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/")]
35 #![forbid(unsafe_code)]
42 use core::cmp::Ordering;
44 use core::ops::{Neg, Add, Sub, Mul, Div, Rem};
45 use core::ops::{AddAssign, SubAssign, MulAssign, DivAssign, RemAssign};
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> {
79 impl<T> StatusAnd<T> {
80 pub fn map<F: FnOnce(T) -> U, U>(self, f: F) -> StatusAnd<U> {
90 ($status:ident|=, $e:expr) => {
92 $crate::StatusAnd { status, value } => {
98 ($status:ident=, $e:expr) => {
100 $crate::StatusAnd { status, value } => {
108 /// Category of internally-represented number.
109 #[derive(Copy, Clone, PartialEq, Eq, Debug)]
117 /// IEEE-754R 4.3: Rounding-direction attributes.
118 #[derive(Copy, Clone, PartialEq, Eq, Debug)]
129 fn neg(self) -> Round {
131 Round::TowardPositive => Round::TowardNegative,
132 Round::TowardNegative => Round::TowardPositive,
133 Round::NearestTiesToEven | Round::TowardZero | Round::NearestTiesToAway => self,
138 /// A signed type to represent a floating point number's unbiased exponent.
139 pub type ExpInt = i16;
141 // \c ilogb error results.
142 pub const IEK_INF: ExpInt = ExpInt::max_value();
143 pub const IEK_NAN: ExpInt = ExpInt::min_value();
144 pub const IEK_ZERO: ExpInt = ExpInt::min_value() + 1;
146 #[derive(Copy, Clone, PartialEq, Eq, Debug)]
147 pub struct ParseError(pub &'static str);
149 /// A self-contained host- and target-independent arbitrary-precision
150 /// floating-point software implementation.
152 /// `apfloat` uses significand bignum integer arithmetic as provided by functions
153 /// in the `ieee::sig`.
155 /// Written for clarity rather than speed, in particular with a view to use in
156 /// the front-end of a cross compiler so that target arithmetic can be correctly
157 /// performed on the host. Performance should nonetheless be reasonable,
158 /// particularly for its intended use. It may be useful as a base
159 /// implementation for a run-time library during development of a faster
160 /// target-specific one.
162 /// All 5 rounding modes in the IEEE-754R draft are handled correctly for all
163 /// implemented operations. Currently implemented operations are add, subtract,
164 /// multiply, divide, fused-multiply-add, conversion-to-float,
165 /// conversion-to-integer and conversion-from-integer. New rounding modes
166 /// (e.g., away from zero) can be added with three or four lines of code.
168 /// Four formats are built-in: IEEE single precision, double precision,
169 /// quadruple precision, and x87 80-bit extended double (when operating with
170 /// full extended precision). Adding a new format that obeys IEEE semantics
171 /// only requires adding two lines of code: a declaration and definition of the
174 /// All operations return the status of that operation as an exception bit-mask,
175 /// so multiple operations can be done consecutively with their results or-ed
176 /// together. The returned status can be useful for compiler diagnostics; e.g.,
177 /// inexact, underflow and overflow can be easily diagnosed on constant folding,
178 /// and compiler optimizers can determine what exceptions would be raised by
179 /// folding operations and optimize, or perhaps not optimize, accordingly.
181 /// At present, underflow tininess is detected after rounding; it should be
182 /// straight forward to add support for the before-rounding case too.
184 /// The library reads hexadecimal floating point numbers as per C99, and
185 /// correctly rounds if necessary according to the specified rounding mode.
186 /// Syntax is required to have been validated by the caller.
188 /// It also reads decimal floating point numbers and correctly rounds according
189 /// to the specified rounding mode.
191 /// Non-zero finite numbers are represented internally as a sign bit, a 16-bit
192 /// signed exponent, and the significand as an array of integer limbs. After
193 /// normalization of a number of precision P the exponent is within the range of
194 /// the format, and if the number is not denormal the P-th bit of the
195 /// significand is set as an explicit integer bit. For denormals the most
196 /// significant bit is shifted right so that the exponent is maintained at the
197 /// format's minimum, so that the smallest denormal has just the least
198 /// significant bit of the significand set. The sign of zeros and infinities
199 /// is significant; the exponent and significand of such numbers is not stored,
200 /// but has a known implicit (deterministic) value: 0 for the significands, 0
201 /// for zero exponent, all 1 bits for infinity exponent. For NaNs the sign and
202 /// significand are deterministic, although not really meaningful, and preserved
203 /// in non-conversion operations. The exponent is implicitly all 1 bits.
205 /// `apfloat` does not provide any exception handling beyond default exception
206 /// handling. We represent Signaling NaNs via IEEE-754R 2008 6.2.1 should clause
207 /// by encoding Signaling NaNs with the first bit of its trailing significand
213 /// Some features that may or may not be worth adding:
215 /// Optional ability to detect underflow tininess before rounding.
217 /// New formats: x87 in single and double precision mode (IEEE apart from
218 /// extended exponent range) (hard).
220 /// New operations: sqrt, nexttoward.
225 + FromStr<Err = ParseError>
234 + Add<Output = StatusAnd<Self>>
235 + Sub<Output = StatusAnd<Self>>
236 + Mul<Output = StatusAnd<Self>>
237 + Div<Output = StatusAnd<Self>>
238 + Rem<Output = StatusAnd<Self>> {
239 /// Total number of bits in the in-memory format.
242 /// Number of bits in the significand. This includes the integer bit.
243 const PRECISION: usize;
245 /// The largest E such that 2<sup>E</sup> is representable; this matches the
246 /// definition of IEEE 754.
247 const MAX_EXP: ExpInt;
249 /// The smallest E such that 2<sup>E</sup> is a normalized number; this
250 /// matches the definition of IEEE 754.
251 const MIN_EXP: ExpInt;
256 /// Positive Infinity.
257 const INFINITY: Self;
259 /// NaN (Not a Number).
260 // FIXME(eddyb) provide a default when qnan becomes const fn.
263 /// Factory for QNaN values.
264 // FIXME(eddyb) should be const fn.
265 fn qnan(payload: Option<u128>) -> Self;
267 /// Factory for SNaN values.
268 // FIXME(eddyb) should be const fn.
269 fn snan(payload: Option<u128>) -> Self;
271 /// Largest finite number.
272 // FIXME(eddyb) should be const (but FloatPair::largest is nontrivial).
273 fn largest() -> Self;
275 /// Smallest (by magnitude) finite number.
276 /// Might be denormalized, which implies a relative loss of precision.
277 const SMALLEST: Self;
279 /// Smallest (by magnitude) normalized finite number.
280 // FIXME(eddyb) should be const (but FloatPair::smallest_normalized is nontrivial).
281 fn smallest_normalized() -> Self;
285 fn add_r(self, rhs: Self, round: Round) -> StatusAnd<Self>;
286 fn sub_r(self, rhs: Self, round: Round) -> StatusAnd<Self> {
287 self.add_r(-rhs, round)
289 fn mul_r(self, rhs: Self, round: Round) -> StatusAnd<Self>;
290 fn mul_add_r(self, multiplicand: Self, addend: Self, round: Round) -> StatusAnd<Self>;
291 fn mul_add(self, multiplicand: Self, addend: Self) -> StatusAnd<Self> {
292 self.mul_add_r(multiplicand, addend, Round::NearestTiesToEven)
294 fn div_r(self, rhs: Self, round: Round) -> StatusAnd<Self>;
296 // This is not currently correct in all cases.
297 fn ieee_rem(self, rhs: Self) -> StatusAnd<Self> {
301 v = unpack!(status=, v / rhs);
302 if status == Status::DIV_BY_ZERO {
303 return status.and(self);
306 assert!(Self::PRECISION < 128);
309 let x = unpack!(status=, v.to_i128_r(128, Round::NearestTiesToEven, &mut false));
310 if status == Status::INVALID_OP {
311 return status.and(self);
315 let mut v = unpack!(status=, Self::from_i128(x));
316 assert_eq!(status, Status::OK); // should always work
319 v = unpack!(status=, v * rhs);
320 assert_eq!(status - Status::INEXACT, Status::OK); // should not overflow or underflow
323 v = unpack!(status=, self - v);
324 assert_eq!(status - Status::INEXACT, Status::OK); // likewise
327 status.and(v.copy_sign(self)) // IEEE754 requires this
332 /// C fmod, or llvm frem.
333 fn c_fmod(self, rhs: Self) -> StatusAnd<Self>;
334 fn round_to_integral(self, round: Round) -> StatusAnd<Self>;
336 /// IEEE-754R 2008 5.3.1: nextUp.
337 fn next_up(self) -> StatusAnd<Self>;
339 /// IEEE-754R 2008 5.3.1: nextDown.
341 /// *NOTE* since nextDown(x) = -nextUp(-x), we only implement nextUp with
342 /// appropriate sign switching before/after the computation.
343 fn next_down(self) -> StatusAnd<Self> {
344 (-self).next_up().map(|r| -r)
347 fn abs(self) -> Self {
348 if self.is_negative() { -self } else { self }
350 fn copy_sign(self, rhs: Self) -> Self {
351 if self.is_negative() != rhs.is_negative() {
359 fn from_bits(input: u128) -> Self;
360 fn from_i128_r(input: i128, round: Round) -> StatusAnd<Self> {
362 Self::from_u128_r(input.wrapping_neg() as u128, -round).map(|r| -r)
364 Self::from_u128_r(input as u128, round)
367 fn from_i128(input: i128) -> StatusAnd<Self> {
368 Self::from_i128_r(input, Round::NearestTiesToEven)
370 fn from_u128_r(input: u128, round: Round) -> StatusAnd<Self>;
371 fn from_u128(input: u128) -> StatusAnd<Self> {
372 Self::from_u128_r(input, Round::NearestTiesToEven)
374 fn from_str_r(s: &str, round: Round) -> Result<StatusAnd<Self>, ParseError>;
375 fn to_bits(self) -> u128;
377 /// Converts a floating point number to an integer according to the
378 /// rounding mode. In case of an invalid operation exception,
379 /// deterministic values are returned, namely zero for NaNs and the
380 /// minimal or maximal value respectively for underflow or overflow.
381 /// If the rounded value is in range but the floating point number is
382 /// not the exact integer, the C standard doesn't require an inexact
383 /// exception to be raised. IEEE-854 does require it so we do that.
385 /// Note that for conversions to integer type the C standard requires
386 /// round-to-zero to always be used.
388 /// The *is_exact output tells whether the result is exact, in the sense
389 /// that converting it back to the original floating point type produces
390 /// the original value. This is almost equivalent to `result == Status::OK`,
391 /// except for negative zeroes.
392 fn to_i128_r(self, width: usize, round: Round, is_exact: &mut bool) -> StatusAnd<i128> {
394 if self.is_negative() {
396 // Negative zero can't be represented as an int.
399 let r = unpack!(status=, (-self).to_u128_r(width, -round, is_exact));
401 // Check for values that don't fit in the signed integer.
402 if r > (1 << (width - 1)) {
403 // Return the most negative integer for the given width.
405 Status::INVALID_OP.and(-1 << (width - 1))
407 status.and(r.wrapping_neg() as i128)
410 // Positive case is simpler, can pretend it's a smaller unsigned
411 // integer, and `to_u128` will take care of all the edge cases.
412 self.to_u128_r(width - 1, round, is_exact).map(
417 fn to_i128(self, width: usize) -> StatusAnd<i128> {
418 self.to_i128_r(width, Round::TowardZero, &mut true)
420 fn to_u128_r(self, width: usize, round: Round, is_exact: &mut bool) -> StatusAnd<u128>;
421 fn to_u128(self, width: usize) -> StatusAnd<u128> {
422 self.to_u128_r(width, Round::TowardZero, &mut true)
425 fn cmp_abs_normal(self, rhs: Self) -> Ordering;
427 /// Bitwise comparison for equality (QNaNs compare equal, 0!=-0).
428 fn bitwise_eq(self, rhs: Self) -> bool;
430 // IEEE-754R 5.7.2 General operations.
432 /// Implements IEEE minNum semantics. Returns the smaller of the 2 arguments if
433 /// both are not NaN. If either argument is a NaN, returns the other argument.
434 fn min(self, other: Self) -> Self {
437 } else if other.is_nan() {
439 } else if other.partial_cmp(&self) == Some(Ordering::Less) {
446 /// Implements IEEE maxNum semantics. Returns the larger of the 2 arguments if
447 /// both are not NaN. If either argument is a NaN, returns the other argument.
448 fn max(self, other: Self) -> Self {
451 } else if other.is_nan() {
453 } else if self.partial_cmp(&other) == Some(Ordering::Less) {
460 /// IEEE-754R isSignMinus: Returns whether the current value is
463 /// This applies to zeros and NaNs as well.
464 fn is_negative(self) -> bool;
466 /// IEEE-754R isNormal: Returns whether the current value is normal.
468 /// This implies that the current value of the float is not zero, subnormal,
469 /// infinite, or NaN following the definition of normality from IEEE-754R.
470 fn is_normal(self) -> bool {
471 !self.is_denormal() && self.is_finite_non_zero()
474 /// Returns `true` if the current value is zero, subnormal, or
477 /// This means that the value is not infinite or NaN.
478 fn is_finite(self) -> bool {
479 !self.is_nan() && !self.is_infinite()
482 /// Returns `true` if the float is plus or minus zero.
483 fn is_zero(self) -> bool {
484 self.category() == Category::Zero
487 /// IEEE-754R isSubnormal(): Returns whether the float is a
489 fn is_denormal(self) -> bool;
491 /// IEEE-754R isInfinite(): Returns whether the float is infinity.
492 fn is_infinite(self) -> bool {
493 self.category() == Category::Infinity
496 /// Returns `true` if the float is a quiet or signaling NaN.
497 fn is_nan(self) -> bool {
498 self.category() == Category::NaN
501 /// Returns `true` if the float is a signaling NaN.
502 fn is_signaling(self) -> bool;
506 fn category(self) -> Category;
507 fn is_non_zero(self) -> bool {
510 fn is_finite_non_zero(self) -> bool {
511 self.is_finite() && !self.is_zero()
513 fn is_pos_zero(self) -> bool {
514 self.is_zero() && !self.is_negative()
516 fn is_neg_zero(self) -> bool {
517 self.is_zero() && self.is_negative()
520 /// Returns `true` if the number has the smallest possible non-zero
521 /// magnitude in the current semantics.
522 fn is_smallest(self) -> bool {
523 Self::SMALLEST.copy_sign(self).bitwise_eq(self)
526 /// Returns `true` if the number has the largest possible finite
527 /// magnitude in the current semantics.
528 fn is_largest(self) -> bool {
529 Self::largest().copy_sign(self).bitwise_eq(self)
532 /// Returns `true` if the number is an exact integer.
533 fn is_integer(self) -> bool {
534 // This could be made more efficient; I'm going for obviously correct.
535 if !self.is_finite() {
538 self.round_to_integral(Round::TowardZero).value.bitwise_eq(
543 /// If this value has an exact multiplicative inverse, return it.
544 fn get_exact_inverse(self) -> Option<Self>;
546 /// Returns the exponent of the internal representation of the Float.
548 /// Because the radix of Float is 2, this is equivalent to floor(log2(x)).
549 /// For special Float values, this returns special error codes:
551 /// NaN -> \c IEK_NAN
553 /// Inf -> \c IEK_INF
555 fn ilogb(self) -> ExpInt;
557 /// Returns: self * 2<sup>exp</sup> for integral exponents.
558 /// Equivalent to C standard library function `ldexp`.
559 fn scalbn_r(self, exp: ExpInt, round: Round) -> Self;
560 fn scalbn(self, exp: ExpInt) -> Self {
561 self.scalbn_r(exp, Round::NearestTiesToEven)
564 /// Equivalent to C standard library function with the same name.
566 /// While the C standard says exp is an unspecified value for infinity and nan,
567 /// this returns INT_MAX for infinities, and INT_MIN for NaNs (see `ilogb`).
568 fn frexp_r(self, exp: &mut ExpInt, round: Round) -> Self;
569 fn frexp(self, exp: &mut ExpInt) -> Self {
570 self.frexp_r(exp, Round::NearestTiesToEven)
574 pub trait FloatConvert<T: Float>: Float {
575 /// Converts a value of one floating point type to another.
576 /// The return value corresponds to the IEEE754 exceptions. *loses_info
577 /// records whether the transformation lost information, i.e., whether
578 /// converting the result back to the original type will produce the
579 /// original value (this is almost the same as return `value == Status::OK`,
580 /// but there are edge cases where this is not so).
581 fn convert_r(self, round: Round, loses_info: &mut bool) -> StatusAnd<T>;
582 fn convert(self, loses_info: &mut bool) -> StatusAnd<T> {
583 self.convert_r(Round::NearestTiesToEven, loses_info)
587 macro_rules! float_common_impls {
588 ($ty:ident<$t:tt>) => {
589 impl<$t> Default for $ty<$t> where Self: Float {
590 fn default() -> Self {
595 impl<$t> ::core::str::FromStr for $ty<$t> where Self: Float {
596 type Err = ParseError;
597 fn from_str(s: &str) -> Result<Self, ParseError> {
598 Self::from_str_r(s, Round::NearestTiesToEven).map(|x| x.value)
602 // Rounding ties to the nearest even, by default.
604 impl<$t> ::core::ops::Add for $ty<$t> where Self: Float {
605 type Output = StatusAnd<Self>;
606 fn add(self, rhs: Self) -> StatusAnd<Self> {
607 self.add_r(rhs, Round::NearestTiesToEven)
611 impl<$t> ::core::ops::Sub for $ty<$t> where Self: Float {
612 type Output = StatusAnd<Self>;
613 fn sub(self, rhs: Self) -> StatusAnd<Self> {
614 self.sub_r(rhs, Round::NearestTiesToEven)
618 impl<$t> ::core::ops::Mul for $ty<$t> where Self: Float {
619 type Output = StatusAnd<Self>;
620 fn mul(self, rhs: Self) -> StatusAnd<Self> {
621 self.mul_r(rhs, Round::NearestTiesToEven)
625 impl<$t> ::core::ops::Div for $ty<$t> where Self: Float {
626 type Output = StatusAnd<Self>;
627 fn div(self, rhs: Self) -> StatusAnd<Self> {
628 self.div_r(rhs, Round::NearestTiesToEven)
632 impl<$t> ::core::ops::Rem for $ty<$t> where Self: Float {
633 type Output = StatusAnd<Self>;
634 fn rem(self, rhs: Self) -> StatusAnd<Self> {
639 impl<$t> ::core::ops::AddAssign for $ty<$t> where Self: Float {
640 fn add_assign(&mut self, rhs: Self) {
641 *self = (*self + rhs).value;
645 impl<$t> ::core::ops::SubAssign for $ty<$t> where Self: Float {
646 fn sub_assign(&mut self, rhs: Self) {
647 *self = (*self - rhs).value;
651 impl<$t> ::core::ops::MulAssign for $ty<$t> where Self: Float {
652 fn mul_assign(&mut self, rhs: Self) {
653 *self = (*self * rhs).value;
657 impl<$t> ::core::ops::DivAssign for $ty<$t> where Self: Float {
658 fn div_assign(&mut self, rhs: Self) {
659 *self = (*self / rhs).value;
663 impl<$t> ::core::ops::RemAssign for $ty<$t> where Self: Float {
664 fn rem_assign(&mut self, rhs: Self) {
665 *self = (*self % rhs).value;