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/nightly-rustc/")]
35 #![forbid(unsafe_code)]
42 use core::cmp::Ordering;
44 use core::ops::{Add, Div, Mul, Neg, Rem, Sub};
45 use core::ops::{AddAssign, DivAssign, MulAssign, RemAssign, SubAssign};
46 use core::str::FromStr;
49 /// IEEE-754R 7: Default exception handling.
51 /// UNDERFLOW or OVERFLOW are always returned or-ed with INEXACT.
53 pub struct Status: u8 {
55 const INVALID_OP = 0x01;
56 const DIV_BY_ZERO = 0x02;
57 const OVERFLOW = 0x04;
58 const UNDERFLOW = 0x08;
64 #[derive(Copy, Clone, PartialEq, Eq, PartialOrd, Ord, Debug)]
65 pub struct StatusAnd<T> {
71 pub fn and<T>(self, value: T) -> StatusAnd<T> {
72 StatusAnd { status: self, value }
76 impl<T> StatusAnd<T> {
77 pub fn map<F: FnOnce(T) -> U, U>(self, f: F) -> StatusAnd<U> {
78 StatusAnd { status: self.status, value: f(self.value) }
84 ($status:ident|=, $e:expr) => {
86 $crate::StatusAnd { status, value } => {
92 ($status:ident=, $e:expr) => {
94 $crate::StatusAnd { status, value } => {
102 /// Category of internally-represented number.
103 #[derive(Copy, Clone, PartialEq, Eq, Debug)]
111 /// IEEE-754R 4.3: Rounding-direction attributes.
112 #[derive(Copy, Clone, PartialEq, Eq, Debug)]
123 fn neg(self) -> Round {
125 Round::TowardPositive => Round::TowardNegative,
126 Round::TowardNegative => Round::TowardPositive,
127 Round::NearestTiesToEven | Round::TowardZero | Round::NearestTiesToAway => self,
132 /// A signed type to represent a floating point number's unbiased exponent.
133 pub type ExpInt = i16;
135 // \c ilogb error results.
136 pub const IEK_INF: ExpInt = ExpInt::MAX;
137 pub const IEK_NAN: ExpInt = ExpInt::MIN;
138 pub const IEK_ZERO: ExpInt = ExpInt::MIN + 1;
140 #[derive(Copy, Clone, PartialEq, Eq, Debug)]
141 pub struct ParseError(pub &'static str);
143 /// A self-contained host- and target-independent arbitrary-precision
144 /// floating-point software implementation.
146 /// `apfloat` uses significand bignum integer arithmetic as provided by functions
147 /// in the `ieee::sig`.
149 /// Written for clarity rather than speed, in particular with a view to use in
150 /// the front-end of a cross compiler so that target arithmetic can be correctly
151 /// performed on the host. Performance should nonetheless be reasonable,
152 /// particularly for its intended use. It may be useful as a base
153 /// implementation for a run-time library during development of a faster
154 /// target-specific one.
156 /// All 5 rounding modes in the IEEE-754R draft are handled correctly for all
157 /// implemented operations. Currently implemented operations are add, subtract,
158 /// multiply, divide, fused-multiply-add, conversion-to-float,
159 /// conversion-to-integer and conversion-from-integer. New rounding modes
160 /// (e.g., away from zero) can be added with three or four lines of code.
162 /// Four formats are built-in: IEEE single precision, double precision,
163 /// quadruple precision, and x87 80-bit extended double (when operating with
164 /// full extended precision). Adding a new format that obeys IEEE semantics
165 /// only requires adding two lines of code: a declaration and definition of the
168 /// All operations return the status of that operation as an exception bit-mask,
169 /// so multiple operations can be done consecutively with their results or-ed
170 /// together. The returned status can be useful for compiler diagnostics; e.g.,
171 /// inexact, underflow and overflow can be easily diagnosed on constant folding,
172 /// and compiler optimizers can determine what exceptions would be raised by
173 /// folding operations and optimize, or perhaps not optimize, accordingly.
175 /// At present, underflow tininess is detected after rounding; it should be
176 /// straight forward to add support for the before-rounding case too.
178 /// The library reads hexadecimal floating point numbers as per C99, and
179 /// correctly rounds if necessary according to the specified rounding mode.
180 /// Syntax is required to have been validated by the caller.
182 /// It also reads decimal floating point numbers and correctly rounds according
183 /// to the specified rounding mode.
185 /// Non-zero finite numbers are represented internally as a sign bit, a 16-bit
186 /// signed exponent, and the significand as an array of integer limbs. After
187 /// normalization of a number of precision P the exponent is within the range of
188 /// the format, and if the number is not denormal the P-th bit of the
189 /// significand is set as an explicit integer bit. For denormals the most
190 /// significant bit is shifted right so that the exponent is maintained at the
191 /// format's minimum, so that the smallest denormal has just the least
192 /// significant bit of the significand set. The sign of zeros and infinities
193 /// is significant; the exponent and significand of such numbers is not stored,
194 /// but has a known implicit (deterministic) value: 0 for the significands, 0
195 /// for zero exponent, all 1 bits for infinity exponent. For NaNs the sign and
196 /// significand are deterministic, although not really meaningful, and preserved
197 /// in non-conversion operations. The exponent is implicitly all 1 bits.
199 /// `apfloat` does not provide any exception handling beyond default exception
200 /// handling. We represent Signaling NaNs via IEEE-754R 2008 6.2.1 should clause
201 /// by encoding Signaling NaNs with the first bit of its trailing significand
207 /// Some features that may or may not be worth adding:
209 /// Optional ability to detect underflow tininess before rounding.
211 /// New formats: x87 in single and double precision mode (IEEE apart from
212 /// extended exponent range) (hard).
214 /// New operations: sqrt, nexttoward.
219 + FromStr<Err = ParseError>
228 + Add<Output = StatusAnd<Self>>
229 + Sub<Output = StatusAnd<Self>>
230 + Mul<Output = StatusAnd<Self>>
231 + Div<Output = StatusAnd<Self>>
232 + Rem<Output = StatusAnd<Self>>
234 /// Total number of bits in the in-memory format.
237 /// Number of bits in the significand. This includes the integer bit.
238 const PRECISION: usize;
240 /// The largest E such that 2<sup>E</sup> is representable; this matches the
241 /// definition of IEEE 754.
242 const MAX_EXP: ExpInt;
244 /// The smallest E such that 2<sup>E</sup> is a normalized number; this
245 /// matches the definition of IEEE 754.
246 const MIN_EXP: ExpInt;
251 /// Positive Infinity.
252 const INFINITY: Self;
254 /// NaN (Not a Number).
255 // FIXME(eddyb) provide a default when qnan becomes const fn.
258 /// Factory for QNaN values.
259 // FIXME(eddyb) should be const fn.
260 fn qnan(payload: Option<u128>) -> Self;
262 /// Factory for SNaN values.
263 // FIXME(eddyb) should be const fn.
264 fn snan(payload: Option<u128>) -> Self;
266 /// Largest finite number.
267 // FIXME(eddyb) should be const (but FloatPair::largest is nontrivial).
268 fn largest() -> Self;
270 /// Smallest (by magnitude) finite number.
271 /// Might be denormalized, which implies a relative loss of precision.
272 const SMALLEST: Self;
274 /// Smallest (by magnitude) normalized finite number.
275 // FIXME(eddyb) should be const (but FloatPair::smallest_normalized is nontrivial).
276 fn smallest_normalized() -> Self;
280 fn add_r(self, rhs: Self, round: Round) -> StatusAnd<Self>;
281 fn sub_r(self, rhs: Self, round: Round) -> StatusAnd<Self> {
282 self.add_r(-rhs, round)
284 fn mul_r(self, rhs: Self, round: Round) -> StatusAnd<Self>;
285 fn mul_add_r(self, multiplicand: Self, addend: Self, round: Round) -> StatusAnd<Self>;
286 fn mul_add(self, multiplicand: Self, addend: Self) -> StatusAnd<Self> {
287 self.mul_add_r(multiplicand, addend, Round::NearestTiesToEven)
289 fn div_r(self, rhs: Self, round: Round) -> StatusAnd<Self>;
291 // This is not currently correct in all cases.
292 fn ieee_rem(self, rhs: Self) -> StatusAnd<Self> {
296 v = unpack!(status=, v / rhs);
297 if status == Status::DIV_BY_ZERO {
298 return status.and(self);
301 assert!(Self::PRECISION < 128);
304 let x = unpack!(status=, v.to_i128_r(128, Round::NearestTiesToEven, &mut false));
305 if status == Status::INVALID_OP {
306 return status.and(self);
310 let mut v = unpack!(status=, Self::from_i128(x));
311 assert_eq!(status, Status::OK); // should always work
314 v = unpack!(status=, v * rhs);
315 assert_eq!(status - Status::INEXACT, Status::OK); // should not overflow or underflow
318 v = unpack!(status=, self - v);
319 assert_eq!(status - Status::INEXACT, Status::OK); // likewise
322 status.and(v.copy_sign(self)) // IEEE754 requires this
327 /// C fmod, or llvm frem.
328 fn c_fmod(self, rhs: Self) -> StatusAnd<Self>;
329 fn round_to_integral(self, round: Round) -> StatusAnd<Self>;
331 /// IEEE-754R 2008 5.3.1: nextUp.
332 fn next_up(self) -> StatusAnd<Self>;
334 /// IEEE-754R 2008 5.3.1: nextDown.
336 /// *NOTE* since nextDown(x) = -nextUp(-x), we only implement nextUp with
337 /// appropriate sign switching before/after the computation.
338 fn next_down(self) -> StatusAnd<Self> {
339 (-self).next_up().map(|r| -r)
342 fn abs(self) -> Self {
343 if self.is_negative() { -self } else { self }
345 fn copy_sign(self, rhs: Self) -> Self {
346 if self.is_negative() != rhs.is_negative() { -self } else { self }
350 fn from_bits(input: u128) -> Self;
351 fn from_i128_r(input: i128, round: Round) -> StatusAnd<Self> {
353 Self::from_u128_r(input.wrapping_neg() as u128, -round).map(|r| -r)
355 Self::from_u128_r(input as u128, round)
358 fn from_i128(input: i128) -> StatusAnd<Self> {
359 Self::from_i128_r(input, Round::NearestTiesToEven)
361 fn from_u128_r(input: u128, round: Round) -> StatusAnd<Self>;
362 fn from_u128(input: u128) -> StatusAnd<Self> {
363 Self::from_u128_r(input, Round::NearestTiesToEven)
365 fn from_str_r(s: &str, round: Round) -> Result<StatusAnd<Self>, ParseError>;
366 fn to_bits(self) -> u128;
368 /// Converts a floating point number to an integer according to the
369 /// rounding mode. In case of an invalid operation exception,
370 /// deterministic values are returned, namely zero for NaNs and the
371 /// minimal or maximal value respectively for underflow or overflow.
372 /// If the rounded value is in range but the floating point number is
373 /// not the exact integer, the C standard doesn't require an inexact
374 /// exception to be raised. IEEE-854 does require it so we do that.
376 /// Note that for conversions to integer type the C standard requires
377 /// round-to-zero to always be used.
379 /// The *is_exact output tells whether the result is exact, in the sense
380 /// that converting it back to the original floating point type produces
381 /// the original value. This is almost equivalent to `result == Status::OK`,
382 /// except for negative zeroes.
383 fn to_i128_r(self, width: usize, round: Round, is_exact: &mut bool) -> StatusAnd<i128> {
385 if self.is_negative() {
387 // Negative zero can't be represented as an int.
390 let r = unpack!(status=, (-self).to_u128_r(width, -round, is_exact));
392 // Check for values that don't fit in the signed integer.
393 if r > (1 << (width - 1)) {
394 // Return the most negative integer for the given width.
396 Status::INVALID_OP.and(-1 << (width - 1))
398 status.and(r.wrapping_neg() as i128)
401 // Positive case is simpler, can pretend it's a smaller unsigned
402 // integer, and `to_u128` will take care of all the edge cases.
403 self.to_u128_r(width - 1, round, is_exact).map(|r| r as i128)
406 fn to_i128(self, width: usize) -> StatusAnd<i128> {
407 self.to_i128_r(width, Round::TowardZero, &mut true)
409 fn to_u128_r(self, width: usize, round: Round, is_exact: &mut bool) -> StatusAnd<u128>;
410 fn to_u128(self, width: usize) -> StatusAnd<u128> {
411 self.to_u128_r(width, Round::TowardZero, &mut true)
414 fn cmp_abs_normal(self, rhs: Self) -> Ordering;
416 /// Bitwise comparison for equality (QNaNs compare equal, 0!=-0).
417 fn bitwise_eq(self, rhs: Self) -> bool;
419 // IEEE-754R 5.7.2 General operations.
421 /// Implements IEEE minNum semantics. Returns the smaller of the 2 arguments if
422 /// both are not NaN. If either argument is a NaN, returns the other argument.
423 fn min(self, other: Self) -> Self {
426 } else if other.is_nan() {
428 } else if other.partial_cmp(&self) == Some(Ordering::Less) {
435 /// Implements IEEE maxNum semantics. Returns the larger of the 2 arguments if
436 /// both are not NaN. If either argument is a NaN, returns the other argument.
437 fn max(self, other: Self) -> Self {
440 } else if other.is_nan() {
442 } else if self.partial_cmp(&other) == Some(Ordering::Less) {
449 /// IEEE-754R isSignMinus: Returns whether the current value is
452 /// This applies to zeros and NaNs as well.
453 fn is_negative(self) -> bool;
455 /// IEEE-754R isNormal: Returns whether the current value is normal.
457 /// This implies that the current value of the float is not zero, subnormal,
458 /// infinite, or NaN following the definition of normality from IEEE-754R.
459 fn is_normal(self) -> bool {
460 !self.is_denormal() && self.is_finite_non_zero()
463 /// Returns `true` if the current value is zero, subnormal, or
466 /// This means that the value is not infinite or NaN.
467 fn is_finite(self) -> bool {
468 !self.is_nan() && !self.is_infinite()
471 /// Returns `true` if the float is plus or minus zero.
472 fn is_zero(self) -> bool {
473 self.category() == Category::Zero
476 /// IEEE-754R isSubnormal(): Returns whether the float is a
478 fn is_denormal(self) -> bool;
480 /// IEEE-754R isInfinite(): Returns whether the float is infinity.
481 fn is_infinite(self) -> bool {
482 self.category() == Category::Infinity
485 /// Returns `true` if the float is a quiet or signaling NaN.
486 fn is_nan(self) -> bool {
487 self.category() == Category::NaN
490 /// Returns `true` if the float is a signaling NaN.
491 fn is_signaling(self) -> bool;
495 fn category(self) -> Category;
496 fn is_non_zero(self) -> bool {
499 fn is_finite_non_zero(self) -> bool {
500 self.is_finite() && !self.is_zero()
502 fn is_pos_zero(self) -> bool {
503 self.is_zero() && !self.is_negative()
505 fn is_neg_zero(self) -> bool {
506 self.is_zero() && self.is_negative()
509 /// Returns `true` if the number has the smallest possible non-zero
510 /// magnitude in the current semantics.
511 fn is_smallest(self) -> bool {
512 Self::SMALLEST.copy_sign(self).bitwise_eq(self)
515 /// Returns `true` if the number has the largest possible finite
516 /// magnitude in the current semantics.
517 fn is_largest(self) -> bool {
518 Self::largest().copy_sign(self).bitwise_eq(self)
521 /// Returns `true` if the number is an exact integer.
522 fn is_integer(self) -> bool {
523 // This could be made more efficient; I'm going for obviously correct.
524 if !self.is_finite() {
527 self.round_to_integral(Round::TowardZero).value.bitwise_eq(self)
530 /// If this value has an exact multiplicative inverse, return it.
531 fn get_exact_inverse(self) -> Option<Self>;
533 /// Returns the exponent of the internal representation of the Float.
535 /// Because the radix of Float is 2, this is equivalent to floor(log2(x)).
536 /// For special Float values, this returns special error codes:
538 /// NaN -> \c IEK_NAN
540 /// Inf -> \c IEK_INF
542 fn ilogb(self) -> ExpInt;
544 /// Returns: self * 2<sup>exp</sup> for integral exponents.
545 /// Equivalent to C standard library function `ldexp`.
546 fn scalbn_r(self, exp: ExpInt, round: Round) -> Self;
547 fn scalbn(self, exp: ExpInt) -> Self {
548 self.scalbn_r(exp, Round::NearestTiesToEven)
551 /// Equivalent to C standard library function with the same name.
553 /// While the C standard says exp is an unspecified value for infinity and nan,
554 /// this returns INT_MAX for infinities, and INT_MIN for NaNs (see `ilogb`).
555 fn frexp_r(self, exp: &mut ExpInt, round: Round) -> Self;
556 fn frexp(self, exp: &mut ExpInt) -> Self {
557 self.frexp_r(exp, Round::NearestTiesToEven)
561 pub trait FloatConvert<T: Float>: Float {
562 /// Converts a value of one floating point type to another.
563 /// The return value corresponds to the IEEE754 exceptions. *loses_info
564 /// records whether the transformation lost information, i.e., whether
565 /// converting the result back to the original type will produce the
566 /// original value (this is almost the same as return `value == Status::OK`,
567 /// but there are edge cases where this is not so).
568 fn convert_r(self, round: Round, loses_info: &mut bool) -> StatusAnd<T>;
569 fn convert(self, loses_info: &mut bool) -> StatusAnd<T> {
570 self.convert_r(Round::NearestTiesToEven, loses_info)
574 macro_rules! float_common_impls {
575 ($ty:ident<$t:tt>) => {
576 impl<$t> Default for $ty<$t>
580 fn default() -> Self {
585 impl<$t> ::core::str::FromStr for $ty<$t>
589 type Err = ParseError;
590 fn from_str(s: &str) -> Result<Self, ParseError> {
591 Self::from_str_r(s, Round::NearestTiesToEven).map(|x| x.value)
595 // Rounding ties to the nearest even, by default.
597 impl<$t> ::core::ops::Add for $ty<$t>
601 type Output = StatusAnd<Self>;
602 fn add(self, rhs: Self) -> StatusAnd<Self> {
603 self.add_r(rhs, Round::NearestTiesToEven)
607 impl<$t> ::core::ops::Sub for $ty<$t>
611 type Output = StatusAnd<Self>;
612 fn sub(self, rhs: Self) -> StatusAnd<Self> {
613 self.sub_r(rhs, Round::NearestTiesToEven)
617 impl<$t> ::core::ops::Mul for $ty<$t>
621 type Output = StatusAnd<Self>;
622 fn mul(self, rhs: Self) -> StatusAnd<Self> {
623 self.mul_r(rhs, Round::NearestTiesToEven)
627 impl<$t> ::core::ops::Div for $ty<$t>
631 type Output = StatusAnd<Self>;
632 fn div(self, rhs: Self) -> StatusAnd<Self> {
633 self.div_r(rhs, Round::NearestTiesToEven)
637 impl<$t> ::core::ops::Rem for $ty<$t>
641 type Output = StatusAnd<Self>;
642 fn rem(self, rhs: Self) -> StatusAnd<Self> {
647 impl<$t> ::core::ops::AddAssign for $ty<$t>
651 fn add_assign(&mut self, rhs: Self) {
652 *self = (*self + rhs).value;
656 impl<$t> ::core::ops::SubAssign for $ty<$t>
660 fn sub_assign(&mut self, rhs: Self) {
661 *self = (*self - rhs).value;
665 impl<$t> ::core::ops::MulAssign for $ty<$t>
669 fn mul_assign(&mut self, rhs: Self) {
670 *self = (*self * rhs).value;
674 impl<$t> ::core::ops::DivAssign for $ty<$t>
678 fn div_assign(&mut self, rhs: Self) {
679 *self = (*self / rhs).value;
683 impl<$t> ::core::ops::RemAssign for $ty<$t>
687 fn rem_assign(&mut self, rhs: Self) {
688 *self = (*self % rhs).value;