largest(self) -> bool {
525 Self::largest().copy_sign(self).bitwise_eq(self)
528 /// Returns `true` if the number is an exact integer.
529 fn is_integer(self) -> bool {
530 // This could be made more efficient; I'm going for obviously correct.
531 if !self.is_finite() {
534 self.round_to_integral(Round::TowardZero).value.bitwise_eq(
539 /// If this value has an exact multiplicative inverse, return it.
540 fn get_exact_inverse(self) -> Option<Self>;
542 /// Returns the exponent of the internal representation of the Float.
544 /// Because the radix of Float is 2, this is equivalent to floor(log2(x)).
545 /// For special Float values, this returns special error codes:
547 /// NaN -> \c IEK_NAN
549 /// Inf -> \c IEK_INF
551 fn ilogb(self) -> ExpInt;
553 /// Returns: self * 2<sup>exp</sup> for integral exponents.
554 fn scalbn_r(self, exp: ExpInt, round: Round) -> Self;
555 fn scalbn(self, exp: ExpInt) -> Self {
556 self.scalbn_r(exp, Round::NearestTiesToEven)
559 /// Equivalent of C standard library function.
561 /// While the C standard says exp is an unspecified value for infinity and nan,
562 /// this returns INT_MAX for infinities, and INT_MIN for NaNs (see `ilogb`).
563 fn frexp_r(self, exp: &mut ExpInt, round: Round) -> Self;
564 fn frexp(self, exp: &mut ExpInt) -> Self {
565 self.frexp_r(exp, Round::NearestTiesToEven)
569 pub trait FloatConvert<T: Float>: Float {
570 /// Converts a value of one floating point type to another.
571 /// The return value corresponds to the IEEE754 exceptions. *loses_info
572 /// records whether the transformation lost information, i.e., whether
573 /// converting the result back to the original type will produce the
574 /// original value (this is almost the same as return `value == Status::OK`,
575 /// but there are edge cases where this is not so).
576 fn convert_r(self, round: Round, loses_info: &mut bool) -> StatusAnd<T>;
577 fn convert(self, loses_info: &mut bool) -> StatusAnd<T> {
578 self.convert_r(Round::NearestTiesToEven, loses_info)
582 macro_rules! float_common_impls {
583 ($ty:ident<$t:tt>) => {
584 impl<$t> Default for $ty<$t> where Self: Float {
585 fn default() -> Self {
590 impl<$t> ::std::str::FromStr for $ty<$t> where Self: Float {
591 type Err = ParseError;
592 fn from_str(s: &str) -> Result<Self, ParseError> {
593 Self::from_str_r(s, Round::NearestTiesToEven).map(|x| x.value)
597 // Rounding ties to the nearest even, by default.
599 impl<$t> ::std::ops::Add for $ty<$t> where Self: Float {
600 type Output = StatusAnd<Self>;
601 fn add(self, rhs: Self) -> StatusAnd<Self> {
602 self.add_r(rhs, Round::NearestTiesToEven)
606 impl<$t> ::std::ops::Sub for $ty<$t> where Self: Float {
607 type Output = StatusAnd<Self>;
608 fn sub(self, rhs: Self) -> StatusAnd<Self> {
609 self.sub_r(rhs, Round::NearestTiesToEven)
613 impl<$t> ::std::ops::Mul for $ty<$t> where Self: Float {
614 type Output = StatusAnd<Self>;
615 fn mul(self, rhs: Self) -> StatusAnd<Self> {
616 self.mul_r(rhs, Round::NearestTiesToEven)
620 impl<$t> ::std::ops::Div for $ty<$t> where Self: Float {
621 type Output = StatusAnd<Self>;
622 fn div(self, rhs: Self) -> StatusAnd<Self> {
623 self.div_r(rhs, Round::NearestTiesToEven)
627 impl<$t> ::std::ops::Rem for $ty<$t> where Self: Float {
628 type Output = StatusAnd<Self>;
629 fn rem(self, rhs: Self) -> StatusAnd<Self> {
634 impl<$t> ::std::ops::AddAssign for $ty<$t> where Self: Float {
635 fn add_assign(&mut self, rhs: Self) {
636 *self = (*self + rhs).value;
640 impl<$t> ::std::ops::SubAssign for $ty<$t> where Self: Float {
641 fn sub_assign(&mut self, rhs: Self) {
642 *self = (*self - rhs).value;
646 impl<$t> ::std::ops::MulAssign for $ty<$t> where Self: Float {
647 fn mul_assign(&mut self, rhs: Self) {
648 *self = (*self * rhs).value;
652 impl<$t> ::std::ops::DivAssign for $ty<$t> where Self: Float {
653 fn div_assign(&mut self, rhs: Self) {
654 *self = (*self / rhs).value;
658 impl<$t> ::std::ops::RemAssign for $ty<$t> where Self: Float {
659 fn rem_assign(&mut self, rhs: Self) {
660 *self = (*self % rhs).value;