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)]
40 use core::cmp::Ordering;
42 use core::ops::{Add, Div, Mul, Neg, Rem, Sub};
43 use core::ops::{AddAssign, DivAssign, MulAssign, RemAssign, SubAssign};
44 use core::str::FromStr;
47 /// IEEE-754R 7: Default exception handling.
49 /// UNDERFLOW or OVERFLOW are always returned or-ed with INEXACT.
51 pub struct Status: u8 {
53 const INVALID_OP = 0x01;
54 const DIV_BY_ZERO = 0x02;
55 const OVERFLOW = 0x04;
56 const UNDERFLOW = 0x08;
62 #[derive(Copy, Clone, PartialEq, Eq, PartialOrd, Ord, Debug)]
63 pub struct StatusAnd<T> {
69 pub fn and<T>(self, value: T) -> StatusAnd<T> {
70 StatusAnd { status: self, value }
74 impl<T> StatusAnd<T> {
75 pub fn map<F: FnOnce(T) -> U, U>(self, f: F) -> StatusAnd<U> {
76 StatusAnd { status: self.status, value: f(self.value) }
82 ($status:ident|=, $e:expr) => {
84 $crate::StatusAnd { status, value } => {
90 ($status:ident=, $e:expr) => {
92 $crate::StatusAnd { status, value } => {
100 /// Category of internally-represented number.
101 #[derive(Copy, Clone, PartialEq, Eq, Debug)]
109 /// IEEE-754R 4.3: Rounding-direction attributes.
110 #[derive(Copy, Clone, PartialEq, Eq, Debug)]
121 fn neg(self) -> Round {
123 Round::TowardPositive => Round::TowardNegative,
124 Round::TowardNegative => Round::TowardPositive,
125 Round::NearestTiesToEven | Round::TowardZero | Round::NearestTiesToAway => self,
130 /// A signed type to represent a floating point number's unbiased exponent.
131 pub type ExpInt = i16;
133 // \c ilogb error results.
134 pub const IEK_INF: ExpInt = ExpInt::MAX;
135 pub const IEK_NAN: ExpInt = ExpInt::MIN;
136 pub const IEK_ZERO: ExpInt = ExpInt::MIN + 1;
138 #[derive(Copy, Clone, PartialEq, Eq, Debug)]
139 pub struct ParseError(pub &'static str);
141 /// A self-contained host- and target-independent arbitrary-precision
142 /// floating-point software implementation.
144 /// `apfloat` uses significand bignum integer arithmetic as provided by functions
145 /// in the `ieee::sig`.
147 /// Written for clarity rather than speed, in particular with a view to use in
148 /// the front-end of a cross compiler so that target arithmetic can be correctly
149 /// performed on the host. Performance should nonetheless be reasonable,
150 /// particularly for its intended use. It may be useful as a base
151 /// implementation for a run-time library during development of a faster
152 /// target-specific one.
154 /// All 5 rounding modes in the IEEE-754R draft are handled correctly for all
155 /// implemented operations. Currently implemented operations are add, subtract,
156 /// multiply, divide, fused-multiply-add, conversion-to-float,
157 /// conversion-to-integer and conversion-from-integer. New rounding modes
158 /// (e.g., away from zero) can be added with three or four lines of code.
160 /// Four formats are built-in: IEEE single precision, double precision,
161 /// quadruple precision, and x87 80-bit extended double (when operating with
162 /// full extended precision). Adding a new format that obeys IEEE semantics
163 /// only requires adding two lines of code: a declaration and definition of the
166 /// All operations return the status of that operation as an exception bit-mask,
167 /// so multiple operations can be done consecutively with their results or-ed
168 /// together. The returned status can be useful for compiler diagnostics; e.g.,
169 /// inexact, underflow and overflow can be easily diagnosed on constant folding,
170 /// and compiler optimizers can determine what exceptions would be raised by
171 /// folding operations and optimize, or perhaps not optimize, accordingly.
173 /// At present, underflow tininess is detected after rounding; it should be
174 /// straight forward to add support for the before-rounding case too.
176 /// The library reads hexadecimal floating point numbers as per C99, and
177 /// correctly rounds if necessary according to the specified rounding mode.
178 /// Syntax is required to have been validated by the caller.
180 /// It also reads decimal floating point numbers and correctly rounds according
181 /// to the specified rounding mode.
183 /// Non-zero finite numbers are represented internally as a sign bit, a 16-bit
184 /// signed exponent, and the significand as an array of integer limbs. After
185 /// normalization of a number of precision P the exponent is within the range of
186 /// the format, and if the number is not denormal the P-th bit of the
187 /// significand is set as an explicit integer bit. For denormals the most
188 /// significant bit is shifted right so that the exponent is maintained at the
189 /// format's minimum, so that the smallest denormal has just the least
190 /// significant bit of the significand set. The sign of zeros and infinities
191 /// is significant; the exponent and significand of such numbers is not stored,
192 /// but has a known implicit (deterministic) value: 0 for the significands, 0
193 /// for zero exponent, all 1 bits for infinity exponent. For NaNs the sign and
194 /// significand are deterministic, although not really meaningful, and preserved
195 /// in non-conversion operations. The exponent is implicitly all 1 bits.
197 /// `apfloat` does not provide any exception handling beyond default exception
198 /// handling. We represent Signaling NaNs via IEEE-754R 2008 6.2.1 should clause
199 /// by encoding Signaling NaNs with the first bit of its trailing significand
205 /// Some features that may or may not be worth adding:
207 /// Optional ability to detect underflow tininess before rounding.
209 /// New formats: x87 in single and double precision mode (IEEE apart from
210 /// extended exponent range) (hard).
212 /// New operations: sqrt, nexttoward.
217 + FromStr<Err = ParseError>
226 + Add<Output = StatusAnd<Self>>
227 + Sub<Output = StatusAnd<Self>>
228 + Mul<Output = StatusAnd<Self>>
229 + Div<Output = StatusAnd<Self>>
230 + Rem<Output = StatusAnd<Self>>
232 /// Total number of bits in the in-memory format.
235 /// Number of bits in the significand. This includes the integer bit.
236 const PRECISION: usize;
238 /// The largest E such that 2<sup>E</sup> is representable; this matches the
239 /// definition of IEEE 754.
240 const MAX_EXP: ExpInt;
242 /// The smallest E such that 2<sup>E</sup> is a normalized number; this
243 /// matches the definition of IEEE 754.
244 const MIN_EXP: ExpInt;
249 /// Positive Infinity.
250 const INFINITY: Self;
252 /// NaN (Not a Number).
253 // FIXME(eddyb) provide a default when qnan becomes const fn.
256 /// Factory for QNaN values.
257 // FIXME(eddyb) should be const fn.
258 fn qnan(payload: Option<u128>) -> Self;
260 /// Factory for SNaN values.
261 // FIXME(eddyb) should be const fn.
262 fn snan(payload: Option<u128>) -> Self;
264 /// Largest finite number.
265 // FIXME(eddyb) should be const (but FloatPair::largest is nontrivial).
266 fn largest() -> Self;
268 /// Smallest (by magnitude) finite number.
269 /// Might be denormalized, which implies a relative loss of precision.
270 const SMALLEST: Self;
272 /// Smallest (by magnitude) normalized finite number.
273 // FIXME(eddyb) should be const (but FloatPair::smallest_normalized is nontrivial).
274 fn smallest_normalized() -> Self;
278 fn add_r(self, rhs: Self, round: Round) -> StatusAnd<Self>;
279 fn sub_r(self, rhs: Self, round: Round) -> StatusAnd<Self> {
280 self.add_r(-rhs, round)
282 fn mul_r(self, rhs: Self, round: Round) -> StatusAnd<Self>;
283 fn mul_add_r(self, multiplicand: Self, addend: Self, round: Round) -> StatusAnd<Self>;
284 fn mul_add(self, multiplicand: Self, addend: Self) -> StatusAnd<Self> {
285 self.mul_add_r(multiplicand, addend, Round::NearestTiesToEven)
287 fn div_r(self, rhs: Self, round: Round) -> StatusAnd<Self>;
289 // This is not currently correct in all cases.
290 fn ieee_rem(self, rhs: Self) -> StatusAnd<Self> {
294 v = unpack!(status=, v / rhs);
295 if status == Status::DIV_BY_ZERO {
296 return status.and(self);
299 assert!(Self::PRECISION < 128);
302 let x = unpack!(status=, v.to_i128_r(128, Round::NearestTiesToEven, &mut false));
303 if status == Status::INVALID_OP {
304 return status.and(self);
308 let mut v = unpack!(status=, Self::from_i128(x));
309 assert_eq!(status, Status::OK); // should always work
312 v = unpack!(status=, v * rhs);
313 assert_eq!(status - Status::INEXACT, Status::OK); // should not overflow or underflow
316 v = unpack!(status=, self - v);
317 assert_eq!(status - Status::INEXACT, Status::OK); // likewise
320 status.and(v.copy_sign(self)) // IEEE754 requires this
325 /// C fmod, or llvm frem.
326 fn c_fmod(self, rhs: Self) -> StatusAnd<Self>;
327 fn round_to_integral(self, round: Round) -> StatusAnd<Self>;
329 /// IEEE-754R 2008 5.3.1: nextUp.
330 fn next_up(self) -> StatusAnd<Self>;
332 /// IEEE-754R 2008 5.3.1: nextDown.
334 /// *NOTE* since nextDown(x) = -nextUp(-x), we only implement nextUp with
335 /// appropriate sign switching before/after the computation.
336 fn next_down(self) -> StatusAnd<Self> {
337 (-self).next_up().map(|r| -r)
340 fn abs(self) -> Self {
341 if self.is_negative() { -self } else { self }
343 fn copy_sign(self, rhs: Self) -> Self {
344 if self.is_negative() != rhs.is_negative() { -self } else { self }
348 fn from_bits(input: u128) -> Self;
349 fn from_i128_r(input: i128, round: Round) -> StatusAnd<Self> {
351 Self::from_u128_r(input.wrapping_neg() as u128, -round).map(|r| -r)
353 Self::from_u128_r(input as u128, round)
356 fn from_i128(input: i128) -> StatusAnd<Self> {
357 Self::from_i128_r(input, Round::NearestTiesToEven)
359 fn from_u128_r(input: u128, round: Round) -> StatusAnd<Self>;
360 fn from_u128(input: u128) -> StatusAnd<Self> {
361 Self::from_u128_r(input, Round::NearestTiesToEven)
363 fn from_str_r(s: &str, round: Round) -> Result<StatusAnd<Self>, ParseError>;
364 fn to_bits(self) -> u128;
366 /// Converts a floating point number to an integer according to the
367 /// rounding mode. In case of an invalid operation exception,
368 /// deterministic values are returned, namely zero for NaNs and the
369 /// minimal or maximal value respectively for underflow or overflow.
370 /// If the rounded value is in range but the floating point number is
371 /// not the exact integer, the C standard doesn't require an inexact
372 /// exception to be raised. IEEE-854 does require it so we do that.
374 /// Note that for conversions to integer type the C standard requires
375 /// round-to-zero to always be used.
377 /// The *is_exact output tells whether the result is exact, in the sense
378 /// that converting it back to the original floating point type produces
379 /// the original value. This is almost equivalent to `result == Status::OK`,
380 /// except for negative zeroes.
381 fn to_i128_r(self, width: usize, round: Round, is_exact: &mut bool) -> StatusAnd<i128> {
383 if self.is_negative() {
385 // Negative zero can't be represented as an int.
388 let r = unpack!(status=, (-self).to_u128_r(width, -round, is_exact));
390 // Check for values that don't fit in the signed integer.
391 if r > (1 << (width - 1)) {
392 // Return the most negative integer for the given width.
394 Status::INVALID_OP.and(-1 << (width - 1))
396 status.and(r.wrapping_neg() as i128)
399 // Positive case is simpler, can pretend it's a smaller unsigned
400 // integer, and `to_u128` will take care of all the edge cases.
401 self.to_u128_r(width - 1, round, is_exact).map(|r| r as i128)
404 fn to_i128(self, width: usize) -> StatusAnd<i128> {
405 self.to_i128_r(width, Round::TowardZero, &mut true)
407 fn to_u128_r(self, width: usize, round: Round, is_exact: &mut bool) -> StatusAnd<u128>;
408 fn to_u128(self, width: usize) -> StatusAnd<u128> {
409 self.to_u128_r(width, Round::TowardZero, &mut true)
412 fn cmp_abs_normal(self, rhs: Self) -> Ordering;
414 /// Bitwise comparison for equality (QNaNs compare equal, 0!=-0).
415 fn bitwise_eq(self, rhs: Self) -> bool;
417 // IEEE-754R 5.7.2 General operations.
419 /// Implements IEEE minNum semantics. Returns the smaller of the 2 arguments if
420 /// both are not NaN. If either argument is a NaN, returns the other argument.
421 fn min(self, other: Self) -> Self {
424 } else if other.is_nan() {
426 } else if other.partial_cmp(&self) == Some(Ordering::Less) {
433 /// Implements IEEE maxNum semantics. Returns the larger of the 2 arguments if
434 /// both are not NaN. If either argument is a NaN, returns the other argument.
435 fn max(self, other: Self) -> Self {
438 } else if other.is_nan() {
440 } else if self.partial_cmp(&other) == Some(Ordering::Less) {
447 /// IEEE-754R isSignMinus: Returns whether the current value is
450 /// This applies to zeros and NaNs as well.
451 fn is_negative(self) -> bool;
453 /// IEEE-754R isNormal: Returns whether the current value is normal.
455 /// This implies that the current value of the float is not zero, subnormal,
456 /// infinite, or NaN following the definition of normality from IEEE-754R.
457 fn is_normal(self) -> bool {
458 !self.is_denormal() && self.is_finite_non_zero()
461 /// Returns `true` if the current value is zero, subnormal, or
464 /// This means that the value is not infinite or NaN.
465 fn is_finite(self) -> bool {
466 !self.is_nan() && !self.is_infinite()
469 /// Returns `true` if the float is plus or minus zero.
470 fn is_zero(self) -> bool {
471 self.category() == Category::Zero
474 /// IEEE-754R isSubnormal(): Returns whether the float is a
476 fn is_denormal(self) -> bool;
478 /// IEEE-754R isInfinite(): Returns whether the float is infinity.
479 fn is_infinite(self) -> bool {
480 self.category() == Category::Infinity
483 /// Returns `true` if the float is a quiet or signaling NaN.
484 fn is_nan(self) -> bool {
485 self.category() == Category::NaN
488 /// Returns `true` if the float is a signaling NaN.
489 fn is_signaling(self) -> bool;
493 fn category(self) -> Category;
494 fn is_non_zero(self) -> bool {
497 fn is_finite_non_zero(self) -> bool {
498 self.is_finite() && !self.is_zero()
500 fn is_pos_zero(self) -> bool {
501 self.is_zero() && !self.is_negative()
503 fn is_neg_zero(self) -> bool {
504 self.is_zero() && self.is_negative()
507 /// Returns `true` if the number has the smallest possible non-zero
508 /// magnitude in the current semantics.
509 fn is_smallest(self) -> bool {
510 Self::SMALLEST.copy_sign(self).bitwise_eq(self)
513 /// Returns `true` if the number has the largest possible finite
514 /// magnitude in the current semantics.
515 fn is_largest(self) -> bool {
516 Self::largest().copy_sign(self).bitwise_eq(self)
519 /// Returns `true` if the number is an exact integer.
520 fn is_integer(self) -> bool {
521 // This could be made more efficient; I'm going for obviously correct.
522 if !self.is_finite() {
525 self.round_to_integral(Round::TowardZero).value.bitwise_eq(self)
528 /// If this value has an exact multiplicative inverse, return it.
529 fn get_exact_inverse(self) -> Option<Self>;
531 /// Returns the exponent of the internal representation of the Float.
533 /// Because the radix of Float is 2, this is equivalent to floor(log2(x)).
534 /// For special Float values, this returns special error codes:
536 /// NaN -> \c IEK_NAN
538 /// Inf -> \c IEK_INF
540 fn ilogb(self) -> ExpInt;
542 /// Returns: self * 2<sup>exp</sup> for integral exponents.
543 /// Equivalent to C standard library function `ldexp`.
544 fn scalbn_r(self, exp: ExpInt, round: Round) -> Self;
545 fn scalbn(self, exp: ExpInt) -> Self {
546 self.scalbn_r(exp, Round::NearestTiesToEven)
549 /// Equivalent to C standard library function with the same name.
551 /// While the C standard says exp is an unspecified value for infinity and nan,
552 /// this returns INT_MAX for infinities, and INT_MIN for NaNs (see `ilogb`).
553 fn frexp_r(self, exp: &mut ExpInt, round: Round) -> Self;
554 fn frexp(self, exp: &mut ExpInt) -> Self {
555 self.frexp_r(exp, Round::NearestTiesToEven)
559 pub trait FloatConvert<T: Float>: Float {
560 /// Converts a value of one floating point type to another.
561 /// The return value corresponds to the IEEE754 exceptions. *loses_info
562 /// records whether the transformation lost information, i.e., whether
563 /// converting the result back to the original type will produce the
564 /// original value (this is almost the same as return `value == Status::OK`,
565 /// but there are edge cases where this is not so).
566 fn convert_r(self, round: Round, loses_info: &mut bool) -> StatusAnd<T>;
567 fn convert(self, loses_info: &mut bool) -> StatusAnd<T> {
568 self.convert_r(Round::NearestTiesToEven, loses_info)
572 macro_rules! float_common_impls {
573 ($ty:ident<$t:tt>) => {
574 impl<$t> Default for $ty<$t>
578 fn default() -> Self {
583 impl<$t> ::core::str::FromStr for $ty<$t>
587 type Err = ParseError;
588 fn from_str(s: &str) -> Result<Self, ParseError> {
589 Self::from_str_r(s, Round::NearestTiesToEven).map(|x| x.value)
593 // Rounding ties to the nearest even, by default.
595 impl<$t> ::core::ops::Add for $ty<$t>
599 type Output = StatusAnd<Self>;
600 fn add(self, rhs: Self) -> StatusAnd<Self> {
601 self.add_r(rhs, Round::NearestTiesToEven)
605 impl<$t> ::core::ops::Sub for $ty<$t>
609 type Output = StatusAnd<Self>;
610 fn sub(self, rhs: Self) -> StatusAnd<Self> {
611 self.sub_r(rhs, Round::NearestTiesToEven)
615 impl<$t> ::core::ops::Mul for $ty<$t>
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>
629 type Output = StatusAnd<Self>;
630 fn div(self, rhs: Self) -> StatusAnd<Self> {
631 self.div_r(rhs, Round::NearestTiesToEven)
635 impl<$t> ::core::ops::Rem for $ty<$t>
639 type Output = StatusAnd<Self>;
640 fn rem(self, rhs: Self) -> StatusAnd<Self> {
645 impl<$t> ::core::ops::AddAssign for $ty<$t>
649 fn add_assign(&mut self, rhs: Self) {
650 *self = (*self + rhs).value;
654 impl<$t> ::core::ops::SubAssign for $ty<$t>
658 fn sub_assign(&mut self, rhs: Self) {
659 *self = (*self - rhs).value;
663 impl<$t> ::core::ops::MulAssign for $ty<$t>
667 fn mul_assign(&mut self, rhs: Self) {
668 *self = (*self * rhs).value;
672 impl<$t> ::core::ops::DivAssign for $ty<$t>
676 fn div_assign(&mut self, rhs: Self) {
677 *self = (*self / rhs).value;
681 impl<$t> ::core::ops::RemAssign for $ty<$t>
685 fn rem_assign(&mut self, rhs: Self) {
686 *self = (*self % rhs).value;